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src="https://crossmark-cdn.crossref.org/widget/v2.0/logos/CROSSMARK_BW_square_no_text.svg" title="" alt=""> </figure> <div id="crossmark-icon"> 检查更新</d我v></一个></d我v><一个rticle class="widget-listing people-also-looked-at side-article-related hidden"> <h5 class="like-h4" data-event="peoplelookedat-link-click">人也看了</h5></一个rt我cle> </aside> <main class=""> <div class="article-section"> <div class="article-container" data-html="True"> <div class="abstract-container"> <div class="article-header-container">   <div class="header-bar-one"> <h2>原始研究的文章</h2></d我v><d我vclass="header-bar-three-container"> <div class="header-bar-three"> 前面。建立环境。,24April 2023<br>秒。结构传感、控制和资产管理<br><gydF4y2Baspan class="volumeInfo">卷9 - 2023 |</gydF4y2Baspan> <a href="https://doi.org/10.3389/fbuil.2023.1143597">https://doi.org/10.3389/fbuil.2023.1143597</一个></d我v></d我v></div> <div class="JournalAbstract"> <a id="h1" name="h1"></a> <h1>变分模型的贝叶斯推理技术更新结构系统与未知噪声统计数据</h1><d我vclass="authors"> <a href="//www.thespel.com/journal/people/u/2250926" class="user-id-2250926"><img class="pr5" src="//www.thespel.com/loop/images/profile/2250926/24" onerror="this.src='https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg'" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba">Mansureh-Sadat Nabiyan</一个><sup>1</gydF4y2Basup> <sup>__</gydF4y2Basup> <a href="//www.thespel.com/journal/people/u/2275823" class="user-id-2275823"><img class="pr5" src="//www.thespel.com/loop/images/profile/2275823/24" onerror="this.src='https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg'" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba">马赫迪·沙里夫</一个><sup>2</gydF4y2Basup> <sup>__</gydF4y2Basup> <a href="//www.thespel.com/journal/people/u/2161834" class="user-id-2161834"><img class="pr5" src="//www.thespel.com/loop/images/profile/2161834/24" onerror="this.src='https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg'" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba">哈米德Ebrahimian</一个><sup>3</gydF4y2Basup>*<gydF4y2Basup>__</gydF4y2Basup> <a href="//www.thespel.com/journal/people/u/244788" class="user-id-244788"><img class="pr5" src="//www.thespel.com/loop/images/profile/244788/24" onerror="this.src='https://f96a1a95aaa960e01625-a34624e694c43cdf8b40aa048a644ca4.ssl.cf2.rackcdn.com/Design/Images/newprofile_default_profileimage_new.jpg'" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba">Babak Moaveni</一个><sup>1</gydF4y2Basup> <sup>__</gydF4y2Basup> </div> <ul class="notes"> <li><span><sup>1</gydF4y2Basup></span>土木与环境工程系,塔夫茨大学,梅德福,或者美国</gydF4y2Bali> <li><span><sup>2</gydF4y2Basup></span>加文·多尔蒂Geosolutions有限公司,都柏林,爱尔兰</gydF4y2Bali> <li><span><sup>3</gydF4y2Basup></span>土木与环境工程系,内华达大学雷诺美国公司</gydF4y2Bali> </ul> <p class="mb15">动态模型的结构和机械系统可以更新以匹配测量数据通过贝叶斯推理过程。然而,古典的性能(节)贝叶斯模型更新方法显著减少时,预先假设模型预测误差的统计特征是违反了。为了克服这一问题,本文提出了一种自适应递归变分贝叶斯估计方法的预测误差的统计特征共同未知模型参数。这种方法提高了模型的准确性和鲁棒性的更新,包括模型预测误差的估计。这种方法演示了使用性能的数值模拟数据获得材料非线性地震激励下的结构框架。结果表明,在非平稳噪声的存在/错误,非自适应的方法无法估计未知的模型参数,而该方法能准确的估计。</p><d我vclass="clear"></div> </div> <div class="JournalFullText"> <a id="h2" name="h2"></a> <h2>1介绍</h2><pcgydF4y2Balass="mb15">贝叶斯模型更新的目标是最小化估计不确定模型参数的实测值和预测值的差异反应(<一个href="//www.thespel.com/journal/articles/10.3389/#B17">Friswell摩特谢德,2013</一个>)。这种技术被广泛用于结构系统识别(<一个href="//www.thespel.com/journal/articles/10.3389/#B9">Behmanesh et al ., 2015</一个>),参数估计(<一个href="//www.thespel.com/journal/articles/10.3389/#B12">Ching et al ., 2006</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B4">Astroza et al ., 2014</一个>),损伤识别(<一个href="//www.thespel.com/journal/articles/10.3389/#B14">Doebling et al ., 1996</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B48">杨et al ., 2006</一个>)和虚拟传感(<一个href="//www.thespel.com/journal/articles/10.3389/#B44">文策尔et al ., 2007</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B29">Nabiyan et al ., 2020</一个>)。然而,贝叶斯模型更新的性能取决于质量的预测误差的先验知识,其中包括建模误差和测量噪声的影响<一个href="//www.thespel.com/journal/articles/10.3389/#B8">贝克和袁,2004年</一个>)。古典(非自适应的)贝叶斯模型更新方法,假设预测误差作为一个静止不动的,零均值高斯白噪声过程。然而,这并非总是如此在实践中,和预测误差通常是一个非平稳、非白人和非高斯过程由于建模误差的影响(<一个href="//www.thespel.com/journal/articles/10.3389/#B34">Sanayei et al ., 2001</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B25">法律和斯图尔特,2012</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>)。非自适应的贝叶斯方法的估计精度可以负面影响在实践中当预测误差偏离了平稳高斯假设(<一个href="//www.thespel.com/journal/articles/10.3389/#B27">用,1972</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B47">徐et al ., 2019</一个>)。</p><pcgydF4y2Balass="mb15">减轻需要先验知识对预测误差模型更新,几种方法称为适应性提出了贝叶斯模型更新方法(<一个href="//www.thespel.com/journal/articles/10.3389/#B1">Akhlaghi et al ., 2017</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B2">Amini德黑兰的et al ., 2020</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B39">歌et al ., 2020 b</一个>)。最适应的贝叶斯模型的更新方法在文献中考虑一个零均值高斯白噪声未知的建模预测误差的协方差矩阵和估计误差协方差矩阵与其他模型参数或状态。<一个href="//www.thespel.com/journal/articles/10.3389/#B51">郑et al。(2018)</一个>开发了一种鲁棒自适应无味卡尔曼滤波(UKF)提高状态估计的精度和鲁棒性与不确定的非线性系统噪声协方差。在这种方法中,首先是非线性系统的状态估计使用一个标准的UKF (<一个href="//www.thespel.com/journal/articles/10.3389/#B46">吴和史密斯,2007</一个>),然后一个covariance-matching方法(<一个href="//www.thespel.com/journal/articles/10.3389/#B27">用,1972</一个>)是用来估计过程噪声和测量噪声的协方差矩阵。<一个href="//www.thespel.com/journal/articles/10.3389/#B3">Astroza et al。(2019)</一个>使用类似的方法,共同估计未知模型参数和预测误差的协方差矩阵的对角元素。<一个href="//www.thespel.com/journal/articles/10.3389/#B20">黄et al。(2020)</一个>建立了一个分层贝叶斯模型结合稀疏贝叶斯学习(<一个href="//www.thespel.com/journal/articles/10.3389/#B11">小费2001</一个>双卡尔曼滤波器。他们的层次模型采用两种推理水平,状态和参数估计和noise-parameter学习。他们认为一个零均值高斯分布的测量噪声协方差矩阵的对角元素都学会了单独从测量数据到当前时间步。<一个href="//www.thespel.com/journal/articles/10.3389/#B49">袁,郭鹤年(2016)</一个>提出了一种贝叶斯概率算法来估计噪声协方差矩阵的扩展卡尔曼滤波器使用最大<egydF4y2Bam>后验</egydF4y2Bam>的方法。他们的方法也适用于与时变非平稳噪声协方差矩阵。<一个href="//www.thespel.com/journal/articles/10.3389/#B39">歌et al。(2020 b)</一个>提出了两种自适应卡尔曼滤波器制定基于covariance-matching技术(<一个href="//www.thespel.com/journal/articles/10.3389/#B27">用,1972</一个>)联合估计未知模型参数和完整的预测误差的协方差矩阵。验证研究显示了性能优越的自适应滤波方法相比,标准UKF的预定义的分布预测错误。提到研究假设为高斯白噪声的预测误差。然而,预测误差的建模错误可能导致非零的意思(<一个href="//www.thespel.com/journal/articles/10.3389/#B34">Sanayei et al ., 2001</一个>)。为了解决这个问题,<一个href="//www.thespel.com/journal/articles/10.3389/#B24">Kontoroupi和史密斯(2016)</一个>提出了一种贝叶斯方法估计偏差(非零均值)预测误差。他们认为,预测误差的均值向量和协方差矩阵是定常和高斯和inverse-Wishart分布,分别。在之前的工作,<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., (2022</一个>)开发了一种两步边际最大<egydF4y2Bam>后验</egydF4y2Bam>(MAP)估计方法找到未知模型参数的点估计和预测误差统计,均值向量和协方差矩阵的预测误差被认为是时变的。</p><pcgydF4y2Balass="mb15">在本文中,我们介绍一个完全不同的数学方法更好的性能,在我们以前的工作相比,(<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>)估计未知模型参数和统计特性(均值向量和协方差矩阵)的预测误差,以及逼近他们的联合后验分布。这个高维联合后验分布的准确计算是棘手的,那么这个过程需要近似(<一个href="//www.thespel.com/journal/articles/10.3389/#B38">Šmidl奎因,2006</一个>)。可以使用两个近似方案:随机或取样方法,如马尔可夫链蒙特卡罗(密度)(<一个href="//www.thespel.com/journal/articles/10.3389/#B10">主教和Nasrabadi, 2006</一个>等)和确定性或变分框架变分贝叶斯(VB) (<一个href="//www.thespel.com/journal/articles/10.3389/#B31">Opper萨阿德,2001</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B7">比尔,2003</一个>)。抽样方法相比,VB易于分析方法在计算上更少的要求(<一个href="//www.thespel.com/journal/articles/10.3389/#B7">比尔,2003</一个>)。VB方法用于这项工作作为一个工具来隔离后分配到单独的组件,可以帮助分析解决问题。VB方法已成功申请联合州和噪声估计在导航、目标跟踪和控制好相关应用程序(<一个href="//www.thespel.com/journal/articles/10.3389/#B22">黄et al ., 2017</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B50">Zhang et al ., 2018</一个>)。在这些应用程序,VB自适应卡尔曼滤波方法用于联合估计零平均预测误差的协方差矩阵和线性的状态(<一个href="//www.thespel.com/journal/articles/10.3389/#B36">Sarkka Nummenmaa, 2009</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B41">太阳et al ., 2012</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B21">黄et al ., 2016</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B22">黄et al ., 2017</一个>)或非线性(<一个href="//www.thespel.com/journal/articles/10.3389/#B35">Sarkka Hartikainen, 2013</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B37">施et al ., 2018</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B40">太阳et al ., 2018</一个>)的状态空间模型。VB方法假设近似联合分布是一些单一的产物——或多变量因素和使用Kullback-Leibler(吉隆坡)分歧最小化之间的差异近似和真正的后部。在本文中,我们介绍一种新的自适应方法基于VB的非线性模型更新方法近似的联合后验分布的未知模型参数和统计特征在每个时间步预测误差。</p><pcgydF4y2Balass="mb15">本文结构如下:第二节提供了模型更新问题陈述。第三节给出了详细的推导提出了VB的估算方法的联合后验分布未知模型参数和预测误差的统计特征。VB的制定方法然后比较与两步边际地图估计方法(<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>在第四节)。在第五节和第六节,该方法由两个模型验证更新案例:一个与时变测量噪声和其他建模误差。结果相比的两步估计方法发表在边际地图<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., (2022</一个>)和一个非自适应的贝叶斯模型更新方法。最后,第七节中给出的结论。</p><一个我d="h3" name="h3"></a> <h2>2模型更新问题陈述</h2><pcgydF4y2Balass="mb15">我们认为测量响应的非线性动态系统(或线性)<我nline-formula id="inf1"> <math id="m1"> <mrow> <mi mathvariant="bold"> y</gydF4y2Bami> </mrow> </math> </inline-formula>及其对应的模型(如。,有限元(FE)]预测<我nline-formula id="inf2"> <math id="m2"> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> </mrow> </mfenced> </mrow> </math> </inline-formula>,在那里<我nline-formula id="inf3"> <math id="m3"> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> </mrow> </math> </inline-formula>是未知模型参数的矢量。参数估计问题<我nline-formula id="inf4"> <math id="m4"> <mrow> <mi> k</gydF4y2Bami> <mo> =</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> <mo> ,</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> <mo> ,</gydF4y2Bamo> <mo> ⋯</gydF4y2Bamo> <mo> ,</gydF4y2Bamo> <mi> N</gydF4y2Bami> </mrow> </math> </inline-formula>可以制定为(<一个href="//www.thespel.com/journal/articles/10.3389/#B19">微积分,2004</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B15">Ebrahimian et al ., 2015</一个>)</p><dgydF4y2Ba我vclass="equationImageholder"> <math id="e1"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> =</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mrow> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> <mo> +</gydF4y2Bamo> <msub> <mi mathvariant="bold"> γ</gydF4y2Bami> <mrow> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e2"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> =</gydF4y2Bamo> <mi mathvariant="bold"> h</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf5"> <math id="m7"> <mrow> <msub> <mi mathvariant="bold"> γ</gydF4y2Bami> <mrow> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </math> </inline-formula>过程噪声和吗<我nline-formula id="inf6"> <math id="m8"> <mrow> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>预测误差。在这项研究中,输入部队被认为是已知的,所以对于符号简洁,依赖模型的预测响应输入力量不是明确情商所示。<一个href="//www.thespel.com/journal/articles/10.3389/#e2">2</一个>。过程噪声都要遵循一个零均值高斯白噪声过程与协方差矩阵<b>gydF4y2Ba问</b>,gydF4y2Ba也就是说,<我nline-formula id="inf7"> <math id="m9"> <mrow> <msub> <mi mathvariant="bold"> γ</gydF4y2Bami> <mrow> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> <mo> ∼</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mn mathvariant="bold"> 0</gydF4y2Bamn> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> 问</gydF4y2Bami> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。在非自适应的贝叶斯模型更新方法,预测误差被认为是一个零均值高斯白噪声过程与一个常数或定常协方差矩阵<b>R</b>,gydF4y2gydF4y2BaBa也就是说,<我nline-formula id="inf8"> <math id="m10"> <mrow> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ∼</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mn mathvariant="bold"> 0</gydF4y2Bamn> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。方程式中定义的参数估计问题。<一个href="//www.thespel.com/journal/articles/10.3389/#e1">1</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e2">2</一个>、非自适应的贝叶斯方法可以用来寻找估计前两个统计未知模型参数的时候(<一个href="//www.thespel.com/journal/articles/10.3389/#B15">Astroza et al ., 2015</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B29">Nabiyan et al ., 2020</一个>)。然而,在自适应贝叶斯模型更新方法,预测误差可以建模为一个非平稳高斯随机过程与未知和时变均值向量<我nline-formula id="inf9"> <math id="m11"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和协方差矩阵<我nline-formula id="inf10"> <math id="m12"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,也就是说,<我nline-formula id="inf11"> <math id="m13"> <mrow> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ∼</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,估计递归和共同的未知向量模型参数<我nline-formula id="inf12"> <math id="m14"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。</p><pcgydF4y2Balass="mb15">在本文中,我们开发了一种新的自适应递推贝叶斯模型更新方法。像其他递归贝叶斯模型更新算法,该方法有两个步骤在每一个时间<我nline-formula id="inf13"> <math id="m15"> <mrow> <mi> k</gydF4y2Bami> </mrow> </math> </inline-formula>:“预测”和“修正”(<一个href="//www.thespel.com/journal/articles/10.3389/#B5">Astroza et al ., 2017</一个>)。在“预测”一步,新的测量<我nline-formula id="inf14"> <math id="m16"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>在时间<我nline-formula id="inf15"> <math id="m17"> <mrow> <mi> k</gydF4y2Bami> </mrow> </math> </inline-formula>还没有给估计过程。因此,之前的估计<我nline-formula id="inf16"> <math id="m18"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf17"> <math id="m19"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf18"> <math id="m20"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>用-上标,通过一个动态模型预测使用的后验估计前一个时间步<egydF4y2Bam>k</egydF4y2Bam>1。情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e1">1</一个>可以用作未知模型参数的动态模型<我nline-formula id="inf19"> <math id="m21"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>(<一个href="//www.thespel.com/journal/articles/10.3389/#B4">Astroza et al ., 2014</一个>;<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>)。预测的先验估计<我nline-formula id="inf20"> <math id="m22"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf21"> <math id="m23"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>中定义的动态模型<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., (2022</一个>可以使用),考虑的遗忘因子参数<我nline-formula id="inf22"> <math id="m24"> <mrow> <mi> ρ</gydF4y2Bami> <mo> ∈</gydF4y2Bamo> <mrow> <mfenced open close="]" separators="|"> <mrow> <mrow> <mfenced open="(" close="" separators="|"> <mrow> <mn> 0 1</gydF4y2Bamn> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>和<我nline-formula id="inf23"> <math id="m25"> <mrow> <msup> <mi> ρ</gydF4y2Bami> <mo> ′</gydF4y2Bamo> </msup> <mo> ∈</gydF4y2Bamo> <mrow> <mfenced open close="]" separators="|"> <mrow> <mrow> <mfenced open="(" close="" separators="|"> <mrow> <mn> 0 1</gydF4y2Bamn> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。