Visualisation of big time series data Rob J Hyndman

Visualisation of big time series data Rob J Hyndman Visualisation of big time series data 1 Visualisation of big time series data Rob J Hyndman with Earo Wang, Nikolay Laptev Yanfei Kang, Kate Smith-Miles Visualisation of big time series data 1 Visualisation of big time series data Rob J Hyndman with Earo Wang, Nikolay Laptev Yanfei Kang, Kate Smith-Miles Visualisation of big time series data 1 Visualisation of big time series data Rob J Hyndman with Earo Wang, Nikolay Laptev Yanfei Kang, Kate Smith-Miles Visualisation of big time series data 1 Visualisation of big time series data Rob J Hyndman with Earo Wang, Nikolay Laptev Yanfei Kang, Kate Smith-Miles Visualisation of big time series data 1 Visualisation of big time series data Rob J Hyndman with Earo Wang, Nikolay Laptev Yanfei Kang, Kate Smith-Miles Visualisation of big time series data 1 Visualisation of big time series data Rob J Hyndman with Earo Wang, Nikolay Laptev Yanfei Kang, Kate Smith-Miles Visualisation of big time series data 1 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 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 0.0 0.2 0.4 0.6 0.8 1.0 How to plot lots of time series? 0.0 0.2 0.4 0.6 0.8 1.0 Time Visualisation of big time series data The problem 3 How to plot lots of time series? Visualisation of big time series data The problem 4 How to plot lots of time series? Visualisation of big time series data The problem 4 How to plot lots of time series? Visualisation of big time series data The problem 4 How to plot lots of time series? Visualisation of big time series data The problem 4 How to plot lots of time series? Visualisation of big time series data The problem 4 Key idea Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985). Examples for time series lag correlation size and direction of trend strength of seasonality timing of peak seasonality spectral entropy Visualisation of big time series data John W Tukey The problem 5 Key idea Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985). Examples for time series lag correlation size and direction of trend strength of seasonality timing of peak seasonality spectral entropy Visualisation of big time series data John W Tukey The problem 5 Key idea Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985). Examples for time series lag correlation size and direction of trend strength of seasonality timing of peak seasonality spectral entropy Visualisation of big time series data John W Tukey The problem 5 Key idea Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985). Examples for time series lag correlation size and direction of trend strength of seasonality timing of peak seasonality spectral entropy Visualisation of big time series data John W Tukey The problem 5 Key idea Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985). Examples for time series lag correlation size and direction of trend strength of seasonality timing of peak seasonality spectral entropy Visualisation of big time series data John W Tukey The problem 5 Key idea Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985). Examples for time series lag correlation size and direction of trend strength of seasonality timing of peak seasonality spectral entropy John W Tukey Called “features” or “characteristics” in the machine learning literature. Visualisation of big time series data The problem 5 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 6 Australian tourism demand Visualisation of big time series data Australian tourism demand 7 Australian tourism demand 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 Visualisation of big time series data Australian tourism demand 7 Visualisation of big time series data BEEOth BEDOth BEEBus BEDBus BEGBus BEFBus BEGOth BEFOth BEEVis BEDVis BEEHol BEDHol BEGVis BEFVis BEGHol BEFHol BECOth BEBOth BECBus BEBBus BECVis BEBVis BECHol BEBHol BDFVis BEAOth BDFOth BEABus BDFBus BEAVis BEAHol BDFHol BDEOth BDDOth BDEBus BDDBus BDEVis BDDVis BDEHol BDDHol BDCOth BDBOth BDCBus BDBBus BDCVis BDBVis BDCHol BDBHol BDAOth BCCOth BDABus BCCBus BDAVis BCCVis BDAHol BCCHol BCBOth BCAOth BCBBus BCABus BCBVis BCAVis BCBHol BCAHol BBAOth BACOth BBABus BACBus BBAVis BACVis BBAHol BACHol BAAVis BABOth BAAOth BABBus BAABus BABVis BABHol BAAHol Domestic tourism demand: Victoria Australian tourism demand 8 An STL decomposition 7.0 6.0 6.1 −0.4 0.0 remainder 5.8 trend 6.4 −0.5 0.5 seasonal 5.0 data Tourism demand for holidays in Peninsula Yt = S t + Tt + R t St is periodic with mean 0 2000 Visualisation of big time series data 2005 2010 time Australian tourism demand 9 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 Qtr 0.5 Q1 Holiday Seasonal Component 1.0 0.0 Q2 Q3 Q4 −0.5 −1.0 BAA BAB BAC BBA BCA BCBBCCBDA BDBBDCBDDBDE BDF BEA