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Time Series Prediction Forecasting the Future and Understanding the Past Santa Fe Institute Proceedings on the Studies in the Sciences of Complexity Edited by Andreas Weingend and Neil Gershenfeld. NIST Complex System Program Perspectives on Standard Benchmark Data

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Time Series PredictionForecasting the Future andUnderstanding the PastSanta Fe Institute Proceedings on the Studies in the Sciences of ComplexityEdited by Andreas Weingend and Neil Gershenfeld

NIST Complex System Program

Perspectives on Standard Benchmark Data

In Quantifying Complex Systems

Vincent Stanford

Complex Systems Test Bed project

August 31, 2007

chaos in nature theory and technology
Chaos in Nature, Theory, and Technology

Rings of Saturn

Lorentz Attractor

Aircraft dynamics at

high angles of attack

time series prediction a santa fe institute competition using standard data sets
Time Series PredictionA Santa Fe Institute competition using standard data sets
  • Santa Fe Institute (SFI) founded in 1984 to “… focus the tools of traditional scientific disciplines and emerging computer resources on … the multidisciplinary study of complex systems…”
  • “This book is the result of an unsuccessful joke. … Out of frustration with the fragmented and anecdotal literature, we made what we thought was a humorous suggestion: run a competition. …no one laughed.”
  • Time series from physics, biology, economics, …, beg the same questions:
    • What happens next?
    • What kind of system produced this time series?
    • How much can we learn about the producing system?
  • Quantitative answers can permit direct comparisons
  • Make some standard data sets in consultation with subject matter experts in a variety of areas.
  • Very NISTY; but we are in a much better position to do this in the age of Google and the Internet.
selecting benchmark data sets for inclusion in the book
Selecting benchmark data setsFor inclusion in the book
  • Subject matter expert advisor group:
    • Biology
    • Economics
    • Astrophysics
    • Numerical Analysis
    • Statistics
    • Dynamical Systems
    • Experimental Physics
the data sets
The Data Sets
  • Far-infrared laser excitation
  • Sleep Apnea
  • Currency exchange rates
  • Particle driven in nonlinear multiple well potentials
  • Variable star data
  • J. S. Bach fugue notes
j s bach benchmark
J.S. Bach benchmark
  • Dynamic, yes.
  • But is it an iterative map?
  • Is it amenable to time delay embedding?
competition tasks
Competition Tasks
  • Predict the withheld continuations of the data sets provided for training and measure errors
  • Characterize the systems as to:
    • Degrees of Freedom
    • Predictability
    • Noise characteristics
    • Nonlinearity of the system
  • Infer a model for the governing equations
  • Describe the algorithms employed
complex time series benchmark taxonomy
Complex Time Series Benchmark Taxonomy
  • Synthetic
  • Nonstationary
  • Stochastic
  • Noisy
  • Long
  • Blind
  • Nonlinear
  • Vector
  • Many trials
  • Discontinuous
  • Switching
  • Catastrophes
  • Episodes
  • Natural
  • Stationary
  • Low dimensional
  • Clean
  • Short
  • Documented
  • Linear
  • Scalar
  • One trial
  • Continuous
time honored linear models
Time honored linear models
  • Auto Regressive Moving Average (ARMA)
  • Many linear estimation techniques based on Least Squares, or Least Mean Squares
  • Power spectra, and Autocorrelation characterize such linear systems
  • Randomness comes only from forcing function x(t)
simple nonlinear systems can exhibit chaotic behavior
Simple nonlinear systemscan exhibit chaotic behavior
  • Spectrum, autocorrelation, characterize linear systems, not these
  • Deterministic chaos looks random to linear analysis methods
  • Logistic map is an early example (Elam 1957).