导致这些动态模型<我nline-formula id="inf24"> <math id="m26"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mrow> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>和<我nline-formula id="inf25"> <math id="m27"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mrow> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>。“修正”步骤,更新之前估计的新测量<我nline-formula id="inf26"> <math id="m28"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>获得后验估计,用<我nline-formula id="inf27"> <math id="m29"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf28"> <math id="m30"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf29"> <math id="m31"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>。进一步更新过程描述如下:</p><pcgydF4y2Balass="mb15">在我们以前的工作(<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>),我们开发了一种两步最大值<egydF4y2Bam>后验</egydF4y2Bam>(MAP)估计方法来估计<我nline-formula id="inf30"> <math id="m32"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf31"> <math id="m33"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf32"> <math id="m34"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>通过最大化联合后验分布<我nline-formula id="inf33"> <math id="m35"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,也就是说,</p><d我vclass="equationImageholder"> <math id="e3"> <mrow> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <munder> <mi> argmax</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </munder> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> <mo> 。</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 3</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">为了解决这张地图问题,我们把问题分成两个迭代MAP估计问题<我nline-formula id="inf34"> <math id="m37"> <mrow> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <munder> <mi> argmax</gydF4y2Bami> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </munder> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> y</gydF4y2Bami> </mrow> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>和<我nline-formula id="inf35"> <math id="m38"> <mrow> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <munder> <mi> argmax</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </munder> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。在本文中,我们的目标是找到整个联合后验分布未知模型参数和噪声,例如,<我nline-formula id="inf36"> <math id="m39"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。然而,这种联合后验分布分析工作是不容易处理,因为变量的数量高,联合分布是高度复杂的。为了克服这个问题,我们使用了VB方法近似这联合后验分布。VB方法和推导细节在下一节中解释。</p><一个我d="h4" name="h4"></a> <h2>3变分贝叶斯(VB)方法</h2><pcgydF4y2Balass="mb15">VB是一个联合分布近似方法<我nline-formula id="inf37"> <math id="m40"> <mrow> <mi> p</gydF4y2Bami> </mrow> </math> </inline-formula>由一个联合分布<我nline-formula id="inf38"> <math id="m41"> <mrow> <mi> 问</gydF4y2Bami> </mrow> </math> </inline-formula>可映像成变量或grouped-variable因素。Kullback-Leibler(吉隆坡)散度标准是被用来制造<我nline-formula id="inf39"> <math id="m42"> <mrow> <mi> 问</gydF4y2Bami> </mrow> </math> </inline-formula>尽可能接近<我nline-formula id="inf40"> <math id="m43"> <mrow> <mi> p</gydF4y2Bami> </mrow> </math> </inline-formula>(<一个href="//www.thespel.com/journal/articles/10.3389/#B10">主教和Nasrabadi, 2006</一个>)。使用VB的方法,我们近似的联合后验分布未知模型参数和预测误差均值向量和协方差矩阵分离这个联合后验分布到两个因素如下:</p><d我vclass="equationImageholder"> <math id="e4"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> ≈</gydF4y2Bamo> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 4</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf41"> <math id="m45"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>和<我nline-formula id="inf42"> <math id="m46"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>是未知的分布,可以通过最小化KL获得正确的——之间的分歧和Eq的左边。<一个href="//www.thespel.com/journal/articles/10.3389/#e4">4</一个>。KL散度被定义为</p><d我vclass="equationImageholder"> <math id="e5"> <mrow> <mi mathvariant="normal"> K</gydF4y2Bami> <mi mathvariant="normal"> l</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mrow> <mfenced open="‖" close="" separators="|"> <mrow> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mrow> <mstyle displaystyle="true"> <mo> ∫</gydF4y2Bamo> </mstyle> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mfrac> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> |</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mi> d</gydF4y2Bami> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mi> d</gydF4y2Bami> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mi> d</gydF4y2Bami> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mo> 。</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 5</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">使用变分法来减少上述KL散度对每个人<我nline-formula id="inf43"> <math id="m48"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>和<我nline-formula id="inf44"> <math id="m49"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>在保持固定将导致另一个方程式。<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>。这个推导中可以找到的细节<一个href="//www.thespel.com/journal/articles/10.3389/#B43">魏因斯托克(1974</一个>);<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#B42">Tzikas et al ., (2008</一个>)。</p><d我vclass="equationImageholder"> <math id="e6"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <msub> <mi> c</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mo> ⁡</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 6</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e7"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <msub> <mi> c</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mo> ⁡</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 7</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf45"> <math id="m52"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mi mathvariant="bold"> x</gydF4y2Bami> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mi> f</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>表示的期望值<我nline-formula id="inf46"> <math id="m53"> <mrow> <mi> f</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>关于<b>x</b>gydF4y2gydF4y2BaBa的概率密度函数<我nline-formula id="inf47"> <math id="m54"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,也就是说,<我nline-formula id="inf48"> <math id="m55"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mi mathvariant="bold"> x</gydF4y2Bami> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mi> f</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mrow> <mstyle displaystyle="true"> <mo> ∫</gydF4y2Bamo> </mstyle> <mrow> <mi> f</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> </mfenced> </mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> </mfenced> </mrow> <mi> d</gydF4y2Bami> <mi mathvariant="bold"> x</gydF4y2Bami> </mrow> </mrow> </mrow> </math> </inline-formula>。条款<我nline-formula id="inf49"> <math id="m56"> <mrow> <msub> <mi> c</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf50"> <math id="m57"> <mrow> <msub> <mi> c</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> </mrow> </math> </inline-formula>表示常量与变量<我nline-formula id="inf51"> <math id="m58"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf52"> <math id="m59"> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </math> </inline-formula>,分别。因为方程式。<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>耦合通过这个词吗<我nline-formula id="inf53"> <math id="m60"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>、分析解决方案不可用。因此,定点迭代算法可以用来找到方程式的近似解。<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>。为此,我们努力扩大方程式的右手边。<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>从他们的内心括号外的通过以下步骤。</p><pcgydF4y2Balass="mb15">的联合分布<我nline-formula id="inf54"> <math id="m61"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>在方程式。<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>可以被分解为</p><d我vclass="equationImageholder"> <math id="e8"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> <mo> ×</gydF4y2Bamo> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 8</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf55"> <math id="m63"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold-italic"> y</gydF4y2Bami> <mi mathvariant="normal"> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>似然函数,<我nline-formula id="inf56"> <math id="m64"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>的先验分布<我nline-formula id="inf57"> <math id="m65"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf58"> <math id="m66"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>之前的联合分布吗<我nline-formula id="inf59"> <math id="m67"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf60"> <math id="m68"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf61"> <math id="m69"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>因为这是一个递归算法,也就是说,在每个时间步,只有新的测量吗<我nline-formula id="inf62"> <math id="m70"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>用于更新参数,<我nline-formula id="inf63"> <math id="m71"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>取决于过去的测量。</p><pcgydF4y2Balass="mb15">在这里,我们进一步扩大右边的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e8">8</一个>。基于情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e2">2</一个>的似然函数<我nline-formula id="inf64"> <math id="m72"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>有一个高斯分布</p><d我vclass="equationImageholder"> <math id="e9"> <mrow> <mtable columnalign="center"> <mtr> <mtd> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> 。</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 9</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">右边的第二个和第三个条件的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e8">8</一个>,它是假定,类似于(<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>),<gydF4y2Ba我nline-formula id="inf65"> <math id="m74"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf66"> <math id="m75"> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </math> </inline-formula>有先验分布的高斯和normal-inverse-Wishart(此类),分别。此类分布是高斯分布(或正常)的产品和一个inverse-Wishart (IW)。选择此类先验分布<我nline-formula id="inf67"> <math id="m76"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>因为它是一个高斯分布的共轭先验均值向量和协方差矩阵未知的可能性。后的共轭性保证相同的函数形式和先验分布(<一个href="//www.thespel.com/journal/articles/10.3389/#B30">欧哈根和福斯特,2004</一个>)。因此,第二和第三项右边的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e8">8</一个>可以写成:</p><d我vclass="equationImageholder"> <math id="e10"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 10</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e11"> <mrow> <mtable columnalign="center"> <mtr> <mtd> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mi> 我</gydF4y2Bami> <mi> W</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mo> ×</gydF4y2Bamo> <mi> 我</gydF4y2Bami> <mi> W</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 11</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf68"> <math id="m79"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>和<我nline-formula id="inf69"> <math id="m80"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>分别是均值向量和协方差矩阵未知的模型参数<我nline-formula id="inf70"> <math id="m81"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>给定的测量<我nline-formula id="inf71"> <math id="m82"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </math> </inline-formula>但不是<我nline-formula id="inf72"> <math id="m83"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。-标代表之前的估计。<我nline-formula id="inf73"> <math id="m84"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf74"> <math id="m85"> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf75"> <math id="m86"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf76"> <math id="m87"> <mrow> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>统计参数的先验估计应用于此类分布。<我nline-formula id="inf77"> <math id="m88"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>之前的预估吗<我nline-formula id="inf78"> <math id="m89"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf79"> <math id="m90"> <mrow> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>是对称正定矩阵。<我nline-formula id="inf80"> <math id="m91"> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>和<我nline-formula id="inf81"> <math id="m92"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>是信心参数和自由度参数,分别。他们是标量参数和满足<我nline-formula id="inf82"> <math id="m93"> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> ></gydF4y2Bamo> <mn> 0</gydF4y2Bamn> </mrow> </math> </inline-formula>和<我nline-formula id="inf83"> <math id="m94"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> ></gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </math> </inline-formula>,在那里<我nline-formula id="inf84"> <math id="m95"> <mrow> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>是测量传感器的数量。</p><pcgydF4y2Balass="mb15">用方程式。<一个href="//www.thespel.com/journal/articles/10.3389/#e9">9</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e10">10</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e11">11</一个>为情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e8">8</一个>,我们获得</p><d我vclass="equationImageholder"> <math id="e12"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ×</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mi> 我</gydF4y2Bami> <mi> W</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> <mo> 。