Automatic algorithms for time series forecasting

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Rob J Hyndman Automaticalgorithms for time series forecasting

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Outline 1 Motivation 2 Forecastingcompetitions 3 Exponential smoothing 4 ARIMA modelling 5 Automatic nonlinear forecasting? 6 Time series with complex seasonality 7 Hierarchical and grouped time series 8 Recent developments Automatic algorithms for time series forecasting Motivation 2

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Motivation Automatic algorithms fortime series forecasting Motivation 3

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Motivation 1 Common inbusiness to have over 1000 products that need forecasting at least monthly. 2 Forecasts are often required by people who are untrained in time series analysis. Speciﬁcations Automatic forecasting algorithms must: ¯ determine an appropriate time series model; ¯ estimate the parameters; ¯ compute the forecasts with prediction intervals. Automatic algorithms for time series forecasting Motivation 4

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Motivation 1 Common inbusiness to have over 1000 products that need forecasting at least monthly. 2 Forecasts are often required by people who are untrained in time series analysis. Speciﬁcations Automatic forecasting algorithms must: ¯ determine an appropriate time series model; ¯ estimate the parameters; ¯ compute the forecasts with prediction intervals. Automatic algorithms for time series forecasting Motivation 4

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Example: Asian sheep Automaticalgorithms for time series forecasting Motivation 5 Numbers of sheep in Asia Year millionsofsheep 1960 1970 1980 1990 2000 2010 250300350400450500550

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Example: Asian sheep Automaticalgorithms for time series forecasting Motivation 5 Automatic ETS forecasts Year millionsofsheep 1960 1970 1980 1990 2000 2010 250300350400450500550

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Example: Cortecosteroid sales Automaticalgorithms for time series forecasting Motivation 6 Monthly cortecosteroid drug sales in Australia Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.4

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Example: Cortecosteroid sales Automaticalgorithms for time series forecasting Motivation 6 Forecasts from ARIMA(3,1,3)(0,1,1)[12] Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.41.6

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Outline 1 Motivation 2 Forecastingcompetitions 3 Exponential smoothing 4 ARIMA modelling 5 Automatic nonlinear forecasting? 6 Time series with complex seasonality 7 Hierarchical and grouped time series 8 Recent developments Automatic algorithms for time series forecasting Forecasting competitions 7

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Makridakis and Hibon(1979) Automatic algorithms for time series forecasting Forecasting competitions 8

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Makridakis and Hibon(1979) Automatic algorithms for time series forecasting Forecasting competitions 8

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Makridakis and Hibon(1979) This was the ﬁrst large-scale empirical evaluation of time series forecasting methods. Highly controversial at the time. Difﬁculties: How to measure forecast accuracy? How to apply methods consistently and objectively? How to explain unexpected results? Common thinking was that the more sophisticated mathematical models (ARIMA models at the time) were necessarily better. If results showed ARIMA models not best, it must be because analyst was unskilled. Automatic algorithms for time series forecasting Forecasting competitions 9

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Makridakis and Hibon(1979) It is amazing to me, however, that after all this exercise in identifying models, transforming and so on, that the autoregressive moving averages come out so badly. I wonder whether it might be partly due to the authors not using the backwards forecasting approach to obtain the initial errors. — W.G. Gilchrist I ﬁnd it hard to believe that Box-Jenkins, if properly applied, can actually be worse than so many of the simple methods . . . these authors are more at home with simple procedures than with Box-Jenkins. — C. Chatﬁeld Automatic algorithms for time series forecasting Forecasting competitions 10

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Makridakis and Hibon(1979) It is amazing to me, however, that after all this exercise in identifying models, transforming and so on, that the autoregressive moving averages come out so badly. I wonder whether it might be partly due to the authors not using the backwards forecasting approach to obtain the initial errors. — W.G. Gilchrist I ﬁnd it hard to believe that Box-Jenkins, if properly applied, can actually be worse than so many of the simple methods . . . these authors are more at home with simple procedures than with Box-Jenkins. — C. Chatﬁeld Automatic algorithms for time series forecasting Forecasting competitions 10

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Consequences of MH(1979) As a result of this paper, researchers started to: ¯ consider how to automate forecasting methods; ¯ study what methods give the best forecasts; ¯ be aware of the dangers of over-ﬁtting; ¯ treat forecasting as a different problem from time series analysis. Makridakis Hibon followed up with a new competition in 1982: 1001 series Anyone could submit forecasts (avoiding the charge of incompetence) Multiple forecast measures used. Automatic algorithms for time series forecasting Forecasting competitions 11

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Consequences of MH(1979) As a result of this paper, researchers started to: ¯ consider how to automate forecasting methods; ¯ study what methods give the best forecasts; ¯ be aware of the dangers of over-ﬁtting; ¯ treat forecasting as a different problem from time series analysis. Makridakis Hibon followed up with a new competition in 1982: 1001 series Anyone could submit forecasts (avoiding the charge of incompetence) Multiple forecast measures used. Automatic algorithms for time series forecasting Forecasting competitions 11

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M-competition Automatic algorithms fortime series forecasting Forecasting competitions 12

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M-competition Main ﬁndings (takenfrom Makridakis Hibon, 2000) 1 Statistically sophisticated or complex methods do not necessarily provide more accurate forecasts than simpler ones. 2 The relative ranking of the performance of the various methods varies according to the accuracy measure being used. 3 The accuracy when various methods are being combined outperforms, on average, the individual methods being combined and does very well in comparison to other methods. 4 The accuracy of the various methods depends upon the length of the forecasting horizon involved. Automatic algorithms for time series forecasting Forecasting competitions 13

