# Modified Hypograph Index of univariate functional dataset

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### Description

This function computes the Modified Hypograph Index (MEI) of elements of a univariate functional dataste.

### Usage

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

MHI( X(t) ) = \frac{1}{N} ∑_{i=1}^N \tilde{λ}( X(t) ≥q X_i(t) ),

where \tilde{λ}(\cdot) is the normalised Lebesgue measure over I=[a,b], that is \tilde{λ(A)} = λ( A ) / ( b - a ).

### Value

The function returns a vector containing the values of MHI 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.

HI, MEI, EI, 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 ) MHI( fD ) MHI( Data )