</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 12</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">高斯分布(或正常)和inverse-Wishart (IW)分布比例以下表达式:</p><d我vclass="equationImageholder"> <math id="e13_a"> <mrow> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> Σ</gydF4y2Bami> </mrow> </mfenced> </mrow> <mo> ∝</gydF4y2Bamo> <msup> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mi mathvariant="bold"> Σ</gydF4y2Bami> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mn> 1</gydF4y2Bamn> <mo> /</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> </mrow> </mfenced> </mrow> </mrow> </msup> <mo> ⁡</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mi mathvariant="bold"> μ</gydF4y2Bami> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mi mathvariant="bold"> Σ</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> x</gydF4y2Bami> <mo> −</gydF4y2Bamo> <mi mathvariant="bold"> μ</gydF4y2Bami> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> </mrow> </math> <label>(才十三岁,一个)</gydF4y2Balabel> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e13_b"> <mrow> <mi> 我</gydF4y2Bami> <mi> W</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> Σ</gydF4y2Bami> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <mi> v</gydF4y2Bami> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> V</gydF4y2Bami> </mrow> </mfenced> </mrow> <mo> ∝</gydF4y2Bamo> <msup> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mi mathvariant="bold"> Σ</gydF4y2Bami> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mrow> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> v</gydF4y2Bami> <mo> +</gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> +</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </mfenced> </mrow> <mo> /</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> </mrow> </mrow> </msup> <mo> ⁡</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi mathvariant="normal"> t</gydF4y2Bami> <mi mathvariant="normal"> r</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="bold"> V</gydF4y2Bami> <msup> <mi mathvariant="bold"> Σ</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> </mrow> </math> <label>(13 b)</gydF4y2Balabel> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf85"> <math id="m99"> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mo> ∙</gydF4y2Bamo> </mrow> </mfenced> </mrow> </math> </inline-formula>代表行列式和<我nline-formula id="inf86"> <math id="m100"> <mrow> <mi mathvariant="normal"> t</gydF4y2Bami> <mi mathvariant="normal"> r</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mo> ∙</gydF4y2Bamo> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>一个矩阵表示的痕迹。符号”<我nline-formula id="inf87"> <math id="m101"> <mrow> <mo> ∝</gydF4y2Bamo> </mrow> </math> </inline-formula>“代表”成正比”是用于情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e13_a">13</一个>,规范条款将被忽略。</p><pcgydF4y2Balass="mb15">用情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e12">12</一个>和正常的定义和信息战分布在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e13_a">13</一个>,这个术语<我nline-formula id="inf88"> <math id="m102"> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,这是用于方程式<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>可以扩展如下:</p><d我vclass="equationImageholder"> <math id="e14"> <mrow> <mtable columnalign="center"> <mtr> <mtd> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <mrow> <mo> (</gydF4y2Bamo> </mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> <msup> <mrow> <mo> )</gydF4y2Bamo> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> +</gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> +</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> <mn> 2</gydF4y2Bamn> </mfrac> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi mathvariant="normal"> t</gydF4y2Bami> <mi mathvariant="normal"> r</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <msub> <mi> c</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 14</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf89"> <math id="m104"> <mrow> <msub> <mi> c</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> </mrow> </math> </inline-formula>对所有变量代表一个常数<我nline-formula id="inf90"> <math id="m105"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf91"> <math id="m106"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。</p><pcgydF4y2Balass="mb15">现在,拥有的扩张<我nline-formula id="inf92"> <math id="m107"> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e14">14</一个>在方程式,我们计算期望术语。<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>。从每学期获得期望的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e14">14</一个>,期望在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba我nline-formula id="inf93"> <math id="m108"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,可以表示如下:</p><d我vclass="equationImageholder"> <math id="e15"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <msub> <mi> c</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 15</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf94"> <math id="m110"> <mrow> <msub> <mi> c</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>是期望的总和情商的右边的所有条款。<一个href="//www.thespel.com/journal/articles/10.3389/#e14">14</一个>,除了第二和第四条款,并与尊重是恒定的<我nline-formula id="inf95"> <math id="m111"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。应该注意的是,第四项情商的期望。<一个href="//www.thespel.com/journal/articles/10.3389/#e14">14</一个>等于本身,因为它不取决于什么<我nline-formula id="inf96"> <math id="m112"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf97"> <math id="m113"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。</p><pcgydF4y2Balass="mb15">使用<一个href="//www.thespel.com/journal/articles/10.3389/#SM1">补充附录引理S1</一个>在附录中,期望在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e15">15</一个>可以评估</p><d我vclass="equationImageholder"> <math id="e16"> <mtable class="aligned"> <mtr> <mtd columnalign="left"> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo> =</gydF4y2Bamo> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mrow> <mo> (</gydF4y2Bamo> </mrow> <mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <mi mathvariant="normal"> t</gydF4y2Bami> <mi mathvariant="normal"> r</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mo> ×</gydF4y2Bamo> <msub> <mrow> <mi mathvariant="normal"> c</gydF4y2Bami> <mi mathvariant="normal"> o</gydF4y2Bami> <mi mathvariant="normal"> v</gydF4y2Bami> </mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi mathvariant="normal"> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi mathvariant="normal"> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mo> ]</gydF4y2Bamo> </mrow> </mrow> <mo> 。</gydF4y2Bamo> </mtd> </mtr> </mtable> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 16</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">现在,我们考虑<我nline-formula id="inf98"> <math id="m115"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>作为此类分布,即<我nline-formula id="inf99"> <math id="m116"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> |</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> /</gydF4y2Bamo> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mi> 我</gydF4y2Bami> <mi> W</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="|" close="" separators="|"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,加上上标表示后验估计。因此,用均值和协方差<我nline-formula id="inf100"> <math id="m117"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,也就是说,<我nline-formula id="inf101"> <math id="m118"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>和<我nline-formula id="inf102"> <math id="m119"> <mrow> <msub> <mrow> <mi mathvariant="normal"> c</gydF4y2Bami> <mi mathvariant="normal"> o</gydF4y2Bami> <mi mathvariant="normal"> v</gydF4y2Bami> </mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi mathvariant="normal"> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi mathvariant="normal"> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> /</gydF4y2Bamo> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e16">16</一个>,以期望对<我nline-formula id="inf103"> <math id="m120"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,我们将</p><d我vclass="equationImageholder"> <math id="e17"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <mfrac> <mrow> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> <mo> 。</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 17</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">应该注意的是,在推导情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e17">17</一个>,我们使用<我nline-formula id="inf104"> <math id="m122"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> </mrow> </math> </inline-formula>(<一个href="//www.thespel.com/journal/articles/10.3389/#B18">Granstrom Orguner, 2011</一个>),<gydF4y2Ba我nline-formula id="inf105"> <math id="m123"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>信息战分布表示的意思吗<我nline-formula id="inf106"> <math id="m124"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,可以计算如下(<一个href="//www.thespel.com/journal/articles/10.3389/#B30">欧哈根和福斯特,2004</一个>)。</p><d我vclass="equationImageholder"> <math id="e18"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mfrac> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> −</gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </mfrac> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 18</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf107"> <math id="m126"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ></gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> +</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </math> </inline-formula>(<一个href="//www.thespel.com/journal/articles/10.3389/#B30">欧哈根和福斯特,2004</一个>)。</p><pcgydF4y2Balass="mb15">在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e6">6</一个>,<gydF4y2Ba我nline-formula id="inf108"> <math id="m127"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>指数成正比的吗<我nline-formula id="inf109"> <math id="m128"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </msub> <mrow> <mfenced open="[" close="" separators="|"> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。所以,用情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e17">17</一个>为情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e15">15</一个>和Eq的指数。<一个href="//www.thespel.com/journal/articles/10.3389/#e15">15</一个>,可以看出</p><d我vclass="equationImageholder"> <math id="e19"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ∝</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> ×</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> 。</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 19</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">通过线性化<我nline-formula id="inf110"> <math id="m130"> <mrow> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e19">19</一个>由一阶泰勒展开式<我nline-formula id="inf111"> <math id="m131"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,也就是说,<我nline-formula id="inf112"> <math id="m132"> <mrow> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ≃</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,在那里<我nline-formula id="inf113"> <math id="m133"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>是铁的灵敏度矩阵模型对吗<我nline-formula id="inf114"> <math id="m134"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>在<我nline-formula id="inf115"> <math id="m135"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula><b>,</b>gydF4y2Ba也就是说,<我nline-formula id="inf116"> <math id="m136"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msub> <mrow> <mfenced open close="|" separators="|"> <mrow> <mfrac> <mrow> <mo> ∂</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> <mrow> <mo> ∂</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> =</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </msub> </mrow> </math> </inline-formula>,情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e19">19</一个>收益率</p><d我vclass="equationImageholder"> <math id="e20"> <mtable class="aligned"> <mtr> <mtd columnalign="left"> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ∝</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo> ×</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mrow> <mrow> <mo> (</gydF4y2Bamo> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mo> ×</gydF4y2Bamo> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> <mrow> <mo> )</gydF4y2Bamo> </mrow> </mrow> <mo> 。</gydF4y2Bamo> </mrow> </mtd> </mtr> </mtable> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 20.</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">第一个在右边的情商指数项。<一个href="//www.thespel.com/journal/articles/10.3389/#e20">20.</一个>显示了一个高斯分布<我nline-formula id="inf117"> <math id="m138"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>忽略了标准化的术语。通过使用<一个href="//www.thespel.com/journal/articles/10.3389/#SM1">补充附录引理S2</一个>在附录中,右边的第二个指数项情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e20">20.</一个>也代表了一个高斯分布<我nline-formula id="inf118"> <math id="m139"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。因此,情商的右边。<一个href="//www.thespel.com/journal/articles/10.3389/#e20">20.</一个>是两个高斯分布的产物,基于<一个href="//www.thespel.com/journal/articles/10.3389/#SM1">补充附录引理S3</一个>在附录中,结果在一个高斯分布,即:<我nline-formula id="inf119"> <math id="m140"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> |</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。获得这个高斯分布的参数,我们匹配条件在左边和右边的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e20">20.</一个>,从而导致</p><d我vclass="equationImageholder"> <math id="e21_a"> <mrow> <mfenced open close=")" separators="|"> <mrow> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> +</gydF4y2Bamo> <msub> <mi mathvariant="bold"> K</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mrow> </mfenced> </mrow> <mspace width="5em"></mspace> </math> <label>(21日)</gydF4y2Balabel> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e21_b"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> K</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <msub> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> y</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <msubsup> <mi mathvariant="bold"> K</gydF4y2Bami> <mi> k</gydF4y2Bami> <mi> T</gydF4y2Bami> </msubsup> </mrow> <mspace width="5em"></mspace> </math> <label>(21 b)</gydF4y2Balabel> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf120"> <math id="m143"> <mrow> <msub> <mi mathvariant="bold"> K</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> =</gydF4y2Bamo> <msub> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> y</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> </mrow> </math> </inline-formula>,<我nline-formula id="inf121"> <math id="m144"> <mrow> <msub> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> =</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> </mrow> </math> </inline-formula>,<我nline-formula id="inf122"> <math id="m145"> <mrow> <msub> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> y</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> =</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> −</gydF4y2Bamo> </msubsup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <mo> +</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>。