# multiMBD: (Modified) Band Depth for multivariate functional data In roahd: Robust Analysis of High Dimensional Data

## Description

These functions compute the Band Depth (BD) and Modified Band Depth (MBD) of elements of a multivariate functional dataset.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 multiMBD(Data, weights = "uniform", manage_ties = FALSE) ## S3 method for class 'mfData' multiMBD(Data, weights = "uniform", manage_ties = FALSE) ## Default S3 method: multiMBD(Data, weights = "uniform", manage_ties = FALSE) multiBD(Data, weights = "uniform") ## S3 method for class 'mfData' multiBD(Data, weights = "uniform") ## Default S3 method: multiBD(Data, weights = "uniform") 

## Arguments

 Data specifies the the multivariate functional dataset. It is either an object of class mfData or a list of 2-dimensional matrices having as rows the elements of that component and as columns the measurements of the functional data over the grid. weights either a set of weights (of the same length of Data ) or the string "uniform" specifying that a set of uniform weights (of value 1 / L, where L is the number of dimensions of the functional dataset and thus the length of Data) is to be used. manage_ties a logical flag specifying whether the check for ties and the relative treatment is to be carried out while computing the MBDs in each dimension. It is directly passed to MBD.

## Details

Given a multivariate functional dataset composed of N elements with L components each, \mathbf{X_1} =( X^1_1(t), X^2_1(t), …, X^L_1(t)), and a set of L non-negative weights,

w_1, w_2, …, w_L, \qquad ∑_{i=1}^L w_i = 1,

these functions compute the BD and MBD of each element of the functional dataset, namely:

BD( \mathbf{X_j} ) = ∑_{i=1}^{L} w_i BD( X^i_j ), \quad \forall j = 1, … N.

MBD( \mathbf{X_j} ) = ∑_{i=1}^{L} w_i MBD( X^i_j ), \quad \forall j = 1, … N.

## Value

The function returns a vector containing the depths of each element of the multivariate functional dataset.

## References

Ieva, F. and Paganoni, A. M. (2013). Depth measures for multivariate functional data, Communications in Statistics: Theory and Methods, 41, 1265-1276.

Tarabelloni, N., Ieva, F., Biasi, R. and Paganoni, A. M. (2015). Use of Depth Measure for Multivariate Functional Data in Disease Prediction: An Application to Electrocardiograph Signals, International Journal of Biostatistics, 11.2, 189-201.

MBD, BD, toListOfValues, mfData
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 N = 20 P = 1e3 grid = seq( 0, 10, length.out = P ) # Generating an exponential covariance function to be used to simulate gaussian # functional data Cov = exp_cov_function( grid, alpha = 0.2, beta = 0.8 ) # First component of the multivariate guassian functional dataset Data_1 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov ) # First component of the multivariate guassian functional dataset Data_2 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov ) mfD = mfData( grid, list( Data_1, Data_2 ) ) multiBD( mfD, weights = 'uniform' ) multiMBD( mfD, weights = 'uniform', manage_ties = TRUE ) multiBD( mfD, weights = c( 1/3, 2/3 )) multiMBD( mfD, weights = c( 1/3, 2/3 ), manage_ties = FALSE ) multiBD( list( Data_1, Data_2 ), weights = 'uniform') multiMBD( list( Data_1, Data_2 ), weights = 'uniform', manage_ties = TRUE ) multiBD( list( Data_1, Data_2 ), weights = c( 1/3, 2/3 )) multiMBD( list( Data_1, Data_2 ), weights = c( 1/3, 2/3 ), manage_ties = FALSE )