Area-under-curve order relation between univariate functional data

This function implements an order relation between univariate functional data based on the area-under-curve relation, that is to say a pre-order relation obtained by comparing the area-under-curve of two different functional data.

area_ordered(fData, gData)

Arguments

fData

the first univariate functional dataset containing elements to be compared, in form of fData object.

gData

the second univariate functional dataset containing elements to be compared , in form of fData object.

Value

The function returns a logical vector of length \(\max(N,M)\)containing the value of the predicate for all the corresponding elements.

Details

Given a univariate functional dataset, \(X_1(t), X_2(t), \ldots, X_N(t)\) and another functional dataset \(Y_1(t),\) \(Y_2(t), \ldots, Y_M(t)\) defined over the same compact interval \(I=[a,b]\), the function computes the area-under-curve (namely, the integral) in both the datasets, and checks whether the first ones are lower or equal than the second ones.

By default the function tries to compare each \(X_i(t)\) with the corresponding \(Y_i(t)\), thus assuming \(N=M\), but when either \(N=1\) or \(M=1\), the comparison is carried out cycling over the dataset with fewer elements. In all the other cases (\(N\neq M,\) and either \(N \neq 1\) or \(M \neq 1\)) the function stops.

References

Valencia, D., Romo, J. and Lillo, R. (2015). A Kendall correlation coefficient for functional dependence, Universidad Carlos III de Madrid technical report, http://EconPapers.repec.org/RePEc:cte:wsrepe:ws133228.

See also

area_under_curve, fData

Examples

P = 1e3

grid = seq( 0, 1, length.out = P )

Data_1 = matrix( c( 1 * grid,
                    2 *  grid ),
                 nrow = 2, ncol = P, byrow = TRUE )

Data_2 = matrix( 3 * ( 0.5 - abs( grid - 0.5 ) ),
                 nrow = 1, byrow = TRUE )

Data_3 = rbind( Data_1, Data_1 )


fD_1 = fData( grid, Data_1 )
fD_2 = fData( grid, Data_2 )
fD_3 = fData( grid, Data_3 )

area_ordered( fD_1, fD_2 )
#> [1]  TRUE FALSE

area_ordered( fD_2, fD_3 )
#> [1] FALSE  TRUE FALSE  TRUE