This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Click here to see
To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

Estimate Time-Invariant State-Space Model

This example shows how to generate data from a known model, specify a state-space model containing unknown parameters corresponding to the data generating process, and then fit the state-space model to the data.

Suppose that a latent process is this AR(1) process


where ut is Gaussian with mean 0 and standard deviation 1.

Generate a random series of 100 observations from xt, assuming that the series starts at 1.5.

T = 100;
ARMdl = arima('AR',0.5,'Constant',0,'Variance',1);
x0 = 1.5;
rng(1); % For reproducibility
x = simulate(ARMdl,T,'Y0',x0);

Suppose further that the latent process is subject to additive measurement error as indicated in the equation


where εt is Gaussian with mean 0 and standard deviation 0.1.

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

y = x + 0.1*randn(T,1);

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


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

A = NaN;

Specify the state-disturbance-loading coefficient matrix.

B = NaN;

Specify the measurement-sensitivity coefficient matrix.

C = 1;

Specify the observation-innovation coefficient matrix

D = NaN;

Specify the state-space model using the coefficient matrices. Also, specify the initial state mean, variance, and distribution (which is stationary).

Mean0 = 0;
Cov0 = 10;
StateType = 0;
Mdl = ssm(A,B,C,D,'Mean0',Mean0,'Cov0',Cov0,'StateType',StateType);

Mdl is an ssm model. Verify that the model is correctly specified using the display in the Command Window.

Pass the observations to estimate to estimate the parameter. Set a starting value for the parameter to params0. σ1 and σ2 must be positive, so set the lower bound constraints using the 'lb' name-value pair argument. Specify that the lower bound of ϕ is -Inf.

params0 = [0.9; 0.5; 0.1];
EstMdl = estimate(Mdl,y,params0,'lb',[-Inf; 0; 0])
Method: Maximum likelihood (fmincon)
Sample size: 100
Logarithmic  likelihood:     -140.532
Akaike   info criterion:      287.064
Bayesian info criterion:      294.879
      |     Coeff      Std Err   t Stat    Prob  
 c(1) | 0.45425       0.19870   2.28611  0.02225 
 c(2) | 0.89013       0.30359   2.93205  0.00337 
 c(3) | 0.38750       0.57858   0.66975  0.50302 
      |  Final State   Std Dev   t Stat    Prob  
 x(1) | 1.52989       0.35621   4.29498  0.00002 
EstMdl = 
State-space model type: ssm

State vector length: 1
Observation vector length: 1
State disturbance vector length: 1
Observation innovation vector length: 1
Sample size supported by model: Unlimited

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

State equation:
x1(t) = (0.45)x1(t-1) + (0.89)u1(t)

Observation equation:
y1(t) = x1(t) + (0.39)e1(t)

Initial state distribution:

Initial state means

Initial state covariance matrix
 x1  10 

State types

EstMdl is an ssm model. The results of the estimation appear in the Command Window, contain the fitted state-space equations, and contain a table of parameter estimates, their standard errors, t statistics, and p-values.

You can use or display, for example the fitted state-transition matrix using dot notation.

ans = 0.4543

Pass EstMdl to forecast to forecast observations, or to simulate to conduct a Monte Carlo study.

See Also

| | |

Related Topics