值得注意的是,情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e21_a">(21日)</一个>和情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e21_b">(21 b)</一个>类似于非自适应的贝叶斯模型更新配方,除了这个词吗<我nline-formula id="inf123"> <math id="m146"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,这是添加在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e21_a">(21日)</一个>考虑非零平均预测误差。</p><pcgydF4y2Balass="mb15">以类似的方式,我们可以评估期望在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>如下。情商的数学期望。<一个href="//www.thespel.com/journal/articles/10.3389/#e14">14</一个>关于<我nline-formula id="inf124"> <math id="m147"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>导致</p><d我vclass="equationImageholder"> <math id="e22"> <mtable class="aligned"> <mtr> <mtd columnalign="left"> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> +</gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> +</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> <mn> 2</gydF4y2Bamn> </mfrac> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi mathvariant="normal"> t</gydF4y2Bami> <mi mathvariant="normal"> r</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <msub> <mi> c</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> </mtd> </mtr> </mtable> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 22</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="noindent">在哪里<我nline-formula id="inf125"> <math id="m149"> <mrow> <msub> <mi> c</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> </mrow> </math> </inline-formula>的和预期的第三,第四,最后情商的右边。<一个href="//www.thespel.com/journal/articles/10.3389/#e14">14</一个>和是恒定的<我nline-formula id="inf126"> <math id="m150"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf127"> <math id="m151"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。</p><pcgydF4y2Balass="mb15">现在,考虑<我nline-formula id="inf128"> <math id="m152"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> |</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>和线性化<我nline-formula id="inf129"> <math id="m153"> <mrow> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>由一阶泰勒展开式<我nline-formula id="inf130"> <math id="m154"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,也就是说,<我nline-formula id="inf131"> <math id="m155"> <mrow> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ≃</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,在那里<我nline-formula id="inf132"> <math id="m156"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>的灵敏度矩阵模型对吗<我nline-formula id="inf133"> <math id="m157"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>在<我nline-formula id="inf134"> <math id="m158"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,期望项右边的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e22">22</一个>可以得到如下使用引理1在附录中。</p><d我vclass="equationImageholder"> <math id="e23"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <mi mathvariant="normal"> t</gydF4y2Bami> <mi mathvariant="normal"> r</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> </mrow> </mfenced> </mrow> <mo> 。</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 23</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">基于情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e7">7</一个>,<gydF4y2Ba我nline-formula id="inf135"> <math id="m160"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>指数成正比的吗<我nline-formula id="inf136"> <math id="m161"> <mrow> <msub> <mi mathvariant="normal"> E</gydF4y2Bami> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </msub> <mo> (</gydF4y2Bamo> <mi> ln</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。用情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e23">23</一个>为情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e22">22</一个>和Eq的指数。<一个href="//www.thespel.com/journal/articles/10.3389/#e22">22</一个>,<gydF4y2Ba我nline-formula id="inf137"> <math id="m162"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>可以找到如下:</p><d我vclass="equationImageholder"> <math id="e24"> <mtable class="aligned"> <mtr> <mtd columnalign="left"> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> ∝</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <msup> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mrow> <mn> 1</gydF4y2Bamn> <mo> /</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> </mrow> </msup> </mrow> </mfrac> <mo> ×</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mrow> <mo> (</gydF4y2Bamo> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <mo> ×</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> <mrow> <mo> )</gydF4y2Bamo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo> ×</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <msup> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <mn> 1</gydF4y2Bamn> <mo> /</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> </mrow> </msup> </mrow> </mfrac> <mo> ×</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> ×</gydF4y2Bamo> <msup> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mrow> <mo> −</gydF4y2Bamo> <mrow> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> +</gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> +</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </mfenced> </mrow> <mo> /</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> </mrow> </mrow> </msup> <mo> ×</gydF4y2Bamo> <mi> 经验值</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mo> −</gydF4y2Bamo> <mfrac> <mrow> <mn> 1</gydF4y2Bamn> </mrow> <mrow> <mn> 2</gydF4y2Bamn> </mrow> </mfrac> <mi mathvariant="normal"> t</gydF4y2Bami> <mi mathvariant="normal"> r</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> +</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> </mrow> </mfenced> </mrow> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> 。</gydF4y2Bamo> </mtd> </mtr> </mtable> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 24</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">右边的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e24">24</一个>包括两个高斯分布的乘积<我nline-formula id="inf138"> <math id="m164"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和一个IW分布<我nline-formula id="inf139"> <math id="m165"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。这两个高斯分布的产品导致了高斯分布基于引理3在附录中。因此,情商的右边。<一个href="//www.thespel.com/journal/articles/10.3389/#e24">24</一个>导致此类分布,这是一个产品的正态分布<我nline-formula id="inf140"> <math id="m166"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和一个IW分布<我nline-formula id="inf141"> <math id="m167"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。用<我nline-formula id="inf142"> <math id="m168"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mo> =</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> |</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <mfrac> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mi> 我</gydF4y2Bami> <mi> W</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> |</gydF4y2Bamo> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>在左边的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e24">24</一个>,利用引理4在附录中,此类的四个参数分布可以通过匹配得到如下条款以左边和右边的情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e24">24</一个>。</p><d我vclass="equationImageholder"> <math id="e25_a"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mfrac> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mrow> <mn> 1</gydF4y2Bamn> <mo> +</gydF4y2Bamo> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> +</gydF4y2Bamo> <mfrac> <mn> 1</gydF4y2Bamn> <mrow> <mn> 1</gydF4y2Bamn> <mo> +</gydF4y2Bamo> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo> ,</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> </math> <label>(25)</gydF4y2Balabel> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e25_b"> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> <mo> +</gydF4y2Bamo> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> </math> <label>(25 b)</gydF4y2Balabel> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e25_c"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> <mo> +</gydF4y2Bamo> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> ,</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> </math> <label>(25度)</gydF4y2Balabel> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e25_d"> <mrow> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mo> +</gydF4y2Bamo> <mfrac> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <mrow> <mn> 1</gydF4y2Bamn> <mo> +</gydF4y2Bamo> <msubsup> <mi> λ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfrac> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <msub> <mi mathvariant="bold"> h</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mo> −</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <mo> +</gydF4y2Bamo> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <mo> 。</gydF4y2Bamo> </mrow> <mspace width="5em"></mspace> </math> <label>(25 d)</gydF4y2Balabel> <div class="clear"></div> </div> <p class="mb15">现在有<我nline-formula id="inf143"> <math id="m173"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>作为一个近似<我nline-formula id="inf144"> <math id="m174"> <mrow> <mi> p</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> |</gydF4y2Bamo> <msub> <mi mathvariant="bold"> y</gydF4y2Bami> <mrow> <mn> 1</gydF4y2Bamn> <mo> :</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>,我们可以评估的期望<我nline-formula id="inf145"> <math id="m175"> <mrow> <msub> <mi> 问</gydF4y2Bami> <mi mathvariant="bold"> θ</gydF4y2Bami> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <msub> <mi> 问</gydF4y2Bami> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi mathvariant="bold"> R</gydF4y2Bami> </mrow> </msub> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>代表点的估计未知模型参数和噪声。因此,<我nline-formula id="inf146"> <math id="m176"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf147"> <math id="m177"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf148"> <math id="m178"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e21_a">(21日)</一个>,情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e25_a">(25)</一个>,(Eq。<一个href="//www.thespel.com/journal/articles/10.3389/#e18">18</一个>),分别可以被认为是估计<我nline-formula id="inf149"> <math id="m179"> <mrow> <msub> <mi mathvariant="bold"> θ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> </mrow> </math> </inline-formula> <inline-formula id="inf150"> <math id="m180"> <mrow> <msub> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,<我nline-formula id="inf151"> <math id="m181"> <mrow> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e21_a">(21日)</一个>,情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e25_a">(25)</一个>,(Eq。<一个href="//www.thespel.com/journal/articles/10.3389/#e18">18</一个>)是耦合方程,可以反复使用定点迭代算法解决。在每次迭代中,模型预测响应和灵敏度矩阵需要更新基于未知模型参数更新<我nline-formula id="inf152"> <math id="m182"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>;然而,在每个时间步长计算灵敏度矩阵可以计算要求。减少执行时间,我们可以用之前的灵敏度矩阵<我nline-formula id="inf153"> <math id="m183"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>在情商。<一个href="//www.thespel.com/journal/articles/10.3389/#e25_d">(25 d)</一个>。自收敛标准没有改变,使用<我nline-formula id="inf154"> <math id="m184"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>而不是<我nline-formula id="inf155"> <math id="m185"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>对最终的评估结果没有影响。本文用有限差分法计算灵敏度矩阵在每个时间步。该算法的递归VB联合模型和噪声识别提出了<一个href="//www.thespel.com/journal/articles/10.3389/#F1">图1</一个>。这个框架可以使用任何有限元建模与仿真平台,如OpenSees用于这项研究。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 图1</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g001.jpg" name="Figure1" target="_blank"><img id="F1" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g001.jpg"></a> <p><b>图1</b>gydF4y2Ba。递归算法贝叶斯(VB)变分方法。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <a id="h5" name="h5"></a> <h2>4 VB方法的比较与两步边际地图估计方法</h2><pcgydF4y2Balass="mb15">在本节中,我们比较了提出了变分贝叶斯(VB)方法与两步边际地图最近开发的估计方法<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al。(2022)</一个>进行联合估计未知模型参数和均值向量和协方差矩阵的预测误差。两种方法的配方密切相似。有两个主要的两种方法之间的差异,解释如下。首先,在边际地图两步估计的方法,作为信息战分布模式<我nline-formula id="inf201"> <math id="m231"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>点估计,而在VB方法,信息战的均值分布分配<我nline-formula id="inf202"> <math id="m232"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="normal"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>(见<一个href="//www.thespel.com/journal/articles/10.3389/#T1">表1</一个>)。这种差异的原因是地图的方法是使用前的方法,而使用的预期值是后者。值得注意的是,模式和信息战分布不一致(<一个href="//www.thespel.com/journal/articles/10.3389/#B30">欧哈根和福斯特,2004</一个>)。第二,使用不同的方程来计算这个词<我nline-formula id="inf203"> <math id="m233"> <mrow> <msubsup> <mi mathvariant="normal"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>所示<一个href="//www.thespel.com/journal/articles/10.3389/#T1">表1</一个>。用来计算方程<我nline-formula id="inf204"> <math id="m234"> <mrow> <msubsup> <mi mathvariant="normal"> V</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>在VB方法中有一个额外的术语,例如,<我nline-formula id="inf205"> <math id="m235"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> </mrow> </math> </inline-formula>。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 表1</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-t001.jpg" name="Table1" target="_blank"><img id="T1" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-t001.jpg"></a> <p><b>表1</b>gydF4y2Ba。差异提出了VB法和边际地图两步估计法。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <p class="mb15">在大多数应用程序中,这两个有小差异影响的结果,因为以下原因。首先,模式之间的差异和信息战分布随时间减少的价值<我nline-formula id="inf212"> <math id="m242"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>增加时间在分母上的特征方程<我nline-formula id="inf213"> <math id="m243"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>。其次,正如未知模型参数的协方差矩阵<我nline-formula id="inf214"> <math id="m244"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>减少通过贝叶斯估计(<一个href="//www.thespel.com/journal/articles/10.3389/#B33">歌et al ., 2020 a</一个>),这些额外的术语对结果的影响也可以忽略不计。