The analysis of marked point patterns evolving through space and time

A maximum pseudo-likelihood approach has previously been developed for fitting pairwise interaction models to patterns generated by growth-interaction processes that are sampled at fixed time points. This approach is now extended, not only by estimating the parameters of the process through time, but also by employing least squares estimation since likelihood based approaches are much more computationally demanding. First, simple stochastic models are used to demonstrate that least squares methods are as powerful as likelihood-based approaches, as well as being mathematically and computationally simpler. The algorithm generates simulations of the deterministic growth-interaction and stochastic immigration-death process, and through these the parameter estimates are determined. Logistic and linear growth are then combined with (symmetric) disc-interaction and (asymmetric) area-interaction processes, and between them these generate a variety of mark-point spatial structures. A robustness study shows that the procedure works well in that the presence, structure and strength of a growth-interaction process can be determined even when an incorrectly presumed model is employed. Thus, the technique is likely to prove to be very useful in general practical applications where the underlying process generating mechanism is almost certain to be unknown. Finally, the procedure is applied to the analysis of a new Swedish pine forest data set for which tree location and diameter at breast height were recorded in 1985, 1990 and 1996.

[1]  E. Renshaw,et al.  The evolution of a single-paired immigration death process , 2008 .

[2]  Olle Häggström,et al.  Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes , 1999 .

[3]  Anders Brix,et al.  Space‐time Multi Type Log Gaussian Cox Processes with a View to Modelling Weeds , 2001 .

[4]  Richard G. Cornell.  Statistical Methods for Cancer Studies , 1984 .

[5]  P. Guttorp,et al.  Space-time estimation of grid-cell hourly ozone levels for assessment of a deterministic model , 1998, Environmental and Ecological Statistics.

[6]  Eric Renshaw,et al.  The evolution and measurement of a population of pairs , 1995 .

[7]  Sylvia Richardson,et al.  A hierarchical model for space–time surveillance data on meningococcal disease incidence , 2003 .

[8]  Carlos. Comas Rodriguez.  Modelling forest dynamics through the development of spatial and temporal marked point processes , 2005 .

[9]  P. Guttorp,et al.  A space-time analysis of ground-level ozone data , 1994 .

[10]  J. R. Wallis,et al.  Some ecological consequences of a computer model of forest growth , 1972 .

[11]  Gavin J. Gibson,et al.  Estimating parameters in stochastic compartmental models using Markov chain methods , 1998 .

[12]  E. Renshaw,et al.  Applying the saddlepoint approximation to bivariate stochastic processes. , 2000, Mathematical biosciences.

[13]  Eric Renshaw,et al.  The evolution of a batch-immigration death process subject to counts , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[14]  C. Wikle.  Hierarchical Models in Environmental Science , 2003 .

[15]  F. J. Alonso,et al.  The Kriged Kalman filter , 1998 .

[16]  T. Mattfeldt.  Stochastic Geometry and Its Applications , 1996 .

[17]  E. Renshaw,et al.  Gibbs point processes for studying the development of spatial-temporal stochastic processes , 2001 .

[18]  T. Pukkala.  Predicting diameter growth in even-aged Scots pine stands with a spatial and non-spatial model. , 1989 .

[19]  S. Pacala,et al.  Forest models defined by field measurements: I. The design of a northeastern forest simulator , 1993 .

[20]  A. V. Vecchia.  Estimation and model identification for continuous spatial processes , 1988 .

[21]  Dietrich Stoyan,et al.  Marked Point Processes in Forest Statistics , 1992, Forest Science.

[22]  N. Cressie,et al.  A dimension-reduced approach to space-time Kalman filtering , 1999 .

[23]  P. Diggle,et al.  Spatiotemporal prediction for log‐Gaussian Cox processes , 2001 .

[24]  Peter J. Diggle,et al.  Detecting dependence between marks and locations of marked point processes , 2004 .

[25]  C. Wikle.  Spatio-temporal Methods in Climatology , 2002 .

