基于核加权类对准则的高光谱影像特征提取

刘敬; 李青妍; 刘逸

doi:10.37188/OPE.20212906.1397

English Version Light学术出版中心

您当前的位置：

首页 >

文章列表页 >

基于核加权类对准则的高光谱影像特征提取

信息科学 | 更新时间：2021-07-03

- 基于核加权类对准则的高光谱影像特征提取
- Spectral feature extraction of hyperspectral remote sensing images based on kernel class pair-weighted criterion
- 光学精密工程
- 作者机构：
  
  1.西安邮电大学电子工程学院，陕西西安 710121
  2.重庆邮电大学计算机科学与技术学院，重庆 400065
  3.西安电子科技大学电子工程学院，陕西西安 710071
- 作者简介：
  
  [ "刘　敬（1975—），女，安徽宿州人，博士，副教授，硕士生导师，2004年于西安电子科技大学获硕士学位，2009年于西安电子科技大学获博士学位，主要从事高光谱遥感影像光谱特征提取、地物分类、和解混方面的研究。E-mail：zyhalj1975@163.com" ]
  
  刘　敬（1975—），女，安徽宿州人，博士，副教授，硕士生导师，2004年于西安电子科技大学获硕士学位，2009年于西安电子科技大学获博士学位，主要从事高光谱遥感影像光谱特征提取、地物分类、和解混方面的研究。E-mail：zyhalj1975@163.com
  [ "李青妍（1995—），女，山西运城人，博士研究生，2020年于西安邮电大学获得硕士学位，现为重庆邮电大学博士研究生，主要从事遥感影像特征提取和地物分类方面的研究。E-mail：labcqy@163.com" ]
  
  李青妍（1995—），女，山西运城人，博士研究生，2020年于西安邮电大学获得硕士学位，现为重庆邮电大学博士研究生，主要从事遥感影像特征提取和地物分类方面的研究。E-mail：labcqy@163.com
  [ "刘　逸（1976—），男，安徽宿州人，博士，讲师。2002年于西安电子科技大学获硕士学位，2013年于西安电子科技大学获博士学位，主要从事计算智能，高光谱遥感影像地物分类方面的研究。E-mail：yiliu@xidian.edu.cn" ]
  
  刘　逸（1976—），男，安徽宿州人，博士，讲师。2002年于西安电子科技大学获硕士学位，2013年于西安电子科技大学获博士学位，主要从事计算智能，高光谱遥感影像地物分类方面的研究。E-mail：yiliu@xidian.edu.cn
- 基金信息：
  
  国家自然科学基金(61672405);陕西省自然科学基金(2018JM4018);国家自然科学基金(62077038);陕西省自然科学基金(2021JM-459)
- DOI：10.37188/OPE.20212906.1397 中图分类号： TP391
- 纸质出版日期：2021-06-15，
  
  收稿日期：2020-07-06，
  
  修回日期：2020-09-05，
扫描看全文
引用本文
刘敬,李青妍,刘逸.基于核加权类对准则的高光谱影像特征提取[J].光学精密工程,2021,29(06):1397-1405.

LIU Jing,LI Qing-yan,LIU Yi.Spectral feature extraction of hyperspectral remote sensing images based on kernel class pair-weighted criterion[J].Optics and Precision Engineering,2021,29(06):1397-1405.
刘敬,李青妍,刘逸.基于核加权类对准则的高光谱影像特征提取[J].光学精密工程,2021,29(06):1397-1405. DOI： 10.37188/OPE.20212906.1397.

LIU Jing,LI Qing-yan,LIU Yi.Spectral feature extraction of hyperspectral remote sensing images based on kernel class pair-weighted criterion[J].Optics and Precision Engineering,2021,29(06):1397-1405. DOI： 10.37188/OPE.20212906.1397.

