Structural (state-space) time-series model decomposes a series into unobserved level, slope, and seasonal components with a Kalman filter, then forecasts ahead with a model-based prediction band.
Rather than differencing the series (as ARIMA does), a structural model writes it as a sum of unobserved components — a stochastic level, an optional slope (trend), and an optional dummy-variable seasonal — each driven by its own random shock. The variances of those shocks are estimated by Gaussian maximum likelihood via the Kalman filter's prediction-error decomposition; a state smoother then recovers the most likely path of each component over the whole sample.
A zero estimated variance is informative: it collapses a component to a deterministic form (e.g. a fixed slope or a constant seasonal pattern). Forecasts iterate the fitted state forward, and the prediction interval widens with the horizon as the accumulated component uncertainty compounds.
One numeric series (row order = time) + model type (local level / local linear trend / basic structural) + seasonal period (for the seasonal model) + forecast horizon h + interval level.
Estimated component variances, log-likelihood and AIC, smoothed level/slope/seasonal component charts, and a history-plus-forecast fan chart with a table of point forecasts and bounds.
The smoothed level (and slope) show the underlying trend with noise and season removed; the seasonal component shows the repeating pattern. A near-zero component variance means that piece is effectively deterministic over the sample.
Fit is by numerical optimization, so component variances match a reference state-space fit to practical precision (~1e-3); the log-likelihood and forecasts agree much more tightly.