Visualizing and forecasting big time series data

1. Rob J Hyndman
Visualizing and forecasting
big time series data
Victoria: scaled

2. Outline
1 Examples of big time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Application: Australian labour market
6 Fast computation tricks
7 hts package for R
8 References
Visualising and forecasting big time series data Examples of big time series 2

3. 1. Australian tourism demand
Visualising and forecasting big time series data Examples of big time series 3

4. 1. Australian tourism demand
Visualising and forecasting big time series data Examples of big time series 3
Quarterly data on visitor night from
1998:Q1 – 2013:Q4
From: National Visitor Survey, based on
annual interviews of 120,000 Australians
aged 15+, collected by Tourism Research
Australia.
Split by 7 states, 27 zones and 76 regions
(a geographical hierarchy)
Also split by purpose of travel
Holiday
Visiting friends and relatives (VFR)
Business
Other
304 bottom-level series

5. 2. Labour market participation
Australia and New Zealand Standard
Classification of Occupations
8 major groups
43 sub-major groups
97 minor groups
– 359 unit groups
* 1023 occupations
Example: statistician
2 Professionals
22 Business, Human Resource and Marketing
Professionals
224 Information and Organisation Professionals
2241 Actuaries, Mathematicians and Statisticians
224113 Statistician
Visualising and forecasting big time series data Examples of big time series 4

6. 2. Labour market participation
Australia and New Zealand Standard
Classification of Occupations
8 major groups
43 sub-major groups
97 minor groups
– 359 unit groups
* 1023 occupations
Example: statistician
2 Professionals
22 Business, Human Resource and Marketing
Professionals
224 Information and Organisation Professionals
2241 Actuaries, Mathematicians and Statisticians
224113 Statistician
Visualising and forecasting big time series data Examples of big time series 4

7. 3. PBS sales
Visualising and forecasting big time series data Examples of big time series 5

8. 3. PBS sales
ATC drug classification
A Alimentary tract and metabolism
B Blood and blood forming organs
C Cardiovascular system
D Dermatologicals
G Genito-urinary system and sex hormones
H Systemic hormonal preparations, excluding sex hormones
and insulins
J Anti-infectives for systemic use
L Antineoplastic and immunomodulating agents
M Musculo-skeletal system
N Nervous system
P Antiparasitic products, insecticides and repellents
R Respiratory system
S Sensory organs
V Various
Visualising and forecasting big time series data Examples of big time series 6

9. 3. PBS sales
ATC drug classification
A Alimentary tract and metabolism14 classes
A10 Drugs used in diabetes84 classes
A10B Blood glucose lowering drugs
A10BA Biguanides
A10BA02 Metformin
Visualising and forecasting big time series data Examples of big time series 7

10. 4. Spectacle sales
Visualising and forecasting big time series data Examples of big time series 8
Monthly sales data from 2000 – 2014
Provided by a large spectacle manufacturer
Split by brand (26), gender (3), price range
(6), materials (4), and stores (600)
About a million bottom-level series

11. 4. Spectacle sales
Visualising and forecasting big time series data Examples of big time series 8
Monthly sales data from 2000 – 2014
Provided by a large spectacle manufacturer
Split by brand (26), gender (3), price range
(6), materials (4), and stores (600)
About a million bottom-level series

12. 4. Spectacle sales
Visualising and forecasting big time series data Examples of big time series 8
Monthly sales data from 2000 – 2014
Provided by a large spectacle manufacturer
Split by brand (26), gender (3), price range
(6), materials (4), and stores (600)
About a million bottom-level series

13. 4. Spectacle sales
Visualising and forecasting big time series data Examples of big time series 8
Monthly sales data from 2000 – 2014
Provided by a large spectacle manufacturer
Split by brand (26), gender (3), price range
(6), materials (4), and stores (600)
About a million bottom-level series

14. Hierarchical time series
A hierarchical time series is a collection of
several time series that are linked together in
a hierarchical structure.
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Examples
Net labour turnover
Pharmaceutical sales
Tourism by state and region
Visualising and forecasting big time series data Examples of big time series 9

15. Hierarchical time series
A hierarchical time series is a collection of
several time series that are linked together in
a hierarchical structure.
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Examples
Net labour turnover
Pharmaceutical sales
Tourism by state and region
Visualising and forecasting big time series data Examples of big time series 9

16. Hierarchical time series
A hierarchical time series is a collection of
several time series that are linked together in
a hierarchical structure.
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Examples
Net labour turnover
Pharmaceutical sales
Tourism by state and region
Visualising and forecasting big time series data Examples of big time series 9

17. Hierarchical time series
A hierarchical time series is a collection of
several time series that are linked together in
a hierarchical structure.
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Examples
Net labour turnover
Pharmaceutical sales
Tourism by state and region
Visualising and forecasting big time series data Examples of big time series 9

18. Grouped time series
A grouped time series is a collection of time
series that can be grouped together in a
number of non-hierarchical ways.
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
Examples
Tourism by state and purpose of travel
Glasses by brand and store
Visualising and forecasting big time series data Examples of big time series 10

19. Grouped time series
A grouped time series is a collection of time
series that can be grouped together in a
number of non-hierarchical ways.
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
Examples
Tourism by state and purpose of travel
Glasses by brand and store
Visualising and forecasting big time series data Examples of big time series 10

20. Grouped time series
A grouped time series is a collection of time
series that can be grouped together in a
number of non-hierarchical ways.
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
Examples
Tourism by state and purpose of travel
Glasses by brand and store
Visualising and forecasting big time series data Examples of big time series 10

21. Outline
1 Examples of big time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Application: Australian labour market
6 Fast computation tricks
7 hts package for R
8 References
Visualising and forecasting big time series data Time series visualisation 11

22. Victorian tourism dataBAAHolBABHol
BAAVisBABVis
BAABusBABBus
BAAOthBABOth
BACHolBBAHol
BACVisBBAVis
BACBusBBABus
BACOthBBAOth
BCAHolBCBHol
BCAVisBCBVis
BCABusBCBBus
BCAOthBCBOth
BCCHolBDAHol
BCCVisBDAVis
BCCBusBDABus
BCCOthBDAOth
BDBHolBDCHol
BDBVisBDCVis
BDBBusBDCBus
BDBOthBDCOth
BDDHolBDEHol
BDDVisBDEVis
BDDBusBDEBus
BDDOthBDEOth
BDFHolBEAHol
BDFVisBEAVis
BDFBusBEABus
BDFOthBEAOth
BEBHolBECHol
BEBVisBECVis
BEBBusBECBus
BEBOthBECOth
BEDHolBEEHol
BEDVisBEEVis
BEDBusBEEBus
BEDOthBEEOth
BEFHolBEGHol
BEFVisBEGVis
BEFBusBEGBus
BEFOthBEGOth
Visualising and forecasting big time series data Time series visualisation 12

23. Kite diagrams
000
Line graph profile
Duplicate & flip
around the hori-
zontal axis
Fill the colour
Visualising and forecasting big time series data Time series visualisation 13

24. Kite diagrams: Victorian tourism
20002010
Holiday
20002010
VFR
20002010
Business
20002010
BAA
BAB
BAC
BBA
BCA
BCB
BCC
BDA
BDB
BDC
BDD
BDE
BDF
BEA
BEB
BEC
BED
BEE
BEF
Other
BEG
Victoria
Visualising and forecasting big time series data Time series visualisation 14

25. Kite diagrams: Victorian tourism
Visualising and forecasting big time series data Time series visualisation 14

26. Kite diagrams: Victorian tourism
20002010
Holiday
20002010
VFR
20002010
Business
20002010
BAA
BAB
BAC
BBA
BCA
BCB
BCC
BDA
BDB
BDC
BDD
BDE
BDF
BEA
BEB
BEC
BED
BEE
BEF
Other
BEG
Victoria: scaled
Visualising and forecasting big time series data Time series visualisation 14

27. An STL decomposition
STL decomposition of tourism demand
for holidays in Peninsula
5.06.07.0
data
−0.50.5
seasonal
5.86.16.4
trend
−0.40.0
2000 2005 2010
remainder
Visualising and forecasting big time series data Time series visualisation 15

28. Seasonal stacked bar chart
Place positive values above the origin
while negative values below the origin
Map the bar length to the magnitude
Encode quarters by colours
Visualising and forecasting big time series data Time series visualisation 16

29. Seasonal stacked bar chart
Place positive values above the origin
while negative values below the origin
Map the bar length to the magnitude
Encode quarters by colours
−1.0
−0.5
0.0
0.5
1.0
Holiday
BAA BABBACBBABCABCBBCCBDABDBBDCBDDBDEBDF BEA BEBBECBEDBEE BEFBEG
Regions
SeasonalComponent
Qtr
Q1
Q2
Q3
Q4
Visualising and forecasting big time series data Time series visualisation 16

30. Seasonal stacked bar chart: VIC
Visualising and forecasting big time series data Time series visualisation 17

31. Seasonal stacked bar chart: VIC
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
HolidayVFRBusinessOther
BAABABBACBBABCABCBBCCBDABDBBDCBDDBDEBDFBEABEBBECBEDBEEBEFBEG
Regions
SeasonalComponent
Qtr
Q1
Q2
Q3
Q4
Visualising and forecasting big time series data Time series visualisation 17

32. Corrgram of remainder
Visualising and forecasting big time series data Time series visualisation 18
Compute the correlations
among the remainder
components
Render both the sign and
magnitude using a colour
mapping of two hues
Order variables according to
the first principal component of
the correlations.

