Automatic time series forecasting
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This document outlines an automatic time series forecasting method. It discusses the motivation for automatic forecasting, including that many businesses require forecasts for thousands of products on a monthly basis and forecasts are often needed by non-experts. It reviews important early forecasting competitions from the 1970s and 1980s that helped establish best practices in evaluating and comparing forecasting methods. The competitions found that simpler methods can often outperform more complex ones, like ARIMA models, and that combining different forecasting methods can improve accuracy. This helped establish automatic forecasting as an area of study.
![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 time series forecasting Exponential smoothing 34](https://image.slidesharecdn.com/9kynqj03rc9on1l3se4u-signature-4cb3c528949477dc14dfa9d6f32e28b2c3568de2c2cef34e68a08609fce531a6-poli-150108024900-conversion-gate02/85/Automatic-time-series-forecasting-98-320.jpg)
![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 time series forecasting Exponential smoothing 34](https://image.slidesharecdn.com/9kynqj03rc9on1l3se4u-signature-4cb3c528949477dc14dfa9d6f32e28b2c3568de2c2cef34e68a08609fce531a6-poli-150108024900-conversion-gate02/85/Automatic-time-series-forecasting-99-320.jpg)
![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 time series forecasting Exponential smoothing 34](https://image.slidesharecdn.com/9kynqj03rc9on1l3se4u-signature-4cb3c528949477dc14dfa9d6f32e28b2c3568de2c2cef34e68a08609fce531a6-poli-150108024900-conversion-gate02/85/Automatic-time-series-forecasting-100-320.jpg)
![Auto ARIMA
Automatic time series forecasting ARIMA modelling 49
Forecasts from ARIMA(3,1,3)(0,1,1)[12]
Year
Totalscripts(millions)
1995 2000 2005 2010
0.40.60.81.01.21.4](https://image.slidesharecdn.com/9kynqj03rc9on1l3se4u-signature-4cb3c528949477dc14dfa9d6f32e28b2c3568de2c2cef34e68a08609fce531a6-poli-150108024900-conversion-gate02/85/Automatic-time-series-forecasting-130-320.jpg)
![Auto ARIMA
fit - auto.arima(h02)
fcast - forecast(fit)
plot(fcast)
Automatic time series forecasting ARIMA modelling 50
Forecasts from ARIMA(3,1,3)(0,1,1)[12]
Year
Totalscripts(millions)
1995 2000 2005 2010
0.40.60.81.01.21.4](https://image.slidesharecdn.com/9kynqj03rc9on1l3se4u-signature-4cb3c528949477dc14dfa9d6f32e28b2c3568de2c2cef34e68a08609fce531a6-poli-150108024900-conversion-gate02/85/Automatic-time-series-forecasting-131-320.jpg)
![Auto ARIMA
fit
Series: h02
ARIMA(3,1,3)(0,1,1)[12]
Coefficients:
ar1 ar2 ar3 ma1 ma2 ma3
-0.3648 -0.0636 0.3568 -0.4850 0.0479 -0.353
s.e. 0.2198 0.3293 0.1268 0.2227 0.2755 0.212
sma1
-0.5931
s.e. 0.0651
sigma^2 estimated as 0.002706: log likelihood=290.25
AIC=-564.5 AICc=-563.71 BIC=-538.48
Automatic time series forecasting ARIMA modelling 51](https://image.slidesharecdn.com/9kynqj03rc9on1l3se4u-signature-4cb3c528949477dc14dfa9d6f32e28b2c3568de2c2cef34e68a08609fce531a6-poli-150108024900-conversion-gate02/85/Automatic-time-series-forecasting-132-320.jpg)
Automatic time series forecasting
- 1. Rob J Hyndman Automatic time series forecasting
- 2. Outline 1 Motivation 2 Forecasting competitions 3 Forecasting the PBS 4 Exponential smoothing 5 ARIMA modelling 6 Automatic nonlinear forecasting? 7 Time series with complex seasonality 8 Recent developments Automatic time series forecasting Motivation 2
- 3. Motivation Automatic time series forecasting Motivation 3
- 4. Motivation Automatic time series forecasting Motivation 3
- 5. Motivation Automatic time series forecasting Motivation 3
- 6. Motivation Automatic time series forecasting Motivation 3
- 7. Motivation Automatic time series forecasting Motivation 3
- 8. Motivation 1 Common in business 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 time series forecasting Motivation 4
- 9. Motivation 1 Common in business 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 time series forecasting Motivation 4
- 10. Example: Asian sheep Automatic time series forecasting Motivation 5 Numbers of sheep in Asia Year millionsofsheep 1960 1970 1980 1990 2000 2010 250300350400450500550
- 11. Example: Asian sheep Automatic time series forecasting Motivation 5 Automatic ETS forecasts Year millionsofsheep 1960 1970 1980 1990 2000 2010 250300350400450500550
- 12. Example: Cortecosteroid sales Automatic 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
- 13. Example: Cortecosteroid sales Automatic time series forecasting Motivation 6 Automatic ARIMA forecasts Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.4
