Tid: 28 februari 2011 kl 15.15-17.00.

Plats : Seminarierummet 3721, Institutionen för matematik, KTH, Lindstedts väg 25. Karta!

Föredragshållare: Oleg Seleznjev, Umeå universitet

Titel: Multivariate piecewise linear interpolation of a random field

Abstract

Let a random field defined on a d-dimensional unit cube and X(t), t ∈ [0; 1]^d, with finite second moment be observed at a finite number of points. Suppose further that the points are vertices of hyperrectangles, and that the hyperrectangles are generated by a grid in the cube. At any unsampled point we approximate the value of the field by a piecewise linear multivariate interpolator, which is a natural extension of a one dimensional piecewise linear interpolator. The approximation accuracy is measured by the integrated mean squared error. Following Berman (1974) we extend the concept of local stationarity for random fields and focus on fields satisfying this condition. Approximation of stochastic processes from this class is studied in our previous works (see, e.g., Seleznjev, 2000; Abramowicz and Seleznjev, 2011). For q.m. (quadratic mean) continuous locally stationary random fields we derive the exact asymptotic behavior of the approximation error. A method is proposed for determining the asymptotically optimal knots (sample points) allocation between the mesh dimensions. Additionally, for q.m. continuous and differentiable Hölder continuous fields we determine asymptotical upper bounds for the approximation error. These results can be used in various problems in numerical analysis of random functions, in environmental and geosciences, in  image processing, and in simulation studies.

References

•  Abramowicz, K. and Seleznjev, O.  (2011). Spline approximation of a random process with singularity. J.  Statist.  Plann.  Inference 141, 1333--1342.
•  Berman, S.M.  (1974). Sojourns and extremes of Gaussian process. Ann.  Probab.  2, 999--1026.   (1980)
• Seleznjev, O.  (2000). Spline approximation of random processes and design problems. J.  Statist.  Plann  Inference 84, 249--262.

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