STA 961 Syllabus

STA961: Statistical Stochastic Processes

Prof:	Robert L. Wolpert	wolpert@stat.duke.edu	OH: Mon 4:30-5:30pm, 211c Old Chem
TA:	Yun Yang	yy84@stat.duke.edu	OH: Mon 7:00-9:00pm, 211a Old Chem
Class:			Tue/Thu 11:45a-1:00p, Soc Sci 124
Text:	Rasmussen & Williams,	Gaussian Processes for Machine Learning	Free on-line version
Opt'l:	Santner, Williams & Notz,	The Design and Analysis of Computer Experiments
	Reed & Simon,	Functional Analysis, Volume I
	Meyn & Tweedie,	Markov Chains and Stochastic Stability
	Michael Stein,	Interpolation of Spatial Data

Syllabus

Week Topic Homework

I. Brownian Bridge and Related Topics Problems Due

Jan 09 Intro: Probability distributions & Function spaces

Jan 14-16 Motivation: Donsker & Kolmogorov-Smirnov

Jan 21-23 Dirichlet Sobolev spaces & Brownian Bridge hw1 Feb 18

Jan 28-30 Free Sobolev & O-U, Karhunen-Loève

Feb 04-06 No class (SAMSI Workshop: Topological Analysis)

Feb 11 Reflection and Maxima of the Brownian Bridge

Feb 13 No class--- Snow Day. Homework postponed.

II. Gaussian Processes and Computer Emulation

Feb 18-20 Isotropic Covariance Functions & Gaussian Procs

Feb 25-27 Sacks/Welch/Mitchell & Kennedy/O'Hagan & c.

Mar 04-06 Prediction & Inference for GPs (PCA notes)

--- Spring Break (Mar 09-16) ---

Mar 18 Multiple Regression, GPs, and PCA

Mar 20 Case Study: Heavy Ion Collisions (Chris C-S)

III. Exotic Time Series

Mar 25-27 Examples: Gaussian AR(p) & Linear B/D PPs hw2 Apr 10

Apr 01-03 No class (SIAM/ASA Conference in Savannah)

Apr 08-10 Six different AR(1)-like Gamma Processes notes1 notes2

Apr 15 Integer-valued AR(1)-like Processes or α-Stables hw3 Apr 24

Week	Topic	Homework
	I. Brownian Bridge and Related Topics	Problems	Due
Jan 09	Intro: Probability distributions & Function spaces
Jan 14-16	Motivation: Donsker & Kolmogorov-Smirnov
Jan 21-23	Dirichlet Sobolev spaces & Brownian Bridge	hw1	Feb 18
Jan 28-30	Free Sobolev & O-U, Karhunen-Loève
Feb 04-06	No class (SAMSI Workshop: Topological Analysis)
Feb 11	Reflection and Maxima of the Brownian Bridge
Feb 13	No class--- Snow Day. Homework postponed.
	II. Gaussian Processes and Computer Emulation
Feb 18-20	Isotropic Covariance Functions & Gaussian Procs
Feb 25-27	Sacks/Welch/Mitchell & Kennedy/O'Hagan & c.
Mar 04-06	Prediction & Inference for GPs (PCA notes)
	--- Spring Break (Mar 09-16) ---
Mar 18	Multiple Regression, GPs, and PCA
Mar 20	Case Study: Heavy Ion Collisions (Chris C-S)
	III. Exotic Time Series
Mar 25-27	Examples: Gaussian AR(p) & Linear B/D PPs	hw2	Apr 10
Apr 01-03	No class (SIAM/ASA Conference in Savannah)
Apr 08-10	Six different AR(1)-like Gamma Processes	notes1	notes2
Apr 15	Integer-valued AR(1)-like Processes or α-Stables	hw3	Apr 24

Spring 2014 Edition

This year STA 961 will focus primarily on some basics and on two special topics:

Basics of Statical SPs: Donsker's Theorem and the Brownian Bridge; Karhunen-Loève; Likelihood Ratios in sequential procedures as martingales and random walks; some function space background (L₂ Sobolev theory, Hilbert & Banach spaces), probability distributions on function spaces.
Inference and prediction for Gaussian Processes and Random Fields, and in particular on building GP emulators for modeling complex computer output;
Exotic time series with Infinitely Divisible distributions including integer-valued autocorrelated AR(p)-like processes with Poisson or Negative Binomial marginals, useful for modeling autocorrelated count data, and positive-valued processes with Gamma or α-Stable marginals, useful for modeling correlated positive quantities or for use at intermediate stages of hierarchical models (e.g., for building more flexible Bayesian semi- or non-parametric models than is possible with Dirichlet processes).

Familiarity with a wide range of probability distributions (at the level of STA 230 or, better, STA 831) and the multivariate normal distribution (at the level of STA 210 or, better, STA 721) are assumed. Measure-theoretic probability (e.g. STA 711) is helpful but not required. There will be negligible overlap with topics covered in the Spring 2012 course STA 357 (Topics in Stochastic Processes) or STA 942 (Time Series & Forecasting).

STA 961 Course Description

STA 961: Statistical Stochastic Processes

Topics in the theory and modelling aspects of Stochastic Processes central to Bayesian statistical analysis. Topics vary from year to year but may include some of:

Convergence theory of Markov chains in general state spaces;
Convergence rates and mixing times for Markov chains;
Stochastic approximation and adaptive estimators;
Inference and prediction for Gaussian Processes and Random Fields;
Modeling and Inference with Continuous-time Jump Processes;
Theory, Application, and inference for Lévy Random Flights;
Inference for Diffusions;
Exotic time series with Infinitely Divisible distributions;
Functional Data Analysis.