%% Simulate Multiplicative ARIMA Models
% This example shows how to simulate sample paths from a multiplicative seasonal ARIMA model using simulate. 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);
res = infer(EstMdl,y);

%
%% Simulate Airline Passenger Counts.
% Use the fitted model to simulate 25 realizations of airline passenger counts over a 60-month (5-year) horizon. Use the observed series and inferred residuals as presample data.

rng('default')
Ysim = simulate(EstMdl,60,'NumPaths',25,'Y0',y,'E0',res);
mn = mean(Ysim,2);

figure
plot(y,'k')
hold on
plot(T+1:T+60,Ysim,'Color',[.85,.85,.85]);
h = plot(T+1:T+60,mn,'k--','LineWidth',2);
xlim([0,T+60])
title('Simulated Airline Passenger Counts')
legend(h,'Simulation Mean','Location','NorthWest')
hold off

% The simulated forecasts show growth and seasonal periodicity similar to the observed series.
%
%% Estimate the Probability of a Future Event.
% Use simulations to estimate the probability that log airline passenger counts will meet or exceed the value 7 sometime during the next 5 years. Calculate the Monte Carlo error associated with the estimated probability.

rng default
Ysim = simulate(EstMdl,60,'NumPaths',1000,'Y0',y,'E0',res);

g7 = sum(Ysim >= 7) > 0;
phat = mean(g7)
err = sqrt(phat*(1-phat)/1000)

% There is approximately a 39% chance that the (log) number of airline passengers will meet or exceed 7 in the next 5 years. The Monte Carlo standard error of the estimate is about 0.02.
%
%% Plot the Distribution of Passengers at a Future Time.
% Use the simulations to plot the distribution of (log) airline passenger counts 60 months into the future.

figure
histogram(Ysim(60,:),10)
title('Distribution of Passenger Counts in 60 months')

% Copyright 2012 The MathWorks, Inc.