- 14. Outline 1 Motivation 2 Forecasting competitions 3 Forecasting the PBS 4 Exponential smoothing 5 ARIMA modelling 6 Automatic nonlinear forecasting? 7 Time series with complex seasonality 8 Recent developments Automatic time series forecasting Forecasting competitions 7
- 15. Makridakis and Hibon (1979) Automatic time series forecasting Forecasting competitions 8
- 16. Makridakis and Hibon (1979) Automatic time series forecasting Forecasting competitions 8
- 17. 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 time series forecasting Forecasting competitions 9
- 18. Makridakis and Hibon (1979) I do not believe that it is very fruitful to attempt to classify series according to which forecasting techniques perform “best”. The performance of any particular technique when applied to a particular series depends essentially on (a) the model which the series obeys; (b) our ability to identify and fit this model correctly and (c) the criterion chosen to measure the forecasting accuracy. — M.B. Priestley . . . the paper suggests the application of normal scientific experimental design to forecasting, with measures of unbiased testing of forecasts against subsequent reality, for success or failure. A long overdue reform. — F.H. Hansford-Miller Automatic time series forecasting Forecasting competitions 10
- 19. Makridakis and Hibon (1979) I do not believe that it is very fruitful to attempt to classify series according to which forecasting techniques perform “best”. The performance of any particular technique when applied to a particular series depends essentially on (a) the model which the series obeys; (b) our ability to identify and fit this model correctly and (c) the criterion chosen to measure the forecasting accuracy. — M.B. Priestley . . . the paper suggests the application of normal scientific experimental design to forecasting, with measures of unbiased testing of forecasts against subsequent reality, for success or failure. A long overdue reform. — F.H. Hansford-Miller Automatic time series forecasting Forecasting competitions 10
- 20. Makridakis and Hibon (1979) Modern man is fascinated with the subject of forecasting — W.G. Gilchrist 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 Automatic time series forecasting Forecasting competitions 11
- 21. Makridakis and Hibon (1979) Modern man is fascinated with the subject of forecasting — W.G. Gilchrist 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 Automatic time series forecasting Forecasting competitions 11
- 22. Makridakis and Hibon (1979) I find it hard to believe that Box-Jenkins, if properly applied, can actually be worse than so many of the simple methods — C. Chatfield Why do empirical studies sometimes give different answers? It may depend on the selected sample of time series, but I suspect it is more likely to depend on the skill of the analyst and on their individual interpretations of what is meant by Method X. — C. Chatfield . . . these authors are more at home with simple procedures than with Box-Jenkins. — C. Chatfield Automatic time series forecasting Forecasting competitions 12
- 23. Makridakis and Hibon (1979) I find it hard to believe that Box-Jenkins, if properly applied, can actually be worse than so many of the simple methods — C. Chatfield Why do empirical studies sometimes give different answers? It may depend on the selected sample of time series, but I suspect it is more likely to depend on the skill of the analyst and on their individual interpretations of what is meant by Method X. — C. Chatfield . . . these authors are more at home with simple procedures than with Box-Jenkins. — C. Chatfield Automatic time series forecasting Forecasting competitions 12
- 24. Makridakis and Hibon (1979) I find it hard to believe that Box-Jenkins, if properly applied, can actually be worse than so many of the simple methods — C. Chatfield Why do empirical studies sometimes give different answers? It may depend on the selected sample of time series, but I suspect it is more likely to depend on the skill of the analyst and on their individual interpretations of what is meant by Method X. — C. Chatfield . . . these authors are more at home with simple procedures than with Box-Jenkins. — C. Chatfield Automatic time series forecasting Forecasting competitions 12
- 25. Makridakis and Hibon (1979) There is a fact that Professor Priestley must accept: empirical evidence is in disagreement with his theoretical arguments. — S. Makridakis M. Hibon Dr Chatfield expresses some personal views about the first author . . . It might be useful for Dr Chatfield to read some of the psychological literature quoted in the main paper, and he can then learn a little more about biases and how they affect prior probabilities. — S. Makridakis M. Hibon Automatic time series forecasting Forecasting competitions 13
