LDA-线性判别分析

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LDA 线性判别分析
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<p>1. <wbr>LDA是什么</p>
<p><wbr><wbr><wbr>线性判别式分析（Linear Discriminant Analysis），简称为LDA。也称为Fisher线性判别（Fisher Linear Discriminant，FLD），是模式识别的经典<a href="http://lib.csdn.net/base/datastructure" class="replace_word" title="算法与数据结构知识库" target="_blank" style="color:#df3434; font-weight:bold;">算法</a>，在1996年由Belhumeur引入模式识别和<a href="http://lib.csdn.net/base/ai" class="replace_word" title="人工智能知识库" target="_blank" style="color:#df3434; font-weight:bold;">人工智能</a>领域。</p>
<p><wbr><wbr><wbr><span style="font-family:宋体">基本思想是将高维的模式样本投影到最佳鉴别矢量空间，以达到抽取分类信息和压缩特征空间维数的效果，投影后保证模式样本在新的子空间有<strong>最大的类间距离</strong>和<strong>最小的类内距离</strong>，即模式在该空间中有最佳的可分离性。</span></p>
<p><span style="font-family:宋体"><wbr><wbr><wbr></span>LDA与前面介绍过的PCA都是常用的降维技术。<strong>PCA主要是从特征的协方差角度，去找到比较好的投影方式</strong>。<strong>LDA更多的是考虑了标注</strong>，<strong>即希望投影后不同类别之间数据点的距离更大，同一类别的数据点更紧凑</strong>。</p>
<p><wbr><wbr><wbr><wbr>下面给出一个例子，说明LDA的目标。</p>
<p><wbr><wbr><wbr><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102010103238.png" width="331" height="215" style="width:264px; height:185px">
<wbr><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102010120697.png" width="363" height="215" style="width:310px; height:183px"></p>
<p><wbr><wbr><wbr>       可以看到两个类别，一个绿色类别，一个红色类别。左图是两个类别的原始数据，现在要求将数据从二维降维到一维。直接投影到x1轴或者x2轴，不同类别之间会有重复，导致分类效果下降。右图映射到的直线就是用LDA方法计算得到的，可以看到，红色类别和绿色类别在映射之后之间的距离是最大的，而且每个类别内部点的离散程度是最小的（或者说聚集程度是最大的）。</p>
<p><wbr><wbr><wbr>这里还有一个LDA的例子： <wbr><wbr></p>
<p><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014250451.png" width="257" height="249"><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014251376.png" width="272" height="242" style="width:299px; height:240px"></p>
<p> </p>
<p><wbr><wbr><wbr>下图是一个LDA和PCA区别的例子：</p>
<p><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102010174967.png" width="309" height="282"></p>
<p><wbr><wbr><wbr>       <wbr>两个类别，class1的点都是圆圈，class2的点都是十字。图中有两条直线，斜率在1左右的这条直线是PCA选择的映射直线，斜率在 -1左右的这条直线是LDA选择的映射直线。其余不在这两条直线上的点是原始数据点。可以看到由于LDA考虑了“类别”这个信息（即标注），映射后，可以很好的将class1和class2的点区分开。</p>
<p>2. LDA的一些说明</p>
<p><wbr><wbr><wbr>   第一，降维后的维度是多少？</p>
<p><wbr><wbr><wbr>     PCA降维是直接和数据维度相关的，比如原始数据是n维的，那么PCA后，可以任意选取1维、2维，一直到n维都行（当然是对应特征值大的那些）。LDA 降维是直接和类别的个数相关的，与数据本身的维度没关系，比如原始数据是n维的，一共有C个类别，那么LDA降维之后，一般就是1维，2维到C-1维进行选择（当然对应的特征值也是最大的一些），举个例子，假设图象分类，两个类别正例反例，每个图象10000维特征，那么LDA之后，就只有1维特征，并且这维特征的分类能力最好。<br>
<wbr><wbr><wbr>PS：对于很多两类分类的情况，LDA之后就剩下1维，找到分类效果最好的一个阈值貌似就可以了。</p>
<p><wbr><wbr><wbr>   第二，投影的坐标系是否正交</p>
<p><wbr><wbr><wbr>    PCA投影的坐标系都是正交的，而LDA根据类别的标注，关注分类能力，因此不保证投影到的坐标系是正交的（一般都不正交）。</p>
<p>3. LDA计算过程（两个类别）</p>
<p><wbr><wbr><wbr>这一节，主要讨论只有两个类别的数据，LDA的计算。现在要找到一个向量w，将数据x投影到w上去之后，得到新的数据y。第一，为了实现投影后的两个类别的距离较远，用映射后两个类别的均值差的绝对值来度量。第二，为了实现投影后，每个类内部数据点比较聚集，用投影后每个类别的方差来度量。</p>
<p><wbr><wbr><wbr><wbr>类别i的均值： <wbr><wbr></p>
<p><wbr><br>
 <wbr> <wbr> <img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014454997.png" width="108" height="46" style="width:92px; height:46px">
</p>
<p><br>
 <wbr> <wbr> <wbr> 类别i投影后的均值（实际上等于mi的投影）：</p>
<p><br>
 <wbr> <wbr> <wbr> <img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014472583.png" width="353" height="46" style="width:287px; height:43px">
</p>
<p><br>
 <wbr> <wbr> <wbr> 投影后均值差的绝对值：</p>
