Grouped time-series forecasting: Application to regional infant mortality counts

Transcription

1 Grouped time-series forecasting: Application to regional infant mortality counts Han Lin Shang and Peter W. F. Smith University of Southampton

2 Motivation 1 Multiple time series can be disaggregated by hierarchical/grouped structure

3 Motivation 1 Multiple time series can be disaggregated by hierarchical/grouped structure 2 Hyndman, Ahmed, Athanasopoulos and Shang (2010, CSDA) considered four hierarchical methods, but did not consider the construction of prediction interval for hierarchical/grouped time series

4 Motivation 1 Multiple time series can be disaggregated by hierarchical/grouped structure 2 Hyndman, Ahmed, Athanasopoulos and Shang (2010, CSDA) considered four hierarchical methods, but did not consider the construction of prediction interval for hierarchical/grouped time series 3 Present a parametric bootstrap method to construct prediction interval

5 Motivation 1 Multiple time series can be disaggregated by hierarchical/grouped structure 2 Hyndman, Ahmed, Athanasopoulos and Shang (2010, CSDA) considered four hierarchical methods, but did not consider the construction of prediction interval for hierarchical/grouped time series 3 Present a parametric bootstrap method to construct prediction interval 4 Apply to infant mortality forecasting

6 Data Consider regional infant mortality counts from 1933 to 2003, available in the hts package Darwin Northern Territory Queensland Western Australia Perth South Australia New South Wales Brisbane Australia Sydney Canberra Melbourne Capital Territory Adelaide Victoria Tasmania Hobart

7 Data 1 Hierarchical structure is expressed below Level Number of series Australia 1 Gender 2 State 8 Gender State 16 Total 27 2 Since multiple time series can be disaggregated by state first or gender first, our data are called grouped time series 3 Forecast regional infant mortality count from 2004 to 2013

8 Hierarchical tree Total Male Female VIC NSW QLD SA WA ACT NT TAS VIC NSW QLD SA WA ACT NT TAS Figure: A two level hierarchical tree diagram.

9 Bottom-up method 1 Generate base (or independent) forecasts for each series at the bottom level

10 Bottom-up method 1 Generate base (or independent) forecasts for each series at the bottom level 2 Aggregate these upwards to produce revised forecasts

11 Bottom-up method 1 Generate base (or independent) forecasts for each series at the bottom level 2 Aggregate these upwards to produce revised forecasts 3 E.g., ȲMale,h = Ȳ Male,h VIC Ȳ NT Male,h, Ȳ Total,h = ȲMale,h + ȲFemale,h, where h represents horizon

12 Bottom-up method 1 Generate base (or independent) forecasts for each series at the bottom level 2 Aggregate these upwards to produce revised forecasts 3 E.g., ȲMale,h = Ȳ Male,h VIC Ȳ NT Male,h, Ȳ Total,h = ȲMale,h + ȲFemale,h, where h represents horizon 4 Base forecasts = Revised forecasts

13 Bottom-up in action Level 0 Level 2 total nsw wa vic qld sa nt actot tas Level 1 Level 3 female nsw_f nsw_m male vic_f vic_m qld_f qld_m sa_f sa_m wa_f nt_f actot_f tas_f wa_m nt_m actot_m tas_m

14 Point forecast accuracy: data design 1 For series in the bottom level, select optimal exponential smoothing model based on information criterion, such as AIC (by defualt) or BIC

15 Point forecast accuracy: data design 1 For series in the bottom level, select optimal exponential smoothing model based on information criterion, such as AIC (by defualt) or BIC 2 Re-estimate the parameters of model using a rolling window approach, with the initial fitting period (1933 to 1993)

16 Point forecast accuracy: data design 1 For series in the bottom level, select optimal exponential smoothing model based on information criterion, such as AIC (by defualt) or BIC 2 Re-estimate the parameters of model using a rolling window approach, with the initial fitting period (1933 to 1993) 3 Forecasts are produced for one- to ten-step-ahead

17 Point forecast accuracy: data design 1 For series in the bottom level, select optimal exponential smoothing model based on information criterion, such as AIC (by defualt) or BIC 2 Re-estimate the parameters of model using a rolling window approach, with the initial fitting period (1933 to 1993) 3 Forecasts are produced for one- to ten-step-ahead 4 Iterate the process, by increasing the sample size of training period by one year until 2003