然而,额外的术语<我nline-formula id="inf215"> <math id="m245"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mi> k</gydF4y2Bami> </mrow> <mo> +</gydF4y2Bamo> </msubsup> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi mathvariant="bold"> C</gydF4y2Bami> <mi> k</gydF4y2Bami> <mo> −</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> </mrow> </math> </inline-formula>在VB方法提高了估计<我nline-formula id="inf216"> <math id="m246"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mi> k</gydF4y2Bami> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>在早期的迭代中,进而导致更好的未知模型参数估计,特别是在最初的预测误差协方差矩阵的估计是不选中。在下一节中,比较这两种方法的评估结果通过两个数值案例研究。</p><一个我d="h6" name="h6"></a> <h2>5例1:原有1-bay钢框架考虑时变时刻测量噪声</h2><pcgydF4y2Balass="mb15">在本节中,该方法的性能评价时应用到数值模型原有1-bay钢框架结构在地震荷载作用下。估计结果与边际地图的两步估计方法(<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>)和一个非自适应的贝叶斯模型更新方法(<一个href="//www.thespel.com/journal/articles/10.3389/#B15">Ebrahimian et al ., 2015</一个>)。故事高度和海湾的宽度是3.5米和6.0米,分别如图所示<一个href="//www.thespel.com/journal/articles/10.3389/#F2">图2一个</一个>。框架的几何和材料特性类似于那些用于我们的以前的工作(<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>),可以找到更详细的考虑的案例研究。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 图2</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g002.jpg" name="Figure2" target="_blank"><img id="F2" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g002.jpg"></a> <p><b>图2</b>gydF4y2Ba。<b>(gydF4y2Ba一)</b>gydF4y2Ba原有1-bay钢框架和时刻<b>(B)</b>0gydF4y2gydF4y2BaBa°组件洛马普列塔地震地面加速度时间历史的记录在洛斯盖多斯站(<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>)。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <p class="mb15">开发框架的数值模型在OpenSees (<一个href="//www.thespel.com/journal/articles/10.3389/#B26">麦肯纳,2000</一个>)。为此,部队梁柱单元与集成点7用于列和梁。单纤维用于表示每个法兰梁和柱截面,而10纤维用于离散化他们的网。单轴Giuffre-Menegotto-Pinto (GMP)材料模型(<一个href="//www.thespel.com/journal/articles/10.3389/#B16">Filippou et al ., 1983</一个>)和主要参数</p><d我vclass="equationImageholder"> <math id="equ1"> <mrow> <msup> <mi mathvariant="bold"> θ</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msup> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msubsup> <mi> E</gydF4y2Bami> <mi> c</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi> F</gydF4y2Bami> <mrow> <mi> y</gydF4y2Bami> <mi> c</gydF4y2Bami> </mrow> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi> b</gydF4y2Bami> <mi> c</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi> E</gydF4y2Bami> <mi> b</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi> F</gydF4y2Bami> <mrow> <mi> y</gydF4y2Bami> <mi> b</gydF4y2Bami> </mrow> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <msubsup> <mi> b</gydF4y2Bami> <mi> b</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mn> 200年</gydF4y2Bamn> <mtext> </mtext> <mi mathvariant="normal"> G</gydF4y2Bami> <mi mathvariant="normal"> P</gydF4y2Bami> <mi mathvariant="normal"> 一个</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mn> 350年</gydF4y2Bamn> <mtext> </mtext> <mi mathvariant="normal"> 米</gydF4y2Bami> <mi mathvariant="normal"> P</gydF4y2Bami> <mi mathvariant="normal"> 一个</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mn> 0.08</gydF4y2Bamn> <mo> ,</gydF4y2Bamo> <mn> 200年</gydF4y2Bamn> <mtext> </mtext> <mi mathvariant="normal"> G</gydF4y2Bami> <mi mathvariant="normal"> P</gydF4y2Bami> <mi mathvariant="normal"> 一个</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mn> 250年</gydF4y2Bamn> <mtext> </mtext> <mi mathvariant="normal"> 米</gydF4y2Bami> <mi mathvariant="normal"> P</gydF4y2Bami> <mi mathvariant="normal"> 一个</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mn> 0.05</gydF4y2Bamn> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> </mrow> </math> <div class="clear"></div> </div> <p class="noindent">用于模型钢纤维和模拟名义/真实结构的动态响应,在哪里<我nline-formula id="inf217"> <math id="m248"> <mrow> <mi> E</gydF4y2Bami> <mo> =</gydF4y2Bamo> </mrow> </math> </inline-formula>杨氏模量,<我nline-formula id="inf218"> <math id="m249"> <mrow> <msub> <mi> F</gydF4y2Bami> <mi> y</gydF4y2Bami> </msub> <mo> =</gydF4y2Bamo> </mrow> </math> </inline-formula>屈服应力,<我nline-formula id="inf219"> <math id="m250"> <mrow> <mi> b</gydF4y2Bami> <mo> =</gydF4y2Bamo> </mrow> </math> </inline-formula>加工硬化率。前三个参数用下标“<我nline-formula id="inf220"> <math id="m251"> <mrow> <mi> c</gydF4y2Bami> </mrow> </math> </inline-formula>”是为列,最后三人用下标”<我nline-formula id="inf221"> <math id="m252"> <mrow> <mi> b</gydF4y2Bami> </mrow> </math> </inline-formula>“用于光束。一个节点质量<我nline-formula id="inf222"> <math id="m253"> <mrow> <mo> =</gydF4y2Bamo> <mn> 80000年</gydF4y2Bamn> <mtext> </mtext> <mi mathvariant="normal"> k</gydF4y2Bami> <mi mathvariant="normal"> g</gydF4y2Bami> </mrow> </math> </inline-formula>,黑圆圈所示<一个href="//www.thespel.com/journal/articles/10.3389/#F2">图2一个</一个>在梁柱节点,每个故事被认为是代表死亡和生活质量。模型阻尼能量耗散,瑞利阻尼和阻尼比2%被认为是第一个结构的两种振动模式。</p><pcgydF4y2Balass="mb15">模拟测量数据,框架结构是激动的洛马普列塔地震(0°组件在洛斯盖多斯车站),如图所示<一个href="//www.thespel.com/journal/articles/10.3389/#F2">图2 b</一个>。然后,水平绝对加速度响应时间的历史每层(如图所示的黑匣子<一个href="//www.thespel.com/journal/articles/10.3389/#F2">图2一个</一个>)提取和人工测量噪声污染导致模拟测量数据。测量噪音被认为是一个非平稳高斯随机过程与时变均值向量<我nline-formula id="inf223"> <math id="m254"> <mrow> <msubsup> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> </mrow> </math> </inline-formula>和协方差矩阵<我nline-formula id="inf224"> <math id="m255"> <mrow> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> </mrow> </math> </inline-formula>如下:值得注意的是,非平稳噪声的时变均值和协方差是一个常见的问题在实际的监控,和假定的正弦形式均值和协方差的假设,考虑可行性研究。</p><d我vclass="equationImageholder"> <math id="e26"> <mrow> <msubsup> <mi mathvariant="bold"> μ</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mn> 1.97</gydF4y2Bamn> <mo> ,</gydF4y2Bamo> <mn> 5.41</gydF4y2Bamn> <mo> ,</gydF4y2Bamo> <mn> 7.50</gydF4y2Bamn> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> <mo> ×</gydF4y2Bamo> <mi> 罪</gydF4y2Bami> <mo> ⁡</gydF4y2Bamo> <mo> ⁡</gydF4y2Bamo> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mfrac> <mrow> <mn> 4</gydF4y2Bamn> <mi> π</gydF4y2Bami> </mrow> <mi> N</gydF4y2Bami> </mfrac> <mi> k</gydF4y2Bami> </mrow> </mfenced> </mrow> <mo> ×</gydF4y2Bamo> <msup> <mn> 10</gydF4y2Bamn> <mrow> <mo> −</gydF4y2Bamo> <mn> 2</gydF4y2Bamn> </mrow> </msup> <mi> g</gydF4y2Bami> </mrow> <mo> ,</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 26</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <div class="equationImageholder"> <math id="e27"> <mrow> <msubsup> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> =</gydF4y2Bamo> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mtable columnalign="center"> <mtr> <mtd> <mn> 1.66</gydF4y2Bamn> </mtd> <mtd> <mn> 0</gydF4y2Bamn> </mtd> <mtd> <mn> 0</gydF4y2Bamn> </mtd> </mtr> <mtr> <mtd> <mn> 0</gydF4y2Bamn> </mtd> <mtd> <mn> 2.88</gydF4y2Bamn> </mtd> <mtd> <mn> 0</gydF4y2Bamn> </mtd> </mtr> <mtr> <mtd> <mn> 0</gydF4y2Bamn> </mtd> <mtd> <mn> 0</gydF4y2Bamn> </mtd> <mtd> <mn> 7.65</gydF4y2Bamn> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> <mo> ×</gydF4y2Bamo> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi> 罪</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mfrac> <mrow> <mi> π</gydF4y2Bami> </mrow> <mrow> <mi> N</gydF4y2Bami> </mrow> </mfrac> <mi> k</gydF4y2Bami> </mrow> </mfenced> </mrow> <mo> +</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </mfenced> </mrow> <mn> 2</gydF4y2Bamn> </msup> <mo> ×</gydF4y2Bamo> <msup> <mn> 10</gydF4y2Bamn> <mrow> <mo> −</gydF4y2Bamo> <mn> 4</gydF4y2Bamn> </mrow> </msup> <msup> <mi> g</gydF4y2Bami> <mn> 2</gydF4y2Bamn> </msup> </mrow> <mo> 。</gydF4y2Bamo> <mspace width="5em"></mspace> <mo stretchy="false"> (</gydF4y2Bamo> <mn> 27</gydF4y2Bamn> <mo stretchy="false"> )</gydF4y2Bamo> </math> <div class="clear"></div> </div> <p class="mb15">我们的目标是估计未知模型参数<我nline-formula id="inf225"> <math id="m258"> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <msub> <mi> E</gydF4y2Bami> <mi> c</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi> F</gydF4y2Bami> <mrow> <mi> y</gydF4y2Bami> <mi> c</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi> b</gydF4y2Bami> <mi> c</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi> E</gydF4y2Bami> <mi> b</gydF4y2Bami> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi> F</gydF4y2Bami> <mrow> <mi> y</gydF4y2Bami> <mi> b</gydF4y2Bami> </mrow> </msub> <mo> ,</gydF4y2Bamo> <msub> <mi> b</gydF4y2Bami> <mi> b</gydF4y2Bami> </msub> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> </mrow> </math> </inline-formula>和比较他们的真实值<我nline-formula id="inf226"> <math id="m259"> <mrow> <msup> <mi mathvariant="bold"> θ</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msup> </mrow> </math> </inline-formula>。的初始估计未知模型参数选择及其协方差矩阵<我nline-formula id="inf227"> <math id="m260"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mn> 0.7</gydF4y2Bamn> <msubsup> <mi> E</gydF4y2Bami> <mi> c</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 0.7</gydF4y2Bamn> <msubsup> <mi> F</gydF4y2Bami> <mrow> <mi> y</gydF4y2Bami> <mi> c</gydF4y2Bami> </mrow> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 1.2</gydF4y2Bamn> <msubsup> <mi> b</gydF4y2Bami> <mi> c</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 0.8</gydF4y2Bamn> <msubsup> <mi> E</gydF4y2Bami> <mi> b</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 1.3</gydF4y2Bamn> <msubsup> <mi> F</gydF4y2Bami> <mrow> <mi> y</gydF4y2Bami> <mi> b</gydF4y2Bami> </mrow> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 0.8</gydF4y2Bamn> <msubsup> <mi> b</gydF4y2Bami> <mi> b</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> </mrow> </math> </inline-formula>和<我nline-formula id="inf228"> <math id="m261"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mn> 0</gydF4y2Bamn> </mrow> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mi mathvariant="normal"> d</gydF4y2Bami> <mi mathvariant="normal"> 我</gydF4y2Bami> <mi mathvariant="normal"> 一个</gydF4y2Bami> <mi mathvariant="normal"> g</gydF4y2Bami> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mn> 0.2</gydF4y2Bamn> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mn> 2</gydF4y2Bamn> </msup> </mrow> </math> </inline-formula>,分别。最初的均值向量和协方差矩阵的预测误差(测量噪声),分别承担<我nline-formula id="inf229"> <math id="m262"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mn mathvariant="bold"> 0</gydF4y2Bamn> </mrow> </math> </inline-formula>;<我nline-formula id="inf230"> <math id="m263"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msup> <mn> 10</gydF4y2Bamn> <mrow> <mo> −</gydF4y2Bamo> <mn> 5</gydF4y2Bamn> </mrow> </msup> <msub> <mi mathvariant="bold"> 我</gydF4y2Bami> <mrow> <mn> 3</gydF4y2Bamn> <mo> ×</gydF4y2Bamo> <mn> 3</gydF4y2Bamn> </mrow> </msub> </mrow> </math> </inline-formula>,在那里<我nline-formula id="inf231"> <math id="m264"> <mrow> <msub> <mi mathvariant="bold"> 我</gydF4y2Bami> <mrow> <mn> 3</gydF4y2Bamn> <mo> ×</gydF4y2Bamo> <mn> 3</gydF4y2Bamn> </mrow> </msub> </mrow> </math> </inline-formula>是单位矩阵。选择其他此类的初始参数分布<我nline-formula id="inf232"> <math id="m265"> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </math> </inline-formula>,<我nline-formula id="inf233"> <math id="m266"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mn> 4.1</gydF4y2Bamn> </mrow> </math> </inline-formula>,<我nline-formula id="inf234"> <math id="m267"> <mrow> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> −</gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </mfenced> </mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf235"> <math id="m268"> <mrow> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> =</gydF4y2Bamo> <mn> 3</gydF4y2Bamn> </mrow> </math> </inline-formula>。过程噪声协方差矩阵被选中<我nline-formula id="inf236"> <math id="m269"> <mrow> <mi mathvariant="bold"> 问</gydF4y2Bami> <mo> =</gydF4y2Bamo> <mi mathvariant="normal"> d</gydF4y2Bami> <mi mathvariant="normal"> 我</gydF4y2Bami> <mi mathvariant="normal"> 一个</gydF4y2Bami> <mi mathvariant="normal"> g</gydF4y2Bami> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msup> <mn> 10</gydF4y2Bamn> <mrow> <mo> −</gydF4y2Bamo> <mn> 4</gydF4y2Bamn> </mrow> </msup> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mn> 2</gydF4y2Bamn> </msup> </mrow> </math> </inline-formula>和遗忘因子参数用于定义的动态模型预测误差的均值向量和协方差矩阵被假定<我nline-formula id="inf237"> <math id="m270"> <mrow> <mi> ρ</gydF4y2Bami> <mo> =</gydF4y2Bamo> <mn> 0.95</gydF4y2Bamn> </mrow> </math> </inline-formula>和<我nline-formula id="inf238"> <math id="m271"> <mrow> <msup> <mi> ρ</gydF4y2Bami> <mo> ′</gydF4y2Bamo> </msup> <mo> =</gydF4y2Bamo> <mn> 0.95</gydF4y2Bamn> </mrow> </math> </inline-formula>分别基于我们的研究中,参数估计的结果是可以接受的<我nline-formula id="inf239"> <math id="m272"> <mrow> <mn> 0.8</gydF4y2Bamn> <mo> ≤</gydF4y2Bamo> <mi> ρ</gydF4y2Bami> <mo> <</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </math> </inline-formula>和<我nline-formula id="inf240"> <math id="m273"> <mrow> <mn> 0.7</gydF4y2Bamn> <mo> ≤</gydF4y2Bamo> <msup> <mi> ρ</gydF4y2Bami> <mo> ′</gydF4y2Bamo> </msup> <mo> <</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </math> </inline-formula>。然而,选择较低的值<我nline-formula id="inf241"> <math id="m274"> <mrow> <mi> ρ</gydF4y2Bami> </mrow> </math> </inline-formula>和<我nline-formula id="inf242"> <math id="m275"> <mrow> <msup> <mi> ρ</gydF4y2Bami> <mo> ′</gydF4y2Bamo> </msup> </mrow> </math> </inline-formula>可能恶化性能模型的更新过程。评估过程的敏感性其他过滤调优参数(如初始模型参数值,<我nline-formula id="inf243"> <math id="m276"> <mrow> <mi mathvariant="bold"> 问</gydF4y2Bami> </mrow> </math> </inline-formula>),已提出的另一项研究的主题<一个href="//www.thespel.com/journal/articles/10.3389/#B6">Astroza et al ., (2019 b</一个>)。</p><pcgydF4y2Balass="mb15">现在,拟议的VB方法应用于联合估计未知模型参数向量<我nline-formula id="inf244"> <math id="m277"> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> </mrow> </math> </inline-formula>和预测误差的均值向量和协方差矩阵。在这个验证研究,结果比较与两步边际地图估计方法(<一个href="//www.thespel.com/journal/articles/10.3389/#B28">Nabiyan et al ., 2022</一个>)和非自适应的贝叶斯模型更新方法(<一个href="//www.thespel.com/journal/articles/10.3389/#B15">Ebrahimian et al ., 2015</一个>当使用相同的初始值)。