Presentation on theme: "Time-Series Analysis J. C. (Clint) Sprott"— Presentation transcript:
1 Time-Series Analysis J. C. (Clint) Sprott 3/31/2017J. C. (Clint) SprottDepartment of PhysicsUniversity of Wisconsin - MadisonWorkshop presented at the2004 SCTPLS Annual Conferenceat Marquette Universityon July 15, 2004Entire presentation available on WWW
3 MotivationMany 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.
4 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.
5 Precautions More art than science No sure-fire methods Easy to fool yourselfMany published false claimsMust use multiple testsConclusions seldom definitiveCompare with surrogate dataMust ask the right questions“Is it chaos?” too simplistic
7 Examples Weather data Climate data Tide levels Seismic waves Cepheid variable starsSunspotsFinancial marketsEcological fluctuationsEKG and EEG data…
8 (Non-)Time Series Core samples Terrain features Sequence of letters in written textNotes in a musical compositionBases in a DNA moleculeHeartbeat intervalsDripping faucetNecker cube flipsEye fixations during a visual task...
9 Methods Linear (traditional) Nonlinear (chaotic) Fourier Analysis AutocorrelationARMALPC …Nonlinear (chaotic)State space reconstructionCorrelation dimensionLyapunov exponentPrinciple component analysisSurrogate data …
12 Typical Experimental Data 3/31/2017Typical Experimental Data5xNot usually shown in textbooksCould be:Plasma fluctuationsStock market dataMeteorological dataEEG or EKGEcological dataetc...Until recently, no hope of detailed understandingCould be an example of deterministic chaos-5Time500
18 Time-Delayed Embedding Space Plot x(t) vs. x(t-), x(t-2), x(t-3), …Embedding dimension is # of delaysMust choose and dim carefullyOrbit does not fill the spaceDiffiomorphic to actual orbitDim of orbit = min # of variablesx(t) can be any measurement fcn
25 Surrogate Data Original time series Shuffled surrogate Phase randomized
26 General Strategy Verify integrity of the data Test for stationarity Look at return maps, etc.Look at autocorrelation functionLook at power spectrumCalculate correlation dimensionCalculate Lyapunov exponentCompare with surrogate data setsConstruct modelsMake predictions from models
28 Types of Attractors Examples: simple damped pendulum 3/31/2017Types of AttractorsLimit CycleFixed PointFocusNodeTorusStrange AttractorExamples:simple damped pendulumdriven mass on a springinner tube
29 Strange Attractors Occur in infinite variety (like snowflakes) 3/31/2017Strange AttractorsLimit set as t Set of measure zeroBasin of attractionFractal structurenon-integer dimensionself-similarityinfinite detailChaotic dynamicssensitivity to initial conditionstopological transitivitydense periodic orbitsAesthetic appealOccur in infinite variety (like snowflakes)Produced many millions, looked at over 100,000Like pornography, know it when you see it