depth.simplicialVolume {ddalpha} | R Documentation

## Calculate Simplicial Volume Depth

### Description

Calculates the simpicial volume depth of points w.r.t. a multivariate data set.

### Usage

    depth.simplicialVolume(x, data, exact = F, k = 0.05, seed = 0)

### 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.
`exact` | `exact=F` (by default) implies the approximative algorithm, considering `k` simplices, `exact=T` implies the exact algorithm.
`k` | Number (k>1) or portion (if 0<k<1) of simplices that are considered if `exact=F`. If k>1, then the algorithmic complexity is polynomial in d but is independent of the number of observations in `data`, given k. If 0<k<1, then the algorithmic complexity is exponential in the number of observations in `data`, but the calculation precision stays approximately the same.
`seed` | The random seed. The default value `seed=0` makes no changes.

### Details

Calculates Oja depth (also: Simplicial volume depth). At first the Oja outlyingness function `O(x,data)` is calculated as the average of the volumes of simplices built on d data points and the measurement point `x` (Oja, 1983).

Zuo and Serfling (2000) proposed Oja depth based on the Oja outlyingness function as `1/(1 + O(x,data)/S)`, where S is a square root of the determinant of `cov(data)`, which makes the depth function affine-invariant.

### Value

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

### References

Oja, H. (1983). Descriptive statistics for multivariate distributions. Statistics & Probability Letters 1 327–332.

Zuo, Y.J. and Serfling, R. (2000). General notions of statistical depth function. The Annals of Statistics 28 461–482.

### 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.spatial` for calculation of spatial depth.

`depth.zonoid` for calculation of zonoid depth.

`depth.potential` for calculation of data potential.

### Examples

    # 3-dimensional normal distribution
    data <- mvrnorm(20, rep(0, 3),
                    matrix(c(1, 0, 0,
                             0, 2, 0,
                             0, 0, 1),
                           nrow = 3))
    x <- mvrnorm(10, rep(1, 3),
                 matrix(c(1, 0, 0,
                          0, 1, 0,
                          0, 0, 1),
                        nrow = 3))

    #exact
    depths <- depth.simplicialVolume(x, data, exact = TRUE)
    cat("Depths: ", depths, "\n")

    #approximative
    depths <- depth.simplicialVolume(x, data, exact = FALSE, k = 0.2)
    cat("Depths: ", depths, "\n")

[Package ddalpha version 1.3.1 Index]