如何计算R中的马氏距离？

在本文中，我们将在 R 编程语言中计算马氏距离。

马氏距离用于计算多元距离度量空间中两点或向量之间的距离，这是一种涉及多个变量的统计分析。首先，我们需要一个数据框。

示例：创建数据框

R

set.seed(700)
score_1 <− rnorm(20,12,1)
score_2 <− rnorm(20,11,12)
score_3 <− rnorm(20,15,23)
score_4 <− rnorm(20,16,3)
  
df <− data.frame(score_1, score_2, score_3, score_4)
df

R

mahalanobis(df, colMeans(df), cov(df))

R

# create new column for Mahalanobis distances
df$mahalnobis<- mahalanobis(df, colMeans(df), cov(df))
df

R

# create new column for p-value 
df$pvalue <- pchisq(df$mahalnobis, df=3)
df

输出：

score_1     score_2    score_3   score_4
1  11.91218  20.3843568  68.179655 12.864159
2  11.77103  13.5718323 -30.953642 15.241168
3  11.91570  29.9250800  42.570528  7.179686
4  10.25905  10.7594514  17.879960 19.639647
5  13.01343  15.7463448   3.185857 12.776482
6  11.78211  14.9688992  31.368892 16.043620
7  13.51328  10.5017826  58.985715 14.701817
8  11.10565  20.4965614   6.806652 15.876947
9  11.20834  12.7588547  10.461229 16.991393
10 11.10233 -10.3961351  18.082209 15.258644
11 12.34732  -0.8615359  57.411750 13.400421
12 12.08361  15.0248600 -17.853098 13.999682
13 12.86457  -6.1221908  23.184838 20.389762
14 10.58871  17.1000715  20.900155 12.560962
15 10.74134   6.3728076  39.173259 17.865589
16 11.20248   8.8909128  24.696939 14.384012
17 12.89797  34.8522136  10.035498 14.975053
18 11.37993  14.4232355  28.129197 16.395271
19 11.78309  14.9324201  23.584362 14.765245
20 12.77480  30.7969171  -9.635902 10.203178

mahalanobis()函数用于计算 R 中的 Mahalanobis 距离。它是内置类型。

Syntax: mahalanobis(Data , center, cov)

where:

Data: matrix or vector of data
center: mean vector
cov: covariance matrix

编程需要懂一点英语

示例：计算马氏距离

R

mahalanobis(df, colMeans(df), cov(df))

输出：

4.46866714558536 4.61260586529474 7.41513071619846 5.21448589688871

2.84292222223026 0.673116763926688 6.04984394951585 1.72865361097932

1.03750690527476 7.21856549018804 4.85579110162481 2.90808365141091

7.57223884458172 3.27702692226183 2.68208130355785 0.916110244005359

6.79796970070888 0.829693729587342 0.0356208551487593 4.86388508103035

编程需要懂一点英语

计算每一行的 Mahalanobis

基于 Mahalanobis 距离，我们发现一些距离远高于其他距离，为了确定其具有统计学意义，我们需要计算 p 值。

示例：计算每一行的马氏距离

R

# create new column for Mahalanobis distances
df$mahalnobis<- mahalanobis(df, colMeans(df), cov(df))
df

输出：

score_1     score_2    score_3   score_4
1  11.91218  20.3843568  68.179655 12.864159
2  11.77103  13.5718323 -30.953642 15.241168
3  11.91570  29.9250800  42.570528  7.179686
4  10.25905  10.7594514  17.879960 19.639647
5  13.01343  15.7463448   3.185857 12.776482
6  11.78211  14.9688992  31.368892 16.043620
7  13.51328  10.5017826  58.985715 14.701817
8  11.10565  20.4965614   6.806652 15.876947
9  11.20834  12.7588547  10.461229 16.991393
10 11.10233 -10.3961351  18.082209 15.258644
11 12.34732  -0.8615359  57.411750 13.400421
12 12.08361  15.0248600 -17.853098 13.999682
13 12.86457  -6.1221908  23.184838 20.389762
14 10.58871  17.1000715  20.900155 12.560962
15 10.74134   6.3728076  39.173259 17.865589
16 11.20248   8.8909128  24.696939 14.384012
17 12.89797  34.8522136  10.035498 14.975053
18 11.37993  14.4232355  28.129197 16.395271
19 11.78309  14.9324201  23.584362 14.765245
20 12.77480  30.7969171  -9.635902 10.203178

计算 p 值

每个距离的 p 值计算为具有 k-1（k = 变量数）度的 Mahalanobis 距离的卡方统计量。

pchisq()函数用于计算累积卡方密度。

Syntax: pchisq(vec, df)

Parameters:

vec: Vector of x-values
df: Degree of Freedom

编程需要懂一点英语

示例：计算 p 值

R

# create new column for p-value 
df$pvalue <- pchisq(df$mahalnobis, df=3)
df

输出：

通常，小于 0.001 的 p 值被认为是异常值。在这种情况下，所有 p 值都大于 0.001