localdepth: Local depth
In localdepth: Local Depth

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/localdepth.R

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

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)

`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

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.

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

Claudio Agostinelli and Mario Romanazzi

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.

quantile.localdepth, plot.localdepth

# 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)