depth.betaSkeleton {ddalpha} | R Documentation

## Calculate Beta-Skeleton Depth

### Description

Calculates the beta-skeleton depth of points w.r.t. a multivariate data set.

### Usage

    depth.betaSkeleton(x, data, beta = 2, distance = "Lp", Lp.p = 2,
                       mah.estimate = "moment", mah.parMcd = 0.75)

### Arguments

`x` | Matrix of objects (numerical vector as one object) whose depth is to be calculated; each row contains a d-variate point. Should have the same dimension as `data`.
`data` | Matrix of data where each row contains a d-variate point, w.r.t. which the depth is to be calculated.
`beta` | The paremeter defining the positionning of the balls' centers, see Yang and Modarres (2017) for details. By default (together with other arguments) equals `2`, which corresponds to the lens depth, see Liu and Modarres (2011).
`distance` | A character string defining the distance to be used for determining inclusion of a point into the lens (influence region), see Yang and Modarres (2017) for details. Possibilities are `"Lp"` for the Lp-metric (default) or `"Mahalanobis"` for the Mahalanobis distance adjustment.
`Lp.p` | A non-negative number defining the distance's power equal `2` by default (Euclidean distance); is used only when `distance = "Lp"`.
`mah.estimate` | A character string specifying which estimates to use when calculating sample covariance matrix; can be `"none"`, `"moment"` or `"MCD"`, determining whether traditional moment or Minimum Covariance Determinant (MCD) (see `covMcd`) estimates for mean and covariance are used. By default `"moment"` is used. Is used only when `distance = "Mahalanobis"`.
`mah.parMcd` | The value of the argument `alpha` for the function `covMcd`; is used when `distance = "Mahalanobis"` and `mah.estimate = "MCD"`.

### Details

Calculates the beta-skeleton depth, see Yang and Modarres (2017). Its particular case, lens depth, see Liu and Modarres (2011), is obtained when `beta = 2`, `distance = "Lp"` and `Lp.p = 2` (default settings). For tne example of the lens depth, the depth of an observation `x` is calculated as the portion of lens containing `x`, with lens being an intersection of two closed balls centered at two sample's points each having radius equal to the distance between these two points.

### Value

Numerical vector of depths, one for each row in `x`; or one depth value if `x` is a numerical vector.

### References

Liu, Z. and Modarres, R. (2011). Lens data depth and median. Journal of Nonparametric Statistics 23(4) 1063–1074.

Yang, M. and Modarres, R. (2017). β-skeleton depth functions and medians. Commmunications in Statistics - Theory and Methods to appear.

### See Also

`depth.halfspace` for calculation of the Tukey depth.

`depth.Mahalanobis` for calculation of Mahalanobis depth.

`depth.projection` for calculation of projection depth.

`depth.simplicial` for calculation of simplicial depth.

`depth.simplicialVolume` for calculation of simplicial volume depth.

`depth.spatial` for calculation of spatial depth.

`depth.zonoid` for calculation of zonoid depth.

`depth.potential` for calculation of data potential.

### Examples

    # 5-dimensional normal distribution
    data <- mvrnorm(1000, rep(0, 5),
                    matrix(c(1, 0, 0, 0, 0,
                             0, 2, 0, 0, 0,
                             0, 0, 3, 0, 0,
                             0, 0, 0, 2, 0,
                             0, 0, 0, 0, 1),
                    nrow = 5))
    x <- mvrnorm(10, rep(1, 5),
                 matrix(c(1, 0, 0, 0, 0,
                          0, 1, 0, 0, 0,
                          0, 0, 1, 0, 0,
                          0, 0, 0, 1, 0,
                          0, 0, 0, 0, 1),
                 nrow = 5))

    depths <- depth.betaSkeleton(x, data)
    cat("Depths:", depths, "\n")

[Package ddalpha version 1.3.9 Index]