Advanced✓ Mathematical
◆ The PatternHidden states evolving under noise
State-space models assume a hidden state evolves over time according to a transition equation, and we observe it with noise through an observation equation. The Kalman filter optimally estimates the hidden state. ETS and ARIMA can both be written in state-space form.
State: xt = F·xt−1 + wt | Obs: yt = H·xt + vt
The Kalman filter recursively estimates xt from noisy observations yt, balancing model prediction with measurement update.
// Interactive — Kalman filter tracking
Process Noise20
Observation Noise50
# Python — Kalman-based structural model from statsmodels.tsa.statespace.structural import UnobservedComponents model = UnobservedComponents( series, level='local linear trend', seasonal=12 ).fit() forecast = model.get_forecast(steps=24)
Pattern bridge: The Kalman filter is the time-series equivalent of Bayesian updating — prior × likelihood = posterior, applied recursively at each time step. In markets, the hidden state is the true “fair value” obscured by market noise.