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State-Space Model
The general modeling framework in which an unobserved state evolves over time according to a Markov-style state equation, and observations are generated from that hidden state via a separate observation equation, the umbrella class containing hidden Markov models, the Kalman filter's linear-Gaussian setting, and particle-filtered nonlinear models as special cases.
In Plain English
A state-space model is the general two-layer template underlying an entire family of filtering and estimation problems: somewhere there's a true, hidden state evolving over time, and all you get to see are noisy or indirect observations generated from that state. Which specific algorithm you use to actually estimate the hidden state, exact discrete-state HMM machinery, the Kalman filter's exact linear-Gaussian formulas, or a particle filter's Monte Carlo approximation, depends entirely on what kind of state-space model you're dealing with: discrete states, continuous linear-Gaussian states, or fully general nonlinear/non-Gaussian states.
Definition
A state-space model consists of a state equation describing how a hidden state evolves, generally for some (possibly nonlinear) function and process noise , satisfying the Markov property, together with an observation equation linking the hidden state to the observed data via some (possibly nonlinear) function and observation noise . This general template unifies several of this category's central models as special cases: a hidden Markov model is a state-space model with a discrete hidden state; the model solved by the Kalman filter is a state-space model with linear and Gaussian ; and the model solved by a particle filter is a state-space model with general (nonlinear, non-Gaussian) . This general, theory-facing framework should be distinguished from state-space representation, the specific time-series modeling technique of rewriting a particular applied model (an ARMA process, a structural local-level/trend model) into this two-equation form in order to apply Kalman-filter-based estimation, that entry describes one specific technique built on top of the general framework defined here.
Properties
- Recognizing the shared state-space-model structure underneath the hidden Markov model, the Kalman filter, and the particle filter is what makes these three seemingly separate algorithms into one coherent family: different choices of state type (discrete vs. continuous) and dynamics (linear-Gaussian vs. general) simply call for a different, matched estimation algorithm.
- This general framework and state-space representation (the time-series technique) are closely related but answer different questions: this entry defines the class of models and their filtering algorithms broadly, while state-space representation is specifically about rewriting a given time series model, like ARMA, into this form so Kalman-filter machinery becomes available for estimation.
- Beyond the three headline algorithms in this category, state-space models underpin an enormous range of applied sequential-estimation problems: GPS navigation, econometric structural models with time-varying parameters, and epidemiological compartment models with noisy case-count observations are all naturally posed as state-space models.
At a Glance