Hypograph Index of univariate functional dataset

This function computes the Hypograph Index (HI) of elements of a univariate functional dataset.

HI(Data)

# S3 method for fData
HI(Data)

# S3 method for default
HI(Data)

Arguments

Data: either an fData object or a matrix-like dataset of functional data (e.g. fData$values), with observations as rows and measurements over grid points as columns.

Value

The function returns a vector containing the values of HI for each element of the functional dataset provided in Data.

Details

Given a univariate functional dataset, $X_1(t), X_2(t), \ldots, X_N(t)$, defined over a compact interval $I=[a,b]$, this function computes the HI, i.e.:

$$HI( X(t) ) = \frac{1}{N} \sum_{i=1}^N I( G( X_i(t) ) \subset hyp( X(t) ) ) = \frac{1}{N} \sum_{i=1}^N I( X_i(t) \leq X(t), \ \ \forall t \in I), $$

where $G(X_i(t))$ indicates the graph of $X_i(t)$, $hyp( X(t))$ indicates the hypograph of $X_i(t)$.

References

Lopez-Pintado, S. and Romo, J. (2012). A half-region depth for functional data, Computational Statistics and Data Analysis, 55, 1679-1695.

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

Examples


N = 20
P = 1e2

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

C = exp_cov_function( grid, alpha = 0.2, beta = 0.3 )

Data = generate_gauss_fdata( N,
                             centerline = sin( 2 * pi * grid ),
                             C )
fD = fData( grid, Data )

HI( fD )
#>  [1] 0.20 0.30 0.25 0.25 0.40 0.05 0.85 0.75 0.10 0.10 0.60 0.85 0.55 0.10 0.70
#> [16] 0.10 0.20 0.35 0.25 0.75

HI( Data )
#>  [1] 0.20 0.30 0.25 0.25 0.40 0.05 0.85 0.75 0.10 0.10 0.60 0.85 0.55 0.10 0.70
#> [16] 0.10 0.20 0.35 0.25 0.75