Department of Mathematical Modelling

Technical University of Denmark Building 321/305 DK-2800 Lyngby Denmark

Telephone: +45 4588 1433 Telefax : +45 4588 2673 E-mail : imm@imm.dtu.dk

Analysis of irregularly distributed points

Karsten Hartelius

Summary

The present thesis is on the analysis of irregularly distributed points.

The main part of the thesis is concerned with enterpolating and restoration of irregularly distribyted points. The least squares methods of kriging and Kalman filtering and the Bayesian restoration method of iterated conditional modes are applied to this problem.

The Kalman filter is described as a powerfull tool for modelling two-dimensional data. Motivated by the development of the reduced update Kalman filter we propose a reduced update Kalman smoother which offers considerable computa- tional savings.

Kriging is described as a robust estimator which may be appled straightfor- wardly to a wide range of point patterns and processes when the correlation structure is known.

We give a qualitative and quantitative comparison of kriging, Kalman filter and iterated conditional modes. The Kalman filter have in a case study on fusion of maps with different spatial resolution, shown to provide a powerful modelling of autocorrelated noise structures.

The Kalman filter have shown to be superior to ordinary kriging in precision and computational speed. Simple kriging has same statistical properties as the Kalman filter, but the usage of simple kriging may lead to ill-conditioned matrices when applied to highly irregularly distributed points.

Adaptive Kalman filter schemes are investigated. A new parallel Kalman filter algorithm based on windowing technique gives good results in a case study on the Igallico satellite scene and represents an interesting contextuel classifier. Extended Kalman filtering on the other hand seems to be well suited for interpolation in gradually changing environments.

Bayesian restoration is applied to a point matching problem, which consists of matching a grid to an image of (irregularly) distributed point observa- tions. We present an extension to an existing grid model, which is based on a combined line- and point-process. A pseudolikelihood estimator for the parameters is introduced, which is defined in terms of the semivariance structure. The developed models have been applied to a case study on hybridisation analysis, which comprise matching a grid to an arrayed set of DNA- clones spotted onto a hybridisation filter.

The line process has proven to perform a satisfactorly modelling of shifted fields (subgrids) in the hybridisation grid, and a two-staged hierarchical grid matching scheme which was designed to firstly locate the overall positions of "node-blocks" in the grid and secondly locate the individual positions of grid nodes has proven to work.

IMM Ph.D Thesis 21, 1996

Last modi fied April 18, 1997

For further information, please contact, Finn Kuno Christensen, IMM, Bldg. 321, DTU
Phone: (+45) 4588 1433. Fax: (+45) 4588 2673, E-mail: fkc@imm.dtu.dk

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