Implicitly Create State-Space Model Containing Regression Component

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This example shows how to implicitly create a state-space model that contains a regression component in the observation equation. The state model is an ARMA(1,1).

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. Specify the regression component by deflating the observations within the function. Symbolically, the model is:

$$\begin{array}{l}
\left[ {\begin{array}{*{20}{c}}
{{x_{1,t}}}\\
{{x_{2,t}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{{\phi _1}}&{{\theta _1}}\\
0&0
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_{1,t - 1}}}\\
{{x_{2,t - 1}}}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
{{\sigma _1}}\\
1
\end{array}} \right]{u_{1t}}\\
{y_t} - \beta {z_t} = a{x_{1,t}} + {\sigma _2}{\varepsilon _t}.
\end{array}$$


% Copyright 2015 The MathWorks, Inc.

function [A,B,C,D,Mean0,Cov0,StateType,DeflateY] = regressionParamMap(params,y,z)
% State-space model with a regression component 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
% an ARMA(1,1).
    varu1 = exp(params(3)); % Positive variance constraint
    vare1 = exp(params(4));
    A = [params(1) params(2); 0 0];
    B = [sqrt(varu1); 1]; 
    C = [1 0];
    D = sqrt(vare1);
    Mean0 = [0.5 0.5];
    Cov0 = eye(2);
    StateType = [0 0];
    DeflateY = y - params(5)*z;
end

Save this code as a file named regressionParamMap on your MATLAB® path.

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

Mdl = ssm(@(params)regressionParamMap(params,y,z));

ssm implicitly creates the state-space model. Usually, you cannot verify implicitly defined state-space models.

Before creating the model, ensure that the data y and z exist in your workspace.

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