A = [a1, a2, ..., an]
B = [b1, b2, ..., bn]

ave_a = (a1 + a2 +...+ an)/n
ave_b = (b1 + b2 +...+ bn)/m

dev_a = [a1, a2, ..., an] - ave_a
dev_b = [b1, b2, ..., bn] - ave_b

cov_ab = ave(dev_a x dev_b)
cov_ba = ave(dev_b x dev_a)

import numpy as np
import matplotlib.pyplot as mp
a = np.random.randint(1, 30, 10)
b = np.random.randint(1, 30, 10)
#平均值
ave_a = np.mean(a)
ave_b = np.mean(b)
#离差
dev_a = a - ave_a
dev_b = b - ave_b
#协方差
cov_ab = np.mean(dev_a*dev_b)
cov_ba = np.mean(dev_b*dev_a)
print(\'a与b数组：\', a, b)
print(\'a与b样本方差：\', np.sum(dev_a**2)/(len(dev_a)-1), np.sum(dev_b**2)/(len(dev_b)-1))
print(\'a与b协方差：\',cov_ab, cov_ba)
#绘图，查看两条图线的相关性
mp.figure(\'COV LINES\', facecolor=\'lightgray\')
mp.title(\'COV LINES\', fontsize=16)
mp.xlabel(\'x\', fontsize=14)
mp.ylabel(\'y\', fontsize=14)
x = np.arange(0, 10)
#a,b两条线
mp.plot(x, a, color=\'dodgerblue\', label=\'Line1\')
mp.plot(x, b, color=\'limegreen\', label=\'Line2\')
#a,b两条线的平均线
mp.plot([0, 9], [ave_a, ave_a], color=\'dodgerblue\', linestyle=\'--\', alpha=0.7, linewidth=3)
mp.plot([0, 9], [ave_b, ave_b], color=\'limegreen\', linestyle=\'--\', alpha=0.7, linewidth=3)
mp.grid(linestyle=\'--\', alpha=0.5)
mp.legend()
mp.tight_layout()
mp.show()

# a组样本 与 b组样本做对照后的相关系数
cov_ab/(std_a x std_b)
# b组样本 与 a组样本做对照后的相关系数
cov_ba/(std_b x std_a)
# a样本与a样本作对照   b样本与b样本做对照   二者必然相等
cov_ab/(std_a x std_b)=cov_ba/(std_b x std_a)

若相关系数越接近于0，越表示两组样本越不相关。
若相关系数越接近于1，越表示两组样本正相关。
若相关系数越接近于-1，越表示两组样本负相关。

print(\'相关系数：\', cov_ab/(np.std(a)*np.std(b)), cov_ba/(np.std(a)*np.std(b)))

# 相关矩阵
numpy.corrcoef(a, b)    
# 相关矩阵的分子矩阵 
# [[a方差，ab协方差], [ba协方差, b方差]]
numpy.cov(a, b)        

# 协方差
import numpy as np
import matplotlib.pyplot as mp
import datetime as dt
import matplotlib.dates as md
def dmy2ymd(dmy):
  """
  把日月年转年月日
  :param day:
  :return:
  """
  dmy = str(dmy, encoding=\'utf-8\')
  t = dt.datetime.strptime(dmy, \'%d-%m-%Y\')
  s = t.date().strftime(\'%Y-%m-%d\')
  return s
dates, bhp_closing_prices = \
  np.loadtxt(\'bhp.csv\',
             delimiter=\',\',
             usecols=(1, 6),
             unpack=True,
             dtype=\'M8[D],f8\',
             converters={1: dmy2ymd})  # 日月年转年月日
vale_closing_prices = \
  np.loadtxt(\'vale.csv\',
             delimiter=\',\',
             usecols=(6,),
             unpack=True)  # 因为日期一样,所以此处不读日期
# print(dates)
# 绘制收盘价的折现图
mp.figure(\'COV\', facecolor=\'lightgray\')
mp.title(\'COV\', fontsize=18)
mp.xlabel(\'Date\', fontsize=14)
mp.ylabel(\'Price\', fontsize=14)
mp.grid(linestyle=":")
# 设置刻度定位器
# 每周一一个主刻度,一天一个次刻度
ax = mp.gca()
ma_loc = md.WeekdayLocator(byweekday=md.MO)
ax.xaxis.set_major_locator(ma_loc)
ax.xaxis.set_major_formatter(md.DateFormatter(\'%Y-%m-%d\'))
ax.xaxis.set_minor_locator(md.DayLocator())
# 修改dates的dtype为md.datetime.datetiem
dates = dates.astype(md.datetime.datetime)
mp.plot(dates, bhp_closing_prices,
        color=\'dodgerblue\',
        linewidth=2,
        linestyle=\'-\',
        alpha=0.8,
        label=\'BHP Closing Price\')
mp.plot(dates, vale_closing_prices,
        color=\'orangered\',
        linewidth=2,
        linestyle=\'-\',
        alpha=0.8,
        label=\'VALE Closing Price\')
# 计算两只股票收盘价的协方差
# 求均值
mean_bhp = np.mean(bhp_closing_prices)
mean_vale = np.mean(vale_closing_prices)
# 求离差
dev_bhp = bhp_closing_prices - mean_bhp
dev_vale = vale_closing_prices - mean_vale
# 协方差(bhp的离差乘以vale的离差)
cov = (dev_bhp * dev_vale).mean()  # 用的是总体的数据
print(\'cov:\', cov)  # cov: 3.135577333333333
# 两条图线的相关性
k = cov / np.std(bhp_closing_prices * np.std(vale_closing_prices))
print(k)  # 0.8664988296368301
# 相关系数 = a方差/(a标准差*a标准差)
k = np.corrcoef(bhp_closing_prices, vale_closing_prices)
print(k)
"""
[[1.         0.86649883]
 [0.86649883 1.        ]]
"""
# q秋两组数据的协方差
covm = np.cov(bhp_closing_prices, vale_closing_prices)  # 用的是 样本的数据
print(covm)
"""
[[8.53844379 3.24370069]
 [3.24370069 1.64122023]]
"""
mp.legend()
mp.gcf().autofmt_xdate()
mp.show()