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  12. 鐢熺墿鍥惧儚鍘荤浉鍏冲垎鏋愬紩鎿?鍦ㄨ繖閲屾彃鍏ュ浘鐗囨弿杩? /></li></ol> <h3>Python鐧藉寲</h3> <p>鐧藉寲鍙樻崲鎴栫悆闈㈠彉鎹㈡槸涓€绉嶇嚎鎬у彉鎹紝瀹冨皢鍏锋湁宸茬煡鍗忔柟宸煩闃电殑闅忔満鍙橀噺鍚戦噺杞崲涓轰竴缁勫崗鏂瑰樊涓哄崟浣嶇煩闃电殑鏂板彉閲忥紝杩欐剰鍛崇潃瀹冧滑涓嶇浉鍏充笖姣忎釜鍙橀噺鐨勬柟宸负 1銆傝繖绉嶅彉鎹㈣绉颁负鈥滅櫧鍖栤€濓紝鍥犱负瀹冨皢杈撳叆鍚戦噺鏇存敼涓虹櫧鍣０鍚戦噺銆?/p> <p>鍏朵粬鍑犱釜杞彉涓庣櫧鍖栧瘑鍒囩浉鍏筹細</p> <ul><li>鍘荤浉鍏冲彉鎹粎鍒犻櫎鐩稿叧鎬э紝浣嗕繚鎸佹柟宸笉鍙樸€?/li><li>鏍囧噯鍖栧彉鎹㈠皢鏂瑰樊璁剧疆涓?1锛屼絾淇濇寔鐩稿叧鎬т笉鍙樸€?/li><li>鐫€鑹插彉鎹㈠皢鐧借壊闅忔満鍙橀噺鍚戦噺鍙樻崲涓哄叿鏈夋寚瀹氬崗鏂瑰樊鐭╅樀鐨勯殢鏈哄悜閲忋€?/li></ul> <p>鍋囪 <span ><span ><span > X X </span><span ><span ><span ></span><span >X</span></span></span></span></span> 鏄竴涓殢鏈猴紙鍒楋級鍚戦噺锛屽叿鏈夐潪濂囧紓鍗忔柟宸煩闃?<span ><span ><span > 危 \Sigma </span><span ><span ><span ></span><span >危</span></span></span></span></span> 鍜屽钩鍧囧€?0 銆傜劧鍚庯紝浣跨敤婊¤冻鏉′欢 <span ><span ><span > W T W = 危 鈭? 1 W^{ T } W=\Sigma^{-1} </span><span ><span ><span ></span><span ><span >W</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >T</span></span></span></span></span></span></span></span></span><span >W</span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span >危</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >鈭?/span><span >1</span></span></span></span></span></span></span></span></span></span></span></span></span> 鐨勭櫧鍖栫煩闃?<span ><span ><span > W W </span><span ><span ><span ></span><span >W</span></span></span></span></span> 杩涜鍙樻崲 <span ><span ><span > Y = W X Y=W X </span><span ><span ><span ></span><span >Y</span><span ></span><span >=</span><span ></span></span><span ><span ></span><span >W</span><span >X</span></span></span></span></span>锛岀敓鎴愬叿鏈夊崟浣嶅瑙掑崗鏂瑰樊鐨勭櫧鍖栭殢鏈哄悜閲?<span ><span ><span > Y Y </span><span ><span ><span ></span><span >Y</span></span></span></span></span>銆?/p> <p>鏈夋棤闄愬涓彲鑳界殑鐧藉寲鐭╅樀<span ><span ><span > W W </span><span ><span ><span ></span><span >W</span></span></span></span></span>閮芥弧瓒充笂杩版潯浠躲€傚父鐢ㄧ殑閫夋嫨鏄?span ><span ><span > W = 危 鈭? 1 / 2 W=\Sigma^{-1 / 2} </span><span ><span ><span ></span><span >W</span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span >危</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >鈭?/span><span >1/2</span></span></span></span></span></span></span></span></span></span></span></span></span>锛堥浂鐩镐綅鍒嗛噺鍒嗘瀽鐧藉寲锛夛紝<span ><span ><span > W = L T W=L^T </span><span ><span ><span ></span><span >W</span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span >L</span><span ><span ><span ><span ><span ><span ></span><span ><span >T</span></span></span></span></span></span></span></span></span></span></span></span>锛屽叾涓?span ><span ><span > L L </span><span ><span ><span ></span><span >L</span></span></span></span></span>鏄?span ><span ><span > 危 鈭? 1 \Sigma^{-1} </span><span ><span ><span ></span><span ><span >危</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >鈭?/span><span >1</span></span></span></span></span></span></span></span></span></span></span></span></span>鐨凜holesky鍒嗚В锛?Cholesky 鐧藉寲锛夛紝鎴?<span ><span ><span > 危 \Sigma </span><span ><span ><span ></span><span >危</span></span></span></span></span> 鐨勭壒寰佺郴缁燂紙涓绘垚鍒嗗垎鏋愮櫧鍖栵級銆?/p> <p>鍙互閫氳繃鐮旂┒ <span ><span ><span > X X </span><span ><span ><span ></span><span >X</span></span></span></span></span> 鍜?<span ><span ><span > Y Y </span><span ><span ><span ></span><span >Y</span></span></span></span></span> 鐨勪簰鍗忔柟宸拰浜掔浉鍏虫潵閫夊嚭鏈€浣崇櫧鍖栧彉鎹€?渚嬪锛屽湪鍘熷 <span ><span ><span > X X </span><span ><span ><span ></span><span >X</span></span></span></span></span> 鍜岀櫧鍖栧悗鐨?<span ><span ><span > Y Y </span><span ><span ><span ></span><span >Y</span></span></span></span></span> 涔嬮棿瀹炵幇鏈€澶у垎閲忕浉鍏虫€х殑鍞竴鏈€浼樼櫧鍖栧彉鎹㈡槸鐢辩櫧鍖栫煩闃?<span ><span ><span > W = P 鈭? 1 / 2 V 鈭? 1 / 2 W=P^{-1 / 2} V^{-1 / 2} </span><span ><span ><span ></span><span >W</span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span >P</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >鈭?