CovLP: CovLp

Description

Weighted by $L^p$ depth (outlyingness) multivariate location and scatter estimators.

Usage

CovLP(x, p = 1, a = 1, b = 1)

Arguments

The data as a matrix or data frame. If it is a matrix or data frame, then each row is viewed as one multivariate observation.

The parameter of the weighted $L^p$ depth

parameter of a simple weight function w=a*x+b

Value

loc: Robust Estimate of Location:
cov: Robust Estimate of Covariance:
Returns depth weighted covariance matrix.

Details

Using depth function one can define a depth-weighted location and scatter estimators. In case of location estimator we have $$L(F)={\int{{x}{{w}_{1}}(D({x},F))dF({x})}}/{{{w}_{1}}(D({x},F))dF({x})},$$ Subsequently, a depth-weighted scatter estimator is defined as $$S(F)=\frac{\int{({x}-L(F)){{({x}-L(F))}^{T}}{{w}_{2}}(D({x},F))dF({x})}}{\int{{{w}_{2}}(D({x},F))dF({x})}},$$ where ${{w}_{2}}(\cdot )$ is a suitable weight function that can be different from ${{w}_{1}}(\cdot )$ .

The DepthProc package offers these estimators for weighted ${L}^{p}$ depth. Note that $L(\cdot )$ and $S(\cdot )$ include multivariate versions of trimmed means and covariance matrices. Their sample counterparts take the form $${{T}_{WD}}({{{X}}^{n}})={\sum\limits_{i=1}^{n}{{{d}_{i}}{{X}_{i}}}}/{\sum\limits_{i=1}^{n}{{{d}_{i}}}} ,$$ $$DIS({{{X}}^{n}})=\frac{\sum\limits_{i=1}^{n}{{{d}_{i}}\left( {{{X}}_{i}}-{{T}_{WD}}({{{X}}^{n}}) \right){{\left( {{{X}}_{i}}-{{T}_{WD}}({{{X}}^{n}}) \right)}^{T}}}}{\sum\limits_{i=1}^{n}{{{d}_{i}}}},$$ where ${{d}_{i}}$ are sample depth weights, ${{w}_{1}}(x)={{w}_{2}}(x)=x$ .

Examples

Run this code

x = mvrnorm(n = 100, mu = c(0,0), Sigma = 3*diag(2))
 cov_x = CovLP(x,1,1,1)

 # EXAMPLE 2
 data(under5.mort,inf.mort,maesles.imm)
 data1990 = na.omit(cbind(under5.mort[,1],inf.mort[,1],maesles.imm[,1]))
 CovLP(data1990)