Post on 29-May-2020
Hierarchical Forecasting 1
2017 Beijing Workshop onForecasting
HierarchicalForecasting
Rob J Hyndman
robjhyndman.com/beijing2017
Outline
1 Hierarchical and grouped time series
2 Forecast reconciliation
3 Fast computational tricks
Hierarchical Forecasting Hierarchical and grouped time series 2
Labour market participationAustralia and New Zealand Standard Classification ofOccupations
8 major groups43 sub-major groups
97 minor groups– 359 unit groups
* 1023 occupations
Example: statistician2 Professionals
22 Business, Human Resource and Marketing Professionals224 Information and Organisation Professionals
2241 Actuaries, Mathematicians and Statisticians224113 Statistician
Hierarchical Forecasting Hierarchical and grouped time series 3
Labour market participationAustralia and New Zealand Standard Classification ofOccupations
8 major groups43 sub-major groups
97 minor groups– 359 unit groups
* 1023 occupations
Example: statistician2 Professionals
22 Business, Human Resource and Marketing Professionals224 Information and Organisation Professionals
2241 Actuaries, Mathematicians and Statisticians224113 Statistician
Hierarchical Forecasting Hierarchical and grouped time series 3
Australian tourism demand
Hierarchical Forecasting Hierarchical and grouped time series 4
Australian tourism demand
Hierarchical Forecasting Hierarchical and grouped time series 4
Quarterly data on visitor night from 1998:Q1 –2013:Q4From: National Visitor Survey, based on annualinterviews of 120,000 Australians aged 15+,collected by Tourism Research Australia.Split by 7 states, 27 zones and 76 regions (ageographical hierarchy)Also split by purpose of travel
HolidayVisiting friends and relatives (VFR)BusinessOther
304 bottom-level series
3. PBS sales
Hierarchical Forecasting Hierarchical and grouped time series 5
3. PBS salesATC drug classification
A Alimentary tract and metabolismB Blood and blood forming organsC Cardiovascular systemD DermatologicalsG Genito-urinary system and sex hormonesH Systemic hormonal preparations, excluding sex hormones and in-
sulinsJ Anti-infectives for systemic useL Antineoplastic and immunomodulating agentsM Musculo-skeletal systemN Nervous systemP Antiparasitic products, insecticides and repellentsR Respiratory systemS Sensory organsV Various
Hierarchical Forecasting Hierarchical and grouped time series 6
3. PBS salesATC drug classification
A Alimentary tract and metabolism14 classes
A10 Drugs used in diabetes84 classes
A10B Blood glucose lowering drugs
A10BA Biguanides
A10BA02 Metformin
Hierarchical Forecasting Hierarchical and grouped time series 7
Spectacle sales
Hierarchical Forecasting Hierarchical and grouped time series 8
Monthly UK sales data from 2000 – 2014Provided by a large spectacle manufacturerSplit by brand (26), gender (3), price range (6),materials (4), and stores (600)About 1 million bottom-level series
Spectacle sales
Hierarchical Forecasting Hierarchical and grouped time series 8
Monthly UK sales data from 2000 – 2014Provided by a large spectacle manufacturerSplit by brand (26), gender (3), price range (6),materials (4), and stores (600)About 1 million bottom-level series
Spectacle sales
Hierarchical Forecasting Hierarchical and grouped time series 8
Monthly UK sales data from 2000 – 2014Provided by a large spectacle manufacturerSplit by brand (26), gender (3), price range (6),materials (4), and stores (600)About 1 million bottom-level series
Spectacle sales
Hierarchical Forecasting Hierarchical and grouped time series 8
Monthly UK sales data from 2000 – 2014Provided by a large spectacle manufacturerSplit by brand (26), gender (3), price range (6),materials (4), and stores (600)About 1 million bottom-level series
Hierarchical time seriesA hierarchical time series is a collection of several timeseries that are linked together in a hierarchical structure.
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
ExamplesLabour turnover by occupationPharmaceutical salesTourism by state and region
Hierarchical Forecasting Hierarchical and grouped time series 9
Hierarchical time seriesA hierarchical time series is a collection of several timeseries that are linked together in a hierarchical structure.
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
ExamplesLabour turnover by occupationPharmaceutical salesTourism by state and region
Hierarchical Forecasting Hierarchical and grouped time series 9
Hierarchical time seriesA hierarchical time series is a collection of several timeseries that are linked together in a hierarchical structure.
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
ExamplesLabour turnover by occupationPharmaceutical salesTourism by state and region
Hierarchical Forecasting Hierarchical and grouped time series 9
Hierarchical time seriesA hierarchical time series is a collection of several timeseries that are linked together in a hierarchical structure.
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
ExamplesLabour turnover by occupationPharmaceutical salesTourism by state and region
Hierarchical Forecasting Hierarchical and grouped time series 9
Grouped time seriesA grouped time series is a collection of time series thatcan be grouped together in a number of non-hierarchicalways.
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
ExamplesLabour turnover by occupation and stateSpectacle sales by brand, gender, stores, etc.Tourism by state and purpose of travel
Hierarchical Forecasting Hierarchical and grouped time series 10
Grouped time seriesA grouped time series is a collection of time series thatcan be grouped together in a number of non-hierarchicalways.
