高数学习笔记之范数与距离度量(python实现)

x_norm=np.linalg.norm(x, ord=None, axis=None, keepdims=False)  　

import numpy as np 
from numpy import linalg

x = np.array([3,4])
# 向量范数（默认参数ord=None，axis=None，keepdims=False）
print('L1=\n',linalg.norm(x,ord=1))
print('L2=\n',linalg.norm(x))
print('L∞=\n',linalg.norm(x,ord=np.inf))

y = np.array([
        [0,3,4],
        [1,6,4]])

# 矩阵范数
print('矩阵1范数=\n',linalg.norm(y,ord=1))
print('矩阵2范数=\n',linalg.norm(y))
print('矩阵∞范数=\n',linalg.norm(y,ord=np.inf))

print('矩阵每个行向量求向量的1范数：',linalg.norm(y,ord=1,axis=1,keepdims=True)) 

L1=
 7.0
L2=
 5.0
L∞=
 4.0
矩阵1范数=
 9.0
矩阵2范数=
 8.831760866327848
矩阵∞范数=
 11.0
矩阵每个行向量求向量的1范数： [[ 7.]
 [11.]]

import numpy as np 
vector1 = np.mat([1,2,3])
vector2 = np.mat([4,5,6])
print('欧氏距离:\n',np.sqrt((vector1-vector2)*(vector1-vector2).T))

欧氏距离:
 [[5.19615242]]

import numpy as np 
vector1 = np.mat([1,2,3])
vector2 = np.mat([4,5,6])
print('曼哈顿距离:\n',np.sum(np.abs(vector1-vector2)))

曼哈顿距离:
 9

import numpy as np 
vector1 = np.mat([1,2,3])
vector2 = np.mat([4,7,5])
print('切比雪夫距离:\n',np.abs(vector1-vector2).max())

切比雪夫距离:
 5

import numpy as np 
from numpy import linalg
#方法1
vector1 = np.mat([1,2,3])
vector2 = np.mat([4,7,5])
cosV12 = np.dot(vector1,vector2.T)/(linalg.norm(vector1)*linalg.norm(vector2))
print('夹角余弦:\n',cosV12)
#方法2
vector1 = [1,2,3]
vector2 = [4,7,5]
cosV12 = np.dot(vector1,vector2)/(linalg.norm(vector1)*linalg.norm(vector2))
print('夹角余弦:\n',cosV12)
#区别：方法1向量为matrix格式，方法2为list

夹角余弦:
 [[0.92966968]]

import numpy as np 
from numpy import linalg
'''
np.nonzero的用法，返回非零元素的位置(某行某列)
vector = np.mat([[1,1,0,1,0,1,0,0,1],
                 [1,0,1,1,0,1,0,0,1]])
smstr = np.nonzero(vector)
print(np.array(smstr))
print(np.array(smstr).ndim)
print(smstr)
'''
vector1 = np.mat([1,1,0,1,0,1,0,0,1])
vector2 = np.mat([0,1,1,0,0,0,1,1,1])
smstr = np.nonzero(vector1-vector2)
#print(np.array(smstr))
#print(np.array(smstr).ndim)
print('汉明距离:\n',np.shape(smstr[0])[0])

'''
[[0 0 0 0 0 1 1 1 1 1]
 [0 1 3 5 8 0 2 3 5 8]]
2
 (array([0, 0, 0, 0, 0, 1, 1, 1, 1, 1], dtype=int64), array([0, 1, 3, 5, 8, 0, 2, 3, 5, 8], dtype=int64))
'''
汉明距离:
 6

import numpy as np 
import scipy.spatial.distance as dist
matV = np.mat([[1,1,0,1,0,1,0,0,1],
              [0,1,1,0,0,0,1,1,1]])
print('杰卡德距离:\n',dist.pdist(matV,'jaccard'))

杰卡德距离:
 [0.75]

import numpy as np 
from numpy import linalg

featuremat = np.mat([np.random.randint(0,9,3),
                     np.random.randint(0,9,3),
                     np.random.randint(0,9,3)])
print(featuremat)
# 计算均值
mv1 = np.mean(featuremat[0])
mv2 = np.mean(featuremat[1])
# 计算两列标准差
dv1 = np.std(featuremat[0])
dv2 = np.std(featuremat[1])
# 相关系数和相关距离
corrf = np.mean(np.multiply(featuremat[0]-mv1,featuremat[1]-mv2)/(dv1*dv2))
print('二维相关系数',corrf)
print('相关距离=',1-corrf)
print('***'*15)
# 使用numpy进行相关系数计算
print('多维相关系数=\n',np.corrcoef(featuremat))
# 使用numpy进行协方差矩阵计算
print('多维协方差=\n',np.cov(featuremat))

[[4 2 3]
 [6 6 7]
 [4 5 0]]
二维相关系数 0.0
相关距离= 1.0
*********************************************
多维相关系数=
 [[ 1.          0.         -0.18898224]
 [ 0.          1.         -0.98198051]
 [-0.18898224 -0.98198051  1.        ]]
多维协方差=
 [[ 1.          0.         -0.5       ]
 [ 0.          0.33333333 -1.5       ]
 [-0.5        -1.5         7.        ]]

import numpy as np 
from numpy import linalg

featuremat = np.mat([np.random.randint(0,9,3),
                     np.random.randint(0,9,3)])

print(featuremat)

covinv = linalg.inv(np.cov(featuremat))
tp = featuremat.T[0]-featuremat.T[1]
distma = np.sqrt(np.dot(np.dot(tp,covinv),tp.T))
print(distma)

[[4 2 8]
 [4 2 7]]
[[2.]]