Area-under-curve order relation between univariate functional data

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Description

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.

Usage

1

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.

Details

Given a univariate functional dataset, X_1(t), X_2(t), …, X_N(t) and another functional dataset Y_1(t), Y_2(t), …, 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.

Value

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

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

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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 )

area_ordered( fD_2, fD_3 )

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