1. Visualisation of big time series data Visualisation of big time series data 1 Rob J Hyndman 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 trafc 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, nance, 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, nance, etc. Over 10,000 simulated series from various chaotic and stochastic models. Visualisation of big time series data The problem 4 12. FRED: research.stlouisfed.org/fred2/ Visualisation of big time series data The problem 5 13. Quandl: www.quandl.com Visualisation of big time series data The problem 6 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 0.0 0.2 0.4 0.6 0.8 1.0 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 trafc 5 What next? Visualisation of big time series data Australian tourism demand 10 44. Australian tourism demand Visualisation of big time series data Australian tourism demand 11 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 = 0 + 11(t) + 22(t) + et where k(t) is a kth-degree orthogonal polynomial in time t. To separate the linearity ( 1) and curvature ( 2). 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 + 53. Trend analysis Visualisation of big time series data Australian tourism demand 17 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 rst 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 trafc 5 What next? Visualisation of big time series data M3 competition data 20 59. M3 forecasting competition Visualisation of big time series data M3 competition data 21 60. M3 forecasting competition Visualisation of big time series data M3 competition data 21 61. M3 forecasting competition The M3-Competition is a nal 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, nance 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 62. M3 forecasting competition Visualisation of big time series data M3 competition data 23 q 63. Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 Var(Rt) Var(YtTt) Strength of trend: 1 Var(Rt) Var(YtSt) Spectral entropy: H = fy() log fy()d, where fy() 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 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(YtTt) Strength of trend: 1 Var(Rt) Var(YtSt) Spectral entropy: H = fy() log fy()d, where fy() 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 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(YtTt) Strength of trend: 1 Var(Rt) Var(YtSt) Spectral entropy: H = fy() log fy()d, where fy() 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 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(YtTt) Strength of trend: 1 Var(Rt) Var(YtSt) Spectral entropy: H = fy() log fy()d, where fy() 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 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(YtTt) Strength of trend: 1 Var(Rt) Var(YtSt) Spectral entropy: H = fy() log fy()d, where fy() 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 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(YtTt) Strength of trend: 1 Var(Rt) Var(YtSt) Spectral entropy: H = fy() log fy()d, where fy() 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 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(YtTt) Strength of trend: 1 Var(Rt) Var(YtSt) Spectral entropy: H = fy() log fy()d, where fy() 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 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 78. 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 q q q q q q qq qq q q q q q q q q q q q q q qq q qq q qqq q q qq q q qq q qq qq q qqq q q q q qq qqqq q q qq q q qq q q q q q q qq q qq q q q q q q q q q q qq q q q q q qq q q qq qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q qq qqq q q q q q qq q qqq qqqq q qq q q q q qq q qq qq qqqq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q qq q qq q q q qqqq q qqq q qqq qq q qq qq qqqqqqqqqq q q q q q q qq q qq q qq q q q q qq q q qqq qq qqq q q qqqqqq q q q q qq qq qqqq q q qq q qq q q qqqq q q q q qqq qq q q qqqqq qqq qq q q q q q q q q q q q qq q qqq q qqq qqq q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qqqq q q q qqq q q q qq q q q q qqqqq q qqqq q q q qq q q qq qq q q qq q q q q qqqq q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q qq q q qq q q q q qqq q q q qq qq q q q q q qq qqq q qq q q qqq q q q qq q q qq qqqqqqq qq q q q q qqqqqq qqq q q q q q q q q q q q q q q qqq q q q q q qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q qq q q q qqq q q q