挖掘建模⑤—因子分析与python实现

Python介绍、 Unix & Linux & Window & Mac 平台安装更新 Python3 及VSCode下Python环境配置配置
python基础知识及数据分析工具安装及简单使用(Numpy/Scipy/Matplotlib/Pandas/StatsModels/Scikit-Learn/Keras/Gensim))
数据探索（数据清洗）①——数据质量分析（对数据中的缺失值、异常值和一致性进行分析）
数据探索（数据清洗）②—Python对数据中的缺失值、异常值和一致性进行处理
数据探索（数据集成、数据变换、数据规约）③—Python对数据规范化、数据离散化、属性构造、主成分分析 降维
数据探索（数据特征分析）④—Python分布分析、对比分析、统计量分析、期性分析、贡献度分析、相关性分析
挖掘建模①—分类与预测
挖掘建模②—Python实现预测
挖掘建模③—聚类分析(包括相关性分析、雷达图等)及python实现
挖掘建模④—关联规则及Apriori算法案例与python实现
挖掘建模⑤—因子分析与python实现

因子分析python实现

读取数据导入包

import pandas as pd
import numpy as np
import math as math
import numpy as np
from numpy import *
from scipy.stats import bartlett
from factor_analyzer import *
import numpy.linalg as nlg
from sklearn.cluster import KMeans
from matplotlib import cm
import matplotlib.pyplot as plt

df = pd.read_csv(f'ks/工作簿1.csv', encoding='gbk')
df2 = df.copy()
sj_count = df[['id']].count()  # 数据个数31
del df2['id']
del df2['2018年']
del df2['province']
column_count = len(df2.columns)  # 字段（列）21

皮尔森相关系数

# 皮尔森相关系数
df2_corr = df2.corr()
print("\n相关系数:\n", df2_corr)

绘制热力图

# 热力图
cmap = cm.Blues
# cmap = cm.hot_r
fig = plt.figure()
ax = fig.add_subplot(111)
map = ax.imshow(df2_corr, interpolation='nearest',
                cmap=cmap, vmin=0, vmax=1)
plt.title('correlation coefficient--headmap')
ax.set_yticks(range(len(df2_corr.columns)))
ax.set_yticklabels(df2_corr.columns)
ax.set_xticks(range(len(df2_corr)))
ax.set_xticklabels(df2_corr.columns)
plt.colorbar(map)
plt.show()

KMO测度

def kmo(dataset_corr):
    corr_inv = np.linalg.inv(dataset_corr)
    nrow_inv_corr, ncol_inv_corr = dataset_corr.shape
    A = np.ones((nrow_inv_corr, ncol_inv_corr))
    for i in range(0, nrow_inv_corr, 1):
        for j in range(i, ncol_inv_corr, 1):
            A[i, j] = -(corr_inv[i, j]) / \
                (math.sqrt(corr_inv[i, i] * corr_inv[j, j]))
            A[j, i] = A[i, j]
    dataset_corr = np.asarray(dataset_corr)
    kmo_num = np.sum(np.square(dataset_corr)) - \
        np.sum(np.square(np.diagonal(A)))
    kmo_denom = kmo_num + np.sum(np.square(A)) - \
        np.sum(np.square(np.diagonal(A)))
    kmo_value = kmo_num / kmo_denom
    return kmo_value
    
# KMO测度
# KMO值：0.9以上非常好；0.8以上好；0.7一般；0.6差；0.5很差；0.5以下不能接受；
# 巴特利球形检验的值范围在0-1，越接近1，使用因子分析效果越好。
print("\nKMO测度:", kmo(df2_corr))

巴特利特球形检验

# 巴特利特球形检验
df2_corr1 = df2_corr.values
print("\n巴特利特球形检验:", bartlett(df2_corr1[0], df2_corr1[1], df2_corr1[2], df2_corr1[3], df2_corr1[4],
                              df2_corr1[5], df2_corr1[6], df2_corr1[7],df2_corr1[8],df2_corr1[9],df2_corr1[10],
                              df2_corr1[11],df2_corr1[12],df2_corr1[13],df2_corr1[14],df2_corr1[15],df2_corr1[16],
                              df2_corr1[17],df2_corr1[18],df2_corr1[19],df2_corr1[20]))

