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)

sprott@physics.wisc.edu (my

email)