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
ExamplesLabour turnover by occupation and stateSpectacle sales by brand, gender, stores, etc.Tourism by state and purpose of travel
Hierarchical Forecasting Hierarchical and grouped time series 10
Grouped time seriesA grouped time series is a collection of time series thatcan be grouped together in a number of non-hierarchicalways.
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
ExamplesLabour turnover by occupation and stateSpectacle sales by brand, gender, stores, etc.Tourism by state and purpose of travel
Hierarchical Forecasting Hierarchical and grouped time series 10
Grouped time seriesA grouped time series is a collection of time series thatcan be grouped together in a number of non-hierarchicalways.
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
ExamplesLabour turnover by occupation and stateSpectacle sales by brand, gender, stores, etc.Tourism by state and purpose of travel
Hierarchical Forecasting Hierarchical and grouped time series 10
The problem
1 How to forecast time series at all nodessuch that the forecasts add up in the sameway as the original data?
2 Can we exploit relationships between theseries to improve the forecasts?
Hierarchical Forecasting Hierarchical and grouped time series 11
The problem
1 How to forecast time series at all nodessuch that the forecasts add up in the sameway as the original data?
2 Can we exploit relationships between theseries to improve the forecasts?
Hierarchical Forecasting Hierarchical and grouped time series 11
The solution
1 Forecast all series at all levels of aggregationusing an automatic forecasting algorithm(e.g., ets, auto.arima, . . . )
2 Reconcile the resulting forecasts so they addup correctly using least squaresoptimization (i.e., find closest reconciledforecasts to the original forecasts).
3 This is all available in the hts package in R.
Hierarchical Forecasting Hierarchical and grouped time series 12
The solution
1 Forecast all series at all levels of aggregationusing an automatic forecasting algorithm(e.g., ets, auto.arima, . . . )
2 Reconcile the resulting forecasts so they addup correctly using least squaresoptimization (i.e., find closest reconciledforecasts to the original forecasts).
3 This is all available in the hts package in R.
Hierarchical Forecasting Hierarchical and grouped time series 12
The solution
1 Forecast all series at all levels of aggregationusing an automatic forecasting algorithm(e.g., ets, auto.arima, . . . )
2 Reconcile the resulting forecasts so they addup correctly using least squaresoptimization (i.e., find closest reconciledforecasts to the original forecasts).
3 This is all available in the hts package in R.
Hierarchical Forecasting Hierarchical and grouped time series 12
Hierarchical time series
Total
A B C
Hierarchical Forecasting Hierarchical and grouped time series 13
yt : observed aggregate of allseries at time t.
yX,t : observation on series X at timet.
bt : vector of all series at bottomlevel in time t.
Hierarchical time series
Total
A B C
Hierarchical Forecasting Hierarchical and grouped time series 13
yt : observed aggregate of allseries at time t.
yX,t : observation on series X at timet.
bt : vector of all series at bottomlevel in time t.
Hierarchical time series
Total
A B C
yt = [yt, yA,t, yB,t, yC,t]′ =
1 1 11 0 00 1 00 0 1
yA,tyB,tyC,t
Hierarchical Forecasting Hierarchical and grouped time series 13
yt : observed aggregate of allseries at time t.
yX,t : observation on series X at timet.
bt : vector of all series at bottomlevel in time t.
Hierarchical time series
Total
A B C
yt = [yt, yA,t, yB,t, yC,t]′ =
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
yA,tyB,tyC,t
Hierarchical Forecasting Hierarchical and grouped time series 13
yt : observed aggregate of allseries at time t.
yX,t : observation on series X at timet.
bt : vector of all series at bottomlevel in time t.
Hierarchical time series
Total
A B C
yt = [yt, yA,t, yB,t, yC,t]′ =
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
yA,tyB,tyC,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Hierarchical and grouped time series 13
yt : observed aggregate of allseries at time t.
yX,t : observation on series X at timet.
bt : vector of all series at bottomlevel in time t.
Hierarchical time series
Total
A B C
yt = [yt, yA,t, yB,t, yC,t]′ =
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
yA,tyB,tyC,t
︸ ︷︷ ︸
btyt = Sbt
Hierarchical Forecasting Hierarchical and grouped time series 13
yt : observed aggregate of allseries at time t.
yX,t : observation on series X at timet.
bt : vector of all series at bottomlevel in time t.
Hierarchical time seriesTotal
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =
ytyA,tyB,tyC,tyAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 11 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Hierarchical and grouped time series 14
Hierarchical time seriesTotal
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =
ytyA,tyB,tyC,tyAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 11 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Hierarchical and grouped time series 14
Hierarchical time seriesTotal
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =
ytyA,tyB,tyC,tyAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 11 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Hierarchical and grouped time series 14
yt = Sbt
Grouped dataAX AY A
BX BY B
X Y Total
yt =
ytyA,tyB,tyX,tyY,tyAX,tyAY,tyBX,tyBY,t
=
1 1 1 11 1 0 00 0 1 11 0 1 00 1 0 11 0 0 00 1 0 00 0 1 00 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyBX,tyBY,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Hierarchical and grouped time series 15
Grouped dataAX AY A
BX BY B
X Y Total
yt =
ytyA,tyB,tyX,tyY,tyAX,tyAY,tyBX,tyBY,t
=
1 1 1 11 1 0 00 0 1 11 0 1 00 1 0 11 0 0 00 1 0 00 0 1 00 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyBX,tyBY,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Hierarchical and grouped time series 15
Grouped dataAX AY A
BX BY B
X Y Total
yt =
ytyA,tyB,tyX,tyY,tyAX,tyAY,tyBX,tyBY,t
=
1 1 1 11 1 0 00 0 1 11 0 1 00 1 0 11 0 0 00 1 0 00 0 1 00 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyBX,tyBY,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Hierarchical and grouped time series 15
yt = Sbt
Hierarchical and grouped time series
Every collection of time series with aggregationconstraints can be written as
yt = Sbt
whereyt is vector of all series at time tbt is vector of the most disaggregated series at time tS is “summing matrix” containing the aggregationconstraints.
