Random fields: spectral representation, smoothness, excursions. Fall 2014

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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

 

Gaussian fields, stationarity, isotropy, cosine fields, orthogonal expansions

ATW  Ch 2, pp. 30-39

Slides: RF1

Wednesday   5/11

 MVL15

Spectral representation of covariance function,  differentiability,  white noise

ATW  Ch 2, pp. 36-47

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

 

ATW  Ch 2, pp. 42-56

Slides:  RF2

Tuesday      18/11

 MVL14

spectral representation for isotropic fieds, continuity

ATW  Ch 2, pp. 53-67

Slides:  RF2, RF3

Wednesday 19/11

 MVL15

 continuity, differentiability

 ATW  Ch 2, pp. 60-70

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

ATW  Ch 2, pp.  70-78

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

  MVL15
  Rice's formula, Slepian models

 ATW Ch 2, pp. 78-82, Ch 4, pp.  194-214

  RF6

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:

RF 1

RF 2

RF 3

RF 4

RF 5

RF6

nonstationary

nonstationary references

RF-excercises1

RF-excercises2

RF-excercises3