Functional summary statistics for the Johnson–Mehl model

Abstract

The Johnson–Mehl germination-growth model is a spatio-temporal point process model which among other things have been used for the description of neurotransmitters datasets. However, for such datasets parametric Johnson–Mehl models fitted by maximum likelihood have yet not been evaluated by means of functional summary statistics. This paper therefore invents four functional summary statistics adapted to the Johnson–Mehl model, with two of them based on the second-order properties and the other two on the nuclei-boundary distances for the associated Johnson–Mehl tessellation. The functional summary statistics theoretical properties are investigated, non-parametric estimators are suggested, and their usefulness for model checking is examined in a simulation study. The functional summary statistics are also used for checking fitted parametric Johnson–Mehl models for a neurotransmitters dataset.

Keywords:

• germination-growth process
• model checking
• neurotransmitters
• pair correlation function
• spatio-temporal point process
• tessellation
• typical cell

Acknowledgements

We thank Sung Nok Chiu for providing the neurotransmitters dataset studied in Section 4.2. We are grateful to an anonymous referee for useful comments.

Funding

Supported by the Danish Council for Independent Research – Natural Sciences, grant 12-124675, ‘Mathematical and Statistical Analysis of Spatial Data’, and by the Center for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation.