做数学建模不得不会的数据特征分析---相关性分析

相关性分析是分析连续变量之间的线性相关程度的强弱，我们可以通过图来初步判断，当然了比较权威的是通过Pearson相关系数（皮尔逊相关系数） / Sperman秩相关系数（斯皮尔曼相关系数）来判断

引入相关模块

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats
% matplotlib inline

（1）图示初判

我们生成三组数据，data1为0-100的随机数并从小到大排列，data2为0-50的随机数并从小到大排列，data3为0-500的随机数并从大到小排列

data1 = pd.Series(np.random.rand(50)*100).sort_values()
data2 = pd.Series(np.random.rand(50)*50).sort_values()
data3 = pd.Series(np.random.rand(50)*500).sort_values(ascending = False)
#

绘制散点图看一下data1与data2之间的关系

fig = plt.figure(figsize = (10,4))
ax1 = fig.add_subplot(1,2,1)
ax1.scatter(data1, data2)
plt.grid()

通过散点图我们可以看出，data1与data2基本是呈现线性正相关的
再来绘制散点图观察一下data1余data3之间的关系

ax2 = fig.add_subplot(1,2,2)
ax2.scatter(data1, data3)
plt.grid()

基本呈现线性负相关
还有一种比较常用的方法是通过散点图矩阵来初步判断多变量之间的关系

data = pd.DataFrame(np.random.randn(200,4)*100, columns = ['A','B','C','D'])
pd.scatter_matrix(data,figsize=(8,8),c = 'k',marker = '+',diagonal='hist',alpha = 0.8,range_padding=0.1)
data.head()

数据展示：
矩阵散点图:
(2)计算Pearson相关系数
生成样本数据

data1 = pd.Series(np.random.rand(100)*100).sort_values()
data2 = pd.Series(np.random.rand(100)*50).sort_values()
data = pd.DataFrame({'value1':data1.values,'value2':data2.values})
print(data.head())

样本数据展示：

正态性检验：

u1,u2 = data['value1'].mean(),data['value2'].mean()  # 计算均值
std1,std2 = data['value1'].std(),data['value2'].std()  # 计算标准差
print('value1正态性检验：\n',stats.kstest(data['value1'], 'norm', (u1, std1)))
print('value2正态性检验：\n',stats.kstest(data['value2'], 'norm', (u2, std2)))
print('------')

制作Pearson相关系数求值表

data['(x-u1)*(y-u2)'] = (data['value1'] - u1) * (data['value2'] - u2)
data['(x-u1)**2'] = (data['value1'] - u1)**2
data['(y-u2)**2'] = (data['value2'] - u2)**2
print(data.head())

求出Pearson相关系数

r = data['(x-u1)*(y-u2)'].sum() / (np.sqrt(data['(x-u1)**2'].sum() * data['(y-u2)**2'].sum()))
print('Pearson相关系数为：%.4f' % r)

|r| > 0.8 → 高度线性相关

当然了强大的Python提供了函数直接计算Pearson相关系数
生成样本数据：

data1 = pd.Series(np.random.rand(100)*100).sort_values()
data2 = pd.Series(np.random.rand(100)*50).sort_valdata:image/svg+xml;base64,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data = pd.DataFrame({'value1':data1.values,'value2':data2.values})
print(data.head())

求出Pearson相关系数

data.corr()
# pandas相关性方法：data.corr(method='pearson', min_periods=1) → 直接给出数据字段的相关系数矩阵
# method默认pearson

（3）求Sperman秩相关系数
生成样本数据：

data = pd.DataFrame({'智商':[106,86,100,101,99,103,97,113,112,110],'每周看电视小时数':[7,0,27,50,28,29,20,12,6,17]})
print(data)

“智商”、“每周看电视小时数”重新按照从小到大排序，并设定秩次index

data.sort_values('智商', inplace=True)
data['range1'] = np.arange(1,len(data)+1)
data.sort_values('每周看电视小时数', inplace=True)
data['range2'] = np.arange(1,len(data)+1)
print(data)

求出di，di2

data['d'] = data['range1'] - data['range2']
data['d2'] = data['d']**2
print(data)

求出rs

n = len(data)
rs = 1 - 6 * (data['d2'].sum()) / (n * (n**2 - 1))
print('Pearson相关系数为：%.4f' % rs)

上面真是太麻烦了，我们还是试着用Python中自带函数来实现吧

data = pd.DataFrame({'智商':[106,86,100,101,99,103,97,113,112,110],'每周看电视小时数':[7,0,27,50,28,29,20,12,6,17]})
print(data)
print('------')
# 创建样本数据
data.corr(method='spearman')
# pandas相关性方法：data.corr(method='pearson', min_periods=1) → 直接给出数据字段的相关系数矩阵
# method默认pearson

ok~

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