- 26. Makridakis and Hibon (1979) There is a fact that Professor Priestley must accept: empirical evidence is in disagreement with his theoretical arguments. — S. Makridakis M. Hibon Dr Chatfield expresses some personal views about the first author . . . It might be useful for Dr Chatfield to read some of the psychological literature quoted in the main paper, and he can then learn a little more about biases and how they affect prior probabilities. — S. Makridakis M. Hibon Automatic time series forecasting Forecasting competitions 13
- 27. 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 time series forecasting Forecasting competitions 14
- 28. 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 time series forecasting Forecasting competitions 14
- 29. M-competition Automatic time series forecasting Forecasting competitions 15
- 30. M-competition Main findings (taken from 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 time series forecasting Forecasting competitions 16
- 31. M3 competition Automatic time series forecasting Forecasting competitions 17
- 32. 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 time series forecasting Forecasting competitions 18
- 33. 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 time series forecasting Forecasting competitions 19
- 34. M3 results (recalculated) Method MAPE 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 time series forecasting Forecasting competitions 20
- 35. M3 results (recalculated) Method MAPE 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 time series forecasting Forecasting competitions 20 ® Calculations do not match published paper. ® Some contestants apparently submitted multiple entries but only best ones published.
- 36. Outline 1 Motivation 2 Forecasting competitions 3 Forecasting the PBS 4 Exponential smoothing 5 ARIMA modelling 6 Automatic nonlinear forecasting? 7 Time series with complex seasonality 8 Recent developments Automatic time series forecasting Forecasting the PBS 21
- 37. Forecasting the PBS Automatic time series forecasting Forecasting the PBS 22
- 38. Forecasting the PBS The Pharmaceutical Benefits Scheme (PBS) is the Australian government drugs subsidy scheme. Many drugs bought from pharmacies are subsidised to allow more equitable access to modern drugs. The cost to government is determined by the number and types of drugs purchased. Currently nearly 1% of GDP ($14 billion). The total cost is budgeted based on forecasts of drug usage. Automatic time series forecasting Forecasting the PBS 23
- 39. Forecasting the PBS The Pharmaceutical Benefits Scheme (PBS) is the Australian government drugs subsidy scheme. Many drugs bought from pharmacies are subsidised to allow more equitable access to modern drugs. The cost to government is determined by the number and types of drugs purchased. Currently nearly 1% of GDP ($14 billion). The total cost is budgeted based on forecasts of drug usage. Automatic time series forecasting Forecasting the PBS 23
- 40. Forecasting the PBS The Pharmaceutical Benefits Scheme (PBS) is the Australian government drugs subsidy scheme. Many drugs bought from pharmacies are subsidised to allow more equitable access to modern drugs. The cost to government is determined by the number and types of drugs purchased. Currently nearly 1% of GDP ($14 billion). The total cost is budgeted based on forecasts of drug usage. Automatic time series forecasting Forecasting the PBS 23
- 41. Forecasting the PBS The Pharmaceutical Benefits Scheme (PBS) is the Australian government drugs subsidy scheme. Many drugs bought from pharmacies are subsidised to allow more equitable access to modern drugs. The cost to government is determined by the number and types of drugs purchased. Currently nearly 1% of GDP ($14 billion). The total cost is budgeted based on forecasts of drug usage. Automatic time series forecasting Forecasting the PBS 23
- 42. Forecasting the PBS In 2001: $4.5 billion budget, under-forecasted by $800 million. Thousands of products. Seasonal demand. Subject to covert marketing, volatile products, uncontrollable expenditure. Although monthly data available for 10 years, data are aggregated to annual values, and only the first three years are used in estimating the forecasts. All forecasts being done with the FORECAST function in MS-Excel! Automatic time series forecasting Forecasting the PBS 24
