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
Agenda

Introductory lecture

Hands-on tutorial

Strange attractors

– Break –

Individual exploration

Closing comments
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.
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 statespace 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 timeseries 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.
Precautions

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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
Applications

Prediction

Noise reduction

Scientific insight

Control
Examples
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Weather data
Climate data
Tide levels
Seismic waves
Cepheid variable stars
Sunspots
Financial markets
Ecological fluctuations
EKG and EEG data
…
(Non-)Time Series
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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
...
Methods

Linear (traditional)
Fourier Analysis
 Autocorrelation
 ARMA
 LPC …


Nonlinear (chaotic)
State space reconstruction
 Correlation dimension
 Lyapunov exponent
 Principle component analysis
 Surrogate data …

Resources
Hierarchy of Dynamical
Behaviors
Typical Experimental Data
5
x
-5
0
Time
500
Stationarity
Detrending
Detrended
Case Study
First Return Map
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
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Diffiomorphic to actual orbit
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Dim of orbit = min # of variables
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x(t) can be any measurement fcn
Measurement Functions
Hénon map:
Xn+1 = 1 – 1.4X2 + 0.3Yn
Yn+1 = Xn
Correlation Dimension
N(r)  rD2
D2 = dlogN(r)/dlogr
Inevitable Ambiguity
Lyapunov Exponent
 Rn =  R0
n
e
 = <ln|Rn/R0|>
Principal Component Analysis
x(t)
State-space Prediction
Surrogate Data
Original time series
Shuffled surrogate
Phase randomized
General Strategy
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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
Tutorial using CDA
Types of Attractors
Fixed Point
Focus
Limit Cycle
Node
Torus
Strange Attractor
Strange Attractors
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Limit set as t  
Set of measure zero
Basin of attraction
Fractal structure
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Chaotic dynamics
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non-integer dimension
self-similarity
infinite detail
sensitivity to initial conditions
topological transitivity
dense periodic orbits
Aesthetic appeal
Individual Exploration
using CDA
Practical Considerations

Calculation speed
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Required number of data points
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Required precision of the data
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Noisy data
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Multivariate data
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Filtered data
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Missing data
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Nonuniformly sampled data
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Nonstationary data
Some General High-Dimensional
Models
N
Fourier Series: x(t )  ao   a cosit  b sin it
i
i1 i
N
Linear Autoregression: x(t )  ao   a x(t  i)  noise
i
i1
(ARMA, LPC, MEM…)




N
N
Nonlinear Autogression: x(t )  ao   x(t  i) a   a x(t  j)
ij

 i


i

1
j

1
(Polynomial Map)


Neural Network:
N
D
x(t )  bo   b tanh  a x(t  j)
i1 i
j1 ij
Artificial Neural Network
Summary

Nature is complex
but

Simple models
may suffice
References

http://sprott.physics.wisc.edu/lec
tures/tsa.ppt (this presentation)

http://sprott.physics.wisc.edu/cd
a.htm (Chaos Data Analyzer)

sprott@physics.wisc.edu (my
email)