Hierarchical Forecasting Hierarchical and grouped time series 16
Outline
1 Hierarchical and grouped time series
2 Forecast reconciliation
3 Fast computational tricks
Hierarchical Forecasting Forecast reconciliation 17
Hierarchical/grouped time seriesForecasts should be “coherent”, unbiased, minimumvariance.
Existing methods:ã Bottom-upã Top-downã Middle-out
How to compute forecast intervals?
Most research is concerned about relativeperformance of existing methods.
Hierarchical Forecasting Forecast reconciliation 18
Hierarchical/grouped time seriesForecasts should be “coherent”, unbiased, minimumvariance.
Existing methods:ã Bottom-upã Top-downã Middle-out
How to compute forecast intervals?
Most research is concerned about relativeperformance of existing methods.
Hierarchical Forecasting Forecast reconciliation 18
Hierarchical/grouped time seriesForecasts should be “coherent”, unbiased, minimumvariance.
Existing methods:ã Bottom-upã Top-downã Middle-out
How to compute forecast intervals?
Most research is concerned about relativeperformance of existing methods.
Hierarchical Forecasting Forecast reconciliation 18
Hierarchical/grouped time seriesForecasts should be “coherent”, unbiased, minimumvariance.
Existing methods:ã Bottom-upã Top-downã Middle-out
How to compute forecast intervals?
Most research is concerned about relativeperformance of existing methods.
Hierarchical Forecasting Forecast reconciliation 18
Hierarchical/grouped time seriesForecasts should be “coherent”, unbiased, minimumvariance.
Existing methods:ã Bottom-upã Top-downã Middle-out
How to compute forecast intervals?
Most research is concerned about relativeperformance of existing methods.
Hierarchical Forecasting Forecast reconciliation 18
Hierarchical/grouped time seriesForecasts should be “coherent”, unbiased, minimumvariance.
Existing methods:ã Bottom-upã Top-downã Middle-out
How to compute forecast intervals?
Most research is concerned about relativeperformance of existing methods.
Hierarchical Forecasting Forecast reconciliation 18
Hierarchical/grouped time seriesForecasts should be “coherent”, unbiased, minimumvariance.
Existing methods:ã Bottom-upã Top-downã Middle-out
How to compute forecast intervals?
Most research is concerned about relativeperformance of existing methods.
Hierarchical Forecasting Forecast reconciliation 18
Top-down method
Hierarchical Forecasting Forecast reconciliation 19
AdvantagesWorks well in presenceof low counts.Single forecastingmodel easy to buildProvides reliableforecasts for aggregatelevels.
DisadvantagesLoss of information,especially individualseries dynamics.Distribution of forecaststo lower levels can bedifficultNo prediction intervals
Top-down method
Hierarchical Forecasting Forecast reconciliation 19
AdvantagesWorks well in presenceof low counts.Single forecastingmodel easy to buildProvides reliableforecasts for aggregatelevels.
DisadvantagesLoss of information,especially individualseries dynamics.Distribution of forecaststo lower levels can bedifficultNo prediction intervals
Top-down method
Hierarchical Forecasting Forecast reconciliation 19
AdvantagesWorks well in presenceof low counts.Single forecastingmodel easy to buildProvides reliableforecasts for aggregatelevels.
DisadvantagesLoss of information,especially individualseries dynamics.Distribution of forecaststo lower levels can bedifficultNo prediction intervals
Top-down method
Hierarchical Forecasting Forecast reconciliation 19
AdvantagesWorks well in presenceof low counts.Single forecastingmodel easy to buildProvides reliableforecasts for aggregatelevels.
DisadvantagesLoss of information,especially individualseries dynamics.Distribution of forecaststo lower levels can bedifficultNo prediction intervals
Top-down method
Hierarchical Forecasting Forecast reconciliation 19
AdvantagesWorks well in presenceof low counts.Single forecastingmodel easy to buildProvides reliableforecasts for aggregatelevels.
DisadvantagesLoss of information,especially individualseries dynamics.Distribution of forecaststo lower levels can bedifficultNo prediction intervals
Top-down method
Hierarchical Forecasting Forecast reconciliation 19
AdvantagesWorks well in presenceof low counts.Single forecastingmodel easy to buildProvides reliableforecasts for aggregatelevels.
DisadvantagesLoss of information,especially individualseries dynamics.Distribution of forecaststo lower levels can bedifficultNo prediction intervals
Bottom-up method
Hierarchical Forecasting Forecast reconciliation 20
AdvantagesNo loss of information.Better capturesdynamics of individualseries.