33. Corrgram of remainder: VIC
Visualising and forecasting big time series data Time series visualisation 19
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
BEEHolBEFOthBEEOthBDEOthBEBOthBEABusBEFBusBDCOthBACHolBEBBusBEAVisBBAHolBDEHolBABOthBAAVisBAAHolBDCHolBBABusBCBHolBEGBusBDDVisBABVisBDAVisBEAOthBDFHolBEEBusBAAOthBACOthBDAOthBDEBusBCBOthBACBusBEBVisBACVisBCAOthBEFVisBCBVisBEDHolBEGOthBDBHolBABBusBEBHolBDFBusBECHolBCAHolBDBOthBEAHolBDCBusBECVisBDBVisBCCHolBBAVisBABHolBBAOthBCCOthBCBBusBCCVisBEGVisBDDHolBECOthBDCVisBAABusBCCBusBECBusBCAVisBDFVisBEGHolBDDOthBEDOthBEDVisBDDBusBDEVisBEFHolBEEVisBDBBusBDABusBDAHolBCABusBDFOthBEDBus
BEEHolBEFOthBEEOthBDEOthBEBOthBEABusBEFBusBDCOthBACHolBEBBusBEAVisBBAHolBDEHolBABOthBAAVisBAAHolBDCHolBBABusBCBHolBEGBusBDDVisBABVisBDAVisBEAOthBDFHolBEEBusBAAOthBACOthBDAOthBDEBusBCBOthBACBusBEBVisBACVisBCAOthBEFVisBCBVisBEDHolBEGOthBDBHolBABBusBEBHolBDFBusBECHolBCAHolBDBOthBEAHolBDCBusBECVisBDBVisBCCHolBBAVisBABHolBBAOthBCCOthBCBBusBCCVisBEGVisBDDHolBECOthBDCVisBAABusBCCBusBECBusBCAVisBDFVisBEGHolBDDOthBEDOthBEDVisBDDBusBDEVisBEFHolBEEVisBDBBusBDABusBDAHolBCABusBDFOthBEDBus

34. Corrgram of remainder: VIC
Visualising and forecasting big time series data Time series visualisation 19
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
BDAHol
BDDHol
BEBHol
BEFHol
BECHol
BEDHol
BDFHol
BCCHol
BDCHol
BCAHol
BEAHol
BEGHol
BBAHol
BAAHol
BABHol
BDBHol
BDEHol
BACHol
BCBHol
BEEHol
BDAHol
BDDHol
BEBHol
BEFHol
BECHol
BEDHol
BDFHol
BCCHol
BDCHol
BCAHol
BEAHol
BEGHol
BBAHol
BAAHol
BABHol
BDBHol
BDEHol
BACHol
BCBHol
BEEHol

35. Corrgram of remainder: TAS
Visualising and forecasting big time series data Time series visualisation 20
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
FCAHol
FBBHol
FBAHol
FAAHol
FCBHol
FCAVis
FBBVis
FAAVis
FCBBus
FAAOth
FCAOth
FBBOth
FBABus
FBAOth
FCBVis
FCABus
FBAVis
FCBOth
FBBBus
FAABus
FCAHol
FBBHol
FBAHol
FAAHol
FCBHol
FCAVis
FBBVis
FAAVis
FCBBus
FAAOth
FCAOth
FBBOth
FBABus
FBAOth
FCBVis
FCABus
FBAVis
FCBOth
FBBBus
FAABus

36. Principal components decomposition
Visualising and forecasting big time series data Time series visualisation 21
−25−15−55
PC1
−50510
PC2
−50510
2000 2005 2010
PC3
Time
First three PCs

37. Principal components decomposition
Visualising and forecasting big time series data Time series visualisation 21
−25−20−15−10−505
Season plot: PC1
Month
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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

38. Principal components decomposition
Visualising and forecasting big time series data Time series visualisation 21
−50510
Season plot: PC2
Month
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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

39. Principal components decomposition
Visualising and forecasting big time series data Time series visualisation 21
−50510
Season plot: PC3
Month
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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

40. Principal components decomposition
Visualising and forecasting big time series data Time series visualisation 22
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−0.15 −0.10 −0.05 0.00 0.05
−0.100.000.050.100.150.20
Loading 1
Loading2
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NSW
VIC
QLD
SA
TAS
NT
WA

41. Principal components decomposition
Visualising and forecasting big time series data Time series visualisation 22
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−0.15 −0.10 −0.05 0.00 0.05
−0.100.000.050.100.150.20
Loading 1
Loading2
q
q
q
q
Hol
Vis
Bus
Oth

42. Feature analysis
Summarize each time series with a feature
vector:
strength of trend
summer seasonality
winter seasonality
Box-Pierce statistic on remainder of STL
Lumpiness (variance of annual variances of
remainder)
Do PCA on feature matrix
Visualising and forecasting big time series data Time series visualisation 23

43. Feature analysis
Summarize each time series with a feature
vector:
strength of trend
summer seasonality
winter seasonality
Box-Pierce statistic on remainder of STL
Lumpiness (variance of annual variances of
remainder)
Do PCA on feature matrix
Visualising and forecasting big time series data Time series visualisation 23

44. Feature analysis
Summarize each time series with a feature
vector:
strength of trend
summer seasonality
winter seasonality
Box-Pierce statistic on remainder of STL
Lumpiness (variance of annual variances of
remainder)
Do PCA on feature matrix
Visualising and forecasting big time series data Time series visualisation 23

45. Feature analysis
Summarize each time series with a feature
vector:
strength of trend
summer seasonality
winter seasonality
Box-Pierce statistic on remainder of STL
Lumpiness (variance of annual variances of
remainder)
Do PCA on feature matrix
Visualising and forecasting big time series data Time series visualisation 23

46. Feature analysis
Summarize each time series with a feature
vector:
strength of trend
summer seasonality
winter seasonality
Box-Pierce statistic on remainder of STL
Lumpiness (variance of annual variances of
remainder)
Do PCA on feature matrix
Visualising and forecasting big time series data Time series visualisation 23

47. Feature analysis
Summarize each time series with a feature
vector:
strength of trend
summer seasonality
winter seasonality
Box-Pierce statistic on remainder of STL
Lumpiness (variance of annual variances of
remainder)
Do PCA on feature matrix
Visualising and forecasting big time series data Time series visualisation 23

48. Feature analysis
Summarize each time series with a feature
vector:
strength of trend
summer seasonality
winter seasonality
Box-Pierce statistic on remainder of STL
Lumpiness (variance of annual variances of
remainder)
Do PCA on feature matrix
Visualising and forecasting big time series data Time series visualisation 23

49. Feature analysis
Visualising and forecasting big time series data Time series visualisation 24
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trend
summer
wintercorr
lum
py
−2
0
2
−5.0 −2.5 0.0 2.5
PC1 (39.1% explained var.)
PC2(23.6%explainedvar.)
groups
q
q
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Bus
Hol
Oth
Vis

50. Feature analysis
Visualising and forecasting big time series data Time series visualisation 24
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trend
summer
wintercorr
lum
py
−2
0
2
−5.0 −2.5 0.0 2.5
PC1 (39.1% explained var.)
PC2(23.6%explainedvar.)
groups
q
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NSW
NT
QLD
SA
TAS
VIC
WA

51. Outline
1 Examples of big time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Application: Australian labour market
6 Fast computation tricks
7 hts package for R
8 References
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 25

52. Hierarchical/grouped time series
Forecasts should be “aggregate
consistent”, unbiased, minimum variance.
Existing methods:
¢ Bottom-up
¢ Top-down
¢ Middle-out
How to compute forecast intervals?
Most research is concerned about relative
performance of existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26

53. Hierarchical/grouped time series
Forecasts should be “aggregate
consistent”, unbiased, minimum variance.
Existing methods:
¢ Bottom-up
¢ Top-down
¢ Middle-out
How to compute forecast intervals?
Most research is concerned about relative
performance of existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26