基于核加权类对准则的高光谱影像特征提取 Spectral feature extraction of hyperspectral remote sensing images based on kernel class pair-weighted criterion 1 corresp first-author E-mail： jingliu@xupt.edu.cn E-mail： jingliu@xupt.edu.cn 刘　敬（1975—），女，安徽宿州人，博士，副教授，硕士生导师，2004年于西安电子科技大学获硕士学位，2009年于西安电子科技大学获博士学位，主要从事高光谱遥感影像光谱特征提取、地物分类、和解混方面的研究。E-mail：zyhalj1975@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422822&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422824&type= 22.01333618 32.00404358 刘　敬（1975—），女，安徽宿州人，博士，副教授，硕士生导师，2004年于西安电子科技大学获硕士学位，2009年于西安电子科技大学获博士学位，主要从事高光谱遥感影像光谱特征提取、地物分类、和解混方面的研究。E-mail： zyhalj1975@163.com zyhalj1975@163.com 1 2 李青妍（1995—），女，山西运城人，博士研究生，2020年于西安邮电大学获得硕士学位，现为重庆邮电大学博士研究生，主要从事遥感影像特征提取和地物分类方面的研究。E-mail：labcqy@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422832&type= 22.01332855 32.00398254 李青妍（1995—），女，山西运城人，博士研究生，2020年于西安邮电大学获得硕士学位，现为重庆邮电大学博士研究生，主要从事遥感影像特征提取和地物分类方面的研究。E-mail： labcqy@163.com labcqy@163.com 3 刘　逸（1976—），男，安徽宿州人，博士，讲师。2002年于西安电子科技大学获硕士学位，2013年于西安电子科技大学获博士学位，主要从事计算智能，高光谱遥感影像地物分类方面的研究。E-mail：yiliu@xidian.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422837&type= 22.01334000 32.00404358 刘　逸（1976—），男，安徽宿州人，博士，讲师。2002年于西安电子科技大学获硕士学位，2013年于西安电子科技大学获博士学位，主要从事计算智能，高光谱遥感影像地物分类方面的研究。E-mail： yiliu@xidian.edu.cn yiliu@xidian.edu.cn 西安邮电大学电子工程学院，陕西西安 710121 西安邮电大学电子工程学院，陕西西安 710121 School of Electronic Engineering， Xi’an University of Posts and Telecommunications， Xi’an 710121， China School of Electronic Engineering， Xi’an University of Posts and Telecommunications， Xi’an 710121， China 重庆邮电大学计算机科学与技术学院，重庆 400065 重庆邮电大学计算机科学与技术学院，重庆 400065 School of Computer Science & Technology， Chongqing University of Posts and Telecommunications， Chongqing 400065， China School of Computer Science & Technology， Chongqing University of Posts and Telecommunications， Chongqing 400065， China 西安电子科技大学电子工程学院，陕西西安 710071 西安电子科技大学电子工程学院，陕西西安 710071 School of Electronic Engineering， Xidian University， Xi’an 710071， China School of Electronic Engineering， Xidian University， Xi’an 710071， China 针对高光谱遥感影像中相似光谱的不同地物与野类同时存在时，提取有效的非线性可分性特征的问题，提出一种核加权类对准则。首先，推导出类对形式的核线性判别分析准则，即核类对准则，将核类间和类内散布矩阵均表示为类对形式。然后，提出核加权类对准则，依据核空间中各类对的可分性分别对各类对的核类间和类内散布矩阵进行加权，使得各类对的可分性均衡地保留在特征子空间中。采用K近邻分类器和最小距离分类器评估特征提取的效果。基于两个实测高光谱遥感影像的实验结果均表明：相比原空间法、核线性判别分析方法和kernel weighted pairwise Fisher准则，所提核加权类对准则在降维的同时，通过提高可分性小的类对的识别率来提高整体地物识别率。 To extract efficient nonlinear discriminant features when foreign objects， with similar spectra and outlier classes， are present in hyperspectral remote sensing images （HRSIs）， a kernel class pair-weighted （KCP-weighted） criterion is proposed. First， we derive a class pair form of the kernel linear discriminant analysis （KLDA） criterion， viz. the kernel class pair （KCP） criterion， in which the kernel-between-class and kernel-within-class scatter matrices are both expressed in the form of class pairs. Then， the KCP-weighted criterion is proposed to weight the kernel-between-class and kernel-within-class scatter matrices of each class pair according to their separability in a kernel space. The KCP-weighted criterion can ensure that the separabilities of class pairs are balanced in the KCP-weighted feature subspace. Finally， the K-nearest neighbor and minimum distance classifiers are used to evaluate the feature extraction performance. Experimental results of two real HRSIs show that， compared with the original space and KLDA methods as well as the kernel weighted pairwise Fisher criterion， the presented KCP-weighted criterion can effectively improve the overall terrain classification rate while reducing the dimensionality of the data. 核线性判别分析核加权类对准则特征提取高光谱遥感影像 kernel linear discriminant analysis kernel class pair-weighted criterion feature extraction hyperspectral remote sensing images 1 引　言 1 高光谱遥感影像具有维数高［ 1 1 - 4 4 1-4 ］、冗余信息多的特性［ 5 5 - 6 6 5-6 ］，同时地物的光谱有“同物异谱”、“异物同谱”等现象，这些因素使得高光谱遥感影像具有非线性分布的特点，给特征提取带来困难。高光谱遥感影像非线性可分性特征的提取，可以通过核方法解决。Baudat G.和Anouar F.提出核线性判别分析（KLDA），引入核函数计算高维核空间的内积，将线性判别分析推广至非线性判别分析，以提取非线性的可分性特征［ 7 7 ］。但当高光谱遥感影像中相似光谱的不同地物与野类同时存在时，KLDA往往会使得可分性大的类对容易分，而可分性小的类对难分，进一步影响整体地物分类精度。Dai等人对KLDA进行改进，通过对核类间散布矩阵进行加权，以解决野类影响特征提取的问题，但该方法未考虑到类内散布对特征提取结果的影响［ 8 8 ］。Lu等人采用迭代方法对KLDA进行改进，基于类对形式将所有类对的散度累加和作为新的目标函数，令各类对的散度最大使得目标函数最大化［ 9 9 ］。但该方法需要对所有类对分别进行迭代计算，计算量大。Liu等人提出加权类对（CP-weighted）准则，利用类对可分性测度，分别对各类对的类间散布与类内散布进行加权［ 10 10 ］。但该方法仅能提取线性可分性特征，不能提取非线性可分性特征。为解决高光谱遥感影像中相似光谱的不同地物与野类同时存在时，提取有效的非线性可分性特征的问题，本文提出一种核加权类对（KCP-weighted）准则。首先推导出类对形式的KLDA准则，即核类对（KCP）准则。然后，依据核空间中各类对的可分性分别对各类对的核类间和类内散布矩阵进行加权，增加可分性小的类对对特征方向的贡献，抑制可分性大的类对对特征方向的影响。KCP-weighted方法的优点：1.同时考虑到核空间中各类对的类间、类内散布对特征方向的影响，可以更好地捕捉原始数据中存在的非线性可分性特征；2.在核空间中利用类对的可分性测度，增加可分性小的类对对特征方向的贡献，降低可分性大的影响，使得特征子空间中各类对的非线性可分性特征得以均衡保留。本文采用两个实测遥感影像的实验结果表明：相比原空间法、KLDA法、核Fisher类对加权（KWP-Fisher）法，所提KCP-weighted法能提取出更合理的非线性可分性特征，通过提高可分性小的类对的分类结果来提高整体识别率。 2 KCP-weighted算法思想 1 针对高光谱遥感影像中相似光谱的不同地物与野类同时存在时，提取有效的非线性可分性特征的问题，本文首先推导出类对形式的KLDA准则，即KCP准则。然后，依据核空间中各类对的可分性分别对各类对的核类间和类内散布矩阵进行加权，提出KCP-weighted准则。 2.1　KCP 准则 2 给定样本的原始空间维数为 <math id="M1"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422885&type= 1.86266661 2.79399991 ，样本类别数为 <math id="M2"><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423085&type= 2.62466669 2.79399991 ，训练样本数为 <math id="M3"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422851&type= 2.87866688 2.87866688 ，训练样本集为 <math id="M4"><mi mathvariant="bold-italic">X</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422854&type= 30.81866837 4.91066647 ，第 <math id="M5"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423289&type= 1.10066664 2.79399991 类的训练样本的个数为 <math id="M6"><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422863&type= 3.64066648 3.80999994 。其中，原始空间中第 <math id="M7"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423289&type= 1.10066664 2.79399991 类的第 <math id="M8"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423103&type= 1.86266661 3.64066648 个样本为 <math id="M9"><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422877&type= 4.82600021 4.06400013 ， <math id="M10"><mi>q</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422881&type= 26.50066566 4.91066647 。将原始 <math id="M11"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422885&type= 1.86266661 2.79399991 维空间的样本 <math id="M12"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422891&type= 2.87866688 3.72533321 映射到高维核空间 <math id="M13"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422907&type= 2.53999996 2.79399991 后记为 <math id="M14"><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422912&type= 8.38199997 4.65666676 。 <math id="M15"><mo><</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422924&type= 27.43199921 4.65666676 表示在高维核空间 <math id="M16"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422907&type= 2.53999996 2.79399991 中向量 <math id="M17"><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422912&type= 8.38199997 4.65666676 与向量 <math id="M18"><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422916&type= 8.38199997 4.65666676 的内积。核函数 <math id="M19"><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422931&type= 12.78466606 4.48733330 与 <math id="M20"><mo><</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422924&type= 27.43199921 4.65666676 的关系可以表示为： <math id="M21"><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo><</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422927&type= 44.02666473 4.65666676 。本文核函数 <math id="M22"><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422931&type= 12.78466606 4.48733330 采用RBF核函数，即： <math id="M23"><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="|"><mrow><mo>-</mo><mrow><mrow><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422938&type= 53.25533295 7.02733374 。 KLDA引入核函数计算高维核空间的内积，将线性判别分析推广至非线性判别分析，可提取非线性可分性特征，其准则函数为： <math display="block" id="M24"><msup><mrow><mi>J</mi></mrow><mrow><mi>ϕ</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi>t</mi><mi>r</mi><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422942&type= 51.47733307 9.99066734 ，满足使 <math id="M25"><msup><mrow><mi>J</mi></mrow><mrow><mi>ϕ</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422944&type= 16.00200081 4.82600021 最大的 <math id="M26"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422948&type= 9.56733322 3.80999994 的列向量由 <math id="M27"><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422952&type= 14.13933277 5.16466665 对应的前 <math id="M28"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423157&type= 2.20133328 2.79399991 个最大特征值对应的特征向量组成。 <math id="M29"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423037&type= 4.06400013 3.97933316 为核类内散布矩阵，如式（1）式（1）所示。即： <math id="M30"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><mi>ϕ</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><mi>ϕ</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422963&type= 71.79733276 9.73666668 。其中， <math id="M31"><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422966&type= 2.62466669 4.57200003 为第 <math id="M32"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423289&type= 1.10066664 2.79399991 类的先验概率， <math id="M33"><mi>ϕ</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422973&type= 10.24466610 5.67266655 为核空间中第 <math id="M34"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423289&type= 1.10066664 2.79399991 类的第 <math id="M35"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423103&type= 1.86266661 3.64066648 个样本， <math id="M36"><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423108&type= 4.40266657 3.97933316 为核空间中各类样本的均值向量，可利用 <math id="M37"><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><mi>ϕ</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422989&type= 32.68133163 9.73666668 解得。 <math id="M38"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423004&type= 3.64066648 3.97933316 为核类间散布矩阵，即： <math id="M39"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>ϕ</mi></mrow></msup></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>ϕ</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422994&type= 55.62599945 8.38199997 。核类间散布矩阵的类对形式 <math id="M40"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423002&type= 5.50333309 4.31799984 可以重新写为［ 11 11 ］： <math id="M41"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>-</mo><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>ϕ</mi></mrow></msup></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>ϕ</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422999&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17422998&type= 57.48867035 8.38199997 。在核空间中，核类间散布矩阵 <math id="M42"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423004&type= 3.64066648 3.97933316 与各类对的核类间散布矩阵之和 <math id="M43"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423002&type= 5.50333309 4.31799984 一致。核类间散布矩阵 <math id="M44"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423004&type= 3.64066648 3.97933316 可以写为各类对的核类间散布矩阵之和 <math id="M45"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423006&type= 17.35666656 8.89000034 的形式。其中， <math id="M46"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423008&type= 4.40266657 4.31799984 为核空间中第 <math id="M47"><mi>i</mi><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423384&type= 2.20133328 3.64066648 类对的核类间散布矩阵。因此，式（1）式（1）可以进一步定义为 <math display="block" id="M48"><msup><mrow><mi>J</mi></mrow><mrow><mi>ϕ</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi>t</mi><mi>r</mi><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423012&type= 59.09733200 22.35199928 ，其中， <math id="M49"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423015&type= 29.80266571 5.58799982 ， <math id="M50"><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423017&type= 22.94466591 3.97933316 。依据文献［ 12 12 ］，式（2）式（2）为Kernel Pairwise Fisher准则。