[26]  Peter J. Diggle,et al.  Competition for Light in a Plant Monoculture Modelled as a Spatial Stochastic Process , 1981 .

[27]  E G Knox,et al.  The Detection of Space‐Time Interactions , 1964 .

[28]  C Gössl,et al.  Bayesian Spatiotemporal Inference in Functional Magnetic Resonance Imaging , 2001, Biometrics.

[29]  Dietrich Stoyan,et al.  The use of marked point processes in ecological and environmental forest studies , 1995, Environmental and Ecological Statistics.

[30]  R. Häggkvist,et al.  Second-order analysis of space-time clustering , 1995, Statistical methods in medical research.

[31]  Phaedon C. Kyriakidis,et al.  Geostatistical Space–Time Models: A Review , 1999 .

[32]  Bradley P. Carlin,et al.  Hierarchical Spatio-Temporal Mapping of Disease Rates , 1997 .

[33]  Christopher K. Wikle.  A Kernel-Based Spectral Approach for Spatiotemporal Dynamic Models , 2002 .

[34]  Richard H. Jones,et al.  Models for Continuous Stationary Space-Time Processes , 1997 .

[35]  B. Hambly.  Fractals, random shapes, and point fields , 1994 .

[36]  Ke Xu,et al.  A Kernel-Based Spatio-Temporal Dynamical Model for Nowcasting Weather Radar Reflectivities , 2005 .

[37]  J. Møller,et al.  Statistical Inference and Simulation for Spatial Point Processes , 2003 .

[38]  Eric Renshaw.  Modelling biological populations in space and time , 1990 .

[39]  Jonathan R. Stroud,et al.  Dynamic models for spatiotemporal data , 2001 .

[40]  Y. MacNab,et al.  Autoregressive Spatial Smoothing and Temporal Spline Smoothing for Mapping Rates , 2001, Biometrics.

[41]  E. Renshaw,et al.  Can Markov chain Monte Carlo be usefully applied to stochastic processes with hidden birth times , 1998 .

[42]  K. Hopcraft,et al.  Distinguishing population processes by external monitoring , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[43]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .

[44]  George R. Staebler,et al.  Growth and spacing in an even-aged stand of Douglas-fir , 1951 .

[45]  Dietrich Stoyan,et al.  On Variograms in Point Process Statistics , 1996 .

[46]  W. D. Ray.  Spatial Time Series , 2008, Encyclopedia of GIS.

[47]  Anders Brix,et al.  Space-time multi type log Gaussian Cox processes with a view modeling weed data , 1998 .

[48]  Gavin J. Gibson,et al.  Estimating likelihoods for spatio-temporal models using importance sampling , 2003, Stat. Comput..

[49]  Klaus Von Gadow,et al.  Modelling Forest Development , 1998, Forestry Sciences.

[50]  C. Wikle,et al.  A Kernel-Based Spatio-Temporal Dynamical Model for Nowcasting Radar Precipitation Bill , 2022 .

[51]  T. Pukkala.  Prediction of tree diameter and height in a Scots pine stand as a function of the spatial pattern of trees. , 1989 .

[52]  D. Stoyan,et al.  Recent applications of point process methods in forestry statistics , 2000 .

[53]  Timothy G. Gregoire,et al.  Modelling Longitudinal and Spatially Correlated Data , 1997 .

[54]  Gavin J. Gibson,et al.  Inference for immigration-death processes with single and paired immigrants , 2001 .

[55]  Andrew B. Lawson,et al.  Statistical Methods in Spatial Epidemiology , 2001 .

[56]  Timo Pukkala,et al.  Effect of spatial pattern of trees on the growth of Norway spruce stand. , 1991 .

[57]  John A. Silander,et al.  Delineating forest canopy species in the northeastern united states using multi-temporal TM imagery , 1998 .

[58]  P. Brown,et al.  Blur‐generated non‐separable space–time models , 2000 .

[59]  Patrick Brown,et al.  Space–time calibration of radar rainfall data , 2001 .

[60]  ScienceDirect.  Computational statistics & data analysis , 1983 .