biloop_transform: Biloop Transformation

In dcorBSS: Distance-Correlation Based Methods for Blind Source Separation and Dependence Analysis

Biloop Transformation

Description

Apply the biloop transformation to a numeric vector or matrix.

Usage

biloop_transform(
  X,
  c1 = 4,
  c2 = 4,
  do_scale = FALSE,
  location = NULL,
  scale = NULL
)

Arguments

X	A numeric vector or matrix. Rows correspond to observations and columns to variables.
c1	Positive scaling constant used before the hyperbolic tangent transformation. See details.
c2	Positive scaling constant applied to the first transformed coordinate. See details.
do_scale	Logical; if TRUE, standardize each variable before applying the biloop transformation.
location	Optional centering vector used when do_scale = TRUE. Defaults to stats::median(). Ignored otherwise.
scale	Optional scaling vector used when do_scale = TRUE. Defaults to stats::mad(). Ignored otherwise.

Details

The transformation maps each univariate variable to a two-dimensional nonlinear embedding x \mapsto (u(x), v(x)) based on two tangential circles. The data may first be robustly standardized (advised). For multivariate input, the transformation is applied columnwise and the transformed columns are concatenated.

u(x) and v(x) are defined as:

u(x) = \begin{cases} c_2(1 + \cos(2\pi \tanh(x/c_1) + \pi)), & x \ge 0 \\ -c_2(1 + \cos(2\pi \tanh(x/c_1) - \pi)), & x < 0 \end{cases}

v(x) = \sin(2\pi \tanh(x/c_1))

By default, no scaling is performed. If do_scale = TRUE and neither location nor scale is supplied, the columns are centered using stats::median() and scaled using stats::mad(). Alternatively, user-supplied location and scale vectors can be provided.

The biloop transformation can be useful when computing dependence measures on heavy-tailed or contaminated data.

Value

A numeric matrix with nrow(X) rows and 2 * ncol(X) columns. If X is a vector, the result has two columns.

References

Leyder, S., Raymaekers, J., and Rousseeuw, P. J. (2026). Robust Distance Covariance. International Statistical Review, 94, 1–25. doi:10.1111/insr.70005

Examples

x <- rnorm(10)
biloop_transform(x)

X <- cbind(rnorm(10), rt(10, df = 3))
colnames(X) <- c("A","B")
biloop_transform(X, do_scale = TRUE)

# Robust scaling using median and Qn from robustbase
if (requireNamespace("robustbase", quietly = TRUE)) {
  loc <- apply(X, 2, median)
  scl <- apply(X, 2, robustbase::Qn)
  biloop_transform(X, do_scale = TRUE, location = loc, scale = scl)
}

# Robust distance correlation after the biloop transformation
if (requireNamespace("robustbase", quietly = TRUE) &&
    requireNamespace("energy", quietly = TRUE)) {
  x <- rt(100, df = 3)
  y <- x^2 + rnorm(100, sd = 0.2)
  loc_x <- median(x)
  scl_x <- robustbase::Qn(x)
  loc_y <- median(y)
  scl_y <- robustbase::Qn(y)

  xb <- biloop_transform(x, do_scale = TRUE, location = loc_x, scale = scl_x)
  yb <- biloop_transform(y, do_scale = TRUE, location = loc_y, scale = scl_y)
  energy::dcor(xb, yb)
}

dcorBSS documentation built on Nov. 10, 2026, 5:07 p.m.