J. C. (Clint) SprottDepartment of Physics
University of Wisconsin - Madison
Workshop presented at the
2004 SCTPLS Annual Conference
at Marquette University
on July 15, 2004
Time-Series Analysis
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.
GoalsThis 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.
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
Applications
Prediction
Noise reduction
Scientific insight
Control
Examples Weather data Climate data Tide levels Seismic waves Cepheid variable stars Sunspots Financial markets Ecological fluctuations EKG and EEG data …
(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 ...
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
Time0 500
x
5
-5
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
Diffiomorphic to actual orbit
Dim of orbit = min # of variables
x(t) can be any measurement fcn
Measurement Functions
Hénon map: Xn+1 = 1 – 1.4X2 + 0.3Yn
Yn+1 = Xn
Correlation Dimension
D2 = dlogN(r)/dlogr
N(r) rD2
Inevitable Ambiguity
Lyapunov Exponent
= <ln|Rn/R0|>
Rn = R0en
Principal Component Analysisx(t)
State-space Prediction
Surrogate Data
Original time series
Shuffled surrogate
Phase randomized
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
Tutorial using CDA
Types of AttractorsFixed Point Limit Cycle
Torus Strange Attractor
Focus Node
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
Individual Exploration using CDA
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
Some General High-Dimensional Models
tiibtiN
i iaatx sincos1
o)(
noise)(1
o)(
itN
ixiaatx
)(1
)(1
o)( jtxN
j ijaiaitN
ixatx
)(1
tanh1
o)( jtxD
j ijaN
i ibbtx
Fourier Series:
Linear Autoregression:
Nonlinear Autogression:
Neural Network:
(ARMA, LPC, MEM…)
(Polynomial Map)
Artificial Neural Network
Summary
Nature is complex
Simple models may suffice
but
References
http://sprott.physics.wisc.edu/lec
tures/tsa.ppt
(this presentation)
http://sprott.physics.wisc.edu/cd
a.htm
(Chaos Data Analyzer)
email)
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