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M3 competition Automatic algorithmsfor time series forecasting Forecasting competitions 14

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Makridakis and Hibon(2000) “The M3-Competition is a final attempt by the authors to settle the accuracy issue of various time series methods. . . The extension involves the inclusion of more methods/ researchers (in particular in the areas of neural networks and expert systems) and more series.” 3003 series All data from business, demography, finance and economics. Series length between 14 and 126. Either non-seasonal, monthly or quarterly. All time series positive. MH claimed that the M3-competition supported the findings of their earlier work. However, best performing methods far from “simple”. Automatic algorithms for time series forecasting Forecasting competitions 15

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Makridakis and Hibon(2000) Best methods: Theta A very confusing explanation. Shown by Hyndman and Billah (2003) to be average of linear regression and simple exponential smoothing with drift, applied to seasonally adjusted data. Later, the original authors claimed that their explanation was incorrect. Forecast Pro A commercial software package with an unknown algorithm. Known to ﬁt either exponential smoothing or ARIMA models using BIC. Automatic algorithms for time series forecasting Forecasting competitions 16

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M3 results (recalculated) MethodMAPE sMAPE MASE Theta 17.42 12.76 1.39 ForecastPro 18.00 13.06 1.47 ForecastX 17.35 13.09 1.42 Automatic ANN 17.18 13.98 1.53 B-J automatic 19.13 13.72 1.54 Automatic algorithms for time series forecasting Forecasting competitions 17

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M3 results (recalculated) MethodMAPE sMAPE MASE Theta 17.42 12.76 1.39 ForecastPro 18.00 13.06 1.47 ForecastX 17.35 13.09 1.42 Automatic ANN 17.18 13.98 1.53 B-J automatic 19.13 13.72 1.54 Automatic algorithms for time series forecasting Forecasting competitions 17 ® Calculations do not match published paper. ® Some contestants apparently submitted multiple entries but only best ones published.

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Outline 1 Motivation 2 Forecastingcompetitions 3 Exponential smoothing 4 ARIMA modelling 5 Automatic nonlinear forecasting? 6 Time series with complex seasonality 7 Hierarchical and grouped time series 8 Recent developments Automatic algorithms for time series forecasting Exponential smoothing 18

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M N,N: Simple exponential smoothing Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M N,N: Simple exponential smoothing A,N: Holt’s linear method Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M N,N: Simple exponential smoothing A,N: Holt’s linear method Ad,N: Additive damped trend method Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M N,N: Simple exponential smoothing A,N: Holt’s linear method Ad,N: Additive damped trend method M,N: Exponential trend method Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M N,N: Simple exponential smoothing A,N: Holt’s linear method Ad,N: Additive damped trend method M,N: Exponential trend method Md,N: Multiplicative damped trend method Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M N,N: Simple exponential smoothing A,N: Holt’s linear method Ad,N: Additive damped trend method M,N: Exponential trend method Md,N: Multiplicative damped trend method A,A: Additive Holt-Winters’ method Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M N,N: Simple exponential smoothing A,N: Holt’s linear method Ad,N: Additive damped trend method M,N: Exponential trend method Md,N: Multiplicative damped trend method A,A: Additive Holt-Winters’ method A,M: Multiplicative Holt-Winters’ method Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M There are 15 separate exp. smoothing methods. Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M There are 15 separate exp. smoothing methods. Each can have an additive or multiplicative error, giving 30 separate models. Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M There are 15 separate exp. smoothing methods. Each can have an additive or multiplicative error, giving 30 separate models. Only 19 models are numerically stable. Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M There are 15 separate exp. smoothing methods. Each can have an additive or multiplicative error, giving 30 separate models. Only 19 models are numerically stable. Multiplicative trend models give poor forecasts leaving 15 models. Automatic algorithms for time series forecasting Exponential smoothing 19

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation E T S : ExponenTial Smoothing Examples: A,N,N: Simple exponential smoothing with additive errors A,A,N: Holt’s linear method with additive errors M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors Automatic algorithms for time series forecasting Exponential smoothing 20

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation E T S : ExponenTial Smoothing Examples: A,N,N: Simple exponential smoothing with additive errors A,A,N: Holt’s linear method with additive errors M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors Automatic algorithms for time series forecasting Exponential smoothing 20

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation E T S : ExponenTial Smoothing ↑ Trend Examples: A,N,N: Simple exponential smoothing with additive errors A,A,N: Holt’s linear method with additive errors M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors Automatic algorithms for time series forecasting Exponential smoothing 20

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation E T S : ExponenTial Smoothing ↑ Trend Seasonal Examples: A,N,N: Simple exponential smoothing with additive errors A,A,N: Holt’s linear method with additive errors M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors Automatic algorithms for time series forecasting Exponential smoothing 20

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation E T S : ExponenTial Smoothing ↑ Error Trend Seasonal Examples: A,N,N: Simple exponential smoothing with additive errors A,A,N: Holt’s linear method with additive errors M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors Automatic algorithms for time series forecasting Exponential smoothing 20

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation E T S : ExponenTial Smoothing ↑ Error Trend Seasonal Examples: A,N,N: Simple exponential smoothing with additive errors A,A,N: Holt’s linear method with additive errors M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors Automatic algorithms for time series forecasting Exponential smoothing 20

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Exponential smoothing methods SeasonalComponent Trend N A M Component (None) (Additive) (Multiplicative) N (None) N,N N,A N,M A (Additive) A,N A,A A,M Ad (Additive damped) Ad,N Ad,A Ad,M M (Multiplicative) M,N M,A M,M Md (Multiplicative damped) Md,N Md,A Md,M General notation E T S : ExponenTial Smoothing ↑ Error Trend Seasonal Examples: A,N,N: Simple exponential smoothing with additive errors A,A,N: Holt’s linear method with additive errors M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors Automatic algorithms for time series forecasting Exponential smoothing 20 Innovations state space models ¯ All ETS models can be written in innovations state space form (IJF, 2002). ¯ Additive and multiplicative versions give the same point forecasts but different prediction intervals.