54. Hierarchical/grouped time series
Forecasts should be “aggregate
consistent”, unbiased, minimum variance.
Existing methods:
¢ Bottom-up
¢ Top-down
¢ Middle-out
How to compute forecast intervals?
Most research is concerned about relative
performance of existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26

55. Hierarchical/grouped time series
Forecasts should be “aggregate
consistent”, unbiased, minimum variance.
Existing methods:
¢ Bottom-up
¢ Top-down
¢ Middle-out
How to compute forecast intervals?
Most research is concerned about relative
performance of existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26

56. Hierarchical/grouped time series
Forecasts should be “aggregate
consistent”, unbiased, minimum variance.
Existing methods:
¢ Bottom-up
¢ Top-down
¢ Middle-out
How to compute forecast intervals?
Most research is concerned about relative
performance of existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26

57. Hierarchical/grouped time series
Forecasts should be “aggregate
consistent”, unbiased, minimum variance.
Existing methods:
¢ Bottom-up
¢ Top-down
¢ Middle-out
How to compute forecast intervals?
Most research is concerned about relative
performance of existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26

58. Hierarchical/grouped time series
Forecasts should be “aggregate
consistent”, unbiased, minimum variance.
Existing methods:
¢ Bottom-up
¢ Top-down
¢ Middle-out
How to compute forecast intervals?
Most research is concerned about relative
performance of existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26

59. Top-down method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
Advantages
Works well in
presence of low
counts.
Single forecasting
model easy to
build
Provides reliable
forecasts for
aggregate levels.
Disadvantages
Loss of information,
especially
individual series
dynamics.
Distribution of
forecasts to lower
levels can be
difficult
No prediction
intervals

60. Top-down method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
Advantages
Works well in
presence of low
counts.
Single forecasting
model easy to
build
Provides reliable
forecasts for
aggregate levels.
Disadvantages
Loss of information,
especially
individual series
dynamics.
Distribution of
forecasts to lower
levels can be
difficult
No prediction
intervals

61. Top-down method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
Advantages
Works well in
presence of low
counts.
Single forecasting
model easy to
build
Provides reliable
forecasts for
aggregate levels.
Disadvantages
Loss of information,
especially
individual series
dynamics.
Distribution of
forecasts to lower
levels can be
difficult
No prediction
intervals

62. Top-down method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
Advantages
Works well in
presence of low
counts.
Single forecasting
model easy to
build
Provides reliable
forecasts for
aggregate levels.
Disadvantages
Loss of information,
especially
individual series
dynamics.
Distribution of
forecasts to lower
levels can be
difficult
No prediction
intervals

63. Top-down method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
Advantages
Works well in
presence of low
counts.
Single forecasting
model easy to
build
Provides reliable
forecasts for
aggregate levels.
Disadvantages
Loss of information,
especially
individual series
dynamics.
Distribution of
forecasts to lower
levels can be
difficult
No prediction
intervals

64. Top-down method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
Advantages
Works well in
presence of low
counts.
Single forecasting
model easy to
build
Provides reliable
forecasts for
aggregate levels.
Disadvantages
Loss of information,
especially
individual series
dynamics.
Distribution of
forecasts to lower
levels can be
difficult
No prediction
intervals

65. Bottom-up method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 28
Advantages
No loss of
information.
Better captures
dynamics of
individual series.
Disadvantages
Large number of
series to be
forecast.
Constructing
forecasting models
is harder because
of noisy data at
bottom level.
No prediction
intervals

66. Bottom-up method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 28
Advantages
No loss of
information.
Better captures
dynamics of
individual series.
Disadvantages
Large number of
series to be
forecast.
Constructing
forecasting models
is harder because
of noisy data at
bottom level.
No prediction
intervals

67. Bottom-up method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 28
Advantages
No loss of
information.
Better captures
dynamics of
individual series.
Disadvantages
Large number of
series to be
forecast.
Constructing
forecasting models
is harder because
of noisy data at
bottom level.
No prediction
intervals

68. Bottom-up method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 28
Advantages
No loss of
information.
Better captures
dynamics of
individual series.
Disadvantages
Large number of
series to be
forecast.
Constructing
forecasting models
is harder because
of noisy data at
bottom level.
No prediction
intervals

69. Bottom-up method
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 28
Advantages
No loss of
information.
Better captures
dynamics of
individual series.
Disadvantages
Large number of
series to be
forecast.
Constructing
forecasting models
is harder because
of noisy data at
bottom level.
No prediction
intervals

70. The BLUF approach
Hyndman et al (CSDA 2011) proposed a new
statistical framework for forecasting
hierarchical time series which:
1 provides point forecasts that are
consistent across the hierarchy;
2 allows for correlations and interaction
between series at each level;
3 provides estimates of forecast uncertainty
which are consistent across the hierarchy;
4 allows for ad hoc adjustments and
inclusion of covariates at any level.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 29

71. The BLUF approach
Hyndman et al (CSDA 2011) proposed a new
statistical framework for forecasting
hierarchical time series which:
1 provides point forecasts that are
consistent across the hierarchy;
2 allows for correlations and interaction
between series at each level;
3 provides estimates of forecast uncertainty
which are consistent across the hierarchy;
4 allows for ad hoc adjustments and
inclusion of covariates at any level.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 29

72. The BLUF approach
Hyndman et al (CSDA 2011) proposed a new
statistical framework for forecasting
hierarchical time series which:
1 provides point forecasts that are
consistent across the hierarchy;
2 allows for correlations and interaction
between series at each level;
3 provides estimates of forecast uncertainty
which are consistent across the hierarchy;
4 allows for ad hoc adjustments and
inclusion of covariates at any level.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 29

73. The BLUF approach
Hyndman et al (CSDA 2011) proposed a new
statistical framework for forecasting
hierarchical time series which:
1 provides point forecasts that are
consistent across the hierarchy;
2 allows for correlations and interaction
between series at each level;
3 provides estimates of forecast uncertainty
which are consistent across the hierarchy;
4 allows for ad hoc adjustments and
inclusion of covariates at any level.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 29

74. Hierarchical data
Total
A B C
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 30
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.

75. Hierarchical data
Total
A B C
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 30
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.

76. Hierarchical data
Total
A B C
yt = [Yt, YA,t, YB,t, YC,t] =




1 1 1
1 0 0
0 1 0
0 0 1






YA,t
YB,t
YC,t


Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 30
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.

77. Hierarchical data
Total
A B C
yt = [Yt, YA,t, YB,t, YC,t] =




1 1 1
1 0 0
0 1 0
0 0 1




S


YA,t
YB,t
YC,t


Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 30
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.

78. Hierarchical data
Total
A B C
yt = [Yt, YA,t, YB,t, YC,t] =




1 1 1
1 0 0
0 1 0
0 0 1




S


YA,t
YB,t
YC,t


Bt
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 30
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.

79. Hierarchical data
Total
A B C
yt = [Yt, YA,t, YB,t, YC,t] =




1 1 1
1 0 0
0 1 0
0 0 1




S


YA,t
YB,t
YC,t


Bt
yt = SBt
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 30
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.

80. Hierarchical data
Total
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =












Yt
YA,t
YB,t
YC,t
YAX,t
YAY,t
YAZ,t
YBX,t
YBY,t
YBZ,t
YCX,t
YCY,t
YCZ,t












=












1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 0 0 0 1 1 1
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1












S







YAX,t
YAY,t
YAZ,t
YBX,t
YBY,t
YBZ,t
YCX,t
YCY,t
YCZ,t







Bt
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 31

81. Hierarchical data
Total
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =












Yt
YA,t
YB,t
YC,t
YAX,t
YAY,t
YAZ,t
YBX,t
YBY,t
YBZ,t
YCX,t
YCY,t
YCZ,t












=












1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 0 0 0 1 1 1
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1












S







YAX,t
YAY,t
YAZ,t
YBX,t
YBY,t
YBZ,t
YCX,t
YCY,t
YCZ,t







Bt
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 31

82. Hierarchical data
Total
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =












Yt
YA,t
YB,t
YC,t
YAX,t
YAY,t
YAZ,t
YBX,t
YBY,t
YBZ,t
YCX,t
YCY,t
YCZ,t












=












1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 0 0 0 1 1 1
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1





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S
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Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 31
yt = SBt

83. Grouped data
AX AY A
BX BY B
X Y Total
yt =

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Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 32

84. Grouped data
AX AY A
BX BY B
X Y Total
yt =
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Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 32

85. Grouped data
AX AY A
BX BY B
X Y Total
yt =








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1 1 0 0
0 0 1 1
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0 1 0 1
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YBX,t
YBY,t

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Bt
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 32
yt = SBt

86. Forecasting notation
Let ˆyn(h) be vector of initial h-step forecasts,
made at time n, stacked in same order as yt.
(They may not add up.)
Hierarchical forecasting methods of the form:
˜yn(h) = SPˆyn(h)
for some matrix P.
P extracts and combines base forecasts
ˆyn(h) to get bottom-level forecasts.
S adds them up
Revised reconciled forecasts: ˜yn(h).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 33

