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

CovLP(x, pdim = 2, la = 1, lb = 1)

## Arguments

x 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 $${L} ^ {p} dim$$ depth parameter of a simple weight function $$w = ax + b$$ parameter of a simple weight function $$w = ax + b$$

## 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, $${{w}_{1}}(x) = {{w}_{2}}(x) = x$$.

depthContour and depthPersp for depth graphics.

## Examples

# EXAMPLE 1
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)#>
#> Call:
#> CovLP(x = data1990)
#> -> Method:  Depth Weighted Estimator
#>
#> Robust Estimate of Location:
#>   49.7  41.9  83.0
#>
#> Robust Estimate of Covariance:
#>       [,1]    [,2]    [,3]
#> [1,]  3010.1  1771.8  -499.6
#> [2,]  1771.8  1079.8  -295.4
#> [3,]  -499.6  -295.4   252.7