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);
ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0 0 NaN NaN MA{1} -0.37716 0.066794 -5.6466 1.6364e-08 SMA{12} -0.57238 0.085439 -6.6992 2.0952e-11 Variance 0.0012634 0.00012395 10.193 2.1406e-24
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)
phat = 0.3910
err = sqrt(phat*(1-phat)/1000)
err = 0.0154
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')
See Also
arima
| estimate
| infer
| simulate
Related Examples
- Specify Multiplicative ARIMA Model
- Estimate Multiplicative ARIMA Model
- Forecast Multiplicative ARIMA Model
- Check Fit of Multiplicative ARIMA Model