# HI: Hypograph Index of univariate functional dataset In roahd: Robust Analysis of High Dimensional Data

## Description

This function computes the Hypograph Index (HI) of elements of a univariate functional dataset.

## Usage

 1 2 3 4 5 6 7 HI(Data) ## S3 method for class 'fData' HI(Data) ## Default S3 method: HI(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 HI, i.e.:

HI( X(t) ) = \frac{1}{N} ∑_{i=1}^N I( G( X_i(t) ) \subset hyp( X(t) ) ) = \frac{1}{N} ∑_{i=1}^N I( X_i(t) ≤q X(t), \ \ \forall t \in I),

where G(X_i(t)) indicates the graph of X_i(t), hyp( X(t)) indicates the hypograph of X_i(t).

## Value

The function returns a vector containing the values of HI 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.

MHI, EI, MEI, fData
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 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 ) HI( fD ) HI( Data )