Lecture notes. Written from the material of Berkeley Stat 153, licensed CC BY 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

19. State Space Models¶

What this covers¶

Every model so far in the course — regression, AR, ARIMA — fits a series $y_t$ directly. This chapter asks what happens when $y_t$ is not the quantity of interest at all, but a noisy, partial measurement of some other process $x_t$ that we never see. It builds the state space model (also called the dynamic linear model, DLM) that formalizes this, works through the linear Gaussian version in detail, and follows one worked biomedical example — monitoring a bone marrow transplant from three blood markers with 40% missing data — from raw variables to fitted parameters. It assumes the AR(1) and VAR-style vector generalization of AR, and the language of white noise and covariance matrices; the reading behind it is Shumway and Stoffer, Ch. 5.5 and 6–6.1.

From a fixed coefficient to a hidden process¶

Ordinary regression writes $y_t = X\beta + \epsilon$ and treats $\beta$ as a fixed unknown constant, estimated once from the data. The natural generalization is to let $\beta$ drift over time: $\beta_t = \beta_{t-1} + \eta_t$, with $y_t = X\beta_t + \epsilon$. The moment $\beta$ is allowed to be time-varying and only inferred through $y_t$, this is already a state space model — the "state" is the time-varying coefficient $\beta_t$, which is never observed directly.

A second route to the same idea starts from an AR(1) process, $x_t = \phi x_{t-1} + w_t$. Suppose $x_t$ itself is not observed, and all that is available is a noisy version of it, $y_t = x_t + v_t$. The pair of equations — a state equation $x_t = \phi x_{t-1} + w_t$ describing how the hidden signal evolves, and an observation equation $y_t = x_t + v_t$ describing how it is measured — is a state space model.

Both examples share the same two organizing principles, stated generally:

There is a hidden, or latent, state process $x_t$. It is a Markov process: conditional on the present state $x_t$, the future $\{x_s : s>t\}$ and the past $\{x_s : s<t\}$ are independent.
The observations $y_t$ are independent given the states $x_t$. Any dependence between the observations — the reason $y_1$ tells you something about $y_5$ — is entirely inherited through the states; nothing in the observation process itself carries memory.

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The state chain runs along the top, hidden; each observation below depends only on the state directly above it. Correlation between $y_{t-1}$ and $y_{t+1}$ exists only because it is carried through $x_t$.

A building block: from AR to VAR¶

Before assembling the general model, it helps to have the vector generalization of AR(1) in hand. Everything discussed under AR so far has been a single series. When $k$ series influence each other, the vector autoregressive (VAR) model writes

\[x_t = \alpha + \Phi x_{t-1} + w_t,\]

where $x_t$ is now a $k$-vector and $\Phi$ is a $k\times k$ transition matrix expressing how all of $x_{t-1}$'s components feed into $x_t$. A state space model is a further generalization of this: it keeps a VAR-like equation for the hidden state, but adds a second equation for how that state is actually measured.

The linear Gaussian model¶

The linear Gaussian state space model — also called the dynamic linear model (DLM) — pairs a vector-autoregressive state equation

\[x_t = \Phi x_{t-1} + \Upsilon u_t + w_t, \qquad w_t \overset{iid}{\sim} N_p(0, Q), \qquad x_0 \sim N_p(\mu_0, \sigma_0),\]

with an observation equation

\[y_t = A_t x_t + \Gamma u_t + v_t, \qquad v_t \sim N_q(0, R).\]

Reading off each piece:

$x_t$ is $p$-dimensional and hidden. $\Phi$, a $p\times p$ matrix, is the state transition parameter: it describes the internal dynamics, how the state at $t-1$ produces the state at $t$.
$u_t$ is an $r$-dimensional fixed input series — covariates, experimental conditions, interventions, anything supplied rather than learned from the data.
$\Upsilon$ ($p\times r$) controls how those inputs move the state; $w_t$ is the state noise, covariance $Q$.
$y_t$ is $q$-dimensional data — $q$ can be larger or smaller than $p$, the state dimension — and is a linearly transformed, noisy view of $x_t$.
$A_t$, a $q\times p$ measurement (observation) matrix, picks out which linear combinations of the state are actually measured. Letting $A_t$ depend on $t$ is what lets the model handle missing data cleanly: for a time point where a measurement is missing, drop the corresponding row of $A_t$.
$v_t$ is measurement noise, covariance $R$; $\Gamma$ ($q\times r$) lets inputs affect the observation directly, bypassing the state.