同理，将核类内散布矩阵 <math id="M51"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423037&type= 4.06400013 3.97933316 分解为类对形式，各类对的核平均类内散布矩阵为 <math id="M52"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423021&type= 29.71800041 5.24933338 。在高维核空间中，类对形式的核类内散布矩阵之和 <math id="M53"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423034&type= 17.86466599 8.89000034 与核类内散布矩阵 <math id="M54"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423037&type= 4.06400013 3.97933316 ，依据函数表达与字母无关的特性满足： <math display="block" id="M55"><mtable><mtr><mtd><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><mi>C</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mfenced></mrow></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mfenced><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow><mo>+</mo><mfrac><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>-</mo></mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow><mo>=</mo></mtd></mtr><mtr><mtd><mi>C</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow><mo>=</mo><mfenced separators="|"><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423029&type= 67.39466858 43.34933472 <math id="M56"><mfenced separators="|"><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423031&type= 11.93799973 3.97933316 为常量，不会影响核空间中最终提取的特征的方向。因此，用各类对的核类内散布矩阵之和 <math id="M57"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423034&type= 17.86466599 8.89000034 代替核类内散布矩阵 <math id="M58"><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423037&type= 4.06400013 3.97933316 ，KLDA的准则函数可以进一步合理扩展为： <math display="block" id="M59"><msup><mrow><mi>J</mi></mrow><mrow><mi>ϕ</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi>t</mi><mi>r</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423039&type= 64.43132782 22.94466591 . 我们将式（4）式（4）定义为KCP准则，即类对形式的KLDA准则，KCP准则为KLDA方法的合理扩展。最大化KCP准则的最优特征向量为 <math id="M60"><msup><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423041&type= 44.61933517 11.09133339 前 <math id="M61"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423157&type= 2.20133328 2.79399991 个最大特征值对应的特征向量。 2.2　KCP-weighted准则 2 通过采用核空间中各类对的类心距离作为可分性测度，分别对各类对的核类内、类间散布矩阵加权，增加可分性小的类对对特征方向的贡献，减小可分性大的类对对特征方向的影响。则式（4）式（4）可以进一步定义为： <math display="block" id="M62"><mtable><mtr><mtd><msup><mrow><mi>J</mi></mrow><mrow><mi>ϕ</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mi>t</mi><mi>r</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423045&type= 51.90066528 29.63333130 . 我们定义式（5）式（5）为KCP-weighted准则，且寻找使得KCP-weighted准则最大的特征子空间。 <math id="M63"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423047&type= 9.82133293 3.80999994 为KCP-weighted准则的映射矩阵。式（5）式（5）中， <math id="M64"><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423065&type= 3.30200005 4.74133301 为核空间中各类对的权重，计算过程如下：①计算各类对所占的比重。计算核空间各类对类心欧式距离 <math id="M65"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423051&type= 3.47133350 3.72533321 与各类对类心欧式距离之和的比值，即： <math id="M66"><mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mo>/</mo><mrow><mo>∑</mo><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423053&type= 14.39333248 5.24933338 ；②令： <math id="M67"><msub><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mrow><mrow><mo>∑</mo><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mrow><mo>/</mo><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423055&type= 23.19866753 5.50333309 ；③归一化 <math id="M68"><msub><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423059&type= 3.30200005 5.58799982 。将 <math id="M69"><msub><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423059&type= 3.30200005 5.58799982 与 <math id="M70"><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423061&type= 12.95400047 5.67266655 的比值作为归一化结果，即： <math id="M71"><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mrow><mrow><msub><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mo>/</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423063&type= 28.10933495 6.18066692 ，使得权重 <math id="M72"><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423065&type= 3.30200005 4.74133301 的取值范围为0到1。可分性大的类对被赋予小权重，而可分性小的类对被赋予大权重，从而实现增加可分性小的类对对特征方向的贡献，减小可分性大的类对对特征方向的影响。 KCP-weighted方法的最优特征向量为 <math id="M73"><msup><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423067&type= 51.81600189 11.09133339 前 <math id="M74"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423157&type= 2.20133328 2.79399991 个最大特征值对应的特征向量。但因映射函数 <math id="M75"><mi>ϕ</mi><mfenced separators="|"><mrow><mo>⋅</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423071&type= 6.51933336 3.97933316 的形式无法给出，需要引入核函数进行计算： <math id="M76"><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo><</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423073&type= 44.02666473 4.65666676 。对于 <math id="M77"><mi>u</mi><mo>,</mo><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423075&type= 5.50333309 3.21733332 两类任意训练样本，建立 <math id="M78"><msub><mrow><mi>N</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>×</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423077&type= 12.61533356 3.80999994 维的内积矩阵 <math id="M79"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423079&type= 4.91066647 3.72533321 ，即 <math id="M80"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>v</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mtable columnalign="left"><mtr><mtd><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>u</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>v</mi></mrow></msub><mo> </mo></mtd></mtr></mtable></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423081&type= 28.61733437 7.53533268 。其中， <math id="M81"><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>k</mi><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>u</mi></mrow><mrow><mfenced separators="|"><mrow><mi>i</mi></mrow></mfenced></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>v</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mo><</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>u</mi></mrow><mrow><mfenced separators="|"><mrow><mi>i</mi></mrow></mfenced></mrow></msubsup><mo stretchy="false">)</mo><mo>⋅</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>v</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup><mo stretchy="false">)</mo><mo>></mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423083&type= 58.08132935 5.33400011 。