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Motivation Data Method Result Conclusion
Grouped time-series forecasting:Application to regional infant mortality counts
Han Lin Shang and Peter W. F. SmithUniversity of Southampton
Motivation Data Method Result Conclusion
Motivation
1 Multiple time series can be disaggregated byhierarchical/grouped structure
2 Hyndman, Ahmed, Athanasopoulos and Shang (2010, CSDA)considered four hierarchical methods, but did not consider theconstruction of prediction interval for hierarchical/groupedtime series
3 Present a parametric bootstrap method to construct predictioninterval
4 Apply to infant mortality forecasting
Motivation Data Method Result Conclusion
Motivation
1 Multiple time series can be disaggregated byhierarchical/grouped structure
2 Hyndman, Ahmed, Athanasopoulos and Shang (2010, CSDA)considered four hierarchical methods, but did not consider theconstruction of prediction interval for hierarchical/groupedtime series
3 Present a parametric bootstrap method to construct predictioninterval
4 Apply to infant mortality forecasting
Motivation Data Method Result Conclusion
Motivation
1 Multiple time series can be disaggregated byhierarchical/grouped structure
2 Hyndman, Ahmed, Athanasopoulos and Shang (2010, CSDA)considered four hierarchical methods, but did not consider theconstruction of prediction interval for hierarchical/groupedtime series
3 Present a parametric bootstrap method to construct predictioninterval
4 Apply to infant mortality forecasting
Motivation Data Method Result Conclusion
Motivation
1 Multiple time series can be disaggregated byhierarchical/grouped structure
2 Hyndman, Ahmed, Athanasopoulos and Shang (2010, CSDA)considered four hierarchical methods, but did not consider theconstruction of prediction interval for hierarchical/groupedtime series
3 Present a parametric bootstrap method to construct predictioninterval
4 Apply to infant mortality forecasting
Motivation Data Method Result Conclusion
Data
Consider regional infant mortality counts from 1933 to 2003,available in the hts package
Western Australia
South Australia
Northern Territory
Queensland
New South Wales
VictoriaTasmania
Capital Territory
Perth
Adelaide
Darwin
Brisbane
Sydney
Melbourne
Hobart
Canberra
Australia
Motivation Data Method Result Conclusion
Data
1 Hierarchical structure is expressed below
Level Number of seriesAustralia 1Gender 2State 8Gender × State 16Total 27
2 Since multiple time series can be disaggregated by state firstor gender first, our data are called grouped time series
3 Forecast regional infant mortality count from 2004 to 2013
Motivation Data Method Result Conclusion
Hierarchical tree
Total
Male
VIC NSW QLD SA WA ACT NT TAS
Female
VIC NSW QLD SA WA ACT NT TAS
Figure: A two level hierarchical tree diagram.
Motivation Data Method Result Conclusion
Bottom-up method
1 Generate base (or independent) forecasts for each series at thebottom level
2 Aggregate these upwards to produce revised forecasts3 E.g., YMale,h = Y VIC
Male,h + ... + Y NTMale,h,
YTotal,h = YMale,h + YFemale,h, where h represents horizon4 Base forecasts = Revised forecasts
Motivation Data Method Result Conclusion
Bottom-up method
1 Generate base (or independent) forecasts for each series at thebottom level
2 Aggregate these upwards to produce revised forecasts
3 E.g., YMale,h = Y VICMale,h + ... + Y NT
Male,h,YTotal,h = YMale,h + YFemale,h, where h represents horizon
4 Base forecasts = Revised forecasts
Motivation Data Method Result Conclusion
Bottom-up method
1 Generate base (or independent) forecasts for each series at thebottom level
2 Aggregate these upwards to produce revised forecasts3 E.g., YMale,h = Y VIC
Male,h + ... + Y NTMale,h,
YTotal,h = YMale,h + YFemale,h, where h represents horizon
4 Base forecasts = Revised forecasts
Motivation Data Method Result Conclusion
Bottom-up method
1 Generate base (or independent) forecasts for each series at thebottom level
2 Aggregate these upwards to produce revised forecasts3 E.g., YMale,h = Y VIC
Male,h + ... + Y NTMale,h,
YTotal,h = YMale,h + YFemale,h, where h represents horizon4 Base forecasts = Revised forecasts
Motivation Data Method Result Conclusion
Bottom-up in action
Level 0
1940 1960 1980 2000