/span><span >1/2</span></span></span></span></span></span></span></span></span><span ><span >V</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >鈭?/span><span >1/2</span></span></span></span></span></span></span></span></span></span></span></span></span>锛屽叾涓?<span ><span ><span > P P </span><span ><span ><span ></span><span >P</span></span></span></span></span> 鏄浉鍏崇煩闃碉紝<span ><span ><span > V V </span><span ><span ><span ></span><span >V</span></span></span></span></span> 鏄柟宸煩闃点€?/p> <h4>鍗忔柟宸煩闃?/h4> <p><span ><span ><span ><span > x variance聽 = 鈭? i = 1 n ( x i 鈭? x mean聽 ) 2 n x_{\text {variance }}=\frac{\sum_{i=1}^n\left(x_i-x_{\text {mean }}\right)^2}{n} </span><span ><span ><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >variance聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span ></span><span ><span ><span ><span ><span ><span ></span><span ><span >n</span></span></span><span ><span ></span><span ></span></span><span ><span ></span><span ><span ><span >鈭?/span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >i</span><span >=</span><span >1</span></span></span></span><span ><span ></span><span ><span >n</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span ><span ><span >(</span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span >i</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >鈭?/span><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >mean聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span >)</span></span><span ><span ><span ><span ><span ><span ></span><span ><span >2</span></span></span></span></span></span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span><span ></span></span></span></span></span></span></span></p> <p><span ><span ><span ><span > x covariance聽 = 鈭? i = 1 n ( x i 鈭? x mean聽 ) ( y i 鈭? y mean聽 ) n x_{\text {covariance }}=\frac{\sum_{i=1}^n\left(x_i-x_{\text {mean }}\right)\left(y_i-y_{\text {mean }}\right)}{n} </span><span ><span ><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >covariance聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span ></span><span ><span ><span ><span ><span ><span ></span><span ><span >n</span></span></span><span ><span ></span><span ></span></span><span ><span ></span><span ><span ><span >鈭?/span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >i</span><span >=</span><span >1</span></span></span></span><span ><span ></span><span ><span >n</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span ><span >(</span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span >i</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >鈭?/span><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >mean聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span >)</span></span><span ></span><span ><span >(</span><span ><span >y</span><span ><span ><span ><span ><span ><span ></span><span ><span >i</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >鈭?/span><span ></span><span ><span >y</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >mean聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span >)</span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span><span ></span></span></span></span></span></span></span></p> <p><span ><span ><span ><span > x variance聽 = 鈭? i = 1 n ( x i 鈭? x mean聽 ) 2 n 鈭? 1 x_{\text {variance }}=\frac{\sum_{i=1}^n\left(x_i-x_{\text {mean }}\right)^2}{n-1} </span><span ><span ><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >variance聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span ></span><span ><span ><span ><span ><span ><span ></span><span ><span >n</span><span ></span><span >鈭?/span><span ></span><span >1</span></span></span><span ><span ></span><span ></span></span><span ><span ></span><span ><span ><span >鈭?