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.

1 | ```
area_ordered(fData, gData)
``` |

`fData` |
the first univariate functional dataset containing elements to
be compared, in form of |

`gData` |
the second univariate functional dataset containing elements to
be compared , in form of |

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.

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

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`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | ```
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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