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ETS state spacemodel xt−1 εt yt Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt yt+1 εt+1 Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt yt+1 εt+1 xt+1 Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 xt+2 Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 xt+2 yt+3 εt+3 Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 xt+2 yt+3 εt+3 xt+3 Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 xt+2 yt+3 εt+3 xt+3 yt+4 εt+4 Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal)

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ETS state spacemodel xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 xt+2 yt+3 εt+3 xt+3 yt+4 εt+4 Automatic algorithms for time series forecasting Exponential smoothing 21 State space model xt = (level, slope, seasonal) Estimation Compute likelihood L from ε1, ε2, . . . , εT. Optimize L wrt model parameters.

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Innovations state spacemodels Let xt = ( t, bt, st, st−1, . . . , st−m+1) and εt iid ∼ N(0, σ2 ). yt = h(xt−1) + k(xt−1)εt Observation equation µt et xt = f(xt−1) + g(xt−1)εt State equation Additive errors: k(xt−1) = 1. yt = µt + εt. Multiplicative errors: k(xt−1) = µt. yt = µt(1 + εt). εt = (yt − µt)/µt is relative error. Automatic algorithms for time series forecasting Exponential smoothing 22

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Innovations state spacemodels All models can be written in state space form. Additive and multiplicative versions give same point forecasts but different prediction intervals. Estimation L∗ (θ, x0) = n log n t=1 ε2 t /k2 (xt−1) + 2 n t=1 log |k(xt−1)| = −2 log(Likelihood) + constant Minimize wrt θ = (α, β, γ, φ) and initial states x0 = ( 0, b0, s0, s−1, . . . , s−m+1). Automatic algorithms for time series forecasting Exponential smoothing 23

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Innovations state spacemodels All models can be written in state space form. Additive and multiplicative versions give same point forecasts but different prediction intervals. Estimation L∗ (θ, x0) = n log n t=1 ε2 t /k2 (xt−1) + 2 n t=1 log |k(xt−1)| = −2 log(Likelihood) + constant Minimize wrt θ = (α, β, γ, φ) and initial states x0 = ( 0, b0, s0, s−1, . . . , s−m+1). Automatic algorithms for time series forecasting Exponential smoothing 23 Q: How to choose between the 15 useful ETS models?

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Cross-validation Traditional evaluation Automatic algorithmsfor time series forecasting Exponential smoothing 24 q q q q q q q q q q q q q q q q q q q q q q q q q time Training data Test data

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Cross-validation Traditional evaluation Standard cross-validation Automaticalgorithms for time series forecasting Exponential smoothing 24 q q q q q q q q q q q q q q q q q q q q q q q q q time Training data Test data q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qqq qq q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q q q q q q q q q qqqq q q q q q q q q q q q q q q q q q q q q q qq q qq q

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Cross-validation Traditional evaluation Standard cross-validation Timeseries cross-validation Automatic algorithms for time series forecasting Exponential smoothing 24 q q q q q q q q q q q q q q q q q q q q q q q q q time Training data Test data q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qqq qq q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q q q q q q q q q qqqq q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q

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Cross-validation Traditional evaluation Standard cross-validation Timeseries cross-validation Automatic algorithms for time series forecasting Exponential smoothing 24 q q q q q q q q q q q q q q q q q q q q q q q q q time Training data Test data q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qqq qq q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q q q q q q q q q qqqq q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q

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Cross-validation Traditional evaluation Standard cross-validation Timeseries cross-validation Automatic algorithms for time series forecasting Exponential smoothing 24 q q q q q q q q q q q q q q q q q q q q q q q q q time Training data Test data q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qqq qq q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q q q q q q q q q qqqq q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Also known as “Evaluation on a rolling forecast origin”

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Akaike’s Information Criterion AIC= −2 log(L) + 2k where L is the likelihood and k is the number of estimated parameters in the model. This is a penalized likelihood approach. If L is Gaussian, then AIC ≈ c + T log MSE + 2k where c is a constant, MSE is from one-step forecasts on training set, and T is the length of the series. Minimizing the Gaussian AIC is asymptotically equivalent (as T → ∞) to minimizing MSE from one-step forecasts on test set via time series cross-validation. Automatic algorithms for time series forecasting Exponential smoothing 25

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Akaike’s Information Criterion AIC= −2 log(L) + 2k Corrected AIC For small T, AIC tends to over-ﬁt. Bias-corrected version: AICC = AIC + 2(k+1)(k+2) T−k Bayesian Information Criterion BIC = AIC + k[log(T) − 2] BIC penalizes terms more heavily than AIC Minimizing BIC is consistent if there is a true model. Automatic algorithms for time series forecasting Exponential smoothing 26

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What to use? Choice:AIC, AICc, BIC, CV-MSE CV-MSE too time consuming for most automatic forecasting purposes. Also requires large T. As T → ∞, BIC selects true model if there is one. But that is never true! AICc focuses on forecasting performance, can be used on small samples and is very fast to compute. Empirical studies in forecasting show AIC is better than BIC for forecast accuracy. Automatic algorithms for time series forecasting Exponential smoothing 27

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ets algorithm inR Automatic algorithms for time series forecasting Exponential smoothing 28 Based on Hyndman, Koehler, Snyder Grose (IJF 2002): Apply each of 15 models that are appropriate to the data. Optimize parameters and initial values using MLE. Select best method using AICc. Produce forecasts using best method. Obtain prediction intervals using underlying state space model.

