Python——因子分析（KMO檢驗和Bartlett's球形檢驗）

因子分析用Python做的一個典型例子

一、實驗目的

採用合適的數據分析方法對下面的題進行解答

二、實驗要求

採用因子分析方法，根據48位應聘者的15項指標得分，選出6名最優秀的應聘者。

三、代碼

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
def main():
    df=pd.read_csv("./data/applicant.csv")
    # print(df)
    df2=df.copy()
    print("\n原始數據:\n",df2)
    del df2['ID']
    # print(df2)

    # 皮爾森相關係數
    df2_corr=df2.corr()
    print("\n相關係數:\n",df2_corr)

    #熱力圖
    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測度
    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

    print("\nKMO測度:", kmo(df2_corr))

    # 巴特利特球形檢驗
    df2_corr1 = df2_corr.values
    print("\n巴特利特球形檢驗:", 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]))

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

    # 求公因子個數m,使用前m個特徵值的比重大於85%的標準，選出了公共因子是五個
    for m in range(1, 15):
        if eig['eig_value'][:m].sum() / eig['eig_value'].sum() >= 0.85:
            print("\n公因子個數:", m)
            break

    # 因子載荷陣
    A = np.mat(np.zeros((15, 5)))
    i = 0
    j = 0
    while i < 5:
        j = 0
        while j < 15:
            A[j:, i] = sqrt(eig_value[i]) * eigvector[j, i]
            j = j + 1
        i = i + 1
    a = pd.DataFrame(A)
    a.columns = ['factor1', 'factor2', 'factor3', 'factor4', 'factor5']
    a.index = df2_corr.columns
    print("\n因子載荷陣\n", a)
    fa = FactorAnalyzer(n_factors=5)
    fa.loadings_ = a
    # print(fa.loadings_)
    print("\n特殊因子方差:\n", fa.get_communalities())  # 特殊因子方差，因子的方差貢獻度 ，反映公共因子對變量的貢獻
    var = fa.get_factor_variance()  # 給出貢獻率
    print("\n解釋的總方差（即貢獻率）:\n", var)

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

    # 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)
    factor_score.columns = ['factor1', 'factor2', 'factor3', 'factor4', 'factor5']
    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應試者的五個因子得分：\n",pd.DataFrame(fa_t_score))

    # 綜合得分
    wei = [[0.50092], [0.137087], [0.097055], [0.079860], [0.049277]]
    fa_t_score = np.dot(fa_t_score, wei) / 0.864198
    fa_t_score = pd.DataFrame(fa_t_score)
    fa_t_score.columns = ['綜合得分']
    fa_t_score.insert(0, 'ID', range(1, 49))
    print("\n綜合得分：\n", fa_t_score)
    print("\n綜合得分：\n", fa_t_score.sort_values(by='綜合得分', ascending=False).head(6))

    plt.figure()
    ax1=plt.subplot(111)
    X=fa_t_score['ID']
    Y=fa_t_score['綜合得分']
    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()

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

if __name__ == '__main__':
    main()