Linear-Gaussian State-Space Models

Summary

A state-space model pairs a state transition (dynamic) equation describing how a hidden Markov state ${\mathbf{x}}_k$ evolves with an observation (measurement) equation describing how each measurement ${\mathbf{y}}_k$ depends on the current state. In the linear-Gaussian case both equations are linear with additive Gaussian noise, so the entire model is specified by the matrices ${\mathbf{A}}_{k-1},{\mathbf{H}}_k$, the noise covariances ${\mathbf{Q}}_{k-1},{\mathbf{R}}_k$, and a Gaussian prior $\mathcal{N}({\mathbf{m}}_0,{\mathbf{P}}_0)$. This is the model the Kalman filter solves exactly.

Overview

Särkkä builds filtering on the general probabilistic state-space model (a hidden Markov model with continuous state), then specializes to the linear-Gaussian case. The general form makes the two structural assumptions — Markov dynamics and conditionally independent measurements — explicit; the linear-Gaussian form adds the closed-form algebra.

Main Content

Definition: Probabilistic state-space model (Särkkä Def. 4.1)

A probabilistic state-space model (a.k.a. non-linear filtering model) is a sequence of conditional distributions

$${\mathbf{x}}_k\sim p({\mathbf{x}}_k\mid {\mathbf{x}}_{k-1}),{\mathbf{y}}_k\sim p({\mathbf{y}}_k\mid {\mathbf{x}}_k),k=1,2,\dots \text{\hspace{1em}(4.1)}$$

where

• ${\mathbf{x}}_k\in {\mathbb{R}}^n$ is the state at time step $k$ (hidden);
• ${\mathbf{y}}_k\in {\mathbb{R}}^m$ is the measurement at time step $k$ (observed);
• $p({\mathbf{x}}_k\mid {\mathbf{x}}_{k-1})$ is the dynamic model (how the state evolves stochastically);
• $p({\mathbf{y}}_k\mid {\mathbf{x}}_k)$ is the measurement model (distribution of measurements given the state).

Markov and conditional-independence assumptions (Särkkä Props. 4.1–4.2)

The model is Markovian in two senses:

• The state is a Markov sequence — the future is independent of the past given the present: $$p({\mathbf{x}}_k\mid {\mathbf{x}}_{1:k-1},{\mathbf{y}}_{1:k-1})=p({\mathbf{x}}_k\mid {\mathbf{x}}_{k-1}).\text{\hspace{1em}(4.2)}$$
• Measurements are conditionally independent given the corresponding state: $$p({\mathbf{y}}_k\mid {\mathbf{x}}_{1:k},{\mathbf{y}}_{1:k-1})=p({\mathbf{y}}_k\mid {\mathbf{x}}_k).\text{\hspace{1em}(4.4)}$$

These two properties are exactly what makes the predict/update recursion of The Kalman Filter valid. They also factor the joint prior and likelihood:

$$p({\mathbf{x}}_{0:T})=p({\mathbf{x}}_0)\prod _{k=1}^{T}p({\mathbf{x}}_k\mid {\mathbf{x}}_{k-1}),p({\mathbf{y}}_{1:T}\mid {\mathbf{x}}_{0:T})=\prod _{k=1}^{T}p({\mathbf{y}}_k\mid {\mathbf{x}}_k).\text{\hspace{1em}(4.7–4.8)}$$

Definition: Linear-Gaussian state-space model (Särkkä §4.3, Eq. 4.17–4.18)

The linear-Gaussian (linear filtering) model is

$${\mathbf{x}}_k={\mathbf{A}}_{k-1}{\mathbf{x}}_{k-1}+{\mathbf{q}}_{k-1},{\mathbf{q}}_{k-1}\sim \mathcal{N}(0,{\mathbf{Q}}_{k-1})\text{(state / transition equation)}$$ $${\mathbf{y}}_k={\mathbf{H}}_k{\mathbf{x}}_k+{\mathbf{r}}_k,{\mathbf{r}}_k\sim \mathcal{N}(0,{\mathbf{R}}_k)\text{(observation / measurement equation)}\text{\hspace{1em}(4.17)}$$

with prior ${\mathbf{x}}_0\sim \mathcal{N}({\mathbf{m}}_0,{\mathbf{P}}_0)$. Equivalently, in density form:

$$p({\mathbf{x}}_k\mid {\mathbf{x}}_{k-1})=\mathcal{N}({\mathbf{x}}_k\mid {\mathbf{A}}_{k-1}{\mathbf{x}}_{k-1},{\mathbf{Q}}_{k-1}),p({\mathbf{y}}_k\mid {\mathbf{x}}_k)=\mathcal{N}({\mathbf{y}}_k\mid {\mathbf{H}}_k{\mathbf{x}}_k,{\mathbf{R}}_k).\text{\hspace{1em}(4.18)}$$

