Fixed-lag online data assimilation

Generic reconstructing the hidden state at each time from the observed record up to a fixed number of steps ahead of it. The closed-form method is used for a cgns_model; a general stochastic_model is out of scope in this release.

Usage

aci_online(model, obs, lag, ...)

# S3 method for class 'cgns_model'
aci_online(
  model,
  obs,
  lag,
  filter = NULL,
  conditional = NULL,
  init = NULL,
  regularize = NULL,
  force_validate = FALSE,
  ...
)

# S3 method for class 'stochastic_model'
aci_online(model, obs, lag, ...)

Arguments

model: A cgns_model or stochastic_model object.
obs: An observed trajectory, or anything as_obs() accepts.
lag: Number of future steps each estimate may condition on. A non-negative whole number, or Inf for the whole record. No default: the lag is the argument the function exists for, and defaulting it invites the full-lag result to be mistaken for aci_smoother().
...: Arguments passed to methods.
filter: Optional precomputed filter path; recomputed when NULL. It must be the explicit single-step filter, which is the discretization the Theorem 3 recursions are exact for.
conditional: Optional aci_conditional_spec selecting a conditional ACI reduction; see aci_conditional().
init: Optional list with the initial hidden mean and cov.
regularize: Covariance policy for this call; see aci_filter(). One record covers the whole call, and is returned in meta$regularization.
force_validate: FALSE (the default) lets a filter that aci_filter() produced for this same run, unaltered since, skip per-step re-validation, as in aci_smoother().

Value

An assimilation path of kind "online", carrying meta$lag, the per-anchor meta$lag_effective, meta$saturated and meta$scheme. Its kind is what keeps it out of the places a complete smoother is required: lag_table(smoother = ) rejects it with aci_error_dims.

Details

lag is the number of future observations each estimate may condition on: the estimate at index j uses the observed record through index j + lag, and saturates at the end of the record. lag = 0 returns the filter moments unchanged. lag = Inf returns the complete Theorem 3 posterior given the whole record.

Methods (by class)

aci_online(cgns_model): Closed-form fixed-lag online Theorem 3 smoother for a conditional-Gaussian model.
aci_online(stochastic_model): Classed not-implemented condition for a general (non-CGNS) stochastic model.

Scheme

aci_online() computes the discrete Theorem 3 posterior: the exact conditional law of the hidden state given the observed increments on the sampling grid under the explicit single-step discretization. aci_smoother() integrates the continuous backward smoothing equations with an Euler step of the same size. These are two discretizations of the same continuous-time object and they agree only to first order in the step, so at full lag aci_online() does not reproduce aci_smoother(). The gap grows with the length of the record; it is not a constant offset. On the packaged ENSO partition (l = 3, dt = 0.005) the smoothed means differ by up to 1.89e-02 against a mean scale of 0.388 over 401 steps, and by up to 9.58e-02 against a scale of 2.22 over 4001 steps. The resulting ACI values differ by up to 0.104 against a scale of 1.093 at 401 steps and 0.482 against 2.347 at 4001. lag_table() and the causal influence range estimators use the discrete scheme throughout, which is why lt_diag() and aci()$aci can differ by that same amount. meta$scheme records which scheme produced a path: "theorem3_discrete" here and on the lag table's reference smoother, "backward_ode_euler" on aci_smoother().

The one exact boundary is the other end: at lag = 0 the returned moments are the filter moments, unchanged value for value.

References

Andreou, M., Chen, N. and Li, Y. (2026). An adaptive online smoother with closed-form solutions and information-theoretic lag selection for conditional Gaussian nonlinear systems. Journal of Nonlinear Science 36(4), 71. doi:10.1007/s00332-026-10271-x

Examples

m <- aci_dyad_model()
sim <- simulate(m, seed = 1, t_end = 2, dt = 0.01)
ob <- as_obs(sim)
aci_online(m, ob, lag = 5)
#> Warning: No init$cov supplied; using a diffuse prior. Discard an initial burn-in window when interpreting results.
#> <da_path_gaussian> kind = online, l = 1, N+1 = 201