DisadvantagesLarge number of seriesto be forecast.Constructing forecastingmodels is harderbecause of noisy data atbottom level.No prediction intervals
Bottom-up method
Hierarchical Forecasting Forecast reconciliation 20
AdvantagesNo loss of information.Better capturesdynamics of individualseries.
DisadvantagesLarge number of seriesto be forecast.Constructing forecastingmodels is harderbecause of noisy data atbottom level.No prediction intervals
Bottom-up method
Hierarchical Forecasting Forecast reconciliation 20
AdvantagesNo loss of information.Better capturesdynamics of individualseries.
DisadvantagesLarge number of seriesto be forecast.Constructing forecastingmodels is harderbecause of noisy data atbottom level.No prediction intervals
Bottom-up method
Hierarchical Forecasting Forecast reconciliation 20
AdvantagesNo loss of information.Better capturesdynamics of individualseries.
DisadvantagesLarge number of seriesto be forecast.Constructing forecastingmodels is harderbecause of noisy data atbottom level.No prediction intervals
Bottom-up method
Hierarchical Forecasting Forecast reconciliation 20
AdvantagesNo loss of information.Better capturesdynamics of individualseries.
DisadvantagesLarge number of seriesto be forecast.Constructing forecastingmodels is harderbecause of noisy data atbottom level.No prediction intervals
Forecasting notation
Let yn(h) be vector of initial h-step forecasts, made attime n, stacked in same order as yt.(In general, they will not “add up”.)
Reconciled forecasts must be of the form:yn(h) = SPyn(h)
for some matrix P.
P extracts and combines base forecasts yn(h) to getbottom-level forecasts.S adds them up
Hierarchical Forecasting Forecast reconciliation 21
Forecasting notation
Let yn(h) be vector of initial h-step forecasts, made attime n, stacked in same order as yt.(In general, they will not “add up”.)
Reconciled forecasts must be of the form:yn(h) = SPyn(h)
for some matrix P.
P extracts and combines base forecasts yn(h) to getbottom-level forecasts.S adds them up
Hierarchical Forecasting Forecast reconciliation 21
Forecasting notation
Let yn(h) be vector of initial h-step forecasts, made attime n, stacked in same order as yt.(In general, they will not “add up”.)
Reconciled forecasts must be of the form:yn(h) = SPyn(h)
for some matrix P.
P extracts and combines base forecasts yn(h) to getbottom-level forecasts.S adds them up
Hierarchical Forecasting Forecast reconciliation 21
Forecasting notation
Let yn(h) be vector of initial h-step forecasts, made attime n, stacked in same order as yt.(In general, they will not “add up”.)
Reconciled forecasts must be of the form:yn(h) = SPyn(h)
for some matrix P.
P extracts and combines base forecasts yn(h) to getbottom-level forecasts.S adds them up
Hierarchical Forecasting Forecast reconciliation 21
Forecasting notation
Let yn(h) be vector of initial h-step forecasts, made attime n, stacked in same order as yt.(In general, they will not “add up”.)
Reconciled forecasts must be of the form:yn(h) = SPyn(h)
for some matrix P.
P extracts and combines base forecasts yn(h) to getbottom-level forecasts.S adds them up
Hierarchical Forecasting Forecast reconciliation 21
Bottom-up forecasts
yn(h) = SPyn(h)
Bottom-up forecasts are obtained using
P = [0 | I] ,
where 0 is null matrix and I is identity matrix.Pmatrix extracts only bottom-level forecasts fromyn(h)S adds them up to give the bottom-up forecasts.
Hierarchical Forecasting Forecast reconciliation 22
Bottom-up forecasts
yn(h) = SPyn(h)
Bottom-up forecasts are obtained using
P = [0 | I] ,
where 0 is null matrix and I is identity matrix.Pmatrix extracts only bottom-level forecasts fromyn(h)S adds them up to give the bottom-up forecasts.
Hierarchical Forecasting Forecast reconciliation 22
Bottom-up forecasts
yn(h) = SPyn(h)
Bottom-up forecasts are obtained using
P = [0 | I] ,
where 0 is null matrix and I is identity matrix.Pmatrix extracts only bottom-level forecasts fromyn(h)S adds them up to give the bottom-up forecasts.
Hierarchical Forecasting Forecast reconciliation 22
Top-down forecasts
yn(h) = SPyn(h)
Top-down forecasts are obtained using
P = [p | 0]
where p = [p1, p2, . . . , pmK ]′ is a vector of proportions that
sum to one.P distributes forecasts of the aggregate to the lowestlevel series.Different methods of top-down forecasting lead todifferent proportionality vectors p.
Hierarchical Forecasting Forecast reconciliation 23
Top-down forecasts
yn(h) = SPyn(h)
Top-down forecasts are obtained using
P = [p | 0]
where p = [p1, p2, . . . , pmK ]′ is a vector of proportions that
sum to one.P distributes forecasts of the aggregate to the lowestlevel series.Different methods of top-down forecasting lead todifferent proportionality vectors p.
Hierarchical Forecasting Forecast reconciliation 23
Top-down forecasts
yn(h) = SPyn(h)
Top-down forecasts are obtained using
P = [p | 0]
where p = [p1, p2, . . . , pmK ]′ is a vector of proportions that
sum to one.P distributes forecasts of the aggregate to the lowestlevel series.Different methods of top-down forecasting lead todifferent proportionality vectors p.
