spml.reg: Circular or angular regression

In Directional: A Collection of Functions for Directional Data Analysis

Circular or angular regression

Description

Regression with circular dependent variable and Euclidean or categorical independent variables.

Usage

spml.reg(y, x, rads = TRUE, xnew = NULL, seb = FALSE, tol = 1e-6, maxiters = 100)
circpurka.reg(y, x, rads = TRUE, xnew = NULL, tol = 1e-6, maxiters = 300)
cipc.reg(y, x, rads = TRUE, xnew = NULL, tol = 1e-6, maxiters = 100)
gcpc.reg(y, x, rads = TRUE, xnew = NULL)

Arguments

y	The dependent variable, a numerical vector, it can be in radians or degrees, or a numerical matrix with two columns containing eigenvectors.
x	The independent variable(s). Can be Euclidean or categorical (factor variables).
rads	If the dependent variable is expressed in rads, this should be TRUE and FALSE otherwise.
xnew	The new values of some independent variable(s) whose circular values you want to predict. Can be Euclidean or categorical. If you have no new x values, leave it NULL (default).
seb	a boolean variable. If TRUE, the standard error of the coefficients will be be returned. Set to FALSE in case of simulation studies or in other cases such as a forward regression setting for example. In these cases, it can save some time.
tol	The tolerance value to terminate the Newton-Raphson algorithm.
maxiters	The maximum number of iterations allowed in the Newton-Raphson algorithm.

Details

For the spml.reg(), the Newton-Raphson algorithm is fitted in this regression as described in Presnell et al. (1998). For the cipc.reg(), the Newton-Raphson algorithm is fitted in this regression as described in Tsagris and Alenazy (2023). Note that the cipc.reg() is the same as the wrapped Cauchy regression. For the circpurka.reg() a variant of the Newton-Raphson algorithm is used, followed by the optim() function is employed. For the gcpc.reg() the optim() and the optimise() functions are being used.

Value

A list including:

runtime	The runtime of the procedure.
iters	The number of iterations required until convergence of the Newton-Raphson algorithm.
beta	The regression coefficients.
seb	The standard errors of the coefficients.
loglik	The value of the maximised log-likelihood.
est	The fitted values expressed in radians if the obsereved data are in radians and in degrees otherwise. If xnew is not NULL, i.e. if you have new x values, then the predicted values of y will be returned.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

References

Presnell B., Morrison S. P. and Littell Ramon C. (1998). Projected multivariate linear models for directional data. Journal of the American Statistical Association, 93(443): 1068–1077.

Purkayastha S. (1991). A Rotationally Symmetric Directional Distribution: Obtained through Maximum Likelihood Characterization. The Indian Journal of Statistics, Series A, 53(1): 70–83

Tsagris M. and Alzeley O. (2025). Circular and spherical projected Cauchy distributions: A Novel Framework for Circular and Directional Data Modeling. Australian & New Zealand Journal of Statistics, 67(1): 77–103. https://arxiv.org/pdf/2302.02468.pdf

Alzeley O. and Tsagris M. (2026). On the generalized circular projected Cauchy distribution. Mathematics, 14(11): 1934. https://www.mdpi.com/2227-7390/14/11/1934

See Also

circlin.cor, circ.cor1, circ.cor2, spher.cor, spher.reg

Examples

x <- rnorm(100)
z <- cbind(3 + 2 * x, 1 -3 * x)
y <- cbind( rnorm(100,z[ ,1], 1), rnorm(100, z[ ,2], 1) )
y <- y / sqrt( rowSums(y^2) )
y <- ( atan( y[, 2] / y[, 1] ) + pi * I(y[, 1] < 0) ) %% (2 * pi)
a <- spml.reg(y, x, rads = TRUE, xnew = x)
b <- cipc.reg(y, x, rads = TRUE, xnew = x)

Directional documentation built on Sept. 2, 2026, 5:09 p.m.