Suppose that a latent process is an AR(1) model. The state equation is

where $$u_t$$ is Gaussian with mean 0 and standard deviation 1.

Generate a random series of 100 observations from $$x_t$$, 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. The observation equation is

where $${\epsilon }_t$$ is Gaussian with mean 0 and standard deviation 0.75. Together, the latent process and observation equations compose a state-space model.

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

y = x + 0.75*randn(T,1);

Specify the four coefficient matrices.

A = 0.5;
B = 1;
C = 1;
D = 0.75;

Specify the state-space model using the coefficient matrices.

Mdl = ssm(A,B,C,D)

Mdl = 
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) = (0.50)x1(t-1) + u1(t)

Observation equation:
y1(t) = x1(t) + (0.75)e1(t)

Initial state distribution:

Initial state means
 x1 
  0 

Initial state covariance matrix
     x1   
 x1  1.33 

State types
     x1     
 Stationary 

Mdl is an ssm model. Verify that the model is correctly specified using the display in the Command Window. The software infers that the state process is stationary. Subsequently, the software sets the initial state mean and covariance to the mean and variance of the stationary distribution of an AR(1) model.

Simulate one path each of states and observations. Specify that the paths span 100 periods.

simX = simsmooth(Mdl,y);

simX is a 100-by-1 vector of simulated states.

Plot the true state values with the simulated states.

figure;
plot(1:T,x,'-k',1:T,simX,':r','LineWidth',2);
title 'True State Values and Simulated States';
xlabel 'Period';
ylabel 'State';
legend({'True state values','Simulated state values'});

By default, simulate simulates one path for each state in the state-space model. To conduct a Monte Carlo study, specify to simulate a large number of paths using the 'NumPaths' name-value pair argument.

The simsmooth function draws random samples from the distribution of smoothed states, or the distribution of a state given all of the data and parameters. This is the definition of posterior distribution of a state. Suppose that a latent process is an AR(1). The state equation is

where $$u_t$$ is Gaussian with mean 0 and standard deviation 1.

Generate a random series of 100 observations from $$x_t$$, 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. The observation equation is

where $${\epsilon }_t$$ is Gaussian with mean 0 and standard deviation 0.75. Together, the latent process and observation equations compose a state-space model.

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

y = x + 0.75*randn(T,1);

Specify the four coefficient matrices.

A = 0.5;
B = 1;
C = 1;
D = 0.75;

Specify the state-space model using the coefficient matrices.

Mdl = ssm(A,B,C,D);

Smooth the states of the state space model.

xsmooth = smooth(Mdl,y);

Draw 1000 paths from the posterior distribution of $$x_1$$.

N = 1000;
SimX = simsmooth(Mdl,y,'NumPaths',N);

SimX is a 100-by- 1-by- 1000 array. Rows correspond to periods, columns correspond to individual states, and leaves correspond to separate paths.

Because SimX has a singleton dimension, collapse it so that its leaves correspond to the columns using squeeze.

SimX = squeeze(SimX);

Compute the mean, standard deviation, and 95% confidence intervals of the state at each period.

xbar = mean(SimX,2);
xstd = std(SimX,[],2);
ci = [xbar - 1.96*xstd, xbar + 1.96*xstd];

Plot the smoothed states, and the means and 95% confidence intervals of the draws at each period.

figure;
plot(xsmooth,'k','LineWidth',2);
hold on;
plot(xbar,'--r','LineWidth',2);
plot(1:T,ci(:,1),'--r',1:T,ci(:,2),'--r');
legend('Smoothed states','Simulation Mean','95% CIs');
title('Smooth States and Simulation Statistics');
xlabel('Period')