%% Forecast Observations of Time-Invariant Diffuse State-Space Model
% Suppose that a latent process is a random walk. Subsequently, the state equation is
% x_t = x_{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.

T = 100;
x0 = 1.5;
rng(1); % For reproducibility
u = randn(T,1);
x = cumsum([x0;u]);
x = x(2:end);

% 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 = 1;
B = 1;
C = 1;
D = 0.75;

%
% Create the diffuse state-space model using the coefficient matrices. Specify that the initial state distribution is diffuse.

Mdl = dssm(A,B,C,D,'StateType',2)

% Mdl is an dssm model. Verify that the model is correctly specified using the display in the Command Window.
%
% Forecast observations 10 periods into the future, and estimate the mean squared errors of the forecasts.

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

% The forecast intervals flare out because the process is nonstationary.
% Copyright 2015 The MathWorks, Inc.