# MBD: Modified Band Depth for univariate functional data In roahd: Robust Analysis of High Dimensional Data

## Description

This function computes the Modified Band Depth (MBD) of elements of a functional dataset.

## Usage

 1 2 3 4 5 6 7 MBD(Data, manage_ties = FALSE) ## S3 method for class 'fData' MBD(Data, manage_ties = FALSE) ## Default S3 method: MBD(Data, manage_ties = FALSE) 

## Arguments

 Data either a 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. manage_ties a logical flag specifying whether a check for ties and relative treatment must be carried out or not (default is FALSE).

## 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 sample MBD of each element with respect to the other elements of the dataset, i.e.:

MBD( X( t ) ) = {N \choose 2 }^{-1} ∑_{1 ≤q i_1 < i_2 ≤q N} \tilde{λ}\big( {t : \min( X_{i_1}(t), X_{i_2}(t) ) ≤q X(t) ≤q \max( X_{i_1}(t), X_{i_2}(t) ) } \big),

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

See the References section for more details.

## Value

The function returns a vector containing the values of MBD for the given dataset.

## References

Lopez-Pintado, S. and Romo, J. (2009). On the Concept of Depth for Functional Data, Journal of the American Statistical Association, 104, 718-734.

Lopez-Pintado, S. and Romo. J. (2007). Depth-based inference for functional data, Computational Statistics & Data Analysis 51, 4957-4968.

BD, MBD_relative, BD_relative, fData
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 grid = seq( 0, 1, length.out = 1e2 ) D = matrix( c( 1 + sin( 2 * pi * grid ), 0 + sin( 4 * pi * grid ), 1 - sin( pi * ( grid - 0.2 ) ), 0.1 + cos( 2 * pi * grid ), 0.5 + sin( 3 * pi + grid ), -2 + sin( pi * grid ) ), nrow = 6, ncol = length( grid ), byrow = TRUE ) fD = fData( grid, D ) MBD( fD ) MBD( D )