qq q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq q q qq q qq q q qq q qqq q q q q q q q q q q q qq q q qq q q q q qq q q q q qq q q qq q q qq q q q qq qqq q qq q q qq q q q q qqqq q qq q q q qqqq qq q q q qqq qqqq q q qqq q qqq q q qq q qqq q q qq q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qqqq q qqq q qqq q qqq q q q q qq q qq q q q q q q q qq q q q q q q q qq q qq q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q qq qq q q qqqq q q qqq q q q q q q q q qq q qqqq q qq qq q q q qq qq q q q q q q q qq q q q q q q qq q q q q q q qq q qqq q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q qq q q q q q qq q q q q q qq q q q q q q q q qq qq q q qq q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q 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qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q q q q qq q q q q q q qqqq q q q qqqq q q qq qq q q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qqq qqqq qqqq q qq qqqqqq q qqqq q q q q q q q q q q qqq q q q q q q q q q q qqqq q q q q q q q qq q q qq q q qq q qq q q q q qq qqq q q q q q q q q qq q qq q q q qqq q q q q q q qq q q q q q q q qq qqq q q qq q q q q q qq q qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qqqq q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q q q q q q q q q q q q q q qq q qq qq q qq q qq q q qqq q qq qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q qqq q qq q q qq qq q q qqqq q q q q q q q q q q qqq q q q q q qqqq qqqq q q q q q q q q q q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q qq qq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q q qqq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q qq q q q q q qq qq qqq qqq q q q q q q qqqqqqqq q q qqqqqqqqqqqqqqqqq q qq q q q q qqqqq qqq q q qqqqq q q q qqqq q q q q q q q qq q q q qqqqq q qqqqq q qq qqq qqqq q qq q qq q q q qq q qq 3 2 1 0 1 2 3 2 0 2 4 PC1 PC2 Feature calculation Principal component decomposition 79. Feature space of M3 data Visualisation of big time series data M3 competition data 28 q q q q q q qq qq q q q q q q q q q q q q q qq q qq q qqq q q qq q q qq q qq qq q qqq q q q q qq qqqq q q qq q q qq q q q q q q qq q qq q q q q q q q q q q qq q q q q q qq q q qq qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q qq qqq q q q q q qq q qqq qqqq q qq q q q q qq q qq qq qqqq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q qq q qq q q q qqqq q qqq q qqq qq q qq qq qqqqqqqqqq q q q q q q qq q qq q qq q q q q qq q q qqq qq qqq q q qqqqqq q q q q qq qq qqqq q q qq q qq q q qqq q q q q q qqq qq q q qqqqq qqq qq q q q q q q q q q q q qq q qqq q qqq qqq q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qqqq q q q qqq q q q qq q q q q qqqqq q qqqq q q q qq q q qq qq q q qq q q q q qq qq q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q qq q q qq q q q q qqq q q q qq qq q q q q q qq qqq q qq q q qqq q q q qq q q qq qqqqqq q qq q q q q qqqqqq qqq q q q q q q q q q q q q q q qqq q q q q q qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qqq q q qq q q q qqq q q q qq q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq q q qq q qq q q qq q qqq q q q q q q q q q q q qq q q qq q q q q qq q q q q qq q q qq q q qq q q q qq qqq q qq q q qq q q q q qqqq q qq q q q qqqq qq q q q qqq qqqq q q qqq q qqq q q qq q qqq q q qq q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qqqq q qqq q qqq q qqq q q q q qq q qq q q q q q q q qq q q q q q q q qq q qq q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q qq qq q q qqqq q q qqq q q q q q q q q qq q qqqq q qq qq q q q qq qq q q q q q q q qq q q q q q q qq q q q q q q qq q qqq q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q qq q q q q q qq q q q q q qq q q q q q q q q qq qq q q qq q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q q q q q q q qq qqq q qq q q q q q q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q q q q qq q q q q q q qqqq q q q qqqq q q qq qq q q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qqq qqqq qqqq q qq qqqqqq q qqqq q q q q q q q q q q qqq q q q q q q q q q q qqqq q q q q q q q qq q q qq q q qq q qq q q q q qq qqq q q q q q q q q qq q qq q q q qqq q q q q q q qq q q q q q q q qq qqq q q qq q q q q q qq q qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qqqq q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q q q q q q q q q q q q q q qq q qq qq q qq q qq q q qq q q qq qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q qqq q qq q q qq qq q q qqqq q q q q q q q q q q qqq q q q q q qqqq qqqq q q q q q q q q q q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q qq qq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q q qqq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q qq q q q q q qq qq qqq qqq q q q q q q qqqqqqqq q q qqqqqqqqqqqqqqqqq q qq q q q q qqqqq qqq q q qqqqq q q q qqqq q q q q q q q qq q q q qqqqq q qqqqq q qq qqq qqqq q qq q qq q q q qq q qq 3 2 1 0 1 2 3 2 0 2 4 PC1 PC2 First two PCs explain 68% of variation. 