Glossary ======== .. glossary:: :sorted: 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 :math:`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 :class:`~pgmuvi.multiband_ls_significance.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 :math:`k(\mathbf{x}, \mathbf{x}')` that defines the covariance between the GP function values at inputs :math:`\mathbf{x}` and :math:`\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): :math:`P = 1 / \mu_q`. 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. 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. 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. RBF kernel Squared-exponential kernel A smooth kernel defined by :math:`k(r) = \exp(-r^2 / 2\ell^2)` where :math:`\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. R-hat :math:`\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. Separable kernel A multi-dimensional kernel of the form :math:`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.