Weighted by \(L^p\) depth (outlyingness) multivariate location and scatter estimators.
CovLP(x, pdim = 2, la = 1, lb = 1)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 \(L^pdim\) depth
parameter of a simple weight function w=a*x+b
parameter of a simple weight function w=a*x+b
loc: Robust Estimate of Location:
cov: Robust Estimate of Covariance:
Returns depth weighted covariance matrix.
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, \( {{w}_{1}}(x)={{w}_{2}}(x)=x \) .
depthContour and depthPersp for depth graphics.
x = mvrnorm(n = 100, mu = c(0,0), Sigma = 3*diag(2))
cov_x = CovLP(x, 2, 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)
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