Talk given at the ACEMS big data workshop in Brisbane, 23 February 2015

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

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

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

4. 1. Australian tourism demand
Visualising and forecasting big time series data Examples of biggish 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 biggish 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 biggish time series 4

7. 3. Spectacle sales
Visualising and forecasting big time series data Examples of biggish time series 5
Monthly UK 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 1 million bottom-level series

8. 3. Spectacle sales
Visualising and forecasting big time series data Examples of biggish time series 5
Monthly UK 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 1 million bottom-level series

9. 3. Spectacle sales
Visualising and forecasting big time series data Examples of biggish time series 5
Monthly UK 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 1 million bottom-level series

10. 3. Spectacle sales
Visualising and forecasting big time series data Examples of biggish time series 5
Monthly UK 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 1 million bottom-level series

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

12. 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 7

13. 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 8

14. Kite diagrams: Victorian tourism
Visualising and forecasting big time series data Time series visualisation 8

15. Kite diagrams: Victorian tourism
Visualising and forecasting big time series data Time series visualisation 8

16. 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 8

17. 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 9

18. 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 10

19. 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 10

20. Seasonal stacked bar chart: VIC
Visualising and forecasting big time series data Time series visualisation 11

21. Corrgram of remainder
Visualising and forecasting big time series data Time series visualisation 12
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.

22. Corrgram of remainder: VIC
Visualising and forecasting big time series data Time series visualisation 13
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
BEEHolBEFOthBEEOthBDEOthBEBOthBEABusBEFBusBDCOthBACHolBEBBusBEAVisBBAHolBDEHolBABOthBAAVisBAAHolBDCHolBBABusBCBHolBEGBusBDDVisBABVisBDAVisBEAOthBDFHolBEEBusBAAOthBACOthBDAOthBDEBusBCBOthBACBusBEBVisBACVisBCAOthBEFVisBCBVisBEDHolBEGOthBDBHolBABBusBEBHolBDFBusBECHolBCAHolBDBOthBEAHolBDCBusBECVisBDBVisBCCHolBBAVisBABHolBBAOthBCCOthBCBBusBCCVisBEGVisBDDHolBECOthBDCVisBAABusBCCBusBECBusBCAVisBDFVisBEGHolBDDOthBEDOthBEDVisBDDBusBDEVisBEFHolBEEVisBDBBusBDABusBDAHolBCABusBDFOthBEDBus
BEEHolBEFOthBEEOthBDEOthBEBOthBEABusBEFBusBDCOthBACHolBEBBusBEAVisBBAHolBDEHolBABOthBAAVisBAAHolBDCHolBBABusBCBHolBEGBusBDDVisBABVisBDAVisBEAOthBDFHolBEEBusBAAOthBACOthBDAOthBDEBusBCBOthBACBusBEBVisBACVisBCAOthBEFVisBCBVisBEDHolBEGOthBDBHolBABBusBEBHolBDFBusBECHolBCAHolBDBOthBEAHolBDCBusBECVisBDBVisBCCHolBBAVisBABHolBBAOthBCCOthBCBBusBCCVisBEGVisBDDHolBECOthBDCVisBAABusBCCBusBECBusBCAVisBDFVisBEGHolBDDOthBEDOthBEDVisBDDBusBDEVisBEFHolBEEVisBDBBusBDABusBDAHolBCABusBDFOthBEDBus

23. Corrgram of remainder: TAS
Visualising and forecasting big time series data Time series visualisation 14
−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

24. Feature analysis
Summarize each time series with a feature
vector:
strength of trend
lumpiness (variance of annual variances of
remainder)
strength of seasonality
size of seasonal peak
size of seasonal trough
ACF1
linearity of trend
curvature of trend
spectral entropy
Do PCA on feature matrix
Visualising and forecasting big time series data Time series visualisation 15

25. Feature analysis
Visualising and forecasting big time series data Time series visualisation 16
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season
spike
entropy
fo.acf
peak
trough
linearity
curvature
−2
0
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−2 0 2
standardized PC1 (43.2% explained var.)
standardizedPC2(18.6%explainedvar.)