BEB BEC BED BEE BEF BEG Regions Visualisation of big time series data Australian tourism demand 10 Seasonal stacked bar chart: VIC Visualisation of big time series data Australian tourism demand 11 Seasonal stacked bar chart: VIC Visualisation of big time series data Australian tourism demand 11 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 Holiday VFR Qtr Q1 Q2 Q3 Q4 Business Seasonal Component Seasonal stacked bar chart: VIC Other BAABABBACBBABCABCBBCCBDABDBBDCBDDBDEBDFBEABEBBECBEDBEEBEFBEG Regions Visualisation of big time series data Australian tourism demand 11 Trend analysis Linearity: the long-term direction and strength of trend. Curvature: the “changing direction” of trend. Estimate by regression: Tt = β̂0 + β̂1 φ1 (t ) + βˆ2 φ2 (t ) + et where φk (t ) is a kth-degree orthogonal polynomial in time t. To separate the linearity (β̂1 ) and curvature (β̂2 ). Visualisation of big time series data Australian tourism demand 12 4 3 2 1 0 Holiday 4 3 2 1 0 VFR 4 3 2 1 0 Business Trend Linearity Trend analysis Other 4 3 2 1 0 Direction − + BAA BAB BAC BBA BCA BCBBCC BDA BDBBDCBDDBDE BDF BEA BEB BEC BED BEE BEF BEG Regions Visualisation of big time series data Australian tourism demand 13 Trend analysis Visualisation of big time series data Australian tourism demand 13 BEEHol BEFOth BEEOth BDEOth BEBOth BEABus BEFBus BDCOth BACHol BEBBus BEAVis BBAHol BDEHol BABOth BAAVis BAAHol BDCHol BBABus BCBHol BEGBus BDDVis BABVis BDAVis BEAOth BDFHol BEEBus BAAOth BACOth BDAOth BDEBus BCBOth BACBus BEBVis BACVis BCAOth BEFVis BCBVis BEDHol BEGOth BDBHol BABBus BEBHol BDFBus BECHol BCAHol BDBOth BEAHol BDCBus BECVis BDBVis BCCHol BBAVis BABHol BBAOth BCCOth BCBBus BCCVis BEGVis BDDHol BECOth BDCVis BAABus BCCBus BECBus BCAVis BDFVis BEGHol BDDOth BEDOth BEDVis BDDBus BDEVis BEFHol BEEVis BDBBus BDABus BDAHol BCABus BDFOth BEDBus Corrgram of remainder BEEHol BEFOth BEEOth BDEOth BEBOth BEABus BEFBus BDCOth BACHol BEBBus BEAVis BBAHol BDEHol BABOth BAAVis BAAHol BDCHol BBABus BCBHol BEGBus BDDVis BABVis BDAVis BEAOth BDFHol BEEBus BAAOth BACOth BDAOth BDEBus BCBOth BACBus BEBVis BACVis BCAOth BEFVis BCBVis BEDHol BEGOth BDBHol BABBus BEBHol BDFBus BECHol BCAHol BDBOth BEAHol BDCBus BECVis BDBVis BCCHol BBAVis BABHol BBAOth BCCOth BCBBus BCCVis BEGVis BDDHol BECOth BDCVis BAABus BCCBus BECBus BCAVis BDFVis BEGHol BDDOth BEDOth BEDVis BDDBus BDEVis BEFHol BEEVis BDBBus BDABus BDAHol BCABus BDFOth BEDBus Visualisation of big time series data 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 Australian tourism demand 14 BEEHol BEFOth BEEOth BDEOth BEBOth BEABus BEFBus BDCOth BACHol BEBBus BEAVis BBAHol BDEHol BABOth BAAVis BAAHol BDCHol BBABus BCBHol BEGBus BDDVis BABVis BDAVis BEAOth BDFHol BEEBus BAAOth BACOth BDAOth BDEBus BCBOth BACBus BEBVis BACVis BCAOth BEFVis BCBVis BEDHol BEGOth BDBHol BABBus BEBHol BDFBus BECHol BCAHol BDBOth BEAHol BDCBus BECVis BDBVis BCCHol BBAVis BABHol BBAOth BCCOth BCBBus BCCVis BEGVis BDDHol BECOth BDCVis BAABus BCCBus BECBus BCAVis BDFVis BEGHol BDDOth BEDOth BEDVis BDDBus BDEVis BEFHol BEEVis BDBBus BDABus BDAHol BCABus BDFOth BEDBus Corrgram of remainder BEEHol BEFOth BEEOth BDEOth BEBOth BEABus BEFBus BDCOth BACHol BEBBus BEAVis BBAHol BDEHol BABOth BAAVis BAAHol BDCHol BBABus BCBHol BEGBus BDDVis BABVis BDAVis BEAOth BDFHol BEEBus BAAOth BACOth BDAOth BDEBus BCBOth BACBus BEBVis BACVis BCAOth BEFVis BCBVis BEDHol BEGOth BDBHol BABBus BEBHol BDFBus BECHol BCAHol BDBOth BEAHol BDCBus BECVis BDBVis BCCHol BBAVis BABHol BBAOth BCCOth BCBBus BCCVis BEGVis BDDHol BECOth BDCVis BAABus BCCBus BECBus BCAVis BDFVis BEGHol BDDOth BEDOth BEDVis BDDBus BDEVis BEFHol BEEVis BDBBus BDABus BDAHol BCABus BDFOth BEDBus Compute the correlations among the remainder components 1 0.8 0.6 Render both the sign and magnitude using a colour mapping0.4 0.2 of two hues Order variables according to the first principal component of the correlations. Visualisation of big time series data 0 −0.2 −0.4 −0.6 −0.8 −1 Australian tourism demand 14 BEEHol BCBHol BACHol BDEHol BDBHol BABHol BAAHol BBAHol BEGHol BEAHol BCAHol BDCHol BCCHol BDFHol BEDHol BECHol BEFHol BEBHol BDDHol BDAHol Corrgram of remainder BDAHol 1 BDDHol 0.8 BEBHol BEFHol 0.6 BECHol BEDHol 0.4 BDFHol BCCHol 0.2 BDCHol BCAHol 0 BEAHol BEGHol −0.2 BBAHol BAAHol −0.4 BABHol BDBHol −0.6 BDEHol BACHol −0.8 BCBHol BEEHol −1 Visualisation of big time series data Australian tourism demand 14 FAABus FBBBus FCBOth FBAVis FCABus FCBVis FBAOth FBABus FBBOth FCAOth FAAOth FCBBus FAAVis FBBVis FCAVis FCBHol FAAHol FBAHol FBBHol FCAHol Corrgram of remainder: TAS FCAHol 1 FBBHol 0.8 FBAHol FAAHol 0.6 FCBHol FCAVis 0.4 FBBVis FAAVis 0.2 FCBBus FAAOth 0 FCAOth FBBOth −0.2 FBABus FBAOth −0.4 FCBVis FCABus −0.6 FBAVis FCBOth −0.8 FBBBus FAABus −1 Visualisation of big time series