In the analysis of spatial point patterns, an important role is played by statistical tests based on simulation envelopes, such as envelope simulations Ripley's K function. Recent ecological literature has correctly pointed out a common error in interpretation envelopes. However, this led to widespread belief that themselves are invalid. On contrary, envelope‐based correct procedures, under appropriate conditions. paper, we explain principles Monte Carlo and their interpretation, canvas...

In Geographical Information Systems, spatial point pattern data are often analysed by dividing space into pixels, recording the presence or absence of points in each pixel, and fitting a logistic regression. We study weaknesses this approach, propose improvements, demonstrate an application to prospective geology Western Australia. Models based on different pixel grids incompatible (a ‘change-of-support’ problem) unless pixels very small. On fine grid, regression is approximately Poisson...

Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further processes are as limits sequences auto-logistic schemes.

SUMMARY From certain points of view, the range probability models currently available for describing joint behaviour two point processes is rather limited. In this paper we explore structure some further and apply our results to statistical analysis bivariate spatial patterns.

Newman (1970) introduced an interesting new class of point processes which he called Gauss-Poisson. They are characterized, in the most general case, by two measures. We determine necessary and sufficient conditions on these measures for resulting process to be well defined, proceed a systematic study its properties. These include stationarity, ergodicity, infinite divisibility. mention connections with other classes some statistical results. Our basic approach is through probability...

We study a bivariate stochastic process { X ( t )} = Z ))} , where E is continuous-time Markov chain describing the environment and Z(t of interest. In context which motivated this study, models gating behaviour single ion channel. It assumed that given channel with infinitesimal generator at time dependent on ) not . derive necessary sufficient conditions for to be reversible, showing then its equilibrium distribution has product form reflects independence state special case when controls...

Though stochastic models are widely used to describe single ion channel behaviour, statistical inference based on them has received little con­sideration. This paper describes techniques of inference, in particular likelihood methods, suitable for Markov incorporating limited time resolution by means a discrete detection limit. To simplify the analysis, attention is restricted two-state models, although methods have more general applicability. Non-uniqueness mean open-time and closed-time...

Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further processes are as limits sequences auto-logistic schemes.

We develop statistical methods for analysing a pattern of points on region the sphere, including intensity modelling and estimation, summary functions such as K function, point process models, model‐fitting techniques. The are demonstrated by dataset giving sky positions galaxies. Copyright © 2016 John Wiley & Sons, Ltd.

We consider a semi-Markov process with finite state space, partitioned into two classes termed ‘open' and ‘closed'. It is possible to observe only which class the in. show that complete information concerning aggregated contained in an embedded Markov renewal process, whose parameters, moments equilibrium behaviour are determined. Such processes have found considerable application stochastic modelling of single ion channels. In setting there time interval omission, i.e. brief sojourns either...

Summary This paper reviews some interesting but scattered results that are known about correlation in bivariate Poisson distributions and processes presents new results. A particular concern both contexts is with examples relating to negative correlation.

Stochastic models of ion channels have been based largely on Markov theory where individual states and transition rates must be specified, sojourn-time densities for each state are constrained to exponential. This study presents an approach random-sum methods alternating-renewal theory, allowing grouped into classes provided the successive sojourn times in a given class independent identically distributed. Under these conditions form special case. The utility is illustrated by considering...

Membrane patches usually contain several ion channels of a given type. However, most the stochastic modelling on which data analysis (in particular, estimation kinetic constants) is currently based, relates to single channel rather than multiple channels. Attempts circumvent this problem experimentally by recording under conditions where activity low are restrictive and can introduce bias; moreover, possibly important information how multichannel systems behave will be missed. We have...

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by common partitioning the state space. The interaction assumed to arise from dependence some transition rates for given at specified time on states other subsystems that time. With two classes, labelled 0 and 1, superposition process arising counts number latter class. Key structure results theory processes are summarized. These then applied also processes. In particular, we consider...

A simple, widely applicable method is described for determining factorial moments of N̂ t , the number occurrences in (0, ] some event defined terms an underlying Markov renewal process, and asymptotic expressions these as → ∞. The moment formulae combine to yield expression probability generating function thereby further properties such counts. developed by considering counting processes associated with events that are determined states at two successive renewals a which it both simplifies...