Symbol glossary (used throughout this folder):

• ${\mathbf{A}}_{k-1}$ ($n\times n$): transition matrix of the dynamic model (step $k-1\to k$).
• ${\mathbf{H}}_k$ ($m\times n$): measurement-model matrix mapping state to observation space.
• ${\mathbf{q}}_{k-1}$, ${\mathbf{Q}}_{k-1}$ ($n\times n$): process noise and its covariance.
• ${\mathbf{r}}_k$, ${\mathbf{R}}_k$ ($m\times m$): measurement noise and its covariance.
• ${\mathbf{m}}_0$, ${\mathbf{P}}_0$: prior mean and covariance of the initial state.
• ${\mathbf{q}}_{k-1}\perp {\mathbf{r}}_k$ and both white (independent across time).

Time-invariant special case. When the matrices do not depend on $k$ (${\mathbf{A}}_{k-1}=\mathbf{A}$, ${\mathbf{H}}_k=\mathbf{H}$, etc.) the model is linear time-invariant (LTI) — the usual setting for BSTS-style structural models, where each component contributes fixed blocks.

Examples

Gaussian random walk + noise (local level model; Särkkä Ex. 4.1)

Scalar state, the simplest non-trivial state-space model:

$$x_k=x_{k-1}+q_{k-1},q_{k-1}\sim \mathcal{N}(0,Q);y_k=x_k+r_k,r_k\sim \mathcal{N}(0,R).$$

Here $A=1$, $H=1$. The hidden signal $x_k$ random-walks; we see it through additive noise. This is the local level component of a structural time series — see Local Linear Trend and Seasonality. Filtered in The Kalman Filter.

Constant-velocity car tracking (Särkkä Ex. 3.6 / 4.3)

A continuous-time Wiener-velocity model discretized to a 4-D state $\mathbf{x}=(x_1,x_2,{\dot{x}}_1,{\dot{x}}_2)$ (position + velocity). With sampling interval $\Delta t$:

$$\mathbf{A}=\left(\begin{array}{cccc}1& 0& \Delta t& 0\\ 0& 1& 0& \Delta t\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),\mathbf{H}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right),$$ $$\mathbf{Q}=\left(\begin{array}{cccc}\frac{q_1^c\Delta t^3}{3}& 0& \frac{q_1^c\Delta t^2}{2}& 0\\ 0& \frac{q_2^c\Delta t^3}{3}& 0& \frac{q_2^c\Delta t^2}{2}\\ \frac{q_1^c\Delta t^2}{2}& 0& q_1^c\Delta t& 0\\ 0& \frac{q_2^c\Delta t^2}{2}& 0& q_2^c\Delta t\end{array}\right),\mathbf{R}=\left(\begin{array}{cc}{\sigma }_1^2& 0\\ 0& {\sigma }_2^2\end{array}\right)$$

where $q_1^c,q_2^c$ are the continuous-time process-noise spectral densities. Only position is measured; velocity is inferred. Demonstrates the modular block structure that carries over to structural time-series models.

Connections

• State-Space Models and the Kalman Filter - Overview — where this model sits in the pipeline
• The Kalman Filter — exact filtering solution for this model
• The RTS Smoother — exact smoothing solution for this model
• Marginal Likelihood via the Kalman Filter — learning $\mathbf{A},\mathbf{H},\mathbf{Q},\mathbf{R}$ from data
• Bayesian Structural Time-Series Model — applied BSTS form (observation + state equations 2.1–2.2 map directly onto 4.17)
• Local Linear Trend and Seasonality — concrete trend/seasonal blocks for $\mathbf{A},\mathbf{Q},\mathbf{H}$

See Also

• Carryover Effects and Distributed Lags — autoregressive/distributed-lag dynamics expressible in state-space form (cf. Särkkä Ex. 3.4, $x_k=\sum a_i{x}_{k-i}+q$)
• Single Marketing Time Series — ARIMA models have equivalent state-space representations
• Hilbert Space Gaussian Processes — temporal GPs as linear-Gaussian state-space models