机器学习原理与实战 | PCA降维实践

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

1. PCA介绍

1.1 概念

思想:

机器学习原理与实战 | PCA降维实践

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 代表当时坐标轴,即 'get current axis'
ax.spines['right'].set_color('none')            # 躲藏坐标轴
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,
             arrowprops=dict(arrowstyle="->", connectionstyle="arc3,rad=.2"))
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"))

机器学习原理与实战 | PCA降维实践
图中正变量方形的点是原始数据经过预处理后(归一化、缩放)的数据,圆形的点是从一维康复到二维人脸识别概念股后的数据算法。同时,咱们画出主成分特算法导论征向量u1,u2 。依据上图,来介绍几个有意思的结论:首要,圆形的点实际上就是方形的点在向量u1,u2 所在直线算法是指什么上的投影。所谓PCA数据康复,并不是真实的康复,仅仅把降维后的坐标转换为原坐标系中的坐标而已。针对咱们的比如,仅仅把由向量u1,u2 决议的一维测试仪坐标系中的坐标转换为原始二维坐标系中的坐标。其次,主成分特征向量u1,u2是相互垂直的。再次,方形点和圆形点算法分析的目的是之间的距离,就是P人脸识别身份认证系统CA数据降维后的差变量名的命名规则错。

1.2 降维及康复示意图

plt.figure(figsize=(8, 8), dpi=144)
plt.title('Physcial meanings of PCA')
ymin = xmin = -1
ymax = xmax = 1
plt.xlim(xmin, xmax)
plt.ylim(ymin, ymax)
ax = plt.gca()                                  # gca 代表当时坐标轴,即 'get current axis'
ax.spines['right'].set_color('none')            # 躲藏坐标轴
ax.spines['top'].set_color('none')
plt.scatter(norm[:, 0], norm[:, 1], marker='s', c='b')
plt.scatter(Z[:, 0], Z[:, 1], marker='o', c='r')
plt.arrow(0, 0, U[0][0], U[1][0], color='r', linestyle='-')
plt.arrow(0, 0, U[0][1], U[1][1], color='r', linestyle='--')
plt.annotate(r'$U_{reduce} = u^{(1)}$',
             xy=(U[0][0], U[1][0]), xycoords='data',
             xytext=(U_reduce[0][0] + 0.2, U_reduce[1][0] - 0.1), fontsize=10,
             arrowprops=dict(arrowstyle="->", connectionstyle="arc3,rad=.2"))
plt.annotate(r'$u^{(2)}$',
             xy=(U[0][1], U[1][1]), xycoords='data',
             xytext=(U[0][1] + 0.2, U[1][1] - 0.1), fontsize=10,
             arrowprops=dict(arrowstyle="->", connectionstyle="arc3,rad=.2"))
plt.annotate(r'raw data',
             xy=(norm[0][0], norm[0][1]), xycoords='data',
             xytext=(norm[0][0] + 0.2, norm[0][1] - 0.2), fontsize=10,
             arrowprops=dict(arrowstyle="->", connectionstyle="arc3,rad=.2"))
plt.annotate(r'projected data',
             xy=(Z[0][0], Z[0][1]), xycoords='data',
             xytext=(Z[0][0] + 0.2, Z[0][1] - 0.1), fontsize=10,
             arrowprops=dict(arrowstyle="->", connectionstyle="arc3,rad=.2"))
Text(0.03390904029252009, -0.28050757997562326, 'projected data')

机器学习原理与实战 | PCA降维实践

2. PCA 算法模仿

2.1 Numpy完成

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 包完成

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. 实测试仪例: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函数能够帮助咱们截取中心部分,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)

查看部分图人脸识别失败怎么解决

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)
# 这儿显现两个人的各5张图画
shownum = 40
# 提取前k个人的名字
title = target_names[:int(shownum/10)]
j = 1
# 每个人的10张图画主题曲前面的5张来展现
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

机器学习原理与实战 | PCA降维实践

提取悉数40人的第一张图画,变量的定义并进行展现

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')

机器学习原理与实战 | PCA降维实践
区分数据集

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,))

运用svm来完成人脸辨认

from sklearn.svm import SVC
# 指定SVC的class_weight参数,让SVC模型能依据练习样本的数量来均衡地调整权重
clf = SVC(class_weight='balanced')
# 练习
clf.fit(X_train, y_train)
# 核算得分
trainscore = clf.score(X_train,y_train)
testscore = clf.score(X_test,y_test)
print("trainscore:{},testscore:{}".format(trainscore, testscore))
# 猜测
y_pred = clf.predict(X_test)
trainscore:1.0,testscore:0.975

显现图画测验集图画

# 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>

机器学习原理与实战 | PCA降维实践

猜测是正确的,能够发现svm的猜测作用非常好

y_test[1] == y_pred[1]
True

其中人脸识别软件PCA模型的explained_variance_rat数据处理包括哪些内容io变量能够获取经PCA处理后的数据复原率

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

现在运用的测试抑郁程度的问卷是4096个特征,现在运用PCA对特征进行降维,再查看图画的变化;

from sklearn.decomposition import PCA
# 原图展现
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')
# 降维后的图片展现
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,中心过程有损耗
    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')

机器学习原理与实战 | PCA降维实践
机器学习原理与实战 | PCA降维实践

能够看见降维后的人脸逐渐模糊,从4096特征维度讲到140维度还是能够坚持脸部的大部分特征

zhuanlan.zhihu.com/p/271969151 关于 fit人脸识别解除方法教程(), transform()算法设计与分析, fit_transform()人脸识别一直失败原因差异,这篇博客有介绍

必须先用测试fit_transform(trainData),之后数据处理英文再transform(testData)。假如直接transfor人脸识别软件m(testData),程序会报错

假如fit_transfrom(trainData)后,运用fit_transform(tes算法的有穷性是指tData)而不transform(testData),测试抑郁症的20道题尽管也能归一化,可是两个结果不是在同一个“规范”下的,具有明显差变量的定义异。也就是咱们需要用处理练习集的归一化过程来处理测验集,保证有相同的数据处理

from sklearn.svm import SVC
# 设定多降到的维度
pca = PCA(n_components=140)
# 先运用练习集对进行练习与归一化处理
X_train_pca = pca.fit_transform(X_train)
# 然后对测验采用练习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模型能依据练习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进行练习
clf.fit(X_train_pca, y_train)
# 核算得分
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

运用GridSearchCV来进一步挑选

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_)
# 核算得分
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

能够看见作用还是非常不错的

import pandas as pd
result = pd.DataFrame()
result['pred'] = y_pred
result['true'] = y_test
result['compares'] = y_pred==y_test
result.head(10)
.dataframe tbody tr th:only-of-type {算法的五个特性 vertical-align: middle; } .datafr人脸识别用照片可以识别吗ame tbody tr th { vertical-align: top; } .dataf变量类型有哪些rame thead th { text-align: right; }
predtruecompares
01818True
100True
266True
33131True
41010True
52727True
63636True
73232True
82929Tr变量泵ue
93333True

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