data Australian tourism demand 15 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 16 M3 forecasting competition 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 17 M3 forecasting competition ● Visualisation of big time series data M3 competition data 18 Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Strength of trend: 1 − Var(Rt ) Var(Yt −Tt ) Var(Rt ) Var(Yt −St ) Rπ Spectral entropy: H = − −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 19 Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Strength of trend: 1 − Var(Rt ) Var(Yt −Tt ) Var(Rt ) Var(Yt −St ) Rπ Spectral entropy: H = − −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 19 Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Strength of trend: 1 − Var(Rt ) Var(Yt −Tt ) Var(Rt ) Var(Yt −St ) Rπ Spectral entropy: H = − −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 19 Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Strength of trend: 1 − Var(Rt ) Var(Yt −Tt ) Var(Rt ) Var(Yt −St ) Rπ Spectral entropy: H = − −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 19 Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Strength of trend: 1 − Var(Rt ) Var(Yt −Tt ) Var(Rt ) Var(Yt −St ) Rπ Spectral entropy: H = − −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 19 Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Strength of trend: 1 − Var(Rt ) Var(Yt −Tt ) Var(Rt ) Var(Yt −St ) Rπ Spectral entropy: H = − −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 19 Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Strength of trend: 1 − Var(Rt ) Var(Yt −Tt ) Var(Rt ) Var(Yt −St ) Rπ Spectral entropy: H = − −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 19 Candidate features 3000 1000 N0001 5000 Seasonality 1978 1980 1982 1984 1986 1988 N2602 0 10000 20000 1976 1982 1984 1986 6000 10000 1980 2000 N1906 1978 1984 1986 Visualisation of big time series data 1988 1990 M3 competition data 1992 20 Candidate features 4000 2000 N0125 6000 Trend 1980 1982 1984 1986 1988 5000 1978 3000 N1978 1976 1984 1986 1988 1990 1992 1000 N0546 4000 7000 1982 1975 Visualisation of big time series data 1980 1985 M3 competition data 20 Candidate features N0001 5800 6000 6200 ACF1 1988 1989 1990 N2658 3000 5000 7000 1987 1988 1989 1990 1991 8000 7000 N2409 9000 1987 1984 1986 Visualisation of big time series data 1988 1990 M3 competition data 1992 20 Candidate features 4000 2500 N2487 5500 Spectral entropy 1968 1970 1972 1974 4500 1966 3000 N0794 1964 1988 1990 1992 2000 N0121 2400 2800 1986 1976 1978 Visualisation of big time series data 1980 1982 1984 M3 competition data 1986 1988 20 Candidate features 4000 2000 N0002 6000 Box Cox 1978 1980 1982 1984 1986 1988 7000 5000 N0468 9000 1976 1970 1975 1980 1985 6000 1965 2000 N0354 1960 1960 1965 1970 Visualisation of big time series data 1975 1980 M3 competition data 1985 20 Candidate features 0.4 0.8 2 6 10 0.0 0.4 0.8 0.9 0.0 0.0 0.6 0.5 SpecEntr 0.0 0.6 Trend 8 Season 0.6 2 Freq 0.0 0.6 −0.4 ACF Lambda 0.5 0.7 0.9 0.0 0.4 Visualisation of big time series data 0.8 −0.4 0.2 0.8 M3 competition data 21 Dimension reduction for time series ● Visualisation of big time series data M3 competition data 22 Dimension reduction for time series 0.4 0.8 2 6 10 0.0 0.4 0.8 0.9 0.0 0.0 0.6 0.5 SpecEntr 0.0 0.6 Trend Season 8 Feature calculation 0.6 2 Freq 0.0 0.6 −0.4 ACF Lambda 0.5 0.7 0.9 0.0 0.4 0.8 −0.4 0.2 0.8 ● Visualisation of big time series data M3 competition data 22 Dimension reduction for time series 0.4 0.8 2 6 10 0.0 0.4 0.8 0.9 0.0 0.0 0.6 0.5 SpecEntr 0.0 0.6 Trend Season 8 Feature calculation 0.6 2 Freq 0.0 0.6 −0.4 ACF 0.5 3 2 PC2 1 0 −1 −2 −3 0.7 0.9 0.0 0.4 Principal Lambda component decomposition 0.8 −0.4 0.2 0.8 ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ● ●● ●● ●● ●● ●● ● ●● ● ● ●● ●●● ● ●● ● ●● ●● ● ● ● ● ● ●● ●● ● ●● ● ●● ●●●● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●●● ●● ● ● 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space of M3 data First two PCs explain 68% of variation. 3 2 PC2 1 0 −1 −2 −3 ● ● ● ● ●● ● ●●● ●● ●●● ● ● ●● ●● ●● ●● ●● ●● ● ●● ● ● ●● ●●● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●●●●● ● ● ● ●●●● ● ● ● ● ●● ●● ● ● ● ●●●●● ● ●● ●● ●● ● ● ● ● ● ●● ●● ● ● ●● ● ●●● ●● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ●●●● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ●● ● ●● ●● ●●●●●● ●● ●●● ● ●●●●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ●● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●●● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ●● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ●● ● ●● ● ● ●● ● ● 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time series data M3 competition data 23 Feature space of M3 data ACF 3 2 PC2 1 0 −1 −2 −3 ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ●● ●● ●● ●●● ●● ● ●●●● ● ●● ●●● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ●●● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ●●● ● ●● ● ● ● ● ● ●● ●● ● ● ●●● ●● ● ● ● ●●● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ●●●● ●● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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Compute minimum MASE from all three methods Visualisation of big time series data M3 competition data 24 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 24 Predictability 2000 4000 6000 8000 10000 Theta 1975 1980 Visualisation of big time series data 1985 1990 M3 competition data 25 Predictability 2000 4000 6000 8000 10000 ETS 1975 1980 Visualisation of big time series data 1985 1990 M3 competition data 25 Predictability 2000 4000 6000 8000 10000 AR 1975 1980 Visualisation of big time series data 1985 1990 M3 competition data 25 Predictability 3000 4000 5000 6000 Theta 1980 1982 1984 Visualisation of big time series data 1986 1988 1990 M3 competition data 1992 26 Predictability 3000 4000 5000 6000 ETS 1980 1982 1984 Visualisation of big time series data 1986 1988 1990 M3 competition data 1992 26 Predictability 3000 4000 5000 6000 STL−AR 1980 1982 1984 Visualisation of big time series data 1986 1988 1990 M3 competition data 1992 26 Predictability 6000 6500 7000 7500 8000 Theta 1984 1986 Visualisation of big time series data 1988 1990 1992 M3 competition data 1994 27 Predictability 6000 6500 7000 7500 8000 ETS 1984 1986 Visualisation of big time series data 1988 1990 1992 M3 competition data 1994 27 Predictability 6000 6500 7000 7500 8000 STL−AR 1984 1986 Visualisation of big time series data 1988 1990 1992 M3 competition data 1994 27 Predictability Low 3 PC2 1 0 −1 −2 −3 ● ●●● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●●● ● ● ● ●● ●●● ● ●● ● ● ●● ●● ● ● ● ● ● ●● ●● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● 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2 PC1 4 6 −2 Monthly data 4 6 2 ● ●● ● ●●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ●● ● ● ●● ● ●● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ●● ●●● ● ● ●●●● ●● ● ●●● ● ●● ● ● ● ●● ● ●●● ●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ●● ●●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●●●● ●● ● ● ●●●● ● ● ●●●● ●● ● ● ●●●●●● ●● ● ●●● ● ●● ●● ● ● ●●● ●● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ●● ● ●●● ● ●● ●● ●●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ●●● ● ●● ● ● ●● ● ● ●● ● ● ● ●● ●● ● ●● ●● ●● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●●● ●● ●● ●● ● ●● ●● ● ●● ● ● ●● ● ● ● ● ●●● ●● ● ●● ●● ● ●●●●●● ● ● ●●● ● ●● ● ● ● ● ● ●● ●●● ● ● ● ●●●● ●● ● ● ●● ● ●● ●● ● ● ●● ●● ● ● ●●● ●● ● ● ● ●● ●●● ●● ● ● ●● ● ●● ●● ● ● ● ●●●● ●● ●●●● ●●● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ● ●●● ●●●● ● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ●● ●●● ● ● ● ● ● ●● ● ● ●● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● 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●●● ●● ●● ● ●●● ●● ● ● ● ● ● ● Best Ets NoDiff Stlm−ar Theta ● ● ● PC2 PC2 2 PC1 4 2 −2 0 Monthly data 4 0 ●● ●●●●● ● ●●● ●●● ● ●●● ●● ●● ● ● ● ● ● 0 ● −2 −4 ● ●● ● ●●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ●● ● ● ●● ● ●● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ●● ●●● ● ● ●●●● ●● ● ●●● ● ●● ● ● ● ●● ● ●●● ●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ●● ●●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●●●● ●● ● ● ●●●● ● ● ●●●● ●● ● ● ●●●●●● ●● ● ●●● ● ●● ●● ● ● ●●● ●● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ●● ● ●●● ● ●● ●● ●●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ●●● ● ●● ● ● ●● ● ● ●● ● ● ● ●● ●● ● ●● ●● ●● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●●● ●● ●● ●● ● ●● ●● ● ●● ● ● ●● ● ● ● ● ●●● ●● ● ●● ●● ● ●●●●●● ● ● ●●● ● ●● ● ● ● ● ● ●● ●●● ● ● ● ●●●● ●● ● ● ●● ● ●● ●● ● ● ●● ●● ● ● ●●● ●● ● ● ● ●● ●●● ●● ● ● ●● ● ●● ●● ● ● ● ●●●● ●● ●●●● ●●● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ● ●●● ●●●● ● ● ● ● ● 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● ● ●● ●● ● ● ●●● ● ●● ●●●● ●●● ● ● ●● ●● ●● ● ● ●●●●● ●●●●● ● ● ●● ● ●● ● ●● ● ●● ● ●●●● ● ● ● ● ●●● ●●● ● ● ● ●●● ● ● ● ● ● ●●● ●● ●● ● ●●● ●● ● ● ● ● ● ● Best Ets NoDiff Stlm−ar ● ● ● −4 −2 0 2 PC1 4 6 Actual Visualisation of big time series data −2 0 2 PC1 4 6 SVM prediction M3 competition data 29 Generating new time series We can use the feature space to: å Generate new time series with similar features to existing series å Generate new time series where there are “holes” in the feature space. Let {PC1 , PC2 , . . . , PCn } be a “population” of time series of specified length and period. Genetic algorithm uses a process of selection, crossover and mutation to evolve the population towards a target point Ti . p