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Exponential smoothing Automatic algorithmsfor time series forecasting Exponential smoothing 29 Forecasts from ETS(M,A,N) Year millionsofsheep 1960 1970 1980 1990 2000 2010 300400500600

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Exponential smoothing fit -ets(livestock) fcast - forecast(fit) plot(fcast) Automatic algorithms for time series forecasting Exponential smoothing 30 Forecasts from ETS(M,A,N) Year millionsofsheep 1960 1970 1980 1990 2000 2010 300400500600

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Exponential smoothing Automatic algorithmsfor time series forecasting Exponential smoothing 31 Forecasts from ETS(M,N,M) Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.41.6

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Exponential smoothing fit -ets(h02) fcast - forecast(fit) plot(fcast) Automatic algorithms for time series forecasting Exponential smoothing 32 Forecasts from ETS(M,N,M) Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.41.6

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Exponential smoothing fit ETS(M,N,M) Smoothingparameters: alpha = 0.4597 gamma = 1e-04 Initial states: l = 0.4501 s = 0.8628 0.8193 0.7648 0.7675 0.6946 1.2921 1.3327 1.1833 1.1617 1.0899 1.0377 0.9937 sigma: 0.0675 AIC AICc BIC -115.69960 -113.47738 -69.24592 Automatic algorithms for time series forecasting Exponential smoothing 33

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M3 comparisons Method MAPEsMAPE MASE Theta 17.42 12.76 1.39 ForecastPro 18.00 13.06 1.47 ForecastX 17.35 13.09 1.42 Automatic ANN 17.18 13.98 1.53 B-J automatic 19.13 13.72 1.54 ETS 17.38 13.13 1.43 Automatic algorithms for time series forecasting Exponential smoothing 34

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Exponential smoothing Automatic algorithmsfor time series forecasting Exponential smoothing 35

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Exponential smoothing Automatic algorithmsfor time series forecasting Exponential smoothing 35 www.OTexts.org/fpp

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Exponential smoothing Automatic algorithmsfor time series forecasting Exponential smoothing 35

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Outline 1 Motivation 2 Forecastingcompetitions 3 Exponential smoothing 4 ARIMA modelling 5 Automatic nonlinear forecasting? 6 Time series with complex seasonality 7 Hierarchical and grouped time series 8 Recent developments Automatic algorithms for time series forecasting ARIMA modelling 36

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ARIMA models yt−1 yt−2 yt−3 yt Inputs Output Automaticalgorithms for time series forecasting ARIMA modelling 37

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ARIMA models yt−1 yt−2 yt−3 εt yt Inputs Output Automaticalgorithms for time series forecasting ARIMA modelling 37 Autoregression (AR) model

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ARIMA models yt−1 yt−2 yt−3 εt εt−1 εt−2 yt Inputs Output Automaticalgorithms for time series forecasting ARIMA modelling 37 Autoregression moving average (ARMA) model

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ARIMA models yt−1 yt−2 yt−3 εt εt−1 εt−2 yt Inputs Output Automaticalgorithms for time series forecasting ARIMA modelling 37 Autoregression moving average (ARMA) model Estimation Compute likelihood L from ε1, ε2, . . . , εT. Use optimization algorithm to maximize L.

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ARIMA models yt−1 yt−2 yt−3 εt εt−1 εt−2 yt Inputs Output Automaticalgorithms for time series forecasting ARIMA modelling 37 Autoregression moving average (ARMA) model Estimation Compute likelihood L from ε1, ε2, . . . , εT. Use optimization algorithm to maximize L. ARIMA model Autoregression moving average (ARMA) model applied to differences.

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ARIMA modelling Automatic algorithmsfor time series forecasting ARIMA modelling 38

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Auto ARIMA Automatic algorithmsfor time series forecasting ARIMA modelling 39 Forecasts from ARIMA(0,1,0) with drift Year millionsofsheep 1960 1970 1980 1990 2000 2010 250300350400450500550

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Auto ARIMA fit -auto.arima(livestock) fcast - forecast(fit) plot(fcast) Automatic algorithms for time series forecasting ARIMA modelling 40 Forecasts from ARIMA(0,1,0) with drift Year millionsofsheep 1960 1970 1980 1990 2000 2010 250300350400450500550

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Auto ARIMA Automatic algorithmsfor time series forecasting ARIMA modelling 41 Forecasts from ARIMA(3,1,3)(0,1,1)[12] Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.4

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Auto ARIMA fit -auto.arima(h02) fcast - forecast(fit) plot(fcast) Automatic algorithms for time series forecasting ARIMA modelling 42 Forecasts from ARIMA(3,1,3)(0,1,1)[12] Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.4

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Auto ARIMA fit Series:h02 ARIMA(3,1,3)(0,1,1)[12] Coefficients: ar1 ar2 ar3 ma1 ma2 ma3 sma1 -0.3648 -0.0636 0.3568 -0.4850 0.0479 -0.353 -0.5931 s.e. 0.2198 0.3293 0.1268 0.2227 0.2755 0.212 0.0651 sigma^2 estimated as 0.002706: log likelihood=290.25 AIC=-564.5 AICc=-563.71 BIC=-538.48 Automatic algorithms for time series forecasting ARIMA modelling 43

111.
How does auto.arima()work? A non-seasonal ARIMA process φ(B)(1 − B)d yt = c + θ(B)εt Need to select appropriate orders p, q, d, and whether to include c. Automatic algorithms for time series forecasting ARIMA modelling 44 Algorithm choices driven by forecast accuracy.

