Create State-Space Model with Unknown Parameters

Explicitly Create State-Space Model Containing Unknown Parameters

This example shows how to create a time-invariant, state-space model containing unknown parameter values using ssm.

Define a state-space model containing two dependent MA(1) states, and an additive-error observation model. Symbolically, the equation is

$$\left[\begin{array}{c}{x}_{t,1}\\ x_{t,2}\\ x_{t,3}\\ x_{t,4}\end{array}\right]=\left[\begin{array}{cccc}0& {\theta }_1& {\lambda }_1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& {\theta }_3\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{t-1,1}\\ x_{t-1,2}\\ x_{t-1,3}\\ x_{t-1,4}\end{array}\right]+\left[\begin{array}{cc}{\sigma }_1& 0\\ 1& 0\\ 0& {\sigma }_2\\ 0& 1\end{array}\right]\left[\begin{array}{c}{u}_{t,1}\\ u_{t,2}\end{array}\right]$$

$$y_t=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}{x}_{t,1}\\ x_{t,2}\\ x_{t,3}\\ x_{t,4}\end{array}\right]+\left[\begin{array}{cc}{\sigma }_3& 0\\ 0& {\sigma }_4\end{array}\right]\left[\begin{array}{c}{\epsilon }_{t,1}\\ {\epsilon }_{t,2}\end{array}\right].$$

Note that the states $$x_{t,1}$$ and $$x_{t,3}$$ are the two dependent MA(1) processes. The states $$x_{t,2}$$ and $$x_{t,4}$$ help construct the lag-one, MA effects. For example, $$x_{t,2}$$ picks up the first disturbance ($$u_{t,1}$$), and $$x_{t,1}$$ picks up $$x_{t-1,2}=u_{t-1,1}$$. In all, $$x_{t,1}={\lambda }_1{x}_{t-1,3}+u_{t,1}+{\theta }_1{u}_{t-1,1}$$, which is an MA(1) with $$x_{t-1,3}$$ as an input.

Specify the state-transition coefficient matrix. Use NaN values to indicate unknown parameters.

A = [0 NaN NaN 0; 0 0 0 0; 0 0 0 NaN; 0 0 0 0];

Specify the state-disturbance-loading coefficient matrix.

B = [NaN 0; 1 0; 0 NaN; 0 1];

Specify the measurement-sensitivity coefficient matrix.

C = [1 0 0 0; 0 0 1 0];

Specify the observation-innovation coefficient matrix.

D = [NaN 0; 0 NaN];

Use ssm to define the state-space model.

Mdl = ssm(A,B,C,D)

Mdl = 
State-space model type: ssm

State vector length: 4
Observation vector length: 2
State disturbance vector length: 2
Observation innovation vector length: 2
Sample size supported by model: Unlimited
Unknown parameters for estimation: 7

State variables: x1, x2,...
State disturbances: u1, u2,...
Observation series: y1, y2,...
Observation innovations: e1, e2,...
Unknown parameters: c1, c2,...

State equations:
x1(t) = (c1)x2(t-1) + (c2)x3(t-1) + (c4)u1(t)
x2(t) = u1(t)
x3(t) = (c3)x4(t-1) + (c5)u2(t)
x4(t) = u2(t)

Observation equations:
y1(t) = x1(t) + (c6)e1(t)
y2(t) = x3(t) + (c7)e2(t)

Initial state distribution:

Initial state means are not specified.
Initial state covariance matrix is not specified.
State types are not specified.

Mdl is an ssm model containing unknown parameters. A detailed summary of Mdl prints to the Command Window. It is good practice to verify that the state and observations equations are correct.

Pass Mdl and data to estimate to estimate the unknown parameters.

Implicitly Create Time-Invariant State-Space Model

This example shows how to create a time-invariant state-space model by passing a parameter-mapping function describing the model to ssm (that is, implicitly create a state-space model). The state model is AR(1) model. The states are observed with bias, but without random error. Set the initial state mean and variance, and specify that the state is stationary.

Write a function that specifies how the parameters in params map to the state-space model matrices, the initial state values, and the type of state.

% Copyright 2015 The MathWorks, Inc.

function [A,B,C,D,Mean0,Cov0,StateType] = timeInvariantParamMap(params)
% Time-invariant state-space model parameter mapping function example. This
% function maps the vector params to the state-space matrices (A, B, C, and
% D), the initial state value and the initial state variance (Mean0 and
% Cov0), and the type of state (StateType). The state model is AR(1)
% without observation error.
    varu1 = exp(params(2)); % Positive variance constraint
    A = params(1);
    B = sqrt(varu1);
    C = params(3);
    D = [];
    Mean0 = 0.5;
    Cov0 = 100;
    StateType = 0;
end

Save this code as a file named timeInvariantParamMap to a folder on your MATLAB® path.

Create the state-space model by passing the function timeInvariantParamMap as a function handle to ssm.

Mdl = ssm(@timeInvariantParamMap);

The software implicitly defines the state-space model. Usually, you cannot verify state-space models that you implicitly define.

Mdl is an ssm model object containing unknown parameters. You can estimate the unknown parameters by passing Mdl and response data to estimate.