%% Specify Multiplicative ARIMA Model
% This example shows how to specify a seasonal ARIMA model using arima. The time series is monthly international airline passenger numbers from 1949 to 1960.
%% Load the airline passenger data.
% Load the airline data set, and then plot the natural log of the monthly passenger totals.

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

figure
plot(dates,y)
xlim([1,T])
datetick('x','mmmyy')
axis tight
title('Log Airline Passengers')
ylabel('(thousands)')

% The data look nonstationary, with a linear trend and seasonal periodicity.
%
%% Plot the seasonally integrated series.
% Calculate the differenced series, (1 - L)(1 - {L^{12}}){y_t}, where y_t is the original log-transformed data. Plot the differenced series.

A1 = LagOp({1,-1},'Lags',[0,1]);
A12 = LagOp({1,-1},'Lags',[0,12]);
dY = filter(A1*A12,y);

figure
plot(dY)
title('Differenced Log Airline Passengers')

% The differenced series appears stationary.
%
%% Plot the sample autocorrelation function (ACF).

figure
autocorr(dY,'NumLags',50)

% The sample ACF of the differenced series shows significant autocorrelation at lags that are multiples of 12. There is also potentially significant autocorrelation at smaller lags.
%
%% Specify a seasonal ARIMA model.
% Box, Jenkins, and Reinsel suggest the multiplicative seasonal model,
% (1 - L)(1 - {L^{12}}){y_t} = (1 - {\theta _1L})(1 - {\Theta _{12}}L^{12}){\varepsilon _t},
% for this data set (Box et al., 1994).
% Specify this model.

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

% The property P is equal to 13, corresponding to the sum of the nonseasonal and seasonal differencing degrees (1 + 12). The property Q is also equal to 13, corresponding to the sum of the degrees of the nonseasonal and seasonal MA polynomials (1 + 12). Parameters that need to be estimated have value NaN.
% References:
% Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
% Copyright 2015 The MathWorks, Inc.