State Space Model Calculator

State equation (hidden process):

Observation equation (measured series):

Kalman predict:

Kalman update:

When smoothing is enabled, the calculator applies an RTS backward pass to produce x̂ₜ|T.

1. Paste your observations as comma-separated values or one per line.
2. Choose a model: set F and H to represent dynamics and measurement scaling.
3. Set Q for process variability and R for measurement noise.
4. Provide x₀ and P₀ to encode your prior belief.
5. Optional: enable smoothing to estimate the full-state path using all data.
6. Press Submit to see results, charts, and export buttons above.

Model structure and interpretation

A state space model separates an observed series from a latent state that evolves over time. The transition coefficient F controls persistence, while the observation coefficient H links the hidden level to measurements. Larger absolute F produces longer memory and smoother dynamics, and smaller absolute F produces quicker mean reversion. With F=1 and H=1, the setup becomes a local level model that treats the state as a drifting baseline.

Choosing process and observation noise

Process noise Q describes genuine movement in the hidden state, while observation noise R represents measurement error. When Q is high relative to R, the filter follows new readings rapidly. When R is high relative to Q, the filter discounts noisy points and relies more on the predicted state path. A practical rule is to start with Q near the variance of short-term changes and R near the variance of measurement scatter.

Initialization and stability checks

Initial state x0 and variance P0 encode prior belief at time one. A small P0 forces early estimates to stay near x0, while a larger P0 allows rapid adaptation. Stable calculations also require R to be positive and innovation variance S to remain above numerical tolerance. If you include missing values marked as NA, the filter will skip updates and carry forward the prediction.

Forecasting, smoothing, and diagnostics

The prediction x̂t|t−1 provides one-step forecasts that can be compared against yt using innovations vt. The log-likelihood summarizes how plausible the data are under the chosen parameters, and the calculator reports informal AIC and BIC for comparison across alternative settings. Enabling RTS smoothing adds x̂t|T, which uses all observations to refine earlier states, reduce lag, and often tighten uncertainty when the series is dense.

Practical uses and reporting outputs

This calculator supports trend extraction, sensor fusion, and nowcasting in finance, operations, and engineering. Exported tables include gains Kt and variances Pt that document uncertainty, not only point estimates. Report RMSE and MAE of innovations to communicate residual size consistently across runs. When tuning, aim for innovations centered near zero and avoid spiky Kt values unless your process changes.

What does the transition coefficient F represent?

F controls how strongly the hidden state carries forward from one step to the next. Values near 1 imply high persistence, while values closer to 0 imply faster forgetting. Negative values can model alternating behavior, but they may be harder to interpret.

How should I pick Q and R if I have no prior estimates?

Start with R near the variance of measurement noise and Q near the variance of short-term state changes. Then adjust: increase R if the filter overreacts to spikes, or increase Q if estimates lag behind real shifts.

What happens when I enter NA for some observations?

Missing observations skip the update step. The calculator uses the prediction as the filtered estimate for that time, then continues normally at the next available measurement. This is useful for irregular sampling or sensor dropouts.

Why is the Kalman gain K sometimes large?

K grows when prediction uncertainty is high relative to observation uncertainty. A large gain means the filter trusts the new measurement strongly. If K is consistently extreme, reconsider Q, R, or whether the model structure matches the data.

What is the difference between filtered and smoothed states?

Filtered states use information up to the current time only, so they can lag during sharp changes. Smoothed states use the full series and revise earlier estimates backward, often producing cleaner paths and smaller uncertainty for past periods.

How do I compare two model settings objectively?

Compare log-likelihood and the reported AIC/BIC using the same dataset. Prefer higher likelihood and lower AIC/BIC, but also inspect innovations for bias and the state path for plausibility. Good fit should not produce unstable variances.