Optimize: Fitness (PCj ) = − (|PCj − Ti |2 ). Initial population random with some series in neighbourhood of Ti . Visualisation of big time series data M3 competition data 30 Generating new time series We can use the feature space to: å Generate new time series with similar features to existing series å Generate new time series where there are “holes” in the feature space. Let {PC1 , PC2 , . . . , PCn } be a “population” of time series of specified length and period. Genetic algorithm uses a process of selection, crossover and mutation to evolve the population towards a target point Ti . p Optimize: Fitness (PCj ) = − (|PCj − Ti |2 ). Initial population random with some series in neighbourhood of Ti . Visualisation of big time series data M3 competition data 30 Generating new time series We can use the feature space to: å Generate new time series with similar features to existing series å Generate new time series where there are “holes” in the feature space. Let {PC1 , PC2 , . . . , PCn } be a “population” of time series of specified length and period. Genetic algorithm uses a process of selection, crossover and mutation to evolve the population towards a target point Ti . p Optimize: Fitness (PCj ) = − (|PCj − Ti |2 ). Initial population random with some series in neighbourhood of Ti . Visualisation of big time series data M3 competition data 30 Generating new time series We can use the feature space to: å Generate new time series with similar features to existing series å Generate new time series where there are “holes” in the feature space. Let {PC1 , PC2 , . . . , PCn } be a “population” of time series of specified length and period. Genetic algorithm uses a process of selection, crossover and mutation to evolve the population towards a target point Ti . p Optimize: Fitness (PCj ) = − (|PCj − Ti |2 ). Initial population random with some series in neighbourhood of Ti . Visualisation of big time series data M3 competition data 30 Generating new time series We can use the feature space to: å Generate new time series with similar features to existing series å Generate new time series where there are “holes” in the feature space. Let {PC1 , PC2 , . . . , PCn } be a “population” of time series of specified length and period. Genetic algorithm uses a process of selection, crossover and mutation to evolve the population towards a target point Ti . p Optimize: Fitness (PCj ) = − (|PCj − Ti |2 ). Initial population random with some series in neighbourhood of Ti . Visualisation of big time series data M3 competition data 30 Evolving new time series 3 ● ● ● ● ● 2 1 ●●●● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●●● ● ●●●● ● ●● ● ● ● ● ● ● ●● ●●● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ●● ● ● ● ● ● ●● ● ●●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ●●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ●● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● 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● ● ● ● ●● ●● ●●●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● PC2 ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●●●●●●●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ●●● ● ● ● ● ● ● ● ●● ● ●●● ●● ● ● ●●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●● ●● ● ●● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● PC2 2 ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ● ●●● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 Visualisation of big time series data 0 2 PC1 4 −4 6 M3 competition data −2 33 Evolving new time series Evolved yearly data 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● PC2 ● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ● ●● ● ● ●●●●●● ●● ● ● ●● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ●●● ● ● ●● ●●●●●● ● ● ●● ●● ●● ●●● ● ● ● ●● ● ● ● ●●● ● ●● ● ●● ● ● ●● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●●●●● ●● ● ●●● ● ● ●●● ●● ● ●●●●● ● ● ●●● ● ●● ● 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time series data 0 2 PC1 4 6 M3 competition data 33 −4 −4 Evolving new time series PC1 −2 0 2 4 6 −2 Evolved quarterly data E 4 4 ● 0 −2 2 PC2 PC2 2 ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ●● ●●●● ● ●● ●● ● ● ●●● ●● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ●● ● ●● ●● ● ● ●● ● ●● ● ●● ●● ●●●● ● ● ● ● ● ● ● ● ● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ●● ●● ● ●●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●●●● ●●●● ●● ● ●● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ●●●● ● ●●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ●●● ● ● ● ● ●● ●● ● ● ●● ● ●● ● ● ●● ●● ●● ● ● ● ● ● ●●●● ● ●● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●●● ● ●●●● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 