> For the complete documentation index, see [llms.txt](https://docs.algenta.ai/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.algenta.ai/machine-learning/ml-gaussian-processes.md).

# Gaussian process regression

Gaussian process regression with RBF and Matern kernels, exact posterior mean and variance.

`gaussian_process` fits exact zero-mean Gaussian processes on flat row-major feature matrices: covariance construction, its own Cholesky factorization, posterior mean/variance prediction, and log marginal likelihood. It replaces `sklearn.gaussian_process` (`GaussianProcessRegressor` with RBF / Matern nu = 1.5 kernels) and basic GPy usage, with zero Python imports. Fitting is deterministic and O(n³) — sub-millisecond up to n ≈ 200, with n ≈ 2000 the practical exact-GP cap.

## The modules

| Module             | What it does                                                                                                                                                                                                                                                                                         |
| ------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `gaussian_process` | exact zero-mean Gaussian process regression on flat row-major feature matrices — covariance construction, own Cholesky factorization, posterior mean/variance, log marginal likelihood — replacing `sklearn.gaussian_process`'s `GaussianProcessRegressor` (RBF / Matern nu=1.5) and basic GPy usage |

## Functions

| Function                                                        | What it does                                                                                                                           |
| --------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------- |
| `kernel_rbf(r, lengthscale, signal_var)`                        | RBF (squared-exponential) covariance `signal_var · exp(-r²/(2·lengthscale²))` at Euclidean distance r.                                 |
| `kernel_matern32(r, lengthscale, signal_var)`                   | Matern nu = 3/2 covariance `signal_var · (1 + √3·r/l) · exp(-√3·r/l)`.                                                                 |
| `gp_fit(x, y, dim, lengthscale, signal_var, noise_var, kernel)` | Fits an exact GP: builds `K + noise_var·I`, Cholesky-factorizes, solves `alpha = K⁻¹y`. `kernel` is `KERNEL_RBF` or `KERNEL_MATERN32`. |
| `gp_predict(x_train, fit, x_new)`                               | Posterior mean `k*ᵀ alpha` and latent variance `signal_var − ‖L⁻¹k*‖²` (clamped at 0) per query row; O(n²) per point.                  |
| `gp_loglik(fit, y)`                                             | Log marginal likelihood `-½ yᵀalpha − Σ log L_ii − n/2 · log 2π` of the training targets.                                              |

## Guarantees and failure modes

* **Deterministic.** Refitting identical inputs reproduces `alpha` bit-for-bit (asserted in the tests). If the factorization hits a non-positive pivot (e.g. `noise_var = 0` with duplicate rows), it retries on a deterministic escalating jitter ladder (1e-10, 1e-8, 1e-6 scaled by `signal_var + noise_var`), recorded in `fit.jitter` (0.0 normally).
* Predicted variance is the **latent-function** variance and excludes observation noise; add `fit.noise_var` for a noisy-observation predictive variance. Far from all training points the posterior reverts to the prior: mean → 0, variance → `signal_var` (tested to 1e-6).
* Invalid input (empty data, length mismatch, `lengthscale <= 0`, `signal_var <= 0`, `noise_var < 0`, unknown kernel) returns `ok = False` with empty fields; `gp_loglik` returns NaN on a bad fit or wrong-length `y`. No exceptions on the money path.

## Verification

Tests pin the numerics against closed-form references:

* Kernel values: `kernel_rbf(1, 1, 1) = exp(-0.5) = 0.6065306597126334` and `kernel_matern32(1, 1, 1) = (1+√3)·e^{-√3} = 0.4833577245965077`, both to 1e-12; `r = 0` returns exactly `signal_var`.
* 2-point toy (x = \[0, 1], y = \[1, 2], RBF, l = 1, sv = 1, noise = 0.1) against the analytic 2×2 inverse: `alpha = [-0.13425787820991109, 1.8922104722202504]` (1e-9), prediction at 0.5 has mean 1.5513877191046792 and variance 0.08727009545489339 (1e-9), and the log marginal likelihood equals the closed form −3.5770425527832885 (1e-8; Mojo's `std.math` log/exp are \~1e-10-accurate approximations).
* n = 1 closed forms: loglik −1.8270838991417502 and prediction at the training point `sv/(sv+noise) · y = 1.2` exactly (1e-12).
* Interpolation sanity for both kernels: mean ≈ `y_i` at training points with variance ≈ 0 (tol 1e-3), including duplicate rows and 2-D inputs.

**Next:** promote what you trained with [statistical model comparison](/machine-learning/ml-model-comparison.md), or browse the [engine catalog](/engine-runtime/catalog.md).


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation by asking a question.

Perform an HTTP GET request on the following URL with the `ask` and `goal` query parameters:

```
GET https://docs.algenta.ai/machine-learning/ml-gaussian-processes.md?ask=<question>&goal=<user_goal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is what the user is ultimately trying to achieve, the reason they need the answer. Sharing it helps GitBook give you a better, more relevant answer. A goal is most helpful when it describes the outcome the user wants rather than restating the question. For example, with `ask=how do I create an API token`, a goal like `automate deployments from our CI pipeline` lets GitBook tailor the answer to that use case.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.