通过 numpy 和 scikit-learn 计算 PCA 实例

使用 scikit-learn 计算过程

In [64]: data
Out[64]: 
matrix([[2.5, 0.5, 2.2, 1.9, 3.1, 2.3, 2. , 1. , 1.5, 1.1],
        [2.4, 0.7, 2.9, 2.2, 3. , 2.7, 1.6, 1.1, 1.6, 0.9]])

In [65]: from sklearn.decomposition import PCA

In [66]: pca=PCA(n_components=1)

In [69]: newData = pca.fit_transform(data.T)

In [70]: newData
Out[70]: 
array([[-0.82797019],
       [ 1.77758033],
       [-0.99219749],
       [-0.27421042],
       [-1.67580142],
       [-0.9129491 ],
       [ 0.09910944],
       [ 1.14457216],
       [ 0.43804614],
       [ 1.22382056]])

使用 numpy 计算的过程

In [11]: data
Out[11]: 
array([[2.5, 0.5, 2.2, 1.9, 3.1, 2.3, 2. , 1. , 1.5, 1.1],
       [2.4, 0.7, 2.9, 2.2, 3. , 2.7, 1.6, 1.1, 1.6, 0.9]])

In [12]: data = np.mat(data)

In [13]: data
Out[13]: 
matrix([[2.5, 0.5, 2.2, 1.9, 3.1, 2.3, 2. , 1. , 1.5, 1.1],
        [2.4, 0.7, 2.9, 2.2, 3. , 2.7, 1.6, 1.1, 1.6, 0.9]])

In [16]: data = data - data.mean(axis=1)

In [17]: data
Out[17]: 
matrix([[ 0.69, -1.31,  0.39,  0.09,  1.29,  0.49,  0.19, -0.81, -0.31, -0.71],
        [ 0.49, -1.21,  0.99,  0.29,  1.09,  0.79, -0.31, -0.81, -0.31, -1.01]])

#求协方差矩阵
In [31]: data_cov = np.cov(data_mean)

In [32]: data_cov
Out[32]: 
array([[0.61655556, 0.61544444],
       [0.61544444, 0.71655556]])

#求特征向量和特征值
In [34]: tzz,tzxl = np.linalg.eig(data_cov)
Out[33]: 
(array([0.0490834 , 1.28402771]), 
array([[-0.73517866, -0.6778734 ],
        [ 0.6778734 , -0.73517866]]))

In [35]: tzz
Out[35]: array([0.0490834 , 1.28402771])

#选取特征值最大的特征向量，（注意特征向量是 列，不是行）
In [54]: xl = tzxl.T[1]

In [55]: xl
Out[55]: array([-0.6778734 , -0.73517866])

In [63]: data_mean.T.__mul__(np.mat(xl).T)
Out[63]: 
matrix([[-0.82797019],
        [ 1.77758033],
        [-0.99219749],
        [-0.27421042],
        [-1.67580142],
        [-0.9129491 ],
        [ 0.09910944],
        [ 1.14457216],
        [ 0.43804614],
        [ 1.22382056]])