/span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >i</span><span >=</span><span >1</span></span></span></span><span ><span ></span><span ><span >n</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span ><span ><span >(</span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span >i</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >鈭?/span><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >mean聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span >)</span></span><span ><span ><span ><span ><span ><span ></span><span ><span >2</span></span></span></span></span></span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span><span ></span></span></span></span></span></span></span></p> <p><span ><span ><span ><span > x covariance聽 = 鈭? i = 1 n ( x i 鈭? x mean聽 ) ( y i 鈭? y mean聽 ) n 鈭? 1 x_{\text {covariance }}=\frac{\sum_{i=1}^n\left(x_i-x_{\text {mean }}\right)\left(y_i-y_{\text {mean }}\right)}{n-1} </span><span ><span ><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >covariance聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span ></span><span ><span ><span ><span ><span ><span ></span><span ><span >n</span><span ></span><span >鈭?/span><span ></span><span >1</span></span></span><span ><span ></span><span ></span></span><span ><span ></span><span ><span ><span >鈭?/span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >i</span><span >=</span><span >1</span></span></span></span><span ><span ></span><span ><span >n</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span ><span >(</span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span >i</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >鈭?/span><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >mean聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span >)</span></span><span ></span><span ><span >(</span><span ><span >y</span><span ><span ><span ><span ><span ><span ></span><span ><span >i</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >鈭?/span><span ></span><span ><span >y</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >mean聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span >)</span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span><span ></span></span></span></span></span></span></span></p> <p>鍏朵腑锛?/p> <ul><li><span ><span ><span > x variance聽 : x _{\text {variance }}: </span><span ><span ><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >variance聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >:</span></span></span></span></span> 鏄壒寰佺殑鏂瑰樊</li><li><span ><span ><span > x covariance聽 x_{\text {covariance }} </span><span ><span ><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span >covariance聽</span></span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span></span></span></span></span> 锛氭槸鐗瑰緛 <span ><span ><span > x x </span><span ><span ><span ></span><span >x</span></span></span></span></span> 鍜?<span ><span ><span > y y </span><span ><span ><span ></span><span >y</span></span></span></span></span> 涔嬮棿鐨勫崗鏂瑰樊</li><li><span ><span ><span > x i x_i </span><span ><span ><span ></span><span ><span >x</span><span ><span ><span ><span ><span ><span ></span><span ><span >i</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span></span></span></span></span> 鍜?<span ><span ><span > y i y_i </span><span ><span ><span ></span><span ><span >y</span><span ><span ><span ><span ><span ><span ></span><span ><span >i</span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span></span></span></span></span> 锛氭槸鐗瑰緛 <span ><span ><span > x x </span><span ><span ><span ></span><span >x</span></span></span></span></span> 鍜?<span ><span ><span > y y </span><span ><span ><span ></span><span >y</span></span></span></span></span> 鐨勫崟鐙暟鎹偣</li><li><span ><span ><span > 危 \Sigma </span><span ><span ><span ></span><span >危</span></span></span></span></span> 锛氳〃绀哄€肩殑鎬诲拰</li><li>n 锛氭槸鏌愪釜鐗瑰緛鐨勮瀵熸鏁?