Area under curve of elements of univariate functional data

This method computes the (signed) area under the curve of elements of a univariate functional dataset, namely, their integral.

area_under_curve(fData)

Arguments

fData

the functional dataset containing elements whose areas under the curve have to be computed, in form of fData object.

Value

The function returns a numeric vector containing the values of areas under the curve for all the elements of the functional dataset fData.

Details

Given a univariate functional dataset, \(X_1(t), X_2(t), \ldots, X_N(t)\), defined over a compact interval \(I=[a,b]\) and observed on an evenly spaced 1D grid \([a = t_0, t_1, \ldots, t_{P-1} = b] \subset I\), the function computes:

$$ \sum_{i=1}^{P-2} \frac{X(t_{i+1}) - X(t_{i-1})}{2} h \approx \int_a^b X(t) dt,$$

where \(h = t_1 - t_0\).

See also

area_ordered, fData

Examples

P = 1e3
grid = seq( 0, 1, length.out = P )

fD = fData( grid,
            matrix( c( sin( 2 * pi * grid ),
                       cos( 2 * pi * grid ),
                       4 * grid * ( 1 - grid ) ),
                    nrow = 3, ncol = P, byrow = TRUE ) )
plot( fD )


area_under_curve( fD )
#> [1]  2.976474e-18 -4.824700e-17  6.666660e-01