LDA-線性判別分析
阿新 • • 發佈：2019-01-28
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LDA 線性判別分析
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<p>1. <wbr>LDA是什麼</p>
<p><wbr><wbr><wbr>線性判別式分析（Linear Discriminant Analysis），簡稱為LDA。也稱為Fisher線性判別（Fisher Linear Discriminant，FLD），是模式識別的經典<a href="http://lib.csdn.net/base/datastructure" class="replace_word" title="演算法與資料結構知識庫" target="_blank" style="color:#df3434; font-weight:bold;">演算法</a>，在1996年由Belhumeur引入模式識別和<a href="http://lib.csdn.net/base/ai" class="replace_word" title="人工智慧知識庫" target="_blank" style="color:#df3434; font-weight:bold;">人工智慧</a>領域。</p>
<p><wbr><wbr><wbr><span style="font-family:宋體">基本思想是將高維的模式樣本投影到最佳鑑別向量空間，以達到抽取分類資訊和壓縮特徵空間維數的效果，投影后保證模式樣本在新的子空間有<strong>最大的類間距離</strong>和<strong>最小的類內距離</strong>，即模式在該空間中有最佳的可分離性。</span></p>
<p><span style="font-family:宋體"><wbr><wbr><wbr></span>LDA與前面介紹過的PCA都是常用的降維技術。<strong>PCA主要是從特徵的協方差角度，去找到比較好的投影方式</strong>。<strong>LDA更多的是考慮了標註</strong>，<strong>即希望投影后不同類別之間資料點的距離更大，同一類別的資料點更緊湊</strong>。</p>
<p><wbr><wbr><wbr><wbr>下面給出一個例子，說明LDA的目標。</p>
<p><wbr><wbr><wbr><img title="LDA <wbr>線性判別分析" 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>線性判別分析" 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> 可以看到兩個類別，一個綠色類別，一個紅色類別。左圖是兩個類別的原始資料，現在要求將資料從二維降維到一維。直接投影到x1軸或者x2軸，不同類別之間會有重複，導致分類效果下降。右圖對映到的直線就是用LDA方法計算得到的，可以看到，紅色類別和綠色類別在對映之後之間的距離是最大的，而且每個類別內部點的離散程度是最小的（或者說聚集程度是最大的）。</p>
<p><wbr><wbr><wbr>這裡還有一個LDA的例子： <wbr><wbr></p>
<p><img title="LDA <wbr>線性判別分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014250451.png" width="257" height="249"><img title="LDA <wbr>線性判別分析" 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>下圖是一個LDA和PCA區別的例子：</p>
<p><img title="LDA <wbr>線性判別分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102010174967.png" width="309" height="282"></p>
<p><wbr><wbr><wbr> <wbr>兩個類別，class1的點都是圓圈，class2的點都是十字。圖中有兩條直線，斜率在1左右的這條直線是PCA選擇的對映直線，斜率在 -1左右的這條直線是LDA選擇的對映直線。其餘不在這兩條直線上的點是原始資料點。可以看到由於LDA考慮了“類別”這個資訊（即標註），對映後，可以很好的將class1和class2的點區分開。</p>
<p>2. LDA的一些說明</p>
<p><wbr><wbr><wbr> 第一，降維後的維度是多少？</p>
<p><wbr><wbr><wbr> PCA降維是直接和資料維度相關的，比如原始資料是n維的，那麼PCA後，可以任意選取1維、2維，一直到n維都行（當然是對應特徵值大的那些）。LDA 降維是直接和類別的個數相關的，與資料本身的維度沒關係，比如原始資料是n維的，一共有C個類別，那麼LDA降維之後，一般就是1維，2維到C-1維進行選擇（當然對應的特徵值也是最大的一些），舉個例子，假設圖象分類，兩個類別正例反例，每個圖象10000維特徵，那麼LDA之後，就只有1維特徵，並且這維特徵的分類能力最好。<br>
<wbr><wbr><wbr>PS：對於很多兩類分類的情況，LDA之後就剩下1維，找到分類效果最好的一個閾值貌似就可以了。</p>
<p><wbr><wbr><wbr> 第二，投影的座標系是否正交</p>
<p><wbr><wbr><wbr> PCA投影的座標系都是正交的，而LDA根據類別的標註，關注分類能力，因此不保證投影到的座標系是正交的（一般都不正交）。</p>
<p>3. LDA計算過程（兩個類別）</p>
<p><wbr><wbr><wbr>這一節，主要討論只有兩個類別的資料，LDA的計算。現在要找到一個向量w，將資料x投影到w上去之後，得到新的資料y。第一，為了實現投影后的兩個類別的距離較遠，用對映後兩個類別的均值差的絕對值來度量。第二，為了實現投影后，每個類內部資料點比較聚集，用投影后每個類別的方差來度量。</p>
<p><wbr><wbr><wbr><wbr>類別i的均值： <wbr><wbr></p>
<p><wbr><br>
<wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 類別i投影后的均值（實際上等於mi的投影）：</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 投影后均值差的絕對值：</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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（這裡的y是類別i中資料投影后的資料，即y=w_t * x）：</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 目標優化函式為：</p>
<p><br>
<wbr> <wbr> <wbr> <wbr><img title="LDA <wbr>線性判別分析" 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> 下面通過展開m'和s'，定義S_B和S_W： </p>
<p><wbr> <wbr> <wbr> <wbr></p>
<p><wbr><wbr><wbr><img title="LDA <wbr>線性判別分析" 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>線性判別分析" 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>線性判別分析" 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>優化目標J(w)改寫如下，這樣是為了方便推匯出計算w的方法。</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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>推導過程忽略了，最後推導結果如下：</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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>假設資料是n維特徵，m個數據，分類個數為2。那麼Sw實際上是每個類別的協方差矩陣之和，每個類別的協方差矩陣都是n*n的，因此Sw是n*n的，m1-m2是n*1的。計算得到的w是n*1的，即w將維特徵對映為1維。</p>
<p>PS：這裡不用糾結與Sw的協方差矩陣形式，實際上這是拆拆開來w和w_t後的結果，實際上w*Sw*w_t後，仍舊是一個數值，即表示兩個類的投影后方差之和。</p>
<p>4. LDA計算過程（多個類別）</p>
<p><wbr><wbr><wbr> 對於S_w，就是“兩個類別的協方差矩陣之和”變成了“多個類別的協方差矩陣之和”。</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 對於S_b，以前是“兩個類別的均值之差的絕對值”，現在多個類別了，怎麼計算呢？計算任意兩個類別之間的均值之差的絕對值之和？這樣對於N個類別，需要 C(N,2)次計算，這可能是一個方法，不過LDA使用的方法，計算每個類別的均值與所有類別均值的差異，並且對每個類別的資料量進行加權處理。下式中，m為所有類別的均值，m_i為類別i的均值，n_i為類別i的資料量。</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 對於n維特徵、C個類別、m個樣本的資料，是將n維資料對映為C-1維。即所求的w是一個n*(c-1)的矩陣。S_w是一個n*n的矩陣（沒有除以樣本數的協方差矩陣之和），S_B是一個C*C的矩陣，實際上S_B矩陣的的秩最多是C-1，這是因為n_i*(m_i-m)這c向量實際上是線性相關的，因為他們的和是均值m的常數倍。這導致後面求解w，實際上是求出C-1個向量一個構成的一個w。</p>
<p><wbr><wbr><wbr>具體計算忽略。</p>
<p><wbr><wbr><wbr>下面就是一個三個類別的例子： <wbr><br>
<wbr><wbr><wbr><img title="LDA <wbr>線性判別分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102112445054.png" width="341" height="389" style="width:338px; height:354px">
</p>
<p>4. LDA其他變種</p>
<p><wbr><wbr><wbr>如果原始資料投影后，仍舊不能很好的分開，那麼Kernel LDA是一個解決方法。</p>
<p><wbr><wbr><wbr>LDA在計算量上與資料的維度有關，2DLDA能夠大大減小LDA計算量。</p>
<p>5. LDA的一些問題</p>
<p><wbr><wbr><wbr>第一，LDA最多投影出C-1維特徵，如果需要更多特徵，則要引入其他方法。</p>
<p><wbr><wbr><wbr>第二，LDA假設資料服從單峰高斯分佈，比如下面的複雜<a href="http://lib.csdn.net/base/datastructure" class="replace_word" title="演算法與資料結構知識庫" target="_blank" style="color:#df3434; font-weight:bold;">資料結構</a>。<br>
<wbr><wbr><wbr><img title="LDA <wbr>線性判別分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102112430329.png" width="623" height="188">
</p>
