Research On Log Gaussian COX Process Model

There are a large number of spatial point process data in many fields,such as geology,biology and medicine et.Al.Studying the intensity and direction of the points in these data are very interesting.And the density of the points is described by the intensity basically,that means the number of points in the unit area.The spatial point process models are divided into many types by their different assumptions of the intensities.And the Log Gaussian Cox process model is one of the commonly used spatial point process models,which assumes that the logarithm of the intensity is a Gaussian process.However,there are many situations in practice that the logarithm of the intensity is a non-Gaussian process.Thus the Log Gaussian Cox process model is no longer suitable for these situations.In this paper,the Log Gaussian Cox process model is improved by assuming that the logarithm of the intensity is a Gaussian Mixture process,and the improved model is named as Log Gaussian Mixture Cox process model.Since referring to the Gaussian Mixture process,studying the Log Gaussian Cox process model from two cases whether the members of the Gaussian Mixture process are correlated mutually or not.It has been proved that the n_thh order product density of the Log Gaussian Mixture Cox process model is determined by its intensity and pairwise correlation function.Besides,ergodicity of the Log Gaussian Mixture Cox process has been proved in the case where the Gaussian Mixture process members are not correlated mutually.Since the original Minimum Contrast Estimation and Maximum Likelihood Estimation methods are not suitable to the improved model,especially in the case where the Gaussian process members are not correlated,the minimum contrast estimation cannot estimate all parameters of the model,therefore,the calculation formulas regarding minimum contrast estimation and maximum likelihood estimation for estimating the improved model parameters are derived respectively.During the case that Gaussian Mixture process members are uncorrelated,the two estimation methods are combined to estimate the parameters of the model.In addition,the number of members of the Gaussian Mixture process is determined by the AIC?Akaike information criterion?.The expressions of the three statistical inference functions F,G,and L functions in the Log Gaussian Mixture Cox process model are given.Finally,the four examples,the Swedish pine dataset,the Japanese black pine dataset,the Sichuan three-level town dataset and the cell dataset,are used to evaluate the fitting effects of improved and unimproved models respectively.The Log Gaussian Mixture Cox process model,where Gaussian Mixture process members are uncorrelated,are respectively used to fit the Swedish pine and Japanese black pine data sets,estimating the model parameters by combined with minimum contrast estimation and maximum likelihood estimation.For the Sichuan province towns and cell datasets,the applied models are them improved model,where Gaussian Mixture process members are correlated.For the Sichuan province towns data set,the maximum likelihood estimation method is used to estimate the models parameters,and in the cell data set,the models parameters are estimated by using the minimum contrast estimation method.During these four examples,AIC is applied for model selection,and F,G,and L functions are used to analyze the effects of models fitting data.The results show that the Log Gaussian Mixture Cox process model performs better in the four examples.It indicates that the improved model is practical and more flexible than the Log Gaussian Cox process model.