<p><br>
 <wbr> <wbr> <wbr> <img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014492130.png" width="216" height="43" style="width:158px; height:24px">
</p>
<p><br>
 <wbr> <wbr> <wbr> 投影后的方差（这里的y是类别i中数据投影后的数据，即y=w_t * x）：</p>
<p><br>
 <wbr> <wbr> <wbr> <img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014503457.png" width="129" height="32" style="width:114px; height:29px">
</p>
<p><br>
 <wbr> <wbr> <wbr> 目标优化函数为：</p>
<p><br>
 <wbr> <wbr> <wbr> <wbr><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014524146.png" width="148" height="56" style="width:126px; height:48px"></p>
<p><br>
 <wbr> <wbr> <wbr> 下面通过展开m'和s'，定义S_B和S_W： </p>
<p><wbr> <wbr> <wbr> <wbr></p>
<p><wbr><wbr><wbr><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102111540310.png" width="421" height="57" style="width:284px; height:55px">
</p>
<p> </p>
<p><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102111555950.png" width="449" height="41" style="width:255px; height:23px"></p>
<p> <wbr> <wbr> <wbr>  </p>
<p><wbr><wbr><wbr><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102111563767.png" width="444" height="65" style="width:335px; height:51px">
</p>
<p><wbr><wbr><wbr>优化目标J(w)改写如下，这样是为了方便推导出计算w的方法。</p>
<p><br>
 <wbr> <wbr> <wbr> <img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014561448.png" width="188" height="64" style="width:124px; height:41px">
</p>
<p><br>
 <wbr> <wbr> <wbr> <wbr>推导过程忽略了，最后推导结果如下：</p>
<p><br>
 <wbr> <wbr> <wbr> <img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102112004569.png" width="226" height="42" style="width:192px; height:34px">
</p>
<p> </p>
<p><wbr><wbr><wbr>假设数据是n维特征，m个数据，分类个数为2。那么Sw实际上是每个类别的协方差矩阵之和，每个类别的协方差矩阵都是n*n的，因此Sw是n*n的，m1-m2是n*1的。计算得到的w是n*1的，即w将维特征映射为1维。</p>
<p>PS：这里不用纠结与Sw的协方差矩阵形式，实际上这是拆拆开来w和w_t后的结果，实际上w*Sw*w_t后，仍旧是一个数值，即表示两个类的投影后方差之和。</p>
<p>4. LDA计算过程（多个类别）</p>
<p><wbr><wbr><wbr>     对于S_w，就是“两个类别的协方差矩阵之和”变成了“多个类别的协方差矩阵之和”。</p>
<p><br>
 <wbr> <wbr> <wbr> <img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102015035933.png" width="107" height="66" style="width:82px; height:44px">
<br>
<wbr><wbr><wbr></p>
<p>      对于S_b，以前是“两个类别的均值之差的绝对值”，现在多个类别了，怎么计算呢？计算任意两个类别之间的均值之差的绝对值之和？这样对于N个类别，需要 C(N,2)次计算，这可能是一个方法，不过LDA使用的方法，计算每个类别的均值与所有类别均值的差异，并且对每个类别的数据量进行加权处理。下式中，m为所有类别的均值，m_i为类别i的均值，n_i为类别i的数据量。</p>
<p><br>
 <wbr> <wbr> <wbr> <img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102015133280.png" width="389" height="58" style="width:262px; height:45px">
</p>
<p><wbr><wbr><wbr></p>
<p>        对于n维特征、C个类别、m个样本的数据，是将n维数据映射为C-1维。即所求的w是一个n*(c-1)的矩阵。S_w是一个n*n的矩阵（没有除以样本数的协方差矩阵之和），S_B是一个C*C的矩阵，实际上S_B矩阵的的秩最多是C-1，这是因为n_i*(m_i-m)这c向量实际上是线性相关的，因为他们的和是均值m的常数倍。这导致后面求解w，实际上是求出C-1个向量一个构成的一个w。</p>
<p><wbr><wbr><wbr>具体计算忽略。</p>
<p><wbr><wbr><wbr>下面就是一个三个类别的例子： <wbr><br>
<wbr><wbr><wbr><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102112445054.png" width="341" height="389" style="width:338px; height:354px">
</p>
<p>4. LDA其他变种</p>
<p><wbr><wbr><wbr>如果原始数据投影后，仍旧不能很好的分开，那么Kernel LDA是一个解决方法。</p>
<p><wbr><wbr><wbr>LDA在计算量上与数据的维度有关，2DLDA能够大大减小LDA计算量。</p>
<p>5. LDA的一些问题</p>
<p><wbr><wbr><wbr>第一，LDA最多投影出C-1维特征，如果需要更多特征，则要引入其他方法。</p>
<p><wbr><wbr><wbr>第二，LDA假设数据服从单峰高斯分布，比如下面的复杂<a href="http://lib.csdn.net/base/datastructure" class="replace_word" title="算法与数据结构知识库" target="_blank" style="color:#df3434; font-weight:bold;">数据结构</a>。<br>