2000
3000
4000
5000
total
1940 1960 1980 2000
500
1500
2500
Level 1femalemale
1940 1960 1980 2000
050
010
0020
00
Level 2nswvicqldsa
wantactottas
1940 1960 1980 2000
020
060
010
00Level 3
nsw_fvic_fqld_fsa_fwa_fnt_factot_ftas_f
nsw_mvic_mqld_msa_mwa_mnt_mactot_mtas_m
Motivation Data Method Result Conclusion
Point forecast accuracy: data design
1 For series in the bottom level, select optimal exponentialsmoothing model based on information criterion, such as AIC(by defualt) or BIC
2 Re-estimate the parameters of model using a rolling windowapproach, with the initial fitting period (1933 to 1993)
3 Forecasts are produced for one- to ten-step-ahead4 Iterate the process, by increasing the sample size of training
period by one year until 20035 This gives us 10 one-step-ahead forecasts, 9 two-step-ahead
forecasts, ..., and 1 ten-step-ahead forecast6 The advantage of rolling window approach is to assess forecast
accuracy for each horizon
Motivation Data Method Result Conclusion
Point forecast accuracy: data design
1 For series in the bottom level, select optimal exponentialsmoothing model based on information criterion, such as AIC(by defualt) or BIC
2 Re-estimate the parameters of model using a rolling windowapproach, with the initial fitting period (1933 to 1993)
3 Forecasts are produced for one- to ten-step-ahead4 Iterate the process, by increasing the sample size of training
period by one year until 20035 This gives us 10 one-step-ahead forecasts, 9 two-step-ahead
forecasts, ..., and 1 ten-step-ahead forecast6 The advantage of rolling window approach is to assess forecast
accuracy for each horizon
Motivation Data Method Result Conclusion
Point forecast accuracy: data design
1 For series in the bottom level, select optimal exponentialsmoothing model based on information criterion, such as AIC(by defualt) or BIC
2 Re-estimate the parameters of model using a rolling windowapproach, 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 trainingperiod by one year until 2003
5 This gives us 10 one-step-ahead forecasts, 9 two-step-aheadforecasts, ..., and 1 ten-step-ahead forecast
6 The advantage of rolling window approach is to assess forecastaccuracy for each horizon
Motivation Data Method Result Conclusion
Point forecast accuracy: data design
1 For series in the bottom level, select optimal exponentialsmoothing model based on information criterion, such as AIC(by defualt) or BIC
2 Re-estimate the parameters of model using a rolling windowapproach, with the initial fitting period (1933 to 1993)
3 Forecasts are produced for one- to ten-step-ahead4 Iterate the process, by increasing the sample size of training
period by one year until 2003
5 This gives us 10 one-step-ahead forecasts, 9 two-step-aheadforecasts, ..., and 1 ten-step-ahead forecast
6 The advantage of rolling window approach is to assess forecastaccuracy for each horizon
Motivation Data Method Result Conclusion
Point forecast accuracy: data design
1 For series in the bottom level, select optimal exponentialsmoothing model based on information criterion, such as AIC(by defualt) or BIC
2 Re-estimate the parameters of model using a rolling windowapproach, with the initial fitting period (1933 to 1993)
3 Forecasts are produced for one- to ten-step-ahead4 Iterate the process, by increasing the sample size of training
period by one year until 20035 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 forecastaccuracy for each horizon
Motivation Data Method Result Conclusion
Point forecast accuracy: data design
1 For series in the bottom level, select optimal exponentialsmoothing model based on information criterion, such as AIC(by defualt) or BIC
2 Re-estimate the parameters of model using a rolling windowapproach, with the initial fitting period (1933 to 1993)
3 Forecasts are produced for one- to ten-step-ahead4 Iterate the process, by increasing the sample size of training
period by one year until 20035 This gives us 10 one-step-ahead forecasts, 9 two-step-ahead
forecasts, ..., and 1 ten-step-ahead forecast6 The advantage of rolling window approach is to assess forecast
accuracy for each horizon
Motivation Data Method Result Conclusion
Point forecast accuracy: evaluation
