Modified Band Depth for univariate functional data

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Description

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

Usage

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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.

See Also

BD, MBD_relative, BD_relative, fData

Examples

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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 )

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