depthLocal: Local depth
In DepthProc: Statistical Depth Functions for Multivariate Analysis

Description Usage Arguments Details References Examples

Computes local version of depth according to proposals of Paindaveine and Van Bever - see referencess.

1 2	depthLocal(u, X, beta = 0.5, depth1 = "Projection", depth2 = depth1, name = "X", ...)

`u`	Numerical vector or matrix whose depth is to be calculated. Dimension has to be the same as that of the observations.
`X`	The data as a matrix, data frame. If it is a matrix or data frame, then each row is viewed as one multivariate observation.
`beta`	cutoff value for neighbourhood
`depth1`	depth method for symmetrised data
`depth2`	depth method for calculation depth of given point
`name`	name for this data set - it will be used on plots.
`...`	additional parameters passed to depth1 and depth2

A successful concept of local depth was proposed by Paidaveine and Van Bever (2012) . For defining a neighbourhood of a point authors proposed using idea of symmetrisation of a distribution (a sample) with respect to a point in which depth is calculated. In their approach instead of a distribution {P}^{X} , a distribution {{P}_{x}}=1/2{{P}^{X}}+1/2{{P}^{2x-X}} is used. For any β \in [0,1] , let us introduce the smallest depth region bigger or equal to β ,

{R}^{β }(F)=\bigcap\limits_{α \in A(β )}{{{D}_{α }}}(F),

where A(β )=≤ft\{ α ≥ 0:P≤ft[ {{D}_{α }}(F) \right]≥ β \right\} . Then for a locality parameter β we can take a neighbourhood of a point x as R_{x}^{β }(P) .

Formally, let D(\cdot,P) be a depth function. Then the local depth with the locality parameter β and w.r.t. a point x is defined as

L{{D}^{β }}(z,P):z\to D(z,P_{x}^{β }),

where P_{x}^{β }(\cdot )=P≤ft( \cdot |R_{x}^{β }(P) \right) is cond. distr. of P conditioned on R_{x}^{β }(P) .

Paindaveine, D., Van Bever, G. (2013) From depth to local depth : a focus on centrality. Journal of the American Statistical Association 105, 1105-1119 (2013).

## Not run: 
# EXAMPLE 1
data = mvrnorm(100, c(0,5), diag(2)*5)
#by default depth2 = depth1
depthLocal(data, data, depth1 = "LP")
depthLocal(data, data, depth1 = "LP", depth2 = "Projection")
## Depthcontour
depthContour(data, method = "Local", depth1 = "LP")
# EXAMPLE 2
data(inf.mort,maesles.imm)
data1990=na.omit(cbind(inf.mort[,1],maesles.imm[,1]))
depthContour(data1990, method = "Local", depth1 = "LP",beta=0.3)

#EXAMPLE 3
Sigma1 = matrix(c(10,3,3,2),2,2)
X1 = mvrnorm(n= 8500, mu= c(0,0),Sigma1)
Sigma2 = matrix(c(10,0,0,2),2,2)
X2 = mvrnorm(n= 1500, mu= c(-10,6),Sigma2)
BALLOT=rbind(X1,X2)

train <- sample(1:10000, 100)
data<-BALLOT[train,]
depthContour(data, method = "Local", depth1 = "Projection",beta=0.3)

## End(Not run)