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How does auto.arima()work? A non-seasonal ARIMA process φ(B)(1 − B)d yt = c + θ(B)εt Need to select appropriate orders p, q, d, and whether to include c. Hyndman Khandakar (JSS, 2008) algorithm: Select no. differences d via KPSS unit root test. Select p, q, c by minimising AICc. Use stepwise search to traverse model space, starting with a simple model and considering nearby variants. Automatic algorithms for time series forecasting ARIMA modelling 44 Algorithm choices driven by forecast accuracy.

113.
How does auto.arima()work? A non-seasonal ARIMA process φ(B)(1 − B)d yt = c + θ(B)εt Need to select appropriate orders p, q, d, and whether to include c. Hyndman Khandakar (JSS, 2008) algorithm: Select no. differences d via KPSS unit root test. Select p, q, c by minimising AICc. Use stepwise search to traverse model space, starting with a simple model and considering nearby variants. Automatic algorithms for time series forecasting ARIMA modelling 44 Algorithm choices driven by forecast accuracy.

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How does auto.arima()work? A seasonal ARIMA process Φ(Bm )φ(B)(1 − B)d (1 − Bm )D yt = c + Θ(Bm )θ(B)εt Need to select appropriate orders p, q, d, P, Q, D, and whether to include c. Hyndman Khandakar (JSS, 2008) algorithm: Select no. differences d via KPSS unit root test. Select D using OCSB unit root test. Select p, q, P, Q, c by minimising AICc. Use stepwise search to traverse model space, starting with a simple model and considering nearby variants. Automatic algorithms for time series forecasting ARIMA modelling 45

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M3 comparisons Method MAPEsMAPE MASE Theta 17.42 12.76 1.39 ForecastPro 18.00 13.06 1.47 B-J automatic 19.13 13.72 1.54 ETS 17.38 13.13 1.43 AutoARIMA 19.12 13.85 1.47 Automatic algorithms for time series forecasting ARIMA modelling 46

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Outline 1 Motivation 2 Forecastingcompetitions 3 Exponential smoothing 4 ARIMA modelling 5 Automatic nonlinear forecasting? 6 Time series with complex seasonality 7 Hierarchical and grouped time series 8 Recent developments Automatic algorithms for time series forecasting Automatic nonlinear forecasting? 47

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Automatic nonlinear forecasting AutomaticANN in M3 competition did poorly. Linear methods did best in the NN3 competition! Very few machine learning methods get published in the IJF because authors cannot demonstrate their methods give better forecasts than linear benchmark methods, even on supposedly nonlinear data. Some good recent work by Kourentzes and Crone on automated ANN for time series. Watch this space! Automatic algorithms for time series forecasting Automatic nonlinear forecasting? 48

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Outline 1 Motivation 2 Forecastingcompetitions 3 Exponential smoothing 4 ARIMA modelling 5 Automatic nonlinear forecasting? 6 Time series with complex seasonality 7 Hierarchical and grouped time series 8 Recent developments Automatic algorithms for time series forecasting Time series with complex seasonality 49

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Examples Automatic algorithms fortime series forecasting Time series with complex seasonality 50 US finished motor gasoline products Weeks Thousandsofbarrelsperday 1992 1994 1996 1998 2000 2002 2004 6500700075008000850090009500

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Examples Automatic algorithms fortime series forecasting Time series with complex seasonality 50 Number of calls to large American bank (7am−9pm) 5 minute intervals Numberofcallarrivals 100200300400 3 March 17 March 31 March 14 April 28 April 12 May

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Examples Automatic algorithms fortime series forecasting Time series with complex seasonality 50 Turkish electricity demand Days Electricitydemand(GW) 2000 2002 2004 2006 2008 10152025

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TBATS model TBATS Trigonometric termsfor seasonality Box-Cox transformations for heterogeneity ARMA errors for short-term dynamics Trend (possibly damped) Seasonal (including multiple and non-integer periods) Automatic algorithm described in AM De Livera, RJ Hyndman, and RD Snyder (2011). “Forecasting time series with complex seasonal patterns using exponential smoothing”. Journal of the American Statistical Association 106(496), 1513–1527. Automatic algorithms for time series forecasting Time series with complex seasonality 51

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TBATS model yt =observation at time t y (ω) t = (yω t − 1)/ω if ω = 0; log yt if ω = 0. y (ω) t = t−1 + φbt−1 + M i=1 s (i) t−mi + dt t = t−1 + φbt−1 + αdt bt = (1 − φ)b + φbt−1 + βdt dt = p i=1 φidt−i + q j=1 θjεt−j + εt s (i) t = ki j=1 s (i) j,t Automatic algorithms for time series forecasting Time series with complex seasonality 52 s (i) j,t = s (i) j,t−1 cos λ (i) j + s ∗(i) j,t−1 sin λ (i) j + γ (i) 1 dt s (i) j,t = −s (i) j,t−1 sin λ (i) j + s ∗(i) j,t−1 cos λ (i) j + γ (i) 2 dt

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TBATS model yt =observation at time t y (ω) t = (yω t − 1)/ω if ω = 0; log yt if ω = 0. y (ω) t = t−1 + φbt−1 + M i=1 s (i) t−mi + dt t = t−1 + φbt−1 + αdt bt = (1 − φ)b + φbt−1 + βdt dt = p i=1 φidt−i + q j=1 θjεt−j + εt s (i) t = ki j=1 s (i) j,t Automatic algorithms for time series forecasting Time series with complex seasonality 52 s (i) j,t = s (i) j,t−1 cos λ (i) j + s ∗(i) j,t−1 sin λ (i) j + γ (i) 1 dt s (i) j,t = −s (i) j,t−1 sin λ (i) j + s ∗(i) j,t−1 cos λ (i) j + γ (i) 2 dt Box-Cox transformation