- 43. Forecasting the PBS In 2001: $4.5 billion budget, under-forecasted by $800 million. Thousands of products. Seasonal demand. Subject to covert marketing, volatile products, uncontrollable expenditure. Although monthly data available for 10 years, data are aggregated to annual values, and only the first three years are used in estimating the forecasts. All forecasts being done with the FORECAST function in MS-Excel! Automatic time series forecasting Forecasting the PBS 24
- 44. Forecasting the PBS In 2001: $4.5 billion budget, under-forecasted by $800 million. Thousands of products. Seasonal demand. Subject to covert marketing, volatile products, uncontrollable expenditure. Although monthly data available for 10 years, data are aggregated to annual values, and only the first three years are used in estimating the forecasts. All forecasts being done with the FORECAST function in MS-Excel! Automatic time series forecasting Forecasting the PBS 24
- 45. Forecasting the PBS In 2001: $4.5 billion budget, under-forecasted by $800 million. Thousands of products. Seasonal demand. Subject to covert marketing, volatile products, uncontrollable expenditure. Although monthly data available for 10 years, data are aggregated to annual values, and only the first three years are used in estimating the forecasts. All forecasts being done with the FORECAST function in MS-Excel! Automatic time series forecasting Forecasting the PBS 24
- 46. Forecasting the PBS In 2001: $4.5 billion budget, under-forecasted by $800 million. Thousands of products. Seasonal demand. Subject to covert marketing, volatile products, uncontrollable expenditure. Although monthly data available for 10 years, data are aggregated to annual values, and only the first three years are used in estimating the forecasts. All forecasts being done with the FORECAST function in MS-Excel! Automatic time series forecasting Forecasting the PBS 24
- 47. PBS data Automatic time series forecasting Forecasting the PBS 25 Total cost: A03 concession safety net group Time $thousands 1995 2000 2005 020040060080010001200
- 48. PBS data Automatic time series forecasting Forecasting the PBS 25 Total cost: A05 general copayments group Time $thousands 1995 2000 2005 050100150200
- 49. PBS data Automatic time series forecasting Forecasting the PBS 25 Total cost: D01 general copayments group Time $thousands 1995 2000 2005 0100200300400500600700
- 50. PBS data Automatic time series forecasting Forecasting the PBS 25 Total cost: S01 general copayments group Time $thousands 1995 2000 2005 050010001500
- 51. PBS data Automatic time series forecasting Forecasting the PBS 25 Total cost: R03 general copayments group Time $thousands 1995 2000 2005 10002000300040005000
- 52. Outline 1 Motivation 2 Forecasting competitions 3 Forecasting the PBS 4 Exponential smoothing 5 ARIMA modelling 6 Automatic nonlinear forecasting? 7 Time series with complex seasonality 8 Recent developments Automatic time series forecasting Exponential smoothing 26
- 53. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 27
- 54. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 27
- 55. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 27
- 56. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 27
- 57. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 27
- 58. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 27
- 59. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 27
- 60. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 27
- 61. Exponential smoothing methods Seasonal Component 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 exponential smoothing methods. Automatic time series forecasting Exponential smoothing 27
- 62. Exponential smoothing methods Seasonal Component 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 exponential smoothing methods. Each can have an additive or multiplicative error, giving 30 separate models. Automatic time series forecasting Exponential smoothing 27
- 63. Exponential smoothing methods Seasonal Component 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 exponential smoothing methods. Each can have an additive or multiplicative error, giving 30 separate models. Only 19 models are numerically stable. Automatic time series forecasting Exponential smoothing 27
- 64. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 28
- 65. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 28
- 66. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 28
- 67. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 28
- 68. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 28
- 69. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 28
- 70. Exponential smoothing methods Seasonal Component 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 time series forecasting Exponential smoothing 28 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.
