Applied Mathematicsematics

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Additional resources for Annals of Pure and Applied Logic, Volume 152, Issues 1-3, Pages 1-180 (March 2008)

Example text

Briefly, for any r > 0, 1 − F (r) is the probability that X has no points within distance r from an arbitrary fixed point in Rd , 1 − G(r) is the conditional probability that X has no further points within distance r from a typical point in X, and J(r) = (1 − G(r))/(1 − F (r)) (defined for F (r) < 1; see van Lieshout & Baddeley 1996). For a stationary Cox process, 1 − F (r) = E exp − Z(x)dx x , ≤r and 1 − G(r) = E exp − Z(x)dx Z(0) x ≤r In the special case of a homogeneous Poisson process, ρ. 1 − F (r) = 1 − G(r) = exp − ρrd π d/2 /Γ(1 + d/2) .

The left cluster is much less convincing (in the sense of statistical significance), because there are few streamlines pointing towards its peak. Of course it must be kept in mind that statistical significance is necessarily one sided. Streamlines give strong evidence of presence of a feature, but lack of streamlines only indicates the evidence is not strong enough to be sure, and does not prove absence of a cluster. e. 3. This is h = 4, which was chosen for presentation purposes, after viewing the full scale-space.

For example, the scale-space is easily presented as an overlay in one dimension, and a movie in two dimensions, but it is less clear how to present a three dimensional scalespace. g. gradient and curvature) should have their statistical significance displayed, and how they should be visualized. CHAPTER 3 Statistical Inference for Cox Processes J. P. 1 Introduction This chapter is concerned with statistical inference for a large class of point process models, which were studied in a seminal paper by Cox (1955) under the name doubly stochastic Poisson processes, but are today usually called Cox processes.

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