87. Forecasting notation
Let ˆyn(h) be vector of initial h-step forecasts,
made at time n, stacked in same order as yt.
(They may not add up.)
Hierarchical forecasting methods of the form:
˜yn(h) = SPˆyn(h)
for some matrix P.
P extracts and combines base forecasts
ˆyn(h) to get bottom-level forecasts.
S adds them up
Revised reconciled forecasts: ˜yn(h).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 33

88. Forecasting notation
Let ˆyn(h) be vector of initial h-step forecasts,
made at time n, stacked in same order as yt.
(They may not add up.)
Hierarchical forecasting methods of the form:
˜yn(h) = SPˆyn(h)
for some matrix P.
P extracts and combines base forecasts
ˆyn(h) to get bottom-level forecasts.
S adds them up
Revised reconciled forecasts: ˜yn(h).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 33

89. Forecasting notation
Let ˆyn(h) be vector of initial h-step forecasts,
made at time n, stacked in same order as yt.
(They may not add up.)
Hierarchical forecasting methods of the form:
˜yn(h) = SPˆyn(h)
for some matrix P.
P extracts and combines base forecasts
ˆyn(h) to get bottom-level forecasts.
S adds them up
Revised reconciled forecasts: ˜yn(h).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 33

90. Forecasting notation
Let ˆyn(h) be vector of initial h-step forecasts,
made at time n, stacked in same order as yt.
(They may not add up.)
Hierarchical forecasting methods of the form:
˜yn(h) = SPˆyn(h)
for some matrix P.
P extracts and combines base forecasts
ˆyn(h) to get bottom-level forecasts.
S adds them up
Revised reconciled forecasts: ˜yn(h).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 33

91. Forecasting notation
Let ˆyn(h) be vector of initial h-step forecasts,
made at time n, stacked in same order as yt.
(They may not add up.)
Hierarchical forecasting methods of the form:
˜yn(h) = SPˆyn(h)
for some matrix P.
P extracts and combines base forecasts
ˆyn(h) to get bottom-level forecasts.
S adds them up
Revised reconciled forecasts: ˜yn(h).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 33

92. Bottom-up forecasts
˜yn(h) = SPˆyn(h)
Bottom-up forecasts are obtained using
P = [0 | I] ,
where 0 is null matrix and I is identity matrix.
P matrix extracts only bottom-level
forecasts from ˆyn(h)
S adds them up to give the bottom-up
forecasts.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 34

93. Bottom-up forecasts
˜yn(h) = SPˆyn(h)
Bottom-up forecasts are obtained using
P = [0 | I] ,
where 0 is null matrix and I is identity matrix.
P matrix extracts only bottom-level
forecasts from ˆyn(h)
S adds them up to give the bottom-up
forecasts.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 34

94. Bottom-up forecasts
˜yn(h) = SPˆyn(h)
Bottom-up forecasts are obtained using
P = [0 | I] ,
where 0 is null matrix and I is identity matrix.
P matrix extracts only bottom-level
forecasts from ˆyn(h)
S adds them up to give the bottom-up
forecasts.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 34

95. Top-down forecasts
˜yn(h) = SPˆyn(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 lowest level series.
Different methods of top-down forecasting
lead to different proportionality vectors p.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 35

96. Top-down forecasts
˜yn(h) = SPˆyn(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 lowest level series.
Different methods of top-down forecasting
lead to different proportionality vectors p.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 35

97. Top-down forecasts
˜yn(h) = SPˆyn(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 lowest level series.
Different methods of top-down forecasting
lead to different proportionality vectors p.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 35

98. General properties: bias
˜yn(h) = SPˆyn(h)
Assume: base forecasts ˆyn(h) are unbiased:
E[ˆyn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let ˆBn(h) be bottom level base forecasts
with βn(h) = E[ˆBn(h)|y1, . . . , yn].
Then E[ˆyn(h)] = Sβn(h).
We want the revised forecasts to be unbiased:
E[˜yn(h)] = SPSβn(h) = Sβn(h).
Result will hold provided SPS = S.
True for bottom-up, but not for any top-down
method or middle-out method.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 36

99. General properties: bias
˜yn(h) = SPˆyn(h)
Assume: base forecasts ˆyn(h) are unbiased:
E[ˆyn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let ˆBn(h) be bottom level base forecasts
with βn(h) = E[ˆBn(h)|y1, . . . , yn].
Then E[ˆyn(h)] = Sβn(h).
We want the revised forecasts to be unbiased:
E[˜yn(h)] = SPSβn(h) = Sβn(h).
Result will hold provided SPS = S.
True for bottom-up, but not for any top-down
method or middle-out method.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 36

100. General properties: bias
˜yn(h) = SPˆyn(h)
Assume: base forecasts ˆyn(h) are unbiased:
E[ˆyn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let ˆBn(h) be bottom level base forecasts
with βn(h) = E[ˆBn(h)|y1, . . . , yn].
Then E[ˆyn(h)] = Sβn(h).
We want the revised forecasts to be unbiased:
E[˜yn(h)] = SPSβn(h) = Sβn(h).
Result will hold provided SPS = S.
True for bottom-up, but not for any top-down
method or middle-out method.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 36

101. General properties: bias
˜yn(h) = SPˆyn(h)
Assume: base forecasts ˆyn(h) are unbiased:
E[ˆyn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let ˆBn(h) be bottom level base forecasts
with βn(h) = E[ˆBn(h)|y1, . . . , yn].
Then E[ˆyn(h)] = Sβn(h).
We want the revised forecasts to be unbiased:
E[˜yn(h)] = SPSβn(h) = Sβn(h).
Result will hold provided SPS = S.
True for bottom-up, but not for any top-down
method or middle-out method.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 36

102. General properties: bias
˜yn(h) = SPˆyn(h)
Assume: base forecasts ˆyn(h) are unbiased:
E[ˆyn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let ˆBn(h) be bottom level base forecasts
with βn(h) = E[ˆBn(h)|y1, . . . , yn].
Then E[ˆyn(h)] = Sβn(h).
We want the revised forecasts to be unbiased:
E[˜yn(h)] = SPSβn(h) = Sβn(h).
Result will hold provided SPS = S.
True for bottom-up, but not for any top-down
method or middle-out method.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 36

103. General properties: bias
˜yn(h) = SPˆyn(h)
Assume: base forecasts ˆyn(h) are unbiased:
E[ˆyn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let ˆBn(h) be bottom level base forecasts
with βn(h) = E[ˆBn(h)|y1, . . . , yn].
Then E[ˆyn(h)] = Sβn(h).
We want the revised forecasts to be unbiased:
E[˜yn(h)] = SPSβn(h) = Sβn(h).
Result will hold provided SPS = S.
True for bottom-up, but not for any top-down
method or middle-out method.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 36

104. General properties: bias
˜yn(h) = SPˆyn(h)
Assume: base forecasts ˆyn(h) are unbiased:
E[ˆyn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let ˆBn(h) be bottom level base forecasts
with βn(h) = E[ˆBn(h)|y1, . . . , yn].
Then E[ˆyn(h)] = Sβn(h).
We want the revised forecasts to be unbiased:
E[˜yn(h)] = SPSβn(h) = Sβn(h).
Result will hold provided SPS = S.
True for bottom-up, but not for any top-down
method or middle-out method.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 36

105. General properties: bias
˜yn(h) = SPˆyn(h)
Assume: base forecasts ˆyn(h) are unbiased:
E[ˆyn(h)|y1, . . . , yn] = E[yn+h|y1, . . . , yn]
Let ˆBn(h) be bottom level base forecasts
with βn(h) = E[ˆBn(h)|y1, . . . , yn].
Then E[ˆyn(h)] = Sβn(h).
We want the revised forecasts to be unbiased:
E[˜yn(h)] = SPSβn(h) = Sβn(h).
Result will hold provided SPS = S.
True for bottom-up, but not for any top-down
method or middle-out method.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 36

106. General properties: variance
˜yn(h) = SPˆyn(h)
Let variance of base forecasts ˆyn(h) be given
by
Σh = Var[ˆyn(h)|y1, . . . , yn]
Then the variance of the revised forecasts is
given by
Var[˜yn(h)|y1, . . . , yn] = SPΣhP S .
This is a general result for all existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 37

107. General properties: variance
˜yn(h) = SPˆyn(h)
Let variance of base forecasts ˆyn(h) be given
by
Σh = Var[ˆyn(h)|y1, . . . , yn]
Then the variance of the revised forecasts is
given by
Var[˜yn(h)|y1, . . . , yn] = SPΣhP S .
This is a general result for all existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 37