- 71. ETS state space model xt−1 εt yt Automatic time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 72. ETS state space model xt−1 εt yt xt Automatic time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 73. ETS state space model xt−1 εt yt xt yt+1 εt+1 Automatic time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 74. ETS state space model xt−1 εt yt xt yt+1 εt+1 xt+1 Automatic time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 75. ETS state space model xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 Automatic time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 76. ETS state space model xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 xt+2 Automatic time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 77. ETS state space model xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 xt+2 yt+3 εt+3 Automatic time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 78. ETS state space model xt−1 εt yt xt yt+1 εt+1 xt+1 yt+2 εt+2 xt+2 yt+3 εt+3 xt+3 Automatic time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 79. ETS state space model 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 time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal)
- 80. ETS state space model 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 time series forecasting Exponential smoothing 29 State space model xt = (level, slope, seasonal) Estimation Compute likelihood L from ε1, ε2, . . . , εT. Optimize L wrt model parameters.
- 81. Innovations state space models 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 time series forecasting Exponential smoothing 30
- 82. Innovations state space models 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 time series forecasting Exponential smoothing 31
- 83. Innovations state space models 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 time series forecasting Exponential smoothing 31
- 84. Innovations state space models 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 time series forecasting Exponential smoothing 31
- 85. Innovations state space models 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 time series forecasting Exponential smoothing 31
- 86. Innovations state space models 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 time series forecasting Exponential smoothing 31
- 87. Innovations state space models 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 time series forecasting Exponential smoothing 31 Q: How to choose between the 19 different ETS models?
- 88. Cross-validation Traditional evaluation Automatic time series forecasting Exponential smoothing 32 q 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
- 89. Cross-validation Traditional evaluation Standard cross-validation Automatic time series forecasting Exponential smoothing 32 q 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
- 90. Cross-validation Traditional evaluation Standard cross-validation Time series cross-validation Automatic time series forecasting Exponential smoothing 32 q 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
- 91. Cross-validation Traditional evaluation Standard cross-validation Time series cross-validation Automatic time series forecasting Exponential smoothing 32 q 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
- 92. Cross-validation Traditional evaluation Standard cross-validation Time series cross-validation Automatic time series forecasting Exponential smoothing 32 q 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”
- 93. 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 time series forecasting Exponential smoothing 33
- 94. 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 time series forecasting Exponential smoothing 33
- 95. 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 time series forecasting Exponential smoothing 33
- 96. 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 time series forecasting Exponential smoothing 33
- 97. 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 time series forecasting Exponential smoothing 33
- 98. 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 time series forecasting Exponential smoothing 34
- 99. 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 time series forecasting Exponential smoothing 34
- 100. 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 time series forecasting Exponential smoothing 34
- 101. 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 time series forecasting Exponential smoothing 35
- 102. 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 time series forecasting Exponential smoothing 35
- 103. 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 time series forecasting Exponential smoothing 35
- 104. 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 time series forecasting Exponential smoothing 35
- 105. 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 time series forecasting Exponential smoothing 35
- 106. ets algorithm in R Automatic time series forecasting Exponential smoothing 36 Based on Hyndman, Koehler, Snyder Grose (IJF 2002): Apply each of 19 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.
- 107. ets algorithm in R Automatic time series forecasting Exponential smoothing 36 Based on Hyndman, Koehler, Snyder Grose (IJF 2002): Apply each of 19 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.
- 108. ets algorithm in R Automatic time series forecasting Exponential smoothing 36 Based on Hyndman, Koehler, Snyder Grose (IJF 2002): Apply each of 19 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.
- 109. ets algorithm in R Automatic time series forecasting Exponential smoothing 36 Based on Hyndman, Koehler, Snyder Grose (IJF 2002): Apply each of 19 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.
