Random fields: spectral representation,
smoothness, excursions. Fall 2014
First Lecture:
Wednesday, Oct 29 13:00 – 14:45 in MVL15
Lectures and problem sessions: Tuesdays 13:00 – 14:45 and Wednesdays 13:00 – 14:45
The
course will cover the basics theory for random fields, with emphasis on
stationary and differentiable Gaussian fields. It will consist of
lectures, and problem sessions where
participants present solution to selected problems from the course book.
Course literature:
An unfinished manuscript
“Applications of RANDOM FIELDS AND GEOMETRY: Foundations and Case Studies” by Robert Adler, Jonathan Taylor,
and Keith Worsley. A copy will be sent to all on the
course address list.
Complementary literature:
“Level sets and extrema
of random processes and fields” by Jean-Marc Azais
and Mario Wschebor, Wiley, 2009
“Asymptotic Methods in the Theory of
Gaussian Processes” by Vladimir Piterbarg, American Mathematical Society, ser.
Translations of Mathematical Monographs, Vol. 148, 1995
“Random fields and Geometry” by
Robert Adler and Jonathan Taylor, Springer 2007
"Spectral domain"
by Monserrat Fuentes and Brian Reich, in Handbook of Spatial
Statistics, eds. Gelfand, Diggle, Fuentes, Guttorp, p. 57-78, CRC
press, 2010
Course content:
•
Kolmogorov existence theorem, separable
processes, measurable processes
• Stationarity and isotropy
• Spectral representations
• Exceedance sets
• Rice formula
• Slepian models
Examination:
• Oral/written exam: individual
discussion of solutions to a couple of problems
• Solutions of exercises during the
course
|
Date |
Content |
Literature |
|
|
|
|
|
Wednesday 29/10 MVL15 |
introduction,
notation, Kolmogorov existence theorem, separability,
measurability |
ATW Ch 1 (read on your own), Ch 2,
pp. 23-30 Slides: RF1 |
|
Tuesday 4/11 MVL14 |
|
Slides: RF1 |
|
Wednesday 5/11 MVL15 |
Spectral representation of covariance function, differentiability, white noise |
Slides: RF1, RF2 |
|
Tuesday 11/11 MVL14 |
Excercise session |
2.8.1, 2.8.2, 2.8.3, 2.8.4, 2.8.5 |
|
Wednesday 12/11 MVL15 |
differentiability, white noise, spectral representation of fields |
Slides: RF2 |
|
Tuesday 18/11 MVL14 |
spectral representation for isotropic fieds, continuity |
Slides: RF2, RF3 |
|
Wednesday 19/11 MVL15 |
continuity, differentiability |
Slides: RF3 |
|
Tuesday 25/11 MVL14 |
Excercise session |
2.8.7, 2.8.9, 2.8.11, 2.8.12 |
|
Wednesday 26/11 MVL15 |
Spectral estimation |
"Spectral domain", Fuentes & Reich RF4 |
|
Tuesday 2/12 MVL14 |
Excercise session |
|
|
Wednesday 3/12 MVL15 |
Borel, Slepian inequality |
RF5 |
|
Tuesday 9/12 MVL14 |
Cancelled |
|
|
Wednesday 10/12 MVL22 Note! |
Excercise session |
|
|
Tuesday 16/12 MVL14 |
Cancelled |
|
|
Wednesday 17/12 MVL15 |
Cancelled |
|
| Thursday 18/12, 13:15 -15:00 |
Non-stationary spatial structures |
Guest lecture by Peter Guttorp nonstationary, nonstationary_references |
|
Tuesday 13/1 |
Rice's formula, Slepian models |
ATW Ch 2, pp. 78-82, Ch 4, pp. 194-214 |
|
Wednesday 15/1 (starts after intro to GRF course), MVL15 |
Rice's formula, Slepian models |
ATW Ch 2, pp. 78-82, Ch 4, pp. 194-214 RF6 |
| Monday 19/1, 13-15 | Exercise session |
|
|
Wednesday, 28/1, 13-15 |
WAFO |
Guest lecture & excercise session, by Igor Rychlik Bring your own PC! |
Slides: