介绍
利用主成分分析原理，将原来的变量重新组合成一组互相无关的几个综合变量，而这些变量尽可能的保留原有的信息，
从而达到降维的目的，低维的数据让人更易进行可视化分析，以便观察数据的结构分布。
# satisfaction_level ：对公司的满意程度
# last_evaluation ：对公司的评价
# number_project ：做过项目的数量
# average_montly_hours ：每月工作时长
# time_spend_company ：每天在公司的时间
# Work_accident ：工作差错
# promotion_last_5years ：五年内有没有提升
1.导入数据
import numpy as np
import pandas as pd
HR_comma_sep=pd.read_csv('./HR_comma_sep.csv')
HR_comma_sep.head()
| satisfaction_level | last_evaluation | number_project | average_montly_hours | time_spend_company | Work_accident | left | promotion_last_5years | sales | salary | |
| 0 | 0.38 | 0.53 | 2 | 157 | 3 | 0 | 1 | 0 | sales | low |
| 1 | 0.80 | 0.86 | 5 | 262 | 6 | 0 | 1 | 0 | sales | medium |
| 2 | 0.11 | 0.88 | 7 | 272 | 4 | 0 | 1 | 0 | sales | medium |
| 3 | 0.72 | 0.87 | 5 | 223 | 5 | 0 | 1 | 0 | sales | low |
| 4 | 0.37 | 0.52 | 2 | 159 | 3 | 0 | 1 | 0 | sales | low |
2.数据探索
HR_comma_sep.shape
(14999, 10)
HR_comma_sep.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 14999 entries, 0 to 14998
Data columns (total 10 columns):
# Column Non-Null Count Dtype
--- ------ -------------- -----
0 satisfaction_level 14999 non-null float64
1 last_evaluation 14999 non-null float64
2 number_project 14999 non-null int64
3 average_montly_hours 14999 non-null int64
4 time_spend_company 14999 non-null int64
5 Work_accident 14999 non-null int64
6 left 14999 non-null int64
7 promotion_last_5years 14999 non-null int64
8 sales 14999 non-null object
9 salary 14999 non-null object
dtypes: float64(2), int64(6), object(2)
memory usage: 1.1+ MB
HR_comma_sep.describe().T
| count | mean | std | min | 25% | 50% | 75% | max | |
| satisfaction_level | 14999.0 | 0.612834 | 0.248631 | 0.09 | 0.44 | 0.64 | 0.82 | 1.0 |
| last_evaluation | 14999.0 | 0.716102 | 0.171169 | 0.36 | 0.56 | 0.72 | 0.87 | 1.0 |
| number_project | 14999.0 | 3.803054 | 1.232592 | 2.00 | 3.00 | 4.00 | 5.00 | 7.0 |
| average_montly_hours | 14999.0 | 201.050337 | 49.943099 | 96.00 | 156.00 | 200.00 | 245.00 | 310.0 |
| time_spend_company | 14999.0 | 3.498233 | 1.460136 | 2.00 | 3.00 | 3.00 | 4.00 | 10.0 |
| Work_accident | 14999.0 | 0.144610 | 0.351719 | 0.00 | 0.00 | 0.00 | 0.00 | 1.0 |
| left | 14999.0 | 0.238083 | 0.425924 | 0.00 | 0.00 | 0.00 | 0.00 | 1.0 |
| promotion_last_5years | 14999.0 | 0.021268 | 0.144281 | 0.00 | 0.00 | 0.00 | 0.00 | 1.0 |
import numpy as np
import pandas as pd
import matplotlib as mpl
import matplotlib.pyplot as plt
import seaborn as sns
get_ipython().run_line_magic('matplotlib', 'inline')
correlation = HR_comma_sep.corr()
plt.figure(figsize=(10,10))
sns.heatmap(correlation, vmax=1,
square=True,annot=True,cmap='cubehelix')
plt.title('Correlation between different fearures')
Text(0.5, 1, 'Correlation between different fearures')
output_9_1.png
3.模型开发
HR_comma_sep_X=HR_comma_sep.drop(labels=['sales','salary','left'],axis=1)
HR_comma_sep_X.head()
| satisfaction_level | last_evaluation | number_project | average_montly_hours | time_spend_company | Work_accident | promotion_last_5years | |
| 0 | 0.38 | 0.53 | 2 | 157 | 3 | 0 | 0 |
| 1 | 0.80 | 0.86 | 5 | 262 | 6 | 0 | 0 |
| 2 | 0.11 | 0.88 | 7 | 272 | 4 | 0 | 0 |
| 3 | 0.72 | 0.87 | 5 | 223 | 5 | 0 | 0 |
| 4 | 0.37 | 0.52 | 2 | 159 | 3 | 0 | 0 |
from sklearn.preprocessing import StandardScaler
HR_comma_sep_X_std = StandardScaler().fit_transform(HR_comma_sep_X)
mean_vec = np.mean(HR_comma_sep_X_std, axis=0)
cov_mat = (HR_comma_sep_X_std - mean_vec).T.dot((HR_comma_sep_X_std - mean_vec)) / (HR_comma_sep_X_std.shape[0]-1)
print('Covariance matrix \n%s' %cov_mat)
Covariance matrix
