bwCcv: Complete Cross-Validation for Circular Data

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

Complete Cross-Validation for Circular Data

Description

This function calculates the optimal smoothing parameter (bandwidth) for circular data using the complete cross-validation (CCV) method (see \Sexpr[results=rd]{tools:::Rd_expr_doi("10.59170/stattrans-2024-024")}).

Usage

bwCcv(x, 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.
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. Also used to detect convergence near boundaries: if the optimal value is within tol of lower or upper, a warning is issued suggesting interval adjustment. Default is 0.1.

Details

The complete cross-validation (CCV) method is an alternative for bandwidth selection, originally proposed by Jones (1991) for linear densities. Its adaptation to the circular setting was recently studied by Hasilová et al. (2024).

The method uses functionals T_m defined as:

T_m(\kappa) = \frac{(-1)^m}{n(n-1)}\sum_{i=1}^n\sum_{j \neq i}^n K_{\kappa}^{(2m)}(\theta_{i} - \theta_{j})

where K_{\kappa}^{(2m)} is the (2m)-th derivative of K_{\kappa}.

The CCV criterion can be expressed as:

CCV(\kappa) = R(f(\kappa)) - T_0(\kappa) + \frac{1}{2}\bar{\sigma}_h^2 T_1(\kappa) + \frac{1}{24}(\eta_{2}^4(K_{\kappa}) - \eta_{4}(K_{\kappa}))T_2(\kappa)

where \eta_{j}(K_{\kappa}) denotes the j-th moment of the kernel.

For the von Mises kernel, the explicit CCV criterion becomes:

CCV(\kappa) = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n (K_{\kappa} * K_{\kappa})(\theta_i - \theta_j) - T_0(\kappa) + \frac{A_1(\kappa)}{2\kappa}T_1(\kappa) + \frac{2A_1^2(\kappa) - A_2(\kappa)}{8\kappa^2}T_2(\kappa)

where A_k(\kappa) = I_k(\kappa)/I_0(\kappa) is the ratio of modified Bessel functions.

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

Hasilová, K., Horová, I., Valis, D., & Zámečník, S. (2024). A comprehensive exploration of complete cross-validation for circular data. Statistics in Transition New Series, 25(3):1–12. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.59170/stattrans-2024-024")}

See Also

bwScv, bwLscvg, bwTs

Examples

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

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

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