18 Point forecast accuracy: data design 1 For series in the bottom level, select optimal exponential smoothing model based on information criterion, such as AIC (by defualt) or BIC 2 Re-estimate the parameters of model using a rolling window approach, with the initial fitting period (1933 to 1993) 3 Forecasts are produced for one- to ten-step-ahead 4 Iterate the process, by increasing the sample size of training period by one year until This gives us 10 one-step-ahead forecasts, 9 two-step-ahead forecasts,..., and 1 ten-step-ahead forecast

19 Point forecast accuracy: data design 1 For series in the bottom level, select optimal exponential smoothing model based on information criterion, such as AIC (by defualt) or BIC 2 Re-estimate the parameters of model using a rolling window approach, with the initial fitting period (1933 to 1993) 3 Forecasts are produced for one- to ten-step-ahead 4 Iterate the process, by increasing the sample size of training period by one year until This gives us 10 one-step-ahead forecasts, 9 two-step-ahead forecasts,..., and 1 ten-step-ahead forecast 6 The advantage of rolling window approach is to assess forecast accuracy for each horizon

20 Point forecast accuracy: evaluation To compare point forecast accuracy between the base and bottom-up forecasts for all series, calculate mean absolute percentage error, MAPE h = n+(10 h) 1 (11 h) m i=n m j=1 Y t+h,j Ŷt+h,j Y t+h,j where m represents the total number of time series in the hierarchy, and h = 1, 2,..., 10,

21 Point forecast result Level 0 Level 1 Level 2 Level 3 Base BU Base BU Base BU Base BU Mean Bottom-up method outperforms the independent (base) forecasts (without group structure) at the top two levels, not the state level

22 Construction of interval forecasts 1 Provide pointwise interval forecasts for assessing uncertainty

23 Construction of interval forecasts 1 Provide pointwise interval forecasts for assessing uncertainty 2 Proposed method fits within the framework of parametric bootstrapping

24 Construction of interval forecasts 1 Provide pointwise interval forecasts for assessing uncertainty 2 Proposed method fits within the framework of parametric bootstrapping 3 Draw bootstrap samples from the fitted exponential smoothing model for each series at the bottom level

25 Construction of interval forecasts 1 Provide pointwise interval forecasts for assessing uncertainty 2 Proposed method fits within the framework of parametric bootstrapping 3 Draw bootstrap samples from the fitted exponential smoothing model for each series at the bottom level 4 For each bootstrap sample, we construct group structure and obtain point forecasts

26 Construction of interval forecasts 1 Provide pointwise interval forecasts for assessing uncertainty 2 Proposed method fits within the framework of parametric bootstrapping 3 Draw bootstrap samples from the fitted exponential smoothing model for each series at the bottom level 4 For each bootstrap sample, we construct group structure and obtain point forecasts 5 Based on bootstrapped forecasts, we assess the variability of point forecasts by constructing prediction interval

27 Construction of interval forecasts 1 Provide pointwise interval forecasts for assessing uncertainty 2 Proposed method fits within the framework of parametric bootstrapping 3 Draw bootstrap samples from the fitted exponential smoothing model for each series at the bottom level 4 For each bootstrap sample, we construct group structure and obtain point forecasts 5 Based on bootstrapped forecasts, we assess the variability of point forecasts by constructing prediction interval 6 Computationally, the simulate.ets function in the forecast package was used

28 Demonstration of interval forecasts Present 80% pointwise prediction interval of the regional infant mortality counts from 2004 to 2013 at the top two levels Count Total Count Male Female Year Year (a) Level 0 (b) Level 1 Infant mortality counts will continue to decrease in future. The variability of male forecasts is higher than female ones

29 Interval forecast accuracy 1 Given a sample path [Y 1,..., Y n ] where Y t is a column vector of values across the entire hierarchy, we constructed the h-step-ahead interval forecasts

30 Interval forecast accuracy 1 Given a sample path [Y 1,..., Y n ] where Y t is a column vector of values across the entire hierarchy, we constructed the h-step-ahead interval forecasts 2 Let L n+h n (p) and U n+h n (p) be the lower and upper bounds, where p symbolizes the nominal coverage probability

