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.
extreme_rank_length(
dts,
type = c("two_sided", "one_sided_left", "one_sided_right")
)A numeric vector containing the depth of each curve
A matrix or data frame of size \(n\) observations/curves by \(p\) domain/evaluation points.
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.
Oluwasegun Ojo
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) tools:::Rd_expr_doi("10.1016/j.csda.2020.106960") and Myllymäki et al. (2017) tools:::Rd_expr_doi("10.1111/rssb.12172") for more details.
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.
dt3 <- simulation_model3()
erld <- extreme_rank_length(dt3$data)
Run the code above in your browser using DataLab