Chaos and Time-Series Analysis

Physics 505: Topics in Physics

Fall 2000 - 2 credits

Time & Place: 3:30-5:10 p.m., Tuesdays, 1313 Sterling Hall

Instructor: J. C. Sprott, 3285 Chamberlin, 263-4449, sprott@juno.physics.wisc.edu

Prerequisites: Consent of instructor (calculus and programming experience useful)

Text: Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers by Robert C. Hilborn, Second Edition (Oxford University Press, 2000)

This course is an introduction to the exciting new developments in chaos and related topics in nonlinear dynamics, including the detection and quantification of chaos in experimental data. Emphasis will be on physical concepts rather than mathematical proofs and derivations. Fifteen, 100-minute lectures as listed below will include demonstrations, computer animations, slides, and videos. Homework will consist of weekly programming assignments, and so access to a computer (any type) and some programming experience (any language) is assumed. The course will be taught at a level accessible to graduate and advanced undergraduate students in all fields of science and engineering. For further information, see http://sprott.physics.wisc.edu/phys505/.

Course Outline:

Introduction and Overview

One-Dimensional Maps

Dynamical Systems Theory

Chaotic Dissipative Flows

Iterated Maps

Strange Attractors

Stability and Bifurcations

Hamiltonian Chaos

Lyapunov Exponents and Entropy

Nonlinear Prediction and Noise Reduction

Fractals

Calculation of Fractal Dimension

Multifractals

Non-Attracting Chaotic Sets

Spatiotemporal Chaos and Complexity

Statistical Model Building with Reproducing Kernel Spaces and Splines

T Th 4:00-5:15 Fall 2000 CS 1207 709 NOT required

This course is about various aspects of multivariate function estimation and statistical model building given scattered, noisy direct, and indirect data, mostly via the use of smoothing spline and reproducing kernel Hilbert space (rkhs) techniques. Use of public software will be included. Upon completing the course the student should be able to apply modern multivariate smoothing spline and related multivariate function estimation methods to medical, environmental, atmospheric, and economic data sets. Open problems will also also be discussed.

No prior knowledge of Hilbert space is required. Students who already have a background in rkhs (from Statistics 840) may take the course and do special reading assignments while the introductory material is covered.

1. Background, introduction to the theory of reproducing kernel Hilbert spaces and the theory of function estimation via variational/regularization/penalty methods. Smoothing splines on the circle, the plane, the unit cube and other domains. Polynomial, thin plate, regression, partial, hybrid, and ANOVA splines; radial basis functions. Imposition of side conditions and the insertion of auxiliary information.

2. Adaptive estimation of multiple smoothing and tuning parameters and the bias-variance tradeoff. Generalized cross validation, unbiased risk, maximum likelihood and generalized approximate crossvalidation estimates. Degrees of freedom for signal. Applications in penalized likelihood, Tihonov regularization, and related penalty methods. Model selection methods and confidence intervals.

3. Numerical methods for medium sized to very large data sets. The randomized trace estimate for degrees of freedom for signal. Early termination of iterative methods as a form of regularization.

4. Applications in machine learning, (support vector machines), medicine (risk factor modeling), environmental data analysis (time-space models), economics, meteorology (data assimilation), remote sensing (ill-posed inverse problems), and merging of observations and dynamical systems models (as in climate and numerical weather prediction) will be discussed, according to the interests of the class.

Prerequisites: - Statistics Majors, mathematical maturity evidenced by readiness to take 709, and either multivariate analysis, or, some exposure to Hilbert spaces, or cons. instr. Those unfamiliar with Hilbert spaces will be asked to read the first 33 pages of Akhiezer and Glazman, Theory of Linear Operators in Hilbert Spaces, vol. I at the beginning of the course. Graduate students in CS, AOS and other physical sciences, engineering, economics and biostatistics may find some of the techniques studied here useful and are welcome to sit in, or, take the course for credit if they have exposure to linear algebra, sufficient math background to read Akhiezer and Glazman, and are familiar with the basic properties of the multivariate normal distribution, as found, e. g. in Anderson, Multivariate Analysis, or Wilks, Mathematical Statistics. Otherwise, the development will be self-contained. If in doubt, please contact the instructor by e-mail (wahba@stat.wisc.edu) or come to the first class. This will be a seminar-type course. There will be no sit-down exams. Students taking the course for credit will be expected to do one or two computer projects studying the behavior of some of the methods discussed on simulated or experimental data, and one or two projects in an area of application of their choice from 4. with a possible project being the presentation of a lecture in class on a recent paper or recent resarch. Text: Spline Models for Observational Data, G. Wahba, SIAM (1990), and recent papers, tba. Note: Wahba(1990) is on reserve at Wendt, Akhiezer and Glazman is on reserve in Wendt and the Math Library.