108. General properties: variance
˜yn(h) = SPˆyn(h)
Let variance of base forecasts ˆyn(h) be given
by
Σh = Var[ˆyn(h)|y1, . . . , yn]
Then the variance of the revised forecasts is
given by
Var[˜yn(h)|y1, . . . , yn] = SPΣhP S .
This is a general result for all existing methods.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 37

109. BLUF via trace minimization
Theorem
For any P satisfying SPS = S, then
min
P
= trace[SPΣhP S ]
has solution
P = (S Σ†
hS)−1
S Σ†
h.
Σ†
h is generalized inverse of Σh.
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 38

110. BLUF via trace minimization
Theorem
For any P satisfying SPS = S, then
min
P
= trace[SPΣhP S ]
has solution
P = (S Σ†
hS)−1
S Σ†
h.
Σ†
h is generalized inverse of Σh.
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 38

111. BLUF via trace minimization
Theorem
For any P satisfying SPS = S, then
min
P
= trace[SPΣhP S ]
has solution
P = (S Σ†
hS)−1
S Σ†
h.
Σ†
h is generalized inverse of Σh.
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 38

112. Optimal combination forecasts
˜yn(h) = SPˆyn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Σ†
h is generalized inverse of Σh.
Var[˜yn(h)|y1, . . . , yn] = S(S Σ†
hS)−1
S
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 39

113. Optimal combination forecasts
˜yn(h) = SPˆyn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Initial forecasts
Σ†
h is generalized inverse of Σh.
Var[˜yn(h)|y1, . . . , yn] = S(S Σ†
hS)−1
S
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 39

114. Optimal combination forecasts
˜yn(h) = SPˆyn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Initial forecasts
Σ†
h is generalized inverse of Σh.
Var[˜yn(h)|y1, . . . , yn] = S(S Σ†
hS)−1
S
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 39

115. Optimal combination forecasts
˜yn(h) = SPˆyn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Initial forecasts
Σ†
h is generalized inverse of Σh.
Var[˜yn(h)|y1, . . . , yn] = S(S Σ†
hS)−1
S
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 39

116. Optimal combination forecasts
˜yn(h) = SPˆyn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Initial forecasts
Σ†
h is generalized inverse of Σh.
Var[˜yn(h)|y1, . . . , yn] = S(S Σ†
hS)−1
S
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 39

117. Optimal combination forecasts
˜yn(h) = SPˆyn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Initial forecasts
Σ†
h is generalized inverse of Σh.
Var[˜yn(h)|y1, . . . , yn] = S(S Σ†
hS)−1
S
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 39

118. Optimal combination forecasts
˜yn(h) = SPˆyn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Initial forecasts
Σ†
h is generalized inverse of Σh.
Var[˜yn(h)|y1, . . . , yn] = S(S Σ†
hS)−1
S
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 39

119. Optimal combination forecasts
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Solution 1: OLS
Assume εh ≈ SεB,h where εB,h is the
forecast error at bottom level.
Then Σh ≈ SΩhS where Ωh = Var(εB,h).
If Moore-Penrose generalized inverse used,
then (S Σ†
hS)−1
S Σ†
h = (S S)−1
S .
˜yn(h) = S(S S)−1
S ˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 40

120. Optimal combination forecasts
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Solution 1: OLS
Assume εh ≈ SεB,h where εB,h is the
forecast error at bottom level.
Then Σh ≈ SΩhS where Ωh = Var(εB,h).
If Moore-Penrose generalized inverse used,
then (S Σ†
hS)−1
S Σ†
h = (S S)−1
S .
˜yn(h) = S(S S)−1
S ˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 40

121. Optimal combination forecasts
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Solution 1: OLS
Assume εh ≈ SεB,h where εB,h is the
forecast error at bottom level.
Then Σh ≈ SΩhS where Ωh = Var(εB,h).
If Moore-Penrose generalized inverse used,
then (S Σ†
hS)−1
S Σ†
h = (S S)−1
S .
˜yn(h) = S(S S)−1
S ˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 40

122. Optimal combination forecasts
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Solution 1: OLS
Assume εh ≈ SεB,h where εB,h is the
forecast error at bottom level.
Then Σh ≈ SΩhS where Ωh = Var(εB,h).
If Moore-Penrose generalized inverse used,
then (S Σ†
hS)−1
S Σ†
h = (S S)−1
S .
˜yn(h) = S(S S)−1
S ˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 40

123. Optimal combination forecasts
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Solution 1: OLS
Assume εh ≈ SεB,h where εB,h is the
forecast error at bottom level.
Then Σh ≈ SΩhS where Ωh = Var(εB,h).
If Moore-Penrose generalized inverse used,
then (S Σ†
hS)−1
S Σ†
h = (S S)−1
S .
˜yn(h) = S(S S)−1
S ˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 40

124. Optimal combination forecasts
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Solution 1: OLS
Assume εh ≈ SεB,h where εB,h is the
forecast error at bottom level.
Then Σh ≈ SΩhS where Ωh = Var(εB,h).
If Moore-Penrose generalized inverse used,
then (S Σ†
hS)−1
S Σ†
h = (S S)−1
S .
˜yn(h) = S(S S)−1
S ˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 40

125. Optimal combination forecasts
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 41
˜yn(h) = S(S S)−1
S ˆyn(h)Total
A B C

126. Optimal combination forecasts
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 41
˜yn(h) = S(S S)−1
S ˆyn(h)Total
A B C
Weights:
S(S S)−1
S =




0.75 0.25 0.25 0.25
0.25 0.75 −0.25 −0.25
0.25 −0.25 0.75 −0.25
0.25 −0.25 −0.25 0.75





127. Optimal combination forecasts
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Weights: S(S S)−1
S =






















0.69 0.23 0.23 0.23 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
0.23 0.58 −0.17 −0.17 0.19 0.19 0.19 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06
0.23 −0.17 0.58 −0.17 −0.06 −0.06 −0.06 0.19 0.19 0.19 −0.06 −0.06 −0.06
0.23 −0.17 −0.17 0.58 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 0.19 0.19 0.19
0.08 0.19 −0.06 −0.06 0.73 −0.27 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02
0.08 0.19 −0.06 −0.06 −0.27 0.73 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02
0.08 0.19 −0.06 −0.06 −0.27 −0.27 0.73 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02
0.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 0.73 −0.27 −0.27 −0.02 −0.02 −0.02
0.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 0.73 −0.27 −0.02 −0.02 −0.02
0.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 −0.27 0.73 −0.02 −0.02 −0.02
0.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 0.73 −0.27 −0.27
0.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 0.73 −0.27
0.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 −0.27 0.73






















Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 42

128. Optimal combination forecasts
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Weights: S(S S)−1
S =






















0.69 0.23 0.23 0.23 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
0.23 0.58 −0.17 −0.17 0.19 0.19 0.19 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06
0.23 −0.17 0.58 −0.17 −0.06 −0.06 −0.06 0.19 0.19 0.19 −0.06 −0.06 −0.06
0.23 −0.17 −0.17 0.58 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 0.19 0.19 0.19
0.08 0.19 −0.06 −0.06 0.73 −0.27 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02
0.08 0.19 −0.06 −0.06 −0.27 0.73 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02
0.08 0.19 −0.06 −0.06 −0.27 −0.27 0.73 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02
0.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 0.73 −0.27 −0.27 −0.02 −0.02 −0.02
0.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 0.73 −0.27 −0.02 −0.02 −0.02
0.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 −0.27 0.73 −0.02 −0.02 −0.02
0.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 0.73 −0.27 −0.27
0.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 0.73 −0.27
0.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 −0.27 0.73






















Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 42

129. Features
Covariates can be included in initial forecasts.
Adjustments can be made to initial forecasts
at any level.
Very simple and flexible method. Can work
with any hierarchical or grouped time series.
SPS = S so reconciled forcasts are unbiased.
Conceptually easy to implement: OLS on
base forecasts.
Weights are independent of the data and of
the covariance structure of the hierarchy.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 43

130. Features
Covariates can be included in initial forecasts.
Adjustments can be made to initial forecasts
at any level.
Very simple and flexible method. Can work
with any hierarchical or grouped time series.
SPS = S so reconciled forcasts are unbiased.
Conceptually easy to implement: OLS on
base forecasts.
Weights are independent of the data and of
the covariance structure of the hierarchy.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 43

131. Features
Covariates can be included in initial forecasts.
Adjustments can be made to initial forecasts
at any level.
Very simple and flexible method. Can work
with any hierarchical or grouped time series.
SPS = S so reconciled forcasts are unbiased.
Conceptually easy to implement: OLS on
base forecasts.
Weights are independent of the data and of
the covariance structure of the hierarchy.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 43

