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

## Description

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

## Usage

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

## Arguments

 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

## Details

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) .

## References

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).

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ## 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) 

DepthProc documentation built on May 31, 2017, 4:49 a.m.