<p>5. 參考</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">線性判別分析(Linear Discriminant Analysis, LDA）演算法分析</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樓 <span class="user"><a class="username" href="/wh13689831545" target="_blank">wh13689831545</a> <span class="ptime">2016-06-28 13:56發表</span> <a href="#reply" class="cmt_btn reply" title="回覆">[回覆]</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="舉報">[舉報]</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，請問這個lda演算法是特徵提取演算法，提取的是人臉的什麼特徵呢，謝謝</dd></dl><dl class="comment_item comment_topic" id="comment_item_5638395"><dt class="comment_head" floor="2">2樓 <span class="user"><a class="username" href="/huoxiaohuo2" target="_blank">huoxiaohuo2</a> <span class="ptime">2015-11-24 22:53發表</span> <a href="#reply" class="cmt_btn reply" title="回覆">[回覆]</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="舉報">[舉報]</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">請問如果不知道這個先驗的分類，只是給你一堆資料怎麼進行分類？</dd></dl><dl class="comment_item comment_topic" id="comment_item_3027869"><dt class="comment_head" floor="1">1樓 <span class="user"><a class="username" href="/luo451848919" target="_blank">luo451848919</a> <span class="ptime">2013-10-25 22:25發表</span> <a href="#reply" class="cmt_btn reply" title="回覆">[回覆]</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="舉報">[舉報]</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">你說S_b的維數是c*c,明顯不對啊。應該是n*n的，不信你乘一乘，還有就是S_b的秩為什麼最多是c-1啊？謝謝你的答覆！</dd></dl><div class="clear"></div></div>
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LDA 線性判別分析
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<span class="link_postdate">2012-09-26 14:34</span>
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<p>1. <wbr>LDA是什麼</p>
<p><wbr><wbr><wbr>線性判別式分析（Linear Discriminant Analysis），簡稱為LDA。也稱為Fisher線性判別（Fisher Linear Discriminant，FLD），是模式識別的經典<a href="http://lib.csdn.net/base/datastructure" class="replace_word" title="演算法與資料結構知識庫" target="_blank" style="color:#df3434; font-weight:bold;">演算法</a>，在1996年由Belhumeur引入模式識別和<a href="http://lib.csdn.net/base/ai" class="replace_word" title="人工智慧知識庫" target="_blank" style="color:#df3434; font-weight:bold;">人工智慧</a>領域。</p>
<p><wbr><wbr><wbr><span style="font-family:宋體">基本思想是將高維的模式樣本投影到最佳鑑別向量空間，以達到抽取分類資訊和壓縮特徵空間維數的效果，投影后保證模式樣本在新的子空間有<strong>最大的類間距離</strong>和<strong>最小的類內距離</strong>，即模式在該空間中有最佳的可分離性。</span></p>
<p><span style="font-family:宋體"><wbr><wbr><wbr></span>LDA與前面介紹過的PCA都是常用的降維技術。<strong>PCA主要是從特徵的協方差角度，去找到比較好的投影方式</strong>。<strong>LDA更多的是考慮了標註</strong>，<strong>即希望投影后不同類別之間資料點的距離更大，同一類別的資料點更緊湊</strong>。</p>
<p><wbr><wbr><wbr><wbr>下面給出一個例子，說明LDA的目標。</p>
<p><wbr><wbr><wbr><img title="LDA <wbr>線性判別分析" 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>線性判別分析" 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> 可以看到兩個類別，一個綠色類別，一個紅色類別。左圖是兩個類別的原始資料，現在要求將資料從二維降維到一維。直接投影到x1軸或者x2軸，不同類別之間會有重複，導致分類效果下降。右圖對映到的直線就是用LDA方法計算得到的，可以看到，紅色類別和綠色類別在對映之後之間的距離是最大的，而且每個類別內部點的離散程度是最小的（或者說聚集程度是最大的）。</p>
