kernel_matrix: Compute Kernel Matrix Between Input Locations

In NBKP: Beta Kernel Process Modeling with Negative Binomial Response

Compute Kernel Matrix Between Input Locations

Description

Computes the kernel matrix between two sets of input locations using a specified kernel function. Supports both isotropic and anisotropic lengthscales. Available kernels include the Gaussian, Matérn 5/2, and Matérn 3/2.

Usage

kernel_matrix(
  X,
  Xprime = NULL,
  theta = 0.1,
  kernel = c("gaussian", "matern52", "matern32"),
  anisotropic = TRUE
)

Arguments

X	A numeric matrix (or vector) of input locations with shape n \times d.
Xprime	An optional numeric matrix of input locations with shape m \times d. If NULL (default), it is set to X, resulting in a symmetric matrix.
theta	A positive numeric value or vector specifying the kernel lengthscale(s). If a scalar, the same lengthscale is applied to all input dimensions. If a vector, it must be of length d, corresponding to anisotropic scaling.
kernel	A character string specifying the kernel function. Must be one of "gaussian", "matern32", or "matern52".
anisotropic	Logical. If TRUE (default), theta is interpreted as a vector of per-dimension lengthscales. If FALSE, theta is treated as a scalar.

Details

Let \mathbf{x} and \mathbf{x}' denote two input points. The scaled distance is defined as

r = \left\| \frac{\mathbf{x} - \mathbf{x}'}{\boldsymbol{\theta}} \right\|_2.

The available kernels are defined as:

• Gaussian:

  k(\mathbf{x}, \mathbf{x}') = \exp(-r^2)

• Matérn 5/2:

  k(\mathbf{x}, \mathbf{x}') = \left(1 + \sqrt{5} r + \frac{5}{3} r^2 \right) \exp(-\sqrt{5} r)

• Matérn 3/2:

  k(\mathbf{x}, \mathbf{x}') = \left(1 + \sqrt{3} r \right) \exp(-\sqrt{3} r)

The function performs consistency checks on input dimensions and automatically broadcasts theta when it is a scalar.

Value

A numeric matrix of size n \times m, where each element K_{ij} gives the kernel similarity between input X_i and X'_j.

References

Zhao J, Qing K, Xu J (2025). BKP: An R Package for Beta Kernel Process Modeling. arXiv. https://doi.org/10.48550/arxiv.2508.10447.

Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press.

Examples

# Basic usage with default Xprime = X
X <- matrix(runif(20), ncol = 2)
K1 <- kernel_matrix(X, theta = 0.2, kernel = "gaussian")

# Anisotropic lengthscales with Matérn 5/2
K2 <- kernel_matrix(X, theta = c(0.1, 0.3), kernel = "matern52")

# Isotropic Matérn 3/2
K3 <- kernel_matrix(X, theta = 1, kernel = "matern32", anisotropic = FALSE)

# Use Xprime different from X
Xprime <- matrix(runif(10), ncol = 2)
K4 <- kernel_matrix(X, Xprime, theta = 0.2, kernel = "gaussian")

NBKP documentation built on June 18, 2026, 1:06 a.m.