# 机器学习原理与实战 | PCA降维实践

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

# 1. PCA介绍

## 1.1 概念

dots = np.array([[1, 1.5], [2, 1.5], [3, 3.6], [4, 3.2], [5, 5.5]])
def cross_point(x0, y0):
"""
1. line1: y = x
2. line2: y = -x + b => x = b/2
3. [x0, y0] is in line2 => b = x0 + y0
=> x1 = b/2 = (x0 + y0) / 2
=> y1 = x1
"""
x1 = (x0 + y0) / 2
return x1, x1
plt.figure(figsize=(8, 6), dpi=144)
plt.title('2-dimension to 1-dimension')
plt.xlim(0, 8)
plt.ylim(0, 6)
ax = plt.gca()                                  # gca 代表当前坐标轴，即 'get current axis'
ax.spines['right'].set_color('none')            # 隐藏坐标轴
ax.spines['top'].set_color('none')
plt.scatter(dots[:, 0], dots[:, 1], marker='s', c='b')
plt.plot([0.5, 6], [0.5, 6], '-r')
for d in dots:
x1, y1 = cross_point(d[0], d[1])
plt.plot([d[0], x1], [d[1], y1], '--b')
plt.scatter(x1, y1, marker='o', c='r')
plt.annotate(r'projection point',
xy=(x1, y1), xycoords='data',
xytext=(x1 + 0.5, y1 - 0.5), fontsize=10,
plt.annotate(r'vector $u^{(1)}$',
xy=(4.5, 4.5), xycoords='data',
xytext=(5, 4), fontsize=10,
arrowprops=dict(arrowstyle="->", connectionstyle="arc3,rad=.2"))

## 1.2 降维及恢复示意图

dots = np.array([[1, 1.5], [2, 1.5], [3, 3.6], [4, 3.2], [5, 5.5]])
def cross_point(x0, y0):
"""
1. line1: y = x
2. line2: y = -x + b => x = b/2
3. [x0, y0] is in line2 => b = x0 + y0
=> x1 = b/2 = (x0 + y0) / 2
=> y1 = x1
"""
x1 = (x0 + y0) / 2
return x1, x1
plt.figure(figsize=(8, 6), dpi=144)
plt.title('2-dimension to 1-dimension')
plt.xlim(0, 8)
plt.ylim(0, 6)
ax = plt.gca()                                  # gca 代表当前坐标轴，即 'get current axis'
ax.spines['right'].set_color('none')            # 隐藏坐标轴
ax.spines['top'].set_color('none')
plt.scatter(dots[:, 0], dots[:, 1], marker='s', c='b')
plt.plot([0.5, 6], [0.5, 6], '-r')
for d in dots:
x1, y1 = cross_point(d[0], d[1])
plt.plot([d[0], x1], [d[1], y1], '--b')
plt.scatter(x1, y1, marker='o', c='r')
plt.annotate(r'projection point',
xy=(x1, y1), xycoords='data',
xytext=(x1 + 0.5, y1 - 0.5), fontsize=10,
plt.annotate(r'vector $u^{(1)}$',
xy=(4.5, 4.5), xycoords='data',
xytext=(5, 4), fontsize=10,
1
Text(0.03390904029252009, -0.28050757997562326, 'projected data')


# 2. PCA 算法模拟

## 2.1 Numpy实现

A = np.array([[3, 2000],
[2, 3000],
[4, 5000],
[5, 8000],
[1, 2000]], dtype='float')
# 数据归一化
mean = np.mean(A, axis=0)
norm = A - mean
# 数据缩放
scope = np.max(norm, axis=0) - np.min(norm, axis=0)
norm = norm / scope
norm
array([[ 0.        , -0.33333333],
[-0.25      , -0.16666667],
[ 0.25      ,  0.16666667],
[ 0.5       ,  0.66666667],
[-0.5       , -0.33333333]])
U, S, V = np.linalg.svd(np.dot(norm.T, norm))
U
array([[-0.67710949, -0.73588229],
[-0.73588229,  0.67710949]])
U_reduce = U[:, 0].reshape(2,1)
U_reduce
array([[-0.67710949],
[-0.73588229]])
R = np.dot(norm, U_reduce)
R
array([[ 0.2452941 ],
[ 0.29192442],
[-0.29192442],
[-0.82914294],
[ 0.58384884]])
Z = np.dot(R, U_reduce.T)
Z
array([[-0.16609096, -0.18050758],
[-0.19766479, -0.21482201],
[ 0.19766479,  0.21482201],
[ 0.56142055,  0.6101516 ],
[-0.39532959, -0.42964402]])
np.multiply(Z, scope) + mean
array([[2.33563616e+00, 2.91695452e+03],
[2.20934082e+00, 2.71106794e+03],
[3.79065918e+00, 5.28893206e+03],
[5.24568220e+00, 7.66090960e+03],
[1.41868164e+00, 1.42213588e+03]])