80. Feature space of M3 data Visualisation of big time series data M3 competition data 28 q q q q q q qq qq q q q q q q q q q q q q q qqq qq q qqq q q qq q q qq q qq qq q qqq q q q q qq qqqq q q qq q q qq q q q q q q qq q qq q q q q q q q q q q qq q q q q q qq q q qq qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q qq q q q q q qq qqq q q q q q qq q qqq qqqq q qq q q q q qq q qq qq qqqq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q qq q qq q q q qqqq q qqq q qqq qq q qq qq qqqqqqqqqq q q q qq q qq q qq q qq q q q q qq q q qqq qq qqq q q qqqqqq q q q q qq qq qqqq q q qq q qq q q qqq q q q q q qqq qq q q qqqqq qqq qq q q q q q q q q q q q qq q qqq q qqq qqq q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qqqq q q q qqq q q q qq q q q q qqqqq q qqqq q q q qq q q qq qq q q qq qq q q qq qq q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q qq q q qq q q q q qqqq q q qq qq q q q q q qq qqq q qq q q qqq q q q qq q q qq qqqqqq q qq q q q q qqqqqq qqq q q q q q q q q q q q q q q qqq q q q q q qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qqq q q qq q q q qqq q q q qq q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq q q qq q qq q q qq q qqq q q q q q q q q q q q qq q q qq q q q q qq q q q q qq q q qq q q qq q q q qq qqqq qq q q qq q q q q qqqq q qq q q q qqqq qq q q q qqq qqqq q q qqq q qqq q q qq q qqq q q qq q q qqq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qqqq q qqq q qqq q qqq q q q q qqq qq q q q q q q q qq q q q q q q q qq q qq q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q qq qq q q qqqq q q qqq q q q q q q q q qq q qqqq q qq qq q q q qq qq q q q q q q q qq q q q q q q qq q q qq q q qq q qqq q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q qq q q q q q qq q q q q q qq q q q q q q q q qq qq q q qq q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q qq q q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qqq qq q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q q q q q q q qq qqq q qq q q q q q q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qqq q q q q q q q qq q q q q qq q q q q q q qqqq q q q qqqq q q qq qq q q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qqq qqqqqqqq q qq qqqqqq q qqqq q q q q q q q q q q qqq q q q q q q q q q q qqqq q q q q q q q qq q q qq q q qq q qq q q q q qq qqq q q q q q q q q qq q qq q q q qqq qq q q q q qq q q q q q q q qq qqq q q qq q q q q q qq q qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qqqq q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q q q q q q q q q q q q q q qq q qq qq q qq q qq q q qqq q qq qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q qqq q qq q q qq qq q q qqqq q q q q q q q q q q qqq q q q q q qqqq qqqq q q q q q q q q q q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q qqq qq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q q qqq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q qq q q q q q qq qq qqq qqq q q q q q qqqqqqqqq q q qqqqqqqqqqqqqqqqq qqq q q q q qqqqq qqq q q qqqqq q q q qqqq q q q q q q q qq q q q qqqqq q qqqqq q qq qqq qqqq q qq q qq q q q qq q qq 3 2 1 0 1 2 3 2 0 2 4 PC1 PC2 3 6 9 12 value Freq 81. Feature space of M3 data Visualisation of big time series data M3 competition data 28 q q q q q q qq qq q q q q q q q q q q q q q qqq qq q qqq q q qq q q qq q qq qq q qqq q q q q qq qqqq q q qq q q qq q q q q q q qq q qq q q q q q q q q q q qq q q q q q qq q q qq qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q qq q q q q q qq qqq q q q q q qq q qqq qqqq q qq q q q q qq q qq qq qqqq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q qq q qq q q q qqqq q qqq q qqq qq q qq qq qqqqqqqqqq q q q qq q qq q qq q qq q q q q qq q q qqq qq qqq q q qqqqqq q q q q qq qqqqqq q q qq q qq q q qqq q q q q q qqq qq q q qqqqq qqq qq q q q q q q q q q q q qq q qqq q qqq qqq q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qqqq