To compare point forecast accuracy between the base andbottom-up forecasts for all series, calculate mean absolutepercentage error,
MAPEh =1
(11− h)×m
n+(10−h)∑i=n
m∑j=1
∣∣∣∣∣Yt+h,j − Yt+h,j
Yt+h,j
∣∣∣∣∣ ,where m represents the total number of time series in the hierarchy,and h = 1, 2, . . . , 10
Motivation Data Method Result Conclusion
Point forecast result
Level 0 Level 1 Level 2 Level 3Base BU Base BU Base BU Base BU
1 4.26 5.35 5.59 5.72 14.76 14.03 20.98 20.982 6.25 5.96 7.38 6.23 16.32 16.20 25.50 25.503 8.27 6.51 10.26 6.86 18.95 18.95 30.55 30.554 11.94 10.73 14.71 10.34 22.40 22.11 34.55 34.555 19.02 9.37 16.48 10.47 24.87 25.96 39.58 39.586 16.46 6.16 17.60 6.18 27.75 27.74 41.99 41.997 19.59 9.46 19.55 9.58 31.66 34.43 47.57 47.578 20.30 9.74 24.50 10.03 34.61 39.32 54.78 54.789 28.71 11.62 29.72 12.02 33.41 40.38 52.97 52.9710 32.40 27.55 32.42 26.15 37.66 45.66 61.32 61.32Mean 16.72 10.25 17.82 10.36 26.24 28.48 40.98 40.98
Bottom-up method outperforms the independent (base) forecasts(without group structure) at the top two levels, not the state level
Motivation Data Method Result Conclusion
Construction of interval forecasts
1 Provide pointwise interval forecasts for assessing uncertainty
2 Proposed method fits within the framework of parametricbootstrapping
3 Draw bootstrap samples from the fitted exponential smoothingmodel for each series at the bottom level
4 For each bootstrap sample, we construct group structure andobtain point forecasts
5 Based on bootstrapped forecasts, we assess the variability ofpoint forecasts by constructing prediction interval
6 Computationally, the simulate.ets function in the forecastpackage was used
Motivation Data Method Result Conclusion
Construction of interval forecasts
1 Provide pointwise interval forecasts for assessing uncertainty2 Proposed method fits within the framework of parametric
bootstrapping
3 Draw bootstrap samples from the fitted exponential smoothingmodel for each series at the bottom level
4 For each bootstrap sample, we construct group structure andobtain point forecasts
5 Based on bootstrapped forecasts, we assess the variability ofpoint forecasts by constructing prediction interval
6 Computationally, the simulate.ets function in the forecastpackage was used
Motivation Data Method Result Conclusion
Construction of interval forecasts
1 Provide pointwise interval forecasts for assessing uncertainty2 Proposed method fits within the framework of parametric
bootstrapping3 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 andobtain point forecasts
5 Based on bootstrapped forecasts, we assess the variability ofpoint forecasts by constructing prediction interval
6 Computationally, the simulate.ets function in the forecastpackage was used
Motivation Data Method Result Conclusion
Construction of interval forecasts
1 Provide pointwise interval forecasts for assessing uncertainty2 Proposed method fits within the framework of parametric
bootstrapping3 Draw bootstrap samples from the fitted exponential smoothing
model for each series at the bottom level4 For each bootstrap sample, we construct group structure and
obtain point forecasts
5 Based on bootstrapped forecasts, we assess the variability ofpoint forecasts by constructing prediction interval
6 Computationally, the simulate.ets function in the forecastpackage was used
Motivation Data Method Result Conclusion
Construction of interval forecasts
1 Provide pointwise interval forecasts for assessing uncertainty2 Proposed method fits within the framework of parametric
bootstrapping3 Draw bootstrap samples from the fitted exponential smoothing
model for each series at the bottom level4 For each bootstrap sample, we construct group structure and
obtain point forecasts5 Based on bootstrapped forecasts, we assess the variability of
point forecasts by constructing prediction interval
6 Computationally, the simulate.ets function in the forecastpackage was used
Motivation Data Method Result Conclusion
Construction of interval forecasts
1 Provide pointwise interval forecasts for assessing uncertainty2 Proposed method fits within the framework of parametric
bootstrapping3 Draw bootstrap samples from the fitted exponential smoothing
model for each series at the bottom level4 For each bootstrap sample, we construct group structure and