- 110. Exponential smoothing Automatic time series forecasting Exponential smoothing 37 Forecasts from ETS(M,A,N) Year millionsofsheep 1960 1970 1980 1990 2000 2010 300400500600
- 111. Exponential smoothing fit - ets(livestock) fcast - forecast(fit) plot(fcast) Automatic time series forecasting Exponential smoothing 38 Forecasts from ETS(M,A,N) Year millionsofsheep 1960 1970 1980 1990 2000 2010 300400500600
- 112. Exponential smoothing Automatic time series forecasting Exponential smoothing 39 Forecasts from ETS(M,Md,M) Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.41.6
- 113. Exponential smoothing fit - ets(h02) fcast - forecast(fit) plot(fcast) Automatic time series forecasting Exponential smoothing 40 Forecasts from ETS(M,Md,M) Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.41.6
- 114. Exponential smoothing fit ETS(M,Md,M) Smoothing parameters: alpha = 0.3318 beta = 4e-04 gamma = 1e-04 phi = 0.9695 Initial states: l = 0.4003 b = 1.0233 s = 0.8575 0.8183 0.7559 0.7627 0.6873 1.2884 1.3456 1.1867 1.1653 1.1033 1.0398 0.9893 sigma: 0.0651 AIC AICc BIC -121.97999 -118.68967 -65.57195 Automatic time series forecasting Exponential smoothing 41
- 115. M3 comparisons Method MAPE 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 ETS 18.06 13.38 1.52 Automatic time series forecasting Exponential smoothing 42
- 116. Exponential smoothing Automatic time series forecasting Exponential smoothing 43
- 117. Exponential smoothing Automatic time series forecasting Exponential smoothing 43 www.OTexts.com/fpp
- 118. Exponential smoothing Automatic time series forecasting Exponential smoothing 43
- 119. Outline 1 Motivation 2 Forecasting competitions 3 Forecasting the PBS 4 Exponential smoothing 5 ARIMA modelling 6 Automatic nonlinear forecasting? 7 Time series with complex seasonality 8 Recent developments Automatic time series forecasting ARIMA modelling 44
- 120. ARIMA models yt−1 yt−2 yt−3 yt Inputs Output Automatic time series forecasting ARIMA modelling 45
- 121. ARIMA models yt−1 yt−2 yt−3 εt yt Inputs Output Automatic time series forecasting ARIMA modelling 45 Autoregression (AR) model
- 122. ARIMA models yt−1 yt−2 yt−3 εt εt−1 εt−2 yt Inputs Output Automatic time series forecasting ARIMA modelling 45 Autoregression moving average (ARMA) model
- 123. ARIMA models yt−1 yt−2 yt−3 εt εt−1 εt−2 yt Inputs Output Automatic time series forecasting ARIMA modelling 45 Autoregression moving average (ARMA) model Estimation Compute likelihood L from ε1, ε2, . . . , εT. Use optimization algorithm to maximize L.
- 124. ARIMA models yt−1 yt−2 yt−3 εt εt−1 εt−2 yt Inputs Output Automatic time series forecasting ARIMA modelling 45 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.
