Simulate Time-Varying State-Space Model

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This example shows how to generate data from a known model, fit a state-space model to the data, and then simulate series from the fitted model.

Suppose that a set of latent processes comprises an AR(2) and an MA(1) model. There are 50 periods, and the MA(1) process drops out of the model for the final 25 periods. The state equation for the first 25 periods is

$$\begin{array}{l}
{x_{1,t}} = 0.7{x_{1,t - 1}} - 0.2{x_{1,t - 2}} + {u_{1,t}}\\
{x_{2,t}} = {u_{2,t}} + 0.6{u_{2,t - 1}},
\end{array}$$

and for the last 25 periods, it is

$${x_{1,t}} = 0.7{x_{1,t - 1}} - 0.2{x_{1,t - 2}} + {u_{1,t}},$$

where $u_{1,t}$ and $u_{2,t}$ are Gaussian with mean 0 and standard deviation 1.

Assuming that the series starts at 1.5 and 1, respectively, generate a random series of 50 observations from $x_{1,t}$ and $x_{2,t}$.

T = 50;
ARMdl = arima('AR',{0.7,-0.2},'Constant',0,'Variance',1);
MAMdl = arima('MA',0.6,'Constant',0,'Variance',1);
x0 = [1.5 1; 1.5 1];
rng(1);
x = [simulate(ARMdl,T,'Y0',x0(:,1)),...
    [simulate(MAMdl,T/2,'Y0',x0(:,2));nan(T/2,1)]];

The last 25 values for the simulated MA(1) data are NaN values.

Suppose further that the latent processes are measured using

$${y_t} = 2\left( {{x_{1,t}} + {x_{2,t}}} \right) + {\varepsilon _t},$$

for the first 25 periods, and

$${y_t} = 2{x_{1,t}} + {\varepsilon _t}$$

for the last 25 periods, where $\varepsilon_t$ is Gaussian with mean 0 and standard deviation 1.

Use the random latent state process (x) and the observation equation to generate observations.

y = 2*sum(x','omitnan')'+randn(T,1);

Together, the latent process and observation equations compose a state-space model. Supposing that the coefficients are unknown parameters, the state-space model is

$$\begin{array}{l}
\left[ {\begin{array}{*{20}{c}}
{{x_{1,t}}}\\
{{x_{2,t}}}\\
{{x_{3,t}}}\\
{{x_{4,t}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{{\phi _1}}&{{\phi _2}}&0&0\\
1&0&0&0\\
0&0&0&{{\theta _1}}\\
0&0&0&0
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_{1,t - 1}}}\\
{{x_{2,t - 1}}}\\
{{x_{3,t - 1}}}\\
{{x_{4,t - 1}}}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
1&0\\
0&0\\
0&1\\
0&1
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{u_{1,t}}}\\
{{u_{2,t}}}
\end{array}} \right]\\
{y_t} = a({x_{1,t}} + {x_{3,t}}) + {\varepsilon _t}
\end{array}$$

for the first 25 periods,

$$\begin{array}{l}
\left[ {\begin{array}{*{20}{c}}
{{x_{1,t}}}\\
{{x_{2,t}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{{\phi _1}}&{{\phi _2}}&0&0\\
1&0&0&0
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_{1,t - 1}}}\\
{{x_{2,t - 1}}}\\
{{x_{3,t - 1}}}\\
{{x_{4,t - 1}}}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
1\\
0
\end{array}} \right]{u_{1,t}}\\
{y_t} = b{x_{1,t}} + {\varepsilon _t}
\end{array}$$

for period 26, and

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

for the last 24 periods.

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] = AR2MAParamMap(params,T)
%AR2MAParamMap Time-variant state-space model parameter mapping function
%
% 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). From periods 1 to T/2, the
% state model is an AR(2) and an MA(1) model, and the observation model is
% the sum of the two states. From periods T/2 + 1 to T, the state model is
% just the AR(2) model. 
    A1 = {[params(1) params(2) 0 0; 1 0 0 0; 0 0 0 params(3); 0 0 0 0]};
    B1 = {[1 0; 0 0; 0 1; 0 1]}; 
    C1 = {params(4)*[1 0 1 0]};
    Mean0 = ones(4,1);
    Cov0 = 10*eye(4);
    StateType = [0 0 0 0];
    A2 = {[params(1) params(2) 0 0; 1 0 0 0]};
    B2 = {[1; 0]};
    A3 = {[params(1) params(2); 1 0]};
    B3 = {[1; 0]}; 
    C3 = {params(5)*[1 0]};
    A = [repmat(A1,T/2,1);A2;repmat(A3,(T-2)/2,1)];
    B = [repmat(B1,T/2,1);B2;repmat(B3,(T-2)/2,1)];
    C = [repmat(C1,T/2,1);repmat(C3,T/2,1)];
    D = 1;
end

Save this code as a file named AR2MAParamMap on your MATLAB® path.

Create the state-space model by passing the function AR2MAParamMap as a function handle to ssm.

Mdl = ssm(@(params)AR2MAParamMap(params,T));

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

Pass the observed responses (y) to estimate to estimate the parameters. Specify an arbitrary set of positive initial values for the unknown parameters.

params0 = 0.1*ones(5,1);
EstMdl = estimate(Mdl,y,params0);

Method: Maximum likelihood (fminunc)
Sample size: 50
Logarithmic  likelihood:     -114.957
Akaike   info criterion:      239.913
Bayesian info criterion:      249.473
      |     Coeff       Std Err   t Stat     Prob  
---------------------------------------------------
 c(1) |  0.47870       0.26634    1.79733  0.07229 
 c(2) |  0.00809       0.27179    0.02975  0.97626 
 c(3) |  0.55735       0.80958    0.68844  0.49118 
 c(4) |  1.62679       0.41622    3.90848  0.00009 
 c(5) |  1.90021       0.49563    3.83391  0.00013 
      |                                            
      |   Final State   Std Dev    t Stat    Prob  
 x(1) | -0.81229       0.46815   -1.73511  0.08272 
 x(2) | -0.31449       0.45918   -0.68490  0.49341

EstMdl is an ssm model containing the estimated coefficients. Likelihood surfaces of state-space models might contain local maxima. Therefore, it is good practice to try several initial parameter values, or consider using refine.

Simulate one path of responses, states, state disturbances, and observation innovations from Mdl. Specify that each path has T periods of simulated variants.

[Y,X,U,E]= simulate(EstMdl,T);

  • Y is a T-by-1 vector of simulated observations.

  • X is a T-by-1 cell vector of simulated states. Cells 1 through 25 contain 4-by-1 vectors, and cells 26 through 50 contain 2-by-1 vectors.

  • U is a T-by-1 cell vector of simulated state disturbances. Cells 1 through 25 contain 4-by-1 vectors, and cells 26 through 50 contain 2-by-1 vectors.

  • E is a T-by-1 vector of simulated observation innovations.

Access a cell of the simulated states using cell indexing, for example access cell 5 using X{5}.

simStatesPeriod5 = X{5}

simStatesPeriod5 =

   -0.8660
   -2.2826
   -0.7071
    0.2177

See Also

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