<wbr><wbr><wbr><img title="LDA <wbr>线性判别分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102112430329.png" width="623" height="188">
</p>
<p>5. 参考</p>
<p><wbr><wbr><wbr><a href="http://www.cmlab.csie.ntu.edu.tw/~cyy/learning/tutorials/LDA.pdf">Introduction to LDA</a><br>
<wbr><wbr><wbr><a href="http://www.isip.piconepress.com/publications/reports/isip_internal/1998/linear_discrim_analysis/lda_theory.pdf">Linear Discriminant Analysis - A Brief Tutorial</a><br>
<wbr><wbr>  <wbr><a href="http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_fisher_discriminant.htm">http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_fisher_discriminant.htm</a><br>
<wbr><wbr><wbr><a href="http://blog.csdn.net/warmyellow/article/details/5454943">线性判别分析(Linear Discriminant Analysis, LDA）算法分析</a></p>
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<div id="comment_list"><dl class="comment_item comment_topic" id="comment_item_6100175"><dt class="comment_head" floor="3">3楼 <span class="user"><a class="username" href="/wh13689831545" target="_blank">wh13689831545</a> <span class="ptime">2016-06-28 13:56发表</span> <a href="#reply" class="cmt_btn reply" title="回复">[回复]</a> <span class="comment_manage" style="display:none;" commentid="6100175" username="wh13689831545"> <a href="#quote" class="cmt_btn quote" title="引用">[引用]</a> <a href="#report" class="cmt_btn report" title="举报">[举报]</a></span></span></dt><dd class="comment_userface"><a href="/wh13689831545" target="_blank"><img src="http://avatar.csdn.net/8/6/0/3_wh13689831545.jpg" width="40" height="40"></a></dd><dd class="comment_body">您好，请问这个lda算法是特征提取算法，提取的是人脸的什么特征呢，谢谢</dd></dl><dl class="comment_item comment_topic" id="comment_item_5638395"><dt class="comment_head" floor="2">2楼 <span class="user"><a class="username" href="/huoxiaohuo2" target="_blank">huoxiaohuo2</a> <span class="ptime">2015-11-24 22:53发表</span> <a href="#reply" class="cmt_btn reply" title="回复">[回复]</a> <span class="comment_manage" style="display:none;" commentid="5638395" username="huoxiaohuo2"> <a href="#quote" class="cmt_btn quote" title="引用">[引用]</a> <a href="#report" class="cmt_btn report" title="举报">[举报]</a></span></span></dt><dd class="comment_userface"><a href="/huoxiaohuo2" target="_blank"><img src="http://avatar.csdn.net/1/A/6/3_huoxiaohuo2.jpg" width="40" height="40"></a></dd><dd class="comment_body">请问如果不知道这个先验的分类，只是给你一堆数据怎么进行分类？</dd></dl><dl class="comment_item comment_topic" id="comment_item_3027869"><dt class="comment_head" floor="1">1楼 <span class="user"><a class="username" href="/luo451848919" target="_blank">luo451848919</a> <span class="ptime">2013-10-25 22:25发表</span> <a href="#reply" class="cmt_btn reply" title="回复">[回复]</a> <span class="comment_manage" style="display:none;" commentid="3027869" username="luo451848919"> <a href="#quote" class="cmt_btn quote" title="引用">[引用]</a> <a href="#report" class="cmt_btn report" title="举报">[举报]</a></span></span></dt><dd class="comment_userface"><a href="/luo451848919" target="_blank"><img src="http://avatar.csdn.net/0/5/A/3_luo451848919.jpg" width="40" height="40"></a></dd><dd class="comment_body">你说S_b的维数是c*c,明显不对啊。应该是n*n的，不信你乘一乘，还有就是S_b的秩为什么最多是c-1啊？谢谢你的答复！</dd></dl><div class="clear"></div></div>
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<li>

