# How to calculate mahalanobis distance in R?

The Mahalanobis distance is the relative distance between two cases and the centroid, where centroid can be thought of as an overall mean for multivariate data. We can say that the centroid is the multivariate equivalent of mean. If the mahalanobis distance is zero that means both the cases are very same and positive value of mahalanobis distance represents that the distance between the two variables is large. In R, we can use mahalanobis function to find the malanobis distance.

## Example1

Live Demo

Consider the below data frame −

set.seed(981)
x1<−rnorm(20,5,1)
x2<−rnorm(20,5,0.84)
x3<−rnorm(20,10,1.5)
x4<−rnorm(20,10,3.87)
x5<−rnorm(20,1,0.0025)
df1<−data.frame(x1,x2,x3,x4,x5)
df1

## Output

      x1       x2       x3       x4       x5
1 4.016851 4.749189 10.166216 9.681625 1.0014171
2 5.208083 4.252389 8.886381 8.407824 0.9973355
3 4.000509 5.680469 10.452573 9.799825 0.9996433
4 4.968047 5.572099 12.813119 10.603569 0.9970847
5 5.253632 4.523665 8.961203 6.135956 0.9974229
6 4.556114 5.963955 7.784837 3.701523 0.9965163
7 4.987874 5.372996 10.104144 12.125932 1.0014389
8 6.164940 4.762497 9.826518 17.002388 0.9998966
9 5.497089 5.006558 11.701747 7.392629 1.0013103
10 4.649598 4.620766 11.955838 7.700963 1.0058710
11 4.947477 4.583403 9.431569 13.005483 0.9963742
12 7.074752 5.093332 9.743409 15.232665 1.0006305
13 4.042776 5.117288 9.603592 12.308203 1.0013562
14 5.364624 3.846084 11.919156 12.546169 1.0034000
15 6.079298 4.270361 10.527513 9.828845 0.9971954
16 4.410121 4.783754 8.844011 15.277243 1.0002428
17 4.213869 5.879465 9.651568 4.334237 1.0018883
18 4.142827 5.619082 9.544201 10.336943 0.9978379
19 3.012995 3.713027 11.487735 13.324214 1.0029497
20 5.481955 3.778913 9.074235 10.391055 0.9982697

Finding the mahalanobis distance for rows in df1 −

mahalanobis(df1,colMeans(df1),cov(df1))

## Output

[1] 1.192919 3.207677 2.531851 12.073066 3.664532 6.912468 1.766881
[8] 4.880830 3.652825 6.954114 3.152966 8.433015 2.310850 4.239761
[15] 4.013792 4.358375 5.665279 2.711948 9.063510 4.213342

## Example2

Live Demo

y1<−rpois(20,1)
y2<−rpois(20,3)
y3<−rpois(20,5)
y4<−rpois(20,8)
y5<−rpois(20,12)
y6<−rpois(20,10)
df2<−data.frame(y1,y2,y3,y4,y5,y6)
df2

## Output

y1 y2 y3 y4 y5 y6
1 0 2 4 6 11 10
2 1 6 7 4 9 9
3 1 1 6 13 14 11
4 3 3 9 9 16 9
5 2 3 6 10 9 13
6 0 6 7 13 14 13
7 2 2 7 4 15 7
8 0 2 4 8 14 10
9 2 7 3 7 6 12
10 0 2 6 10 10 9
11 0 5 5 10 8 6
12 2 3 5 7 11 9
13 0 5 3 6 9 7
14 0 2 6 3 13 7
15 1 1 7 10 9 9
16 0 3 3 8 12 11
17 0 3 4 5 13 13
18 1 2 6 14 13 8
19 1 2 4 10 8 7
20 1 5 11 13 12 16

mahalanobis(df2,colMeans(df2),cov(df2))

[1] 2.588021 6.383910 4.101547 8.860628 5.248206 8.669764 6.332766
[8] 3.065049 10.556830 2.882808 6.945220 2.333995 4.171714 5.990775
[15] 5.921976 3.198976 5.971216 5.382210 4.167775 11.226611

## Example3

Live Demo

z1<−runif(20,1,2)
z2<−runif(20,1,4)
z3<−runif(20,1,5)
z4<−runif(20,2,5)
z5<−runif(20,5,10)
df3<−data.frame(z1,z2,z3,z4,z5)
df3

## Output

      z1       z2       z3       z4       z5
1 1.388613 3.591918 4.950430 3.012227 7.646999
2 1.536406 2.346386 4.009326 3.344235 6.804723
3 1.307832 2.156929 1.548907 3.719957 9.647134
4 1.452674 3.659639 4.067904 2.821600 9.042116
5 1.821635 1.581077 1.848880 2.133112 8.606968
6 1.472712 1.853850 2.757099 4.971375 8.195671
7 1.129696 1.007614 3.454963 4.500837 9.512772
8 1.084507 3.509503 3.972340 2.557956 5.070359
9 1.066166 3.487398 3.235659 2.692450 8.566473
10 1.622298 3.285975 3.214168 2.816199 6.811145
11 1.215978 2.695426 4.459403 3.883969 7.015267
12 1.748907 1.855413 1.100227 3.676822 8.668907
13 1.785502 3.365582 1.089094 2.232694 6.207582
14 1.313907 1.010318 2.040431 3.337156 6.281897
15 1.211392 2.821926 3.427129 4.835524 8.469758
16 1.127482 1.589360 4.105524 4.575452 7.425941
17 1.914011 1.015687 1.900738 2.542681 8.710688
18 1.156077 1.237109 1.667345 4.654083 6.764100
19 1.770988 3.685755 4.417545 4.637382 6.155797
20 1.594745 3.750948 1.394754 4.548843 9.902893
mahalanobis(df3,colMeans(df3),cov(df3))
[1] 3.680650 2.011037 3.520353 4.338257 5.095421 2.698317 5.394089 7.190855
[9] 6.030547 1.608436 1.705612 2.770687 7.343208 4.676116 2.461363 3.186534
[17] 6.758622 6.152332 9.599646 8.777917