132. Features
Covariates can be included in initial forecasts.
Adjustments can be made to initial forecasts
at any level.
Very simple and flexible method. Can work
with any hierarchical or grouped time series.
SPS = S so reconciled forcasts are unbiased.
Conceptually easy to implement: OLS on
base forecasts.
Weights are independent of the data and of
the covariance structure of the hierarchy.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 43

133. Features
Covariates can be included in initial forecasts.
Adjustments can be made to initial forecasts
at any level.
Very simple and flexible method. Can work
with any hierarchical or grouped time series.
SPS = S so reconciled forcasts are unbiased.
Conceptually easy to implement: OLS on
base forecasts.
Weights are independent of the data and of
the covariance structure of the hierarchy.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 43

134. Features
Covariates can be included in initial forecasts.
Adjustments can be made to initial forecasts
at any level.
Very simple and flexible method. Can work
with any hierarchical or grouped time series.
SPS = S so reconciled forcasts are unbiased.
Conceptually easy to implement: OLS on
base forecasts.
Weights are independent of the data and of
the covariance structure of the hierarchy.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 43

135. Challenges
Computational difficulties in big
hierarchies due to size of the S matrix and
singular behavior of (S S).
Need to estimate covariance matrix to
produce prediction intervals.
Ignores covariance matrix in computing
point forecasts.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 44
˜yn(h) = S(S S)−1
S ˆyn(h)

136. Challenges
Computational difficulties in big
hierarchies due to size of the S matrix and
singular behavior of (S S).
Need to estimate covariance matrix to
produce prediction intervals.
Ignores covariance matrix in computing
point forecasts.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 44
˜yn(h) = S(S S)−1
S ˆyn(h)

137. Challenges
Computational difficulties in big
hierarchies due to size of the S matrix and
singular behavior of (S S).
Need to estimate covariance matrix to
produce prediction intervals.
Ignores covariance matrix in computing
point forecasts.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 44
˜yn(h) = S(S S)−1
S ˆyn(h)

138. Optimal combination forecasts
Solution 1: OLS
Approximate Σ†
1 by cI.
Solution 2: Rescaling
Suppose we approximate Σ1 by its
diagonal.
Let Λ = diagonal Σ1
−1
contain inverse
one-step forecast variances.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 45
˜yn(h) = S(S Σ†
1S)−1
S Σ†
1ˆyn(h)
˜yn(h) = S(SΛS)−1
SΛˆyn(h)

139. Optimal combination forecasts
Solution 1: OLS
Approximate Σ†
1 by cI.
Solution 2: Rescaling
Suppose we approximate Σ1 by its
diagonal.
Let Λ = diagonal Σ1
−1
contain inverse
one-step forecast variances.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 45
˜yn(h) = S(S Σ†
1S)−1
S Σ†
1ˆyn(h)
˜yn(h) = S(SΛS)−1
SΛˆyn(h)

140. Optimal combination forecasts
Solution 1: OLS
Approximate Σ†
1 by cI.
Solution 2: Rescaling
Suppose we approximate Σ1 by its
diagonal.
Let Λ = diagonal Σ1
−1
contain inverse
one-step forecast variances.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 45
˜yn(h) = S(S Σ†
1S)−1
S Σ†
1ˆyn(h)
˜yn(h) = S(SΛS)−1
SΛˆyn(h)

141. Optimal combination forecasts
Solution 1: OLS
Approximate Σ†
1 by cI.
Solution 2: Rescaling
Suppose we approximate Σ1 by its
diagonal.
Let Λ = diagonal Σ1
−1
contain inverse
one-step forecast variances.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 45
˜yn(h) = S(S Σ†
1S)−1
S Σ†
1ˆyn(h)
˜yn(h) = S(SΛS)−1
SΛˆyn(h)

142. Optimal combination forecasts
Solution 1: OLS
Approximate Σ†
1 by cI.
Solution 2: Rescaling
Suppose we approximate Σ1 by its
diagonal.
Let Λ = diagonal Σ1
−1
contain inverse
one-step forecast variances.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 45
˜yn(h) = S(S Σ†
1S)−1
S Σ†
1ˆyn(h)
˜yn(h) = S(SΛS)−1
SΛˆyn(h)

143. Optimal reconciled forecasts
˜yn(h) = S ˆβn(h) = S(S ΛS)−1
S Λˆyn(h)
Easy to estimate, and places weight where
we have best forecasts.
Ignores covariances.
For large numbers of time series, we need
to do calculation without explicitly forming
S or (SΛS)−1
or SΛ.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 46

144. Optimal reconciled forecasts
˜yn(h) = S ˆβn(h) = S(S ΛS)−1
S Λˆyn(h)
Initial forecasts
Easy to estimate, and places weight where
we have best forecasts.
Ignores covariances.
For large numbers of time series, we need
to do calculation without explicitly forming
S or (SΛS)−1
or SΛ.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 46

145. Optimal reconciled forecasts
˜yn(h) = S ˆβn(h) = S(S ΛS)−1
S Λˆyn(h)
Revised forecasts Initial forecasts
Easy to estimate, and places weight where
we have best forecasts.
Ignores covariances.
For large numbers of time series, we need
to do calculation without explicitly forming
S or (SΛS)−1
or SΛ.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 46

146. Optimal reconciled forecasts
˜yn(h) = S ˆβn(h) = S(S ΛS)−1
S Λˆyn(h)
Revised forecasts Initial forecasts
Easy to estimate, and places weight where
we have best forecasts.
Ignores covariances.
For large numbers of time series, we need
to do calculation without explicitly forming
S or (SΛS)−1
or SΛ.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 46

147. Optimal reconciled forecasts
˜yn(h) = S ˆβn(h) = S(S ΛS)−1
S Λˆyn(h)
Revised forecasts Initial forecasts
Easy to estimate, and places weight where
we have best forecasts.
Ignores covariances.
For large numbers of time series, we need
to do calculation without explicitly forming
S or (SΛS)−1
or SΛ.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 46

148. Outline
1 Examples of big time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Application: Australian labour market
6 Fast computation tricks
7 hts package for R
8 References
Visualising and forecasting big time series data Application: Australian tourism 47

149. Australian tourism
Visualising and forecasting big time series data Application: Australian tourism 48

150. Australian tourism
Visualising and forecasting big time series data Application: Australian tourism 48
Hierarchy:
States (7)
Zones (27)
Regions (82)

151. Australian tourism
Visualising and forecasting big time series data Application: Australian tourism 48
Hierarchy:
States (7)
Zones (27)
Regions (82)
Base forecasts
ETS (exponential
smoothing) models

152. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: Total
Year
Visitornights
1998 2000 2002 2004 2006 2008
600006500070000750008000085000

153. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: NSW
Year
Visitornights
1998 2000 2002 2004 2006 2008
18000220002600030000

154. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: VIC
Year
Visitornights
1998 2000 2002 2004 2006 2008
1000012000140001600018000

155. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: Nth.Coast.NSW
Year
Visitornights
1998 2000 2002 2004 2006 2008
50006000700080009000

156. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: Metro.QLD
Year
Visitornights
1998 2000 2002 2004 2006 2008
800090001100013000

157. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: Sth.WA
Year
Visitornights
1998 2000 2002 2004 2006 2008
400600800100012001400

158. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: X201.Melbourne
Year
Visitornights
1998 2000 2002 2004 2006 2008
40004500500055006000

159. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: X402.Murraylands
Year
Visitornights
1998 2000 2002 2004 2006 2008
0100200300

160. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 49
Domestic tourism forecasts: X809.Daly
Year
Visitornights
1998 2000 2002 2004 2006 2008
020406080100

161. Reconciled forecasts
Visualising and forecasting big time series data Application: Australian tourism 50
Total
2000 2005 2010
650008000095000

162. Reconciled forecasts
Visualising and forecasting big time series data Application: Australian tourism 50
NSW
2000 2005 2010
180002400030000
VIC
2000 2005 2010
100001400018000
QLD
2000 2005 2010
1400020000
Other 2000 2005 2010
1800024000

163. Reconciled forecasts
Visualising and forecasting big time series data Application: Australian tourism 50
Sydney
2000 2005 2010
40007000
OtherNSW
2000 2005 2010
1400022000
Melbourne
2000 2005 2010
40005000
OtherVIC
2000 2005 2010
600012000
GCandBrisbane
2000 2005 2010
60009000
OtherQLD
2000 2005 2010
600012000
Capitalcities
2000 2005 2010
1400020000
Other
2000 2005 2010
55007500

164. Forecast evaluation
Select models using all observations;
Re-estimate models using first 12
observations and generate 1- to
8-step-ahead forecasts;
Increase sample size one observation at a
time, re-estimate models, generate
forecasts until the end of the sample;
In total 24 1-step-ahead, 23
2-steps-ahead, up to 17 8-steps-ahead for
forecast evaluation.
Visualising and forecasting big time series data Application: Australian tourism 51

