Glossary

ARD

Automatic Relevance Determination. A setting that gives each input dimension its own length-scale parameter, allowing the model to determine which dimensions are most informative.

Bayesian inference

A statistical framework in which unknown quantities are treated as random variables and updated with observed data via Bayes’ theorem to produce a posterior distribution.

BIC

Bayesian Information Criterion. A penalised likelihood metric used for model comparison; lower BIC indicates a better balance between fit quality and model complexity.

Cholesky decomposition

A factorisation of a positive definite matrix used internally by GPyTorch to compute GP predictions and marginal likelihoods efficiently.

Coherence time

The timescale over which a quasi-periodic oscillation remains phase-coherent. In the spectral mixture kernel, longer coherence corresponds to narrower bandwidth (smaller mixture_scales).

Constraint

A hard bound placed on a GP hyperparameter, preventing the optimiser from exploring values outside a specified range. Implemented as a GPyTorch Interval constraint in pgmuvi.

Excess variance

F_var

A measure of variability amplitude equal to the excess variance beyond what is expected from measurement noise alone, normalised by the mean flux. Defined as \(F_\mathrm{var} = \sqrt{S^2 - \bar{\sigma^2}} / \bar{x}\).

False alarm probability

FAP

The probability of observing a periodogram peak of a given height by chance if the data contain no periodic signal. Used to assess the statistical significance of a detected period. Available via MultibandLSWithSignificance.

Gaussian process

GP

A probability distribution over functions, fully specified by a mean function and a covariance (kernel) function. Any finite collection of function values has a multivariate Gaussian distribution.

GPyTorch

An open-source Python library for Gaussian process inference built on PyTorch. pgmuvi is built on top of GPyTorch.

Hyperparameter

A parameter of the GP kernel or likelihood (as opposed to a function value). Examples include the frequency, bandwidth, and weight of a spectral mixture component, and the noise variance.

Kernel function

Covariance function

A function \(k(\mathbf{x}, \mathbf{x}')\) that defines the covariance between the GP function values at inputs \(\mathbf{x}\) and \(\mathbf{x}'\). The choice of kernel encodes prior assumptions about the smoothness, periodicity, and other properties of the modelled function.

Lomb–Scargle periodogram

A method for estimating the power spectral density of unevenly sampled time series data. Used in pgmuvi to initialise spectral mixture kernel frequencies before optimisation.

MAP estimation

Maximum a posteriori estimation

An optimisation-based approach that finds the single parameter configuration maximising the posterior probability. Fast but does not provide uncertainty estimates on the parameters.

MCMC

Markov chain Monte Carlo

A class of algorithms for sampling from a probability distribution (the posterior in Bayesian inference). MCMC provides full uncertainty quantification but is more computationally expensive than MAP estimation. MCMC support (via Hamiltonian Monte Carlo) is planned for a future release of pgmuvi; the current version supports MAP estimation only.

Mixture component

One term in a spectral mixture kernel, characterised by a centre frequency, bandwidth, and weight. The number of components is controlled by num_mixtures.

Nyquist period

The shortest variability period that can be reliably detected given the sampling cadence, equal to approximately twice the median inter-observation interval. Signals shorter than the Nyquist period are aliased.

Period

The characteristic timescale of a quasi-periodic signal. In the spectral mixture kernel, the period is the reciprocal of the mixture mean (centre frequency): \(P = 1 / \mu_q\).

Prior

A probability distribution placed on a GP hyperparameter before observing any data. Priors encode domain knowledge (e.g., expected period range) and regularise the posterior during MCMC sampling.

PSD

Power spectral density

A function describing how the variance of a time series is distributed across frequencies. Peaks in the PSD indicate quasi-periodic variability; broad low-frequency power indicates correlated (red) noise.

Quasi-periodic variability

Variability that is approximately periodic but lacks exact phase coherence; the period drifts or the amplitude modulates over time. Well described by a spectral mixture kernel with finite bandwidth.

R-hat

\(\hat{R}\)

A convergence diagnostic for MCMC. Values close to 1.0 (< 1.01 is the standard criterion) indicate that multiple chains have converged to the same distribution.

RBF kernel

Squared-exponential kernel

A smooth kernel defined by \(k(r) = \exp(-r^2 / 2\ell^2)\) where \(\ell\) is the length-scale. Represents smooth, aperiodic variability.

Red noise

Stochastic variability with power concentrated at low frequencies (long timescales). Common in AGN and many other astrophysical sources.

Separable kernel

A multi-dimensional kernel of the form \(k(\mathbf{x}, \mathbf{x}') = k_1(x_1, x_1') \cdot k_2(x_2, x_2')\). pgmuvi’s separable 2D model family (e.g. "2DSeparable", "2DAchromatic") uses product kernels with a temporal component and a wavelength component. The default model="2D" uses a non-separable 2D spectral-mixture kernel instead.

Spectral mixture kernel

SMK

A kernel defined by a mixture of Gaussians in the frequency domain (see Wilson & Adams 2013). Highly flexible; can represent quasi-periodic signals, red noise, and multiple simultaneous periodicities.

Stetson K

A robust index of light-curve variability, sensitive to correlated deviations from the mean flux. Less sensitive to outliers than the chi-square test.

White noise

Stochastic variability that is uncorrelated between observations (flat PSD). Modelled in pgmuvi by the GP likelihood noise parameter.