kernel: 1D Isotropic Kernels.

In CovEsts: Nonparametric Estimators for Covariance Functions

1D Isotropic Kernels.

Description

This function computes one of the isotropic kernels listed below. Unlike kernel_symm, these kernels are only defined when x \geq 0. They are used as kernel multipliers in estimators corrected_est and kernel_est.

Usage

kernel(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.
name	The name of the kernel. Options are: gaussian, exponential, wave, rational_quadratic, spherical, circular, bessel_j, matern, and cauchy.
params	A vector of parameters for the kernel. See the documentation below for the position of the parameters. All kernels have a scale parameter as the first value in the vector.

Details

Gaussian Kernel. The isotropic Gaussian kernel, which is positive-definite for {R}^{d}, d \in N, is defined as

a(x;\theta) = \exp(-x^{2} / \theta).

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

Exponential Kernel. The isotropic exponential kernel, which is positive-definite for {R}^{d}, d \in N, is defined as

a(x;\theta) = \exp(-x / \theta).

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

Isotropic Wave (Cardinal Sine) Kernel. The isotropic wave (cardinal sine) kernel, which is positive-definite for {R}^{d}, d \leq 3, is given by

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

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

Isotropic Rational Quadratic Kernel. The isotropic rational quadratic kernel, which is positive-definite for {R}^{d}, d \in N, is defined as

a(x;\theta) = 1 - \frac{x^{2}}{x^{2} + \theta}.

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

Isotropic Spherical Kernel. The isotropic spherical kernel, which is positive-definite for {R}^{3}, d \leq 3, is given by

a(x;\theta) = \left\{ \begin{array}{ll} 1 - \frac{3}{2}\frac{x}{\theta} + \frac{1}{2}\left( \frac{x}{\theta} \right)^{3}, & x < \theta \\ 0, & \mbox{otherwise} \end{array} . \right.

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

Isotropic Circular Kernel. The isotropic circular kernel, which is positive-definite for {R}^{d}, d \leq 2, is given by

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

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

Isotropic Matérn Kernel. The isotropic Matérn kernel, which is positive-definite for {R}^{d}, d \in N, and when \nu > 0, is defined as

a(x; \theta, \nu) = \left(\sqrt{2\nu} \frac{x}{\theta} \right)^{\nu} \left(2^{\nu - 1} \Gamma(\nu) \right)^{-1} K_{\nu}\left( \sqrt{2\nu} \frac{x}{\theta} \right) ,

where K_{\nu}(\cdot) is the modified Bessel function of the second kind. The params argument is of the form c(\theta, \nu).

Isotropic Bessel Kernel. The isotropic Bessel kernel, which is positive-definite for {R}^{d}, d \in N, and when \nu \geq \frac{d}{2} - 1, is given by

a(x; \theta, \nu) = 2^{\nu} \Gamma(\nu + 1) J_{\nu}(x / \theta) (x / \theta)^{-\nu} ,

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

Isotropic Cauchy Kernel. The isotropic Cauchy kernel, which is positive-definite for {R}^{d}, d \in N, and when 0 < \alpha \leq 2 and \beta \geq 0, is defined by

a(x ; \theta, \alpha, \beta) = (1 + (x / \theta)^{\alpha})^{-(\beta / \alpha)} .

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

Value

A vector or matrix of kernel values.

References

Genton, M. (2001). Classes of Kernels for Machine Learning: A Statistics Perspective. Journal of Machine Learning Research. 2, 299-312. https://doi.org/10.1162/15324430260185646

Table 4.2 in Hristopulos, D. T. (2020). Random Fields for Spatial Data Modeling: A Primer for Scientists and Engineers. Springer. https://doi.org/10.1007/978-94-024-1918-4

Examples

x <- c(0.2, 0.4, 0.6)
theta <- 0.9
kernel(x, "gaussian", c(theta))
kernel(x, "exponential", c(theta))
kernel(x, "wave", c(theta))
kernel(x, "rational_quadratic", c(theta))
kernel(x, "spherical", c(theta))
kernel(x, "circular", c(theta))
nu <- 1
kernel(x, "matern", c(theta, nu))
dim <- 1
kernel(x, "bessel_j", c(theta, nu, dim))
alpha <- 1
beta <- 2
kernel(x, "cauchy", c(theta, alpha, beta))
curve(kernel(x, "gaussian", c(theta)), from = 0, to = 5)
curve(kernel(x, "exponential", c(theta)), from = 0, to = 5)
curve(kernel(x, "wave", c(theta)), from = 0, to = 5)
curve(kernel(x, "rational_quadratic", c(theta)), from = 0, to = 5)
curve(kernel(x, "spherical", c(theta)), from = 0, to = 5)
curve(kernel(x, "circular", c(theta)), from = 0, to = 5)
curve(kernel(x, "matern", c(theta, nu)), from = 0, to = 5)
curve(kernel(x, "bessel_j", c(theta, nu, dim)), from = 0, to = 5)
curve(kernel(x, "cauchy", c(theta, alpha, beta)), from = 0, to = 5)

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