<p><wbr><wbr><wbr>這裡還有一個LDA的例子： <wbr><wbr></p>
<p><img title="LDA <wbr>線性判別分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102014250451.png" width="257" height="249"><img title="LDA <wbr>線性判別分析" 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>下圖是一個LDA和PCA區別的例子：</p>
<p><img title="LDA <wbr>線性判別分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102010174967.png" width="309" height="282"></p>
<p><wbr><wbr><wbr> <wbr>兩個類別，class1的點都是圓圈，class2的點都是十字。圖中有兩條直線，斜率在1左右的這條直線是PCA選擇的對映直線，斜率在 -1左右的這條直線是LDA選擇的對映直線。其餘不在這兩條直線上的點是原始資料點。可以看到由於LDA考慮了“類別”這個資訊（即標註），對映後，可以很好的將class1和class2的點區分開。</p>
<p>2. LDA的一些說明</p>
<p><wbr><wbr><wbr> 第一，降維後的維度是多少？</p>
<p><wbr><wbr><wbr> PCA降維是直接和資料維度相關的，比如原始資料是n維的，那麼PCA後，可以任意選取1維、2維，一直到n維都行（當然是對應特徵值大的那些）。LDA 降維是直接和類別的個數相關的，與資料本身的維度沒關係，比如原始資料是n維的，一共有C個類別，那麼LDA降維之後，一般就是1維，2維到C-1維進行選擇（當然對應的特徵值也是最大的一些），舉個例子，假設圖象分類，兩個類別正例反例，每個圖象10000維特徵，那麼LDA之後，就只有1維特徵，並且這維特徵的分類能力最好。<br>
<wbr><wbr><wbr>PS：對於很多兩類分類的情況，LDA之後就剩下1維，找到分類效果最好的一個閾值貌似就可以了。</p>
<p><wbr><wbr><wbr> 第二，投影的座標系是否正交</p>
<p><wbr><wbr><wbr> PCA投影的座標系都是正交的，而LDA根據類別的標註，關注分類能力，因此不保證投影到的座標系是正交的（一般都不正交）。</p>
<p>3. LDA計算過程（兩個類別）</p>
<p><wbr><wbr><wbr>這一節，主要討論只有兩個類別的資料，LDA的計算。現在要找到一個向量w，將資料x投影到w上去之後，得到新的資料y。第一，為了實現投影后的兩個類別的距離較遠，用對映後兩個類別的均值差的絕對值來度量。第二，為了實現投影后，每個類內部資料點比較聚集，用投影后每個類別的方差來度量。</p>
<p><wbr><wbr><wbr><wbr>類別i的均值： <wbr><wbr></p>
<p><wbr><br>
<wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 類別i投影后的均值（實際上等於mi的投影）：</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 投影后均值差的絕對值：</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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（這裡的y是類別i中資料投影后的資料，即y=w_t * x）：</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 目標優化函式為：</p>
<p><br>
<wbr> <wbr> <wbr> <wbr><img title="LDA <wbr>線性判別分析" 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> 下面通過展開m'和s'，定義S_B和S_W： </p>
<p><wbr> <wbr> <wbr> <wbr></p>
<p><wbr><wbr><wbr><img title="LDA <wbr>線性判別分析" 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>線性判別分析" 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>線性判別分析" 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>優化目標J(w)改寫如下，這樣是為了方便推匯出計算w的方法。</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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>推導過程忽略了，最後推導結果如下：</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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>假設資料是n維特徵，m個數據，分類個數為2。那麼Sw實際上是每個類別的協方差矩陣之和，每個類別的協方差矩陣都是n*n的，因此Sw是n*n的，m1-m2是n*1的。計算得到的w是n*1的，即w將維特徵對映為1維。</p>
<p>PS：這裡不用糾結與Sw的協方差矩陣形式，實際上這是拆拆開來w和w_t後的結果，實際上w*Sw*w_t後，仍舊是一個數值，即表示兩個類的投影后方差之和。</p>
<p>4. LDA計算過程（多個類別）</p>
<p><wbr><wbr><wbr> 對於S_w，就是“兩個類別的協方差矩陣之和”變成了“多個類別的協方差矩陣之和”。</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 對於S_b，以前是“兩個類別的均值之差的絕對值”，現在多個類別了，怎麼計算呢？計算任意兩個類別之間的均值之差的絕對值之和？這樣對於N個類別，需要 C(N,2)次計算，這可能是一個方法，不過LDA使用的方法，計算每個類別的均值與所有類別均值的差異，並且對每個類別的資料量進行加權處理。下式中，m為所有類別的均值，m_i為類別i的均值，n_i為類別i的資料量。</p>