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● ●●● ● ●●●●●●●●● ●● ●●● ● ● ● ● ●● ● ●●●● ● ●● ● ●●● ●●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● −4 6 −2 Visualisation of big time series data 0 2 PC1 4 6 M3 competition data 33 Questions raised Can SVM be used to create a forecast selection routine to give better forecasts? How much do M3 conclusions depend on the particular set of time series involved? Has the M3 data set biased forecast method development? What other features should we consider? What difference does it make? Is PCA the right approach? Perhaps we should use multidimensional scaling? Or something else? Should we use more than 2 PC dimensions? Visualisation of big time series data M3 competition data 34 Questions raised Can SVM be used to create a forecast selection routine to give better forecasts? How much do M3 conclusions depend on the particular set of time series involved? Has the M3 data set biased forecast method development? What other features should we consider? What difference does it make? Is PCA the right approach? Perhaps we should use multidimensional scaling? Or something else? Should we use more than 2 PC dimensions? Visualisation of big time series data M3 competition data 34 Questions raised Can SVM be used to create a forecast selection routine to give better forecasts? How much do M3 conclusions depend on the particular set of time series involved? Has the M3 data set biased forecast method development? What other features should we consider? What difference does it make? Is PCA the right approach? Perhaps we should use multidimensional scaling? Or something else? Should we use more than 2 PC dimensions? Visualisation of big time series data M3 competition data 34 Questions raised Can SVM be used to create a forecast selection routine to give better forecasts? How much do M3 conclusions depend on the particular set of time series involved? Has the M3 data set biased forecast method development? What other features should we consider? What difference does it make? Is PCA the right approach? Perhaps we should use multidimensional scaling? Or something else? Should we use more than 2 PC dimensions? Visualisation of big time series data M3 competition data 34 Questions raised Can SVM be used to create a forecast selection routine to give better forecasts? How much do M3 conclusions depend on the particular set of time series involved? Has the M3 data set biased forecast method development? What other features should we consider? What difference does it make? Is PCA the right approach? Perhaps we should use multidimensional scaling? Or something else? Should we use more than 2 PC dimensions? Visualisation of big time series data M3 competition data 34 Questions raised Can SVM be used to create a forecast selection routine to give better forecasts? How much do M3 conclusions depend on the particular set of time series involved? Has the M3 data set biased forecast method development? What other features should we consider? What difference does it make? Is PCA the right approach? Perhaps we should use multidimensional scaling? Or something else? Should we use more than 2 PC dimensions? Visualisation of big time series data M3 competition data 34 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 Yahoo web traffic 35 Yahoo web-traffic Tens of thousands of time series collected at one-hour intervals over one month. Consisting of several server metrics (e.g. CPU usage and paging views) from many server farms globally. Aim: find unusual (anomalous) time series. Visualisation of big time series data Yahoo web traffic 36 30 15000 25 40 25 0 35 600 200 70 20000 15000 10000 5000 0 50 50 Visualisation of big time series data date memory484 busy369 60 memory413 25 memory147 15 busy200 20 memory429 20 busy50 60 memory460 30 value 20000 40 2014−11−09 2014−11−10 2014−11−11 2014−11−12 2014−11−13 2014−11−14 2014−11−15 2014−11−16 2014−11−17 2014−11−18 2014−11−19 2014−11−20 2014−11−21 2014−11−22 2014−11−23 2014−11−24 2014−11−25 2014−11−26 2014−11−27 2014−11−28 2014−11−29 2014−11−30 2014−12−01 2014−12−02 2014−12−03 2014−12−04 2014−12−05 2014−12−06 2014−12−07 2014−12−08 2014−12−09 2014−12−10 35 busy271 value 45 busy233 2014−11−09 2014−11−10 2014−11−11 2014−11−12 2014−11−13 2014−11−14 2014−11−15 2014−11−16 2014−11−17 2014−11−18 2014−11−19 2014−11−20 2014−11−21 2014−11−22 2014−11−23 2014−11−24 2014−11−25 2014−11−26 2014−11−27 2014−11−28 2014−11−29 2014−11−30 2014−12−01 2014−12−02 2014−12−03 2014−12−04 2014−12−05 2014−12−06 2014−12−07 2014−12−08 2014−12−09 2014−12−10 2014−12−11 2014−12−12 2014−11−09 2014−11−10 2014−11−11 2014−11−12 2014−11−13 2014−11−14 2014−11−15 2014−11−16 2014−11−17 2014−11−18 2014−11−19 2014−11−20 2014−11−21 2014−11−22 2014−11−23 2014−11−24 2014−11−25 2014−11−26 2014−11−27 2014−11−28 2014−11−29 2014−11−30 2014−12−01 2014−12−02 2014−12−03 