定义：X是样本数据，每一行是一个数据，它有m个数据，每个数据有n个特征
$$ X = \begin{bmatrix} X_1^{(1)} && X_1^{(2)} && \cdots && X_1^{(n)} \ X_2^{(1)} && X_2^{(2)} && \cdots && X_2^{(n)} \ \cdots && \cdots && \cdots && \cdots \ X_m^{(1)} && X_m^{(2)} && \cdots && X_m^{(n)} \end{bmatrix} $$

$W_k$是求得的前k个主成分矩阵，每一行是一个主成分的单位方向，它有k个主成分方向，每个主成分的方向有n个维度
$$ X = \begin{bmatrix} W_k^{(1)} && W_1^{(2)} && \cdots && W_1^{(n)} \ W_2^{(1)} && W_2^{(2)} && \cdots && W_2^{(n)} \ \cdots && \cdots && \cdots && \cdots \ W_k^{(1)} && W_k^{(2)} && \cdots && W_k^{(n)} \end{bmatrix} $$

问：如何将样本X从N维转换成K维？
答：降维：把所有样本映射到K个主成分上
$$ X \cdot W_k^T = X_k $$

还原：把降维后的数据还原到原坐标空间
$$ X_k \cdot W_k = X_m $$

还原后的X与原X不同。

把PCA封装成类

import numpy as np

class PCA:
    def __init__(self, n_components):
        """初始化PCA"""
        assert n_components >= 1, "n_components must be valid"
        self.n_components = n_components
        self.components_ = None

    def fit(self, X, eta=0.01, n_iters=1e4):
        """获取数据集的前n个主成分"""
        assert self.n_components <= X.shape[1], "n_components must not be greater than the feature number of X"

        def demean(X):
            return X - np.mean(X, axis=0)

        def f(w, X):
            return np.sum((X.dot(w)**2)) / len(X)

        def df(w, X):
            return X.T.dot(X.dot(w)) * 2. / len(X)

        # 把向量单位化
        def direction(w):
            return w / np.linalg.norm(w)

        def first_component(X, initial_w, eta, n_iters=1e4, epsilon=1e-8):
            w = direction(initial_w)
            cur_iter = 0
            while cur_iter < n_iters:
                gradient = df(w, X)
                last_w = w
                w = w + eta * gradient
                w = direction(w)
                if(abs(f(w, X)) - abs(f(last_w, X)) < epsilon):
                   break
                cur_iter += 1
            return w

        X_pca = demean(X)
        self.components_ = np.empty(shape = (self.n_components, X.shape[1]))
        for i in range(self.n_components):
            initial_w = np.random.random(X.shape[1])
            eta = 0.001
            w = first_component(X_pca, initial_w, eta)
            self.components_[i, :] = w
            X_pca = X_pca - X_pca.dot(w).reshape(-1, 1) * w
        return self

    def transform(self, X):
        """将给定的X，映射到各个主成分分量中"""
        assert X.shape[1] == self.components_.shape[1]
        return X.dot(self.components_.T)

    def inverse_transform(self, X):
        """将给定的X反向映射回原来的特征空间"""
        assert X.shape[1] == self.components_.shape[0]
        return X.dot(self.components_)

    def __repr__(self):
        return "PCA(n_components=%d)" % self.n_components

使用PCA降维

准备数据

import numpy as np
import matplotlib.pyplot as plt

X = np.empty((100,2))
X[:,0] = np.random.uniform(0., 100., size=100)
X[:,1] = 0.75 * X[:, 0] + 3. + np.random.normal(0, 10., size=100)

训练模型1

pca = PCA(n_components=2)
pca.fit(X)

输入：pca.components_
输出：array([[ 0.75366776, 0.65725559], [-0.65723751, 0.75368352]])

训练模型2：降维

pca = PCA(n_components=1)
pca.fit(X)

X_reduction = pca.transform(X)
X_restore = pca.inverse_transform(X_reduction)

输入：X_reduction.shape
输出：(100, 1)

输入：X_restore.shape
输出：(100, 2)

对比原始数据与降维再恢复后的数据

plt.scatter(X[:, 0], X[:, 1], color='b', alpha=0.5)
plt.scatter(X_restore[:, 0], X_restore[:, 1], color='r', alpha=0.5)
plt.show()

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