localdepth (version 0.5-7)

localdepth: Local depth

Description

The function evaluates the depth and the local depth for a set of points with respect to a dataset.

Usage

localdepth(x, y = NULL, tau, use = c("volume", "diameter"),  method = c("simplicial", "ellipsoid", "halfspace", "mahalanobis", "hyperspheresimplicial"),  type = c("exact", "approx"), nsamp = "all", nmax = 1,  tol=10^(-9), dimension=NULL, location=NULL, covariance=NULL)

Arguments

x

numeric; vector, dataframe or matrix. If x is a circular vector, a circular version is used. Avoid ties by wiggling the data. The function only issues a warning for ties

y

numeric; vector, dataframe or matrix with the same number of columns as x, or NULL. If NULL, x is used

tau

numeric; threshold value for the evaluation of the local depth. Use function quantile.localdepth to evaluate tau using a quantile of the size of the objects

use

character; the statistic used to measure the size of the objects. Currently, for method equal to "simplicial" or "ellipsoid" allowed statistics are "volume" and "diameter". For method equal to "mahalanobis" this parameter is not used and the only available statistic is pairwise Mahalanobis' distance

method

character; the type of (local) depth to be evaluated

type

character; how to evaluate membership. Only active for method="simplicial". See details.

nsamp

character or numeric; the number of objects that are considered. If "all", the size of all choose(NROW(x), NCOL(x)+1) objects is evaluated. Otherwise, a simple random sample with replacement of size nsamp is performed from the set of all possible objects.

nmax

numeric; maximum fraction (in the range (0,1]) of objects to be considered when nsamp is not equal to all. If nmax=1 the number of searched objects can reach the number of possible objects (choose(NROW(x), NCOL(x)+1) for simplicial and ellipsoid depth)

tol

numeric; tolerance parameter to be fixed depending on the machine precision. Used to decide membership of points located near to the boundary of the objects

dimension

numeric; only used with method="ellipsoid". It is the squared length of the ellipsoid semimajor axis. If dimension is NULL, it is set to NCOL(x)

location

NULL or a numeric vector; the NCOL(x) means vector used in method equal to "mahalanobis". If NULL, apply(x, 2, mean) is used

covariance

NULL or a numeric matrix; the NCOL(x)*NCOL(x) covariance matrix used in method equal to "mahalanobis". If NULL, cov(x) is used

Value

The function returns an object of class localdepth with the following components:

localdepth

vector of the local depth values for the given tau

depth

vector of the depth values

max.localdepth

max(localdepth)

max.depth

max(depth)

num

vector with two components. num[1] gives the number of objects used for the evaluation of the depth; num[2] is the number of objects used for the evaluation of the local depth

call

match.call() result. Note that this is called from the internal function

tau

value of the corresponding input parameter

use

value of the corresponding input parameter

tol

value of the corresponding input parameter

x

value of the corresponding input parameter

y

value of the corresponding input parameter

type

value of the corresponding input parameter

nsamp

value of the corresponding input parameter

method

value of the corresponding input parameter

Details

With method="simplicial" and type="exact", membership of the points in simplices is evaluated; when type="approx", an approximate membership function is used. See references below.

With method="hyperspheresimplicial" the data matrix are normalize to have vectors of length one, then the (local) angular simplicial depth is evaluated.

References

C. Agostinelli and M. Romanazzi (2007). Local depth of univariate distributions. Working paper n. 1/2007, Dipartimento di Statistica, Universita' Ca' Foscari, Venezia.

C. Agostinelli and M. Romanazzi (2008). Local depth of multidimensional data. Working paper n. 3/2008, Dipartimento di Statistica, Universita' Ca' Foscari, Venezia.

C. Agostinelli and M. Romanazzi (2011). Local depth. Journal of Statistical Planning and Inference, 141:817-830. doi:10.1016/j.jspi.2010.08.001.

C. Agostinelli e M. Romanazzi (2013). Nonparametric analysis of directional data based on data depth. Environmental and Ecological Statistics, 20(2):253-270. doi:10.1007/s10651-012-0218-z.

