kernel

Compute the squared exponential kernel defined as \(k = \exp(-\theta (x - x')^2) + g\) , where \(\theta\) is the lengthscale parameter and \(g\) is a jitter term.
Both isotropic and separable kernels are supported.

Implements maximum likelihood estimation for Gaussian processes, supporting both isotropic and separable models with predictive capabilities. Includes penalized likelihood estimation following Li and Sudjianto (2005, <doi:10.1198/004017004000000671>), with cross-validation guided by decorrelated prediction error (DPE) metric. DPE metric, motivated by Mahalanobis distance, serves as evaluation criteria that accounts for predictive uncertainty in tuning parameter selection (Mutoh, Booth, and Stallrich, 2025, <doi:10.48550/arXiv.2511.18111>). Designed specifically for small datasets.

Ayumi Mutoh

GPpenalty

Penalized Likelihood in Gaussian Processes

kernel function

<dl><dt>x1</dt>
<dd>matrix of input locations</dd>
<dt>theta</dt>
<dd>a scalar or vector specifying the lengthscale parameter. If a vector is provided, a separable kernel function is used.
If a scalar is provided and <code>x1</code> has more than one column, an isotropic kernel is assumed.</dd>
<dt>x2</dt>
<dd>matrix of second input locations. If <code>NULL</code>, distance is computed between <code>x1</code> and itself.</dd>
<dt>g</dt>
<dd>a jitter term. It is added when <code>x2=NULL</code> for computational stability.</dd></dl>

Arguments

kernel — kernel

<dl>

<dt>x1</dt>
<dd>matrix of input locations</dd>


<dt>theta</dt>
<dd>a scalar or vector specifying the lengthscale parameter. If a vector is provided, a separable kernel function is used.
If a scalar is provided and <code>x1</code> has more than one column, an isotropic kernel is assumed.</dd>


<dt>x2</dt>
<dd>matrix of second input locations. If <code>NULL</code>, distance is computed between <code>x1</code> and itself.</dd>


<dt>g</dt>
<dd>a jitter term. It is added when <code>x2=NULL</code> for computational stability.</dd>

</dl>

kernel: kernel

Description

Usage

Value

Arguments

Details

Examples