q q q qqq q q q qq q q q q qqqqq q qqqq q q q qq q q qq qq q q qq qq q q qq qq q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q qq q q qq q q q q qqqq q q qq qq q q q q q qq qqq q qq q q qqq q q q qq q q qq qqqqqq q qq q q q q qqqqqq qqq q q q q q q q q q q q q q q qqq q q q q q qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qqq q q qq q q q qqq q q q qq q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq q q qq q qq q q qq q qqq q q q q q q q q q q q qq q q qq q q q q qq q q q q qq q q qq q q qq q q q qq qqqq qq q q qq q q q q qqqq q qq q q q qqqq qq q q q qqq qqqq q q qqq q qqq q q qq q qqq q q qq q q qqq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qqqq q qqq q qqq q qqq q q q q qqq qq q q q q q q q qq q q q q q q q qq q qq q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q qq qq q q qqqq q q qqq q q q q q q q q qq q qqqq q qq qq q q q qq qq q q q q q q q qq q q q q q q qq q q qq q q qq q qqq q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q qq q q q q q qq q q q q q qq q q q q q q q q qq qq q q qq q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q qq q q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qqq qq q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q q q q q q q qq qqq q qq q q q q q q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qqq q q q q q q q qq q q q q qq q q q q q q qqqq q q q qqqq q q qq qqq q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qqq qqqqqqqq q qq qqqqqq q qqqq q q q q q q q q q q qqq q q q q q q q q q q qqqq q q q q q q q qq q q qq q q qq q qq q q q q qq qqq q q q q q q q q qq q qq q q q qqq qq q q q q qq q q q q q q q qq qqq q q qq q q q q q qq q qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qqqq q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q q q q q q q q q q q q q q qq q qq qq q qq q qq q q qqq q qq qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q qqq q qq q q qq qq q q qqqq q q q q q q q q q q qqq q q q q q qqqq qqqq q q q q q q q q q q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q qqqqq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q q qqq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q qq q q q q q qq qq qqq qqq q q q q q qqqqqqqqq q q qqqqqqqqqqqqqqqqq qqq q q q q qqqqq qqq q q qqqqq q q q qqqq q q q q q q q qq q q q qqqqq q qqqqq q qq qqq qqqq q qq q qq q q q qq q qq 3 2 1 0 1 2 3 2 0 2 4 PC1 PC2 0.00 0.25 0.50 0.75 value Season 82. Feature space of M3 data Visualisation of big time series data M3 competition data 28 q q q q q q qq qq q q q q q q q q q q q q q qqq qq q qqq q q qq q q qq q qq qq q qqq q q q q qq qqqq q q qq q q qq q q q q q q qq q qq q q q q q q q q q q qq q q q q q qq q q qq qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q qq q q q q q qq qqq q q q q q qq q qqq qqqq q qq q q q q qq q qq qq qqqq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q qq q qq q q q qqqq q qqq q qqq qq q qq qq qqqqqqqqqq q q q qq q qq q qq q qq q q q q qq q q qqq qq qqq q q qqqqqq q q q q qq qqqqqq q q qq q qq q q qqq q q q q q qqq qq q q qqqqq qqq qq q q q q q q q q q q q qq q qqq q qqq qqq q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qqqq q q q qqq q q q qq q q q q qqqqq q qqqq q q q qq q q qq qq q q qq qq q q qq qq q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q qq q q qq q q q q qqqq q q qq qq q q q q q qq qqq q qq q q qqq q q q qq q q qq qqqqqq q qq q q q q qqqqqq qqq q q q q q q q q q q q q q q qqq q q q q q qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qqq q q qq q q q qqq q q q qq q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq q q qq q qq q q qq q qqq q q q q q q q q q q q qq q q qq q q q q qq q q q q qq q q qq q q qq q q q qq qqqq qq q q qq q q q q qqqq q qq q q q qqqq qq q q q qqq qqqq q q qqq q qqq q q qq q qqq q q qq q q qqq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qqqq q qqq q qqq q qqq q q q q qqq qq q q q q q q q qq q q q q q q q qq q qq q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q qq qq q q qqqq q q qqq q q q q q q q q qq q qqqq q qq qq q q q qq qq q q q q q q q qq q q q q q q qq q q qq q q qq q qqq q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q qq q q q q q qq q q q q q qq q q q q q q q q qq qq q q qq q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q qq q q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qqq qq q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q q q q q q q qq qqq