如前所述,在非自适应的贝叶斯方法中,一个零均值高斯白噪声与定常假设对角协方差矩阵的预测误差<我nline-formula id="inf245"> <math id="m278"> <mrow> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>,也就是说,<我nline-formula id="inf246"> <math id="m279"> <mrow> <msub> <mi mathvariant="bold"> ω</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> ∼</gydF4y2Bamo> <mi> N</gydF4y2Bami> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mn mathvariant="bold"> 0</gydF4y2Bamn> <mo> ,</gydF4y2Bamo> <msub> <mi mathvariant="bold"> R</gydF4y2Bami> <mi> k</gydF4y2Bami> </msub> <mo> =</gydF4y2Bamo> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。</p><pcgydF4y2Balass="mb15">的估计成分预测误差的均值向量和协方差矩阵(在这个例子中测量噪声)的三种方法(VB,两步边际地图,和非自适应的)与他们的真实值<一个href="//www.thespel.com/journal/articles/10.3389/#F3">图3</一个>和<一个href="//www.thespel.com/journal/articles/10.3389/#F4">图4</一个>,分别。非自适应的贝叶斯模型更新方法,如前所述,不估计均值向量和协方差矩阵的预测误差,所以他们保持不变在评估过程中。然而,VB和两步边际地图估计方法可以准确地跟踪真正的趋势/名义误差均值和协方差。我们可以看到<一个href="//www.thespel.com/journal/articles/10.3389/#F4">图4</一个>相比,两步边际映射方法,VB方法更好的预测误差的协方差矩阵估计的早期步骤,因为额外的术语在前一节中讨论。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 图3</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g003.jpg" name="Figure3" target="_blank"><img id="F3" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g003.jpg"></a> <p><b>图3</b>gydF4y2Ba。比较平均向量的估计组件的测量噪声的三种方法(VB,两步边际地图,和非自适应的)与真实的。应该注意的是,<我nline-formula id="inf247"> <math id="m280"> <mrow> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mtable columnalign="center"> <mtr> <mtd> <msub> <mi mathvariant="bold-italic"> μ</gydF4y2Bami> <mn mathvariant="bold"> 1</gydF4y2Bamn> </msub> </mtd> <mtd> <msub> <mi mathvariant="bold-italic"> μ</gydF4y2Bami> <mn mathvariant="bold"> 2</gydF4y2Bamn> </msub> </mtd> <mtd> <msub> <mi mathvariant="bold-italic"> μ</gydF4y2Bami> <mn mathvariant="bold"> 3</gydF4y2Bamn> </msub> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> <mi mathvariant="bold-italic"> T</gydF4y2Bami> </msup> <mo> 。</gydF4y2Bamo> </mrow> </math> </inline-formula></p> </div> <div class="clear"></div> <div class="DottedLine"></div> <div class="Imageheaders"> 图4</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g004.jpg" name="Figure4" target="_blank"><img id="F4" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g004.jpg"></a> <p><b>图4</b>gydF4y2Ba。比较估计组件的测量噪声协方差矩阵的三种方法(VB,两步边际地图,和非自适应的)与真实的。应该注意的是,<我nline-formula id="inf248"> <math id="m281"> <mrow> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> =</gydF4y2Bamo> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mtable columnalign="center"> <mtr> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 11</gydF4y2Bamn> </msub> </mtd> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 12</gydF4y2Bamn> </msub> </mtd> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 13</gydF4y2Bamn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 12</gydF4y2Bamn> </msub> </mtd> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 22</gydF4y2Bamn> </msub> </mtd> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 23</gydF4y2Bamn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 13</gydF4y2Bamn> </msub> </mtd> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 23</gydF4y2Bamn> </msub> </mtd> <mtd> <msub> <mi mathvariant="bold-italic"> R</gydF4y2Bami> <mn mathvariant="bold"> 33</gydF4y2Bamn> </msub> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mrow> </math> </inline-formula>。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <p class="mb15"><a href="//www.thespel.com/journal/articles/10.3389/#F5">图5</一个>显示的时间历史未知模型参数估计的三种方法:提出了VB的方法,两步边际地图估计方法和非自适应的贝叶斯方法。它可以观察到非自适应的模型更新方法收敛于不正确的未知模型参数,甚至发散。然而,自适应方法估计未知模型参数可以很好。疲软的非自适应的方法表明,估计预测误差显著影响模型预测误差时更新结果时变非零均值和,因此,不应该被忽略。比较VB法边际地图的两步估计法,VB更好的参数估计方法<我nline-formula id="inf249"> <math id="m282"> <mrow> <msub> <mi> b</gydF4y2Bami> <mi> c</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>加工硬化率(列),因为更好的预测误差的协方差矩阵估计在早期的时间步骤。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 图5</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g005.jpg" name="Figure5" target="_blank"><img id="F5" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g005.jpg"></a> <p><b>图5</b>gydF4y2Ba。时间的历史估计模型参数获得的三种方法:提出了VB的方法,两步边际地图估计方法和非自适应的贝叶斯方法。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <a id="h7" name="h7"></a> <h2>6例2:原有3-bay钢框架考虑建模错误时刻</h2><pcgydF4y2Balass="mb15">在第二个研究中,我们研究提出了VB方法在原有的存在建模误差,3-bay钢框架。帧是来自<一个href="//www.thespel.com/journal/articles/10.3389/#B33">歌et al ., (2020 a</一个>)和几何框架部分,并加载所示<一个href="//www.thespel.com/journal/articles/10.3389/#F6">图6</一个>。所有梁、柱宽缘资料并与纤维梁柱单元建模。瑞利阻尼衰减2%的第一和第二模式被认为是结构阻尼。分布的重力负载被认为是在节点集中质量。测量数据使用的钢筋本构模型模拟Giuffre-Menegotto-Pinto (GMP)梁、柱,他们的真实属性的选择与前面的示例相同。框架是激动的洛马普列塔地震所示<一个href="//www.thespel.com/journal/articles/10.3389/#F2">图2 b</一个>,水平绝对加速度反应在每层楼的黑盒<一个href="//www.thespel.com/journal/articles/10.3389/#F6">图6</一个>)记录。然后,1% RMS NSR高斯零均值白噪声添加到这些模拟加速度反应被认为是测量数据。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 图6</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g006.jpg" name="Figure6" target="_blank"><img id="F6" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g006.jpg"></a> <p><b>图6</b>gydF4y2Ba。原有3-bay钢框架。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <p class="mb15">当我们使用GMP本构模型<一个href="//www.thespel.com/journal/articles/10.3389/#F7">图7</一个>模拟测量,一个双线性模型(<一个href="//www.thespel.com/journal/articles/10.3389/#F7">图7 b</一个>)在评估过程中添加显式建模误差的模型更新过程。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 图7</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g007.jpg" name="Figure7" target="_blank"><img id="F7" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g007.jpg"></a> <p><b>图7</b>gydF4y2Ba。材料模型<b>(gydF4y2Ba一)</b>gydF4y2Ba用于模拟响应和GMP<b>(B)</b>gydF4y2gydF4y2BaBa双线性估计过程中使用。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <p class="mb15">模型更新在这个例子中,列和梁的材料属性被认为是未知模型参数,与前面的示例相似。的初始估计未知模型参数选择及其协方差矩阵<我nline-formula id="inf250"> <math id="m283"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msup> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mn> 0.8</gydF4y2Bamn> <msubsup> <mi> E</gydF4y2Bami> <mi> c</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 1.2</gydF4y2Bamn> <msubsup> <mi> F</gydF4y2Bami> <mrow> <mi> y</gydF4y2Bami> <mi> c</gydF4y2Bami> </mrow> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 1.4</gydF4y2Bamn> <msubsup> <mi> b</gydF4y2Bami> <mi> c</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 0.8</gydF4y2Bamn> <msubsup> <mi> E</gydF4y2Bami> <mi> b</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 1.2</gydF4y2Bamn> <msubsup> <mi> F</gydF4y2Bami> <mrow> <mi> y</gydF4y2Bami> <mi> b</gydF4y2Bami> </mrow> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> <mo> ,</gydF4y2Bamo> <mn> 1.4</gydF4y2Bamn> <msubsup> <mi> b</gydF4y2Bami> <mi> b</gydF4y2Bami> <mrow> <mi> t</gydF4y2Bami> <mi> r</gydF4y2Bami> <mi> u</gydF4y2Bami> <mi> e</gydF4y2Bami> </mrow> </msubsup> </mrow> </mfenced> </mrow> <mi> T</gydF4y2Bami> </msup> </mrow> </math> </inline-formula>和<我nline-formula id="inf251"> <math id="m284"> <mrow> <msubsup> <mi mathvariant="bold"> P</gydF4y2Bami> <mrow> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ,</gydF4y2Bamo> <mn> 0</gydF4y2Bamn> </mrow> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mi mathvariant="normal"> d</gydF4y2Bami> <mi mathvariant="normal"> 我</gydF4y2Bami> <mi mathvariant="normal"> 一个</gydF4y2Bami> <mi mathvariant="normal"> g</gydF4y2Bami> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mn> 0.2</gydF4y2Bamn> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mn> 2</gydF4y2Bamn> </msup> </mrow> </math> </inline-formula>,分别。最初的预测误差的均值向量和协方差矩阵是假定<我nline-formula id="inf252"> <math id="m285"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> μ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mn mathvariant="bold"> 0</gydF4y2Bamn> </mrow> </math> </inline-formula>和<我nline-formula id="inf253"> <math id="m286"> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <msup> <mn> 10</gydF4y2Bamn> <mrow> <mo> −</gydF4y2Bamo> <mn> 4</gydF4y2Bamn> </mrow> </msup> <msub> <mi mathvariant="bold"> 我</gydF4y2Bami> <mrow> <mn> 3</gydF4y2Bamn> <mo> ×</gydF4y2Bamo> <mn> 3</gydF4y2Bamn> </mrow> </msub> </mrow> </math> </inline-formula>,分别。如前所述,非自适应的方法不更新这两个参数,认为他们在评估过程中常量。此类分布所需的其他初始参数自适应方法选择<我nline-formula id="inf254"> <math id="m287"> <mrow> <msubsup> <mi> λ</gydF4y2Bami> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </math> </inline-formula>,<我nline-formula id="inf255"> <math id="m288"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mn> 4.1</gydF4y2Bamn> </mrow> </math> </inline-formula>,<我nline-formula id="inf256"> <math id="m289"> <mrow> <msubsup> <mi mathvariant="bold"> V</gydF4y2Bami> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> =</gydF4y2Bamo> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msubsup> <mi> v</gydF4y2Bami> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> <mo> −</gydF4y2Bamo> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> −</gydF4y2Bamo> <mn> 1</gydF4y2Bamn> </mrow> </mfenced> </mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold"> R</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </math> </inline-formula>,<我nline-formula id="inf257"> <math id="m290"> <mrow> <msub> <mi> n</gydF4y2Bami> <mi mathvariant="bold"> y</gydF4y2Bami> </msub> <mo> =</gydF4y2Bamo> <mn> 3</gydF4y2Bamn> </mrow> </math> </inline-formula>。过程噪声协方差矩阵被选中<我nline-formula id="inf258"> <math id="m291"> <mrow> <mi mathvariant="bold"> 问</gydF4y2Bami> <mo> =</gydF4y2Bamo> <mi mathvariant="normal"> d</gydF4y2Bami> <mi mathvariant="normal"> 我</gydF4y2Bami> <mi mathvariant="normal"> 一个</gydF4y2Bami> <mi mathvariant="normal"> g</gydF4y2Bami> <msup> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <msup> <mn> 10</gydF4y2Bamn> <mrow> <mo> −</gydF4y2Bamo> <mn> 4</gydF4y2Bamn> </mrow> </msup> <msubsup> <mover accent="true"> <mi mathvariant="bold"> θ</gydF4y2Bami> <mo> ^</gydF4y2Bamo> </mover> <mn> 0</gydF4y2Bamn> <mo> +</gydF4y2Bamo> </msubsup> </mrow> </mfenced> </mrow> <mn> 2</gydF4y2Bamn> </msup> </mrow> </math> </inline-formula>和遗忘因子参数用于定义的动态模型预测误差的均值向量和协方差矩阵被假定<我nline-formula id="inf259"> <math id="m292"> <mrow> <mi> ρ</gydF4y2Bami> <mo> =</gydF4y2Bamo> <mn> 0.9</gydF4y2Bamn> </mrow> </math> </inline-formula>和<我nline-formula id="inf260"> <math id="m293"> <mrow> <msup> <mi> ρ</gydF4y2Bami> <mo> ′</gydF4y2Bamo> </msup> <mo> =</gydF4y2Bamo> <mn> 0.98</gydF4y2Bamn> </mrow> </math> </inline-formula>。</p><pcgydF4y2Balass="mb15"><a href="//www.thespel.com/journal/articles/10.3389/#F8">图8</一个>显示了三种方法:模型更新结果提出了VB的方法,两步边际地图估计方法和非自适应的贝叶斯方法。可以看出,非自适应的方法不能正确估计未知模型参数的所有参数,除了<我nline-formula id="inf261"> <math id="m294"> <mrow> <msub> <mi> E</gydF4y2Bami> <mi> b</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf262"> <math id="m295"> <mrow> <msub> <mi> E</gydF4y2Bami> <mi> c</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>。测量结构响应不敏感参数<我nline-formula id="inf263"> <math id="m296"> <mrow> <msub> <mi> F</gydF4y2Bami> <mi> y</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf264"> <math id="m297"> <mrow> <mi> b</gydF4y2Bami> </mrow> </math> </inline-formula>,而不是<我nline-formula id="inf265"> <math id="m298"> <mrow> <mi> E</gydF4y2Bami> </mrow> </math> </inline-formula>在估计,这结果<我nline-formula id="inf266"> <math id="m299"> <mrow> <msub> <mi> F</gydF4y2Bami> <mi> y</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf267"> <math id="m300"> <mrow> <mi> b</gydF4y2Bami> </mrow> </math> </inline-formula>更影响预测误差(测量噪声+建模误差)。非自适应的方法限制了预测误差为,固定的协方差矩阵。然而,自适应方法释放这种假设通过更新预测误差的统计参数递归地在每个时间步长,从而更好的对未知模型参数估计。VB方法相比,所提出的自适应两步边际地图估计方法,VB方法改善柱和梁刚度硬化的估算值<我nline-formula id="inf268"> <math id="m301"> <mrow> <msub> <mi> b</gydF4y2Bami> <mi> c</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>和<我nline-formula id="inf269"> <math id="m302"> <mrow> <msub> <mi> b</gydF4y2Bami> <mi> b</gydF4y2Bami> </msub> </mrow> </math> </inline-formula>因为更好的预测误差估计。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 图8</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g008.jpg" name="Figure8" target="_blank"><img id="F8" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g008.jpg"></a> <p><b>图8</b>gydF4y2Ba。时间的历史估计模型参数获得的三种方法:提出了VB的方法,两步边际地图估计方法和非自适应的贝叶斯方法,第二个案例研究。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <p class="mb15">调查的能力更新模型预测建模错误的材料反应,绝对加速度反应在每层和moment-curvature响应第一个故事的核心的基础列使用这三种方法的预测更新模型,与真正的同行相比<一个href="//www.thespel.com/journal/articles/10.3389/#F9">图9</一个>和<一个href="//www.thespel.com/journal/articles/10.3389/#F10">图10</一个>,分别。尽管如此,我们可以看到<一个href="//www.thespel.com/journal/articles/10.3389/#F9">图9</一个>实测值和预测值的差异,加速度响应最小化,和估计参数是有偏见的非自适应的的情况下,和在较小程度上两步映射方法。可以观察到<一个href="//www.thespel.com/journal/articles/10.3389/#F10">图10</一个>,响应的预测都大大提高了自适应方法,及其预测同意真正的moment-curvature响应。比较提出了VB与两步方法映射方法,VB方法更好的预测响应,因为更好的未知模型参数的估计。</p><d我vclass="DottedLine"></div> <div class="Imageheaders"> 图9</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g009.jpg" name="Figure9" target="_blank"><img id="F9" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g009.jpg"></a> <p><b>图9</b>gydF4y2Ba。绝对加速度反应在每层获得的真实模型和更新的使用三种方法:提出了VB的方法,两步边际地图估计方法和非自适应的贝叶斯方法。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <div class="Imageheaders"> 图10</d我v><d我vclass="FigureDesc"> <a href="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g010.jpg" name="Figure10" target="_blank"><img id="F10" alt="www.雷竞技rebatfrontiersin.orggydF4y2Ba" class="lazy" data-src="//www.thespel.com/journal/files/Articles/1143597/fbuil-09-1143597-HTML/image_m/fbuil-09-1143597-g010.jpg"></a> <p><b>图10</b>gydF4y2Ba。Moment-curvature响应内柱的底部通过真正的模型和更新一个使用三种方法:提出了VB的方法,两步边际地图估计方法和非自适应的贝叶斯方法。</p></d我v><d我vclass="clear"></div> <div class="DottedLine"></div> <a id="h8" name="h8"></a> <h2>7结论</h2><pcgydF4y2Balass="mb15">在本文中,我们利用了变分贝叶斯(VB)方法,提出一种自适应变分贝叶斯模型更新方法联合模型和噪声识别。提供了详细的数学推导。演示了该方法的性能通过两个数值案例研究。两个非线性钢帧时刻受到地震激发被使用,在这六个参数描述的本构模型钢梁和列被认为是未知的。在第一个案例研究中,绝对加速度反应在每层楼被高斯噪声污染与时变均值向量和协方差矩阵被认为是测量数据。考虑建模误差在第二个案例研究中,钢铁的本构模型Giuffre-Menegotto-Pinto (GMP)是用于数据模拟,和一个双线性本构模型用于评估过程。