主成分分析，即Principal Component Analysis（PCA），是多元統計中的重要內容，也廣泛應用於機器學習和其它領域。它的主要作用是對高維資料進行降維。PCA把原先的n個特徵用數目更少的k個特徵取代，新特徵是舊特徵的線性組合，這些線性組合最大化樣本方差，儘量使新的k個特徵互不相關。關於PCA的更多介紹，請參考：https://en.wikipedia.org/wiki/Principal_component_analysis.

　　PCA的主要演算法如下：

• 組織資料形式，以便於模型使用；
• 計算樣本每個特徵的平均值；
• 每個樣本資料減去該特徵的平均值（歸一化處理）；
• 求協方差矩陣；
• 找到協方差矩陣的特徵值和特徵向量；
• 對特徵值和特徵向量重新排列（特徵值從大到小排列）；
• 對特徵值求取累計貢獻率；
• 對累計貢獻率按照某個特定比例選取特徵向量集的子集合；
• 對原始資料（第三步後）進行轉換。

　　其中協方差矩陣的分解可以通過按對稱矩陣的特徵向量來，也可以通過分解矩陣的SVD來實現，而在Scikit-learn中，也是採用SVD來實現PCA演算法的。關於SVD的介紹及其原理，可以參考：矩陣的奇異值分解（SVD）（理論）。

　　本文將用三種方法來實現PCA演算法，一種是原始演算法，即上面所描述的演算法過程，具體的計算方法和過程，可以參考：A tutorial on Principal Components Analysis, Lindsay I Smith.

　　用以上三種方法來實現PCA的完整的Python如下：

 1 import numpy as np
 2 from sklearn.decomposition import PCA
 3 import sys
 4 #returns choosing how many main factors
 5 def index_lst(lst, component=0, rate=0):
 
 6     #component: numbers of main factors
 7     #rate: rate of sum(main factors)/sum(all factors)
 8     #rate range suggest: (0.8,1)
 9     #if you choose rate parameter, return index = 0 or less than len(lst)
10     if component and rate:
11         print('Component and rate must choose only one!')
12         sys.exit(0)
13     if not component and not rate:
14         print('Invalid parameter for numbers of components!')
15         sys.exit(0)
16     elif component:
17         print('Choosing by component, components are %s......'%component)
18         return component
19     else:
20         print('Choosing by rate, rate is %s ......'%rate)
21         for i in range(1, len(lst)):
22             if sum(lst[:i])/sum(lst) >= rate:
23                 return i
24         return 0
25 
26 def main():
27     # test data
28     mat = [[-1,-1,0,2,1],[2,0,0,-1,-1],[2,0,1,1,0]]
29     
30     # simple transform of test data
31     Mat = np.array(mat, dtype='float64')
32     print('Before PCA transforMation, data is:\n', Mat)
33     print('\nMethod 1: PCA by original algorithm:')
34     p,n = np.shape(Mat) # shape of Mat 
35     t = np.mean(Mat, 0) # mean of each column
36     
37     # substract the mean of each column
38     for i in range(p):
39         for j in range(n):
40             Mat[i,j] = float(Mat[i,j]-t[j])
41             
42     # covariance Matrix
43     cov_Mat = np.dot(Mat.T, Mat)/(p-1)
44     
45     # PCA by original algorithm
46     # eigvalues and eigenvectors of covariance Matrix with eigvalues descending
47     U,V = np.linalg.eigh(cov_Mat) 
48     # Rearrange the eigenvectors and eigenvalues
49     U = U[::-1]
50     for i in range(n):
51         V[i,:] = V[i,:][::-1]
52     # choose eigenvalue by component or rate, not both of them euqal to 0
53     Index = index_lst(U, component=2)  # choose how many main factors
54     if Index:
55         v = V[:,:Index]  # subset of Unitary matrix
56     else:  # improper rate choice may return Index=0
57         print('Invalid rate choice.\nPlease adjust the rate.')
58         print('Rate distribute follows:')
59         print([sum(U[:i])/sum(U) for i in range(1, len(U)+1)])
60         sys.exit(0)
61     # data transformation
62     T1 = np.dot(Mat, v)
63     # print the transformed data
64     print('We choose %d main factors.'%Index)
65     print('After PCA transformation, data becomes:\n',T1)
66     
67     # PCA by original algorithm using SVD
68     print('\nMethod 2: PCA by original algorithm using SVD:')
69     # u: Unitary matrix,  eigenvectors in columns 
70     # d: list of the singular values, sorted in descending order
71     u,d,v = np.linalg.svd(cov_Mat)
72     Index = index_lst(d, rate=0.95)  # choose how many main factors
73     T2 = np.dot(Mat, u[:,:Index])  # transformed data
74     print('We choose %d main factors.'%Index)
75     print('After PCA transformation, data becomes:\n',T2)
76     
77     # PCA by Scikit-learn
78     pca = PCA(n_components=2) # n_components can be integer or float in (0,1)
79     pca.fit(mat)  # fit the model
80     print('\nMethod 3: PCA by Scikit-learn:')
81     print('After PCA transformation, data becomes:')
82     print(pca.fit_transform(mat))  # transformed data
83             
84 main()

執行以上程式碼，輸出結果為：

　　這說明用以上三種方法來實現PCA都是可行的。這樣我們就能理解PCA的具體實現過程啦~~有興趣的讀者可以用其它語言實現一下哈~~

參考文獻：

1. PCA 維基百科： https://en.wikipedia.org/wiki/Principal_component_analysis.
2. 講解詳細又全面的PCA教程： A tutorial on Principal Components Analysis, Lindsay I Smith.
3. 部落格：矩陣的奇異值分解（SVD）（理論）：http://www.cnblogs.com/jclian91/p/8022426.html.
4. 部落格：主成分分析PCA: https://www.cnblogs.com/zhangchaoyang/articles/2222048.html.
5. Scikit-learn的PCA介紹：http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html.