Kalman-filter log-likelihood of the SIRD-GA state-space model

Runs the predict/update Kalman recursion of the linear-Gaussian state-space representation of the SIRD-GA model and returns the approximate log-likelihood of the observed (partially reported, noisy) infectious counts.

Usage

sird_ga_loglik(
  theta,
  y,
  N,
  Delta = 1,
  n_sub = 10L,
  Q0_diag = 1e-08,
  keep_states = FALSE
)

Arguments

theta

Named numeric vector with elements lambda, gamma, alpha, s0, i0, r0, p, tau2 (all strictly positive except r0, which may be zero; p in (0,1)).

y

Numeric vector of observed infectious head counts \(y_0, y_1, \ldots, y_n\) (not normalised; normalisation by N is performed internally, matching the \(O_N(t_k)=O(t_k)/N\) observation model of the manuscript).

N

Population size.

Delta

Time step between consecutive observations (default 1).

n_sub

Internal quadrature resolution, passed to sird_ga_transition().

Q0_diag

Variance assigned to each component of the initial-state distribution (small, reflects near-certainty about x0).

keep_states

Logical; if TRUE, also return the filtered state trajectory (in normalised units) and the predictive mean on the original (head-count) scale.

Value

If keep_states is FALSE (default), the scalar approximate log-likelihood. Otherwise a list with elements loglik, filtered (matrix of filtered means \(\bar X_k\), normalised units), predicted (matrix of one-step-ahead predicted means \(\hat X_k\), normalised units) and fitted (predictive mean of y, on the original head-count scale, i.e. N * p * predicted[, "i"]).

Details

The state-space system is $$X_k = F_k(\eta,\Delta) + A_{k-1}(\eta,\Delta) X_{k-1} + U_k, \quad U_k \sim N_3(0, Q_k(\eta,\Delta)),$$ $$Y_k/N = p \, X_{k,2} + V_k, \quad V_k \sim N(0, R_k(\theta)),$$ with \(X_k=(S_k,I_k,R_k)/N\), observation operator B = (0, p, 0) and \(R_k(\theta) = N^{-1}(p(1-p)+\tau^2) \, i(\eta,t_k)\), following the observation model of Narci et al. (2021) extended to the three-event SIRD jump process.