估计结果的案例研究表明,该VB-based方法表现良好的时变预测误差(测量噪声和建模误差)。提出VB-based方法相比也最近开发边际地图两步估计法,和非自适应的贝叶斯模型更新方法。结果表明,自适应方法有类似的表现,而非自适应的方法导致显著的有偏估计的负面影响不稳定的预测误差。未来的工作范围扩展算法来估计动态输入,这将导致一个联合input-parameter-noise估计,并验证该算法在实际应用建模错误会导致分歧或重大偏差在常规模型更新算法。</p><一个我d="h9" name="h9"></a> <h2>数据可用性声明</h2><pcgydF4y2Balass="mb15">原始数据支持了本文的结论将由作者没有提供过度的预订。</p><一个我d="h10" name="h10"></a> <h2>作者的贡献</h2><pcgydF4y2Balass="mb0">他和M-SN概念化框架。M-SN和派生女士配方。他和BM验证了推导。M-SN和实现女士制定和分析结果。BM他调查的结果。M-SN女士和发达的第一稿。他和BM修改和编辑。他和BM监督这个项目。</p><一个我d="h11" name="h11"></a> <h2>确认</h2><pcgydF4y2Balass="mb03">第四作者承认部分支持本研究通过美国国家科学基金会拨款1903972。意见、发现和结论本文中表达的是作者的,不一定代表赞助商的意见。</p><一个我d="h12" name="h12"></a> <h2>的利益冲突</h2><pcgydF4y2Balass="mb0">作者是受雇于Gavin女士和Doherty Geosolutions有限公司</p><pcgydF4y2Balass="mb0">其余作者声明,这项研究是在没有进行任何商业或财务关系可能被视为一个潜在的利益冲突。</p><一个我d="h13" name="h13"></a> <h2>出版商的注意</h2><pcgydF4y2Balass="mb15">本文表达的所有索赔仅代表作者,不一定代表的附属组织,或那些出版商编辑和评论员。任何产品,可以评估在这篇文章中,或声称,可能是由其制造商并不能保证或认可的出版商。</p><一个我d="h14" name="h14"></a> <h2>补充材料</h2><pcgydF4y2Balass="mb15" id="SM1">本文的补充材料在网上可以找到:<一个href="//www.thespel.com/journal/articles/10.3389/fbuil.2023.1143597/full">https://www.雷竞技rebatfrontiersin.org/articles/10.3389/fbuil.2023.1143597/full补充材料</一个></p><一个我d="h15" name="h15"></a> <h2>引用</h2><d我vclass="References"> <p class="ReferencesCopy1"><a name="B1" id="B1"></a>Akhlaghi, S。周,N。,Huang, Z. (<span class="conf-date">2017年7月</gydF4y2Baspan>)。自适应调整卡尔曼滤波器的噪声协方差动态状态估计,<gydF4y2Baspan class="conf-name">2017年IEEE电力和能源学会学报大会</gydF4y2Baspan>(<gydF4y2Baspan class="publisher-name">IEEE</gydF4y2Baspan>)。<gydF4y2Baspan class="conf-loc">美国芝加哥</gydF4y2Baspan>。doi: 10.1109 / PESGM.2017.8273755</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1109/PESGM.2017.8273755">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Adaptive+adjustment+of+noise+covariance+in+Kalman+filter+for+dynamic+state+estimation&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B2" id="B2"></a>Amini德黑兰的H。Bakhshi,。,Yang, T. T. (2020). Online probabilistic model class selection and joint estimation of structures for post-disaster monitoring.<em>j .振动器控制</egydF4y2Bam>27日,1860 - 1878。doi: 10.1177 / 1077546320949115</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1177/1077546320949115">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Online+probabilistic+model+class+selection+and+joint+estimation+of+structures+for+post-disaster+monitoring&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B3" id="B3"></a>Astroza, R。,Alessandri, A., and Conte, J. P. (2019). A dual adaptive filtering approach for nonlinear finite element model updating accounting for modeling uncertainty.<em>动力机械。系统。信号处理</egydF4y2Bam>115年,782 - 800。doi: 10.1016 / j.ymssp.2018.06.014</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1016/j.ymssp.2018.06.014">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=A+dual+adaptive+filtering+approach+for+nonlinear+finite+element+model+updating+accounting+for+modeling+uncertainty&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B4" id="B4"></a>Astroza, R。,Ebrahimian, H。,Conte,J. P. (2014). Material parameter identification in distributed plasticity FE models of frame-type structures using nonlinear stochastic filtering.<em>j·英格。动力机械。</egydF4y2Bam>141 (5),4014149。doi: 10.1061 /(第3期)em.1943 - 7889.0000851</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1061/(ASCE)EM.1943-7889.0000851">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Material+parameter+identification+in+distributed+plasticity+FE+models+of+frame-type+structures+using+nonlinear+stochastic+filtering&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B5" id="B5"></a>Astroza, R。,Ebrahimian, H。,Conte,J. P. (2017). “Batch and recursive bayesian estimation methods for nonlinear structural system identification,” in<em>风险和可靠性分析:理论和应用程序。施普林格在可靠性工程系列</egydF4y2Bam>。编辑器p Gardoni(德国柏林:<gydF4y2Baspan class="publisher-name">施普林格</gydF4y2Baspan>),341 - 364。doi: 10.1007 / 978 - 3 - 319 - 52425 - 2 _15</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1007/978-3-319-52425-2_15">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Batch+and+recursive+bayesian+estimation+methods+for+nonlinear+structural+system+identification&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B6" id="B6"></a>Astroza, R。,Ebrahimian, H。,Conte,J. P. (2019b). Performance comparison of Kalman−based filters for nonlinear structural finite element model updating.<em>j .声音。振动器</egydF4y2Bam>438年,520 - 542。doi: 10.1016 / j.jsv.2018.09.023</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1016/j.jsv.2018.09.023">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Performance+comparison+of+Kalman−based+filters+for+nonlinear+structural+finite+element+model+updating&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B7" id="B7"></a>比尔,m . j . (2003)。<egydF4y2Bam>变分近似贝叶斯推理的算法</egydF4y2Bam>:<gydF4y2Baspan class="publisher-name">论文。伦敦大学大学</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">伦敦英国</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Variational+algorithms+for+approximate+Bayesian+inference&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B8" id="B8"></a>贝克,j·L。,Yuen, K.-V. (2004). Model selection using response measurements: Bayesian probabilistic approach.<em>j·英格。动力机械。</egydF4y2Bam>130 (2),192 - 203。doi: 10.1061 /(第3期)0733 - 9399 (2004)130:2 (192)</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1061/(asce)0733-9399(2004)130:2(192)">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Model+selection+using+response+measurements:+Bayesian+probabilistic+approach&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B9" id="B9"></a>Behmanesh,我。,Moaveni B。,lombaert, G., and Papadimitriou, C. (2015). Hierarchical Bayesian model updating for structural identification.<em>动力机械。系统。信号处理</egydF4y2Bam>64年,360 - 376。doi: 10.1016 / j.ymssp.2015.03.026</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1016/j.ymssp.2015.03.026">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Hierarchical+Bayesian+model+updating+for+structural+identification&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B10" id="B10"></a>主教,c . M。,N一个srabadi, N. M. (2006).<em>模式识别和机器学习</egydF4y2Bam>。纽约,纽约,美国:<gydF4y2Baspan class="publisher-name">施普林格</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Pattern+recognition+and+machine+learning&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B11" id="B11"></a>Cesmd, (2019)。工程中心强大的运动数据,cesmd——合作。<一个href="https://strongmotioncenter.org">https://strongmotioncenter.org</一个>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Center+for+engineering+strong+motion+data,+cesmd-+A+cooperative+effort&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B12" id="B12"></a>京,J。,贝克,j·L。,Porter,K。A. (2006). Bayesian state and parameter estimation of uncertain dynamical systems.<em>概率Eng。动力机械。</egydF4y2Bam>21 (1),81 - 96。doi: 10.1016 / j.probengmech.2005.08.003</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1016/j.probengmech.2005.08.003">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Bayesian+state+and+parameter+estimation+of+uncertain+dynamical+systems&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B13" id="B13"></a>Deisenroth, m . P。费萨尔,A。,Ong, C. S. (2020).<em>数学对机器学习</egydF4y2Bam>。<gydF4y2Baspan class="publisher-name">剑桥大学出版社</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">英国剑桥</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Mathematics+for+machine+learning&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B14" id="B14"></a>Doebling, s W。法勒,c R。,' m B。,Shevitz, D. W. (1996).<em>损伤识别和健康监测的结构和机械系统振动的变化特点:一个文献综述</egydF4y2Bam>。美国纳米:<gydF4y2Baspan class="publisher-name">洛斯阿拉莫斯国家实验室</gydF4y2Baspan>。doi: 10.2172/249299</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.2172/249299">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Damage+identification+and+health+monitoring+of+structural+and+mechanical+systems+from+changes+in+their+vibration+characteristics:+A+literature+review&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B15" id="B15"></a>Ebrahimian, H。Astroza, R。,Conte,J. P. (2015). Extended Kalman filter for material parameter estimation in nonlinear structural finite element models using direct differentiation method.<em>Earthq。Eng。结构体。直流发电机。</egydF4y2Bam>44 (10),1495 - 1522。doi: 10.1002 / eqe.2532</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1002/eqe.2532">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Extended+Kalman+filter+for+material+parameter+estimation+in+nonlinear+structural+finite+element+models+using+direct+differentiation+method&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B16" id="B16"></a>Filippou, f . C。波波夫,e . P。,Bertero, V. V. (1983).<em>债券恶化对滞回行为的钢筋混凝土的影响关节。<egydF4y2Bam>座83 - 19</egydF4y2Bam>。地震工程研究中心</egydF4y2Bam>。美国加利福尼亚州伯克利:<gydF4y2Baspan class="publisher-name">加州大学</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Effects+of+bond+deterioration+on+hysteretic+behavior+of+reinforced+concrete+joints.+EERC+83–19.+Earthquake+Engineering+Research+Center&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B17" id="B17"></a>Friswell, M。,米ottershead, J. E. (2013).<em>在结构动力学有限元模型更新</egydF4y2Bam>。<gydF4y2Baspan class="publisher-name">施普林格科学与商业媒体</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">柏林,德国</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Finite+element+model+updating+in+structural+dynamics&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B18" id="B18"></a>Granstrom, K。,Orguner, U. (2011).<em>属性和近似一个矩阵变量的概率密度函数</egydF4y2Bam>。<gydF4y2Baspan class="publisher-name">林雪平大学电子报刊</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">瑞典林雪平</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Properties+and+approximations+of+some+matrix+variate+probability+density+functions&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B19" id="B19"></a>微积分,美国(2004年)。<egydF4y2Bam>卡尔曼滤波和神经网络</egydF4y2Bam>。<gydF4y2Baspan class="publisher-name">约翰•威利父子</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">纽约,纽约,美国</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Kalman+filtering+and+neural+networks&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B20" id="B20"></a>黄,Y。Yu, J。,贝克,j·L。、朱、H。,l我,H. (2020). Novel sparseness-inducing dual Kalman filter and its application to tracking time-varying spatially-sparse structural stiffness changes and inputs.<em>第一版。方法:。动力机械。Eng。</egydF4y2Bam>372年,113411年。doi: 10.1016 / j.cma.2020.113411</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1016/j.cma.2020.113411">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Novel+sparseness-inducing+dual+Kalman+filter+and+its+application+to+tracking+time-varying+spatially-sparse+structural+stiffness+changes+and+inputs&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B21" id="B21"></a>黄,Y。,Zhang, Y., Li, N., and Zhao, L. (2016). Design of sigma-point Kalman filter with recursive updated measurement.<em>Int。j .电路系统。信号的过程。</egydF4y2Bam>35 (5),1767 - 1782。doi: 10.1007 / s00034 - 015 - 0137 - y</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1007/s00034-015-0137-y">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Design+of+sigma-point+Kalman+filter+with+recursive+updated+measurement&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B22" id="B22"></a>黄,Y。,Zhang, Y., Wu, Z., Li, N., and Chambers, J. (2017). A novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices.<em>IEEE T奥特曼。来讲</egydF4y2Bam>63 (2),594 - 601。doi: 10.1109 / TAC.2017.2730480</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1109/TAC.2017.2730480">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=A+novel+adaptive+Kalman+filter+with+inaccurate+process+and+measurement+noise+covariance+matrices&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B23" id="B23"></a>Kollo, T。,VonRosen, D. (2006).<em>先进的多元统计与矩阵</egydF4y2Bam>。<gydF4y2Baspan class="publisher-name">施普林格科学与商业媒体</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">柏林,德国</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Advanced+multivariate+statistics+with+matrices&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B24" id="B24"></a>Kontoroupi, T。,Smyth, A. W. (2016). Online noise identification for joint state and parameter estimation of nonlinear systems.<em>ASCE-ASME j .不确定的风险。Eng。系统。一个文明。Eng。</egydF4y2Bam>2 (3),B4015006。doi: 10.1061 / AJRUA6.0000839</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1061/AJRUA6.0000839">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Online+noise+identification+for+joint+state+and+parameter+estimation+of+nonlinear+systems&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B25" id="B25"></a>法律,k . J。,Stuart, A. M. (2012). Evaluating data assimilation algorithms.<em>星期一,天气牧师。</egydF4y2Bam>140 (11),3757 - 3782。doi: 10.1175 / mwr - d - 11 - 00257.1</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1175/mwr-d-11-00257.1">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Evaluating+data+assimilation+algorithms&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B26" id="B26"></a>麦肯纳,F。,Fenves, G. L., and Scott, M. H. (2000).<em>地震工程仿真的开放系统</egydF4y2Bam>。美国加利福尼亚州伯克利:<gydF4y2Baspan class="publisher-name">加州大学</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Open+system+for+earthquake+engineering+simulation&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B27" id="B27"></a>论著,r (1972)。自适应滤波方法。<egydF4y2Bam>IEEE T奥特曼。来讲</egydF4y2Bam>17 (5),693 - 698。doi: 10.1109 / tac.1972.1100100</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1109/tac.1972.1100100">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Approaches+to+adaptive+filtering&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B28" id="B28"></a>Nabiyan, m . S。,Ebrahimian, H。,Moaveni B。,P一个p一个d我mitriou, C. (2022). Adaptive bayesian inference framework for joint model and noise identification.<em>j·英格。动力机械。</egydF4y2Bam>148 (3),4021165。