1.协方差和协方差矩阵的概念公式
1.1协方差公式
1.2协方差矩阵公式
有数据集={X,Y,Z}，是三维度的数据，即此此数据集中的样例有3个特征
"""
Aim：计算两个维度的协方差covariance
"""
import numpy as np
class CCovariance(object):
'''计算X,Y这俩维度的协方差
'''
def __init__(self, X, Y):
self.X = X
self.Y = Y
self.Covariance_way1()
self.Covariance_way2()
self.Covariance_way3()
def Covariance_way1(self):
'''
协方差公式法计算两个等长向量的协方差convariance
'''
X,Y = np.array(self.X), np.array(self.Y)
meanX, meanY = np.mean(X), np.mean(Y)
n = np.shape(X)[0]
#按照协方差公式计算协方差，Note:分母一定是n-1
covariance = sum(np.multiply(X-meanX, Y-meanY))/(n-1)
print('协方差公式法求得的协方差:', covariance)
return covariance
def Covariance_way2(self):
'''
向量中心化方法计算两个等长向量的协方差convariance
'''
X,Y = np.array(self.X),np.array(self.Y)
n = np.shape(X)[0]
centrX = X-np.mean(X)
centrY = Y-np.mean(Y)
convariance = sum(np.multiply(centrX, centrY))/(n-1)
print('向量中心化方法求得协方差:', convariance)
return convariance
def Covariance_way3(self):
'''
numpy.conv(X,Y)提供的协方差函数求协方差
'''
conv = np.cov(self.X, self.Y)
print('np.cov(X,Y)求得的X的方差:', conv[0,0])
print('np.cov(X,Y)求得的Y的方差:', conv[1,1])
print('np.cov(X,Y)求得的X和Y的协方差:',conv[0,1])
if __name__=='__main__':
X = [10,15,23,11,42,9,11,8,11,21]
Y = [15,46,21,9,45,48,21,5,12,20]
c = CCovariance(X,Y)
协方差矩阵的多种求解Python实现
"""
Aim：计算一个多维度样本的协方差矩阵covariance matrix
Note:协方差矩阵是计算的样本中每个特征之间的协方差，所以协方差矩阵是特征个数阶的对称阵
"""
import numpy as np
class CCovMat(object):
'''计算多维度样本集的协方差矩阵
Note：请保证输入的样本集m×n，m行样例，每个样例n个特征
'''
def __init__(self, samples):
#样本集shpae=(m,n)，m是样本总数，n是样本的特征个数
self.samples = samples
self.covmat1 = [] #保存方法1求得的协方差矩阵
self.covmat2 = [] #保存方法1求得的协方差矩阵
#用方法1计算协方差矩阵
self._calc_covmat1()
#用方法2计算协方差矩阵
self._calc_covmat2()
def _covariance(self, X, Y):
'''
计算两个等长向量的协方差convariance
'''
n = np.shape(X)[0]
X, Y = np.array(X), np.array(Y)
meanX, meanY = np.mean(X), np.mean(Y)
#按照协方差公式计算协方差，Note:分母一定是n-1
cov = sum(np.multiply(X-meanX, Y-meanY))/(n-1)
return cov
def _calc_covmat1(self):
'''
方法1：根据协方差公式和协方差矩阵的概念计算协方差矩阵
'''
S = self.samples #样本集
na = np.shape(S)[1] #特征attr总数
self.covmat1 = np.full((na, na), fill_value=0.) #保存协方差矩阵
for i in range(na):
for j in range(na):
self.covmat1[i,j] = self._covariance(S[:,i], S[:,j])
return self.covmat1
def _calc_covmat2(self):
'''
方法2：先样本集中心化再求协方差矩阵
'''
S = self.samples #样本集
ns = np.shape(S)[0] #样例总数
mean = np.array([np.mean(attr) for attr in S.T]) #样本集的特征均值
print('样本集的特征均值:\n',mean)
centrS = S - mean ##样本集的中心化
print('样本集的中心化(每个元素将去当前维度特征的均值):\n', centrS)
#求协方差矩阵
self.covmat2 = np.dot(centrS.T, centrS)/(ns - 1)
return self.covmat2
def CovMat1(self):
return self.covmat1
def CovMat2(self):
return self.covmat2
if __name__=='__main__':
'10样本3特征的样本集'
samples = np.array([[10, 15, 29],
[15, 46, 13],
[23, 21, 30],
[11, 9, 35],
[42, 45, 11],
[9, 48, 5],
[11, 21, 14],
[8, 5, 15],
[11, 12, 21],
[21, 20, 25]])
cm = CCovMat(samples)
print('样本集(10行3列，10个样例，每个样例3个特征):\n', samples)
print('按照协方差公式求得的协方差矩阵:\n', cm.CovMat1())
print('按照样本集的中心化求得的协方差矩阵:\n', cm.CovMat1())
print('numpy.cov()计算的协方差矩阵:\n', np.cov(samples.T))