Pure seasonal models

1. Pure seasonal models

In this chapter, you will learn about fitting and forecasting seasonal ARIMA models

2. Pure Seasonal Models

We often collect data with known seasonal components such as the monthly Air Passengers time series and the quarterly Johnson & Johnson earnings time series. Both have an annual cycle, 1 cycle every 12 months for the Air Passengers and 1 cycle every 4 quarters for Johnson and Johnson. The letter S denotes the seasonal period.

3. Pure Seasonal Models

Although not realistic, it is instructive to first consider pure seasonal models such as the seasonal AR shown here it is an AR(1) model, but only at the seasonal lag of 12. The model, which might be appropriate for average monthly temperatures, states that what happens this month depends on what happened in the same month last year plus some noise. You see a simulation of 3 years from such a model. Notice, for example, that January is typically small whereas June and July are typically large. The behavior of the ACF and PACF for seasonal models is similar to the nonseasonal model, but things happen only at the seasons, 1S, 2S, 3S, and so on.

4. ACF and PACF of Pure Seasonal Models

Here's an example of the ACF and PACF of a seasonal AR(1) with S equals 12 and of a seasonal MA(1) with S equals 12. Note the similarity with the nonseasonal models, but in these cases, everything happens at the seasonal lags, 1S, 2S, and so on. The Seasonal ARMA case is not shown, but the ACF and the PACF would tail off at the seasons.

5. Let's practice!

Now it's your turn!

This exercise is part of the course

ARIMA Models in R

IntermediateSkill Level

5.0+

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You will investigate the nature of time series data and learn the basics of ARMA models that can explain the behavior of such data. You will learn the basic R commands needed to help set up raw time series data to a form that can be analyzed using ARMA models.

Exercise 1: First things first Exercise 2: Data play Exercise 3: Elements of time series Exercise 4: Stationarity and nonstationarity Exercise 5: Differencing Exercise 6: Detrending data Exercise 7: Dealing with trend and heteroscedasticity Exercise 8: Stationary time series: ARMA Exercise 9: Simulating ARMA models

You will discover the wonderful world of ARMA models and how to fit these models to time series data. You will learn how to identify a model, how to choose the correct model, and how to verify a model once you fit it to data. You will learn how to use R time series commands from the stats and astsa packages.

Exercise 1: AR and MA models Exercise 2: Fitting an AR(1) model Exercise 3: Fitting an AR(2) model Exercise 4: Fitting an MA(1) model Exercise 5: AR and MA together Exercise 6: Fitting an ARMA model Exercise 7: Identify an ARMA model Exercise 8: Model choice and residual analysis Exercise 9: Model choice - I Exercise 10: Model choice - II Exercise 11: Residual analysis - I Exercise 12: Residual analysis - II Exercise 13: ARMA get in

Now that you know how to fit ARMA models to stationary time series, you will learn about integrated ARMA (ARIMA) models for nonstationary time series. You will fit the models to real data using R time series commands from the stats and astsa packages.

Exercise 1: ARIMA - integrated ARMA Exercise 2: ARIMA - plug and play Exercise 3: Simulated ARIMA Exercise 4: Global warming Exercise 5: ARIMA diagnostics Exercise 6: Diagnostics - simulated overfitting Exercise 7: Diagnostics - global temperatures Exercise 8: Forecasting ARIMA Exercise 9: Forecasting simulated ARIMA Exercise 10: Forecasting global temperatures

You will learn how to fit and forecast seasonal time series data using seasonal ARIMA models. This is accomplished using what you learned in the previous chapters and by learning how to extend the R time series commands available in the stats and astsa packages.

Exercise 1: Pure seasonal models

Current Exercise

Exercise 2: P/ACF of pure seasonal models Exercise 3: Fit a pure seasonal model Exercise 4: Mixed seasonal models Exercise 5: Fit a mixed seasonal model Exercise 6: Data analysis - unemployment I Exercise 7: Data analysis - unemployment II Exercise 8: Data analysis - commodity prices Exercise 9: Data analysis - birth rate Exercise 10: Forecasting seasonal ARIMA Exercise 11: Forecasting monthly unemployment Exercise 12: How hard is it to forecast commodity prices?Exercise 13: Congratulations!