2014−12−04 2014−12−05 2014−12−06 2014−12−07 2014−12−08 2014−12−09 2014−12−10 2014−12−11 2014−12−12 Yahoo web-traffic 10000 5000 400 1000 10 500 25000 20000 15000 10000 5000 0 date Yahoo web traffic date 37 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. Change index: Time of maximum KL score Visualisation of big time series data Yahoo web traffic 38 Feature space ACF1: first order autocorrelation = Corr(Yt , Yt−1 ) Strength of trend and seasonality based on STL Trend linearity and curvature Size of seasonal peak and trough Spectral entropy Lumpiness: variance of block variances (block size 24). Spikiness: variances of leave-one-out variances of STL remainders. Level shift: Maximum difference in trimmed means of consecutive moving windows of size 24. Variance change: Max difference in variances of consecutive moving windows of size 24. Flat spots: Discretize sample space into 10 equal-sized intervals. Find max run length in any interval. Number of crossing points of mean line. Kullback-Leibler score: Maximum of R DKL (PkQ) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by kernel density estimators applied to consecutive windows of size 48. 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n x .id ge ure rvat ckuls core s eesss n i ik in spmp u l h 0 troug entro py ots 2 ● fsp ● peak s ritynt ea oi lin cp standardized PC2 (17.3% explained var.) Principal component analysis vlstcA C rhheianftnFg1e d ● ● ● ● −2.5 0.0 2.5 standardized PC1 (28.7% explained var.) 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scor sss ne s ikiine spmp lu ots troug h 2 ● fsp ● peak s ritynt ea oi lin cp standardized PC2 (17.3% explained var.) What is “anomalous” vlstcA C rhheifantnFg1e d ● ● ● ● −2.5 0.0 2.5 standardized PC1 (28.7% explained var.) We need a measure of the “anomalousness” of a time series. 1 Rank points based on their local density. 2 Rank points based on whether they are within α-convex hulls of different radius. Visualisation of big time series data Yahoo web traffic 40 ● ●● ●● ● ●●● ● ● ● ● ●● ●● ● ● ●● ● ● ●●●●●●● ● ●●● ● ● ●● ●● ● ● ● ●● ●●● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●●●●● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● 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scor sss ne s ikiine spmp lu ots troug h 2 ● fsp ● peak s ritynt ea oi lin cp standardized PC2 (17.3% explained var.) What is “anomalous” vlstcA C rhheifantnFg1e d ● ● ● ● −2.5 0.0 2.5 standardized PC1 (28.7% explained var.) We need a measure of the “anomalousness” of a time series. 1 Rank points based on their local density. 2 Rank points based on whether they are within α-convex hulls of different radius. Visualisation of big time series data Yahoo web traffic 40 ● ●● ●● ● ●●● ● ● ● ● ●● ●● ● ● ●● ● ● ●●●●●●● ● ●●● ● ● ●● ●● ● ● ● ●● ●●● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●●●●● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● 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scor sss ne s ikiine spmp lu ots troug h 2 ● fsp ● peak s ritynt ea oi lin cp standardized PC2 (17.3% explained var.) What is “anomalous” vlstcA C rhheifantnFg1e d ● ● ● ● −2.5 0.0 2.5 standardized PC1 (28.7% explained var.) We need a measure of the “anomalousness” of a time series. 1 Rank points based on their local density. 2 Rank points based on whether they are within α-convex hulls of different radius. Visualisation of big time series data Yahoo web traffic 40 Bivariate kernel density n f̂ (x; H) = 1X n KH (x − Xi ) i=1 Xi ∈ a bivariate random sample {X1 , X2 , . . . , Xn } KH (x) is the standard normal kernel function H estimated by minimizing the sum of AMISE Rank points based on f̂ values in 2d PCA space. Visualisation of big time series data Yahoo web traffic 41 Bivariate kernel density n f̂ (x; H) = 1X n KH (x − Xi ) i=1 Xi ∈ a bivariate random sample {X1 , X2 , . . . , Xn } KH (x) is the standard normal kernel function H estimated by minimizing the sum of AMISE Rank points based on f̂ values in 2d PCA space. Visualisation of big time series data Yahoo web traffic 41 −4 −2 2 4 −6 5 1 −8 pc2 0 2 4 6 Bivariate density ranking 3 −5 0 5 pc1 Visualisation of big time series data Yahoo web traffic 42 40000 30000 20000 10000 0 6000 4000 2000 0 50000 40000 30000 20000 10000 0 40000 30000 20000 10000 0 5000 4000 3000 2000 1000 0 S7793 S8494 S10464 S7833 S1715 2015−02−28 2015−03−01 2015−03−02 2015−03−03 2015−03−04 2015−03−05 2015−03−06 2015−03−07 2015−03−08 2015−03−09 2015−03−10 2015−03−11 2015−03−12 2015−03−13 2015−03−14 2015−03−15 2015−03−16 2015−03−17 2015−03−18 2015−03−19 2015−03−20 2015−03−21 2015−03−22 2015−03−23 2015−03−24 2015−03−25 2015−03−26 2015−03−27 2015−03−28 2015−03−29 2015−03−30 2015−03−31 2015−04−01 value Bivariate density ranking date Visualisation of big time series data Yahoo web traffic 42 