/li></ul> <h4>鐗瑰緛鍊煎拰鐗瑰緛鍚戦噺</h4> <p><span ><span ><span ><span > A 鈭? 位 I = 0 A-\lambda I=0 </span><span ><span ><span ></span><span >A</span><span ></span><span >鈭?/span><span ></span></span><span ><span ></span><span >位</span><span >I</span><span ></span><span >=</span><span ></span></span><span ><span ></span><span >0</span></span></span></span></span></span></p> <p><span ><span ><span ><span > A v = 位 v A v=\lambda v </span><span ><span ><span ></span><span >A</span><span >v</span><span ></span><span >=</span><span ></span></span><span ><span ></span><span >位</span><span >v</span></span></span></span></span></span></p> <ul><li>A锛氱壒寰佺殑鍗忔柟宸煩闃?/li><li><span ><span ><span > 位 \lambda </span><span ><span ><span ></span><span >位</span></span></span></span></span> 锛氭槸鐗瑰緛鍊肩煩闃?/li><li>I锛氭槸鍗曚綅鐭╅樀</li><li>v锛氭槸鐗瑰緛鍚戦噺鐨勭煩闃?/li></ul> <h4>涓绘垚鍒嗗垎鏋愮櫧鍖栨柟绋?/h4> <p><span ><span ><span ><span > W P C A = 螞 鈭? 1 2 U T X W_{P C A}=\Lambda^{\frac{-1}{2}} U^T X </span><span ><span ><span ></span><span ><span >W</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >PC</span><span >A</span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >=</span><span ></span></span><span ><span ></span><span ><span >螞</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span ></span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >2</span></span></span></span><span ><span ></span><span ></span></span><span ><span ></span><span ><span ><span >鈭?/span><span >1</span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span><span ></span></span></span></span></span></span></span></span></span></span><span ><span >U</span><span ><span ><span ><span ><span ><span ></span><span ><span >T</span></span></span></span></span></span></span></span><span >X</span></span></span></span></span></span></p> <h4>闆剁浉浣嶅垎閲忓垎鏋愮櫧鍖栨柟绋?/h4> <p><span ><span ><span ><span > W Z C A = U 螞 鈭? 1 2 U T X W_{Z C A}=U \Lambda^{\frac{-1}{2}} U^T X </span><span ><span ><span ></span><span ><span >W</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >ZC</span><span >A</span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span></span><span ></span><span >=</span><span ></span></span><span ><span ></span><span >U</span><span ><span >螞</span><span ><span ><span ><span ><span ><span ></span><span ><span ><span ><span ></span><span ><span ><span ><span ><span ><span ></span><span ><span ><span >2</span></span></span></span><span ><span ></span><span ></span></span><span ><span ></span><span ><span ><span >鈭?/span><span >1</span></span></span></span></span><span >鈥?/span></span><span ><span ><span ></span></span></span></span></span><span ></span></span></span></span></span></span></span></span></span></span><span ><span >U</span><span ><span ><span ><span ><span ><span ></span><span ><span >T</span></span></span></span></span></span></span></span><span >X</span></span></span></span></span></span></p> <h4>Python鐧藉寲</h4> <pre><code >import numpy as np import matplotlib.pyplot as plt from scipy import linalg x = np.array([[1,2,3,4,5], [11,12,13,14,15]]) print('x.shape:', x.shape) xc = x.T - np.mean(x.T, axis=0) xc = xc.T print('xc.shape:', xc.shape, '\n') xcov = np.cov(xc, rowvar=True, bias=True) print('Covariance matrix: \n', xcov, '\n') w, v = linalg.eig(xcov) print( 
    x.shape: (2, 5)
xc.shape: (2, 5) 
 
Covariance matrix: 
 [[2. 2.]
 [2. 2.]] 
 
Eigenvalues:
 [4. 0.] 
 
Eigenvectors:
 [[ 0.70710678 -0.70710678]
 [ 0.70710678  0.70710678]] 
 
Diagonal matrix for inverse square root of Eigenvalues:
 [[5.00000000e-01 0.00000000e+00]
 [0.00000000e+00 4.74531328e+07]]

    馃憠鏇存柊锛氫簹鍥捐法闄?/h3>

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