obtain point forecasts5 Based on bootstrapped forecasts, we assess the variability of
point forecasts by constructing prediction interval6 Computationally, the simulate.ets function in the forecast
package was used
Motivation Data Method Result Conclusion
Demonstration of interval forecasts
Present 80% pointwise prediction interval of the regional infantmortality counts from 2004 to 2013 at the top two levels
Year
Cou
nt
1940 1950 1960 1970 1980 1990 2000
1000
2000
3000
4000
5000
6000 Total
(a) Level 0
1940 1950 1960 1970 1980 1990 200050
010
0015
0020
0025
0030
00
Year
Cou
nt
MaleFemale
(b) Level 1
Infant mortality counts will continue to decrease in future. Thevariability of male forecasts is higher than female ones
Motivation Data Method Result Conclusion
Interval forecast accuracy
1 Given a sample path [Y1, . . . ,Yn] where Yt is a column vectorof values across the entire hierarchy, we constructed theh-step-ahead interval forecasts
2 Let Ln+h|n(p) and Un+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
In+h,j =
{1 if Yn+h,j ∈ [Ln+h|n,j(p), Un+h|n,j(p)]
0 if Yn+h,j /∈ [Ln+h|n,j(p), Un+h|n,j(p)] j = 1, . . . ,m
Motivation Data Method Result Conclusion
Interval forecast accuracy
1 Given a sample path [Y1, . . . ,Yn] where Yt is a column vectorof values across the entire hierarchy, we constructed theh-step-ahead interval forecasts
2 Let Ln+h|n(p) and Un+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
In+h,j =
{1 if Yn+h,j ∈ [Ln+h|n,j(p), Un+h|n,j(p)]
0 if Yn+h,j /∈ [Ln+h|n,j(p), Un+h|n,j(p)] j = 1, . . . ,m
Motivation Data Method Result Conclusion
Interval forecast accuracy
1 Given a sample path [Y1, . . . ,Yn] where Yt is a column vectorof values across the entire hierarchy, we constructed theh-step-ahead interval forecasts
2 Let Ln+h|n(p) and Un+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
In+h,j =
{1 if Yn+h,j ∈ [Ln+h|n,j(p), Un+h|n,j(p)]
0 if Yn+h,j /∈ [Ln+h|n,j(p), Un+h|n,j(p)] j = 1, . . . ,m
Motivation Data Method Result Conclusion
Empirical coverage probability
Empirical coverage probability (ECP) is defined as
ECPh = 1−∑n+(10−h)
l=n
∑mj=1 Il+h,j
m× (11− h), h = 1, . . . , 10
h 1 2 3 4 5 6 7 8 9 10ECP 0.71 0.72 0.75 0.69 0.64 0.73 0.72 0.69 0.72 0.74
Table: Empirical coverage probability at nominal of 0.8
Motivation Data Method Result Conclusion
Hypothesis testing: interval forecast accuracy
1 To test if the ECP differs from the nominal coverageprobability, we performed log likelihood-ratio test statistics(see Christoffersen 1998, for more details)
2 Christoffersen (1998) proposed a test for unconditionalcoverage, a test for independence of indicator sequence, and ajoint test of conditional coverage and independence
3 At the nominal coverage probability of 0.8, log likelihood-ratioare
h 1 2 3 4 5 6 7 8 9 10LR 5.73 4.55 1.87 3.24 9.23 5.28 5.94 4.03 2.55 5.01
Table: Critical value is 5.99 at 95% level of significance
4 At 95% level of significance, only 1 in 10 is greater thancritical value
Motivation Data Method Result Conclusion
Hypothesis testing: interval forecast accuracy
1 To test if the ECP differs from the nominal coverageprobability, we performed log likelihood-ratio test statistics(see Christoffersen 1998, for more details)
2 Christoffersen (1998) proposed a test for unconditionalcoverage, a test for independence of indicator sequence, and ajoint test of conditional coverage and independence
3 At the nominal coverage probability of 0.8, log likelihood-ratioare
h 1 2 3 4 5 6 7 8 9 10LR 5.73 4.55 1.87 3.24 9.23 5.28 5.94 4.03 2.55 5.01
Table: Critical value is 5.99 at 95% level of significance
4 At 95% level of significance, only 1 in 10 is greater thancritical value
Motivation Data Method Result Conclusion
Hypothesis testing: interval forecast accuracy
1 To test if the ECP differs from the nominal coverageprobability, we performed log likelihood-ratio test statistics(see Christoffersen 1998, for more details)
2 Christoffersen (1998) proposed a test for unconditionalcoverage, a test for independence of indicator sequence, and ajoint test of conditional coverage and independence
3 At the nominal coverage probability of 0.8, log likelihood-ratioare