α-convex hulls The space generated by point pairs that can be touched by an empty disc of radius α. α → ∞ gives a convex hull. Points can become isolated when α is small. We rank points based on the value of α when they become isolated. Visualisation of big time series data Yahoo web traffic 43 α-convex hulls The space generated by point pairs that can be touched by an empty disc of radius α. α → ∞ gives a convex hull. Points can become isolated when α is small. We rank points based on the value of α when they become isolated. Visualisation of big time series data Yahoo web traffic 43 α-convex hulls The space generated by point pairs that can be touched by an empty disc of radius α. α → ∞ gives a convex hull. Points can become isolated when α is small. We rank points based on the value of α when they become isolated. Visualisation of big time series data Yahoo web traffic 43 α-convex hulls The space generated by point pairs that can be touched by an empty disc of radius α. α → ∞ gives a convex hull. Points can become isolated when α is small. We rank points based on the value of α when they become isolated. Visualisation of big time series data Yahoo web traffic 43 6 α-convex hull ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ● ●● ● ●●●●●●● ● ● ● ● ● ●●● ● ● ● ● ● ●● ●●● ● ● ● ● ●● ●● ● ●● ● ●● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●●●● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 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2015−03−04 2015−03−05 2015−03−06 2015−03−07 2015−03−08 2015−03−09 2015−03−10 2015−03−11 2015−03−12 2015−03−13 2015−03−14 2015−03−15 2015−03−16 2015−03−17 2015−03−18 2015−03−19 2015−03−20 2015−03−21 2015−03−22 2015−03−23 2015−03−24 2015−03−25 2015−03−26 2015−03−27 2015−03−28 2015−03−29 2015−03−30 2015−03−31 2015−04−01 40000 30000 20000 10000 0 6000 4000 2000 0 50000 40000 30000 20000 10000 0 40000 30000 20000 10000 0 5000 4000 3000 2000 1000 0 2015−02−28 2015−03−01 2015−03−02 2015−03−03 2015−03−04 2015−03−05 2015−03−06 2015−03−07 2015−03−08 2015−03−09 2015−03−10 2015−03−11 2015−03−12 2015−03−13 2015−03−14 2015−03−15 2015−03−16 2015−03−17 2015−03−18 2015−03−19 2015−03−20 2015−03−21 2015−03−22 2015−03−23 2015−03−24 2015−03−25 2015−03−26 2015−03−27 2015−03−28 2015−03−29 2015−03−30 2015−03−31 2015−04−01 value Top 5 anomalous time series date date 47 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 What next? 48 What next? Develop a more comprehensive set of features that are reliable measures and fast to compute. e.g., for finance data. Consider other dimension reduction methods and more than 2 dimensions. Develop dynamic and interactive visualization tools. Make methods available in an R package. Some of the methods are already available in the anomalous package for R on github. Visualisation of big time series data What next? 49 What next? Develop a more comprehensive set of features that are reliable measures and fast to compute. e.g., for finance data. Consider other dimension reduction methods and more than 2 dimensions. Develop dynamic and interactive visualization tools. Make methods available in an R package. Some of the methods are already available in the anomalous package for R on github. Visualisation of big time series data What next? 49 What next? Develop a more comprehensive set of features that are reliable measures and fast to compute. e.g., for finance data. Consider other dimension reduction methods and more than 2 dimensions. Develop dynamic and interactive visualization tools. Make methods available in an R package. Some of the methods are already available in the anomalous package for R on github. Visualisation of big time series data What next? 49 What next? Develop a more comprehensive set of features that are reliable measures and fast to compute. e.g., for finance data. Consider other dimension reduction methods and more than 2 dimensions. Develop dynamic and interactive visualization tools. Make methods available in an R package. Some of the methods are already available in the anomalous package for R on github. Visualisation of big time series data What next? 49 What next? Develop a more comprehensive set of features that are reliable measures and fast to compute. e.g., for finance data. Consider other dimension reduction methods and more than 2 dimensions. Develop dynamic and interactive visualization tools. Make methods available in an R package. Some of the methods are already available in the anomalous package for R on github. Visualisation of big time series data What next? 49 Further information å å å å Papers and R packages: robjhyndman.com Blog: robjhyndman.com/hyndsight Code: github.com/robjhyndman Email: Rob.Hyndman@monash.edu Visualisation of big time series data What next? 50

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