kernel_symm: 1D Isotropic Symmetric Kernels.

In CovEsts: Nonparametric Estimators for Covariance Functions

1D Isotropic Symmetric Kernels.

Description

These functions computes values of kernels that have the properties of symmetric probability distributions. For a kernel a(x), the standardised version is a(x) / \int_{-\infty}^{\infty} a(x) dx, so that the integral is 1. The symmetric kernels are different to kernel and are used in the functions adjusted_est and truncated_est.

Usage

kernel_symm(x, name, params = c(1))

Arguments

x	A vector or matrix of arguments of at least length 1 for which the kernel is computed at. Each value can be negative as well as positive.
name	The name of the kernel. Options are: gaussian, wave, rational_quadratic, bessel_j.
params	A vector of parameters for the kernel. See the documentation below for the position of the parameters. All kernels will have a scale parameter as the first value in the vector.

Details

Symmetric Gaussian Kernel. The symmetric Gaussian kernel is defined as

a(x;\theta) = \sqrt{\pi \theta} \exp(-x^{2} / \theta), \theta > 0.

The params argument is of the form c(\theta).

Symmetric Wave Kernel. The wave (cardinal sine) kernel is given by

a(x;\theta) = \left\{ \begin{array}{ll} (\sqrt{\theta^{2}} \pi)^{-1} \frac{\theta}{x} \sin\left( \frac{x}{\theta} \right), & x \neq 0 \\ 1, & x = 0 \end{array},\right.

where \theta > 0. The params argument is of the form c(\theta)

Symmetric Rational Quadratic Kernel. The symmetric rational quadratic kernel is given by

a(x;\theta) = (\pi \sqrt{\theta})^{-1} \left(1 - \frac{x^{2}}{x^{2} + \theta}\right), \theta > 0.

The params argument is of the form c(\theta)

Symmetric Besesel Kernel. The symmetric Bessel kernel, which is valid when \nu \geq \frac{d}{2} - 1, is given by

a(x; \theta, \nu) = \left(\Gamma\left(\frac{1}{2} + \nu\right)/(2 \sqrt{\pi} \theta \Gamma(1 + \nu))\right) ( 2^{\nu} \Gamma(\nu + 1) J_{\nu}(x / \theta) (x / \theta)^{-\nu}), \,\theta > 0, \nu \geq \frac{d}{2} - 1,

where J_{\nu}(\cdot) is the Bessel function of the first kind and d is the dimension. The params argument is of the form c(\theta, \nu, d).

Value

A vector or matrix of values.

Examples

x <- c(-2, -1, 0, 1, 2)
theta <- 1
kernel_symm(x, "gaussian", c(theta))
kernel_symm(x, "wave", c(theta))
kernel_symm(x, "rational_quadratic", c(theta))
dim <- 1
nu <- 1
kernel_symm(x, "bessel_j", c(theta, nu, dim))
curve(kernel_symm(x, "gaussian", c(theta)), from = -5, to = 5)
curve(kernel_symm(x, "wave", c(theta)), from = -5, to = 5)
curve(kernel_symm(x, "rational_quadratic", c(theta)), from = -5, to = 5)
curve(kernel_symm(x, "bessel_j", c(theta, nu, dim)), from = -5, to = 5)

CovEsts documentation built on Sept. 10, 2025, 10:39 a.m.