MATLAB Examples

Estimate Time-Varying State-Space Model

This example shows how to:

  1. Generate data from a known model.
  2. Create a time-varying, state-space model containing unknown parameters corresponding to the data generating process.
  3. Fit the state-space model to the data.

Suppose that an AR(2) and an MA(1) model comprise a latent process. 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

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

For the last 25 periods, the state equation 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.

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

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 missing.

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. $\varepsilon_t$ is Gaussian with mean 0 and standard deviation 1.

Generate observations using the random latent state process (x) and the observation equation.

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

$$\begin{array}{l}
\left[ {\begin{array}{*{20}{c}}
{{x_{1,t}}}\\
{{x_{2,t}}}\\
{{x_{3,t}}}\\
{{x_{4,t}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{{\phi _1}}&{{\phi _2}}&0&0\\
1&0&0&0\\
0&0&0&{{\theta _1}}\\
0&0&0&0
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_{1,t - 1}}}\\
{{x_{2,t - 1}}}\\
{{x_{3,t - 1}}}\\
{{x_{4,t - 1}}}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
1&0\\
0&0\\
0&1\\
0&1
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{u_{1,t}}}\\
{{u_{2,t}}}
\end{array}} \right]\\
{y_t} = a({x_{1,t}} + {x_{3,t}}) + {\varepsilon _t}
\end{array}{\rm for\;}t = 1,...,25,$$

$$\begin{array}{l}
\left[ {\begin{array}{*{20}{c}}
{{x_{1,t}}}\\
{{x_{2,t}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{{\phi _1}}&{{\phi _2}}&0&0\\
1&0&0&0
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_{1,t - 1}}}\\
{{x_{2,t - 1}}}\\
{{x_{3,t - 1}}}\\
{{x_{4,t - 1}}}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
1\\
0
\end{array}} \right]{u_{1,t}}\\
{y_t} = b{x_{1,t}} + {\varepsilon _t}
\end{array}{\rm for\;}t = 26,$$

$$\begin{array}{l}
\left[ {\begin{array}{*{20}{c}}
{{x_{1,t}}}\\
{{x_{2,t}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{{\phi _1}}&{{\phi _2}}\\
1&0
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_{1,t - 1}}}\\
{{x_{2,t - 1}}}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
1\\
0
\end{array}} \right]{u_{1,t}}\\
{y_t} = b{x_{1,t}} + {\varepsilon _t}
\end{array}{\rm for\;}t = 27,...,50$$

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 in a file named AR2MAParamMap and put it in 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 defines the state-space model. Usually, you cannot verify implicitly defined state-space models.

Pass the observed responses (y) to estimate to estimate the parameters. Specify 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 = 


State-space model type: <a href="matlab: doc ssm">ssm</a>

State vector length: Time-varying
Observation vector length: 1
State disturbance vector length: Time-varying
Observation innovation vector length: 1
Sample size supported by model: 50

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

State equations of period 1, 2, 3,..., 25:
x1(t) = (0.48)x1(t-1) + (8.09e-03)x2(t-1) + u1(t)
x2(t) = x1(t-1)
x3(t) = (0.56)x4(t-1) + u2(t)
x4(t) = u2(t)

State equations of period 26:
x1(t) = (0.48)x1(t-1) + (8.09e-03)x2(t-1) + u1(t)
x2(t) = x1(t-1)

State equations of period 27, 28, 29,..., 50:
x1(t) = (0.48)x1(t-1) + (8.09e-03)x2(t-1) + u1(t)
x2(t) = x1(t-1)


Observation equation of period 1, 2, 3,..., 25:
y1(t) = (1.63)x1(t) + (1.63)x3(t) + e1(t)

Observation equation of period 26, 27, 28,..., 50:
y1(t) = (1.90)x1(t) + e1(t)


Initial state distribution:

Initial state means
 x1  x2  x3  x4 
  1   1   1   1 

Initial state covariance matrix
     x1  x2  x3  x4 
 x1  10  0   0   0  
 x2  0   10  0   0  
 x3  0   0   10  0  
 x4  0   0   0   10 

State types
     x1          x2          x3          x4     
 Stationary  Stationary  Stationary  Stationary 

The estimated parameters are within 1 standard error of their true values, but the standard errors are quite high. 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.