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.

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. 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. 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. M3 forecasting competition
Exploring the feature space of large time series collections M3 competition data 3

5. M3 forecasting competition
Exploring the feature space of large time series collections M3 competition data 3

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. M3 forecasting competition
Exploring the feature space of large time series collections M3 competition data 5
q

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Dimension reduction for time series
Exploring the feature space of large time series collections M3 competition data 9
q

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. 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
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−3
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−2 0 2 4
PC1
PC2
Feature
calculation
Principal
component
decomposition

22. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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PC1
PC2
First two PCs explain 68% of variation.

23. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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−2
−1
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3
−2 0 2 4
PC1
PC2
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9
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24. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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−3
−2
−1
0
1
2
3
−2 0 2 4
PC1
PC2
0.00
0.25
0.50
0.75
value
Season

25. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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−3
−2
−1
0
1
2
3
−2 0 2 4
PC1
PC2
0.25
0.50
0.75
value
Trend

26. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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−3
−2
−1
0
1
2
3
−2 0 2 4
PC1
PC2
0.0
0.5
value
ACF

27. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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−3
−2
−1
0
1
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3
−2 0 2 4
PC1
PC2
0.5
0.6
0.7
0.8
0.9
value
SpecEntr

28. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 10
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−3
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−1
0
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−2 0 2 4
PC1
PC2
0.00
0.25
0.50
0.75
1.00
value
Lambda

29. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 11
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−2 0 2 4
−3−2−10123
PC1
PC2
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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. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 11
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−2 0 2 4
−3−2−10123
PC1
PC2
Yahoo log returns series

32. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 11
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−2 0 2 4
−3−2−10123
PC1
PC2
qYahoo log returns series

33. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 12
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−2 0 2 4
−3−2−10123
PC1
PC2
White noise series

34. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 13
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−2 0 2 4
−3−2−10123
PC1
PC2
Random walk series

35. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 14
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−2 0 2 4
−3−2−10123
PC1
PC2
q
q

36. Feature space of M3 data
Exploring the feature space of large time series collections M3 competition data 14
Time
5 10 15

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. 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. 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
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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. 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