2. Visualisation of
big time series
data
Visualisation of big time series data 1
Rob J Hyndman
with Earo Wang, Nikolay Laptev
Yanfei Kang, Kate Smith-Miles
3. Visualisation of
big time series
data
Visualisation of big time series data 1
Rob J Hyndman
with Earo Wang, Nikolay Laptev
Yanfei Kang, Kate Smith-Miles
4. Visualisation of
big time series
data
Visualisation of big time series data 1
Rob J Hyndman
with Earo Wang, Nikolay Laptev
Yanfei Kang, Kate Smith-Miles
5. Visualisation of
big time series
data
Visualisation of big time series data 1
Rob J Hyndman
with Earo Wang, Nikolay Laptev
Yanfei Kang, Kate Smith-Miles
6. Visualisation of
big time series
data
Visualisation of big time series data 1
Rob J Hyndman
with Earo Wang, Nikolay Laptev
Yanfei Kang, Kate Smith-Miles
7. Visualisation of
big time series
data
Visualisation of big time series data 1
Rob J Hyndman
with Earo Wang, Nikolay Laptev
Yanfei Kang, Kate Smith-Miles
8. Outline
1 The problem
2 Australian tourism demand
3 M3 competition data
4 Yahoo web traffic
5 What next?
Visualisation of big time series data The problem 2
9. Spectacle sales
Visualisation of big time series data The problem 3
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 disaggregated series
10. Fulcher collection
www.comp-engine.org/timeseries
38,190 time series from many sources
Over 20,000 real series from meterology,
medicine, audio, astrophysics, finance, etc.
Over 10,000 simulated series from various
chaotic and stochastic models.
Visualisation of big time series data The problem 4
11. Fulcher collection
www.comp-engine.org/timeseries
38,190 time series from many sources
Over 20,000 real series from meterology,
medicine, audio, astrophysics, finance, etc.
Over 10,000 simulated series from various
chaotic and stochastic models.
Visualisation of big time series data The problem 4
14. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
15. How to plot lots of time series?
Visualisation of big time series data The problem 7
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0.00.20.40.60.81.0
Time
16. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
17. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
18. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
19. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
20. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
21. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
22. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
23. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
24. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
25. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
26. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
27. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
28. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
29. How to plot lots of time series?
Visualisation of big time series data The problem 7
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Time
30. How to plot lots of time series?
Visualisation of big time series data The problem 8
31. How to plot lots of time series?
Visualisation of big time series data The problem 8
32. How to plot lots of time series?
Visualisation of big time series data The problem 8
33. How to plot lots of time series?
Visualisation of big time series data The problem 8
34. How to plot lots of time series?
Visualisation of big time series data The problem 8
35. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Visualisation of big time series data The problem 9
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).
36. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Visualisation of big time series data The problem 9
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).
37. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Visualisation of big time series data The problem 9
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).
38. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Visualisation of big time series data The problem 9
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).
39. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Visualisation of big time series data The problem 9
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).
40. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Visualisation of big time series data The problem 9
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).
41. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Visualisation of big time series data The problem 9
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).
42. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Visualisation of big time series data The problem 9
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).
43. Outline
1 The problem
2 Australian tourism demand
3 M3 competition data
4 Yahoo web traffic
5 What next?
Visualisation of big time series data Australian tourism demand 10
45. Australian tourism demand
Visualisation of big time series data Australian tourism demand 11
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 disaggregated series
46. Domestic tourism demand: VictoriaBAAHolBABHol
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
Visualisation of big time series data Australian tourism demand 12
47. An STL decomposition
Tourism demand for holidays in Peninsula
Yt = St + Tt + Rt St is periodic with mean 0
5.06.07.0
data
−0.50.5
seasonal
5.86.16.4
trend
−0.40.0
2000 2005 2010
remainder
timeVisualisation of big time series data Australian tourism demand 13
48. 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
Visualisation of big time series data Australian tourism demand 14
49. Seasonal stacked bar chart: VIC
Visualisation of big time series data Australian tourism demand 15
50. 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
Visualisation of big time series data Australian tourism demand 15
51. Trend analysis
Linearity: the long-term direction and
strength of trend.
Curvature: the “changing direction” of trend.
Estimate by regression:
Tt = ˆβ0 + ˆβ1φ1(t) + ˆβ2φ2(t) + et
where φk(t) is a kth-degree orthogonal
polynomial in time t.