31 Interval forecast accuracy 1 Given a sample path [Y 1,..., Y n ] where Y t is a column vector of values across the entire hierarchy, we constructed the h-step-ahead interval forecasts 2 Let L n+h n (p) and U n+h n (p) be the lower and upper bounds, where p symbolizes the nominal coverage probability 3 Conditioning on holdout data, the indicator variable is { 1 if Yn+h,j [L I n+h,j = n+h n,j (p), U n+h n,j (p)] 0 if Y n+h,j / [L n+h n,j (p), U n+h n,j (p)] j = 1,..., m

32 Empirical coverage probability Empirical coverage probability (ECP) is defined as ECP h = 1 n+(10 h) m l=n j=1 I l+h,j, h = 1,..., 10 m (11 h) h ECP Table: Empirical coverage probability at nominal of 0.8

33 Hypothesis testing: interval forecast accuracy 1 To test if the ECP differs from the nominal coverage probability, we performed log likelihood-ratio test statistics (see Christoffersen 1998, for more details)

34 Hypothesis testing: interval forecast accuracy 1 To test if the ECP differs from the nominal coverage probability, we performed log likelihood-ratio test statistics (see Christoffersen 1998, for more details) 2 Christoffersen (1998) proposed a test for unconditional coverage, a test for independence of indicator sequence, and a joint test of conditional coverage and independence

35 Hypothesis testing: interval forecast accuracy 1 To test if the ECP differs from the nominal coverage probability, we performed log likelihood-ratio test statistics (see Christoffersen 1998, for more details) 2 Christoffersen (1998) proposed a test for unconditional coverage, a test for independence of indicator sequence, and a joint test of conditional coverage and independence 3 At the nominal coverage probability of 0.8, log likelihood-ratio are h LR Table: Critical value is 5.99 at 95% level of significance

36 Hypothesis testing: interval forecast accuracy 1 To test if the ECP differs from the nominal coverage probability, we performed log likelihood-ratio test statistics (see Christoffersen 1998, for more details) 2 Christoffersen (1998) proposed a test for unconditional coverage, a test for independence of indicator sequence, and a joint test of conditional coverage and independence 3 At the nominal coverage probability of 0.8, log likelihood-ratio are h LR Table: Critical value is 5.99 at 95% level of significance 4 At 95% level of significance, only 1 in 10 is greater than critical value

37 Conclusion 1 Revisited the bottom-up method

38 Conclusion 1 Revisited the bottom-up method 2 Applied it to the regional infant mortality count in Australia

39 Conclusion 1 Revisited the bottom-up method 2 Applied it to the regional infant mortality count in Australia 3 Performed evaluation of point forecast accuracy

40 Conclusion 1 Revisited the bottom-up method 2 Applied it to the regional infant mortality count in Australia 3 Performed evaluation of point forecast accuracy 4 Proposed a parametric bootstrap method to construct prediction interval

41 Conclusion 1 Revisited the bottom-up method 2 Applied it to the regional infant mortality count in Australia 3 Performed evaluation of point forecast accuracy 4 Proposed a parametric bootstrap method to construct prediction interval 5 Performed evaluation of interval forecast accuracy

42 Conclusion 1 Revisited the bottom-up method 2 Applied it to the regional infant mortality count in Australia 3 Performed evaluation of point forecast accuracy 4 Proposed a parametric bootstrap method to construct prediction interval 5 Performed evaluation of interval forecast accuracy 6 Carried out hypothesis testing of interval forecast accuracy

43 Future research 1 Parametric bootstrapping is expected to work for other hierarchical/grouped time series forecasting method, such as top-down methods

44 Future research 1 Parametric bootstrapping is expected to work for other hierarchical/grouped time series forecasting method, such as top-down methods 2 Modeling age-specific mortality counts hierarchically and coherently

45 Future research 1 Parametric bootstrapping is expected to work for other hierarchical/grouped time series forecasting method, such as top-down methods 2 Modeling age-specific mortality counts hierarchically and coherently 3 Extension from mortality count to mortality rate

46 Thank you A draft is available upon request from H.Shang@soton.ac.uk