对于样本类别数为 <math id="M82"><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423085&type= 2.62466669 2.79399991 时，建立 <math id="M83"><mi>N</mi><mo>×</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423087&type= 10.83733368 2.87866688 维内积矩阵 <math id="M84"><mi mathvariant="bold-italic">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423089&type= 2.79399991 2.79399991 ，即 <math id="M85"><mi mathvariant="bold-italic">K</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>v</mi></mrow></msub><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtable columnalign="left"><mtr><mtd><mi>u</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>C</mi><mtext> </mtext></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>C</mi></mtd></mtr></mtable></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423091&type= 28.78666496 7.36600018 。由再生核理论可知，核空间中解向量 <math id="M86"><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423116&type= 4.14866638 3.72533321 位于训练样本张成的子空间中。即： <math display="block" id="M87"><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>α</mi></mrow><mrow><mi>z</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup><mi>ϕ</mi></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">Φ</mi><mi mathvariant="bold-italic">α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423095&type= 38.77733231 8.89000034 ，其中， <math id="M88"><mi mathvariant="bold-italic">Φ</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mi>ϕ</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi>ϕ</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>ϕ</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423097&type= 47.15933609 5.67266655 ， <math id="M89"><msubsup><mrow><mi>α</mi></mrow><mrow><mi>z</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423099&type= 4.48733330 4.06400013 为 <math id="M90"><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423101&type= 3.13266683 3.72533321 的第 <math id="M91"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423103&type= 1.86266661 3.64066648 个分量， <math id="M92"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423162&type= 29.37933159 4.91066647 。将各类样本的均值 <math id="M93"><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423108&type= 4.40266657 3.97933316 映射至 <math id="M94"><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423116&type= 4.14866638 3.72533321 ，有： <math display="block" id="M95"><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>α</mi></mrow><mrow><mi>z</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mo><</mo><mi>ϕ</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfenced><mo>,</mo><mi>ϕ</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced><mo>></mo></mrow></mrow></mfenced></mrow><mo>=</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced open="[" close="" separators="|"><mrow><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced><mo>,</mo><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced><mo>,</mo></mrow></mfenced></mrow><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo></mtd></mtr><mtr><mtd><mtext> </mtext><mfenced open="" close="]" separators="|"><mrow><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced></mrow></mfenced><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>.</mo><mtext> </mtext><mo stretchy="false">(</mo><mn mathvariant="normal">7</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423112&type= 73.32133484 30.22599983 同理，将样本整体均值 <math id="M96"><msup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>ϕ</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423114&type= 4.31799984 3.04800010 映射至 <math id="M97"><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423116&type= 4.14866638 3.72533321 ，有： <math display="block" id="M98"><msubsup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>ϕ</mi></mrow></msup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi>T</mi></mrow></msubsup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ϕ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423118&type= 65.61666107 8.38199997 . 由式（7），式（8）式（8）可知， <math id="M99"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced open="[" close="]" separators="|"><mrow><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced><mo>,</mo><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>j</mi></mrow></mfenced></mrow></msubsup></mrow></mfenced></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423120&type= 75.86132813 9.73666668 ， <math id="M100"><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423124&type= 20.23533440 8.38199997 。令 <math id="M101"><msup><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mi>g</mi></mrow></mfenced></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mi>g</mi></mrow></mfenced></mrow></msubsup><mo stretchy="false">)</mo><mo>,</mo><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mi>g</mi></mrow></mfenced></mrow></msubsup><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423128&type= 61.63733673 5.67266655 <math id="M102"><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mi>g</mi></mrow></mfenced></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mi>g</mi></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423134&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423132&type= 29.71800041 5.67266655 ，有： <math display="block" id="M103"><mtable><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>ϕ</mi></mrow></msubsup></mrow></mrow></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mtext> </mtext><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow></mrow></mfenced><mi mathvariant="bold-italic">A</mi><mo>.