R/band_depths.R
MBD_relative.RdThis function computes Modified Band Depth (BD) of elements of a univariate functional dataset with respect to another univariate functional dataset.
MBD_relative(Data_target, Data_reference)
# S3 method for fData
MBD_relative(Data_target, Data_reference)
# S3 method for default
MBD_relative(Data_target, Data_reference)is the univariate functional dataset, provided either as
an fData object or in matrix
form (N observations as rows and P measurements as columns), whose MBD
have to be computed with respect to the reference dataset.
is the dataset, provided either as an fData
object or in matrix form (N observations
as rows and P measurements as columns), containing the reference
univariate functional data that must be used to compute the MBD of
elements in Data_target. If Data_target is fData, it
must be of class fData.
The function returns a vector containing the MBD of elements in
Data_target with respect to elements in Data_reference.
Given a univariate functional dataset of elements \(X_1(t), X_2(t), \ldots, X_N(t)\), and another univariate functional dataset of elements \(Y_1(t), Y_2(t) \ldots, Y_M(t)\), defined over the same compact interval \(I=[a,b]\), this function computes the MBD of elements of the former with respect to elements of the latter, i.e.:
$$ MBD( X_i( t ) ) = {M \choose 2 }^{-1} \sum_{1 \leq i_1 < i_2 \leq M} \tilde{\lambda}\big( {t : \min( Y_{i_1}(t), Y_{i_2}(t) ) \leq X_i(t) \leq \max( Y_{i_1}(t), Y_{i_2}(t) ) } \big),$$
\(\forall i = 1, \ldots, N\), where \(\tilde{\lambda}(\cdot)\) is the normalized Lebesgue measure over \(I=[a,b]\), that is \(\tilde{\lambda(A)} = \lambda( A ) / ( b - a )\).
MBD, BD, BD_relative,
fData
grid = seq( 0, 1, length.out = 1e2 )
Data_ref = matrix( c( 0 + sin( 2 * pi * grid ),
1 + sin( 2 * pi * grid ),
-1 + sin( 2 * pi * grid ) ),
nrow = 3, ncol = length( grid ), byrow = TRUE )
Data_test_1 = matrix( c( 0.6 + sin( 2 * pi * grid ) ),
nrow = 1, ncol = length( grid ), byrow = TRUE )
Data_test_2 = matrix( c( 0.6 + sin( 2 * pi * grid ) ),
nrow = length( grid ), ncol = 1, byrow = TRUE )
Data_test_3 = 0.6 + sin( 2 * pi * grid )
Data_test_4 = array( 0.6 + sin( 2 * pi * grid ), dim = length( grid ) )
Data_test_5 = array( 0.6 + sin( 2 * pi * grid ), dim = c( 1, length( grid ) ) )
Data_test_6 = array( 0.6 + sin( 2 * pi * grid ), dim = c( length( grid ), 1 ) )
Data_test_7 = matrix( c( 0.5 + sin( 2 * pi * grid ),
-0.5 + sin( 2 * pi * grid ),
1.1 + sin( 2 * pi * grid ) ),
nrow = 3, ncol = length( grid ), byrow = TRUE )
fD_ref = fData( grid, Data_ref )
fD_test_1 = fData( grid, Data_test_1 )
fD_test_2 = fData( grid, Data_test_2 )
fD_test_3 = fData( grid, Data_test_3 )
fD_test_4 = fData( grid, Data_test_4 )
fD_test_5 = fData( grid, Data_test_5 )
fD_test_6 = fData( grid, Data_test_6 )
fD_test_7 = fData( grid, Data_test_7 )
MBD_relative( fD_test_1, fD_ref )
#> [1] 0.6666667
MBD_relative( Data_test_1, Data_ref )
#> [1] 0.6666667
MBD_relative( fD_test_2, fD_ref )
#> [1] 0.6666667
MBD_relative( Data_test_2, Data_ref )
#> [1] 0.6666667
MBD_relative( fD_test_3, fD_ref )
#> [1] 0.6666667
MBD_relative( Data_test_3, Data_ref )
#> [1] 0.6666667
MBD_relative( fD_test_4, fD_ref )
#> [1] 0.6666667
MBD_relative( Data_test_4, Data_ref )
#> [1] 0.6666667
MBD_relative( fD_test_5, fD_ref )
#> [1] 0.6666667
MBD_relative( Data_test_5, Data_ref )
#> [1] 0.6666667
MBD_relative( fD_test_6, fD_ref )
#> [1] 0.6666667
MBD_relative( Data_test_6, Data_ref )
#> [1] 0.6666667
MBD_relative( fD_test_7, fD_ref )
#> [1] 0.6666667 0.6666667 0.0000000
MBD_relative( Data_test_7, Data_ref )
#> [1] 0.6666667 0.6666667 0.0000000