- 125. ARIMA modelling Automatic time series forecasting ARIMA modelling 46
- 126. ARIMA modelling Automatic time series forecasting ARIMA modelling 46
- 127. ARIMA modelling Automatic time series forecasting ARIMA modelling 46
- 128. Auto ARIMA Automatic time series forecasting ARIMA modelling 47 Forecasts from ARIMA(0,1,0) with drift Year millionsofsheep 1960 1970 1980 1990 2000 2010 250300350400450500550
- 129. Auto ARIMA fit - auto.arima(livestock) fcast - forecast(fit) plot(fcast) Automatic time series forecasting ARIMA modelling 48 Forecasts from ARIMA(0,1,0) with drift Year millionsofsheep 1960 1970 1980 1990 2000 2010 250300350400450500550
- 130. Auto ARIMA Automatic time series forecasting ARIMA modelling 49 Forecasts from ARIMA(3,1,3)(0,1,1)[12] Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.4
- 131. Auto ARIMA fit - auto.arima(h02) fcast - forecast(fit) plot(fcast) Automatic time series forecasting ARIMA modelling 50 Forecasts from ARIMA(3,1,3)(0,1,1)[12] Year Totalscripts(millions) 1995 2000 2005 2010 0.40.60.81.01.21.4
- 132. Auto ARIMA fit Series: h02 ARIMA(3,1,3)(0,1,1)[12] Coefficients: ar1 ar2 ar3 ma1 ma2 ma3 -0.3648 -0.0636 0.3568 -0.4850 0.0479 -0.353 s.e. 0.2198 0.3293 0.1268 0.2227 0.2755 0.212 sma1 -0.5931 s.e. 0.0651 sigma^2 estimated as 0.002706: log likelihood=290.25 AIC=-564.5 AICc=-563.71 BIC=-538.48 Automatic time series forecasting ARIMA modelling 51
- 133. 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 time series forecasting ARIMA modelling 52 Algorithm choices driven by forecast accuracy.
- 134. 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 time series forecasting ARIMA modelling 52 Algorithm choices driven by forecast accuracy.
- 135. 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 time series forecasting ARIMA modelling 52 Algorithm choices driven by forecast accuracy.
- 136. 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 AIC. Use stepwise search to traverse model space, starting with a simple model and considering nearby variants. Automatic time series forecasting ARIMA modelling 53
- 137. M3 comparisons Method MAPE sMAPE 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 18.06 13.38 1.52 AutoARIMA 19.04 13.86 1.47 Automatic time series forecasting ARIMA modelling 54
- 138. M3 comparisons Method MAPE sMAPE 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 18.06 13.38 1.52 AutoARIMA 19.04 13.86 1.47 ETS-ARIMA 17.92 13.02 1.44 Automatic time series forecasting ARIMA modelling 54
- 139. M3 conclusions MYTHS Simple methods do better. Exponential smoothing is better than ARIMA. FACTS The best methods are hybrid approaches. ETS-ARIMA (the simple average of ETS-additive and AutoARIMA) is the only fully documented method that is comparable to the M3 competition winners. Automatic time series forecasting ARIMA modelling 55
- 140. M3 conclusions MYTHS Simple methods do better. Exponential smoothing is better than ARIMA. FACTS The best methods are hybrid approaches. ETS-ARIMA (the simple average of ETS-additive and AutoARIMA) is the only fully documented method that is comparable to the M3 competition winners. Automatic time series forecasting ARIMA modelling 55
- 141. M3 conclusions MYTHS Simple methods do better. Exponential smoothing is better than ARIMA. FACTS The best methods are hybrid approaches. ETS-ARIMA (the simple average of ETS-additive and AutoARIMA) is the only fully documented method that is comparable to the M3 competition winners. Automatic time series forecasting ARIMA modelling 55
- 142. M3 conclusions MYTHS Simple methods do better. Exponential smoothing is better than ARIMA. FACTS The best methods are hybrid approaches. ETS-ARIMA (the simple average of ETS-additive and AutoARIMA) is the only fully documented method that is comparable to the M3 competition winners. Automatic time series forecasting ARIMA modelling 55
- 143. M3 conclusions MYTHS Simple methods do better. Exponential smoothing is better than ARIMA. FACTS The best methods are hybrid approaches. ETS-ARIMA (the simple average of ETS-additive and AutoARIMA) is the only fully documented method that is comparable to the M3 competition winners. Automatic time series forecasting ARIMA modelling 55
- 144. Outline 1 Motivation 2 Forecasting competitions 3 Forecasting the PBS 4 Exponential smoothing 5 ARIMA modelling 6 Automatic nonlinear forecasting? 7 Time series with complex seasonality 8 Recent developments Automatic time series forecasting Automatic nonlinear forecasting? 56
- 145. Automatic nonlinear forecasting Automatic ANN 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 (Neurocomputing, 2010) on automated ANN for time series. Watch this space! Automatic time series forecasting Automatic nonlinear forecasting? 57
- 146. Automatic nonlinear forecasting Automatic ANN 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 (Neurocomputing, 2010) on automated ANN for time series. Watch this space! Automatic time series forecasting Automatic nonlinear forecasting? 57
- 147. Automatic nonlinear forecasting Automatic ANN 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 (Neurocomputing, 2010) on automated ANN for time series. Watch this space! Automatic time series forecasting Automatic nonlinear forecasting? 57