</mo><mtext> </mtext><mo stretchy="false">(</mo><mn mathvariant="normal">9</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423136&type= 76.19999695 24.04533195 其中， <math display="block" id="M104"><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></msub></mrow></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><mfenced open="{" close="}" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>q</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>i</mi></mrow><mrow><mfenced separators="|"><mrow><mi>q</mi></mrow></mfenced></mrow></msubsup><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>g</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mi>g</mi></mrow></mfenced></mrow></msubsup><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mi>g</mi></mrow></mfenced></mrow></msubsup><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">m</mi><mo>̂</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></mrow></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423140&type= 102.78533173 33.95133591 则KCP-weighted准则函数可以通过系数矩阵 <math id="M105"><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423144&type= 2.79399991 2.79399991 表示为： <math display="block" id="M106"><msup><mrow><mi>J</mi></mrow><mrow><mi>ϕ</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>K</mi><mi>C</mi><mi>P</mi><mi>W</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mi>r</mi><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mfenced separators="|"><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></mrow></mrow></mfrac></mrow></mfenced><mi mathvariant="bold-italic">A</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423149&type= 61.72200012 20.57399940 ，通过求解 <math id="M107"><msup><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></msub></mrow></mrow></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></msub></mrow></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423153&type= 51.90066528 11.09133339 的前 <math id="M108"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423157&type= 2.20133328 2.79399991 个最大特征值对应的特征向量得到最优系数矩阵 <math id="M109"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423162&type= 29.37933159 4.91066647 ，即可最大化KCP-weighted准则。将原始空间的样本 <math id="M110"><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423167&type= 2.11666679 2.79399991 利用 <math display="block" id="M111"><mtable><mtr><mtd><mi mathvariant="bold-italic">y</mi><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">W</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mi>ϕ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">x</mi></mrow></mfenced><mo>,</mo><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">x</mi></mrow></mfenced><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>k</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>,</mo><mi mathvariant="bold-italic">x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423174&type= 58.41999817 12.19200039 可以映射至KCP-weighted特征子空间。KCP-weighted准则使得特征子空间中各类对的可分性得以均衡保留，通过提高可分性小的类对的分类效果，以提高整体分类精度。 3 KCP-weighted算法流程 1 基于KCP-weighted准则的地物分类详细步骤如下（输入：高光谱数据所有样本，类别标签）：Step1 计算核矩阵。将原始空间中各类样本随机划分为训练样本与测试样本，利用RBF核函数 <math id="M112"><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423184&type= 12.78466606 4.48733330 建立核矩阵；Step2 特征提取。对核空间中的 <math id="M113"><msup><mrow><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></msub></mrow></mrow></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub></mrow></msub></mrow></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423193&type= 51.90066528 11.09133339 进行特征值分解，得到最优系数矩阵 <math id="M114"><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423200&type= 2.79399991 2.79399991 ，并计算各类对的权重 <math id="M115"><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423208&type= 3.30200005 4.74133301 ；Step3 建立模板库。利用式（11）式（11），将原空间样本映射至KCP-weighted特征子空间。并利用核空间中各类的均值与协方差矩阵，建立模板库， <math id="M116"><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423218&type= 2.62466669 2.79399991 为地物类别数；Step4 地物分类。对任意测试样本 <math id="M117"><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423226&type= 2.11666679 2.79399991 ，将原始样本映射至KCP-weighted特征子空间中后记做 <math id="M118"><mi mathvariant="bold-italic">y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423234&type= 1.94733346 3.64066648 。然后，分别利用KNN分类器与MD分类器进行地物分类。 KNN分类器的决策规则为：对于KCP-weighted特征子空间中的任意测试样本 <math id="M119"><mi mathvariant="bold-italic">y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423304&type= 1.94733346 3.64066648 ，判别 <math id="M120"><mi mathvariant="bold-italic">y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423304&type= 1.94733346 3.64066648 属于 <math id="M121"><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>C</mi></mrow></munder><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423258&type= 17.69533348 4.91066647 所决定的类。其中， <math id="M122"><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423267&type= 2.45533323 3.72533321 为 <math id="M123"><mi mathvariant="bold-italic">y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423304&type= 1.94733346 3.64066648 的 <math id="M124"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423452&type= 1.69333339 2.79399991 近邻中属于第 <math id="M125"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423289&type= 1.10066664 2.79399991 类的训练样本数；MD分类器的决策规则为：计算KCP-weighted特征子空间中任意测试样本 <math id="M126"><mi mathvariant="bold-italic">y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423304&type= 1.94733346 3.64066648 到各类训练样本均值的距离，将测试样本 <math id="M127"><mi mathvariant="bold-italic">y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423304&type= 1.94733346 3.64066648 判决为距离各类训练样本均值中，距离最小的均值所在的类。 4 实验结果分析 1 为了验证所提KCP-weighted准则的有效性，分别采用原空间法、KLDA、KWP-Fisher、以及KCP-weighted准则，在相同训练样本和测试样本的条件下进行特征提取。这里采用核空间中各类对的类心欧式距离作为各类对的可分性测度。 4.1　实验数据 2 分别采用Indian Pines整体、Indian Pines子图、和Botswana影像进行实验。数据1为Indian Pines高光谱影像。该数据由AVIRIS传感器对美国印第安纳州Indian Pines实验区成像所得［ 13 13 ］。