## 2.2 sklearn 包实现

from sklearn.decomposition import PCA
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import MinMaxScaler
def std_PCA(**argv):
# MinMaxScaler对数据进行预处理
scaler = MinMaxScaler()
# PCA算法
pca = PCA(**argv)
pipeline = Pipeline([('scaler', scaler),
('pca', pca)])
return pipeline
pca = std_PCA(n_components=1)
R2 = pca.fit_transform(A)
R2
array([[-0.2452941 ],
[-0.29192442],
[ 0.29192442],
[ 0.82914294],
[-0.58384884]])
pca.inverse_transform(R2)
array([[2.33563616e+00, 2.91695452e+03],
[2.20934082e+00, 2.71106794e+03],
[3.79065918e+00, 5.28893206e+03],
[5.24568220e+00, 7.66090960e+03],
[1.41868164e+00, 1.42213588e+03]])

# 3. 实例：pca进行人脸降维

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import fetch_olivetti_faces
# fetch_olivetti_faces函数可以帮助我们截取中间部分，只留下脸部特征
faces = fetch_olivetti_faces(data_home='datasets/')
X = faces.data
y = faces.target
image = faces.images
print("data:{}, label:{}, image:{}".format(X.shape, y.shape, image.shape))
data:(400, 4096), label:(400,), image:(400, 64, 64)

target_names = np.array(["c%d" % i for i in np.unique(y)])
target_names
array(['c0', 'c1', 'c2', 'c3', 'c4', 'c5', 'c6', 'c7', 'c8', 'c9', 'c10',
'c11', 'c12', 'c13', 'c14', 'c15', 'c16', 'c17', 'c18', 'c19',
'c20', 'c21', 'c22', 'c23', 'c24', 'c25', 'c26', 'c27', 'c28',
'c29', 'c30', 'c31', 'c32', 'c33', 'c34', 'c35', 'c36', 'c37',
'c38', 'c39'], dtype='<U3')
plt.figure(figsize=(12, 11), dpi=100)
# 这里显示两个人的各5张图像
shownum = 40
# 提取前k个人的名字
title = target_names[:int(shownum/10)]
j = 1
# 每个人的10张图像主题曲前面的5张来展示
for i in range(shownum):
if i%10 < 5:
plt.subplot(int(shownum/10),5,j)
plt.title("people:"+title[int(i/10)])
plt.imshow(image[i],cmap=plt.cm.gray)
j+=1


subimage = None
for i in range(len(image)):
if i%10 == 0:
if subimage is not None:
#             print("subimage.shape:{},image[i].shape:{}",subimage.shape, image[i].shape)
subimage = np.concatenate((subimage, image[i].reshape(1,64,64)), axis=0)
else:
subimage = image[i].reshape(1,64,64)
plt.figure(figsize=(12,6), dpi=100)
for i in range(subimage.shape[0]):
plt.subplot(int(subimage.shape[0]/10), 10, i+1)
plt.imshow(subimage[i], cmap=plt.cm.gray)
plt.title("name:"+target_names[i])
plt.axis('off')

from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.2, random_state=4)
X_train.shape, X_test.shape, y_train.shape, y_test.shape
((320, 4096), (80, 4096), (320,), (80,))