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TBATS model yt =observation at time t y (ω) t = (yω t − 1)/ω if ω = 0; log yt if ω = 0. y (ω) t = t−1 + φbt−1 + M i=1 s (i) t−mi + dt t = t−1 + φbt−1 + αdt bt = (1 − φ)b + φbt−1 + βdt dt = p i=1 φidt−i + q j=1 θjεt−j + εt s (i) t = ki j=1 s (i) j,t Automatic algorithms for time series forecasting Time series with complex seasonality 52 s (i) j,t = s (i) j,t−1 cos λ (i) j + s ∗(i) j,t−1 sin λ (i) j + γ (i) 1 dt s (i) j,t = −s (i) j,t−1 sin λ (i) j + s ∗(i) j,t−1 cos λ (i) j + γ (i) 2 dt Box-Cox transformation M seasonal periods

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TBATS model yt =observation at time t y (ω) t = (yω t − 1)/ω if ω = 0; log yt if ω = 0. y (ω) t = t−1 + φbt−1 + M i=1 s (i) t−mi + dt t = t−1 + φbt−1 + αdt bt = (1 − φ)b + φbt−1 + βdt dt = p i=1 φidt−i + q j=1 θjεt−j + εt s (i) t = ki j=1 s (i) j,t Automatic algorithms for time series forecasting Time series with complex seasonality 52 s (i) j,t = s (i) j,t−1 cos λ (i) j + s ∗(i) j,t−1 sin λ (i) j + γ (i) 1 dt s (i) j,t = −s (i) j,t−1 sin λ (i) j + s ∗(i) j,t−1 cos λ (i) j + γ (i) 2 dt Box-Cox transformation M seasonal periods global and local trend

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TBATS model yt =observation at time t y (ω) t = (yω t − 1)/ω if ω = 0; log yt if ω = 0. y (ω) t = t−1 + φbt−1 + M i=1 s (i) t−mi + dt t = t−1 + φbt−1 + αdt bt = (1 − φ)b + φbt−1 + βdt dt = p i=1 φidt−i + q j=1 θjεt−j + εt s (i) t = ki j=1 s (i) j,t Automatic algorithms for time series forecasting Time series with complex seasonality 52 s (i) j,t = s (i) j,t−1 cos λ (i) j + s ∗(i) j,t−1 sin λ (i) j + γ (i) 1 dt s (i) j,t = −s (i) j,t−1 sin λ (i) j + s ∗(i) j,t−1 cos λ (i) j + γ (i) 2 dt Box-Cox transformation M seasonal periods global and local trend ARMA error

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TBATS model yt =observation at time t y (ω) t = (yω t − 1)/ω if ω = 0; log yt if ω = 0. y (ω) t = t−1 + φbt−1 + M i=1 s (i) t−mi + dt t = t−1 + φbt−1 + αdt bt = (1 − φ)b + φbt−1 + βdt dt = p i=1 φidt−i + q j=1 θjεt−j + εt s (i) t = ki j=1 s (i) j,t Automatic algorithms for time series forecasting Time series with complex seasonality 52 s (i) j,t = s (i) j,t−1 cos λ (i) j + s ∗(i) j,t−1 sin λ (i) j + γ (i) 1 dt s (i) j,t = −s (i) j,t−1 sin λ (i) j + s ∗(i) j,t−1 cos λ (i) j + γ (i) 2 dt Box-Cox transformation M seasonal periods global and local trend ARMA error Fourier-like seasonal terms

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TBATS model yt =observation at time t y (ω) t = (yω t − 1)/ω if ω = 0; log yt if ω = 0. y (ω) t = t−1 + φbt−1 + M i=1 s (i) t−mi + dt t = t−1 + φbt−1 + αdt bt = (1 − φ)b + φbt−1 + βdt dt = p i=1 φidt−i + q j=1 θjεt−j + εt s (i) t = ki j=1 s (i) j,t Automatic algorithms for time series forecasting Time series with complex seasonality 52 s (i) j,t = s (i) j,t−1 cos λ (i) j + s ∗(i) j,t−1 sin λ (i) j + γ (i) 1 dt s (i) j,t = −s (i) j,t−1 sin λ (i) j + s ∗(i) j,t−1 cos λ (i) j + γ (i) 2 dt Box-Cox transformation M seasonal periods global and local trend ARMA error Fourier-like seasonal terms TBATS Trigonometric Box-Cox ARMA Trend Seasonal

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Examples fit - tbats(gasoline) fcast- forecast(fit) plot(fcast) Automatic algorithms for time series forecasting Time series with complex seasonality 53 Forecasts from TBATS(0.999, {2,2}, 1, {52.1785714285714,8}) Weeks Thousandsofbarrelsperday 1995 2000 2005 70008000900010000

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Examples fit - tbats(callcentre) fcast- forecast(fit) plot(fcast) Automatic algorithms for time series forecasting Time series with complex seasonality 54 Forecasts from TBATS(1, {3,1}, 0.987, {169,5, 845,3}) 5 minute intervals Numberofcallarrivals 0100200300400500 3 March 17 March 31 March 14 April 28 April 12 May 26 May 9 June

136.
Examples fit - tbats(turk) fcast- forecast(fit) plot(fcast) Automatic algorithms for time series forecasting Time series with complex seasonality 55 Forecasts from TBATS(0, {5,3}, 0.997, {7,3, 354.37,12, 365.25,4}) Days Electricitydemand(GW) 2000 2002 2004 2006 2008 2010 10152025

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Outline 1 Motivation 2 Forecastingcompetitions 3 Exponential smoothing 4 ARIMA modelling 5 Automatic nonlinear forecasting? 6 Time series with complex seasonality 7 Hierarchical and grouped time series 8 Recent developments Automatic algorithms for time series forecasting Hierarchical and grouped time series 56

138.
Hierarchical time series Ahierarchical time series is a collection of several time series that are linked together in a hierarchical structure. Total A AA AB AC B BA BB BC C CA CB CC Examples Net labour turnover Tourism by state and region Automatic algorithms for time series forecasting Hierarchical and grouped time series 57

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Hierarchical time series Total AB C Automatic algorithms for time series forecasting Hierarchical and grouped time series 58 Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. bt : vector of all series at bottom level in time t.

