bwTs: Plug-in Method by Tsuruta and Sagae with multiplicative...

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

Plug-in Method by Tsuruta and Sagae with multiplicative method from Terrell and Scott

Description

This function computes the optimal smoothing parameter (bandwidth) for circular data using the plug-in method, introduced by Tsuruta and Sagae (see \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.spl.2017.08.003")}) with the multiplicative method from Terrell and Scott (1980) to form higher-order kernel functions. The method optimally balances the bias-variance tradeoff by minimizing asymptotic mean integrated squared error.

Usage

bwTs(x, verbose = FALSE)

Arguments

x	Data from which the smoothing parameter is to be computed. The object is coerced to a numeric vector in radians using conversion.circular. Can be a numeric vector or an object of class circular.
verbose	Logical indicating whether to print intermediate computational values for debugging purposes. Useful for diagnosing floating-point instability in Bessel-based moment terms. Default is FALSE.

Details

The plug-in approach estimates the optimal bandwidth through the following steps:

1. Apply the multiplicative method from Terrell and Scott (1980) to construct a p-th order kernel function.

2. Derive expression for asymptotic mean integrated squared error (AMISE) expression.

3. Solving for the bandwidth that minimizes the AMISE. The optimal bandwidth for the multiplicative Terrell-Scott method is given by:

   \hat{\kappa}_{TS} = \left[\frac{288}{33 - 16\sqrt{2/5}} \hat{R}_{\hat{\tau}}(m_{VM}) n\right]^{2/9}

   where the computational formula is:

   m_{VM}(\theta) := [2\{f_{VM}^{(2)}\}^2/f_{VM} - 5f_{VM}^{(2)} + 2f_{VM}^{(4)}]/4

   and \hat{R}_{\hat{\tau}}(m_{VM}) is the functional computed under the von Mises assumption using the multiplicative approach. The parameter \hat{\tau} is the MLE estimate of the von Mises concentration parameter used as the initial value.

Value

The computed optimal smoothing parameter, a numeric value derived from the circular version of the multiplicative method for circular kernel density estimation.

References

Tsuruta, Yasuhito & Sagae, Masahiko (2017). Higher order kernel density estimation on the circle. Statistics & Probability Letters, 131:46–50. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.spl.2017.08.003")}

Terrell, George R. & Scott, David W. (1980). On improving convergence rates for nonnegative kernel density estimators. The Annals of Statistics, 8(5):1160–1163.

See Also

bwScv, bwLscvg, bwCcv

Examples

# Example with circular data
library(circular)
set.seed(123)
x <- rvonmises(100, mu = circular(0), kappa = 2)
bw <- bwTs(x)
print(bw)

x <- rwrappednormal(100, mu = circular(1), rho = 0.7)
x <- as.numeric(x)
bw <- bwTs(x)
print(bw)

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