bwLscvg: Generalized Least Squares Cross-Validation for Circular Data

In circularKDE: Recent Methods for Kernel Density Estimation of Circular Data

Generalized Least Squares Cross-Validation for Circular Data

Description

This function computes the optimal smoothing parameter (bandwidth) for circular data using a generalized least squares cross-validation method (see \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s00180-023-01401-0")}).

Usage

bwLscvg(x, g = NULL, lower = 0, upper = 60, tol = 0.1)

Arguments

x	Data from which the smoothing parameter is to be computed. The object is coerced to a numeric vector in radians using circular. Can be a numeric vector or an object of class circular.
g	A numeric scalar that controls the variability in the cross-validation procedure. It influences the scaling in the internal calculations, affecting the bandwidth estimation. It needs to be positive number not equal to 2. Default is 4.
lower	Lower boundary of the interval to be used in the search for the smoothing parameter kappa. Must be a positive numeric value less than upper. Default is 0.
upper	Upper boundary of the interval to be used in the search for the smoothing parameter kappa. Must be a positive numeric value greater than lower. Default is 60.
tol	Convergence tolerance for the optimize function, determining the precision of the optimization process. Default is 0.1.

Details

The generalized least squares cross-validation method (LSCV_g) is an adaptation of the method originally introduced by Zhang for linear data, developed for circular data (see Zamecnik, et.al 2025) to address the finite-sample performance issues of standard LSCV.

The LSCV_g criterion is defined as:

LSCV_g(\kappa) = \frac{1}{n}R(K_{\kappa}) + \frac{1}{n(n-1)} \sum_{i=1}^n \sum_{j \neq i}^n \left(\frac{n-1}{n} (K_{\kappa}*K_{\kappa})(\theta_i-\theta_j) + \frac{2}{g(g-2)} K_{\kappa/g}(\theta_i-\theta_j) - \frac{g-1}{g-2} K_{\kappa/2}(\theta_i-\theta_j)\right)

Using the von Mises kernel, this takes the computational form:

LSCV_g(\kappa) = \frac{1}{2\pi n^2} \sum_{i=1}^n \sum_{j=1}^n \frac{I_0(\kappa \sqrt{2(1+\cos(\theta_i-\theta_j))})}{I_0^2(\kappa)} + \frac{1}{n(n-1)} \sum_{i=1}^n \sum_{j \neq i}^n \left(\frac{2}{g(g-2)} \frac{\exp(\frac{\kappa}{g}\cos(\theta_i-\theta_j))}{2\pi I_0(\kappa/g)} - \frac{g-1}{g-2} \frac{\exp(\frac{\kappa}{2}\cos(\theta_i-\theta_j))}{2\pi I_0(\kappa/2)}\right)

The optimal bandwidth is obtained by minimizing this criterion over the interval [lower, upper].

Value

The computed optimal smoothing parameter kappa, a numeric concentration parameter (analogous to inverse radians) that minimizes the smoothed cross-validation criterion within the interval [lower, upper] and the value of objective function at that point. Higher values indicate sharper, more concentrated kernels and less smoothing; lower values indicate broader kernels and more smoothing. If the optimization fails, a warning is issued.

References

Zámečník, S., Horová, I., & Hasilová, K. (2025). Generalised least square cross-validation for circular data. Communications in Statistics. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s00180-023-01401-0")}

Zhang, J. (2015). Generalized least squares cross-validation in kernel density estimation. Statistica Neerlandica, 69(3), 315-328. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/stan.12061")}

See Also

bwScv, bwFo, bwCcv

Examples

# Example with circular data
library(circular)
set.seed(123)
x <- rwrappednormal(100, mu = circular(2), rho = 0.5)
bw <- bwLscvg(x)
print(round(bw$minimum, 2))

x <- rvonmises(100, mu = circular(0), kappa = 1)
bw <- bwLscvg(x)
print(round(bw$minimum, 2))
plot(density.circular(x, bw = bw$minimum))

circularKDE documentation built on Jan. 7, 2026, 9:06 a.m.