Functions
vector	diagSPD_EQ (real alpha, real rho, real L, int M)

vector	matern_indices (int M, real L)

vector	diagSPD_Matern12 (real alpha, real rho, real L, int M)

vector	diagSPD_Matern32 (real alpha, real rho, real L, int M)

vector	diagSPD_Matern52 (real alpha, real rho, real L, int M)

vector	diagSPD_Periodic (real alpha, real rho, int M)

vector	update_gp (matrix PHI, int M, real L, real alpha, real rho, vector eta, int type, real nu)

vector	apply_gp_term (vector base, int present, int T, int G, int M, real L, int type, real nu, int d, matrix PHI, matrix eta, array[] real rho, array[] real alpha, array[] int flat_idx)

Function Documentation

◆ apply_gp_term()

vector apply_gp_term	(	vector	base,
		int	present,
		int	T,
		int	G,
		int	M,
		real	L,
		int	type,
		real	nu,
		int	d,
		matrix	PHI,
		matrix	eta,
		array[]real	rho,
		array[]real	alpha,
		array[]int	flat_idx )

Add an approximate Gaussian process latent term to a per-observation predictor, with optional integer-order differencing.

Encapsulates the spectral update, per-group evaluation, optional integration, and per-observation gather so module call sites are a single line. When present is 0 the input is returned unchanged. Each group shares the length scale rho and magnitude alpha; per-group spectral coefficients are stacked column-wise in eta.

The differencing order d integrates each group's realisation d times before it enters the predictor:

d = 0: stationary deviations of length T (the basis matrix is T x M). This is the original behaviour.
d >= 1: the basis matrix has T - d rows, so the spectral update yields T - d free values. These fill positions (d + 1):T of a length-T vector whose first d entries are zero; applying cumulative_sum() d times then integrates the series. Because the leading d entries start at zero, they stay zero through every integration pass, anchoring the first d values of the realisation to zero. This leaves the free level (for d = 1) and additionally the free slope (for d >= 2) to the module's fixed effects rather than the GP, which is the identifiability fix. d = 1 matches EpiNow2's non-stationary reproduction-number GP (gp[2:(gp_n + 1)] = noise; cumulative_sum).

Parameters

base	Predictor to which the latent process is added.
present	0 = inert, 1 = active.
T	Number of time points in the integrated series.
G	Number of groups.
M	Number of basis functions.
L	Boundary factor.
type	Kernel type (0 SE, 1 periodic, 2 Matern).
nu	Matern smoothness.
d	Differencing (integration) order, `d >= 0`.
PHI	Basis matrix ((T - d) x M, or (T - d) x 2M periodic).
eta	Spectral coefficients ((2M or M) x G matrix; empty when not present).
rho	Length scale (length-1 array, empty when inert).
alpha	Magnitude (length-1 array, empty when inert).
flat_idx	Per-observation column-major index into the (T x G) latent matrix.

Returns: base + gathered Gaussian process contribution.

Definition at line 161 of file gaussian_process.stan.

◆ diagSPD_EQ()

vector diagSPD_EQ	(	real	alpha,
		real	rho,
		real	L,
		int	M )

Hilbert-space reduced-rank (spectral) approximate Gaussian process.

Adapted from EpiNow2 (https://github.com/epiforecasts/EpiNow2, inst/stan/functions/gaussian_process.stan, MIT licensed, copyright EpiForecasts). Lightly modified for epinowcast: array syntax, naming, and an apply_gp_term() wrapper that matches the apply_arima_residual() per-observation gather used elsewhere in this package.

The approximation follows:

Riutort-Mayol et al. (2023), doi:10.1007/s11222-022-10167-2
https://avehtari.github.io/casestudies/Motorcycle/motorcycle_gpcourse.html

A latent series f over T time points is approximated as f = PHI * (sqrt(S(rho, alpha)) .* eta), where PHI is a fixed (data) matrix of basis functions, S is the spectral density of the chosen kernel evaluated at the basis eigenvalues, and eta are unit-normal spectral coefficients. The per-observation contribution is gathered from the (T x G) latent matrix with a flat column-major index, mirroring the ARIMA path. Spectral density for the squared exponential kernel.

Definition at line 28 of file gaussian_process.stan.

◆ diagSPD_Matern12()

vector diagSPD_Matern12	(	real	alpha,
		real	rho,
		real	L,
		int	M )

Spectral density for the 1/2 Matern (Ornstein-Uhlenbeck) kernel.

Definition at line 46 of file gaussian_process.stan.

◆ diagSPD_Matern32()

vector diagSPD_Matern32	(	real	alpha,
		real	rho,
		real	L,
		int	M )

Spectral density for the 3/2 Matern kernel.

Definition at line 54 of file gaussian_process.stan.

◆ diagSPD_Matern52()

vector diagSPD_Matern52	(	real	alpha,
		real	rho,
		real	L,
		int	M )

Spectral density for the 5/2 Matern kernel.

Definition at line 63 of file gaussian_process.stan.

◆ diagSPD_Periodic()

vector diagSPD_Periodic	(	real	alpha,
		real	rho,
		int	M )

Spectral density for the periodic kernel.

Definition at line 72 of file gaussian_process.stan.

◆ matern_indices()

vector matern_indices	(	int	M,
		real	L )

Squared spectral indices shared by the Matern kernels.

Definition at line 38 of file gaussian_process.stan.

◆ update_gp()

vector update_gp	(	matrix	PHI,
		int	M,
		real	L,
		real	alpha,
		real	rho,
		vector	eta,
		int	type,
		real	nu )

Update a Gaussian process using the spectral densities.