# HRD: Half-Region Depth for univariate functional data In roahd: Robust Analysis of High Dimensional Data

## Description

This function computes the Half-Region Depth (HRD) of elements of a univariate functional dataset.

## Usage

 ```1 2 3 4 5 6 7``` ```HRD(Data) ## S3 method for class 'fData' HRD(Data) ## Default S3 method: HRD(Data) ```

## Arguments

 `Data` either an `fData` object or a matrix-like dataset of functional data (e.g. `fData\$values`), with observations as rows and measurements over grid points as columns.

## Details

Given a univariate functional dataset, X_1(t), X_2(t), …, X_N(t), defined over a compact interval I=[a,b], this function computes the HRD of its elements, i.e.:

HRD(X(t)) = \min( EI( X(t) ), HI(X(t)) ),

where EI(X(t)) indicates the Epigraph Index (EI) of X(t) with respect to the dataset, and HI(X(t)) indicates the Hypograph Index of X(t) with respect to the dataset.

## Value

The function returns a vector containing the values of HRD for each element of the functional dataset provided in `Data`.

## References

Lopez-Pintado, S. and Romo, J. (2012). A half-region depth for functional data, Computational Statistics and Data Analysis, 55, 1679-1695.

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

`MHRD`, `EI`, `HI`, `fData`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```N = 20 P = 1e2 grid = seq( 0, 1, length.out = P ) C = exp_cov_function( grid, alpha = 0.2, beta = 0.3 ) Data = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), C ) fD = fData( grid, Data ) HRD( fD ) HRD( Data ) ```