Logisic map 2.9 < r < 3.99

understanding and learning comments from sfi
Understanding and learningcomments from SFI
  • Weak to Strong models - many parameters to few
  • Data poor to data rich
  • Theory poor to theory rich
  • Weak models progress to strong, e.g. planetary motion:
    • Tycho Brahe: observes and records raw data
    • Kepler: equal areas swept in equal time
    • Newton: universal gravitation, mechanics, and calculus
    • Poincaré: fails to solve three body problem
    • Sussman and Wisdom: Chaos ensues with computational solution!
  • Is that a simplification?
discovering properties of data and inferring complex models
Discovering properties of dataand inferring (complex) models
  • Can’t decompose an output into the product of input and transfer function Y(z)=H(z)X(z) by doing a Z, Laplace, or Fourier transform.
  • Linear Perceptrons were shown to have severe limitations by Minsky and Papert
  • Perceptrons with non-linear threshold logic can solve XOR and many classifications not available with linear version
  • But according to SFI: “Learning XOR is as interesting as memorizing the phone book. More interesting - and more realistic - are real-world problems, such as prediction of financial data.”
  • Many approaches are investigated
time delay embedding differs from traditional experimental measurements
Time delay embeddingDiffers from traditional experimental measurements
  • Provides detailed information about degrees of freedom beyond the scalar measured
  • Rests on probabilistic assumptions - though not guaranteed to be valid for any particular system
  • Reconstructed dynamics are seen through an unknown “smooth transformation”
  • Therefore allows precise questions only about invariants under “smooth transformations”
  • It can still be used for forecasting a time series and “characterizing essential features of the dynamics that produced it”
slide14
Time delay embedding theorems“The most important Phase Space Reconstruction technique is the method of delays”

Vector

Sequence

Scalar

Measurement

  • Assuming the dynamicsf(X) on a V dimensional manifold has a strange attractor A with box counting dimension dA
  • s(X) is a twice differentiable scalar measurement giving{sn}={s(Xn)}
  • M is called the embedding dimension
  •  is generally referred to as the delay, or lag
  • Embedding theorems: if {sn} consists of scalar measurements of the state a dynamical system then, under suitable hypotheses, the time delay embedding {Sn} is a one-to-one transformed image of the {Xn}, provided M > 2dA. (e.g. Takens 1981, Lecture Notes in Mathematics, Springer-Verlag; or Sauer and Yorke, J. of Statistical Physics, 1991)

Time delay

Vectors

time series prediction many different techniques thrown at the data to see if anything sticks
Time series predictionMany different techniques thrown at the data to “see if anything sticks”

Examples:

  • Delay coordinate embedding - Short term prediction by filtered delay coordinates and reconstruction with local linear models of the attractor (T. Sauer).
  • Neural networks with internal delay lines - Performed well on data set A (E. Wan), (M. Mozer)
  • Simple architectures for fast machines - “Know the data and your modeling technique” (X. Zhang and J. Hutchinson)
  • Forecasting pdf’s using HMMs with mixed states - Capturing “Embedology” (A. Frasar and A. Dimiriadis)
  • More…
time series characterization many different techniques thrown at the data to see if anything sticks
Time series characterizationMany different techniques thrown at the data to “see if anything sticks”

Examples:

  • Stochastic and deterministic modeling - Local linear approximation to attractors (M. Kasdagali and A. Weigend)
  • Estimating dimension and choosing time delays - Box counting (F. Pineda and J. Sommerer)
  • Quantifying Chaos using information-theoretic functionals - mutual information and nonlinearity testing.(M. Palus)
  • Statistics for detecting deterministic dynamics - Course grained flow averages (D. Kaplan)
  • More…
what to make of this handbook for the corpus driven study of nonlinear dynamics
What to make of this?Handbook for the corpus driven study of nonlinear dynamics

Very NISTY:

  • Convene a panel of leading researchers
  • Identify areas of interest where improved characterization and predictive measurements can be of assistance to the community
  • Identify standard reference data sets:
    • Development corpra
    • Test sets
  • Develop metrics for prediction and characterization
  • Evaluate participants
  • Is there a sponsor?
  • Are there areas of special importance to communities we know? For example: predicting catastrophic failures of machines from sensors.