https://github.com/elizavetasemenova/kronecker-gp-examples
Examples of Gaussian Processes (GP) regression with non-Gaussian likelihoods and missing values for Kronecker-structured covariance matrices.
https://github.com/elizavetasemenova/kronecker-gp-examples
gaussian-processes kronecker-product stan
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Examples of Gaussian Processes (GP) regression with non-Gaussian likelihoods and missing values for Kronecker-structured covariance matrices.
- Host: GitHub
- URL: https://github.com/elizavetasemenova/kronecker-gp-examples
- Owner: elizavetasemenova
- Created: 2024-12-12T18:24:48.000Z (10 months ago)
- Default Branch: main
- Last Pushed: 2024-12-22T17:48:31.000Z (10 months ago)
- Last Synced: 2025-03-31T03:29:55.939Z (7 months ago)
- Topics: gaussian-processes, kronecker-product, stan
- Language: R
- Homepage:
- Size: 25.4 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Kronecker Gaussian Process - examples in Stan
These examples demonstrate Gaussian Processes (GP) regression with non-Gaussian likelihoods and missing values for Kronecker-structured covariance matrices.
## Key idea
If $\mathbf{x} = (x_1, x_2)$ and the kernel factorizes as:
$$
k((x_1,x_2),(x_1',x_2')) = k_1(x_1,x_1')\,k_2(x_2,x_2'),
$$
then the covariance matrix over grids $X_1 \times X_2$ can be written as:
$$
K = K_2 \otimes K_1.
$$
For three dimensions (e.g., space $\times$ space $\times$ time):
$$
k((x_1,x_2,t),(x_1',x_2',t')) = k_1(x_1,x_1')\,k_2(x_2,x_2')\,k_t(t,t'),
$$
implying:
$$
K = K_t \otimes K_2 \otimes K_1.
$$
This structure enables efficient computations (e.g. Cholesky) at $O(n^{3/d})$ instead of $O(n^3)$.
## Likelihoods
- **Normal**:
$$
y(\mathbf{x}) = f(\mathbf{x}) + \epsilon, \quad \epsilon \sim \mathcal{N}(0,\sigma^2).
$$
- **Poisson**:
$$
y(\mathbf{x}) \sim \text{Poisson}(\exp(f(\mathbf{x}))).
$$
## Examples
- **01**: 2d GP with Normal likelihood, no missingness
- **02**: 2d GP with Poisson likelihood, no missingness
- **03**: 2d GP with Poisson likelihood, with missing values (impute from posterior)
- **04**: 3d GP (e.g. space-time), Poisson likelihood, no missingness, fixed parameters
- **05**: 3d GP with Poisson likelihood, no missingness, parameter inference
- **06**: 3d GP with Poisson likelihood, with missing values (impute from posterior)