马氏距离是两个个案与质心之间的相对距离，质心可以看作是多变量数据的整体平均值。可以说质心是均值的多元等效项。如果马哈拉诺比斯距离为零，则意味着这两种情况都是相同的，并且马哈拉诺比斯距离的正值表示两个变量之间的距离较大。在R中，我们可以使用mahalanobis函数查找malanobis距离。
请看以下数据帧-
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
输出结果
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
在df1中找到行的马哈拉诺比斯距离-
mahalanobis(df1,colMeans(df1),cov(df1))
输出结果
[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
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
输出结果
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
马哈拉诺比斯（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
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
输出结果
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