1. Time-Series Analysis J. C. (Clint) Sprott
3/31/2017
J. C. (Clint) Sprott
Department of Physics
University of Wisconsin - Madison
Workshop presented at the
2004 SCTPLS Annual Conference
at Marquette University
on July 15, 2004
Entire presentation available on WWW

2. Agenda Introductory lecture Hands-on tutorial Strange attractors
– Break –
Individual exploration
Closing comments

3. Motivation
Many quantities in nature fluctuate in time. Examples are the stock market, the weather, seismic waves, sunspots, heartbeats, and plant and animal populations. Until recently it was assumed that such fluctuations are a consequence of random and unpredictable events. With the discovery of chaos, it has come to be understood that some of these cases may be a result of deterministic chaos and hence predictable in the short term and amenable to simple modeling. Many tests have been developed to determine whether a time series is random or chaotic, and if the latter, to quantify the chaos. If chaos is found, it may be possible to improve the short-term predictability and enhance understanding of the governing process.

4. Goals
This workshop will provide examples of time-series data from real systems as well as from simple chaotic models. A variety of tests will be described including linear methods such as Fourier analysis and autoregression, and nonlinear methods using state-space reconstruction. The primary methods for nonlinear analysis include calculation of the correlation dimension and largest Lyapunov exponent, as well as principal component analysis and various nonlinear predictors. Methods for detrending, noise reduction, false nearest neighbors, and surrogate data tests will be explained. Participants will use the "Chaos Data Analyzer" program to analyze a variety of typical time-series records and will learn to distinguish chaos from colored noise and to avoid the many common pitfalls that can lead to false conclusions. No previous knowledge or experience is assumed.

5. Precautions More art than science No sure-fire methods
Easy to fool yourself
Many published false claims
Must use multiple tests
Conclusions seldom definitive
Compare with surrogate data
Must ask the right questions
“Is it chaos?” too simplistic

6. Applications
Prediction
Noise reduction
Scientific insight
Control

7. Examples Weather data Climate data Tide levels Seismic waves
Cepheid variable stars
Sunspots
Financial markets
Ecological fluctuations
EKG and EEG data …

8. (Non-)Time Series Core samples Terrain features
Sequence of letters in written text
Notes in a musical composition
Bases in a DNA molecule
Heartbeat intervals
Dripping faucet
Necker cube flips
Eye fixations during a visual task
...

9. Methods Linear (traditional) Nonlinear (chaotic) Fourier Analysis
Autocorrelation
ARMA
LPC …
Nonlinear (chaotic)
State space reconstruction
Correlation dimension
Lyapunov exponent
Principle component analysis
Surrogate data …

10. Resources

11. Hierarchy of Dynamical Behaviors

12. Typical Experimental Data
3/31/2017
Typical Experimental Data 5 x
Not usually shown in textbooks
Could be:
Plasma fluctuations
Stock market data
Meteorological data
EEG or EKG
Ecological data
etc...
Until recently, no hope of detailed understanding
Could be an example of deterministic chaos -5
Time
500

13. Stationarity

14. Detrending

15. Detrended

16. Case Study

17. First Return Map

18. Time-Delayed Embedding Space
Plot x(t) vs. x(t-), x(t-2), x(t-3), …
Embedding dimension is # of delays
Must choose  and dim carefully
Orbit does not fill the space
Diffiomorphic to actual orbit
Dim of orbit = min # of variables
x(t) can be any measurement fcn

19. Measurement Functions
Xn+1 = 1 – 1.4X Yn
Yn+1 = Xn
Hénon map:

20. Correlation Dimension
N(r)  rD2
D2 = dlogN(r)/dlogr

21. Inevitable Ambiguity

22. Lyapunov Exponent
 Rn =  R0en
 = <ln|Rn/R0|>

23. Principal Component Analysis
x(t)

24. State-space Prediction

25. Surrogate Data Original time series Shuffled surrogate
Phase randomized

26. General Strategy Verify integrity of the data Test for stationarity
Look at return maps, etc.
Look at autocorrelation function
Look at power spectrum
Calculate correlation dimension
Calculate Lyapunov exponent
Compare with surrogate data sets
Construct models
Make predictions from models

27. Tutorial using CDA

28. Types of Attractors Examples: simple damped pendulum
3/31/2017
Types of Attractors
Limit Cycle
Fixed Point
Focus
Node
Torus
Strange Attractor
Examples:
simple damped pendulum
driven mass on a spring
inner tube

29. Strange Attractors Occur in infinite variety (like snowflakes)
3/31/2017
Strange Attractors
Limit set as t  
Set of measure zero
Basin of attraction
Fractal structure
non-integer dimension
self-similarity
infinite detail
Chaotic dynamics
sensitivity to initial conditions
topological transitivity
dense periodic orbits
Aesthetic appeal
Occur in infinite variety (like snowflakes)
Produced many millions, looked at over 100,000
Like pornography, know it when you see it

30. Individual Exploration using CDA

31. Practical Considerations
Calculation speed
Required number of data points
Required precision of the data
Noisy data
Multivariate data
Filtered data
Missing data
Nonuniformly sampled data
Nonstationary data

32. Some General High-Dimensional Models
Fourier Series:
Linear Autoregression:
(ARMA, LPC, MEM…)
Nonlinear Autogression:
(Polynomial Map)
Neural Network:

33. Artificial Neural Network

34. Simple models may suffice
Summary
3/31/2017
Nature is complex
Simple models may suffice
but

35. 3/31/2017
References
tures/tsa.ppt (this presentation)
a.htm (Chaos Data Analyzer)
(my )