To separate the linearity ( ˆβ1) and curvature
( ˆβ2).
Visualisation of big time series data Australian tourism demand 16
52. Trend analysis
Visualisation of big time series data Australian tourism demand 17
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
HolidayVFRBusinessOther
BAA BAB BAC BBABCABCBBCCBDABDBBDCBDDBDEBDF BEA BEBBECBEDBEE BEFBEG
Regions
TrendLinearity
Direction
−
+
54. Corrgram of remainder
Visualisation of big time series data Australian tourism demand 18
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
BEEHolBEFOthBEEOthBDEOthBEBOthBEABusBEFBusBDCOthBACHolBEBBusBEAVisBBAHolBDEHolBABOthBAAVisBAAHolBDCHolBBABusBCBHolBEGBusBDDVisBABVisBDAVisBEAOthBDFHolBEEBusBAAOthBACOthBDAOthBDEBusBCBOthBACBusBEBVisBACVisBCAOthBEFVisBCBVisBEDHolBEGOthBDBHolBABBusBEBHolBDFBusBECHolBCAHolBDBOthBEAHolBDCBusBECVisBDBVisBCCHolBBAVisBABHolBBAOthBCCOthBCBBusBCCVisBEGVisBDDHolBECOthBDCVisBAABusBCCBusBECBusBCAVisBDFVisBEGHolBDDOthBEDOthBEDVisBDDBusBDEVisBEFHolBEEVisBDBBusBDABusBDAHolBCABusBDFOthBEDBus
BEEHolBEFOthBEEOthBDEOthBEBOthBEABusBEFBusBDCOthBACHolBEBBusBEAVisBBAHolBDEHolBABOthBAAVisBAAHolBDCHolBBABusBCBHolBEGBusBDDVisBABVisBDAVisBEAOthBDFHolBEEBusBAAOthBACOthBDAOthBDEBusBCBOthBACBusBEBVisBACVisBCAOthBEFVisBCBVisBEDHolBEGOthBDBHolBABBusBEBHolBDFBusBECHolBCAHolBDBOthBEAHolBDCBusBECVisBDBVisBCCHolBBAVisBABHolBBAOthBCCOthBCBBusBCCVisBEGVisBDDHolBECOthBDCVisBAABusBCCBusBECBusBCAVisBDFVisBEGHolBDDOthBEDOthBEDVisBDDBusBDEVisBEFHolBEEVisBDBBusBDABusBDAHolBCABusBDFOthBEDBus
55. Corrgram of remainder
Visualisation of big time series data Australian tourism demand 18
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
BEEHolBEFOthBEEOthBDEOthBEBOthBEABusBEFBusBDCOthBACHolBEBBusBEAVisBBAHolBDEHolBABOthBAAVisBAAHolBDCHolBBABusBCBHolBEGBusBDDVisBABVisBDAVisBEAOthBDFHolBEEBusBAAOthBACOthBDAOthBDEBusBCBOthBACBusBEBVisBACVisBCAOthBEFVisBCBVisBEDHolBEGOthBDBHolBABBusBEBHolBDFBusBECHolBCAHolBDBOthBEAHolBDCBusBECVisBDBVisBCCHolBBAVisBABHolBBAOthBCCOthBCBBusBCCVisBEGVisBDDHolBECOthBDCVisBAABusBCCBusBECBusBCAVisBDFVisBEGHolBDDOthBEDOthBEDVisBDDBusBDEVisBEFHolBEEVisBDBBusBDABusBDAHolBCABusBDFOthBEDBus
BEEHolBEFOthBEEOthBDEOthBEBOthBEABusBEFBusBDCOthBACHolBEBBusBEAVisBBAHolBDEHolBABOthBAAVisBAAHolBDCHolBBABusBCBHolBEGBusBDDVisBABVisBDAVisBEAOthBDFHolBEEBusBAAOthBACOthBDAOthBDEBusBCBOthBACBusBEBVisBACVisBCAOthBEFVisBCBVisBEDHolBEGOthBDBHolBABBusBEBHolBDFBusBECHolBCAHolBDBOthBEAHolBDCBusBECVisBDBVisBCCHolBBAVisBABHolBBAOthBCCOthBCBBusBCCVisBEGVisBDDHolBECOthBDCVisBAABusBCCBusBECBusBCAVisBDFVisBEGHolBDDOthBEDOthBEDVisBDDBusBDEVisBEFHolBEEVisBDBBusBDABusBDAHolBCABusBDFOthBEDBus
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.
