Python3Numpy——相關性協方差應用

 基本理論

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.

定義

 

生物是一個有機的整體，其各個組成部分都是相關聯的，我們可以通過研究一個生物的牙齒、爪子或者骨骼來復原這個生物。

協方差：

定義：

 

對於離散型隨機變量：

 

對於連續性隨機變量：

 

協方差化簡：

 

當X與Y獨立時， 有Cov(X, Y) = 0

協方差基本性質：

 

隨機變量和的方差與協方差的關系：

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

協方差的有界性

 

相關系數：

定義

 

Python3NumPy關於相關性協方差闡述

導入相關模塊

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

導入數據

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

數據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)

數據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

數據處理：

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

計算bhp_returns和vale_returns的協方差

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

結果：

[[0.00028179 0.00019766]
 [0.00019766 0.00030123]]

取協方差對角線上的元素：

print(covariance.diagonal())

結果：

[0.00028179 0.00030123]

打印協方差矩陣的跡：

print(covariance.trace())

結果：

0.000583023549920278

計算bhp_returns和vale_returns的相關系數：

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

結果：

[[1.00173366 0.70264666]
 [0.70264666 1.0708476 ]]

print(np.corrcoef(bhp_returns, vale_returns))

結果：

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

繪bhp_returns和vale_returns的圖像：

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

 結果：

相關知識點理解

np.diff(a, n=1, axis=-1)

沿着指定軸計算第N維的離散差值 

參數： 

a：輸入矩陣 

n：可選，代表要執行幾次差值 

axis：默認是最后一個 

示例：

import numpy as np
A = np.arange(2 , 14).reshape((3 , 4))
A[1 , 1] = 8
print('A:' , A)
# A: [[ 2 3 4 5]
# [ 6 8 8 9]
# [10 11 12 13]]
print(np.diff(A))
# [[1 1 1]
# [2 0 1]
# [1 1 1]]

從輸出結果可以看出， 其實diff函數就是執行的是后一個元素減去前一個元素。