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 is Gaussian with mean 0 and standard deviation 1.
Generate a random series of 100 observations from , 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 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. and must be positive, so set the lower bound constraints using the
'lb' name-value pair argument. Specify that the lower bound of is
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 x1 0 Initial state covariance matrix x1 x1 10 State types x1 Stationary
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
forecast to forecast observations, or to
simulate to conduct a Monte Carlo study.