Forecasting with ARIMA models

The automatic method in the previous exercise chose an ARIMA(0,1,1) with drift model for the austa data, that is, \(y_t = c + y_{t-1} + \theta e_{t-1} + e_t.\) You will now experiment with various other ARIMA models for the data to see what difference it makes to the forecasts.

The Arima() function can be used to select a specific ARIMA model. Its first argument, order, is set to a vector that specifies the values of \(p\), \(d\) and \(q\). The second argument, include.constant, is a boolean that determines if the constant \(c\), or drift, should be included. Below is an example of a pipe function that would plot forecasts of usnetelec from an ARIMA(2,1,2) model with drift:

> usnetelec %>%
    Arima(order = c(2,1,2), include.constant = TRUE) %>%
    forecast() %>%
    autoplot()

In the examples here, watch for how the different models affect the forecasts and the prediction intervals. The austa data is ready for you to use in your workspace.

Este exercício faz parte do curso

Forecasting in R

Instruções do exercício

Plot forecasts from an ARIMA(0,1,1) model with no drift.
Plot forecasts from an ARIMA(2,1,3) model with drift.
Plot forecasts from an ARIMA(0,0,1) model with a constant.
Plot forecasts from an ARIMA(0,2,1) model with no constant.

Exercício interativo prático

Experimente este exercício completando este código de exemplo.

# Plot forecasts from an ARIMA(0,1,1) model with no drift
austa %>% Arima(order = c(___, ___, ___), include.constant = ___) %>% ___ %>% ___

# Plot forecasts from an ARIMA(2,1,3) model with drift
austa %>% Arima(___, ___) %>% ___ %>% ___

# Plot forecasts from an ARIMA(0,0,1) model with a constant
austa %>% Arima(___, ___) %>% ___ %>% ___

# Plot forecasts from an ARIMA(0,2,1) model with no constant
austa %>% Arima(___, ___) %>% ___ %>% ___

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Este exercício faz parte do curso

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

Exercício atual

Exercise 8: Comparing auto.arima() and ets() on non-seasonal data Exercise 9: Seasonal ARIMA models 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!