Multi-Step Forecasts from a Linear State Model

From a One-Step Rule to a Forecast Horizon

A discrete-time state model describes how a state changes from one time step to the next. A one-step rule may be enough to describe the model, but a forecast usually asks for the state several steps ahead. The calculation is repeated matrix multiplication: each predicted state becomes the input for the next update. Keeping the steps and time indices organized makes it easier to check the arithmetic and to compare predictions with later observations.

Assume throughout that the state is a column vector in \(\mathbb{R}^n\), and that the same real \(n\times n\) matrix \(\mathbf{A}\) is used at every step. The model has no additional input term. Given an initial state \(\mathbf{x}_0\), its recurrence is \(\mathbf{x}_{k+1}=\mathbf{A}\mathbf{x}_k\) for each integer \(k\geq 0\). A finite-horizon forecast is the sequence of states calculated from this rule for a specified, finite number of future steps. It is a model-based prediction, not a guarantee that future observations will match it.

At each step the order of multiplication is determined by the model: the matrix acts on the state column from the left. A forecast table should keep the time index beside each vector. If observed states are available, write them separately as \(\mathbf{y}_k\). The difference \(\mathbf{y}_k-\widehat{\mathbf{x}}_k\) then records how far the observation is from the prediction in each coordinate. This comparison does not by itself explain why a difference occurred.

Computing Successive States

This calculation suggests a general formula: the state at time \(k\) is found by applying \(\mathbf{A}\) a total of \(k\) times to the initial state. The result follows directly from the recurrence, rather than from an assumption about a special kind of transition matrix.

Proof. We use induction on \(k\). At \(k=0\), the formula gives \(\mathbf{A}^0\mathbf{x}_0=\mathbf{I}_n\mathbf{x}_0=\mathbf{x}_0\), as required. Now suppose that \(\mathbf{x}_k=\mathbf{A}^k\mathbf{x}_0\) for some \(k\geq 0\). The recurrence then gives \(\mathbf{x}_{k+1}=\mathbf{A}\mathbf{x}_k=\mathbf{A}(\mathbf{A}^k\mathbf{x}_0)=(\mathbf{A}\mathbf{A}^k)\mathbf{x}_0=\mathbf{A}^{k+1}\mathbf{x}_0\), where the last equality is the definition of the next matrix power. This proves the formula at \(k+1\). By induction it holds for every \(k\geq 0\). \(\square\)

The formula can be used when a forecast is required at a particular horizon, while repeated one-step multiplication is often convenient for constructing a complete table. These are two ways to carry out the same model calculation. The recurrence also determines the forecast uniquely: once the starting state is fixed, each next state is specified as the product of \(\mathbf{A}\) and the current state.

Starting a Forecast at a Later Time

Forecasts need not always start at time \(0\). Suppose a state \(\mathbf{x}_s\) is available at some time \(s\), and we want to forecast \(h\) steps beyond it. The same transition matrix is applied \(h\) times. This makes it possible to create a fresh forecast from a later starting state without recomputing the earlier rows.

Proof. Fix \(s\) and use induction on \(h\). When \(h=0\), \(\mathbf{A}^0\mathbf{x}_s=\mathbf{I}_n\mathbf{x}_s=\mathbf{x}_s\). Suppose \(\mathbf{x}_{s+h}=\mathbf{A}^h\mathbf{x}_s\). Applying the recurrence at time \(s+h\) gives \(\mathbf{x}_{s+h+1}=\mathbf{A}\mathbf{x}_{s+h}=\mathbf{A}\mathbf{A}^h\mathbf{x}_s=\mathbf{A}^{h+1}\mathbf{x}_s\). Therefore the formula holds for \(h+1\), and induction proves it for all \(h\geq 0\). \(\square\)

Comparing Forecasts with Observations

Observed data can be placed beside predictions at matching times. In the example below, the initial state is used to generate a forecast, and observed states are supplied for comparison. Define the observed-minus-predicted difference at time \(k\) to be \(\mathbf{d}_k=\mathbf{y}_k-\widehat{\mathbf{x}}_k\). A positive or negative coordinate indicates the direction of the difference in that coordinate. A zero vector means the observation and prediction agree at that time.

In this comparison the observations at times \(2\) and \(3\) do not equal the model's forecasts. The table reports that difference without changing the model or claiming a cause. If a new forecast is instead started from a later observed state, that is a new starting condition; the resulting forecast should be labelled accordingly.

Common Pitfall: Squaring Entries Instead of the Matrix

Matrix powers use matrix multiplication, not entry-by-entry powers. For the matrix \(\mathbf{A}=\begin{bmatrix}1&1\\0&1\end{bmatrix}\), a tempting but incorrect calculation is to square each entry and conclude \(\mathbf{A}^2=\begin{bmatrix}1&1\\0&1\end{bmatrix}\). That would predict \(\widehat{\mathbf{x}}_2=\begin{bmatrix}3\\1\end{bmatrix}\) from the first worked example, which merely repeats the time-\(1\) state.

The correction is to multiply the matrices in the definition of a power:

This agrees with two successive updates: first \(\mathbf{A}\begin{bmatrix}2\\1\end{bmatrix}=\begin{bmatrix}3\\1\end{bmatrix}\), then \(\mathbf{A}\begin{bmatrix}3\\1\end{bmatrix}=\begin{bmatrix}4\\1\end{bmatrix}\). A second frequent source of error is comparing vectors from different time indices. The forecast and observation must refer to the same time before their difference is interpreted.

Using a Forecast Table Well

A forecast table is useful because it makes both the calculation and the comparison explicit. For a fixed transition matrix and starting state, a reliable workflow is:

The recurrence defines the arithmetic of the forecast, but it does not establish that the model is appropriate for every application or horizon. In practice, the table lets a reader see exactly which predictions follow from the chosen starting state and transition matrix, and where observations begin to differ. Conclusions about whether those differences are important require context beyond the matrix multiplications themselves.