MATLAB Examples

Forecast Time-Varying State-Space Model

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

Suppose that a latent process 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. Subsequently, 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*nansum(x')'+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    0.02976  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.

Forecast observations and states five periods into the future. Also, obtain measures of variability for the forecasts.

numPeriods = 5;
[fY,yMSE,FX,XMSE]= forecast(EstMdl,numPeriods,y);

forecast uses EstMdl.A{end}, ..., EstMdl.D{end} to forecast the state-space model. fY and yMSE are numPeriods-by-1 numeric vectors of forecasted observations and variances of the forecasted observations, respectively. FX and XMSE are numPeriods-by-2 matrices of state forecasts and variances of the state forecasts. The columns indicate the state, and the rows indicate the period. For all output arguments, the last row corresponds to the latest forecast.

Plot the observations, true states, forecasted observations, and state forecasts.

figure;
plot(T-10:T,x(T-10:T,1),'-k',T+1:T+numPeriods,FX(:,1),'-r',...
    T-10:T,y(T-10:T),'--g',T+1:T+numPeriods,fY,'--b',...
    T:T+1,[y(T),fY(1);x(T,1),FX(1,1)]',':k','LineWidth',2);
xlabel('Period')
ylabel('States and Observations')
legend({'True state values','State forecasts',...
    'Observed responses','Forecasted responses'});