Hierarchical Forecasting Forecast reconciliation 23
General properties: bias
yn(h) = SPyn(h)
Assume: base forecasts yn(h) are unbiased:E[yn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let bn(h) be bottom level base forecastswith βn(h) = E[bn(h)|y1, . . . , yn].Then E[yn(h)] = Sβn(h).We want the revised forecasts to be unbiased:E[yn(h)] = SPSβn(h) = Sβn(h).
Reconciled forecast are unbiased if and only if SPS = S.
True for bottom-up, but not top-down or middle-out.Hierarchical Forecasting Forecast reconciliation 24
General properties: bias
yn(h) = SPyn(h)
Assume: base forecasts yn(h) are unbiased:E[yn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let bn(h) be bottom level base forecastswith βn(h) = E[bn(h)|y1, . . . , yn].Then E[yn(h)] = Sβn(h).We want the revised forecasts to be unbiased:E[yn(h)] = SPSβn(h) = Sβn(h).
Reconciled forecast are unbiased if and only if SPS = S.
True for bottom-up, but not top-down or middle-out.Hierarchical Forecasting Forecast reconciliation 24
General properties: bias
yn(h) = SPyn(h)
Assume: base forecasts yn(h) are unbiased:E[yn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let bn(h) be bottom level base forecastswith βn(h) = E[bn(h)|y1, . . . , yn].Then E[yn(h)] = Sβn(h).We want the revised forecasts to be unbiased:E[yn(h)] = SPSβn(h) = Sβn(h).
Reconciled forecast are unbiased if and only if SPS = S.
True for bottom-up, but not top-down or middle-out.Hierarchical Forecasting Forecast reconciliation 24
General properties: bias
yn(h) = SPyn(h)
Assume: base forecasts yn(h) are unbiased:E[yn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let bn(h) be bottom level base forecastswith βn(h) = E[bn(h)|y1, . . . , yn].Then E[yn(h)] = Sβn(h).We want the revised forecasts to be unbiased:E[yn(h)] = SPSβn(h) = Sβn(h).
Reconciled forecast are unbiased if and only if SPS = S.
True for bottom-up, but not top-down or middle-out.Hierarchical Forecasting Forecast reconciliation 24
General properties: bias
yn(h) = SPyn(h)
Assume: base forecasts yn(h) are unbiased:E[yn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let bn(h) be bottom level base forecastswith βn(h) = E[bn(h)|y1, . . . , yn].Then E[yn(h)] = Sβn(h).We want the revised forecasts to be unbiased:E[yn(h)] = SPSβn(h) = Sβn(h).
Reconciled forecast are unbiased if and only if SPS = S.
True for bottom-up, but not top-down or middle-out.Hierarchical Forecasting Forecast reconciliation 24
General properties: bias
yn(h) = SPyn(h)
Assume: base forecasts yn(h) are unbiased:E[yn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let bn(h) be bottom level base forecastswith βn(h) = E[bn(h)|y1, . . . , yn].Then E[yn(h)] = Sβn(h).We want the revised forecasts to be unbiased:E[yn(h)] = SPSβn(h) = Sβn(h).
Reconciled forecast are unbiased if and only if SPS = S.
True for bottom-up, but not top-down or middle-out.Hierarchical Forecasting Forecast reconciliation 24
General properties: bias
yn(h) = SPyn(h)
Assume: base forecasts yn(h) are unbiased:E[yn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let bn(h) be bottom level base forecastswith βn(h) = E[bn(h)|y1, . . . , yn].Then E[yn(h)] = Sβn(h).We want the revised forecasts to be unbiased:E[yn(h)] = SPSβn(h) = Sβn(h).
Reconciled forecast are unbiased if and only if SPS = S.
True for bottom-up, but not top-down or middle-out.Hierarchical Forecasting Forecast reconciliation 24
General properties: bias
yn(h) = SPyn(h)
Assume: base forecasts yn(h) are unbiased:E[yn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let bn(h) be bottom level base forecastswith βn(h) = E[bn(h)|y1, . . . , yn].Then E[yn(h)] = Sβn(h).We want the revised forecasts to be unbiased:E[yn(h)] = SPSβn(h) = Sβn(h).
Reconciled forecast are unbiased if and only if SPS = S.
True for bottom-up, but not top-down or middle-out.Hierarchical Forecasting Forecast reconciliation 24
General properties: variance
yn(h) = SPyn(h)
Let error variance of h-step base forecasts yn(h) be
Σh = Var[yn+h − yn(h) | y1, . . . , yn]
Then the error variance of the corresponding reconciledforecasts is
Var[yn+h − yn(h) | y1, . . . , yn] = SPΣhP′S′
This is a general result for all existing methods.
Hierarchical Forecasting Forecast reconciliation 25
General properties: variance
yn(h) = SPyn(h)
Let error variance of h-step base forecasts yn(h) be
Σh = Var[yn+h − yn(h) | y1, . . . , yn]
Then the error variance of the corresponding reconciledforecasts is
Var[yn+h − yn(h) | y1, . . . , yn] = SPΣhP′S′
This is a general result for all existing methods.
Hierarchical Forecasting Forecast reconciliation 25
Optimal forecast reconciliation
yn(h) = SPyn(h)
Theorem: MinT ReconciliationIf P satisfies SPS = S, then
minP = trace[SPΣhP′S′]has solution P = (S′Σ−1h S)−1S′Σ−1h .