26. Feature analysis
Visualising and forecasting big time series data Time series visualisation 16
0
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600
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BEGBusBCCBusCCAOthCACOth
2000 2005 2010
Time
value

27. Feature analysis
Visualising and forecasting big time series data Time series visualisation 16
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1200
1600
250
500
750
1000
500
1000
1500
2000
1000
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3000
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ADAHolBDCHolBBAHolACAHol
2000 2005 2010
Time
value

28. Feature analysis
Visualising and forecasting big time series data Time series visualisation 16
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NSW NT QLD SA
TAS VIC WA
−2
0
2
4
6
−2
0
2
4
6
−7.5−5.0−2.5 0.0 2.5 5.0−7.5−5.0−2.5 0.0 2.5 5.0−7.5−5.0−2.5 0.0 2.5 5.0
PC1
PC2

29. Feature analysis
Visualising and forecasting big time series data Time series visualisation 16
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Bus Hol
Oth Vis
−2
0
2
4
6
−2
0
2
4
6
−7.5 −5.0 −2.5 0.0 2.5 5.0−7.5 −5.0 −2.5 0.0 2.5 5.0
PC1
PC2

30. Outline
1 Examples of biggish time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Fast computation tricks
6 hts package for R
7 References
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 17

31. 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
Tourism by state and region
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 18

32. 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
Tourism by state and region
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 18

33. 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
Tourism by state and region
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 18

34. Hierarchical time series
Total
A B C
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 19
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.

35. Hierarchical time series
Total
A B C
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 19
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.

36. Hierarchical time series
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 19
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.

37. Hierarchical time series
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 19
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.

38. Hierarchical time series
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 19
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.

39. Hierarchical time series
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 19
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.

40. Hierarchical time series
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 20

41. Hierarchical time series
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 20

42. Hierarchical time series
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 20
yt = SBt

43. 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.)
Reconciled forecasts are 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
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 21

44. 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.)
Reconciled forecasts are 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
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 21

45. 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.)
Reconciled forecasts are 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
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 21

46. 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.)
Reconciled forecasts are 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
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 21

47. 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.)
Reconciled forecasts are 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
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 21

48. 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).
Revised forecasts are unbiased iff SPS = S.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 22

49. 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).
Revised forecasts are unbiased iff SPS = S.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 22

50. 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).
Revised forecasts are unbiased iff SPS = S.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 22

51. 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).
Revised forecasts are unbiased iff SPS = S.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 22

52. 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).
Revised forecasts are unbiased iff SPS = S.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 22

53. 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).
Revised forecasts are unbiased iff SPS = S.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 22

54. 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).
Revised forecasts are unbiased iff SPS = S.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 22

55. 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 .
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 23

56. 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 .
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 23

57. 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 .
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 23

58. 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.
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 24

59. 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.
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 24

60. 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.
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 24

61. 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.
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 24

62. 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.
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 24

63. 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.
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)
Revised forecasts Base forecasts
Equivalent to GLS estimate of regression
ˆyn(h) = Sβn(h) + εh where ε ∼ N(0, Σh).
Problem: Σh hard to estimate.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 24

64. Optimal combination forecasts
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 25
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

65. Optimal combination forecasts
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 25
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

66. Optimal combination forecasts
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 25
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

67. Optimal combination forecasts
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 25
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

68. Optimal combination forecasts
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 25
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

69. Optimal combination forecasts
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 25
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

70. Optimal combination forecasts
Revised forecasts Base forecasts
Solution 2: WLS
Suppose we approximate Σ1 by its
diagonal.
Easy to estimate, and places weight where
we have best forecasts.
Empirically, it gives better forecasts than
other available methods.
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

71. Optimal combination forecasts
Revised forecasts Base forecasts
Solution 2: WLS
Suppose we approximate Σ1 by its
diagonal.
Easy to estimate, and places weight where
we have best forecasts.
Empirically, it gives better forecasts than
other available methods.
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

72. Optimal combination forecasts
Revised forecasts Base forecasts
Solution 2: WLS
Suppose we approximate Σ1 by its
diagonal.
Easy to estimate, and places weight where
we have best forecasts.
Empirically, it gives better forecasts than
other available methods.
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

73. Optimal combination forecasts
Revised forecasts Base forecasts
Solution 2: WLS
Suppose we approximate Σ1 by its
diagonal.
Easy to estimate, and places weight where
we have best forecasts.
Empirically, it gives better forecasts than
other available methods.
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

74. Optimal combination forecasts
Revised forecasts Base forecasts
Solution 2: WLS
Suppose we approximate Σ1 by its
diagonal.
Easy to estimate, and places weight where
we have best forecasts.
Empirically, it gives better forecasts than
other available methods.
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

75. Optimal combination forecasts
Revised forecasts Base forecasts
Solution 2: WLS
Suppose we approximate Σ1 by its
diagonal.
Easy to estimate, and places weight where
we have best forecasts.
Empirically, it gives better forecasts than
other available methods.
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 26
˜yn(h) = S(S Σ†
hS)−1
S Σ†
hˆyn(h)

76. Challenges
Computational difficulties in big
hierarchies due to size of the S matrix and
singular behavior of (S ΛS).
Loss of information in ignoring covariance
matrix in computing point forecasts.
Still need to estimate covariance matrix to
produce prediction intervals.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)