56. Corrgram of remainder
Visualisation of big time series data Australian tourism demand 18
−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
57. Corrgram of remainder: TAS
Visualisation of big time series data Australian tourism demand 19
−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
58. Outline
1 The problem
2 Australian tourism demand
3 M3 competition data
4 Yahoo web traffic
5 What next?
Visualisation of big time series data M3 competition data 20
61. M3 forecasting competition
“The M3-Competition is a final attempt by the authors to
settle the accuracy issue of various time series methods. . .
The extension involves the inclusion of more methods/
researchers (in particular in the areas of neural networks
and expert systems) and more series.”
Makridakis & Hibon, IJF 2000
3003 series
All data from business, demography, finance and
economics.
Series length between 14 and 126.
Either non-seasonal, monthly or quarterly.
All time series positive.
Visualisation of big time series data M3 competition data 22
63. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Visualisation of big time series data M3 competition data 24
64. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Visualisation of big time series data M3 competition data 24
65. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Visualisation of big time series data M3 competition data 24
66. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Visualisation of big time series data M3 competition data 24
67. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Visualisation of big time series data M3 competition data 24
68. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Visualisation of big time series data M3 competition data 24
69. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Visualisation of big time series data M3 competition data 24
70. Candidate features
Visualisation of big time series data M3 competition data 25
Seasonality
N0001
1976 1978 1980 1982 1984 1986 1988
100030005000
N1502
1978 1980 1982 1984 1986
01000020000
N3003
1984 1986 1988 1990 1992
2000600010000
71. Candidate features
Visualisation of big time series data M3 competition data 25
Trend
N0001
1976 1978 1980 1982 1984 1986 1988
200040006000
N1502
1982 1984 1986 1988 1990 1992
30005000
N3003
1975 1980 1985
100040007000
72. Candidate features
Visualisation of big time series data M3 competition data 25
ACF1
N0001
1987 1988 1989 1990
580060006200
N1502
1987 1988 1989 1990 1991
300050007000
N3003
1984 1986 1988 1990 1992
700080009000
73. Candidate features
Visualisation of big time series data M3 competition data 25
Spectral entropy
N0001
1964 1966 1968 1970 1972 1974
250040005500
N1502
1986 1988 1990 1992
30004500
N3003
1976 1978 1980 1982 1984 1986 1988
200024002800
74. Candidate features
Visualisation of big time series data M3 competition data 25
Box Cox
N0005
1976 1978 1980 1982 1984 1986 1988
45006000
N2269
1984 1986 1988 1990 1992
420048005400
N3003
0 10 20 30 40 50 60
350045005500
75. Candidate features
Visualisation of big time series data M3 competition data 26
SpecEntr
0.0 0.4 0.8 2 6 10 0.0 0.4 0.8
0.50.9
0.00.6
Trend
Season
0.00.6
28
Freq
ACF
−0.40.6
0.5 0.7 0.9
0.00.6
0.0 0.4 0.8 −0.4 0.2 0.8
Lambda
76. Dimension reduction for time series
Visualisation of big time series data M3 competition data 27
q
77. Dimension reduction for time series
Visualisation of big time series data M3 competition data 27
q
SpecEntr
0.0 0.4 0.8 2 6 10 0.0 0.4 0.8
0.50.9
0.00.6
Trend
Season
0.00.6
28
Freq
ACF
−0.40.6
0.5 0.7 0.9
0.00.6
0.0 0.4 0.8 −0.4 0.2 0.8
Lambda
Feature
calculation
86. Predictability
Three general forecasting methods:
Theta method Best overall in 2000 M3
competition
ETS Exponential smoothing state
space models
STL-AR AR model applied to seasonally
adjusted series from STL, and
seasonal component forecast
using the seasonal naive method.
Compute minimum MASE from all three methods
Visualisation of big time series data M3 competition data 29
87. Predictability
Three general forecasting methods:
Theta method Best overall in 2000 M3
competition
ETS Exponential smoothing state
space models
STL-AR AR model applied to seasonally
adjusted series from STL, and
seasonal component forecast
using the seasonal naive method.
Compute minimum MASE from all three methods
Visualisation of big time series data M3 competition data 29