q qq q q q q q q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qqq q q q q q q q qq q q q q qq q q q q q q qqqq q q q qqqq q q qq qqq q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qqq qqqqqqqq q qq qqqqqq q qqqq q q q q q q q q q q qqq q q q q q q q q q q qqqq q q q q q q q qq q q qq q q qq q qq q q q q qq qqq q q q q q q q q qq q qq q q q qqq qq q q q q qq q q q q q q q qq qqq q q qq q q q q q qq q qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qqqq q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q q q q q q q q q q q q q q qq q qq qq q qq q qq q q qqq q qq qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q qqq q qq q q qq qq q q qqqq q q q q q q q q q q qqq q q q q q qqqq qqqq q q q q q q q q q q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q qqqqq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q q qqq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q qq q q q q q qq qq qqq qqq q q q q q qqqqqqqqq q q qqqqqqqqqqqqqqqqq qqq q q q q qqqqq qqq q q qqqqq q q q qqqq q q q q q q q qq q q q qqqqq q qqqqq q qq qqq qqqq q qq q qq q q q qq q qq 3 2 1 0 1 2 3 2 0 2 4 PC1 PC2 0.25 0.50 0.75 value Trend 83. Feature space of M3 data Visualisation of big time series data M3 competition data 28 q q q q q q qq qq q q q q q q q q q q q q q qqq qq q qqq q q qq q q qq q qq qq q qqq q q q q qq qqqq q q qq q q qq q q q q q q qq q qq q q q q q q q q q q qq q q q q q qq q q qq qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q qq q q q q q qq qqq q q q q q qq q qqq qqqq q qq q q q q qq q qq qq qqqq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q qq q qq q q q qqqq q qqq q qqq qq q qq qq qqqqqqqqqq q q q qq q qq q qq q qq q q q q qq q q qqq qq qqq q q qqqqqq q q q q qq qq qqqq q q qq q qq q q qqq q q q q q qqq qq q q qqqqq qqq qq q q q q q q q q q q q qq q qqq q qqq qqq q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qqqq q q q qqq q q q qq q q q q qqqqq q qqqq q q q qq q q qq qq q q qq qq q q qq qq q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q qq q q qq q q q q qqqq q q qq qq q q q q q qq qqq q qq q q qqq q q q qq q q qq qqqqqq q qq q q q q qqqqqq qqq q q q q q q q q q q q q q q qqq q q q q q qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qqq q q qq q q q qqq q q q qq q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq q q qq q qq q q qq q qqq q q q q q q q q q q q qq q q qq q q q q qq q q q q qq q q qq q q qq q q q qq qqqq qq q q qq q q q q qqqq q qq q q q qqqq qq q q q qqq qqqq q q qqq q qqq q q qq q qqq q q qq q q qqq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qqqq q qqq q qqq q qqq q q q q qqq qq q q q q q q q qq q q q q q q q qq q qq q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q qq qq q q qqqq q q qqq q q q q q q q q qq q qqqq q qq qq q q q qq qq q q q q q q q qq q q q q q q qq q q qq q q qq q qqq q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q qq q q q q q qq q q q q q qq q q q q q q q q qq qq q q qq q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q qq q q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qqq qq q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q q q q q q q qq qqq q qq q q q q q q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qqq q q q q q q q qq q q q q qq q q q q q q qqqq q q q qqqq q q qq qq q q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qqq qqqqqqqq q qq qqqqqq q qqqq q q q q q q q q q q qqq q q q q q q q q q q qqqq q q q q q q q qq q q qq q q qq q qq q q q q qq qqq q q q q q q q q qq q qq q q q qqq qq q q q q qq q q q q q q q qq qqq q q qq q q q q q qq q qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qqqq q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q q q q q q q q q q q q q q qq q qq qq q qq q qq q q qqq q qq qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q qqq q qq q q qq qq q q qqqq q q q q q q q q q q qqq q q q q q qqqq qqqq q q q q q q q q q q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q qqq qq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q q qqq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q qq q q q q q qq qq qqq qqq q q q q q qqqqqqqqq q q qqqqqqqqqqqqqqqqq qqq q q q q qqqqq qqq q q qqqqq q q q qqqq q q q q q q q qq q q q qqqqq q qqqqq q qq qqq qqqq q qq q qq q q q qq q qq 3 2 1 0 1 2 3 2 0 2 4 PC1 PC2 0.0 0.5 value ACF 84. Feature space of M3 data Visualisation of big time series data M3 competition data 28 q q q q q q qq qq q q q