depthf.hM2 {ddalpha} | R Documentation

## Bivariate h-Mode Depth for Functional Data Based on the L^2 Metric

### Description

The h-mode depth of functional bivariate data (that is, data of the form X:[a,b] \to R^2, or X:[a,b] \to R and the derivative of X) based on the L^2 metric of functions.

### Usage

    depthf.hM2(datafA, datafB, range = NULL, d = 101, q = 0.2)

### Arguments

`datafA` | Bivariate functions whose depth is computed, represented by a multivariate `dataf` object of their arguments (vector), and a matrix with two columns of the corresponding bivariate functional values. `m` stands for the number of functions.
`datafB` | Bivariate random sample functions with respect to which the depth of `datafA` is computed. `datafB` is represented by a multivariate `dataf` object of their arguments (vector), and a matrix with two columns of the corresponding bivariate functional values. `n` is the sample size. The grid of observation points for the functions `datafA` and `datafB` may not be the same.
`range` | The common range of the domain where the functions `datafA` and `datafB` are observed. Vector of length 2 with the left and the right end of the interval. Must contain all arguments given in `datafA` and `datafB`.
`d` | Grid size to which all the functional data are transformed. For depth computation, all functional observations are first transformed into vectors of their functional values of length `d` corresponding to equi-spaced points in the domain given by the interval `range`. Functional values in these points are reconstructed using linear interpolation, and extrapolation.
`q` | The quantile used to determine the value of the bandwidth h in the computation of the h-mode depth. h is taken as the `q`-quantile of all non-zero distances between the functions `B`. By default, this value is set to `q=0.2`, in accordance with the choice of Cuevas et al. (2007).

### Details

The function returns the vectors of sample h-mode depth values. The kernel used in the evaluation is the standard Gaussian kernel, the bandwidth value is chosen as a quantile of the non-zero distances between the random sample curves.

### Value

Three vectors of length `m` of h-mode depth values are returned:

- `hM` the unscaled h-mode depth,

- `hM_norm` the h-mode depth `hM` linearly transformed so that its range is [0,1],

- `hM_norm2` the h-mode depth `FD` linearly transformed by a transformation such that the range of the h-mode depth of `B` with respect to `B` is [0,1]. This depth may give negative values.

### Author(s)

Stanislav Nagy, nagy at karlin.mff.cuni.cz

### References

Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22 (3), 481–496.

### See Also

`depthf.hM`

### Examples

    datafA = dataf.population()$dataf[1:20]
    datafB = dataf.population()$dataf[21:50]

    datafA2 = derivatives.est(datafA,deriv=c(0,1))
    datafB2 = derivatives.est(datafB,deriv=c(0,1))

    depthf.hM2(datafA2,datafB2)

    depthf.hM2(datafA2,datafB2)$hM
    # depthf.hM2(cbind(A2[,,1],A2[,,2]),cbind(B2[,,1],B2[,,2]))$hM
    # the two expressions above should give the same result

[Package ddalpha version 1.3.1 Index]