165. Forecast evaluation
Select models using all observations;
Re-estimate models using first 12
observations and generate 1- to
8-step-ahead forecasts;
Increase sample size one observation at a
time, re-estimate models, generate
forecasts until the end of the sample;
In total 24 1-step-ahead, 23
2-steps-ahead, up to 17 8-steps-ahead for
forecast evaluation.
Visualising and forecasting big time series data Application: Australian tourism 51

166. Forecast evaluation
Select models using all observations;
Re-estimate models using first 12
observations and generate 1- to
8-step-ahead forecasts;
Increase sample size one observation at a
time, re-estimate models, generate
forecasts until the end of the sample;
In total 24 1-step-ahead, 23
2-steps-ahead, up to 17 8-steps-ahead for
forecast evaluation.
Visualising and forecasting big time series data Application: Australian tourism 51

167. Forecast evaluation
Select models using all observations;
Re-estimate models using first 12
observations and generate 1- to
8-step-ahead forecasts;
Increase sample size one observation at a
time, re-estimate models, generate
forecasts until the end of the sample;
In total 24 1-step-ahead, 23
2-steps-ahead, up to 17 8-steps-ahead for
forecast evaluation.
Visualising and forecasting big time series data Application: Australian tourism 51

168. Hierarchy: states, zones, regions
MAPE h = 1 h = 2 h = 4 h = 6 h = 8 Average
Top Level: Australia
Bottom-up 3.79 3.58 4.01 4.55 4.24 4.06
OLS 3.83 3.66 3.88 4.19 4.25 3.94
Scaling (st. dev.) 3.68 3.56 3.97 4.57 4.25 4.04
Level: States
Bottom-up 10.70 10.52 10.85 11.46 11.27 11.03
OLS 11.07 10.58 11.13 11.62 12.21 11.35
Scaling (st. dev.) 10.44 10.17 10.47 10.97 10.98 10.67
Level: Zones
Bottom-up 14.99 14.97 14.98 15.69 15.65 15.32
OLS 15.16 15.06 15.27 15.74 16.15 15.48
Scaling (st. dev.) 14.63 14.62 14.68 15.17 15.25 14.94
Bottom Level: Regions
Bottom-up 33.12 32.54 32.26 33.74 33.96 33.18
OLS 35.89 33.86 34.26 36.06 37.49 35.43
Scaling (st. dev.) 31.68 31.22 31.08 32.41 32.77 31.89
Visualising and forecasting big time series data Application: Australian tourism 52

169. Outline
1 Examples of big time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Application: Australian labour market
6 Fast computation tricks
7 hts package for R
8 References
Visualising and forecasting big time series data Application: Australian labour market 53

170. ANZSCO
Australia and New Zealand Standard
Classification of Occupations
8 major groups
43 sub-major groups
97 minor groups
– 359 unit groups
* 1023 occupations
Example: statistician
2 Professionals
22 Business, Human Resource and Marketing
Professionals
224 Information and Organisation Professionals
2241 Actuaries, Mathematicians and Statisticians
224113 Statistician
Visualising and forecasting big time series data Application: Australian labour market 54

171. ANZSCO
Australia and New Zealand Standard
Classification of Occupations
8 major groups
43 sub-major groups
97 minor groups
– 359 unit groups
* 1023 occupations
Example: statistician
2 Professionals
22 Business, Human Resource and Marketing
Professionals
224 Information and Organisation Professionals
2241 Actuaries, Mathematicians and Statisticians
224113 Statistician
Visualising and forecasting big time series data Application: Australian labour market 54

172. Australian Labour Market data
Visualising and forecasting big time series data Application: Australian labour market 55
Time
Level0
7000900011000
Time
Level1
5001000150020002500
1. Managers
2. Professionals
3. Technicians and trade workers
4. Community and personal services workers
5. Clerical and administrative workers
6. Sales workers
7. Machinery operators and drivers
8. Labourers
Time
Level2
100200300400500600700
Time
Level3
100200300400500600700
Time
Level4
1990 1995 2000 2005 2010
100200300400500

173. Australian Labour Market data
Visualising and forecasting big time series data Application: Australian labour market 55
Time
Level0
7000900011000
Time
Level1
5001000150020002500
1. Managers
2. Professionals
3. Technicians and trade workers
4. Community and personal services workers
5. Clerical and administrative workers
6. Sales workers
7. Machinery operators and drivers
8. Labourers
Time
Level2
100200300400500600700
Time
Level3
100200300400500600700
Time
Level4
1990 1995 2000 2005 2010
100200300400500
Lower three panels
show largest
sub-groups at each
level.

174. Australian Labour Market data
Visualising and forecasting big time series data Application: Australian labour market 55
Time
Level0
7000900011000
Time
Level1
5001000150020002500
1. Managers
2. Professionals
3. Technicians and trade workers
4. Community and personal services workers
5. Clerical and administrative workers
6. Sales workers
7. Machinery operators and drivers
8. Labourers
Time
Level2
100200300400500600700
Time
Level3
100200300400500600700
Time
Level4
1990 1995 2000 2005 2010
100200300400500
Time
Level0
10800112001160012000
Base forecasts
Reconciled forecasts
Time
Level1
680700720740760780800
Time
Level2
140150160170180190200
Time
Level3
140150160170180
Year
Level4
2010 2011 2012 2013 2014 2015
120130140150160

175. Australian Labour Market data
Visualising and forecasting big time series data Application: Australian labour market 55
Time
Level0
7000900011000
Time
Level1
5001000150020002500
1. Managers
2. Professionals
3. Technicians and trade workers
4. Community and personal services workers
5. Clerical and administrative workers
6. Sales workers
7. Machinery operators and drivers
8. Labourers
Time
Level2
100200300400500600700
Time
Level3
100200300400500600700
Time
Level4
1990 1995 2000 2005 2010
100200300400500
Time
Level0
10800112001160012000
Base forecasts
Reconciled forecasts
Time
Level1
680700720740760780800
Time
Level2
140150160170180190200
Time
Level3
140150160170180
Year
Level4
2010 2011 2012 2013 2014 2015
120130140150160
Base forecasts
from auto.arima()
Largest changes
shown for each
level

176. Forecast evaluation (rolling origin)
RMSE h = 1 h = 2 h = 3 h = 4 h = 5 h = 6 h = 7 h = 8 Average
Top level
Bottom-up 74.71 102.02 121.70 131.17 147.08 157.12 169.60 178.93 135.29
OLS 52.20 77.77 101.50 119.03 138.27 150.75 160.04 166.38 120.74
WLS 61.77 86.32 107.26 119.33 137.01 146.88 156.71 162.38 122.21
Level 1
Bottom-up 21.59 27.33 30.81 32.94 35.45 37.10 39.00 40.51 33.09
OLS 21.89 28.55 32.74 35.58 38.82 41.24 43.34 45.49 35.96
WLS 20.58 26.19 29.71 31.84 34.36 35.89 37.53 38.86 31.87
Level 2
Bottom-up 8.78 10.72 11.79 12.42 13.13 13.61 14.14 14.65 12.40
OLS 9.02 11.19 12.34 13.04 13.92 14.56 15.17 15.77 13.13
WLS 8.58 10.48 11.54 12.15 12.88 13.36 13.87 14.36 12.15
Level 3
Bottom-up 5.44 6.57 7.17 7.53 7.94 8.27 8.60 8.89 7.55
OLS 5.55 6.78 7.42 7.81 8.29 8.68 9.04 9.37 7.87
WLS 5.35 6.46 7.06 7.42 7.84 8.17 8.48 8.76 7.44
Bottom Level
Bottom-up 2.35 2.79 3.02 3.15 3.29 3.42 3.54 3.65 3.15
OLS 2.40 2.86 3.10 3.24 3.41 3.55 3.68 3.80 3.25
WLS 2.34 2.77 2.99 3.12 3.27 3.40 3.52 3.63 3.13
Visualising and forecasting big time series data Application: Australian labour market 56

177. Outline
1 Examples of big time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Application: Australian labour market
6 Fast computation tricks
7 hts package for R
8 References
Visualising and forecasting big time series data Fast computation tricks 57

178. Fast computation: hierarchical data
Total
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =












Yt
YA,t
YB,t
YC,t
YAX,t
YAY,t
YAZ,t
YBX,t
YBY,t
YBZ,t
YCX,t
YCY,t
YCZ,t












=












1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 0 0 0 1 1 1
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1












S







YAX,t
YAY,t
YAZ,t
YBX,t
YBY,t
YBZ,t
YCX,t
YCY,t
YCZ,t







Bt
Visualising and forecasting big time series data Fast computation tricks 58
yt = SBt

