Stochastic Reconstruction for Inhomogeneous Point Patterns

Abstract

The stochastic reconstruction approach for point processes aims at producing independent patterns with the same properties as the observed pattern, without specifying any particular model. Instead a so-called energy functional is defined, based on a set of point process summary characteristics. It measures the dissimilarity between the observed pattern (input) and another pattern. The reconstructed pattern (output) is sought iteratively by minimising the energy functional. Hence, the output has approximately the same values of the prescribed summary characteristics as the input pattern. In this paper, we focus on inhomogeneous point patterns and apply formal hypotheses tests to check the quality of reconstructions in terms of the intensity function and morphological properties of the underlying point patterns. We argue that the current version of the algorithm available in the literature for inhomogeneous point processes does not produce outputs with appropriate intensity function. We propose modifications to the algorithm which can remedy this issue.

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Correspondence to Jiří Dvořák.

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We are grateful to the reviewers for their insightful comments and suggestions. We are also grateful to Thorsten Wiegand for the discussion and helpful comments about stochastic reconstruction and to Rasmus P. Waagepetersen for drawing our attention to the simulated tempering approach. This work has been supported by The Charles University Grant Agency, project no. 472217, and The Czech Science Foundation, project no. 19-04412S.

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Koňasová, K., Dvořák, J. Stochastic Reconstruction for Inhomogeneous Point Patterns. Methodol Comput Appl Probab 23, 527–547 (2021). https://doi.org/10.1007/s11009-019-09738-0

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Keywords

  • Stochastic reconstruction
  • Point process
  • Summary characteristics
  • Inhomogeneous process
  • Intensity function

Mathematics Subject Classification (2010)

  • 60G55