Transformations for variance stabilization

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1 FORECASTING USING R Transformations for variance stabilization Rob Hyndman Author, forecast

2 Variance stabilization If the data show increasing variation as the level of the series increases, then a transformation can be useful y 1,...,y n : original observations, w 1,...,w n : transformed observations Mathematical transformations for stabilizing variation Square Root Cube Root Logarithm Inverse w t = y t w t = 3 yt w t =log(y t ) w t = 1/y t Increasing strength

3 Variance stabilization > autoplot(usmelec) + xlab("year") + ylab("") + ggtitle("us monthly net electricity generation")

4 Variance stabilization > autoplot(usmelec^0.5) + xlab("year") + ylab("") + ggtitle("square root electricity generation")

5 Variance stabilization > autoplot(usmelec^ ) + xlab("year") + ylab("") + ggtitle("cube root electricity generation")

6 Variance stabilization > autoplot(log(usmelec)) + xlab("year") + ylab("") + ggtitle("log electricity generation")

7 Variance stabilization > autoplot(-1/usmelec) + xlab("year") + ylab("") + ggtitle("inverse electricity generation")

8 Box-Cox transformations Each of these transformations is close to a member of the family of Box-Cox transformations w t = { log(yt ) λ =0 (y λ t 1)/λ λ 0 λ =1 : No substantive transformation λ = 1 2 : Square root plus linear transformation λ = 1 3 : Cube root plus linear transformation λ =0 : Natural logarithm transformation λ = 1 : Inverse transformation > BoxCox.lambda(usmelec) [1]

9 Back-transformation > usmelec %>% ets(lambda = -0.57) %>% forecast(h = 60) %>% autoplot()

10 FORECASTING USING R Let s practice!

11 FORECASTING USING R ARIMA models

12 ARIMA models Autoregressive (AR) models y t = c + φ 1 y t 1 + φ 2 y t φ p y t p + e t, e t white noise Multiple regression with lagged observations as predictors Moving Average (MA) models y t = c + e t + θ 1 e t 1 + θ 2 e t θ q e t q, e t white noise Multiple regression with lagged errors as predictors Autoregressive Moving Average (ARMA) models y t = c + φ 1 y t φ p y t p + θ 1 e t θ q e t q + e t Multiple regression with lagged observations and lagged errors as predictors ARIMA(p, d, q) models Combine ARMA model with d - lots of differencing

13 US net electricity generation > autoplot(usnetelec) + xlab("year") + ylab("billion kwh") + ggtitle("us net electricity generation")

14 US net electricity generation > fit <- auto.arima(usnetelec) > summary(fit) Series: usnetelec ARIMA(2,1,2) with drift Coefficients: ar1 ar2 ma1 ma2 drift s.e sigma^2 estimated as 2262: log likelihood= AIC=578.7 AICc=580.5 BIC=590.6 Training set error measures: ME RMSE MAE MPE MAPE MASE ACF1 Training set

15 US net electricity generation > fit %>% forecast() %>% autoplot()

16 How does auto.arima() work? Hyndman-Khandakar algorithm: Select number of differences d via unit root tests Select p and q by minimizing AICc Estimate parameters using maximum likelihood estimation Use stepwise search to traverse model space, to save time

17 FORECASTING USING R Let s practice!

18 FORECASTING USING R Seasonal ARIMA models

19 ARIMA models ARIMA (p, d, q) (P, D, Q)m Non-seasonal part of the model Seasonal part of the model d = Number of lag-1 differences p = Number of ordinary AR lags: q = Number of ordinary MA lags: y t 1, y t 2,...,y t p ε t 1, ε t 2,...,ε t q D = Number of seasonal differences P = Number of seasonal AR lags: Q = Number of seasonal MA lags: y t m, y t 2m,...,y t Pm ε t m, ε t 2m..., ε t Qm m = Number of observations per year

20 Example: Monthly retail debit card usage in Iceland > autoplot(debitcards) + xlab("year") + ylab("million ISK") + ggtitle("retail debit card usage in Iceland")

21 Example: Monthly retail debit card usage in Iceland > fit <- auto.arima(debitcards, lambda = 0) > fit Series: debitcards ARIMA(0,1,4)(0,1,1)[12] Box Cox transformation: lambda= 0 Coefficients: ma1 ma2 ma3 ma4 sma s.e sigma^2 estimated as : log likelihood=239.3 AIC= AICc= BIC=-448.6

22 Example: Monthly retail debit card usage in Iceland > fit %>% forecast(h = 36) %>% autoplot() + xlab("year")

23 FORECASTING USING R Let s practice!

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