179. Fast computation: hierarchical data
Total
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
yt =












Yt
YA,t
YAX,t
YAY,t
YAZ,t
YB,t
YBX,t
YBY,t
YBZ,t
YC,t
YCX,t
YCY,t
YCZ,t












=












1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 1 1
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1












S







YAX,t
YAY,t
YAZ,t
YBX,t
YBY,t
YBZ,t
YCX,t
YCY,t
YCZ,t







Bt
Visualising and forecasting big time series data Fast computation tricks 59
yt = SBt

180. Fast computation: hierarchies
Think of the hierarchy as a tree of trees:
Total
T1 T2
. . . TK
Then the summing matrix contains k smaller summing
matrices:
S =






1n1
1n2
· · · 1nK
S1 0 · · · 0
0 S2 · · · 0
...
...
...
...
0 0 · · · SK






where 1n is an n-vector of ones and tree Ti has ni
terminal nodes.
Visualising and forecasting big time series data Fast computation tricks 60

181. Fast computation: hierarchies
Think of the hierarchy as a tree of trees:
Total
T1 T2
. . . TK
Then the summing matrix contains k smaller summing
matrices:
S =






1n1
1n2
· · · 1nK
S1 0 · · · 0
0 S2 · · · 0
...
...
...
...
0 0 · · · SK






where 1n is an n-vector of ones and tree Ti has ni
terminal nodes.
Visualising and forecasting big time series data Fast computation tricks 60

182. Fast computation: hierarchies
SΛS =




S1Λ1S1 0 · · · 0
0 S2Λ2S2 · · · 0
...
... ... ...
0 0 · · · SKΛKSK



+λ0 Jn
λ0 is the top left element of Λ;
Λk is a block of Λ, corresponding to tree Tk;
Jn is a matrix of ones;
n = k nk.
Now apply the Sherman-Morrison formula . . .
Visualising and forecasting big time series data Fast computation tricks 61

183. Fast computation: hierarchies
SΛS =




S1Λ1S1 0 · · · 0
0 S2Λ2S2 · · · 0
...
... ... ...
0 0 · · · SKΛKSK



+λ0 Jn
λ0 is the top left element of Λ;
Λk is a block of Λ, corresponding to tree Tk;
Jn is a matrix of ones;
n = k nk.
Now apply the Sherman-Morrison formula . . .
Visualising and forecasting big time series data Fast computation tricks 61

184. Fast computation: hierarchies
(SΛS)−1
=





(S1Λ1S1)−1
0 · · · 0
0 (S2Λ2S2)−1
· · · 0
...
...
...
...
0 0 · · · (SKΛKSK)−1





−cS0
S0 can be partitioned into K2
blocks, with the (k, )
block (of dimension nk × n ) being
(SkΛkSk)−1
Jnk,n (S Λ S )−1
Jnk,n is a nk × n matrix of ones.
c−1
= λ−1
0 +
k
1nk
(SkΛkSk)−1
1nk
.
Each SkΛkSk can be inverted similarly.
SΛy can also be computed recursively.
Visualising and forecasting big time series data Fast computation tricks 62

185. Fast computation: hierarchies
(SΛS)−1
=





(S1Λ1S1)−1
0 · · · 0
0 (S2Λ2S2)−1
· · · 0
...
...
...
...
0 0 · · · (SKΛKSK)−1





−cS0
S0 can be partitioned into K2
blocks, with the (k, )
block (of dimension nk × n ) being
(SkΛkSk)−1
Jnk,n (S Λ S )−1
Jnk,n is a nk × n matrix of ones.
c−1
= λ−1
0 +
k
1nk
(SkΛkSk)−1
1nk
.
Each SkΛkSk can be inverted similarly.
SΛy can also be computed recursively.
Visualising and forecasting big time series data Fast computation tricks 62
The recursive calculations can be
done in such a way that we never
store any of the large matrices
involved.

186. Fast computation
When the time series are not strictly
hierarchical and have more than two grouping
variables:
Use sparse matrix storage and arithmetic.
Use iterative approximation for inverting
large sparse matrices.
Paige & Saunders (1982)
ACM Trans. Math. Software
Visualising and forecasting big time series data Fast computation tricks 63

187. Fast computation
When the time series are not strictly
hierarchical and have more than two grouping
variables:
Use sparse matrix storage and arithmetic.
Use iterative approximation for inverting
large sparse matrices.
Paige & Saunders (1982)
ACM Trans. Math. Software
Visualising and forecasting big time series data Fast computation tricks 63

188. Fast computation
When the time series are not strictly
hierarchical and have more than two grouping
variables:
Use sparse matrix storage and arithmetic.
Use iterative approximation for inverting
large sparse matrices.
Paige & Saunders (1982)
ACM Trans. Math. Software
Visualising and forecasting big time series data Fast computation tricks 63

189. Outline
1 Examples of big time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Application: Australian labour market
6 Fast computation tricks
7 hts package for R
8 References
Visualising and forecasting big time series data hts package for R 64

190. hts package for R
Visualising and forecasting big time series data hts package for R 65
hts: Hierarchical and grouped time series
Methods for analysing and forecasting hierarchical and grouped
time series
Version: 4.3
Depends: forecast (≥ 5.0)
Imports: SparseM, parallel, utils
Published: 2014-06-10
Author: Rob J Hyndman, Earo Wang and Alan Lee
Maintainer: Rob J Hyndman <Rob.Hyndman at monash.edu>
BugReports: https://github.com/robjhyndman/hts/issues
License: GPL (≥ 2)

191. Example using R
library(hts)
# bts is a matrix containing the bottom level time series
# nodes describes the hierarchical structure
y <- hts(bts, nodes=list(2, c(3,2)))
Visualising and forecasting big time series data hts package for R 66

192. Example using R
library(hts)
# bts is a matrix containing the bottom level time series
# nodes describes the hierarchical structure
y <- hts(bts, nodes=list(2, c(3,2)))
Visualising and forecasting big time series data hts package for R 66
Total
A
AX AY AZ
B
BX BY

193. Example using R
library(hts)
# bts is a matrix containing the bottom level time series
# nodes describes the hierarchical structure
y <- hts(bts, nodes=list(2, c(3,2)))
# Forecast 10-step-ahead using WLS combination method
# ETS used for each series by default
fc <- forecast(y, h=10)
Visualising and forecasting big time series data hts package for R 67

194. forecast.gts function
Usage
forecast(object, h,
method = c("comb", "bu", "mo", "tdgsf", "tdgsa", "tdfp"),
fmethod = c("ets", "rw", "arima"),
weights = c("sd", "none", "nseries"),
positive = FALSE,
parallel = FALSE, num.cores = 2, ...)
Arguments
object Hierarchical time series object of class gts.
h Forecast horizon
method Method for distributing forecasts within the hierarchy.
fmethod Forecasting method to use
positive If TRUE, forecasts are forced to be strictly positive
weights Weights used for "optimal combination" method. When
weights = "sd", it takes account of the standard deviation of
forecasts.
parallel If TRUE, allow parallel processing
num.cores If parallel = TRUE, specify how many cores are going to be
used
Visualising and forecasting big time series data hts package for R 68

195. Outline
1 Examples of big time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Application: Australian labour market
6 Fast computation tricks
7 hts package for R
8 References
Visualising and forecasting big time series data References 69

196. References
RJ Hyndman, RA Ahmed, G Athanasopoulos, and
HL Shang (2011). “Optimal combination forecasts for
hierarchical time series”. Computational statistics &
data analysis 55(9), 2579–2589.
RJ Hyndman, AJ Lee, and E Wang (2014). Fast
computation of reconciled forecasts for hierarchical
and grouped time series. Working paper 17/14.
Department of Econometrics & Business Statistics,
Monash University
RJ Hyndman, AJ Lee, and E Wang (2014). hts:
Hierarchical and grouped time series.
cran.r-project.org/package=hts.
RJ Hyndman and G Athanasopoulos (2014).
Forecasting: principles and practice. OTexts.
OTexts.org/fpp/.
Visualising and forecasting big time series data References 70

197. References
RJ Hyndman, RA Ahmed, G Athanasopoulos, and
HL Shang (2011). “Optimal combination forecasts for
hierarchical time series”. Computational statistics &
data analysis 55(9), 2579–2589.
RJ Hyndman, AJ Lee, and E Wang (2014). Fast
computation of reconciled forecasts for hierarchical
and grouped time series. Working paper 17/14.
Department of Econometrics & Business Statistics,
Monash University
RJ Hyndman, AJ Lee, and E Wang (2014). hts:
Hierarchical and grouped time series.
cran.r-project.org/package=hts.
RJ Hyndman and G Athanasopoulos (2014).
Forecasting: principles and practice. OTexts.
OTexts.org/fpp/.
Visualising and forecasting big time series data References 70
¯ Papers and R code:
robjhyndman.com
¯ Email: Rob.Hyndman@monash.edu