doi: 10.1061 /(第3期)em.1943 - 7889.0002084</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1061/(ASCE)EM.1943-7889.0002084">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Adaptive+bayesian+inference+framework+for+joint+model+and+noise+identification&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B29" id="B29"></a>Nabiyan, m . S。,Khoshnoudian, F., Moaveni, B., and Ebrahimian, H. (2020). Mechanics-based model updating for identification and virtual sensing of an offshore wind turbine using sparse measurements.<em>结构体。Monit来讲。健康。</egydF4y2Bam>28日,e2647。doi: 10.1002 / stc.2647</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1002/stc.2647">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Mechanics-based+model+updating+for+identification+and+virtual+sensing+of+an+offshore+wind+turbine+using+sparse+measurements&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B30" id="B30"></a>欧哈根,一个。,Forster, J. J. (2004).<em>肯德尔的先进的统计理论</egydF4y2Bam>2。2,<gydF4y2Baspan class="publisher-name">阿诺德</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">英国伦敦</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Kendall's+advanced+theory+of+statistics&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B31" id="B31"></a>Opper, M。,Saad, D. (2001).<em>先进的平均场方法:理论与实践</egydF4y2Bam>。<gydF4y2Baspan class="publisher-name">麻省理工学院出版社</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">美国剑桥,马</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Advanced+mean+field+methods:+Theory+and+practice&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B32" id="B32"></a>Paolella, m . s . (2018)。<egydF4y2Bam>线性模型和时间序列分析:回归<egydF4y2Bam>方差分析,ARMA和东南亚四国</egydF4y2Bam></em>。<gydF4y2Baspan class="publisher-name">约翰•威利父子</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">纽约,纽约,美国</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Linear+models+and+time-series+analysis:+Regression,+ANOVA,+ARMA+and+GARCH&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B33" id="B33"></a>歌,M。,Behmanesh,我。,Moaveni B。,P一个p一个d我mitriou, C. (2020). Accounting for modeling errors and inherent structural variability through a hierarchical bayesian model updating approach: An overview.<em>传感器</egydF4y2Bam>20 (14),3874。doi: 10.3390 / s20143874</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://pubmed.ncbi.nlm.nih.gov/32664472/">《公共医学图书馆摘要》</一个>|<gydF4y2Ba一个href="https://doi.org/10.3390/s20143874">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Accounting+for+modeling+errors+and+inherent+structural+variability+through+a+hierarchical+bayesian+model+updating+approach:+An+overview&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B34" id="B34"></a>Sanayei, M。,Arya B。,Santini, E. M., and Wadia-Fascetti, S. (2001). Significance of modeling error in structural parameter estimation.<em>COMPUT-AIDED文明。正无穷。</egydF4y2Bam>16(1),12日到27日。doi: 10.1111 / 0885 - 9507.00210</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1111/0885-9507.00210">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Significance+of+modeling+error+in+structural+parameter+estimation&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B35" id="B35"></a>Sarkka, S。,Hartikainen, J. (<span class="conf-date">2013年9月</gydF4y2Baspan>)。通过变分贝叶斯非线性噪声自适应卡尔曼滤波,<gydF4y2Baspan class="conf-name">学报2013年IEEE国际研讨会在机器学习信号处理(MLSP)</gydF4y2Baspan>,(<gydF4y2Baspan class="publisher-name">IEEE</gydF4y2Baspan>),<gydF4y2Baspan class="conf-loc">南安普顿,英国</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Non-linear+noise+adaptive+Kalman+filtering+via+variational+Bayes&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B36" id="B36"></a>Sarkka, S。,Nummenmaa, A. (2009). Recursive noise adaptive Kalman filtering by variational Bayesian approximations.<em>IEEE T奥特曼。来讲</egydF4y2Bam>54 (3),596 - 600。doi: 10.1109 / TAC.2008.2008348</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1109/TAC.2008.2008348">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Recursive+noise+adaptive+Kalman+filtering+by+variational+Bayesian+approximations&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B37" id="B37"></a>施,Y。,T一个ng,X., Feng, X., Bian, D., and Zhou, X. (2018). Hybrid adaptive cubature Kalman filter with unknown variance of measurement noise.<em>传感器</egydF4y2Bam>18(12),4335年。doi: 10.3390 / s18124335</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://pubmed.ncbi.nlm.nih.gov/30544613/">《公共医学图书馆摘要》</一个>|<gydF4y2Ba一个href="https://doi.org/10.3390/s18124335">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Hybrid+adaptive+cubature+Kalman+filter+with+unknown+variance+of+measurement+noise&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B38" id="B38"></a>Šmidl, V。,问u我nn,A. (2006).<em>变分信号处理中的贝叶斯方法</egydF4y2Bam>。<gydF4y2Baspan class="publisher-name">施普林格科学与商业媒体</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">柏林,德国</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=The+variational+Bayes+method+in+signal+processing&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B39" id="B39"></a>歌,M。,Astroza, R。,Ebrahimian, H。,Moaveni B。,P一个p一个d我mitriou, C. (2020). Adaptive Kalman filters for nonlinear finite element model updating.<em>动力机械。系统。信号处理</egydF4y2Bam>143年,106837年。doi: 10.1016 / j.ymssp.2020.106837</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1016/j.ymssp.2020.106837">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Adaptive+Kalman+filters+for+nonlinear+finite+element+model+updating&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B40" id="B40"></a>太阳,C。,Zhang, Y., Wang, G., and Gao, W. (2018). A new variational Bayesian adaptive extended Kalman filter for cooperative navigation.<em>传感器</egydF4y2Bam>18(8),2538年。doi: 10.3390 / s18082538</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://pubmed.ncbi.nlm.nih.gov/30081473/">《公共医学图书馆摘要》</一个>|<gydF4y2Ba一个href="https://doi.org/10.3390/s18082538">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=A+new+variational+Bayesian+adaptive+extended+Kalman+filter+for+cooperative+navigation&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B41" id="B41"></a>太阳,J。,Zhou, J., and Gu, X. (2012). Variational Bayesian two-stage Kalman filter for systems with unknown inputs.<em>Procedia Eng。</egydF4y2Bam>29日,2265 - 2273。doi: 10.1016 / j.proeng.2012.01.299</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1016/j.proeng.2012.01.299">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Variational+Bayesian+two-stage+Kalman+filter+for+systems+with+unknown+inputs&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B42" id="B42"></a>Tzikas, d G。,l我k一个s, A. C., and Galatsanos, N. P. (2008). The variational approximation for Bayesian inference.<em>IEEE信号的过程。玛格。</egydF4y2Bam>25 (6),131 - 146。doi: 10.1109 / MSP.2008.929620</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1109/MSP.2008.929620">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=The+variational+approximation+for+Bayesian+inference&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B43" id="B43"></a>魏因斯托克,r (1974)。<egydF4y2Bam>变分法:应用物理学和工程学</egydF4y2Bam>。<gydF4y2Baspan class="publisher-name">多佛出版物</gydF4y2Baspan>,<gydF4y2Baspan class="publisher-loc">纽约</gydF4y2Baspan>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Calculus+of+variations:+With+applications+to+physics+and+engineering&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B44" id="B44"></a>文策尔,T。,Burnham, K., Blundell, M., and Williams, R. (2007). Kalman filter as a virtual sensor: Applied to automotive stability systems.<em>反式。本月,量。控制</egydF4y2Bam>29 (2),95 - 115。doi: 10.1177 / 0142331207072990</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1177/0142331207072990">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Kalman+filter+as+a+virtual+sensor:+Applied+to+automotive+stability+systems&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B45" id="B45"></a>吴,j . (2005)。正态分布的一些性质。<一个href="https://cs.nju.edu.cn/_upload/tpl/00/ed/237/template237/paper/Gaussian.pdf">https://cs.nju.edu.cn/_upload/tpl/00/ed/237/template237/paper/Gaussian.pdf</一个>。</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Some+properties+of+the+normal+distribution&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B46" id="B46"></a>吴,M。,Smyth, A. W. (2007). Application of the unscented Kalman filter for real-time nonlinear structural system identification.<em>结构体。Monit来讲。健康。</egydF4y2Bam>14 (7),971 - 990。doi: 10.1002 / stc.186</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1002/stc.186">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Application+of+the+unscented+Kalman+filter+for+real-time+nonlinear+structural+system+identification&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B47" id="B47"></a>徐,D。,Wu,Z., and Huang, Y. (2019). A new adaptive Kalman filter with inaccurate noise statistics.<em>Int。j .电路系统。信号的过程。</egydF4y2Bam>38 (9),4380 - 4404。doi: 10.1007 / s00034 - 019 - 01053 - w</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1007/s00034-019-01053-w">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=A+new+adaptive+Kalman+filter+with+inaccurate+noise+statistics&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B48" id="B48"></a>杨,j . N。林,S。,Huang, H., and Zhou, L. (2006). An adaptive extended Kalman filter for structural damage identification.<em>结构体。Monit来讲。健康。</egydF4y2Bam>13 (4),849 - 867。doi: 10.1002 / stc.84</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1002/stc.84">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=An+adaptive+extended+Kalman+filter+for+structural+damage+identification&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B49" id="B49"></a>袁,k V。,Kuok,S. C. (2016). Online updating and uncertainty quantification using nonstationary output-only measurement.<em>动力机械。系统。信号处理</egydF4y2Bam>66年,62 - 77。doi: 10.1016 / j.ymssp.2015.05.019</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1016/j.ymssp.2015.05.019">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Online+updating+and+uncertainty+quantification+using+nonstationary+output-only+measurement&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B50" id="B50"></a>张,问。,Yang, Y., Xiang, Q., He, Q., Zhou, Z., and Yao, Y. (2018). Noise adaptive Kalman filter for joint polarization tracking and channel equalization using cascaded covariance matching.<em>IEEE光子J。</egydF4y2Bam>10 (1),1 - 11。doi: 10.1109 / JPHOT.2018.2797050</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://doi.org/10.1109/JPHOT.2018.2797050">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Noise+adaptive+Kalman+filter+for+joint+polarization+tracking+and+channel+equalization+using+cascaded+covariance+matching&btnG=">谷歌学术搜索</一个></p></d我v><d我v class="References"> <p class="ReferencesCopy1"><a name="B51" id="B51"></a>郑,B。傅,P。李,B。,Yuan, X. (2018). A robust adaptive unscented Kalman filter for nonlinear estimation with uncertain noise covariance.<em>传感器</egydF4y2Bam>18(3),808年。doi: 10.3390 / s18030808</p><pcgydF4y2Balass="ReferencesCopy2"><a href="https://pubmed.ncbi.nlm.nih.gov/29518960/">《公共医学图书馆摘要》</一个>|<gydF4y2Ba一个href="https://doi.org/10.3390/s18030808">CrossRef全文</一个>|<gydF4y2Ba一个href="https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=A+robust+adaptive+unscented+Kalman+filter+for+nonlinear+estimation+with+uncertain+noise+covariance&btnG=">谷歌学术搜索</一个></p></d我v></div> <div class="thinLineM20"></div> <div class="AbstractSummary"> <p><span>关键词:</gydF4y2Baspan>自适应贝叶斯模型更新,变分贝叶斯技术,噪声识别、模型预测误差、非平稳噪声</p><p><gydF4y2Baspan>引用:</gydF4y2Baspan>Nabiyan M·沙里夫M, Ebrahimian H和Moaveni B(2023)的变分模型的贝叶斯推理技术更新结构系统与未知噪声统计数据。<egydF4y2Bam>前面。建立环境。</egydF4y2Bam>9:1143597。doi: 10.3389 / fbuil.2023.1143597</p><p我d="timestamps"><span>收到:</gydF4y2Baspan>2023年1月13日;<gydF4y2Baspan>接受:</gydF4y2Baspan>2023年3月22日;<br><gydF4y2Baspan>发表:</gydF4y2Baspan>2023年4月24日。</p><d我v><p>编辑:</p><一个href="//www.thespel.com/loop/people/328436/overview">剑李</一个>美国堪萨斯大学</d我v><d我v><p>审核:</p><一个href="//www.thespel.com/loop/people/766824/overview">华平广域网</一个>浙江大学,中国<br><一个href="//www.thespel.com/loop/people/1042785/overview">勇黄</一个>哈尔滨工业大学,中国</d我v><p><span>版权</gydF4y2Baspan>(©2023 Nabiyan, Sharifi Ebrahimian Moaveni。这是一个开放分布式根据文章<一个rel="license" href="http://creativecommons.org/licenses/by/4.0/" target="_blank">知识共享归属许可(CC)。</一个>使用、分发或复制在其他论坛是允许的,提供了原始作者(年代)和著作权人(s)认为,最初发表在这个期刊引用,按照公认的学术实践。没有使用、分发或复制是不符合这些条件的允许。</p><p><gydF4y2Baspan>*通信:</gydF4y2Baspan>哈米德Ebrahimian,<一个href="//www.thespel.com/journal/articles/10.3389/mailto:hebrahimian@unr.edu">hebrahimian@unr.edu</一个></p><p><span><sup>__</gydF4y2Basup></span>这些作者对这项工作同样做出了贡献,分享第一作者</p><d我vclass="clear"></div> </div>   <div class="thin-line-dark"></div> </div> </div> </div> </main> </div> <div class="side-article-subjects only-devices widget-listing people-also-looked-at hidden"> <h5 class="like-h4">人也看了</h5></d我v></d我v></div> </div> <div id="divTemplates"> <div class="modal fade modal-container impact-modal-container" data-backdrop="static" id="impactModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel" aria-hidden="true"></div> <div class="modal fade modal-container supplementary-modal-container" data-backdrop="static" data-keyboard="false" id="supplementaryFilesModal" tabindex="-1" role="dialog" aria-labelledby="mySupplementaryModalLabel" aria-hidden="true" style="display: none; padding-right: 17px;"></div> <div class="modal fade modal-container notifyme-modal-container" data-backdrop="static" id="notifyModal" tabindex="-1" role="dialog" aria-labelledby="Notify on publication" aria-hidden="true"></div> </div> </div> <a id="floating-download-pdf-btn" class="floating-button" data-event="downloadfloating-button-click" href="//www.thespel.com/journal/articles/10.3389/fbuil.2023.1143597/pdf"><span class="floating-button-text" data-event="downloadfloating-button-click">下载</gydF4y2Baspan></a> <frontiers-footer main-domain="frontiersin.org"></frontiers-footer>   </body> </html>

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