State space models and the Kalman filter

Definition A linear state-space model describes a latent state $x_t$ in $\mathbb{R}^n$ and an observation $y_t$ in $\mathbb{R}^m$ by two equations. The state equation and the observation equation are

for $t = 1, 2, \dots$, where the transition matrix $A$ is $n \times n$, the observation matrix $H$ is $m \times n$, and the process noise $w_t$ and the observation noise $v_t$ have mean zero and covariance matrices

which are positive semidefinite of sizes $n \times n$ and $m \times m$. On this page the matrices are known, the initial state is $x_0 \sim \mathcal{N}(m_0, P_0)$, the noises are Gaussian with $w_t \sim \mathcal{N}(0, Q)$ and $v_t \sim \mathcal{N}(0, R)$, and $x_0$, all $w_t$ and all $v_t$ are mutually independent. Known matrices that change with $t$ work the same way date by date. A known intercept, such as a drift $c$ in $x_t = c + A x_{t-1} + w_t$, can be carried as a separate term or by adding a constant component to the state. Correlated process and observation noise and nonlinear equations are outside this lesson.

Definition The state $x_t$ is latent because it is never observed. Only the observations are seen, and $\mathcal{Y}_s$ denotes the information generated by $y_1, \dots, y_s$, with $\mathcal{Y}_0$ containing no observations. Under the model’s probability $\mathbb{P}$, estimating $x_t$ from $\mathcal{Y}_s$ means computing $\mathbb{E}[x_t \mid \mathcal{Y}_s]$ and the conditional covariance of its error. The problem is prediction when $s < t$, filtering when $s = t$ and smoothing when $s > t$. Write

where the superscript minus marks the prediction made before $y_t$ arrives, not a negative value, and $m_0$ and $P_0$ are the initial mean and covariance. Because $w_t$ has mean zero and is independent of $x_{t-1}$ and $y_1, \dots, y_{t-1}$ together, conditioning the state equation on $\mathcal{Y}_{t-1}$ gives the one-step prediction

The prediction error $x_t - m_t^- = A(x_{t-1} - m_{t-1}) + w_t$ is the sum of two pieces that are independent given $\mathcal{Y}_{t-1}$. The first, with covariance $A P_{t-1} A^{\top}$, carries the uncertainty already present through the dynamics, and the second adds the new process noise $Q$.

Definition The innovation at date $t$ is the part of the observation that the prediction did not expect,

where $e_t = x_t - m_t^-$ is the state prediction error. Since $v_t$ is independent of $e_t$, the innovation covariance and the cross-covariance of the state error with the innovation are

The matrix $S_t$ is $m \times m$ and positive semidefinite, and it is invertible whenever $R$ is positive definite. In the Gaussian model, $\nu_t$ given $\mathcal{Y}_{t-1}$ has the distribution $\mathcal{N}(0, S_t)$, and the innovations at different dates are independent. The joint density of $y_1, \dots, y_T$ is therefore the product of the $\mathcal{N}(0, S_t)$ densities evaluated at $\nu_1, \dots, \nu_T$, which is how the likelihood of a state-space model is computed when its parameters are estimated.

Definition Assume $S_t$ is invertible. Among updates of the form $m_t^- + K \nu_t$ with a fixed $n \times m$ matrix $K$, the error $e_t - K \nu_t$ has covariance

The Kalman gain is

and completing the square around it gives

The last term is positive semidefinite for every $K$, so $K_t$ gives the smallest error covariance in the matrix order, and in particular the smallest mean squared error $\operatorname{tr} C(K)$. The updated mean is

In the Gaussian model, $x_t$ and $y_t$ are jointly normal given $\mathcal{Y}_{t-1}$. The conditional mean of a jointly normal vector is affine in the conditioning variable and has the smallest mean squared error among all estimates, so it coincides with this update, and $m_t$ is exactly $\mathbb{E}[x_t \mid \mathcal{Y}_t]$. For a scalar state observed directly, with $H = 1$, $P_t^- > 0$ and $R > 0$, multiplying the prior density $\mathcal{N}(m_t^-, P_t^-)$ by the likelihood of $y_t$ and completing the square in the exponent gives a normal posterior with precision $1/P_t^- + 1/R$ and mean

so the gain $K_t = P_t^-/(P_t^- + R)$ is the share of the total variance that comes from the prediction. With $P_t^-$ fixed, the gain tends to $0$ as $R \to \infty$ and to $1$ as $R \to 0$.

Definition The posterior covariance is the error covariance after the update, which in the Gaussian model is $P_t = \operatorname{Cov}(x_t \mid \mathcal{Y}_t)$. With the Kalman gain,

For an arbitrary gain $K$, the error $(I - KH) e_t - K v_t$ has the Joseph form covariance

a sum of two positive semidefinite terms, one for the prediction error that the update leaves and one for the measurement noise that it lets in. The Joseph form holds for every $K$. The shorter expression $(I - KH) P_t^-$ comes from substituting $K = K_t$, and for other gains it generally gives the wrong covariance. The Joseph form also helps keep a computed covariance symmetric and positive semidefinite under rounding. In the Gaussian model, $P_t$ does not depend on the observed values, so the whole sequence $P_1^-, P_1, P_2^-, P_2, \dots$ can be computed before any data arrive. When $P_t^-$ and $R$ are invertible, the precisions add,

Definition The Kalman filter is the recursion that starts from $m_0$ and $P_0$ and, for $t = 1, 2, \dots$, alternates a prediction and an update,

Under the Gaussian assumptions of the state-space card, and with every $S_t$ invertible, induction on $t$ with the Gaussian update of the gain card shows that the conditional distribution of $x_t$ given $\mathcal{Y}_t$ is exactly $\mathcal{N}(m_t, P_t)$. If $x_0$ and the noises are only mutually uncorrelated with the stated means and covariances, the same recursion gives the best estimate of $x_t$ among those affine in $y_1, \dots, y_t$, with error covariance $P_t$, but the true conditional distribution need not be normal and its mean may differ from $m_t$. Correlated process and observation noise needs extra cross-covariance terms, and nonlinear models need a separate argument or an approximation.