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Hierarchical time series Total AB C Automatic algorithms for time series forecasting Hierarchical and grouped time series 58 Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. bt : vector of all series at bottom level in time t.

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Hierarchical time series Total AB C yt = [Yt, YA,t, YB,t, YC,t] =     1 1 1 1 0 0 0 1 0 0 0 1       YA,t YB,t YC,t   Automatic algorithms for time series forecasting Hierarchical and grouped time series 58 Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. bt : vector of all series at bottom level in time t.

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Hierarchical time series Total AB C yt = [Yt, YA,t, YB,t, YC,t] =     1 1 1 1 0 0 0 1 0 0 0 1     S   YA,t YB,t YC,t   Automatic algorithms for time series forecasting Hierarchical and grouped time series 58 Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. bt : vector of all series at bottom level in time t.

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Hierarchical time series Total AB C yt = [Yt, YA,t, YB,t, YC,t] =     1 1 1 1 0 0 0 1 0 0 0 1     S   YA,t YB,t YC,t   bt Automatic algorithms for time series forecasting Hierarchical and grouped time series 58 Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. bt : vector of all series at bottom level in time t.

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Hierarchical time series Total AB C yt = [Yt, YA,t, YB,t, YC,t] =     1 1 1 1 0 0 0 1 0 0 0 1     S   YA,t YB,t YC,t   bt yt = Sbt Automatic algorithms for time series forecasting Hierarchical and grouped time series 58 Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. bt : vector of all series at bottom level in time t.

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Hierarchical time series Total A AXAY AZ B BX BY BZ C CX CY CZ yt =             Yt YA,t YB,t YC,t YAX,t YAY,t YAZ,t YBX,t YBY,t YBZ,t YCX,t YCY,t YCZ,t             =             1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1             S        YAX,t YAY,t YAZ,t YBX,t YBY,t YBZ,t YCX,t YCY,t YCZ,t        bt Automatic algorithms for time series forecasting Hierarchical and grouped time series 59

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Hierarchical time series Total A AXAY AZ B BX BY BZ C CX CY CZ yt =             Yt YA,t YB,t YC,t YAX,t YAY,t YAZ,t YBX,t YBY,t YBZ,t YCX,t YCY,t YCZ,t             =             1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1             S        YAX,t YAY,t YAZ,t YBX,t YBY,t YBZ,t YCX,t YCY,t YCZ,t        bt Automatic algorithms for time series forecasting Hierarchical and grouped time series 59

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Hierarchical time series Total A AXAY AZ B BX BY BZ C CX CY CZ yt =             Yt YA,t YB,t YC,t YAX,t YAY,t YAZ,t YBX,t YBY,t YBZ,t YCX,t YCY,t YCZ,t             =             1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1             S        YAX,t YAY,t YAZ,t YBX,t YBY,t YBZ,t YCX,t YCY,t YCZ,t        bt Automatic algorithms for time series forecasting Hierarchical and grouped time series 59 yt = Sbt

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Forecasting notation Let ˆyn(h)be vector of initial h-step forecasts, made at time n, stacked in same order as yt. (They may not add up.) Reconciled forecasts are of the form: ˜yn(h) = SPˆyn(h) for some matrix P. P extracts and combines base forecasts ˆyn(h) to get bottom-level forecasts. S adds them up Automatic algorithms for time series forecasting Hierarchical and grouped time series 60

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General properties ˜yn(h) =SPˆyn(h) Forecast bias Assuming the base forecasts ˆyn(h) are unbiased, then the revised forecasts are unbiased iff SPS = S. Forecast variance For any given P satisfying SPS = S, the covariance matrix of the h-step ahead reconciled forecast errors is given by Var[yn+h − ˜yn(h)] = SPWhP S where Wh is the covariance matrix of the h-step ahead base forecast errors. Automatic algorithms for time series forecasting Hierarchical and grouped time series 61

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BLUF via traceminimization Theorem For any P satisfying SPS = S, then min P = trace[SPWhP S ] has solution P = (S W † hS)−1 S W † h. W † h is generalized inverse of Wh. ˜yn(h) = S(S W † hS)−1 S W † h ˆyn(h) Revised forecasts Base forecasts Equivalent to GLS estimate of regression ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Wh). Problem: Wh hard to estimate. Automatic algorithms for time series forecasting Hierarchical and grouped time series 62

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Optimal combination forecasts Revisedforecasts Base forecasts Solution 1: OLS ˜yn(h) = S(S S)−1 S ˆyn(h) Solution 2: WLS Approximate W1 by its diagonal. Assume Wh = khW1. Easy to estimate, and places weight where we have best one-step forecasts. ˜yn(h) = S(S ΛS)−1 S Λˆyn(h) Automatic algorithms for time series forecasting Hierarchical and grouped time series 63 ˜yn(h) = S(S W † hS)−1 S W † h ˆyn(h)