在除去受水汽等其他因素影响的波段后，选取200个光谱波段作为研究对象，空间分辨率为17m，波长范围为0.4~2.5 μm，包括16类地物，该影像大小为145×145。Indian Pines影像如图1 图1 （a）所示。Indian Pines子图包括4类地物，大小为68×86。数据2为Botswana高光谱影像。该数据由EO-1卫星在Botswana获得［ 13 13 ］。在除去受水汽等其他因素影响的波段后，选取145个波段作为研究对象，该影像空间分辨率为30 m，波长为0.4~2.5 μm，包括14类地物。Botswana影像如图1 图1 （b）所示。 4.2　实验结果与分析 2 在实验采用RBF核函数，在特征子空间采用KNN、MD分类器进行分类。随机选取各类样本的20%作为训练样本，其余样本作为测试样本，每个实验取10次结果的平均。在相同的训练样本与测试样本的条件下，KCP-weighted法分类结果分别与原空间法、KLDA法、KWP-Fisher法的分类结果进行比较。AA为平均识别率，即 <math id="M128"><mi>A</mi><mi>A</mi><mo>=</mo><mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>C</mi></mrow></munderover></mstyle><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow><mo>/</mo><mrow><mi>C</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423325&type= 26.58533096 8.89000034 ， <math id="M129"><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423332&type= 4.91066647 3.72533321 为各类识别率。SD为标准差。Kappa为所得Kappa 系数。表1 表1 为Botswana影像在各个特征子空间采用KNN、MD分类器的分类结果，其中， <math id="M130"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423339&type= 1.69333339 2.79399991 为RBF核函数的参数。 2 Classifier 2 Original space method 3 KLDA subspace method 3 KWP - Fisher criterion 3 KCP - weighted criterion AA（%） ±SD（%） Kappa AA（%） ±SD（%） <math id="M131"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa AA（%） ±SD（%） <math id="M132"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa AA（%） ±SD（%） <math id="M133"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa KNN 91.57±0.38 0.90 94.88±0.69 0.28 0.94 93.98±0.68 0.05 0.93 95.60±0.59 0.75 0.95 MD 81.76±0.83 0.80 93.87±0.61 0.11 0.93 92.99±0.59 0.04 0.92 95.03±0.33 0.16 0.94 由表1 表1 可知，在采用KNN分类器， <math id="M134"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423452&type= 1.69333339 2.79399991 为4时，KCP-weighted准则法的平均识别率相比原空间法、KLDA法和KWP-Fisher准则法分别提高了4.03，0.72，1.62个百分点。当采用MD分类器时，所提KCP-weighted准则法的平均识别率相比原空间法、KLDA子空间法和KWP-Fisher准则法分别提高了13.27，1.16，2.04个百分点。其中，KCP-weighted准则法的标准差最小，Kappa系数最大。表2 表2 为Botswana影像核空间中的不相似性测度矩阵。由表2 表2 可知，这里采用 <math id="M135"><mrow><mrow><mi>t</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi>t</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">S</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423377&type= 24.29933357 5.41866684 衡量第 <math id="M136"><mi>i</mi><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423384&type= 2.20133328 3.64066648 类对在核空间中的不相似性，不相似性测度越小，类对的相似性越大。在表2 表2 中类对的不相似性测度的阈值为0.1，相似性大的类对在表2 表2 中加粗显示。 Class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 0 0.804 0.966 1.014 0.967 0.944 0.574 1.001 0.993 1.004 0.983 1.035 1.026 0.885 2 0.804 0 0.208 0.134 0.055 0.031 0.291 0.157 0.094 0.186 0.195 0.130 0.132 0.137 3 0.966 0.208 0 0.238 0.637 0.313 0.361 0.258 0.275 0.141 0.223 0.935 0.474 0.079 4 1.014 0.134 0.238 0 0.267 0.101 0.518 0.073 0.025 0.080 0.211 0.218 0.026 0.151 5 0.967 0.055 0.637 0.267 0 0.047 0.624 0.274 0.193 0.452 0.473 0.093 0.211 0.297 6 0.944 0.031 0.313 0.101 0.047 0 0.476 0.156 0.047 0.219 0.266 0.091 0.085 0.188 7 0.574 0.291 0.361 0.518 0.624 0.476 0 0.508 0.501 0.475 0.453 0.684 0.582 0.265 8 1.001 0.157 0.258 0.073 0.274 0.156 0.508 0 0.126 0.065 0.058 0.180 0.082 0.083 9 0.993 0.094 0.275 0.025 0.193 0.047 0.501 0.126 0 0.151 0.276 0.181 0.036 0.174 10 1.004 0.186 0.141 0.080 0.452 0.219 0.475 0.065 0.151 0 0.086 0.533 0.188 0.093 11 0.983 0.195 0.223 0.211 0.473 0.266 0.453 0.058 0.276 0.086 0 0.602 0.341 0.047 12 1.035 0.130 0.935 0.218 0.093 0.091 0.684 0.180 0.181 0.533 0.602 0 0.127 0.266 13 1.026 0.132 0.474 0.026 0.211 0.085 0.582 0.082 0.036 0.188 0.341 0.127 0 0.191 14 0.885 0.137 0.079 0.151 0.297 0.188 0.265 0.083 0.174 0.093 0.047 0.266 0.191 0 表3 表3 为采用KNN分类器分类的Botswana影像的各类识别结果。由表3 表3 可知，相比KLDA方法、KWP-Fisher准则法，所提KCP-weighted 准则法有更好的识别效果。表中第2，5，6类的识别率均有了明显的提高。从表2 表2 中可知：（5，6）类对，（5，12）类对，以及（6，12）类对的不相似性测度均小于0.1。也就是说，第5类样本不易与其余两类样本分类，而本文所提KCP-weighted准则法能将第5类的识别率相比KLDA法、KWP-Fisher准则法分别提高3.44，4.37个百分点。可见，所提KCP-weighted准则能提取到更合理的特征，通过提高可分性小的类对的分类效果，从而提高整体地物识别率。 Method 1 2 3 4 5 6 7 KLDA subspace method 100.00 97.78 98.61 97.91 86.56 81.67 98.26 KWP-Fisher criterion 99.95 99.26 98.16 96.34 85.63 80.65 97.68 KCP-weighted criterion 99.58 99.26 98.96 97.97 90.00 85.91 98.99 Method 8 9 10 11 12 13 14 KLDA subspace method 98.40 93.98 95.71 95.66 93.72 95.33 94.74 KWP-Fisher criterion 99.07 93.59 92.37 91.43 93.24 93.83 94.47 KCP-weighted criterion 98.52 93.11 95.00 95.70 94.41 94.67 96.32 % 表4 表4 为Indian Pines整体影像在各个特征子空间采用KNN、MD分类器的分类结果。从表4 表4 中可知，本文所提的KCP-weighted准则的平均识别率相较原空间法、KLDA方法和KWP-Fisher准则法，均有明显提高。其中，KNN分类器， <math id="M137"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423452&type= 1.69333339 2.79399991 为6时，KCP-weighted准则法的平均识别率较原空间法、KLDA方法、KWP-Fisher准则法、分别提高了13.18，0.50，2.45个百分点。同理，在MD分类器下识别率也有所提高，KCP-weighted准则的平均识别率较KLDA法、KWP-Fisher法、分别提高了5.26，0.72个百分点。 2 Classifier 2 Original space method 3 KLDA subspace method 3 KWP-Fisher criterion 3 KCP-weighted criterion AA（%） ±SD（%） Kappa AA（%） ±SD（%） <math id="M138"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa AA（%） ±SD（%） <math id="M139"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa AA（%） ±SD（%） <math id="M140"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa KNN 72.42±1.68 0.74 85.10±1.15 0.27 0.84 83.15±2.25 0.03 0.81 85.60±1.03 0.12 0.83 MD 57.37±1.09 0.40 82.27±0.51 5 0.76 86.81±0.62 0.05 0.81 87.53±0.64 0.09 0.81 表5 表5 为Indian Pines子图影像在各个特征子空间采用KNN、MD分类器的分类结果。从表5 表5 可知，所提KCP-weighted准则的平均识别率相比其余三种方法均有明显的提高。在采用KNN分类器分类， <math id="M141"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423452&type= 1.69333339 2.79399991 为7时，KCP-weighted准则的平均识别率相比原空间法、KLDA方法和KWP-Fisher准则法分别提高了6.53，0.52，1.54个百分点。同样，在采用MD分类器进行分类时，相较其余方法，所提KCP-weighted准则法的识别效果也有所提高。此外，KCP-weighted准则法的Kappa系数最大。 