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State-Space Model

Generative model — instantiates Model-Guided Signal Separation

Specifies the target as a hidden state that evolves by known dynamics and is seen only through a noisy observation equation — the source model an estimator later inverts to pull the state back out.

A State-Space Model is the source model at the heart of model-guided separation: it writes down, explicitly, how a hidden state produces what you actually measure. Two equations carry it — a transition equation saying how the latent state evolves from one step to the next, and an observation equation saying how that state is corrupted, mixed, and projected into the noisy signal you record. Its defining contribution is that it commits to the generative story before any filtering happens: the target is named as the state, the dynamics that constrain it are made explicit, and the noise entering both equations is modeled rather than assumed away. It does not itself recover the state — it is the structured model that a recursive estimator such as a Kalman or Particle Filter runs on. Build the model wrong and the cleanest filter in the world will confidently recover the wrong signal.

Example

A water utility wants the true level of a reservoir minute by minute, but its only gauge is a pressure sensor that drifts, spikes, and is jostled by wind-driven surface chop. A State-Space Model frames the problem: the hidden state is the true level and its rate of change; the transition equation says the level can move only through metered inflow minus outflow plus rain, so it cannot physically jump; the observation equation says the sensor reads the true level plus a slow drift term plus zero-mean chop.

That model encodes what the raw reading cannot. A five-centimetre spike between two one-second samples is impossible under the dynamics, so it must be noise, not level. Handed to an estimator, the model is what lets the filter attribute each wiggle to either real inflow or sensor error — but the attribution is only ever as good as the two equations the model wrote down.

How it works

What distinguishes a state-space model from the estimator that runs on it is that it separates structure from inference. Its whole job is the forward generative specification:

  • Name the state — the latent quantities the target consists of, and their dimension.
  • Write the transition — how the state evolves, with process noise absorbing what the dynamics don't capture.
  • Write the observation map — how the state becomes the measurement, with measurement noise for sensor error.
  • Specify the noise covariances — how much to trust the dynamics versus the sensor.

It then hands this structure to an estimator, which does the recursive inversion. The model is deliberately a specification, not a solver.

Tuning parameters

  • State dimension — how much latent structure to track (level alone, or level plus velocity plus drift); richer states capture more but need more information to identify.
  • Process- vs. measurement-noise ratio — raise process noise and the estimate chases the data; lower it and the estimate trusts the model.
  • Linear vs. nonlinear dynamics — fidelity against tractability; linear-Gaussian is cheap, nonlinear needs heavier estimators.
  • Time-invariant vs. time-varying parameters — whether the model itself is allowed to drift.
  • Constraint encoding — hard physical bounds (non-negativity, conservation) baked into the transition.

When it helps, and when it misleads

Its strength is that it makes the generative assumptions explicit and auditable, folds physical constraints directly into the dynamics, and cleanly separates model from estimator so each can be improved on its own.

Model misspecification is the killer: a wrong transition or an understated noise term produces an estimate that is precise and wrong, and the filter's own confidence bands will not warn you. The state is recoverable only if the system is observable — if the observations genuinely constrain it[1]; an unobservable state can be filtered smoothly and still be arbitrary. The classic misuse is to tune the noise covariances until the output matches what you expected to see. The discipline is to check observability, validate against held-out reality, and stress the noise assumptions rather than fitting them to taste.

How it implements the components

State-Space Model fills the source-modeling side of the archetype — the forward generative story, not the inversion:

  • observation_mixture_model — the observation equation is the model of how the hidden state is projected and mixed into the recorded signal.
  • nuisance_and_noise_model — process noise and measurement noise are named and given covariances, not assumed away.
  • domain_constraint — the transition dynamics encode what the system physically can and cannot do, constraining the solution space.

It stops at the model: it does not run the recursive estimation that inverts it — that is the Kalman or Particle Filter — nor check that the latent target is identifiable, which Latent Variable Model foregrounds.

Notes

State-Space Model and Latent Variable Model are close cousins that split on time: a state-space model tracks a hidden state evolving through time under explicit dynamics, while a latent variable model explains cross-sectional covariance among variables with static factors. Reach for the state-space form when the target moves and the dynamics are known; reach for the latent-variable form when the target sits beneath many simultaneous measurements.

References

[1] Observability is the control-theory property that a system's internal state can be reconstructed from its outputs over time. If a state-space model is not observable, no estimator — however good — can recover the hidden state uniquely from the observations. ↩