<a href="/porly/article/details/8020696#comments">LDA 线性判别分析</a>
<p style="margin:0px;"><a href="/wh13689831545" class="user_name">wh13689831545</a>:
您好，请问这个lda算法是特征提取算法，提取的是人脸的什么特征呢，谢谢
</p>
</li>
<li>

<a href="/porly/article/details/8020696#comments">LDA 线性判别分析</a>
<p style="margin:0px;"><a href="/huoxiaohuo2" class="user_name">huoxiaohuo2</a>:
请问如果不知道这个先验的分类，只是给你一堆数据怎么进行分类？
</p>
</li>
<li>

<a href="/porly/article/details/7875888#comments">矩阵范数/谱/条件数</a>
<p style="margin:0px;"><a href="/Runhua_Wang" class="user_name">Runhua_Wang</a>:
看不了图片啊
</p>
</li>
<li>

<a href="/porly/article/details/7877033#comments">Jacobi迭代法 Gauss-Seidel迭代法</a>
<p style="margin:0px;"><a href="/new_man__" class="user_name">new_man__</a>:
两个都是Jacobi迭代……
</p>
</li>
<li>

<a href="/porly/article/details/7877113#comments">幂法求特征值</a>
<p style="margin:0px;"><a href="/ningningchaoge" class="user_name">ningningchaoge</a>:
亲亲，我想做幂法求矩阵特征值，不知道你是否有这方面的详细算法或者代码可以给我借鉴一下吗？对了我做的是...
</p>
</li>
<li>

<a href="/porly/article/details/7874661#comments">强大的矩阵奇异值分解(SVD)及其应用</a>
<p style="margin:0px;"><a href="/zhengjie_1990" class="user_name">zhengjie_1990</a>:
第二次认真读了这篇博文，体会更深刻了
</p>
</li>
<li>

<a href="/porly/article/details/7874661#comments">LDA 线性判别分析</a>
<p style="margin:0px;"><a href="/luo451848919" class="user_name">luo451848919</a>:
你说S_b的维数是c*c,明显不对啊。应该是n*n的，不信你乘一乘，还有就是S_b的秩为什么最多是c...
</p>
</li>
<li>

<a href="/porly/article/details/8009779#comments">js到anction中文乱码问题和超链接中文参数乱码问题</a>
<p style="margin:0px;"><a href="/cxwcccc" class="user_name">cxwcccc</a>:
楼主威武！！
</p>
</li>
<li>

<a href="/porly/article/details/7893049#comments">MIMO信道的信道容量</a>
<p style="margin:0px;"><a href="/yan54" class="user_name">yan54</a>:
好文，谢谢分享
</p>
</li>
<li>

<a href="/porly/article/details/7893049#comments">强大的矩阵奇异值分解(SVD)及其应用</a>
<p style="margin:0px;"><a href="/yidaijiaozi169" class="user_name">yidaijiaozi169</a>:
写的相当不错，非常形象，好理解！
</p>
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