q q q q q q q q q q qqq qq q qqq q q qq q q qq q qq qq q qqq q q q q qq qqqq q q qq q q qq q q q q q q qq q qq q q q q q q q q q q qq q q q q q qq q q qq qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q qq q q q q q qq qqq q q q q q qq q qqq qqqq q qq q q q q qq q qq qq qqqq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q qq q qq q q q qqqq q qqq q qqq qq q qq qq qqqqqqqqqq q q q qq q qq q qq q qq q q q q qq q q qqq qq qqq q q qqqqqq q q q q qq qq qqqq q q qq q qq q q qqq q q q q q qqq qq q q qqqqq qqq qq q q q q q q q q q q q qq q qqq q qqq qqq q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qqqq q q q qqq q q q qq q q q q qqqqq q qqqq q q q qq q q qq qq q q qq qq q q qq qq q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q qq q q qq q q q q qqqq q q qq qq q q q q q qq qqq q qq q q qqq q q q qq q q qq qqqqqq q qq q q q q qqqqqq qqq q q q q q q q q q q q q q q qqq q q q q q qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qqq q q qq q q q qqq q q q qq q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq q q qq q qq q q qq q qqq q q q q q q q q q q q qq q q qq q q q q qq q q q q qq q q qq q q qq q q q qq qqqq qq q q qq q q q q qqqq q qq q q q qqqq qq q q q qqq qqqq q q qqq q qqq q q qq q qqq q q qq q q qqq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qqqq q qqq q qqq q qqq q q q q qqq qq q q q q q q q qq q q q q q q q qq q qq q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q qq qq q q qqqq q q qqq q q q q q q q q qq q qqqq q qq qq q q q qq qq q q q q q q q qq q q q q q q qq q q qq q q qq q qqq q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q qq q q q q q qq q q q q q qq q q q q q q q q qq qq q q qq q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q qq q q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qqq qq q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q q q q q q q qq qqq q qq q q q q q q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qqq q q q q q q q qq q q q q qq q q q q q q qqqq q q q qqqq q q qq qq q q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qqq qqqqqqqq q qq qqqqqq q qqqq q q q q q q q q q q qqq q q q q q q q q q q qqqq q q q q q q q qq q q qq q q qq q qq q q q q qq qqq q q q q q q q q qq q qq q q q qqq qq q q q q qq q q q q q q q qq qqq q q qq q q q q q qq q qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qqqq q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q q q q q q q q q q q q q q qq q qq qq q qq q qq q q qqq q qq qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q qqq q qq q q qq qq q q qqqq q q q q q q q q q q qqq q q q q q qqqq qqqq q q q q q q q q q q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q qqq qq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q q qqq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q qq q q q q q qq qq qqq qqq q q q q q qqqqqqqqq q q qqqqqqqqqqqqqqqqq qqq q q q q qqqqq qqq q q qqqqq q q q qqqq q q q q q q q qq q q q qqqqq q qqqqq q qq qqq qqqq q qq q qq q q q qq q qq 3 2 1 0 1 2 3 2 0 2 4 PC1 PC2 0.5 0.6 0.7 0.8 0.9 value SpecEntr 85. Feature space of M3 data Visualisation of big time series data M3 competition data 28 q q q q q q qq qq q q q q q q q q q q q q q qqq qq q qqq q q qq q q qq q qq qq q qqq q q q q qq qqqq q q qq q q qq q q q q q q qq q qq q q q q q q q q q q qq q q q q q qq q q qq qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q qq q q q q q qq qqq q q q q q qq q qqq qqqq q qq q q q q qq q qq qq qqqq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q qq q qq q q q qqqq q qqq q qqq qq q qq qq qqqqqqqqqq q q q qq q qq q qq q qq q q q q qq q q qqq qq qqq q q qqqqqq q q q q qq qqqqqq q q qq q qq q q qqq q q q q q qqq qq q q qqqqq qqq qq q q q q q q q q q q q qq q qqq q qqq qqq q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qqqq q q q qqq q q q qq q q q q qqqqq q qqqq q q q qq q q qq qq q q qq qq q q qq qq q q q q q q q q q q 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q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qqq q q q q q q q qq q q q q qq q q q q q q qqqq q q q qqqq q q qq qqq q q q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qqq qqqqqqqq q qq qqqqqq q qqqq q q q q q q q q q q qqq q q q q q q q q q q qqqq q q q q q q q qq q q qq q q qq q qq q q q q qq qqq q q q q q q q q qq q qq q q q qqq qq q q q q qq q q q q q q q qq qqq q q qq q q q q q qq q qq q q q q q q q qq q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qqqq