State space and the Kalman filter

Twice now this track has hinted that its models are “really state-space models” — ETS in lesson 2, ARIMA’s likelihood in lesson 3. This lesson opens the box. The state-space view is the track’s unifying formalism, and its crown jewel — the Kalman filter — is arguably the most deployed algorithm of the twentieth century: it navigated Apollo, sits in every GPS receiver and drone, and computes the exact likelihood of every structural time-series model. Better still, you have already derived most of it: it is the Bayesian update lesson running on a moving target.

The model: what you want is not what you see

Two equations, two admissions of ignorance:

with $w_t \sim \mathcal N(0, Q)$, $v_t \sim \mathcal N(0, R)$. The state $x_t$ is what you care about (true demand level, true position, true temperature); $y_t$ is what you get (bookings, GPS pings, a cheap sensor). The simplest instance — the local level model, $x_t = x_{t-1} + w_t$, $y_t = x_t + v_t$ — is already profound: it says the world drifts ($Q$) and the measurement lies ($R$), and asks for the best running belief about the truth. Note it is exactly a random walk observed in noise — ARIMA(0,1,1) in disguise, and SES’s optimal home, as your exercises have twice foreshadowed.

The filter: Bayes, recursively

Maintain a Gaussian belief $x_{t-1} \mid y_{1:t-1} \sim \mathcal N(m_{t-1}, P_{t-1})$ and alternate two steps.

Predict — push the belief through the dynamics; uncertainty grows:

Update — treat the prediction as the prior, the new observation as evidence, and apply the Gaussian-conjugacy algebra of the Bayesian lesson. For the scalar local level it collapses to three unforgettable lines:

The Kalman gain $K_t \in (0,1)$ is a trust dial set by the noise ratio: prediction uncertain relative to the sensor ($P^- \gg R$) ⟹ $K \to 1$, believe the data; sensor noisy ($R \gg P^-$) ⟹ $K \to 0$, believe the model. And “move your estimate a fraction $K$ toward the surprise” is — the third time this shape appears in the track — exponential smoothing, now with its α derived from first principles rather than tuned: in steady state, SES’s optimal α is exactly the converged Kalman gain of the local level model.

Why this is the track’s master formalism

• Every ETS and ARIMA model has a state-space form, and the filter’s one-step prediction errors give the exact Gaussian likelihood — this is literally how statsmodels/forecast compute the MLEs of the last two lessons. One algorithm underwrites the whole classical stack.
• Missing data is trivial: no observation, no update — just predict. Irregular sampling, sensor dropouts, holidays: handled by the same three lines. (Try doing that to a bare ARIMA recursion.)
• Smoothing (the RTS backward pass) refines every past state using the full series — for retrospective analysis; the filter alone is the honest choice for forecasting, since it uses only what was known at the time.
• Structural models: add trend, seasonal and regression states and you get interpretable components with uncertainty — the Bayesian structural time series behind causal-impact analyses.
• Beyond linear-Gaussian: real dynamics bend the assumptions, and the extensions map the terrain — EKF (linearize, the Jacobian doing the work the Matrix Calculus lesson trained), UKF (sample sigma points), and the particle filter (Monte Carlo over states — the 1/√n lesson’s machinery chasing a distribution through time). Each trades exactness for realism; all keep the predict–update heartbeat.

import numpy as np
rng = np.random.default_rng(3)

n, Q, R = 200, 0.05, 1.0
x = np.cumsum(np.sqrt(Q) * rng.standard_normal(n))     # hidden truth
y = x + np.sqrt(R) * rng.standard_normal(n)            # what you see

m, P, est = y[0], 10.0, []
for t in range(n):
    Pp = P + Q                       # predict
    K = Pp / (Pp + R)                # gain
    m = m + K * (y[t] - m)           # update on surprise
    P = (1 - K) * Pp
    est.append(m)

rmse = lambda a: np.sqrt(np.mean((np.array(a) - x) ** 2))
print(rmse(y), rmse(est), K)         # filter beats raw; K ≈ steady state

Exercises