Reconciled forecasts Base forecastsAssume thatΣh = khΣ1 to simplify computations.
Hierarchical Forecasting Forecast reconciliation 26
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
yn(h) = SPyn(h)
Theorem: MinT ReconciliationIf P satisfies SPS = S, then
minP = trace[SPΣhP′S′]has solution P = (S′Σ−1h S)−1S′Σ−1h .
Reconciled forecasts Base forecastsAssume thatΣh = khΣ1 to simplify computations.
Hierarchical Forecasting Forecast reconciliation 26
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
yn(h) = SPyn(h)
Theorem: MinT ReconciliationIf P satisfies SPS = S, then
minP = trace[SPΣhP′S′]has solution P = (S′Σ−1h S)−1S′Σ−1h .
Reconciled forecasts Base forecastsAssume thatΣh = khΣ1 to simplify computations.
Hierarchical Forecasting Forecast reconciliation 26
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
yn(h) = SPyn(h)
Theorem: MinT ReconciliationIf P satisfies SPS = S, then
minP = trace[SPΣhP′S′]has solution P = (S′Σ−1h S)−1S′Σ−1h .
Reconciled forecasts Base forecastsAssume thatΣh = khΣ1 to simplify computations.
Hierarchical Forecasting Forecast reconciliation 26
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 1: OLSAssumeΣ1 ≈ kI.
yn(h) = S(S′S)−1S′yn(h)
Reconciliation does not depend on dataWorks surprisingly well.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 27
yn(h) = S(S′Σ−11 S)−1S′Σ−11 yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 1: OLSAssumeΣ1 ≈ kI.
yn(h) = S(S′S)−1S′yn(h)
Reconciliation does not depend on dataWorks surprisingly well.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 27
yn(h) = S(S′Σ−11 S)−1S′Σ−11 yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 1: OLSAssumeΣ1 ≈ kI.
yn(h) = S(S′S)−1S′yn(h)
Reconciliation does not depend on dataWorks surprisingly well.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 27
yn(h) = S(S′Σ−11 S)−1S′Σ−11 yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 1: OLSAssumeΣ1 ≈ kI.
yn(h) = S(S′S)−1S′yn(h)
Reconciliation does not depend on dataWorks surprisingly well.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 27
yn(h) = S(S′Σ−11 S)−1S′Σ−11 yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 1: OLSAssumeΣ1 ≈ kI.
yn(h) = S(S′S)−1S′yn(h)
Reconciliation does not depend on dataWorks surprisingly well.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 27
yn(h) = S(S′Σ−11 S)−1S′Σ−11 yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 1: OLSAssumeΣ1 ≈ kI.
yn(h) = S(S′S)−1S′yn(h)
Reconciliation does not depend on dataWorks surprisingly well.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 27
yn(h) = S(S′Σ−11 S)−1S′Σ−11 yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 2: WLSApproximateΣ1 by its diagonal.Easy to estimate, and places weight where we havebest forecasts.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 28
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 2: WLSApproximateΣ1 by its diagonal.Easy to estimate, and places weight where we havebest forecasts.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 28
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 2: WLSApproximateΣ1 by its diagonal.Easy to estimate, and places weight where we havebest forecasts.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 28
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 2: WLSApproximateΣ1 by its diagonal.Easy to estimate, and places weight where we havebest forecasts.Still need to estimate covariance matrix to produceprediction intervals.
Hierarchical Forecasting Forecast reconciliation 28
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 3: GLSEstimateΣ1 using shrinkage to the diagonal.Allows for covariances.Difficult to compute for large numbers of timeseries.
Hierarchical Forecasting Forecast reconciliation 29
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 3: GLSEstimateΣ1 using shrinkage to the diagonal.Allows for covariances.Difficult to compute for large numbers of timeseries.
Hierarchical Forecasting Forecast reconciliation 29
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 3: GLSEstimateΣ1 using shrinkage to the diagonal.Allows for covariances.Difficult to compute for large numbers of timeseries.
Hierarchical Forecasting Forecast reconciliation 29
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Optimal forecast reconciliation
Reconciled forecasts Base forecasts
Solution 3: GLSEstimateΣ1 using shrinkage to the diagonal.Allows for covariances.Difficult to compute for large numbers of timeseries.
