%% Forecast Multiplicative ARIMA Model
% This example shows how to forecast a multiplicative seasonal ARIMA model using forecast. The time series is monthly international airline passenger numbers from 1949 to 1960.
%% Load the Data and Estimate a Model.
% Load the data set Data_Airline.

load('Data_Airline.mat')
y = log(Data);
T = length(y);

Mdl = arima('Constant',0,'D',1,'Seasonality',12,...
    'MALags',1,'SMALags',12);
EstMdl = estimate(Mdl,y);

%
%% Forecast Airline Passenger Counts.
% Use the fitted model to generate MMSE forecasts and corresponding mean square errors over a 60-month (5-year) horizon. Use the observed series as presample data. By default, forecast infers presample innovations using the specified model and observations.

[yF,yMSE] = forecast(EstMdl,60,y);
upper = yF + 1.96*sqrt(yMSE);
lower = yF - 1.96*sqrt(yMSE);

figure
plot(y,'Color',[.75,.75,.75])
hold on
h1 = plot(T+1:T+60,yF,'r','LineWidth',2);
h2 = plot(T+1:T+60,upper,'k--','LineWidth',1.5);
plot(T+1:T+60,lower,'k--','LineWidth',1.5)
xlim([0,T+60])
title('Forecast and 95% Forecast Interval')
legend([h1,h2],'Forecast','95% Interval','Location','NorthWest')
hold off

% The MMSE forecast shows airline passenger counts continuing to grow over the forecast horizon. The confidence bounds show that a decline in passenger counts is plausible, however. Because this is a nonstationary process, the width of the forecast intervals grows over time.
%
%% Compare MMSE and Monte Carlo Forecasts.
% Simulate 500 sample paths over the same forecast horizon. Compare the simulation mean to the MMSE forecast.

rng 'default';
res = infer(EstMdl,y);
Ysim = simulate(EstMdl,60,'NumPaths',500,'Y0',y,'E0',res);

yBar = mean(Ysim,2);
simU = prctile(Ysim,97.5,2);
simL = prctile(Ysim,2.5,2);

figure
h1 = plot(yF,'Color',[.85,.85,.85],'LineWidth',5);
hold on
h2 = plot(yBar,'k--','LineWidth',1.5);
xlim([0,60])
plot([upper,lower],'Color',[.85,.85,.85],'LineWidth',5)
plot([simU,simL],'k--','LineWidth',1.5)
title('Comparison of MMSE and Monte Carlo Forecasts')
legend([h1,h2],'MMSE','Monte Carlo','Location','NorthWest')
hold off

% The MMSE forecast and simulation mean are virtually indistinguishable. There are slight discrepancies between the theoretical 95% forecast intervals and the simulation-based 95% forecast intervals.
% Copyright 2015 The MathWorks, Inc.