- 148. Automatic nonlinear forecasting Automatic ANN 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 (Neurocomputing, 2010) on automated ANN for time series. Watch this space! Automatic time series forecasting Automatic nonlinear forecasting? 57
- 149. Automatic nonlinear forecasting Automatic ANN 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 (Neurocomputing, 2010) on automated ANN for time series. Watch this space! Automatic time series forecasting Automatic nonlinear forecasting? 57
- 150. Outline 1 Motivation 2 Forecasting competitions 3 Forecasting the PBS 4 Exponential smoothing 5 ARIMA modelling 6 Automatic nonlinear forecasting? 7 Time series with complex seasonality 8 Recent developments Automatic time series forecasting Time series with complex seasonality 58
- 151. Examples Automatic time series forecasting Time series with complex seasonality 59 US finished motor gasoline products Weeks Thousandsofbarrelsperday 1992 1994 1996 1998 2000 2002 2004 6500700075008000850090009500
- 152. Examples Automatic time series forecasting Time series with complex seasonality 59 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
- 153. Examples Automatic time series forecasting Time series with complex seasonality 59 Turkish electricity demand Days Electricitydemand(GW) 2000 2002 2004 2006 2008 10152025
- 154. TBATS model TBATS Trigonometric terms for 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 time series forecasting Time series with complex seasonality 60
- 155. 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 time series forecasting Time series with complex seasonality 61 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
- 156. 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 time series forecasting Time series with complex seasonality 61 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
- 157. 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 time series forecasting Time series with complex seasonality 61 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
- 158. 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 time series forecasting Time series with complex seasonality 61 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
- 159. 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 time series forecasting Time series with complex seasonality 61 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
- 160. 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 time series forecasting Time series with complex seasonality 61 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
- 161. 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 time series forecasting Time series with complex seasonality 61 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
- 162. Examples fit - tbats(gasoline) fcast - forecast(fit) plot(fcast) Automatic time series forecasting Time series with complex seasonality 62 Forecasts from TBATS(0.999, {2,2}, 1, {52.1785714285714,8}) Weeks Thousandsofbarrelsperday 1995 2000 2005 70008000900010000
- 163. Examples fit - tbats(callcentre) fcast - forecast(fit) plot(fcast) Automatic time series forecasting Time series with complex seasonality 63 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
- 164. Examples fit - tbats(turk) fcast - forecast(fit) plot(fcast) Automatic time series forecasting Time series with complex seasonality 64 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
- 165. Outline 1 Motivation 2 Forecasting competitions 3 Forecasting the PBS 4 Exponential smoothing 5 ARIMA modelling 6 Automatic nonlinear forecasting? 7 Time series with complex seasonality 8 Recent developments Automatic time series forecasting Recent developments 65
- 166. Further competitions 1 2011 tourism 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 time series forecasting Recent developments 66
- 167. Further competitions 1 2011 tourism 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 time series forecasting Recent developments 66
- 168. Further competitions 1 2011 tourism 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 time series forecasting Recent developments 66
- 169. Further competitions 1 2011 tourism 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 time series forecasting Recent developments 66
- 170. Forecasts about forecasting 1 Automatic 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 time series forecasting Recent developments 67
- 171. Forecasts about forecasting 1 Automatic 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 time series forecasting Recent developments 67
- 172. Forecasts about forecasting 1 Automatic 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 time series forecasting Recent developments 67
- 173. Forecasts about forecasting 1 Automatic 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 time series forecasting Recent developments 67
- 174. For further information robjhyndman.com Slides and references for this talk. Links to all papers and books. Links to R packages. A blog about forecasting research. Automatic time series forecasting Recent developments 68