基本理论

Correlation

Are there correlations between variables?

Correlation measures the strength of the linear association between two numerical variables. For example, you could imagine that for children, age correlates with height: the older the child, the taller he or she is. You could reasonably expect to get a straight line or upward curve with a positive slope when you plot age against height.

定义

生物是一个有机的整体，其各个组成部分都是相关联的，我们可以通过研究一个生物的牙齿、爪子或者骨骼来复原这个生物。

协方差：

定义：

对于离散型随机变量：

对于连续性随机变量：

协方差化简：

当X与Y独立时，有Cov(X, Y) = 0

协方差基本性质：

随机变量和的方差与协方差的关系：

D(X +/- Y) = D(X) + D(Y) +/- 2Cov(X, Y)

协方差的有界性

Python3NumPy关于相关性协方差阐述

导入相关模块

import numpy as np
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
import matplotlib.pyplot as plt

导入数据

bhp = np.loadtxt('BHP.csv', delimiter=',', usecols=(6,), unpack=True)

数据BHP.csv文件如下：

BHP	11-02-2011	93.11	94.26	92.9	93.72	1741900
BHP	14-02-2011	94.57	96.23	94.39	95.64	2620800
BHP	15-02-2011	94.45	95.47	93.91	94.56	2461300
BHP	16-02-2011	92.67	93.58	92.56	93.3	3270900
BHP	17-02-2011	92.65	93.98	92.58	93.93	2650200
BHP	18-02-2011	92.34	93	92	92.39	4667300
BHP	22-02-2011	93.14	93.98	91.75	92.11	5359800
BHP	23-02-2011	91.93	92.46	91.05	92.36	7768400
BHP	24-02-2011	92.42	92.71	90.93	91.76	4799100
BHP	25-02-2011	93.48	94.04	92.44	93.91	3448300
BHP	28-02-2011	94.81	95.11	94.1	94.6	4719800
BHP	01-03-2011	95.05	95.2	93.13	93.27	3898900
BHP	02-03-2011	93.89	94.89	93.54	94.43	3727700
BHP	03-03-2011	95.9	96.11	95.18	96.02	3379400
BHP	04-03-2011	96.12	96.44	95.08	95.76	2463900
BHP	07-03-2011	96.51	96.66	94.03	94.47	3590900
BHP	08-03-2011	93.72	94.47	92.9	94.34	3805000
BHP	09-03-2011	92.94	93.13	91.86	92.22	3271700
BHP	10-03-2011	89	89.17	87.93	88.31	5507800
BHP	11-03-2011	88.24	89.8	88.16	89.59	2996800
BHP	14-03-2011	88.17	89.06	87.82	89.02	3434800
BHP	15-03-2011	84.58	87.32	84.35	86.95	5008300
BHP	16-03-2011	86.31	87.28	83.85	84.88	7809799
BHP	17-03-2011	87.32	88.29	86.89	87.38	3947100
BHP	18-03-2011	89.53	89.58	88.05	88.56	3809700
BHP	21-03-2011	90.13	90.16	88.88	89.59	3098200
BHP	22-03-2011	89.5	89.59	88.42	88.71	3500200
BHP	23-03-2011	89.57	90.32	88.85	90.02	4285600
BHP	24-03-2011	90.86	91.35	89.7	91.26	3918800
BHP	25-03-2011	90.42	91.09	90.07	90.67	3632200

vale = np.loadtxt('VALE.csv', delimiter=',', usecols=(6,), unpack=True)

数据VALE.csv文件如下：

VALE	11-02-2011	33.88	34.54	33.63	34.37	18433500
VALE	14-02-2011	34.53	35.29	34.52	35.13	20780700
VALE	15-02-2011	34.89	35.31	34.82	35.14	17756700
VALE	16-02-2011	35.16	35.4	34.81	35.31	16792800
VALE	17-02-2011	35.18	35.6	35.04	35.57	24088300
VALE	18-02-2011	35.31	35.37	34.89	35.03	21286600
VALE	22-02-2011	33.94	34.57	33.36	33.44	28364700
VALE	23-02-2011	33.43	34.12	33.1	33.94	22559300
VALE	24-02-2011	34.3	34.3	33.56	34.21	20591900
VALE	25-02-2011	34.67	34.95	34.05	34.27	20151500
VALE	28-02-2011	34.34	34.51	33.7	34.23	16126000
VALE	01-03-2011	34.39	34.44	33.68	33.76	17282400
VALE	02-03-2011	33.61	34.5	33.57	34.32	15870900
VALE	03-03-2011	34.77	34.89	34.53	34.87	14648200
VALE	04-03-2011	34.67	34.83	34.04	34.5	15330800
VALE	07-03-2011	34.43	34.53	32.97	33.23	25040500
VALE	08-03-2011	33.22	33.7	32.55	33.29	17093000
VALE	09-03-2011	33.23	33.44	32.68	32.88	20026300
VALE	10-03-2011	32.17	32.4	31.68	31.91	30803900
VALE	11-03-2011	31.53	32.42	31.49	32.17	24429900
VALE	14-03-2011	32.03	32.45	31.74	32.44	15525500
VALE	15-03-2011	30.99	31.93	30.79	31.91	24767700
VALE	16-03-2011	31.99	32.03	30.68	31.04	30394153
VALE	17-03-2011	31.44	31.82	31.32	31.51	24035000
VALE	18-03-2011	32.17	32.39	31.98	32.14	19740600
VALE	21-03-2011	32.81	32.85	32.26	32.42	18923700
VALE	22-03-2011	32.13	32.32	31.74	32.25	18934200
VALE	23-03-2011	32.39	32.91	32.22	32.7	18359900
VALE	24-03-2011	32.82	32.94	32.12	32.36	25894100
VALE	25-03-2011	32.26	32.74	31.93	32.34	16688900

数据处理：

bhp_returns = np.diff(bhp) / bhp[:-1]
vale_returns = np.diff(vale) / vale[:-1]

计算bhp_returns和vale_returns的协方差

covariance = np.cov(bhp_returns, vale_returns)
print(covariance)

结果：

[[0.00028179 0.00019766]
 [0.00019766 0.00030123]]

取协方差对角线上的元素：

print(covariance.diagonal())

结果：

[0.00028179 0.00030123]

打印协方差矩阵的迹：

print(covariance.trace())

结果：

0.000583023549920278

计算bhp_returns和vale_returns的相关系数：

print(covariance/((bhp_returns.std()*vale_returns.std())))

结果：

[[1.00173366 0.70264666]
 [0.70264666 1.0708476 ]]

print(np.corrcoef(bhp_returns, vale_returns))

结果：

[[1.         0.67841747]
 [0.67841747 1.        ]]

绘bhp_returns和vale_returns的图像：

t = np.arange(len(bhp_returns))
plot(t, bhp_returns, lw = 1)
plot(t, vale_returns,lw =2)
show()

结果：

Python3Numpy——相关性协方差应用