求特征值和特征向量

# 求特征值和特征向量
eig_value, eigvector = nlg.eig(df2_corr)  # 求矩阵R的全部特征值，构成向量
eig = pd.DataFrame()
eig['names'] = df2_corr.columns
eig['eig_value'] = eig_value
eig.sort_values('eig_value', ascending=False, inplace=True)
print("\n特征值\n：", eig)
eig1 = pd.DataFrame(eigvector)
eig1.columns = df2_corr.columns
eig1.index = df2_corr.columns
print("\n特征向量\n", eig1)

求公因子个数

# 求公因子个数m,使用前m个特征值的比重大于85%的标准，选出了公共因子是六个
for m in range(1, column_count):
    if eig['eig_value'][:m].sum() / eig['eig_value'].sum() >= 0.85:
        print("\n公因子个数:", m)
        break

因子载荷阵、特殊因子方差、解释的总方差（即贡献率）

# 因子载荷阵
A = np.mat(np.zeros((column_count, m)))
i = 0
j = 0
while i < m:
    j = 0
    while j < column_count:
        A[j:, i] = math.sqrt(eig_value[i]) * eigvector[j, i]
        j = j + 1
    i = i + 1
a = pd.DataFrame(A)
factors_list = []
for k in range(0, m):
    factors_list.append('factor'+str(k+1))
# ['factor1', 'factor2', 'factor3', 'factor4', 'factor5']
a.columns = factors_list
a.index = df2_corr.columns
print("\n因子载荷阵\n", a)
fa = FactorAnalyzer(n_factors=5)
fa.loadings_ = a
# print(fa.loadings_)
# 特殊因子方差，因子的方差贡献度 ，反映公共因子对变量的贡献
print("\n特殊因子方差:\n", fa.get_communalities())
var = fa.get_factor_variance()  # 给出贡献率
print("\n解释的总方差（即贡献率）:\n", var)

因子旋转

 # 因子旋转
rotator = Rotator()
b = pd.DataFrame(rotator.fit_transform(fa.loadings_))  # ['factor1', 'factor2', 'factor3', 'factor4', 'factor5']
b.columns = factors_list
b.index = df2_corr.columns
print("\n因子旋转:\n", b)

因子得分

 # 因子得分
X1 = np.mat(df2_corr)
X1 = nlg.inv(X1)
b = np.mat(b)
factor_score = np.dot(X1, b)
factor_score = pd.DataFrame(factor_score)  # ['factor1', 'factor2','factor3', 'factor4', 'factor5']
factor_score.columns = factors_list
factor_score.index = df2_corr.columns
print("\n因子得分：\n", factor_score)
fa_t_score = np.dot(np.mat(df2), np.mat(factor_score))
print("\n应试者的6个因子得分：\n", pd.DataFrame(fa_t_score))

综合得分

 # 综合得分
wei = [[0.411246], [0.205008], [0.081990],
       [0.065977], [0.051158], [0.045077]]
fa_t_score = np.dot(fa_t_score, wei) / 0.860456  # factor6    0.860456

fa_t_score = pd.DataFrame(fa_t_score)
fa_t_score.columns = ['综合得分']
fa_t_score.insert(0, 'ID', range(1, int(sj_count)+1))
print("\n综合得分：\n", fa_t_score)
print("\n综合得分：\n", fa_t_score.sort_values(
    by='综合得分', ascending=False).head(6))

绘制整体综合得分柱状图

plt.figure()
ax1 = plt.subplot(111)
X = fa_t_score['ID']
Y = fa_t_score['综合得分']
plt.bar(X, Y, color="#87CEFA")
# plt.bar(X, Y, color="red")
plt.title('result00')
ax1.set_xticks(range(len(fa_t_score)))
ax1.set_xticklabels(fa_t_score.index)
plt.show()

绘制前五综合得分柱状图

fa_t_score1 = pd.DataFrame()
fa_t_score1 = fa_t_score.sort_values(by='综合得分', ascending=False).head()
ax2 = plt.subplot(111)
X1 = fa_t_score1['ID']
Y1 = fa_t_score1['综合得分']
plt.bar(X1, Y1, color="#87CEFA")
# plt.bar(X1, Y1, color='red')
plt.title('result01')
plt.show()