extreme_rank_length {fdaoutlier}

Compute the Extreme Rank Length Depth.

Description

This function computes the extreme rank length depth (ERLD) of a sample of curves or functions. Functions have to be discretely observed on common domain points. In principle, the ERLD of a function X_i is the proportion of functions in the sample that is considered to be more extreme than X_i, an idea similar to extremal_depth. To determine which functions are more extreme, pointwise ranks of the functions are computed and compared pairwise.

Usage

extreme_rank_length(
  dts,
  type = c("two_sided", "one_sided_left", "one_sided_right")
)

Arguments

dts

A matrix or data frame of size n observations/curves by p domain/evaluation points.

type

A character value. Can be one of "two_sided", "one_sided_left" or "one_sided_right". If "two_sided" is specified, small and large values in dts will be considered extreme. If "one_sided_left" is specified, then only small values in dts are considered to be extreme while for "one_sided_right", only large values in dts are considered to be extreme. "two_sided" is the default. See Details for more details.

Details

There are three possibilities in the (pairwise) comparison of the pointwise ranks of the functions. First possibility is to consider only small values as extreme (when type = "one_sided_left") in which case the raw pointwise ranks r_{ij} are used. The second possibility is to consider only large values as extreme (when type = "one_sided_right") in which case the pointwise ranks used are computed as R_{ij} = n + 1 - r_{ij} where r_{ij} is the raw pointwise rank of the function i at design point j and n is the number of functions in the sample. Third possibility is to consider both small and large values as extreme (when type = "two_sided") in which case the pointwise ranks used is computed as R_{ij} = min(r_ij, n + 1 - r_{ij}). In the computation of the raw pointwise ranks r_{ij}, ties are broken using an average. See Dai et al. (2020) doi:10.1016/j.csda.2020.106960 and Myllymäki et al. (2017) doi:10.1111/rssb.12172 for more details.

Value

A numeric vector containing the depth of each curve

Author(s)

Oluwasegun Ojo

References

Dai, W., Mrkvička, T., Sun, Y., & Genton, M. G. (2020). Functional outlier detection and taxonomy by sequential transformations. Computational Statistics & Data Analysis, 106960.

Myllymäki, M., Mrkvička, T., Grabarnik, P., Seijo, H., & Hahn, U. (2017). Global envelope tests for spatial processes. J. R. Stat. Soc. B, 79:381-404.

Examples

dt3 <- simulation_model3()
erld <- extreme_rank_length(dt3$data)