from sklearn.svm import SVC
# 指定SVC的class_weight参数，让SVC模型能根据训练样本的数量来均衡地调整权重
clf = SVC(class_weight='balanced')
# 训练
clf.fit(X_train, y_train)
# 计算得分
trainscore = clf.score(X_train,y_train)
testscore = clf.score(X_test,y_test)
print("trainscore:{},testscore:{}".format(trainscore, testscore))
# 预测
y_pred = clf.predict(X_test)

from sklearn.svm import SVC
# 指定SVC的class_weight参数，让SVC模型能根据训练样本的数量来均衡地调整权重
clf = SVC(class_weight='balanced')
# 训练
clf.fit(X_train, y_train)
# 计算得分
trainscore = clf.score(X_train,y_train)
testscore = clf.score(X_test,y_test)
print("trainscore:{},testscore:{}".format(trainscore, testscore))
# 预测
y_pred = clf.predict(X_test)

# plt.figure(figsize=(12,6), dpi=100)
plt.subplot(1,1,1)
plt.imshow(X_test[1].reshape(64,64), cmap=plt.cm.gray)
<matplotlib.image.AxesImage at 0x21fb6d83688>

y_test[1] == y_pred[1]
True

from sklearn.decomposition import PCA
pca = PCA(n_components=140)
X_pca = pca.fit_transform(X)
np.sum(pca.explained_variance_ratio_)
0.9585573

from sklearn.decomposition import PCA
# 原图展示
plt.figure(figsize=(12,8), dpi=100)
subimage = faces.images[:5]
for i in range(5):
plt.subplot(1, 5, i+1)
plt.imshow(subimage[i], cmap=plt.cm.gray)
plt.axis('off')
# 降维后的图片展示
k = [140, 75, 37, 19, 8]
plt.figure(figsize=(12,12), dpi=100)
for index in range(len(k)):
pca = PCA(n_components=k[index])
# 进行降维处理
X_pca = pca.fit_transform(X)
# 重新升维，中间过程有损耗
X_invert_pca = pca.inverse_transform(X_pca)
image = X_invert_pca.reshape(-1,64,64)
subimage = image[:5]
for i in range(len(k)):
plt.subplot(len(k), 5, (i+1)+len(k)*index)
plt.imshow(subimage[i], cmap=plt.cm.gray)
#     plt.title("name:"+target_names[i])
plt.axis('off')

https://zhuanlan.zhihu.com/p/271969151 关于 fit(), transform(), fit_transform()区别，这篇博客有介绍

from sklearn.svm import SVC
# 设定多降到的维度
pca = PCA(n_components=140)
# 先使用训练集对进行训练与归一化处理
X_train_pca = pca.fit_transform(X_train)
# 然后对测试采用训练集同样的参数进行归一化处理
X_test_pca = pca.transform(X_test)
# 指定SVC的class_weight参数，让SVC模型能根据训练样本的数量来均衡地调整权重
clf = SVC(class_weight='balanced')
# 用归一化后的数据给svm进行训练
clf.fit(X_train_pca, y_train)
# 计算得分
trainscore = clf.score(X_train_pca,y_train)
testscore = clf.score(X_test_pca,y_test)
print("trainscore:{},testscore:{}".format(trainscore, testscore))
trainscore:1.0,testscore:0.975

from sklearn.model_selection import GridSearchCV
# print("Searching the best parameters for SVC ...")
param_grid = {'C': [1, 5, 10, 50, 100],
'gamma': [0.0001, 0.0005, 0.001, 0.005, 0.01]}
clf = GridSearchCV(SVC(kernel='rbf', class_weight='balanced'), param_grid, verbose=2, n_jobs=4)
clf = clf.fit(X_train_pca, y_train)
print("Best parameters found by grid search:",clf.best_params_)
# 计算得分
trainscore = clf.score(X_train_pca,y_train)
testscore = clf.score(X_test_pca,y_test)
print("trainscore:{},testscore:{}".format(trainscore, testscore))
Fitting 5 folds for each of 25 candidates, totalling 125 fits
Best parameters found by grid search: {'C': 5, 'gamma': 0.005}
trainscore:1.0,testscore:0.9625

import pandas as pd
result = pd.DataFrame()
result['pred'] = y_pred
result['true'] = y_test
result['compares'] = y_pred==y_test
result.head(10)

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