J. C. (Clint) Sprott

1
Time-Series Analysis

• 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

2
Agenda

• Introductory lecture
• Hands-on tutorial
• Strange attractors
• Break
• Individual exploration
• Closing comments

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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.
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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.
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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

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Applications

• Prediction
• Noise reduction
• Scientific insight
• Control

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Examples

• Weather data
• Climate data
• Tide levels
• Seismic waves
• Cepheid variable stars
• Sunspots
• Financial markets
• Ecological fluctuations
• EKG and EEG data

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(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
• ...

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Methods

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

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Resources
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Hierarchy of Dynamical Behaviors
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Typical Experimental Data
5
x
-5
500
Time
0
13
Stationarity
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Detrending
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Detrended
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Case Study
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First Return Map
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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

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Measurement Functions
Xn1 1 1.4X2 0.3Yn Yn1 Xn
Hénon map
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Correlation Dimension
N(r) ? rD2
D2 dlogN(r)/dlogr
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Inevitable Ambiguity
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Lyapunov Exponent
? Rn ? R0e?n
? ltln?Rn/?R0gt
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Principal Component Analysis
x(t)
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State-space Prediction
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Surrogate Data
Original time series
Shuffled surrogate
Phase randomized
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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

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Tutorial using CDA
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Types of Attractors
Limit Cycle
Fixed Point
Focus
Node
Torus
Strange Attractor
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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

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Individual Exploration using CDA
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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

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Some General High-Dimensional Models
Fourier Series
Linear Autoregression
(ARMA, LPC, MEM)
Nonlinear Autogression
(Polynomial Map)
Neural Network
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Artificial Neural Network
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Summary

• Nature is complex
• Simple models may suffice

but
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References

• http//sprott.physics.wisc.edu/lectures/tsa.ppt
  (this presentation)
• http//sprott.physics.wisc.edu/cda.htm (Chaos
  Data Analyzer)
• sprott_at_physics.wisc.edu (my email)