Rearrangement Correlation (r^\#)

Description

Computes the rearrangement correlation coefficient (r^\#), an adjusted version of Pearson's correlation coefficient that addresses the underestimation problem for monotonic dependence. This coefficient captures arbitrary monotone relationships (both linear and nonlinear) while reverting to Pearson's r in linear scenarios. The rearrangement correlation is derived from a tighter inequality than the classical Cauchy-Schwarz inequality, providing sharper bounds and expanded capture range.

Usage

recor(x, y = NULL)

Arguments

Details

The rearrangement correlation coefficient is based on rearrangement inequality theorems that provide tighter bounds than the Cauchy-Schwarz inequality. Mathematically, for samples x and y, it is defined as:

r^\#(x, y) = \frac{s_{x,y}}{\left| s_{x^\uparrow, y^\updownarrow} \right|}

where:

• s_{x,y} is the sample covariance between x and y.

• x^\uparrow is the increasing rearrangement of x.

• y^\updownarrow is either y^\uparrow (increasing rearrangement) if s_{x,y} \geq 0 or y^\downarrow (decreasing rearrangement) if s_{x,y} < 0

Key properties of the rearrangement correlation:

• |r^\#| \leq 1 with equality if and only if x and y are monotone dependent.

• |r^\#| \geq |r| with equality when y is a permutation of ax + b (i.e., linear relationship).

• More accurate than classical coefficients for nonlinear monotone dependence.

• Reverts to Pearson's r for linear relationships and to Spearman's \rho for rank data.

The function automatically handles various input types consistently with stats::cor():

• If x is a vector and y is a vector: returns a single correlation value

• If x is a matrix/data.frame and y is NULL: returns correlation matrix between columns of x

• If x and y are both matrices/data.frames: returns correlation matrix between columns of x and y

Value

A numeric value or matrix containing the rearrangement correlation coefficient(s):

• Single correlation value if x and y are vectors

• Correlation matrix if x is a matrix/data.frame (symmetric if y is NULL, rectangular if y is provided)

References

Ai, X. (2024). Adjust Pearson's r to Measure Arbitrary Monotone Dependence. In Advances in Neural Information Processing Systems (Vol. 37, pp. 37385-37407). https://proceedings.neurips.cc/paper_files/paper/2024/file/41c38a83bd97ba28505b4def82676ba5-Paper-Conference.pdf

See Also

cor for Pearson's correlation coefficient, cor with method = "spearman" for Spearman's rank correlation, cor with method = "kendall" for Kendall's rank correlation, cov for covariance calculation

Examples

# Vector example (perfect linear relationship)
x <- c(1, 2, 3, 4, 5)
y <- c(2, 4, 6, 8, 10)
recor(x, y) # Returns 1.0

# Nonlinear monotone relationship
x <- c(1, 2, 3, 4, 5)
y <- c(1, 8, 27, 65, 125) # y = x^3
recor(x, y) # Higher value than Pearson's r
cor(x, y)

# Matrix example
set.seed(123)
mat <- matrix(rnorm(100), ncol = 5)
colnames(mat) <- LETTERS[1:5]
recor(mat) # 5x5 correlation matrix

# Two matrices
mat1 <- matrix(rnorm(50), ncol = 5)
mat2 <- matrix(rnorm(50), ncol = 5)
recor(mat1, mat2) # 5x5 cross-correlation matrix

# data.frame
recor(iris[, 1:4]) # correlation matrix for iris dataset features