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Exploring the feature space of large time series collections 1
Exploring the
feature space of
large collections
of time se...
Exploring the feature space of large time series collections 1
Exploring the
feature space of
large collections
of time se...
Outline
1 M3 competition data
2 Yahoo web traffic
3 What next?
Exploring the feature space of large time series collections...
M3 forecasting competition
Exploring the feature space of large time series collections M3 competition data 3
M3 forecasting competition
Exploring the feature space of large time series collections M3 competition data 3
M3 forecasting competition
“The M3-Competition is a final attempt by the authors to
settle the accuracy issue of various ti...
M3 forecasting competition
Exploring the feature space of large time series collections M3 competition data 5
q
Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seaso...
Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seaso...
Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seaso...
An STL decomposition
Quarterly visitor nights: Mornington Peninsula
Yt = St + Tt + Rt St is periodic with mean 0
5.06.07.0...
Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Str...
Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Str...
Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Str...
Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Str...
Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Str...
Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Str...
Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Str...
Dimension reduction for time series
Exploring the feature space of large time series collections M3 competition data 9
q
Dimension reduction for time series
Exploring the feature space of large time series collections M3 competition data 9
q
S...
Dimension reduction for time series
Exploring the feature space of large time series collections M3 competition data 9
q
S...
Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 11
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 11
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1990 19...
Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 11
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 11
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 12
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 13
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 14
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Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 14
Time
5 10 15
Outline
1 M3 competition data
2 Yahoo web traffic
3 What next?
Exploring the feature space of large time series collections...
Yahoo web-traffic
Tens of thousands of time series collected at
one-hour intervals over one month.
Consisting of several se...
Yahoo web-traffic
busy233busy271busy50busy200busy369
2014−11−09
2014−11−10
2014−11−11
2014−11−12
2014−11−13
2014−11−14
2014...
Feature space
ACF1: first order autocorrelation = Corr(Yt, Yt−1)
Strength of trend and seasonality based on STL
Trend linea...
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
Exploring the feature space of large collections of time series
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Exploring the feature space of large collections of time series

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It is becoming increasingly common for organizations to collect very large amounts of data over time. Data visualization is essential for exploring and understanding structures and patterns, and to identify unusual observations. However, the sheer quantity of data available challenges current time series visualisation methods.

For example, Yahoo has banks of mail servers that are monitored over time. Many measurements on server performance are collected every hour for each of thousands of servers. We wish to identify servers that are behaving unusually.

Alternatively, we may have thousands of time series we wish to forecast, and we want to be able to identify the types of time series that are easy to forecast and those that are inherently challenging.

I will demonstrate a functional data approach to this problem using a vector of features on each time series, measuring characteristics of the series. For example, the features may include lag correlation, strength of seasonality, spectral entropy, etc. Then we use a principal component decomposition on the features, and plot the first few principal components. This enables us to explore a lower dimensional space and discover interesting structure and unusual observations.

Publié dans : Données & analyses
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Exploring the feature space of large collections of time series