q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q q q q q q q q q q q q q q qq q qq qq q qq q qq q q qqq q qq qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q qqq q qq q q qq qq q q qqqq q q q q q q q q q q qqq q q q q q qqqq qqqq q q q q q q q q q q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q qqqqq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q q qqq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q qq q q q q q qq qq qqq qqq q q q q q qqqqqqqqq q q qqqqqqqqqqqqqqqqq qqq q q q q qqqqq qqq q q qqqqq q q q qqqq q q q q q q q qq q q q qqqqq q qqqqq q qq qqq qqqq q qq q qq q q q qq q qq 3 2 1 0 1 2 3 2 0 2 4 PC1 PC2 0.00 0.25 0.50 0.75 1.00 value Lambda 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 88. Predictability Visualisation of big time series data M3 competition data 30 Theta 1975 1980 1985 1990 200040006000800010000 89. Predictability Visualisation of big time series data M3 competition data 30 ETS 1975 1980 1985 1990 200040006000800010000 90. Predictability Visualisation of big time series data M3 competition data 30 AR 1975 1980 1985 1990 200040006000800010000 91. Predictability Visualisation of big time series data M3 competition data 31 Theta 1980 1982 1984 1986 1988 1990 1992 3000400050006000 92. Predictability Visualisation of big time series data M3 competition data 31 ETS 1980 1982 1984 1986 1988 1990 1992 3000400050006000 93. Predictability Visualisation of big time series data M3 competition data 31 STLAR 1980 1982 1984 1986 1988 1990 1992 3000400050006000 94. Predictability Visualisation of big time series data M3 competition data 32 Theta 1984 1986 1988 1990 1992 1994 60006500700075008000 95. Predictability Visualisation of big time series data M3 competition data 32 ETS 1984 1986 1988 1990 1992 1994 60006500700075008000 96. Predictability Visualisation of big time series data M3 competition data 32 STLAR 1984 1986 1988 1990 1992 1994 60006500700075008000 97. Predictability Visualisation of big time series data M3 competition data 33 q q q q q q qq q q q q q q q q q q q q q q q qq q qq q qq q q q qq q q qq q qq qq q qqq q q q q qq qq qq q q qq q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q qq qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq qqq q q q q q qq q q qq qqqq q q q q q q q qq q qq qq qq qq q q q q q q q q q qqq q q q q q q qq q q q q q qq qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q q q q qq q q q qqq q q qqq q q q q qq q q q q q qqqqqqqqqq q q q q q q qq q q q q q q q q q q q q q q qqq q q qq q q q qqqqqq q q q q qq qq qqqq q q q q q qq q q qqq q q q q q qqq q q q q qqqqq qq q qq q q q q q q q q q q q qq q qqq q qqq qq q q qq q q q q q q qqqqq q q qq q q qq q q qq q q q q q q q q q q q q q q q qqqqq q q q q q q q q q q q q q q q q q q qq q q q q qq q q qqq qq qq q q q qqq q q q q q q q q q q q qqq q qqqq q q q qq q q qq qq q q qq q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q qq q q q q q q q q qqqq q q qq qq q q q q q q q qq q q q q q q q qq q q q qq q q qq q qqq qq q q q q q q q q q qqqq q qq q q q q q q q q q q q q q q qq q q q q q q qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qqq q q qq q q q qqq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq q q qq q qq q q qq q qqq q q q q q q q q q q q qq q q qq q q q q qq q q q q q q q q qq q q qq q q q q q q qq q q q q q qq q q q q q qq q q qq q q q qqqq qq q q q qqq q q qq q q qqq q qqq q q q q 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q q q q q q q q qqq q q q q q q q q q q q q qq q qq qq q qq q q q q q qq q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qqq q qq q q q q q q q qqqq q q q q q q q q q q qqq q q q q q q qq qqqq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q 3 2 1 0 1 2 3 2 0 2 4 PC2 Low q q q q q qq q q q q q q q q q q q q q qq q qq q qq q q qq q q qq q qq qq q qqq q q q qq qq qq q qq q q qq qq q q q q q q qq q q q q q q q q q q q q q q qq qq qq q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q qq q q qq qqqq q q q q q q qq q qq qq qq qq q q q q q qqq q q q q q qq q q q q q q q q q q q qq q q q q qq q q q q q q q qq q q q qqq q q qqq q q q q qq q q q q q qqqqqqqqqq q q q q q q qq q q qq q q q q q q q q qqq q q qq q q q qqqqqq q q q q qq qq qqqq q q q qqq qqq q q qqq qqqqq qq q qq q q q q q q q q qq q qqq q qqq qq qqqqq q q qq q qq q q qq q q q q q q q q q q q q q q 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