depthf.hM {ddalpha}

h-Mode Depth for Functional Data

Description

The h-mode depth of functional real-valued data.

Usage

depthf.hM(datafA, datafB, range = NULL, d = 101, norm = c("C", "L2"),
  q = 0.2)

Arguments

datafA

Functions whose depth is computed, represented by a dataf object of their arguments and functional values. m stands for the number of functions.

datafB

Random sample functions with respect to which the depth of datafA is computed. datafB is represented by a dataf object of their arguments and functional values. n is the sample size. The grid of observation points for the functions datafA and datafB may not be the same.

range

The common range of the domain where the functions datafA and datafB are observed. Vector of length 2 with the left and the right end of the interval. Must contain all arguments given in datafA and datafB.

d

Grid size to which all the functional data are transformed. For depth computation, all functional observations are first transformed into vectors of their functional values of length d corresponding to equi-spaced points in the domain given by the interval range. Functional values in these points are reconstructed using linear interpolation, and extrapolation.

norm

The norm used for the computation of the depth. Two possible choices are implemented: C for the uniform norm of continuous functions, and L2 for the L^2 norm of integrable functions.

q

The quantile used to determine the value of the bandwidth h in the computation of the h-mode depth. h is taken as the q-quantile of all non-zero distances between the functions B. By default, this value is set to q=0.2, in accordance with the choice of Cuevas et al. (2007).

Details

The function returns the vectors of the sample h-mode depth values. The kernel used in the evaluation is the standard Gaussian kernel, the bandwidth value is chosen as a quantile of the non-zero distances between the random sample curves.

Value

A vector of length m of the h-mode depth values.

Author(s)

Stanislav Nagy, [e-mail]

References

Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22 (3), 481–496.

Nagy, S., Gijbels, I. and Hlubinka, D. (2016). Weak convergence of discretely observed functional data with applications. Journal of Multivariate Analysis, 146, 46–62.

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]
depthf.hM(datafA,datafB)
depthf.hM(datafA,datafB,norm="L2")