# How to compare convex hulls.

A convex hull of a set of points is the smallest space of area that enclosed all of these points. If the points are populations or species in a multivariate morphological space, the convex hull is a meassure of disparity, i.e. how much is occupied the morphological space.
In Maubecin et al. 2016, we tested whether floral morphological disparity was significantly larger in recently colonized than in refugium populations of *Calceolaria polyrhiza*. Here I provide the routine to repeat the test.

```
# The function dispar.hull estimates the convex hull of two sets of points, using principal
# components to reduce dimensionality.
# ID is a factor with two levels (identifiers).
# data is a data frame or matrix with at least two numeric variables.
# k is the number of principal compontents to be retained. By defoult k = 2, thus the function
# estimates the convex hull areas. If k equals the number of variables in data, all the
# information in the data is used to estimate the convex hull volumes.
# obs when TRUE (default) the function estimates the observed convex hulls. When FALSE the
# function randomly reshuffled the identifiers.
require(geometry)
dispar.hull <- function(ID, data, k = 2, obs = TRUE){
if(obs == TRUE) ID <- ID else ID <- sample(ID)
id <- unique(ID)
PC <- prcomp(data)$x[,1:k]
D1 <- subset(PC, ID == id[1])
ch1 <- convhulln(D1)
PA1 <- polyarea(D1[ch1[,1],1], D1[ch1[,2],2]); PA1
D2 <- subset(PC, ID == id[2])
ch2 <- convhulln(D2)
PA2 <- polyarea(D2[ch2[,1],1], D2[ch2[,2],2])
res <- c(PA1, PA2)
return(res)
}
# Example
# Simulate some data
set.seed(123)
A <- matrix(rnorm(250, 0, 1), 50, 5)
B <- matrix(rnorm(250, 0.2, 0.35), 50, 5)
M <- as.data.frame(rbind(A, B))
M$type <- c(rep("a", 50), rep("b", 50))
# There is a huge difference in convex hull areas
library(vegan)
PC <- prcomp(M[, 1:5])
plot(PC$x[1:50, 1], PC$x[1:50,2], xlab = "PC1", ylab = "PC2")
points(PC$x[51:100, 1], PC$x[51:100,2], pch = 19)
ordihull(PC, groups = M$type)
# Area of the convex hulls
hull.obs <- dispar.hull(ID = M$type, data=M[, 1:5], obs = TRUE, k = 2)
hull.obs
# Randomization and significance of the difference between convex hulls
pseu.hull <- t(replicate(1000, dispar.hull(ID = M$type, data=M[, 1:5], obs = FALSE, k = 2)))
D <- pseu.hull[, 1] - pseu.hull[, 2]
hist(D); abline(v = hull.obs[1]- hull.obs[2])
P <- length(D[D > hull.obs[1]-hull.obs[2]])/1000
P
```

Written on July 18, 2016