Chapter 5 State Space Models and the Kalman Filter

State space models generally attempt to describe a phenomenon that has two characteristics

  1. There is an underlying system that has a time-varying dynamical relationship, so that the “state” of the system at time \(t\) is related to the state of the system at time \(t-1\). If we know the state of the system at time \(t-1\), then we have everything we need to know in order to make inferences or forcasts about the state at time \(t\).

  2. We cannot observe the true underlying state of the system, but rather we observe a noisy version of it.

These two characteristics lead us to specifying the state equation, which describes how the system evolves from one time point to the next, and the observation equation, which describes how the underlying state is transformed (with noise added) into something that we directly measure.

Suppose there is an initial state \(x_0\sim\mathcal{N}(x_0^0, P_0^0)\). For \(t=1, 2, \dots\) we want to be able to estimate the subsequent states \(x_1, x_2,\dots\). At each time point we observe some data \(y_t\) and we want to incoporate that data into our estimation of \(x_t\).

In the simplest case, we can propose an observation equation \[ y_t = Ax_t + v_t \] where \(v_t\sim\mathcal{N}(0, \sigma^2)\), and a state equation \[ x_t = \theta x_{t-1} + w_t \] where \(w_t\sim\mathcal{N}(0,\tau^2)\). The parameters \(\theta\), \(\tau\), and \(\sigma\) are assumed to be known (you can think of them as tuning parameters) and we want to produce an estimate of \(x_t\) for all \(t\) that are of interest. Recall that the only thing we observe is the sequence \(y_1,y_2,\dots\).

The setting in which a state space model might make the most sense is a dynamical one in which we are trying to estimate the state values \(x_t\) in “real time”, without knowledge of what might come in the future. For example, in guidance and navigation applications with spacecraft, we want to know the position and velocity of the spacecraft as it is traveling through space, taking into account Newton’s laws of motion. Based on our estimate of the position and velocity of the spacecraft, we will need to make decisions about what to do next. Such situations require that we integrate all of the information that is available to us in order to produce the best possible estimate.