Hierarchical Forecasting Forecast reconciliation 29
yn(h) = S(S′Σ−1h S)−1S′Σ−1h yn(h)
Australian tourism
Hierarchical Forecasting Forecast reconciliation 30
Australian tourism
Hierarchical Forecasting Forecast reconciliation 30
Hierarchy:States (7)Zones (27)Regions (82)
Australian tourism
Hierarchical Forecasting Forecast reconciliation 30
Hierarchy:States (7)Zones (27)Regions (82)
Base forecastsETS (exponential smoothing)models
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: Total
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
6000
065
000
7000
075
000
8000
085
000
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: NSW
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
1800
022
000
2600
030
000
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: VIC
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
1000
012
000
1400
016
000
1800
0
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: Nth.Coast.NSW
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
5000
6000
7000
8000
9000
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: Metro.QLD
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
8000
9000
1100
013
000
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: Sth.WA
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
400
600
800
1000
1200
1400
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: X201.Melbourne
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
4000
4500
5000
5500
6000
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: X402.Murraylands
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
010
020
030
0
Base forecasts
Hierarchical Forecasting Forecast reconciliation 31
Domestic tourism forecasts: X809.Daly
Year
Vis
itor
nigh
ts
1998 2000 2002 2004 2006 2008
020
4060
8010
0
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 1● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 2● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 3● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 4● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 5● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Forecast evaluation
Hierarchical Forecasting Forecast reconciliation 32
Training sets Test sets h = 6● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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time
Hierarchy: states, zones, regionsForecast horizon
RMSE h = 1 h = 2 h = 3 h = 4 h = 5 h = 6 Ave
AustraliaBase 1762.04 1770.29 1766.02 1818.82 1705.35 1721.17 1757.28Bottom 1736.92 1742.69 1722.79 1752.74 1666.73 1687.43 1718.22OLS 1747.60 1757.68 1751.77 1800.67 1686.00 1706.45 1741.69WLS 1705.21 1715.87 1703.75 1729.56 1627.79 1661.24 1690.57GLS 1704.64 1715.60 1705.31 1729.04 1626.36 1661.64 1690.43
StatesBase 399.77 404.16 401.92 407.26 395.38 401.17 401.61Bottom 404.29 406.95 404.96 409.02 399.80 401.55 404.43OLS 404.47 407.62 405.43 413.79 401.10 404.90 406.22WLS 398.84 402.12 400.71 405.03 394.76 398.23 399.95GLS 398.84 402.16 400.86 405.03 394.59 398.22 399.95
RegionsBase 93.15 93.38 93.45 93.79 93.50 93.56 93.47Bottom 93.15 93.38 93.45 93.79 93.50 93.56 93.47OLS 93.28 93.53 93.64 94.17 93.78 93.88 93.71WLS 93.02 93.32 93.38 93.72 93.39 93.53 93.39GLS 92.98 93.27 93.34 93.66 93.34 93.46 93.34
Hierarchical Forecasting Forecast reconciliation 33
Outline
1 Hierarchical and grouped time series
2 Forecast reconciliation
3 Fast computational tricks
Hierarchical Forecasting Fast computational tricks 34
Fast computation: hierarchical dataTotal
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =
ytyA,tyB,tyC,tyAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 11 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Fast computational tricks 35
yt = Sbt
Fast computation: hierarchical dataTotal
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =
ytyA,tyAX,tyAY,tyAZ,tyB,tyBX,tyBY,tyBZ,tyC,tyCX,tyCY,tyCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 1 10 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Fast computational tricks 36
yt = Sbt
Fast computation: hierarchical dataTotal
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =
ytyA,tyAX,tyAY,tyAZ,tyB,tyBX,tyBY,tyBZ,tyC,tyCX,tyCY,tyCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 1 10 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Fast computational tricks 36
yt = Sbt
Fast computation: hierarchiesThink of the hierarchy as a tree of trees:
Total
T1 T2 . . . TK
Then the summing matrix contains k smaller summingmatrices:
S =
1′n1 1′n2 · · · 1′nKS1 0 · · · 00 S2 · · · 0... ... . . . ...0 0 · · · SK
where 1n is an n-vector of ones and tree Ti has ni terminalnodes.
Hierarchical Forecasting Fast computational tricks 37
Fast computation: hierarchiesThink of the hierarchy as a tree of trees:
Total
T1 T2 . . . TK
Then the summing matrix contains k smaller summingmatrices:
S =
1′n1 1′n2 · · · 1′nKS1 0 · · · 00 S2 · · · 0... ... . . . ...0 0 · · · SK
where 1n is an n-vector of ones and tree Ti has ni terminalnodes.
Hierarchical Forecasting Fast computational tricks 37
Fast computation: hierarchies
S′ΛS =
S′1Λ1S1 0 · · · 0
0 S′2Λ2S2 · · · 0... ... . . . ...0 0 · · · S′KΛKSK
+ λ0 Jn
λ0 is the top left element ofΛ;Λk is a block ofΛ, corresponding to tree Tk;Jn is a matrix of ones;n =
∑k nk.
Now apply the Sherman-Morrison formula . . .
Hierarchical Forecasting Fast computational tricks 38
Fast computation: hierarchies
S′ΛS =
S′1Λ1S1 0 · · · 0
0 S′2Λ2S2 · · · 0... ... . . . ...0 0 · · · S′KΛKSK
+ λ0 Jn
λ0 is the top left element ofΛ;Λk is a block ofΛ, corresponding to tree Tk;Jn is a matrix of ones;n =
∑k nk.
Now apply the Sherman-Morrison formula . . .
Hierarchical Forecasting Fast computational tricks 38
Fast computation: hierarchies
(S′ΛS)−1 =
(S′1Λ1S1)−1 0 · · · 0
0 (S′2Λ2S2)−1 · · · 0... ... . . . ...0 0 · · · (S′KΛKSK)−1
− cS0
S0 can be partitioned into K2 blocks, with the (k, `) block(of dimension nk × n`) being
(S′kΛkSk)−1Jnk,n`(S′`Λ`S`)−1
Jnk,n` is a nk × n` matrix of ones.c−1 = λ−10 +
∑k
1′nk(S′kΛkSk)−11nk .
Each S′kΛkSk can be inverted similarly.S′Λy can also be computed recursively.