  1. 1. Exploring the feature space of large time series collections 1 Exploring the feature space of large collections of time series Rob J Hyndman
  2. 2. Exploring the feature space of large time series collections 1 Exploring the feature space of large collections of time series Rob J Hyndman with Earo Wang, Nikolay Laptev Yanfei Kang, Kate Smith-Miles
  3. 3. Outline 1 M3 competition data 2 Yahoo web traffic 3 What next? Exploring the feature space of large time series collections M3 competition data 2
  4. 4. M3 forecasting competition Exploring the feature space of large time series collections M3 competition data 3
  5. 5. M3 forecasting competition Exploring the feature space of large time series collections M3 competition data 3
  6. 6. M3 forecasting competition “The M3-Competition is a final attempt by the authors to settle the accuracy issue of various time series methods. . . The extension involves the inclusion of more methods/ researchers (in particular in the areas of neural networks and expert systems) and more series.” Makridakis & Hibon, IJF 2000 3003 series All data from business, demography, finance and economics. Series length between 14 and 126. Either non-seasonal, monthly or quarterly. All time series positive. Exploring the feature space of large time series collections M3 competition data 4
  7. 7. M3 forecasting competition Exploring the feature space of large time series collections M3 competition data 5 q
  8. 8. 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. Exploring the feature space of large time series collections M3 competition data 6 John W Tukey Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985).
  9. 9. 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. Exploring the feature space of large time series collections M3 competition data 6 John W Tukey Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985).
  10. 10. 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. Exploring the feature space of large time series collections M3 competition data 6 John W Tukey Cognostics Computer-produced diagnostics (Tukey and Tukey, 1985).
  11. 11. An STL decomposition Quarterly visitor nights: Mornington 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 timeExploring the feature space of large time series collections M3 competition data 7
  12. 12. Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Var(Rt) Var(Yt−Tt) Strength of trend: 1 − Var(Rt) Var(Yt−St) Spectral entropy: H = − π −π fy(λ) log fy(λ)dλ, where fy(λ) is spectral density of Yt. Low values of H suggest a time series that is easier to forecast (more signal). Autocorrelations: r1, r2, r3, . . . Optimal Box-Cox transformation parameter λ Exploring the feature space of large time series collections M3 competition data 8
  13. 13. Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Var(Rt) Var(Yt−Tt) Strength of trend: 1 − Var(Rt) Var(Yt−St) Spectral entropy: H = − π −π fy(λ) log fy(λ)dλ, where fy(λ) is spectral density of Yt. Low values of H suggest a time series that is easier to forecast (more signal). Autocorrelations: r1, r2, r3, . . . Optimal Box-Cox transformation parameter λ Exploring the feature space of large time series collections M3 competition data 8
  14. 14. Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Var(Rt) Var(Yt−Tt) Strength of trend: 1 − Var(Rt) Var(Yt−St) Spectral entropy: H = − π −π fy(λ) log fy(λ)dλ, where fy(λ) is spectral density of Yt. Low values of H suggest a time series that is easier to forecast (more signal). Autocorrelations: r1, r2, r3, . . . Optimal Box-Cox transformation parameter λ Exploring the feature space of large time series collections M3 competition data 8
  15. 15. Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Var(Rt) Var(Yt−Tt) Strength of trend: 1 − Var(Rt) Var(Yt−St) Spectral entropy: H = − π −π fy(λ) log fy(λ)dλ, where fy(λ) is spectral density of Yt. Low values of H suggest a time series that is easier to forecast (more signal). Autocorrelations: r1, r2, r3, . . . Optimal Box-Cox transformation parameter λ Exploring the feature space of large time series collections M3 competition data 8
  16. 16. Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Var(Rt) Var(Yt−Tt) Strength of trend: 1 − Var(Rt) Var(Yt−St) Spectral entropy: H = − π −π fy(λ) log fy(λ)dλ, where fy(λ) is spectral density of Yt. Low values of H suggest a time series that is easier to forecast (more signal). Autocorrelations: r1, r2, r3, . . . Optimal Box-Cox transformation parameter λ Exploring the feature space of large time series collections M3 competition data 8
  17. 17. Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Var(Rt) Var(Yt−Tt) Strength of trend: 1 − Var(Rt) Var(Yt−St) Spectral entropy: H = − π −π fy(λ) log fy(λ)dλ, where fy(λ) is spectral density of Yt. Low values of H suggest a time series that is easier to forecast (more signal). Autocorrelations: r1, r2, r3, . . . Optimal Box-Cox transformation parameter λ Exploring the feature space of large time series collections M3 competition data 8
  18. 18. Candidate features STL decomposition Yt = St + Tt + Rt Seasonal period Strength of seasonality: 1 − Var(Rt) Var(Yt−Tt) Strength of trend: 1 − Var(Rt) Var(Yt−St) Spectral entropy: H = − π −π fy(λ) log fy(λ)dλ, where fy(λ) is spectral density of Yt. Low values of H suggest a time series that is easier to forecast (more signal). Autocorrelations: r1, r2, r3, . . . Optimal Box-Cox transformation parameter λ Exploring the feature space of large time series collections M3 competition data 8
  19. 19. Dimension reduction for time series Exploring the feature space of large time series collections M3 competition data 9 q
  20. 20. Dimension reduction for time series Exploring the feature space of large time series collections M3 competition data 9 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
  21. 21. Dimension reduction for time series Exploring the feature space of large time series collections M3 competition data 9 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 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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
  22. 22. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 10 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.
  23. 23. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 10 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
  24. 24. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 10 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
  25. 25. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 10 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