So an external input can act on $y_t$ through two different routes: $\Upsilon u_t$ (through the state) or $\Gamma u_t$ (through the observation). The distinction is the difference between something that changes the underlying reality and something that only changes how the reality is reported. Take $x_t$ to be a person's true blood pressure and $y_t$ the reading from a blood pressure cuff. A drug administered at time $t$ belongs in the state equation, because it changes the actual blood pressure. Which nurse took the reading, or which cuff was used, belongs in the observation equation: it changes what the meter reports without touching the underlying physiology. $\Gamma = 0$ is the common choice, when inputs are believed to only drive the state; $\Upsilon = 0$ shows up when modeling a known measurement artifact that has no bearing on the true state at all.

Worked example: monitoring a bone marrow transplant¶

As a concrete case, consider three biomedical markers tracked in a cancer patient who has undergone a bone marrow transplant:

log(white blood cell count) [WBC] — essential for immune function
log(platelet count) [PLT] — essential for blood clotting
hematocrit [HCT] — percentage of red blood cells in total blood volume

Each measures a distinct aspect of bone marrow function, and a transplant is judged successful once the new marrow is producing all three. As with much longitudinal follow-up data, a large fraction of the readings are missing — about 40% here, concentrated after day 35. A state space model can be fit to the three series jointly and used both to estimate those missing values and to track the trajectory; platelet count at 100 days post-transplant is known from prior work to predict long-term survival, giving a concrete reason to want a good estimate of it even on days with no blood draw.

The state equation stacks the three series into a VAR(1):

\[ \begin{pmatrix} x_{t1} \\ x_{t2} \\ x_{t3} \end{pmatrix} = \begin{pmatrix} \phi_{11} & \phi_{12} & \phi_{13} \\ \phi_{21} & \phi_{22} & \phi_{23} \\ \phi_{31} & \phi_{32} & \phi_{33} \end{pmatrix} \begin{pmatrix} x_{t-1,1} \\ x_{t-1,2} \\ x_{t-1,3} \end{pmatrix} + \begin{pmatrix} w_{t1} \\ w_{t2} \\ w_{t3} \end{pmatrix}. \]

This is three stacked regressions. Writing out the first row,

\[x_{t1} = \phi_{11}x_{t-1,1} + \phi_{12}x_{t-1,2} + \phi_{13}x_{t-1,3} + w_{t1},\]

today's log(WBC) is a weighted sum of yesterday's WBC, platelets, and hematocrit, plus noise. The diagonal entries measure self-persistence: $\phi_{11}$ near $1$ means WBC changes slowly and its own recent value strongly predicts tomorrow's; a value near $0$ would say today's value has almost nothing to do with yesterday's, which would be unusual for a blood marker and would more plausibly signal a data problem than genuine dynamics. The off-diagonal entries $\phi_{ij}$ measure cross-influence: how much yesterday's marker $j$ predicts today's marker $i$ — for instance $\phi_{12}$ is the effect of yesterday's platelet count on today's WBC, and $\phi_{21}$ the effect of yesterday's WBC on today's platelet count.

The observation equation is $y_t = A_t x_t + v_t$, where the $3\times 3$ matrix $A_t$ is either the identity or the zero matrix, according to whether a blood sample was actually drawn on day $t$. This is the missing-data mechanism made concrete: a missing day does not require deleting that time point or interpolating by hand, it is handled by zeroing out $A_t$ for that day, and the state equation still propagates an estimate of $x_t$ through it.