h 1 2 3 4 5 6 7 8 9 10LR 5.73 4.55 1.87 3.24 9.23 5.28 5.94 4.03 2.55 5.01
Table: Critical value is 5.99 at 95% level of significance
4 At 95% level of significance, only 1 in 10 is greater thancritical value
Motivation Data Method Result Conclusion
Hypothesis testing: interval forecast accuracy
1 To test if the ECP differs from the nominal coverageprobability, we performed log likelihood-ratio test statistics(see Christoffersen 1998, for more details)
2 Christoffersen (1998) proposed a test for unconditionalcoverage, a test for independence of indicator sequence, and ajoint test of conditional coverage and independence
3 At the nominal coverage probability of 0.8, log likelihood-ratioare
h 1 2 3 4 5 6 7 8 9 10LR 5.73 4.55 1.87 3.24 9.23 5.28 5.94 4.03 2.55 5.01
Table: Critical value is 5.99 at 95% level of significance
4 At 95% level of significance, only 1 in 10 is greater thancritical value
Motivation Data Method Result Conclusion
Conclusion
1 Revisited the bottom-up method
2 Applied it to the regional infant mortality count in Australia3 Performed evaluation of point forecast accuracy4 Proposed a parametric bootstrap method to construct
prediction interval5 Performed evaluation of interval forecast accuracy6 Carried out hypothesis testing of interval forecast accuracy
Motivation Data Method Result Conclusion
Conclusion
1 Revisited the bottom-up method2 Applied it to the regional infant mortality count in Australia
3 Performed evaluation of point forecast accuracy4 Proposed a parametric bootstrap method to construct
prediction interval5 Performed evaluation of interval forecast accuracy6 Carried out hypothesis testing of interval forecast accuracy
Motivation Data Method Result Conclusion
Conclusion
1 Revisited the bottom-up method2 Applied it to the regional infant mortality count in Australia3 Performed evaluation of point forecast accuracy
4 Proposed a parametric bootstrap method to constructprediction interval
5 Performed evaluation of interval forecast accuracy6 Carried out hypothesis testing of interval forecast accuracy
Motivation Data Method Result Conclusion
Conclusion
1 Revisited the bottom-up method2 Applied it to the regional infant mortality count in Australia3 Performed evaluation of point forecast accuracy4 Proposed a parametric bootstrap method to construct
prediction interval
5 Performed evaluation of interval forecast accuracy6 Carried out hypothesis testing of interval forecast accuracy
Motivation Data Method Result Conclusion
Conclusion
1 Revisited the bottom-up method2 Applied it to the regional infant mortality count in Australia3 Performed evaluation of point forecast accuracy4 Proposed a parametric bootstrap method to construct
prediction interval5 Performed evaluation of interval forecast accuracy
6 Carried out hypothesis testing of interval forecast accuracy
Motivation Data Method Result Conclusion
Conclusion
1 Revisited the bottom-up method2 Applied it to the regional infant mortality count in Australia3 Performed evaluation of point forecast accuracy4 Proposed a parametric bootstrap method to construct
prediction interval5 Performed evaluation of interval forecast accuracy6 Carried out hypothesis testing of interval forecast accuracy
Motivation Data Method Result Conclusion
Future research
1 Parametric bootstrapping is expected to work for otherhierarchical/grouped time series forecasting method, such astop-down methods
2 Modeling age-specific mortality counts hierarchically andcoherently
3 Extension from mortality count to mortality rate
Motivation Data Method Result Conclusion
Future research
1 Parametric bootstrapping is expected to work for otherhierarchical/grouped time series forecasting method, such astop-down methods
2 Modeling age-specific mortality counts hierarchically andcoherently
3 Extension from mortality count to mortality rate
Motivation Data Method Result Conclusion
Future research
1 Parametric bootstrapping is expected to work for otherhierarchical/grouped time series forecasting method, such astop-down methods
2 Modeling age-specific mortality counts hierarchically andcoherently
3 Extension from mortality count to mortality rate
Motivation Data Method Result Conclusion
Thank you
A draft is available upon request from [email protected]