distsd {dcortools}

Calculates the distance standard deviation (Edelmann et al. 2020).

Description

Calculates the distance standard deviation (Edelmann et al. 2020).

Usage

distsd(
  X,
  affine = FALSE,
  standardize = FALSE,
  bias.corr = FALSE,
  type.X = "sample",
  metr.X = "euclidean",
  use = "all",
  algorithm = "auto"
)

Arguments

X

contains either the sample or its corresponding distance matrix.

In the first case, X can be provided either as a vector (if one-dimensional), a matrix or a data.frame (if two-dimensional or higher).

In the second case, the input must be a distance matrix corresponding to the sample of interest.

If X is a sample, type.X must be specified as "sample". If X is a distance matrix, type.X must be specified as "distance".

affine

logical; specifies if the affinely invariant distance standard deviation (Dueck et al. 2014) should be calculated or not.

standardize

logical; specifies if X and Y should be standardized dividing each component by its standard deviations. No effect when affine = TRUE.

bias.corr

logical; specifies if the bias corrected version of the sample distance standard deviation (Huo and Szekely 2016) should be calculated.

type.X

For "distance", X is interpreted as a distance matrix. For "sample", X is interpreted as a sample.

metr.X

specifies the metric which should be used to compute the distance matrix for X (ignored when type.X = "distance").

Options are "euclidean", "discrete", "alpha", "minkowski", "gaussian", "gaussauto", "boundsq" or user-specified metrics (see examples).

For "alpha", "minkowski", "gaussian", "gaussauto" and "boundsq", the corresponding parameters are specified via "c(metric, parameter)", e.g. c("gaussian", 3) for a Gaussian metric with bandwidth parameter 3; the default parameter is 2 for "minkowski" and "1" for all other metrics.

See Lyons (2013); Sejdinovic et al. (2013); Bottcher et al. (2018) for details.

use

specifies how to treat missing values. "complete.obs" excludes observations containing NAs, "all" uses all observations.

algorithm

specifies the algorithm used for calculating the distance standard deviation.

"fast" uses an O(n log n) algorithm if the observations are one-dimensional and metr.X and metr.Y are either "euclidean" or "discrete", see also Huo and Szekely (2016).

"memsave" uses a memory saving version of the standard algorithm with computational complexity O(n^2) but requiring only O(n) memory.

"standard" uses the classical algorithm. User-specified metrics always use the classical algorithm.

"auto" chooses the best algorithm for the specific setting using a rule of thumb.

Value

numeric; the distance standard deviation of X.

References

Bottcher B, Keller-Ressel M, Schilling RL (2018). “Detecting independence of random vectors: generalized distance covariance and Gaussian covariance.” Modern Stochastics: Theory and Applications, 3, 353–383.

Dueck J, Edelmann D, Gneiting T, Richards D (2014). “The affinely invariant distance correlation.” Bernoulli, 20, 2305–2330.

Edelmann D, Richards D, Vogel D (2020). “The Distance Standard Deviation.” The Annals of Statistics.. Accepted for publication.

Huo X, Szekely GJ (2016). “Fast computing for distance covariance.” Technometrics, 58(4), 435–447.

Lyons R (2013). “Distance covariance in metric spaces.” The Annals of Probability, 41, 3284–3305.

Sejdinovic D, Sriperumbudur B, Gretton A, Fukumizu K (2013). “Equivalence of distance-based and RKHS-based statistics in hypothesis testing.” The Annals of Statistics, 41, 2263–2291.

Szekely GJ, Rizzo ML, Bakirov NK (2007). “Measuring and testing dependence by correlation of distances.” The Annals of Statistics, 35, 2769–2794.

Szekely GJ, Rizzo ML (2009). “Brownian distance covariance.” The Annals of Applied Statistics, 3, 1236–1265.

Examples

X <- rnorm(100)
distsd(X) # for more examples on the options see the documentation of distcov.