Previsioni con modelli ARIMA

Nel precedente esercizio, il metodo automatico ha scelto per i dati austa un modello ARIMA(0,1,1) con drift, cioè \(y_t = c + y_{t-1} + \theta e_{t-1} + e_t.\) Ora proverai vari altri modelli ARIMA sui dati per vedere come cambiano le previsioni.

La funzione Arima() può essere usata per selezionare un modello ARIMA specifico. Il suo primo argomento, order, è impostato a un vettore che specifica i valori di \(p\), \(d\) e \(q\). Il secondo argomento, include.constant, è un booleano che determina se includere la costante \(c\), ovvero il drift. Di seguito trovi un esempio di pipeline che traccia le previsioni di usnetelec da un modello ARIMA(2,1,2) con drift:

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

Negli esempi qui sotto, osserva come i diversi modelli influenzano le previsioni e gli intervalli di previsione. I dati austa sono già pronti nel tuo workspace.

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Previsioni in R

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Istruzioni dell'esercizio

Traccia le previsioni di un modello ARIMA(0,1,1) senza drift.
Traccia le previsioni di un modello ARIMA(2,1,3) con drift.
Traccia le previsioni di un modello ARIMA(0,0,1) con costante.
Traccia le previsioni di un modello ARIMA(0,2,1) senza costante.

Esercizio pratico interattivo

Prova a risolvere questo esercizio completando il codice di esempio.

# 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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Questo esercizio fa parte del corso

Previsioni 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: Trasformazioni per stabilizzare la varianza Exercise 2: Trasformazioni di Box-Cox per serie temporali Exercise 3: Differenziazione non stagionale per la stazionarietà Exercise 4: Differenziazione stagionale per la stazionarietà Exercise 5: Modelli ARIMA Exercise 6: Modelli ARIMA automatici per serie non stagionali Exercise 7: Previsioni con modelli ARIMA

Esercizio in corso

Exercise 8: Confrontare auto.arima() ed ets() su dati non stagionali Exercise 9: Modelli ARIMA stagionali Exercise 10: Modelli ARIMA automatici per serie temporali stagionali Exercise 11: Esplorare le opzioni di auto.arima()Exercise 12: Confrontare auto.arima() ed ets() su dati stagionali

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!