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Covariance Function

Kernel function that defines the correlation between two input points in a Gaussian process, determining the regularity and structure of the modeled function.

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Matérn Kernel

Family of covariance functions parameterized by a roughness factor ν, offering fine control over the differentiability of the modeled Gaussian process.

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RBF (Gaussian) Kernel

Infinitely differentiable radial basis function covariance, assuming very smooth functions and widely used for standard Gaussian processes.

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Kernel Hyperparameters

Parameters of the covariance function (such as length scale and variance) that control the behavior of the Gaussian process and are optimized by maximum likelihood.

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Length Scale

Kernel hyperparameter determining the distance over which input points are correlated, controlling the variability of the function modeled by the Gaussian process.

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Signal Variance

Kernel hyperparameter representing the vertical standard deviation of the modeled function, controlling the average amplitude of fluctuations in the Gaussian process.

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Observational Noise

Parameter σ² modeling the uncertainty of observations, added to the diagonal of the covariance matrix to handle noisy data in Gaussian processes.

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Conditional Distribution Prediction

Calculation of the posterior distribution of the Gaussian process at a new point, conditioned on existing observations to provide predictive mean and variance.

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Maximum Likelihood Evidence Maximization

Procedure for optimizing Gaussian process hyperparameters by maximizing the marginal log-likelihood of the observed data under the model.

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Karhunen-Loève Theorem

Decomposition of a Gaussian process into a series of orthogonal functions with independent Gaussian coefficients, enabling a compact representation of the process.

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Dot-Product Kernel

Covariance function k(x,x') = σ² + xᵀx' used to model linear or polynomial functions in Gaussian processes.

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Deep Gaussian Process

Extension of Gaussian processes where the covariance function is itself parameterized by a neural network, allowing for complex non-stationary models.

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Sparse Gaussian Process

Computational approximation using inducing points to reduce the cubic complexity O(n³) of standard Gaussian processes for large datasets.

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Cholesky Decomposition

Factorization of the covariance matrix K = LLᵀ used to efficiently solve linear systems and compute the log-likelihood in Gaussian processes.