2 Classifier 2 Original space method 3 KLDA subspace method 3 KWP-Fisher criterion 3 KCP-weighted criterion AA（%） ±SD（%） Kappa AA（%） ±SD（%） <math id="M142"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa AA（%） ±SD（%） <math id="M143"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa AA（%） ±SD（%） <math id="M144"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17423476&type= 1.52400005 2.45533323 Kappa KNN 87.48±0.38 0.81 93.49±0.64 0.029 0.88 92.47±0.56 0.02 0.88 94.01 ± 0.67 0.16 0.91 MD 73.98±0.63 0.55 94.23±0.47 0.027 0.91 94.32±0.24 0.03 0.90 95.13 ± 0.34 0.06 0.92 由Indian Pines整体影像、Indian Pines子图影像、Botswana影像的识别结果可知，KCP-weighted准则法的地物分类结果优于原空间法、KLDA法、KWP-Fisher准则法。这是因为在野类存在的情况下，KLDA会使得特征方向倾向于保留可分性大类对的分类信息，造成特征子空间中可分性小的类对出现重叠，影响可分性小的类对的分类；而KWP-Fisher准则仅考虑各类对的核类间散布矩阵，没有考虑各类对的核类内散布矩阵对特征提取的影响；KCP-weighted准则能同时考虑到核空间中各类对的类间、类内散布对特征方向的影响，利用核空间中类对的可分性测度对各类对的核类内和类间散布矩阵进行加权，使得特征子空间中各类对的非线性可分性特征得以均衡保留。理论分析以及实验结果均表明，本文提出的KCP-weighted准则能在降维的同时，通过提高可分性小的类对的识别结果来提高整体识别率。 5 结　论 1 针对高光谱遥感地物中相似光谱的不同地物与野类同时存在时，提取有效的非线性可分性特征的问题，本文提出了一种基于KCP-weighted准则的高光谱遥感影像非线性特征提取方法。首先，推导出类对形式的KLDA准则，即： KCP准则。然后，依据核空间中各类对的可分性分别对各类对的核类间和类内散布矩阵进行加权，以提取有效的可分性特征。基于实测高光谱遥感数据的实验结果表明：相比原空间法、KLDA法、以及KWP-Fisher法，所提KCP-weighted准则法能在降维的同时，有效提高可分性小的类对的分类结果，从而提高整体地物的分类精度。说明本文所提方法，在高光谱遥感影像中相似光谱的不同地物与野类同时存在时，可提取出有效的非线性可分性特征。参考文献朱福全，王华军，杨丽平，等 . 自适应波段选择与最佳预测顺序的高光谱图像无损压缩［J］. 光学精密工程， 2020 ， 28 （ 7 ）： 1609 - 1617 . 自适应波段选择与最佳预测顺序的高光谱图像无损压缩 ZHU F Q ， WANG H J ， YANG L P ， et al . Hyperspectral image lossless compressionusing adaptive bands selection and optimal prediction sequence ［J］. Opt. Precision Eng. ， 2020 ， 28 （ 7 ）： 1609 - 1617 . （in Chinese） Hyperspectral image lossless compressionusing adaptive bands selection and optimal prediction sequence 黄鸿，李政英，石光耀，等 . 面向高光谱影像分类的多特征流形鉴别嵌入［J］. 光学精密工程， 2019 ， 27 （ 3 ）： 726 - 738 . 面向高光谱影像分类的多特征流形鉴别嵌入 HUANG H ， LI ZH Y ， SHI G Y ， et al . Multi-features manifold discriminant embedding for hyperspectral image classification ［J］. Opt. Precision Eng. ， 2019 ， 27 （ 3 ）： 726 - 738 . （in Chinese） Multi-features manifold discriminant embedding for hyperspectral image classification 黄鸿，唐玉枭，段宇乐 . 半监督多图嵌入的高光谱影像特征提取［J］. 光学精密工程， 2020 ， 28 （ 2 ）： 443 - 456 . 半监督多图嵌入的高光谱影像特征提取 HUANG H ， TANG Y X ， DUAN Y L . Feature extraction of hyperspectral image with semi-supervised multi-graph embedding ［J］. Opt. Precision Eng. ， 2020 ， 28 （ 2 ）： 443 - 456 . （in Chinese） Feature extraction of hyperspectral image with semi-supervised multi-graph embedding 丁国绅，乔延利，易维宁，等 . 基于高光谱图像的改进SIFT特征提取与匹配［J］. 光学精密工程， 2020 ， 28 （ 4 ）： 954 - 962 . 基于高光谱图像的改进SIFT特征提取与匹配 DING G K ， QIAO Y L ， YI W N ， et al . Improved SIFT feature extraction and matching technology based on hyperspectral image ［J］. Opt. Precision Eng. ， 2020 ， 28 （ 4 ）： 954 - 962 . （in Chinese） Improved SIFT feature extraction and matching technology based on hyperspectral image CHEN M L ， WANG Q ， LI X L . Discriminant analysis with graph learning for hyperspectral image classification ［J］. Remote Sensing ， 2018 ， 10 （ 6 ）： 836 . Discriminant analysis with graph learning for hyperspectral image classification HE L ， LI J ， LIU C ， et al . Recent advances on spectral-spatial hyperspectral image classification： an overview and new guidelines ［J］. IEEE Transactions on Geoscience and Remote Sensing ， 2017 ， 56 （ 3 ）： 1579 - 1597 . Recent advances on spectral-spatial hyperspectral image classification： an overview and new guidelines BAUDAT G ， ANOUAR F . Generalized discriminant analysis using a kernel approach ［J］. Neural Computation ， 2000 ， 12 （ 10 ）， 2385 - 2404 . Generalized discriminant analysis using a kernel approach DAI G ， YEUNG D Y ， CHANG H . Extending kernel fisher discriminant analysis with the weighted pairwise Chernoff criterion ［C］. European Conference on Computer Vision ， 2006 ： 308 - 320 . Extending kernel fisher discriminant analysis with the weighted pairwise Chernoff criterion LU F ， LI H . KLDA-an iterative approach to Fisher discriminant analysis ［C］. 2007 IEEE International Conference on Image Processing ， 2007 2（II）： 201 - 204 . KLDA-an iterative approach to Fisher discriminant analysis LIU J ， LI Q Y ， LIU Y . Spectral feature extraction of hyperspectral remote sensing images based on class pair-weighted criterion ［J］. Journal of Applied Remote Sensing ， 2019 ， 13 （ 4 ）： 046504 . Spectral feature extraction of hyperspectral remote sensing images based on class pair-weighted criterion LOOG M . Approximate Pairwise Accuracy Criteria for Multiclass Linear Dimension Reduction ： Generalisations of the Fisher Criterion ［M］. WBBM Report Series 44 ： Delft University Press ， 1999 ： 15 - 20 . Approximate Pairwise Accuracy Criteria for Multiclass Linear Dimension Reduction：Generalisations of the Fisher Criterion LOOG M ， DUIN R P W ， HAEB-UMBACH R . Multiclass linear dimension reduction by weighted pairwise Fisher criteria ［J］. IEEE Transactions on Pattern Analysis & Machine Intelligence ， 2001 ， 23 （ 7 ）： 762 - 766 . Multiclass linear dimension reduction by weighted pairwise Fisher criteria Hyperspectral data set ［OL］.［ 2020-1-9 ］. http：//www.ehu.eus/ccwintco/index.php/ Hyperspectral_ Remote_ Sensing_Scenes http://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes . Hyperspectral data set 10.37188/OPE.20212906.1397 TP391 A 1004-924X（2021）06-1397-09 刘敬,李青妍,刘逸.基于核加权类对准则的高光谱影像特征提取[J].光学精密工程,2021,29(06):1397-1405. 刘敬,李青妍,刘逸.基于核加权类对准则的高光谱影像特征提取[J].光学精密工程,2021,29(06):1397-1405. LIU Jing,LI Qing-yan,LIU Yi.Spectral feature extraction of hyperspectral remote sensing images based on kernel class pair-weighted criterion[J].Optics and Precision Engineering,2021,29(06):1397-1405. LIU Jing,LI Qing-yan,LIU Yi.Spectral feature extraction of hyperspectral remote sensing images based on kernel class pair-weighted criterion[J].Optics and Precision Engineering,2021,29(06):1397-1405. 国家自然科学基金（No.61672405）；陕西省自然科学基金（No.2018JM4018）；国家自然科学基金（No. 62077038）；陕西省自然科学基金（No. 2021JM-459） 2020-07-06 2020-09-05 2021-06-15 2021-07-12 光学精密工程 6 29

浏览量

下载量

CSCD

文章被引用时，请邮件提醒。

提交

工具集

关联资源

SENet生成对抗网络在图像语义描述中的应用

面向高光谱图像分类的半监督双流网络

相位敏感光时域反射仪的信号处理方法综述

基于对比学习方法的遥感影像特征提取与分类

面向高光谱影像分类的深度流形重构置信网络