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    %% Forecast State-Space Model Observations
% This example shows how to forecast observations of a known,
% time-invariant, state-space model.
%%
% Suppose that a latent process is an AR(1).  Subsequently, the state
% equation is 
%
% $$x_t = 0.5x_{t-1} + u_t,$$
%
% 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.

% Copyright 2015 The MathWorks, Inc.

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.  Subsequently, the observation equation is
%
% $$y_t = x_t + \varepsilon_t,$$
%
% where $\varepsilon_t$ 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| 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.
%%
% Forecast the observations 10 periods into the future, and estimate their
% variances.
numPeriods = 10;
[ForecastedY,YMSE] = forecast(Mdl,numPeriods,y);
%%
% Plot the forecasts with the in-sample responses, and 95% Wald-type forecast
% intervals.
ForecastIntervals(:,1) = ForecastedY - 1.96*sqrt(YMSE);
ForecastIntervals(:,2) = ForecastedY + 1.96*sqrt(YMSE);

figure
plot(T-20:T,y(T-20:T),'-k',T+1:T+numPeriods,ForecastedY,'-.r',...
    T+1:T+numPeriods,ForecastIntervals,'-.b',...
    T:T+1,[y(end)*ones(3,1),[ForecastedY(1);ForecastIntervals(1,:)']],':k',...
    'LineWidth',2)
hold on
title({'Observed Responses and Their Forecasts'})
xlabel('Period')
ylabel('Responses')
legend({'Observations','Forecasted observations','95% forecast intervals'},...
    'Location','Best')
hold off