DepthProc (version 1.0.3)

depthLocal: Local depth

Description

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

Usage

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 $\beta \in [0,1]$ , let us introduce the smallest depth region bigger or equal to $\beta$ , $${R}^{\beta }(F)=\bigcap\limits_{\alpha \in A(\beta )}{{{D}_{\alpha }}}(F),$$ where $A(\beta )=\left{ \alpha \ge 0:P\left[ {{D}_{\alpha }}(F) \right]\ge \beta \right}$ . Then for a locality parameter $\beta$ we can take a neighbourhood of a point $x$ as $R_{x}^{\beta }(P)$ .

Formally, let $D(\cdot,P)$ be a depth function. Then the {local depth }with the locality parameter $\beta$ and w.r.t. a point $x$ is defined as $$L{{D}^{\beta }}(z,P):z\to D(z,P_{x}^{\beta }),$$ where $P_{x}^{\beta }(\cdot )=P\left( \cdot |R_{x}^{\beta }(P) \right)$ is cond. distr. of $P$ conditioned on $R_{x}^{\beta }(P)$ .

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

# 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)