<p><br>
<wbr> <wbr> <wbr> <img title="LDA <wbr>線性判別分析" 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> 對於n維特徵、C個類別、m個樣本的資料，是將n維資料對映為C-1維。即所求的w是一個n*(c-1)的矩陣。S_w是一個n*n的矩陣（沒有除以樣本數的協方差矩陣之和），S_B是一個C*C的矩陣，實際上S_B矩陣的的秩最多是C-1，這是因為n_i*(m_i-m)這c向量實際上是線性相關的，因為他們的和是均值m的常數倍。這導致後面求解w，實際上是求出C-1個向量一個構成的一個w。</p>
<p><wbr><wbr><wbr>具體計算忽略。</p>
<p><wbr><wbr><wbr>下面就是一個三個類別的例子： <wbr><br>
<wbr><wbr><wbr><img title="LDA <wbr>線性判別分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102112445054.png" width="341" height="389" style="width:338px; height:354px">
</p>
<p>4. LDA其他變種</p>
<p><wbr><wbr><wbr>如果原始資料投影后，仍舊不能很好的分開，那麼Kernel LDA是一個解決方法。</p>
<p><wbr><wbr><wbr>LDA在計算量上與資料的維度有關，2DLDA能夠大大減小LDA計算量。</p>
<p>5. LDA的一些問題</p>
<p><wbr><wbr><wbr>第一，LDA最多投影出C-1維特徵，如果需要更多特徵，則要引入其他方法。</p>
<p><wbr><wbr><wbr>第二，LDA假設資料服從單峰高斯分佈，比如下面的複雜<a href="http://lib.csdn.net/base/datastructure" class="replace_word" title="演算法與資料結構知識庫" target="_blank" style="color:#df3434; font-weight:bold;">資料結構</a>。<br>
<wbr><wbr><wbr><img title="LDA <wbr>線性判別分析" alt="" src="http://pic002.cnblogs.com/images/2011/79762/2011102112430329.png" width="623" height="188">
</p>
<p>5. 參考</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">線性判別分析(Linear Discriminant Analysis, LDA）演算法分析</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樓 <span class="user"><a class="username" href="/wh13689831545" target="_blank">wh13689831545</a> <span class="ptime">2016-06-28 13:56發表</span> <a href="#reply" class="cmt_btn reply" title="回覆">[回覆]</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="舉報">[舉報]</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，請問這個lda演算法是特徵提取演算法，提取的是人臉的什麼特徵呢，謝謝</dd></dl><dl class="comment_item comment_topic" id="comment_item_5638395"><dt class="comment_head" floor="2">2樓 <span class="user"><a class="username" href="/huoxiaohuo2" target="_blank">huoxiaohuo2</a> <span class="ptime">2015-11-24 22:53發表</span> <a href="#reply" class="cmt_btn reply" title="回覆">[回覆]</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="舉報">[舉報]</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">請問如果不知道這個先驗的分類，只是給你一堆資料怎麼進行分類？</dd></dl><dl class="comment_item comment_topic" id="comment_item_3027869"><dt class="comment_head" floor="1">1樓 <span class="user"><a class="username" href="/luo451848919" target="_blank">luo451848919</a> <span class="ptime">2013-10-25 22:25發表</span> <a href="#reply" class="cmt_btn reply" title="回覆">[回覆]</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="舉報">[舉報]</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">你說S_b的維數是c*c,明顯不對啊。應該是n*n的，不信你乘一乘，還有就是S_b的秩為什麼最多是c-1啊？謝謝你的答覆！</dd></dl><div class="clear"></div></div>
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