This function computes the Modified Band Depth (MBD) of elements of a functional dataset.
MBD(Data, manage_ties = FALSE)
# S3 method for fData
MBD(Data, manage_ties = FALSE)
# S3 method for default
MBD(Data, manage_ties = FALSE)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.
a logical flag specifying whether a check for ties and
relative treatment must be carried out or not (default is FALSE).
The function returns a vector containing the values of MBD for the given dataset.
Given a univariate functional dataset, \(X_1(t), X_2(t), \ldots, X_N(t)\), 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.:
$$ MBD( X( t ) ) = {N \choose 2 }^{-1} \sum_{1 \leq i_1 < i_2 \leq N} \tilde{\lambda}\big( {t : \min( X_{i_1}(t), X_{i_2}(t) ) \leq X(t) \leq \max( X_{i_1}(t), X_{i_2}(t) ) } \big), $$
where \(\tilde{\lambda}(\cdot)\) is the normalized Lebesgue measure over \(I=[a,b]\), that is \(\tilde{\lambda(A)} = \lambda( A ) / ( b - a )\).
See the References section for more details.
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.
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 )
#> [1] 0.4813333 0.5946667 0.6400000 0.5853333 0.6986667 0.3333333
MBD( D )
#> [1] 0.4813333 0.5946667 0.6400000 0.5853333 0.6986667 0.3333333