Seasonal ARIMA models

1. Seasonal ARIMA models

ARIMA models can also handle seasonal time series.

2. ARIMA models

We just need to add seasonal differencing and a whole lot more lagged terms into the model. As you know by now, an ARIMA(p,d,q) model involves d lag-1 differences, p lagged observations and q lagged errors. For a seasonal ARIMA model,

3. ARIMA models

we have another P, D and Q, all uppercase, referring to seasonal versions of these.

4. ARIMA models

D refers to the number of seasonal differences to use, P to the number of seasonally lagged observations, and Q to the number of seasonally lagged errors. Finally, the m on the end indicates the seasonal period - the number of observations in each year of data. As you can imagine, these models can get very complicated to write down. They are also no longer linear, as the seasonal parts get multiplied with the non-seasonal parts of the model. I won't attempt to show you the equations here.

5. Example: Monthly retail debit card usage in Iceland

Let's look at an example of monthly data. This is the total usage for debit cards in Iceland over a 13 year period. There is increasing variation, so we need to use a Box-Cox transformation. To keep it simple, I will use a log transformation and set lambda equals 0. Everything else about the model can be handled automatically using the auto-dot-arima function.

6. Example: Monthly retail debit card usage in Iceland

We apply auto-dot-arima to the data, setting lambda equals 0. The resulting model is an ARIMA(0,1,4)(0,1,1). That is, both seasonal and first differences have been used - indicated by the 1s in the middle slot of each part of the model. It also tells us that 4 lagged errors and 1 seasonally lagged error have been selected. The 0s indicate that no autoregression terms have been used. The rest of the output tells us about the model coefficients and other model information. ARIMA models can be hard to interpret, and all those coefficients don't have a neat explanation in terms of the original data. But they are very powerful models which can handle a very wide range of time series.

7. Example: Monthly retail debit card usage in Iceland

We can pass the model object to the forecast function to get forecasts. I'm forecasting 3 years ahead so you can clearly see the increasing trend and increasing variation due to the Box-Cox transformation. Notice how the seasonal parts of the model have captured the seasonality quite well. A nice feature of seasonal ARIMA models is that they allow the seasonality to change over time. The forecasted seasonal pattern will be most affected by the shape of the seasonality near the end of the series, and not so much on the seasonal patterns at the start of the series. You might wonder where the trend comes from in this model, as there is no drift term here. It turns out that whenever you do two lots of differencing of any kind, the forecasts will have a trend without needing to include the constant. Here we have done both ordinary and seasonal differencing, so there is a trend in the forecasts.

8. Let's practice!

Now it's your turn to try auto-dot-arima on a different seasonal time series.

This exercise is part of the course

Forecasting in R

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The first thing to do in any data analysis task is to plot the data. Graphs enable many features of the data to be visualized, including patterns, unusual observations, and changes over time. The features that are seen in plots of the data must then be incorporated, as far as possible, into the forecasting methods to be used.

Exercise 1: Welcome to Forecasting Using R Exercise 2: Creating time series objects in R Exercise 3: Time series plots Exercise 4: Seasonal plots Exercise 5: Trends, seasonality, and cyclicity Exercise 6: Autocorrelation of non-seasonal time series Exercise 7: Autocorrelation of seasonal and cyclic time series Exercise 8: Match the ACF to the time series Exercise 9: White noise Exercise 10: Stock prices and white noise

In this chapter, you will learn general tools that are useful for many different forecasting situations. It will describe some methods for benchmark forecasting, methods for checking whether a forecasting method has adequately utilized the available information, and methods for measuring forecast accuracy. Each of the tools discussed in this chapter will be used repeatedly in subsequent chapters as you develop and explore a range of forecasting methods.

Exercise 1: Forecasts and potential futures Exercise 2: Naive forecasting methods Exercise 3: Fitted values and residuals Exercise 4: Checking time series residuals Exercise 5: Training and test sets Exercise 6: Evaluating forecast accuracy of non-seasonal methods Exercise 7: Evaluating forecast accuracy of seasonal methods Exercise 8: Do I have a good forecasting model?Exercise 9: Time series cross-validation Exercise 10: Using tsCV() for time series cross-validation

Forecasts produced using exponential smoothing methods are weighted averages of past observations, with the weights decaying exponentially as the observations get older. In other words, the more recent the observation, the higher the associated weight. This framework generates reliable forecasts quickly and for a wide range of time series, which is a great advantage and of major importance to applications in business.

Exercise 1: Exponentially weighted forecasts Exercise 2: Simple exponential smoothing Exercise 3: SES vs naive Exercise 4: Exponential smoothing methods with trend Exercise 5: Holt's trend methods Exercise 6: Exponential smoothing methods with trend and seasonality Exercise 7: Holt-Winters with monthly data Exercise 8: Holt-Winters method with daily data Exercise 9: State space models for exponential smoothing Exercise 10: Automatic forecasting with exponential smoothing Exercise 11: ETS vs seasonal naive Exercise 12: Match the models to the time series Exercise 13: When does ETS fail?

ARIMA models provide another approach to time series forecasting. Exponential smoothing and ARIMA models are the two most widely-used approaches to time series forecasting, and provide complementary approaches to the problem. While exponential smoothing models are based on a description of the trend and seasonality in the data, ARIMA models aim to describe the autocorrelations in the data.

Exercise 1: Transformations for variance stabilization Exercise 2: Box-Cox transformations for time series Exercise 3: Non-seasonal differencing for stationarity Exercise 4: Seasonal differencing for stationarity Exercise 5: ARIMA models Exercise 6: Automatic ARIMA models for non-seasonal time series Exercise 7: Forecasting with ARIMA models Exercise 8: Comparing auto.arima() and ets() on non-seasonal data Exercise 9: Seasonal ARIMA models

Current Exercise

Exercise 10: Automatic ARIMA models for seasonal time series Exercise 11: Exploring auto.arima() options Exercise 12: Comparing auto.arima() and ets() on seasonal data

The time series models in the previous chapters work well for many time series, but they are often not good for weekly or hourly data, and they do not allow for the inclusion of other information such as the effects of holidays, competitor activity, changes in the law, etc. In this chapter, you will look at some methods that handle more complicated seasonality, and you consider how to extend ARIMA models in order to allow other information to be included in the them.

Exercise 1: Dynamic regression Exercise 2: Forecasting sales allowing for advertising expenditure Exercise 3: Forecasting electricity demand Exercise 4: Dynamic harmonic regression Exercise 5: Forecasting weekly data Exercise 6: Harmonic regression for multiple seasonality Exercise 7: Forecasting call bookings Exercise 8: TBATS models Exercise 9: TBATS models for electricity demand Exercise 10: Your future in forecasting!