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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 Time-Series Analysis
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Time-Series Analysis

Dec 31, 2015

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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 – - PowerPoint PPT Presentation
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Page 1: Time-Series Analysis

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

Page 2: Time-Series Analysis

Agenda

Introductory lecture

Hands-on tutorial

Strange attractors

– Break –

Individual exploration

Closing comments

Page 3: Time-Series Analysis

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.

Page 4: Time-Series Analysis

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.

Page 5: Time-Series Analysis

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

Page 6: Time-Series Analysis

Applications

Prediction

Noise reduction

Scientific insight

Control

Page 7: Time-Series Analysis

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

Page 8: Time-Series Analysis

(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 ...

Page 9: Time-Series Analysis

Methods Linear (traditional)

Fourier Analysis Autocorrelation ARMA LPC …

Nonlinear (chaotic) State space reconstruction Correlation dimension Lyapunov exponent Principle component analysis Surrogate data …

Page 10: Time-Series Analysis

Resources

Page 11: Time-Series Analysis

Hierarchy of Dynamical Behaviors

Page 12: Time-Series Analysis

Typical Experimental Data

Time0 500

x

5

-5

Page 13: Time-Series Analysis

Stationarity

Page 14: Time-Series Analysis

Detrending

Page 15: Time-Series Analysis

Detrended

Page 16: Time-Series Analysis

Case Study

Page 17: Time-Series Analysis

First Return Map

Page 18: Time-Series Analysis

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

Page 19: Time-Series Analysis

Measurement Functions

Hénon map: Xn+1 = 1 – 1.4X2 + 0.3Yn

Yn+1 = Xn

Page 20: Time-Series Analysis

Correlation Dimension

D2 = dlogN(r)/dlogr

N(r) rD2

Page 21: Time-Series Analysis

Inevitable Ambiguity

Page 22: Time-Series Analysis

Lyapunov Exponent

= <ln|Rn/R0|>

Rn = R0en

Page 23: Time-Series Analysis

Principal Component Analysisx(t)

Page 24: Time-Series Analysis

State-space Prediction

Page 25: Time-Series Analysis

Surrogate Data

Original time series

Shuffled surrogate

Phase randomized

Page 26: Time-Series Analysis

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

Page 27: Time-Series Analysis

Tutorial using CDA

Page 28: Time-Series Analysis

Types of AttractorsFixed Point Limit Cycle

Torus Strange Attractor

Focus Node

Page 29: Time-Series Analysis

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

Page 30: Time-Series Analysis

Individual Exploration using CDA

Page 31: Time-Series Analysis

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

Page 32: Time-Series Analysis

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)

Page 33: Time-Series Analysis

Artificial Neural Network

Page 34: Time-Series Analysis

Summary

Nature is complex

Simple models may suffice

but

Page 35: Time-Series Analysis

References

http://sprott.physics.wisc.edu/lec

tures/tsa.ppt

(this presentation)

http://sprott.physics.wisc.edu/cd

a.htm

(Chaos Data Analyzer)

[email protected] (my

email)