77. Challenges
Computational difficulties in big
hierarchies due to size of the S matrix and
singular behavior of (S ΛS).
Loss of information in ignoring covariance
matrix in computing point forecasts.
Still need to estimate covariance matrix to
produce prediction intervals.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)

78. Challenges
Computational difficulties in big
hierarchies due to size of the S matrix and
singular behavior of (S ΛS).
Loss of information in ignoring covariance
matrix in computing point forecasts.
Still need to estimate covariance matrix to
produce prediction intervals.
Visualising and forecasting big time series data BLUF: Best Linear Unbiased Forecasts 27
˜yn(h) = S(S ΛS)−1
S Λˆyn(h)

79. Outline
1 Examples of biggish time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Fast computation tricks
6 hts package for R
7 References
Visualising and forecasting big time series data Application: Australian tourism 28

80. Australian tourism
Visualising and forecasting big time series data Application: Australian tourism 29

81. Australian tourism
Visualising and forecasting big time series data Application: Australian tourism 29
Hierarchy:
States (7)
Zones (27)
Regions (82)

82. Australian tourism
Visualising and forecasting big time series data Application: Australian tourism 29
Hierarchy:
States (7)
Zones (27)
Regions (82)
Base forecasts
ETS (exponential
smoothing) models

83. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: Total
Year
Visitornights
1998 2000 2002 2004 2006 2008
600006500070000750008000085000

84. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: NSW
Year
Visitornights
1998 2000 2002 2004 2006 2008
18000220002600030000

85. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: VIC
Year
Visitornights
1998 2000 2002 2004 2006 2008
1000012000140001600018000

86. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: Nth.Coast.NSW
Year
Visitornights
1998 2000 2002 2004 2006 2008
50006000700080009000

87. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: Metro.QLD
Year
Visitornights
1998 2000 2002 2004 2006 2008
800090001100013000

88. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: Sth.WA
Year
Visitornights
1998 2000 2002 2004 2006 2008
400600800100012001400

89. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: X201.Melbourne
Year
Visitornights
1998 2000 2002 2004 2006 2008
40004500500055006000

90. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: X402.Murraylands
Year
Visitornights
1998 2000 2002 2004 2006 2008
0100200300

91. Base forecasts
Visualising and forecasting big time series data Application: Australian tourism 30
Domestic tourism forecasts: X809.Daly
Year
Visitornights
1998 2000 2002 2004 2006 2008
020406080100

92. Reconciled forecasts
Visualising and forecasting big time series data Application: Australian tourism 31
Total
2000 2005 2010
650008000095000

93. Reconciled forecasts
Visualising and forecasting big time series data Application: Australian tourism 31
NSW
2000 2005 2010
180002400030000
VIC
2000 2005 2010
100001400018000
QLD
2000 2005 2010
1400020000
Other 2000 2005 2010
1800024000

94. Reconciled forecasts
Visualising and forecasting big time series data Application: Australian tourism 31
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

95. 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 32

96. 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 32

97. 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 32

98. 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 32

99. 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
WLS 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
WLS 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
WLS 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
WLS 31.68 31.22 31.08 32.41 32.77 31.89
Visualising and forecasting big time series data Application: Australian tourism 33

100. Outline
1 Examples of biggish time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Fast computation tricks
6 hts package for R
7 References
Visualising and forecasting big time series data Fast computation tricks 34

101. 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 35
yt = SBt

102. 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 36
yt = SBt

103. 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 37

104. 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 37

105. 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 38

106. 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 38

107. 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 39

108. 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 39
The recursive calculations can be
done in such a way that we never
store any of the large matrices
involved.

109. Fast computation
A similar algorithm has been developed for
grouped time series with two groups.
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 40

110. Fast computation
A similar algorithm has been developed for
grouped time series with two groups.
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 40

111. Fast computation
A similar algorithm has been developed for
grouped time series with two groups.
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 40

112. Outline
1 Examples of biggish time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Fast computation tricks
6 hts package for R
7 References
Visualising and forecasting big time series data hts package for R 41

113. hts package for R
Visualising and forecasting big time series data hts package for R 42
hts: Hierarchical and grouped time series
Methods for analysing and forecasting hierarchical and grouped
time series
Version: 4.5
Depends: forecast (≥ 5.0), SparseM
Imports: parallel, utils
Published: 2014-12-09
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)

114. 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 43

115. 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 43
Total
A
AX AY AZ
B
BX BY

116. 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 44

117. 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 45

118. Outline
1 Examples of biggish time series
2 Time series visualisation
3 BLUF: Best Linear Unbiased Forecasts
4 Application: Australian tourism
5 Fast computation tricks
6 hts package for R
7 References
Visualising and forecasting big time series data References 46

119. 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 47

120. 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 47
¯ Papers and R code:
robjhyndman.com
¯ Email: Rob.Hyndman@monash.edu