State Space Model (Kalman Filter)

ARIMA can itself be written in state space form, so the state space framework is the more general one. The difference is that state space models work directly with interpretable unobserved components (trend, seasonal, cycle) and estimate them in real time with the Kalman filter, whereas ARIMA parameterises the series through autoregressive and moving-average terms.

It is a recursive algorithm that estimates the hidden state period by period. At each step it predicts the next state, compares the prediction to the new observation to form the innovation, and corrects the estimate by the Kalman gain times that innovation. The same recursions also deliver the likelihood used to fit the model.

The baseline Kalman filter assumes a linear, Gaussian system with well-defined measurement and process noise. For nonlinear or non-Gaussian dynamics you need extensions such as the extended or unscented Kalman filter or simulation-based methods.

Around 30 observations is a practical minimum for a continuous time series. With short series you should keep the component structure simple, because trend, seasonal, and cycle components compete for the same limited information and can become hard to separate.