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Challenges Computational difﬁculties inbig hierarchies due to size of the S matrix and singular behavior of (S ΛS). Loss of information in ignoring covariance matrix in computing point forecasts. Still need to estimate covariance matrix to produce prediction intervals. Automatic algorithms for time series forecasting Hierarchical and grouped time series 64 ˜yn(h) = S(S ΛS)−1 S Λˆyn(h)

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Australian tourism Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 65

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Australian tourism Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 65 Hierarchy: States (7) Zones (27) Regions (82)

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Australian tourism Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 65 Hierarchy: States (7) Zones (27) Regions (82) Base forecasts ETS (exponential smoothing) models

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: Total Year Visitornights 1998 2000 2002 2004 2006 2008 600006500070000750008000085000

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: NSW Year Visitornights 1998 2000 2002 2004 2006 2008 18000220002600030000

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: VIC Year Visitornights 1998 2000 2002 2004 2006 2008 1000012000140001600018000

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: Nth.Coast.NSW Year Visitornights 1998 2000 2002 2004 2006 2008 50006000700080009000

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: Metro.QLD Year Visitornights 1998 2000 2002 2004 2006 2008 800090001100013000

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: Sth.WA Year Visitornights 1998 2000 2002 2004 2006 2008 400600800100012001400

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: X201.Melbourne Year Visitornights 1998 2000 2002 2004 2006 2008 40004500500055006000

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: X402.Murraylands Year Visitornights 1998 2000 2002 2004 2006 2008 0100200300

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Base forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 66 Domestic tourism forecasts: X809.Daly Year Visitornights 1998 2000 2002 2004 2006 2008 020406080100

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Reconciled forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 67 Total 2000 2005 2010 650008000095000

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Reconciled forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 67 NSW 2000 2005 2010 180002400030000 VIC 2000 2005 2010 100001400018000 QLD 2000 2005 2010 1400020000 Other 2000 2005 2010 1800024000

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Reconciled forecasts Automatic algorithmsfor time series forecasting Hierarchical and grouped time series 67 Sydney 2000 2005 2010 40007000 OtherNSW 2000 2005 2010 1400022000 Melbourne 2000 2005 2010 40005000 OtherVIC 2000 2005 2010 600012000 GCandBrisbane 2000 2005 2010 60009000 OtherQLD 2000 2005 2010 600012000 Capitalcities 2000 2005 2010 1400020000 Other 2000 2005 2010 55007500

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Forecast evaluation Select modelsusing all observations; Re-estimate models using ﬁrst 12 observations and generate 1- to 8-step-ahead forecasts; Increase sample size one observation at a time, re-estimate models, generate forecasts until the end of the sample; In total 24 1-step-ahead, 23 2-steps-ahead, up to 17 8-steps-ahead for forecast evaluation. Automatic algorithms for time series forecasting Hierarchical and grouped time series 68

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Hierarchy: states, zones,regions MAPE h = 1 h = 2 h = 4 h = 6 h = 8 Average Top Level: Australia Bottom-up 3.79 3.58 4.01 4.55 4.24 4.06 OLS 3.83 3.66 3.88 4.19 4.25 3.94 WLS 3.68 3.56 3.97 4.57 4.25 4.04 Level: States Bottom-up 10.70 10.52 10.85 11.46 11.27 11.03 OLS 11.07 10.58 11.13 11.62 12.21 11.35 WLS 10.44 10.17 10.47 10.97 10.98 10.67 Level: Zones Bottom-up 14.99 14.97 14.98 15.69 15.65 15.32 OLS 15.16 15.06 15.27 15.74 16.15 15.48 WLS 14.63 14.62 14.68 15.17 15.25 14.94 Bottom Level: Regions Bottom-up 33.12 32.54 32.26 33.74 33.96 33.18 OLS 35.89 33.86 34.26 36.06 37.49 35.43 WLS 31.68 31.22 31.08 32.41 32.77 31.89 Automatic algorithms for time series forecasting Hierarchical and grouped time series 69

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hts package forR Automatic algorithms for time series forecasting Hierarchical and grouped time series 70 hts: Hierarchical and grouped time series Methods for analysing and forecasting hierarchical and grouped time series Version: 4.5 Depends: forecast (≥ 5.0), SparseM Imports: parallel, utils Published: 2014-12-09 Author: Rob J Hyndman, Earo Wang and Alan Lee Maintainer: Rob J Hyndman Rob.Hyndman at monash.edu BugReports: https://github.com/robjhyndman/hts/issues License: GPL (≥ 2)

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Outline 1 Motivation 2 Forecastingcompetitions 3 Exponential smoothing 4 ARIMA modelling 5 Automatic nonlinear forecasting? 6 Time series with complex seasonality 7 Hierarchical and grouped time series 8 Recent developments Automatic algorithms for time series forecasting Recent developments 71

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Further competitions 1 2011tourism forecasting competition. 2 Kaggle and other forecasting platforms. 3 GEFCom 2012: Point forecasting of electricity load and wind power. 4 GEFCom 2014: Probabilistic forecasting of electricity load, electricity price, wind energy and solar energy. Automatic algorithms for time series forecasting Recent developments 72

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Forecasts about forecasting 1Automatic algorithms will become more general — handling a wide variety of time series. 2 Model selection methods will take account of multi-step forecast accuracy as well as one-step forecast accuracy. 3 Automatic forecasting algorithms for multivariate time series will be developed. 4 Automatic forecasting algorithms that include covariate information will be developed. Automatic algorithms for time series forecasting Recent developments 73

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For further information robjhyndman.com Slidesand references for this talk. Links to all papers and books. Links to R packages. A blog about forecasting research. Automatic algorithms for time series forecasting Recent developments 74

Automatic algorithms for time series forecasting

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