## Band Depth for univariate functional data

### Description

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

### Usage

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

### Arguments

 Data either an object of class fData 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), this function computes the sample BD of each element with respect to the other elements of the dataset, i.e.:

BD( X( t ) ) = {N \choose 2 }^{-1} ∑_{1 ≤q i_1 < i_2 ≤q N} I( G(X) \subset B( X_{i_1}, X_{i_2} ) ),

where G(X) is the graphic of X(t), B(X_{i_1},X_{i_2}) is the envelope of X_{i_1}(t) and X_{i_2}(t), and X \in ≤ft\{X_1, …, X_N\right\}.

See the References section for more details.

### Value

The function returns a vector containing the values of BD 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.

MBD, BD_relative, MBD_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 ) BD( fD ) BD( D )