Fitting such a model — by maximum likelihood or by the EM algorithm — means estimating:

$\Phi$: a $3\times 3$ matrix, 9 parameters,
$Q$: the $3\times 3$ symmetric state noise covariance, 6 free parameters (diagonal plus one set of off-diagonals),
$R$: the $3\times 3$ symmetric observation noise covariance, 6 free parameters (often assumed diagonal in practice),
$\mu_0, \sigma_0$: mean and covariance of the initial state, $3 + 6 = 9$ parameters (sometimes fixed from prior knowledge instead of estimated),
$A$: usually fixed at the identity (up to the missing-data zeroing above), though it can be fit.

In Python, statsmodels.tsa.statespace provides the machinery to fit models of this form.

Filtering, prediction, and smoothing¶

Given data $y_{1:s} = \{y_1,\dots,y_s\}$, the task of estimating the hidden state $x_t$ splits into three cases depending on how $s$ compares to $t$:

Filtering ($s = t$): estimate $x_t$ using only the measurements up through time $t$ — the online case, updating the state estimate as each new observation arrives.
Prediction / forecasting ($s < t$): estimate a future state $x_t$ using data only up to an earlier time $s$ — projecting the state forward before the later observations exist.
Smoothing ($s > t$): estimate $x_t$ using the entire dataset, including observations that came after time $t$. Because it uses more information than filtering does, smoothing gives the best estimate of the state at any interior time point — this is exactly what is used to fill in the missing values in the bone marrow example, since data collected on later days sharpens the estimate of what happened on the missing days in between.

Other applications¶

The same two-equation structure recurs across very different measurement problems:

Estimating global temperature $x_t$ from land and sea temperature measurements $y_t$.
Brain-computer interfaces: inferring intended cursor velocity $x_t$ from measurements $y_t$ at 100 electrodes in motor cortex.
Estimating true blood glucose level $x_t$ from intermittent readings $y_t$ from a continuous glucose monitor subject to sensor drift.
Estimating the volatility $x_t$ of the S&P 500 on day $t$ from the observed daily log return $y_t$.

Why use a state space model¶

Set against ARIMA and ordinary regression, a state space model offers three specific advantages:

Missing data is handled naturally — every time point does not need to be observed, and the latent state can still be estimated for the gaps (as in dropping rows of $A_t$ above).
Process noise and measurement noise are separated. ARIMA-type models carry a single innovation term; a state space model distinguishes noise intrinsic to the true underlying signal ($w_t$) from noise introduced by an imperfect sensor ($v_t$) — a distinction real instruments (and real biology) actually have.
Parameters can vary over time, unlike the fixed $\beta$ of a regression model — as in the drifting-coefficient example that opened the chapter.

Sources¶

Slides/lecture notes: 01-state-space-models.md — motivation from regression and AR(1) to the general state space framework, the two organizing principles, and the AR-to-VAR extension.
Slides/lecture notes: 02-linear-gaussian-model.md — the linear Gaussian model (DLM): state and observation equations, and the blood pressure illustration of $\Upsilon$ versus $\Gamma$.
Slides/lecture notes: 03-bone-marrow-transplant-example.md — the bone marrow transplant worked example, parameter counts and fitting, filtering/prediction/smoothing, other applications, and the advantages of state space models. The prediction/smoothing definitions ($s<t$ versus $s>t$) were reconstructed from a markdown rendering artifact in this file that dropped the inequality signs; the reconstruction follows the standard usage in the assigned reading.
Referred to but not supplied: Shumway and Stoffer, Time Series Analysis and Its Applications, Ch. 5.5 and 6–6.1 (the assigned reading for this lecture), and the state-space diagram image linked from 01-state-space-models.md, which is represented here by an original schematic rather than reproduced. No transcript, additional notes, or exercises were supplied for this lecture.

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