Mahalanobis distance with multiple observations per group

Question

I'd like to calculate the Mahalanobis distance among groups of species where:

• i) there are more than two groups (more than two species).
• ii) there are multiple variables (features of such species) to be taken into account.
• iii) there are multiple observations per group (in the dataframe, it means there is more than one row per specie).

I am trying to understand how to run the mahalanobis function in R, under such conditions. This question is similar to:

Mahalanobis distance on R for more than 2 groups

but there, only one variable was used. How could it be done having more than one variable?

Below there is an example, which I believe reproduces my actual data.

Sp. X1  X2  X3
A   0.7 11  215
B   0.8 7   214
B   0.8 6.5 187
C   0.3 4   456
D   0.4 3   111
A   0.1 7   205
A   0.2 7   196
C   0.1 9.3 77
D   0.6 8   135
D   0.8 4   167
B   0.4 6   228
C   0.1 5   214
A   0.4 7   156
C   0.5 2   344

_{Sp. = Specie; X1, X2 and X3 are observed variables.}

In the real dataset, there are more than 50 species and the number of observations varies among them (from 100 rows/specie to 1000).

Answer 1

This is what I've got, using the pairwise.mahalanobis function from the HDMD package:

#data
a = structure(list(Sp = structure(c(1L, 2L, 2L, 3L, 4L, 1L, 1L, 3L,4L, 4L, 2L, 3L, 1L, 3L), .Label = c("A", "B", "C", "D"), class = "factor"), 
                   X1 = c(0.7, 0.8, 0.8, 0.3, 0.4, 0.1, 0.2, 0.1, 0.6, 0.8,0.4, 0.1, 0.4, 0.5), 
                   X2 = c(11, 7, 6.5, 4, 3, 7, 7, 9.3,8, 4, 6, 5, 7, 2), 
                   X3 = c(215L, 214L, 187L, 456L, 111L, 205L,196L, 77L, 135L, 167L, 228L, 214L, 156L, 344L)),
              .Names = c("Sp","X1", "X2", "X3"), 
              row.names = c(NA, -14L),
              class = "data.frame")

library(HDMD) #pairwise.mahalanobis function
library(cluster) #agnes function

group = matrix(a$Sp) #what is being compared
group = t(group[,1]) #prepare for pairwise.mahalanobis function

variables = c("X1","X2","X3") #variables (what is being used for comparison)
variables = as.matrix(a[,variables]) #prepare for pairwise.mahalanobis function

mahala_sq = pairwise.mahalanobis(x=variables, grouping=group) #get squared mahalanobis distances (see mahala_sq$distance).
names = rownames(mahala_sq$means) #capture labels

mahala = sqrt(mahala_sq$distance) #mahalanobis distance
rownames(mahala) = names #set rownames in the dissimilarity matrix
colnames(mahala) = names #set colnames in the dissimilarity matrix

mahala #this is the mahalanobis dissimilarity matrix 

         A        B         C         D
A  0.00000 17.78689  86.83294  62.65437
B 17.78689  0.00000  69.07937  80.31577
C 86.83294 69.07937   0.00000 149.36579
D 62.65437 80.31577 149.36579   0.00000

#This is how I used the dissimilarity matrix to find clusters.
cluster = agnes(mahala,diss=TRUE,keep.diss=FALSE,method="complete") #hierarchical clustering
plot(cluster,which.plots=2) #plot dendrogram