(Modified) Band Depth for multivariate functional data

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

multiMBD(Data, weights = "uniform", manage_ties = FALSE)

# S3 method for mfData
multiMBD(Data, weights = "uniform", manage_ties = FALSE)

# S3 method for default
multiMBD(Data, weights = "uniform", manage_ties = FALSE)

multiBD(Data, weights = "uniform")

# S3 method for mfData
multiBD(Data, weights = "uniform")

# S3 method for default
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.

Value

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

Details

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

$$ w_1, w_2, \ldots, w_L, \qquad \sum_{i=1}^L w_i = 1,$$

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

$$ BD( \mathbf{X_j} ) = \sum_{i=1}^{L} w_i BD( X^i_j ), \quad \forall j = 1, \ldots N.$$

$$ MBD( \mathbf{X_j} ) = \sum_{i=1}^{L} w_i MBD( X^i_j ), \quad \forall j = 1, \ldots N.$$

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.

Examples


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' )
#>  [1] 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
#> [20] 0.1
multiMBD( mfD, weights = 'uniform', manage_ties = TRUE )
#>  [1] 0.3968211 0.3919579 0.3666263 0.4175474 0.3614158 0.3642842 0.4203421
#>  [8] 0.4199632 0.3970421 0.3997579 0.3948158 0.3854737 0.3524158 0.4073263
#> [15] 0.4184368 0.4222526 0.4080526 0.4482632 0.3568474 0.4703579

multiBD( mfD, weights = c( 1/3, 2/3 ))
#>  [1] 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
#> [20] 0.1
multiMBD( mfD, weights = c( 1/3, 2/3 ), manage_ties = FALSE )
#>  [1] 0.4068877 0.3874211 0.3753895 0.4290877 0.3473579 0.3716491 0.4218702
#>  [8] 0.4211579 0.3774912 0.3864491 0.3976737 0.3787965 0.3600561 0.4094737
#> [15] 0.4181860 0.4212035 0.4094737 0.4304035 0.3719719 0.4780000

multiBD( list( Data_1, Data_2 ), weights = 'uniform')
#>  [1] 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
#> [20] 0.1
multiMBD( list( Data_1, Data_2 ), weights = 'uniform', manage_ties = TRUE )
#>  [1] 0.3968211 0.3919579 0.3666263 0.4175474 0.3614158 0.3642842 0.4203421
#>  [8] 0.4199632 0.3970421 0.3997579 0.3948158 0.3854737 0.3524158 0.4073263
#> [15] 0.4184368 0.4222526 0.4080526 0.4482632 0.3568474 0.4703579

multiBD( list( Data_1, Data_2 ), weights = c( 1/3, 2/3 ))
#>  [1] 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
#> [20] 0.1
multiMBD( list( Data_1, Data_2 ), weights = c( 1/3, 2/3 ), manage_ties = FALSE )
#>  [1] 0.4068877 0.3874211 0.3753895 0.4290877 0.3473579 0.3716491 0.4218702
#>  [8] 0.4211579 0.3774912 0.3864491 0.3976737 0.3787965 0.3600561 0.4094737
#> [15] 0.4181860 0.4212035 0.4094737 0.4304035 0.3719719 0.4780000