R: Univariate Integrated and Infimal Depth for Functional Data

depthf.fd1 {ddalpha} | R Documentation

## Univariate Integrated and Infimal Depth for Functional Data

### Description

Usual, and order extended integrated and infimal depths for real-valued functional data based on the halfspace and simplicial depth.

### Usage

    depthf.fd1(datafA, datafB, range = NULL, d = 101, order = 1, approx = 0)

### Arguments

`datafA` | Functions whose depth is computed, represented by a `dataf` object of their arguments and functional values. `m` stands for the number of functions.
`datafB` | Random sample functions with respect to which the depth of `datafA` is computed. `datafB` is represented by a `dataf` object of their arguments and 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, see Nagy et al. (2016).
`order` | The order of the order extended integrated and infimal depths. By default, this is set to `1`, meaning that the usual univariate depths of the functional values are computed. For `order=2` or `3`, the second and the third order extended integrated and infimal depths are computed, respectively.
`approx` | Number of approximations used in the computation of the order extended depth for `order` greater than `1`. For `order=2`, the default value is set to `0`, meaning that the depth is computed at all possible `d^order` combinations of the points in the domain. For `order=3`, the default value is set to `101`. When `approx` is a positive integer, `approx` points are randomly sampled in `[0,1]^order` and at these points the `order`-variate depths of the corresponding functional values are computed.

### Details

The function returns vectors of sample integrated and infimal depth values.

### Value

Four vectors of length `m` of depth values are returned:

- `Simpl_FD` the integrated depth based on the simplicial depth,

- `Half_FD` the integrated depth based on the halfspace depth,

- `Simpl_ID` the infimal depth based on the simplicial depth,

- `Half_ID` the infimal depth based on the halfspace depth.

In addition, two vectors of length `m` of the relative area of smallest depth values is returned:

- `Simpl_IA` the proportions of points at which the depth `Simpl_ID` was attained,

- `Half_IA` the proportions of points at which the depth `Half_ID` was attained.

The values `Simpl_IA` and `Half_IA` are always in the interval [0,1]. They introduce ranking also among functions having the same infimal depth value - if two functions have the same infimal depth, the one with larger infimal area `IA` is said to be less central. For `order=2` and `m=1`, two additional matrices of pointwise depths are also returned:

- `PSD` the matrix of size `d*d` containing the computed pointwise bivariate simplicial depths used for the computation of `Simpl_FD` and `Simpl_ID`,

- `PHD` the matrix of size `d*d` containing the computed pointwise bivariate halfspace depths used for the computation of `Half_FD` and `Half_ID`.

For `order=3`, only `Half_FD` and `Half_ID` are provided.

### Author(s)

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

### References

Nagy, S., Gijbels, I. and Hlubinka, D. (2016). Weak convergence of discretely observed functional data with applications. Journal of Multivariate Analysis, 146, 46–62.

Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. Journal of Computational and Graphical Statistics. To appear.

### See Also

`depthf.fd2`, `infimalRank`

### Examples

    datafA = dataf.population()$dataf[1:20]
    datafB = dataf.population()$dataf[21:50]
    depthf.fd1(datafA,datafB)
    depthf.fd1(datafA,datafB,order=2)
    depthf.fd1(datafA,datafB,order=3,approx=51)

[Package ddalpha version 1.3.4 Index]