本文介绍了12种常用的距离度量原理、优缺点、应用场景，以及基于Numpy和Scipy的Python实现代码。

笔记工具：Notability

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1. 个人笔记

笔记工具：Notability

笔记获取：

1. 公众号： datazero 回复：DM 获取下载地址。（主页左侧边栏扫码）
2. Github：https://github.com/datamonday/BigDataAnalysis
   
   
   

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2. 代码实现

导入必要的包，并构造数据。

import numpy as np
from scipy.spatial.distance import pdist

x = np.random.random(5)
# array([0.75173729, 0.34763686, 0.71927609, 0.24151473, 0.22294162])

y = np.random.random(5)
# array([0.98036113, 0.45482745, 0.87472311, 0.92923963, 0.62922737])

1）闵可夫斯基距离（Minkowski Distance）

# p = 2 ——> 欧氏距离
pdist(xy, metric="minkowski", p=2)

2） 欧氏距离（Euclidean Distance）

# 根据公式求解
np.sqrt(np.sum(np.square(x - y) ) )

# 0.8520305805970781

# 根据scipy库求解
xy = np.vstack([x, y])
pdist(xy, metric="euclidean")

# array([0.85203058])

3） 曼哈顿距离（Manhattan/City Block Distance）

np.sum(np.abs(x - y))

# 1.585272101374208

pdist(xy, metric="cityblock")

# array([1.5852721])

4） 切比雪夫距离（Chebyshev Distance）

np.max(np.abs(x - y))

# 0.6877248997688814

pdist(xy, metric="chebyshev")

# array([0.6877249])

5） 余弦相似度（Cosine Similarity）

np.dot(x, y) / ( np.linalg.norm(x) * np.linalg.norm(y) )

# 0.9232011981703329

1 - pdist(xy, metric="cosine")

# array([0.9232012])

6） 汉明距离（Hamming Distance）

np.mean( x != y )

# 1.0

pdist(xy, metric="hamming")

# array([1.])

7） 杰卡德距离（Jaccard Distance）

molecular = np.double( (x != y).sum() )
denominator = np.double(np.bitwise_or( x != 0, y != 0).sum() )

molecular / denominator

# 1.0

pdist(xy, metric="jaccard")

# array([1.])

8） S Φ rensen-Dice

pdist(xy, metric="dice")

# array([0.])

9） 半正矢距离（Haversine Distance）

"""
计算Ezeiza机场（阿根廷布宜诺斯艾利斯）和戴高乐机场（法国巴黎）之间的距离。
"""

from sklearn.metrics.pairwise import haversine_distances
from math import radians

bsas = [-34.83333, -58.5166646]
paris = [49.0083899664, 2.53844117956]

bsas_in_radians = [radians(_) for _ in bsas]
paris_in_radians = [radians(_) for _ in paris]

result = haversine_distances([bsas_in_radians, paris_in_radians])
# multiply by Earth radius to get kilometers
result * 6371000/1000

输出：

array([[    0.        , 11099.54035582],
       [11099.54035582,     0.        ]])

10） 斜交空间距离（Oblique Space Distance）

11） 兰氏距离（Canberra Distance）

np.sum( np.true_divide( np.abs(x - y), np.abs(x) + np.abs(y) ) )

# 1.4272762731136441

pdist(xy, metric="canberra")

 # array([1.42727627])

12） 马氏距离（Mahalanobis Distance）

• 马氏距离要求样本个数＞维数，此处重新生成样本集：10个样本，2个属性；

• 马氏距离计算两两样本之间的距离，故结果包含： C 10 2 = 45 C^{2}_{10} = 45 C102​=45 个距离分量。

data = np.random.random([10, 2])
data # (10, 2)

array([[0.16057991, 0.03173777],
       [0.04984203, 0.63608966],
       [0.0965663 , 0.54125706],
       [0.14562222, 0.50749436],
       [0.12384608, 0.66895134],
       [0.38362246, 0.96750912],
       [0.66204458, 0.34832719],
       [0.62169272, 0.76812896],
       [0.55320254, 0.59736334],
       [0.53135375, 0.97430267]])

# 求解个维度之间协方差矩阵
S = np.cov(data.T)
# 计算协方差矩阵的逆矩阵
ST = np.linalg.inv(S)
ST

array([[18.39262731, -4.22549979],
       [-4.22549979, 13.68987876]])

n = data.shape[0]
d1 = []

for i in range(0, n):
    for j in range(i + 1, n):
        delta = data[i] - data[j]
        d = np.sqrt( np.dot( np.dot(delta, ST), delta.T) )
        d1.append(d)

print(len(d1))
d1

45
[2.4064983868149823,
 1.9761163000756812,
 1.778448926503528,
 2.404430793536302,
 3.3375019493285927,
 2.1576814382238196,
 2.9094250412104405,
 2.3104822379986585,
 3.426008151540264,
 0.4480137866843753,
 0.7065459737678685,
 0.30815685501580464,
 1.618002146140689,
 3.0847744520164553,
 2.3696411013587313,
 2.2012364723722557,
 2.1104688855720037,
 0.2717792884385083,
 0.4554926318973598,
 1.7230353296945067,
 2.7042335514201556,
 2.183968292942155,
 1.9135693479813816,
 2.1102154148029593,
 0.6287347650129346,
 1.735958615801837,
 2.438573435905017,
 2.012440681515558,
 1.6900976084983395,
 2.048918972209157,
 1.3438862451280977,
 2.862373485815439,
 2.067854534269353,
 1.928870122984677,
 1.8108503250774675,
 2.851523901145603,
 1.409891022070067,
 1.7131869461579778,
 0.6273442013634126,
 1.608018006961265,
 1.1384164544766362,
 2.5238527095532692,
 0.6218088758251554,
 0.94309743685501,
 1.422491582300723]

pdist(data, metric="mahalanobis")

array([2.40649839, 1.9761163 , 1.77844893, 2.40443079, 3.33750195,
       2.15768144, 2.90942504, 2.31048224, 3.42600815, 0.44801379,
       0.70654597, 0.30815686, 1.61800215, 3.08477445, 2.3696411 ,
       2.20123647, 2.11046889, 0.27177929, 0.45549263, 1.72303533,
       2.70423355, 2.18396829, 1.91356935, 2.11021541, 0.62873477,
       1.73595862, 2.43857344, 2.01244068, 1.69009761, 2.04891897,
       1.34388625, 2.86237349, 2.06785453, 1.92887012, 1.81085033,
       2.8515239 , 1.40989102, 1.71318695, 0.6273442 , 1.60801801,
       1.13841645, 2.52385271, 0.62180888, 0.94309744, 1.42249158])

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Reference：

1. 《大数据分析与挖掘》 ch5：聚类算法
2. 数据科学中常见的9种距离度量方法，内含欧氏距离、切比雪夫距离等
3. 9 Distance Measures in Data Science