PCA主成分分析降维实际上是寻求能使样本方差最大的一个低维空间，来最大程度区分样本。例如下面这个二维图表，就可以找一条直线(一维)使得样本点映射在上面方差最大(区分度最大)！

PCA可以用来解决的问题【Andrew Ng曾在讲PCA时提到过】：

1）减少数据因为存储而造成的内存和硬盘的占用

2）加速训练过程

3）高维数据可视化

PCA具体代码

import matplotlib.pyplot as pltfrom sklearn.decomposition import PCAimport numpy as np#随便弄一个四维数组来试验一下DF = np.array([[0,1,4,3],[1,2,8,9],[2,4,16,81],[2,5,20,243],[4,6,24,729]])#样本数不能少于维数#如果你不知道降到几维合适，那就是用mle帮你选取降到几维pca_mle = PCA(n_components='mle').fit(DF) #最大似然法选取特征数量print(pca_mle.explained_variance_ratio_)cumsum = np.cumsum(pca_mle.explained_variance_ratio_.sum()) # 使用累计统计，来表示可解释特征的信息占比和print(cumsum)data_dr = pca_mle.transform(DF)print(data_dr)#每一个维度的数据都进行了零均值化print(data_dr.shape)#作cumsum图评价pca_line = PCA().fit(DF) # 什么参数都不写，代表是 min(x.shape)，一般情况下就是原特征数目了。print(pca_line.explained_variance_ratio_)cumsum = np.cumsum(pca_line.explained_variance_ratio_) # 使用累计统计，来表示可解释特征的信息占比和print(cumsum) #从一维到四维的cumsumplt.plot(range(1, DF.shape[1] + 1), cumsum) #设置x\yplt.xticks(range(1, DF.shape[1] + 1)) # 横坐标轴是整数plt.xlabel('number of components after DR')plt.ylabel('cumulative explained variance')plt.show()

我建议使用Anaconda的Jupyter Notebook来运行，如下：

补充实例：

from sklearn.decomposition import PCAfrom sklearn.datasets import load_digits #8x8的手写数字数据from sklearn.neighbors import KNeighborsClassifierfrom sklearn.model_selection import train_test_splitimport timeX, y = load_digits(return_X_y=True)print('X shape:', X.shape) # 此处会看到X是64维的数据X_train, x_test, y_train, y_test = train_test_split(X, y)tic = time.time()knn_clf = KNeighborsClassifier()knn_clf.fit(X_train, y_train)print (knn_clf.score(x_test, y_test))toc = time.time()print ('Time of method[exec_without_pca] costs:'+str(toc-tic)) print ('----' * 10)tic = time.time()knn_clf = KNeighborsClassifier()pca = PCA(n_components=0.95) # 重构阈值为95%pca.fit(X_train, y_train)X_train_dunction = pca.transform(X_train)X_test_dunction = pca.transform(x_test)knn_clf.fit(X_train_dunction, y_train)print (knn_clf.score(X_test_dunction, y_test))toc = time.time()print ('Time of method[exec_with_pca] costs:'+str(toc-tic)) import matplotlib.pyplot as plt%matplotlib inlinedef draw_graph():pca = PCA(n_components=2)pca.fit(X)X_reduction = pca.transform(X)for i in range(10):plt.scatter(X_reduction[y==i,0], X_reduction[y==i,1], alpha=0.8, label='%s' % i)plt.legend()plt.show()