[[ 1.00006668 0.10502822 -0.14297912 -0.02004945 -0.1008728 0.05870115
0.02560689]
[ 0.10502822 1.00006668 0.34935588 0.33976445 0.1315995 -0.00710476
-0.00868435]
[-0.14297912 0.34935588 1.00006668 0.41723845 0.19679901 -0.00474086
-0.00606436]
[-0.02004945 0.33976445 0.41723845 1.00006668 0.12776343 -0.01014356
-0.00354465]
[-0.1008728 0.1315995 0.19679901 0.12776343 1.00006668 0.00212056
0.06743742]
[ 0.05870115 -0.00710476 -0.00474086 -0.01014356 0.00212056 1.00006668
0.03924805]
[ 0.02560689 -0.00868435 -0.00606436 -0.00354465 0.06743742 0.03924805
1.00006668]]
eig_vals, eig_vecs = np.linalg.eig(cov_mat)
print('Eigenvectors \n%s' %eig_vecs)
print('\nEigenvalues \n%s' %eig_vals)
Eigenvectors
[[-0.08797699 -0.29189921 0.27784886 0.33637135 0.79752505 0.26786864
-0.09438973]
[ 0.50695734 0.30996609 -0.70780994 0.07393548 0.33180877 0.1101505
-0.13499526]
[ 0.5788351 -0.77736008 -0.00657105 -0.19677589 -0.10338032 -0.10336241
-0.02293518]
[ 0.54901653 0.45787675 0.63497294 -0.25170987 0.10388959 -0.01034922
-0.10714981]
[ 0.31354922 0.05287224 0.12200054 0.78782241 -0.28404472 0.04036861
0.42547869]
[-0.01930249 0.04433104 -0.03622859 -0.05762997 0.37489883 -0.8048393
0.45245222]
[ 0.00996933 0.00391698 -0.04873036 -0.39411153 0.10557298 0.50589173
0.75836313]]
Eigenvalues
[1.83017431 0.54823098 0.63363587 0.84548166 1.12659606 0.95598647
1.06036136]
# 构造元组
eig_pairs = [(np.abs(eig_vals[i]), eig_vecs[:,i]) for i in range(len(eig_vals))]
# 对特征值进行排序
eig_pairs.sort(key=lambda x: x[0], reverse=True)
print('Eigenvalues in descending order:')
for i in eig_pairs:
print(i[0])
Eigenvalues in descending order:
1.830174313875499
1.1265960639915473
1.0603613622840846
0.9559864740066265
0.8454816637143464
0.633635874483021
0.5482309765420602
tot = sum(eig_vals)
var_exp = [(i / tot)*100 for i in sorted(eig_vals, reverse=True)]
plt.figure(figsize=(6, 4))
plt.bar(range(7), var_exp, alpha=0.5, align='center', label='individual explained variance')
plt.ylabel('Explained variance ratio')
plt.xlabel('Principal components')
plt.legend(loc='best')
plt.tight_layout()
output_16_0.png
matrix_w = np.hstack((eig_pairs[0][1].reshape(7,1),
eig_pairs[1][1].reshape(7,1)
))
print('Matrix W:\n', matrix_w)
Matrix W:
[[-0.08797699 0.79752505]
[ 0.50695734 0.33180877]
[ 0.5788351 -0.10338032]
[ 0.54901653 0.10388959]
[ 0.31354922 -0.28404472]
[-0.01930249 0.37489883]
[ 0.00996933 0.10557298]]
Y = HR_comma_sep_X_std.dot(matrix_w)
Y
array([[-1.90035018, -1.12083103],
[ 2.1358322 , 0.2493369 ],
[ 3.05891625, -1.68312693],
...,
[-2.0507165 , -1.182032 ],
[ 2.91418496, -1.42752606],
[-1.91543672, -1.17021407]])
from sklearn.decomposition import PCA
pca = PCA().fit(HR_comma_sep_X_std)
plt.plot(np.cumsum(pca.explained_variance_ratio_))
plt.xlim(0,7,1)
plt.xlabel('Number of components')
plt.ylabel('Cumulative explained variance')
Text(0, 0.5, 'Cumulative explained variance')
output_19_1.png
from sklearn.decomposition import PCA
sklearn_pca = PCA(n_components=6)
Y_sklearn = sklearn_pca.fit_transform(HR_comma_sep_X_std)
print(Y_sklearn)
Y_sklearn.shape
[[-1.90035018 -1.12083103 -0.0797787 0.03228437 -0.07256447 0.06063013]
[ 2.1358322 0.2493369 0.0936161 0.50676925 1.2487747 -0.61378158]
[ 3.05891625 -1.68312693 -0.301682 -0.4488635 -1.12495888 0.29066929]
...
[-2.0507165 -1.182032 -0.04594506 0.02441143 -0.01553247 0.24980658]
[ 2.91418496 -1.42752606 -0.36333357 -0.31517759 -0.97107375 0.51444624]
[-1.91543672 -1.17021407 -0.07024077 0.01486762 -0.09545357 0.01773844]]
(14999, 6)
4.总结：
前6个因子方差累计超过90%，因此可以将7维特征空间缩减为6维子空间。