Hierarchical Forecasting Fast computational tricks 39
Fast computation: hierarchies
(S′ΛS)−1 =
(S′1Λ1S1)−1 0 · · · 0
0 (S′2Λ2S2)−1 · · · 0... ... . . . ...0 0 · · · (S′KΛKSK)−1
− cS0
S0 can be partitioned into K2 blocks, with the (k, `) block(of dimension nk × n`) being
(S′kΛkSk)−1Jnk,n`(S′`Λ`S`)−1
Jnk,n` is a nk × n` matrix of ones.c−1 = λ−10 +
∑k
1′nk(S′kΛkSk)−11nk .
Each S′kΛkSk can be inverted similarly.S′Λy can also be computed recursively.
Hierarchical Forecasting Fast computational tricks 39
The recursive calculations can be donein such a way that we never store any ofthe large matrices involved.
Fast computation: grouped dataAX AY AZ ABX BY BZ BCX CY CZ CX Y Z Total
yt =
ytyA,tyB,tyC,tyX,tyY,tyZ,tyAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 11 0 0 1 0 0 1 0 00 1 0 0 1 0 0 1 00 0 1 0 0 1 0 0 11 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
yAX,tyAY,tyAZ,tyBX,tyBY,tyBZ,tyCX,tyCY,tyCZ,t
︸ ︷︷ ︸
bt
Hierarchical Forecasting Fast computational tricks 40
yt = Sbt
Fast computation: grouped data
S =
1′m ⊗ 1′n1′m ⊗ InIm ⊗ 1′nIm ⊗ In
S′ΛS = λ00 Jmn + (ΛR ⊗ Jn) + (Jm ⊗ΛC) +ΛU
ΛR,ΛC andΛU are diagonal matrices correspondingto rows, columns and unaggregated series;λ00 corresponds to aggregate.
Hierarchical Forecasting Fast computational tricks 41
m = number of rowsn = number of columns
Fast computation: grouped data
S =
1′m ⊗ 1′n1′m ⊗ InIm ⊗ 1′nIm ⊗ In
S′ΛS = λ00 Jmn + (ΛR ⊗ Jn) + (Jm ⊗ΛC) +ΛU
ΛR,ΛC andΛU are diagonal matrices correspondingto rows, columns and unaggregated series;λ00 corresponds to aggregate.
Hierarchical Forecasting Fast computational tricks 41
m = number of rowsn = number of columns
Fast computation: grouped data
(SΛS)−1 = A− A1mn1′mnA1/λ00 + 1′mnA1mn
A = Λ−1U −Λ−1U (Jm ⊗ D)Λ−1U − EM−1E′.D is diagonal with elements dj = λ0j/(1+ λ0j
∑i λ−1ij ).
E hasm×m blocks where eij has kth element
(eij)k =
{λ1/2i0 λ
−1ik − λ
1/2i0 λ
−2ik dk, i = j,
−λ1/2j0 λ−1ik λ
−1jk dk, i 6= j.
M ism×m with (i, j) element
(M)ij =
{1+ λi0
∑k λ−1ik − λi0
∑k λ−2ik dk, i = j,
−λ1/2i0 λ1/2j0
∑k λ−1ik λ
−1jk dk, i 6= j.
Hierarchical Forecasting Fast computational tricks 42
Fast computation: grouped data
(SΛS)−1 = A− A1mn1′mnA1/λ00 + 1′mnA1mn
A = Λ−1U −Λ−1U (Jm ⊗ D)Λ−1U − EM−1E′.D is diagonal with elements dj = λ0j/(1+ λ0j
∑i λ−1ij ).
E hasm×m blocks where eij has kth element
(eij)k =
{λ1/2i0 λ
−1ik − λ
1/2i0 λ
−2ik dk, i = j,
−λ1/2j0 λ−1ik λ
−1jk dk, i 6= j.
M ism×m with (i, j) element
(M)ij =
{1+ λi0
∑k λ−1ik − λi0
∑k λ−2ik dk, i = j,
−λ1/2i0 λ1/2j0
∑k λ−1ik λ
−1jk dk, i 6= j.
Hierarchical Forecasting Fast computational tricks 42
Again, the calculations can be done insuch a way that we never store any ofthe large matrices involved.
References
RJ Hyndman, RA Ahmed, G Athanasopoulos and HL Shang (2011). Optimalcombination forecasts for hierarchical time series. Computational Statistics& Data Analysis 55(9), 2579–2589.
RJ Hyndman, A Lee and E Wang (2016). Fast computation of reconciledforecasts for hierarchical and grouped time series. Computational Statistics& Data Analysis 97, 16–32
SL Wickramasuriya, G Athanasopoulos and RJ Hyndman (2015). Forecastinghierarchical and grouped time series through trace minimization. Workingpaper. Dept Econometrics & Business Statistics, Monash University
RJ Hyndman and G Athanasopoulos (2018). Forecasting: principles andpractice. 2nd ed. Melbourne, Australia: OTexts. OTexts.org/fpp2/.
Hierarchical Forecasting Fast computational tricks 43
R packages
Hierarchical Forecasting Fast computational tricks 44
https://github.com/earowang/tsibble
http://pkg.earo.me/sugrrants
https://github.com/mitchelloharawild/fasster
http://pkg.robjhyndman.com/forecast
http://pkg.earo.me/hts