See Also

quantile.localdepth, plot.localdepth

Examples

# Iris Setosa dataset
data(iris)
setosa <- iris[iris$Species=="setosa",-5]
str(setosa)
n <- dim(setosa)[1];p <- dim(setosa)[2]
lab <- paste("U",1:n,sep="")
pairs(setosa)
# Wiggling data with gaussian error
set.seed(1234)
err <- matrix(c(rnorm(n,0,sd(setosa[,1])/1000),
  rnorm(n,0,sd(setosa[,2])/1000),
  rnorm(n,0,sd(setosa[,3])/1000),
  rnorm(n,0,sd(setosa[,4])/1000)),n,p,byrow=FALSE)
setosa1 <- setosa + as.data.frame(err)

## Not run: 
# qd <- quantile.localdepth(setosa1, probs=c(seq(0.01,0.09,0.01),
#   seq(0.1,0.9,0.1)), use="diameter", method="simplicial", nsamp="all", size=TRUE)
# qv <- quantile.localdepth(setosa1, probs=c(seq(0.01,0.09,0.01),
#   seq(0.1,0.9,0.1)), use="volume", method="simplicial", nsamp="all",size=TRUE)
# ## End(Not run)

## Here we explore 10% of simplices (0.1*choose(50,4+1))
qd <- quantile.localdepth(setosa1,
  probs=c(seq(0.01,0.09,0.01),seq(0.1,0.9,0.1)),
  use="diameter", method="simplicial", nsamp=211876, size=TRUE)
qv <- quantile.localdepth(setosa1,
  probs=c(seq(0.01,0.09,0.01),seq(0.1,0.9,0.1)),
  use="volume", method="simplicial", nsamp=211876,size=TRUE)

c(mean(qd$stats),median(qd$stats))
c(mean(qv$stats),median(qv$stats))

round(qd$quantile,2)
round(qv$quantile,8)

boxplot(qd$stats,names="Diameter",xlab="Simplex Size",
  horizontal=TRUE,main="Iris Setosa")
boxplot(qv$stats,names="Volume",xlab="Simplex Size",
  horizontal=TRUE,main="Iris Setosa")

plot(qd$stats,qv$stats,xlab="Simplex Diameter",
  ylab="Simplex Volume", main="Iris Setosa",pch=20)

# Simplicial global/local depth

sldv10 <- localdepth(setosa1,tau=qv$quantile[10],use="volume",
  method="simplicial", type="approx", nsamp="all", nmax=0.2)
str(sldv10)
sd10 <- sldv10$depth       # simplicial depth
sld10 <- sldv10$localdepth # simplicial local depth (volume, 10%)
lab[sd10 == (p+1)/n]        # convex hull vertices (25 points out of 50)
sdmax <- max(sd10);imax <- which.max(sd10);setosa[imax,] # deepest point

# DD-Plot

plot(sldv10, xlab="Simplicial Depth",
  ylab="Local Simplicial Depth", main="Iris Setosa",pch=20,ylim=c(0,1.02))
abline(a=0,b=1,lty="dashed")

## Not run: 
# # Mahalanobis Depth of Iris Versicolor wrt Iris Setosa
# 
# iversicolor <- iris[iris$Species == "versicolor",-5]
# str(iversicolor)
# set.seed(2345)
# err <- matrix(c(rnorm(n,0,sd(iversicolor[,1])/1000),
#   rnorm(n,0,sd(iversicolor[,2])/1000),
#   rnorm(n,0,sd(iversicolor[,3])/1000),
#   rnorm(n,0,sd(iversicolor[,4])/1000)), n,p,byrow=FALSE)
# iversicolor1 <- iversicolor + as.data.frame(err)
# qm <- quantile.localdepth(setosa1,
#   probs=c(seq(0.01,0.09,0.01),seq(0.1,0.9,0.1)),
#   method="mahalanobis",nsamp="all",size=TRUE)
# round(qm$quantile,2)
# vers <- localdepth(x=setosa1, y=iversicolor1, tau=qm$quantile[10],
#   method="mahalanobis", nsamp="all")
# str(vers)
# round(vers$depth,6)
# round(vers$localdepth,6) # identically zero
# ## End(Not run)