# depthLocal: Local depth In zzawadz/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, depth_params1 = list(method = "Projection"), depth_params2 = depth_params1) 

## 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 depth_params1 list of parameters for function depth (method, threads, ndir, la, lb, pdim, mean, cov, exact). depth_params2 as above — default is depth_params1.

## Details

A successful concept of local depth was proposed by Paindaveine 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}} = \frac{ 1 }{ 2 }{{P} ^ {X}} + \frac{ 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.

## 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 26 27 28 29 30 31 32 33 34 35 36 37 ## Not run: # EXAMPLE 1 data <- mvrnorm(100, c(0, 5), diag(2) * 5) # By default depth_params2 = depth_params1 depthLocal(data, data, depth_params1 = list(method = "LP")) depthLocal(data, data, depth_params1 = list(method = "LP"), depth_params2 = list(method = "Projection")) # Depth contour depthContour(data, depth_params = list(method = "Local", depth_params1 = list(method = "LP"))) # EXAMPLE 2 data(inf.mort, maesles.imm) data1990 <- na.omit(cbind(inf.mort[, 1], maesles.imm[, 1])) depthContour(data1990, depth_params = list( method = "Local", depth_params1 = list(method = "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, depth_params = list( method = "Local", beta = 0.3, depth_params1 = list(method = "Projection") )) ## End(Not run) 

zzawadz/DepthProc documentation built on Sept. 27, 2018, 9:11 a.m.