## Multiscale Statistics and Modeling

A section of STAT 551 (Advanced Time Series), Fall 2006

Rice University

This section of the course provides an introduction to the concepts of

Long Range Dependence and statistical self-similarity and develops both,

models and estimators in the context of real world data

Multiscale Statistics, R. Riedi

Classical limit theorems typically assume strong properties such as finite second moments and independence, or at least weak dependence, to arrive at strong results. In real world situations one may be confronted with data for which strong-holds such as the CLT do not seem to be verifiable empirically. A famous example is found with the classical Nil river annual water level. Violations of the CLT can be explained via long range dependence (LRD) or via infinite variance. In this mini course, we will formalize the concept of LRD and give simple time series models with LRD, such as generalized AR processes and fractional Brownian motion. We derive inference tools for LRD. This will lead us to wavelets, a powerful tool in multiscale statistics, to which a short and very intuitive introduction will be given. We will end with a tail estimator based on wavelets which pertains to the issue of infinite variance. If time permits or interest is indicated, cascade processes and Large Deviation Principles will be discussed.

Instructor

Dr. Rudolf Riedi
Duncan Hall 2082, 713 / 348 3020,
Time and Place
MWF 09:00 AM - 9:50 AM Duncan Hall 1044
see also the official Rice for times and the Rice map for how to find the location

#### Office hours:

Monday 1-2 pm pm  (DH 2082), or by appointment

Outline and suggested topics

• Long Range Dependence
• Statistical Self-similarity
• Auto-regressive time series models
• Hierarchical time series models
• Time domail and spectral domain estimators
• Wavelets, wavelet based estimation of LRD and heavy tails
• EXAM (open-notes)

Textbook

• Long Range Dependence (Doukhan et al, editors)

alternative texts will be indicated on request

60%  EXAM
40%  Homework

This doubles as a calendar for the course.
 2006 Covered material Reading: October 2 Overview Slide Talk [pdf] October 4 Overview (Cont) October 6 Statistical Self-similarity Ch. 1, LRD (Doukhan et al eds) October 9 F.d.d. and Consistency Ch. 1 October 11 fBm (Gaussian self-sim process) Ch. 1 October 13 fGn (increments of fBm) Ch. 1 October 16 Recess October 18 LRD and 2nd order self-similarity Ch. 1 October 20 FARIMA (times series model of LRD) Ch. 1 October 23 Spectral Estimation, Wiener-Kinchin Slides [pdf] October 25 Wavelets and spectral estimation Slides [pdf] October 30 Proj 1: Fractional Stochastic Calculus and Stock Markets Slides [ppt] TBA Proj 2+3: Estimation TBA Proj 4+5: LRD in Markets, Simulation

Homework
(tex-source and solutions restricted to Rice University)

 Homework sheet Due date (in class) Solutions Problem Set 1 [ps] [pdf] [tex] Oct 30, 2006 posted Nov 1, 2006

Late Homework Policy

Homework is due at the beginning of class on the due date. After the due date, but before solutions are handed out, homework can be turned in for 50% credit. In this case, please slip your homework under the instructors's office door, or bring it to class. After solutions are handed out, 0% credit will be issued. You are encouraged to work in groups for homeworks but you will hand in your own solution which you are expected to understand.

Any student with a documented disability needing academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain confidential. Students with disabilities should also contact Disabled Student Services in the Ley Student Center.

August 29, 2006 Dr. Rudolf Riedi