  26. 26. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 10 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
  27. 27. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 10 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
  28. 28. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 10 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 1.00 value Lambda
  29. 29. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 11 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 q q 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 q q 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 q q 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 q q 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 q qqq 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 q qq 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 qq qq q qq qqqqqq q q qqq q q q q q q q q q q qqq q q q q q q q q q q qqq q 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 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 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 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 −2 0 2 4 −3−2−10123 PC1 PC2 q q q q
  30. 30. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 11 N1627 1990 1991 1992 1993 1994 N0636 1920 1925 1930 1935 1940 1945 N2981 0 10 20 30 40 50 60 N2699 1984 1986 1988 1990 1992
  31. 31. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 11 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 q q 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 q q 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 q q 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 q q 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 q qqq 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 q qq 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 qq qq q qq qqqqqq q q qqq q q q q q q q q q q qqq q q q q q q q q q q qqq q 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 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 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 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 −2 0 2 4 −3−2−10123 PC1 PC2 Yahoo log returns series
  32. 32. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 11 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 q q 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 q q 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 q q 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 q q 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 q qqq 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 q qq 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 qq qq q qq qqqqqq q q qqq q q q q q q q q q q qqq q q q q q q q q q q qqq q 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 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 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 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 −2 0 2 4 −3−2−10123 PC1 PC2 qYahoo log returns series
  33. 33. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 12 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 q q 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 q q 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 q q 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 q q 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 q qqq 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 q qq 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 qq qq q qq qqqqqq q q qqq q q q q q q q q q q qqq q q q q q q q q q q qqq q 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 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 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 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 −2 0 2 4 −3−2−10123 PC1 PC2 White noise series
  34. 34. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 13 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 q q 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 q q 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 q q 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 q q 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 q qqq 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 q qq 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 qq qq q qq qqqqqq q q qqq q q q q q q q q q q qqq q q q q q q q q q q qqq q 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 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 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 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 −2 0 2 4 −3−2−10123 PC1 PC2 Random walk series
  35. 35. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 14 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 q q 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 q q 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 q q 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 q q 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 q qqq 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 q qq 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 qq qq q qq qqqqqq q q qqq q q q q q q q q q q qqq q q q q q q q q q q qqq q 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 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 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 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 −2 0 2 4 −3−2−10123 PC1 PC2 q q
  36. 36. Feature space of M3 data Exploring the feature space of large time series collections M3 competition data 14 Time 5 10 15
  37. 37. Outline 1 M3 competition data 2 Yahoo web traffic 3 What next? Exploring the feature space of large time series collections Yahoo web traffic 15
  38. 38. 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. Exploring the feature space of large time series collections Yahoo web traffic 16
  39. 39. Yahoo web-traffic busy233busy271busy50busy200busy369 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 date 25 30 35 40 45 20 25 30 35 40 50 60 70 10 15 20 25 50 60 memory460memory429memory147memory413memory484 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 date value 0 5000 10000 15000 20000 200 400 600 0 5000 10000 15000 20000 500 1000 0 5000 10000 15000 20000 25000 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 date value Exploring the feature space of large time series collections Yahoo web traffic 17
  40. 40. 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 DKL(P Q) = 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 Exploring the feature space of large time series collections Yahoo web traffic 18

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