Earthquake modelling at the country level using aggregated spatio-temporal point processes

The goal of this paper is to derive a hazard map for earthquake occurrences in Pakistan from a catalogue that contains spatial coordinates of shallow earthquakes of magnitude 4.5 or larger aggregated over calendar years. We test relative temporal stationarity by the KPSS statistic and use the inhomogeneous J-function to test for inter-point interactions. We then formulate a cluster model, and de-convolve in order to calculate the hazard map, and verify that no particular year has an undue influence on the map. Within the borders of the single country, the KPSS test did not show any deviation from homogeneity in the spatial intensities. The inhomogeneous J-function indicated clustering that could not be attributed to inhomogeneity, and the analysis of aftershocks showed some evidence of two major shocks instead of one during the 2005 Kashmir earthquake disaster. Thus, the spatial point pattern analysis carried out for these data was insightful in various aspects and the hazard map that was obtained may lead to improved measures to protect the population against the disastrous effects of earthquakes

State estimation for temporal point processes

This paper is concerned with combined inference for point processes on the real line observed in a broken interval. For such processes, the classic history-based approach cannot be used. Instead, we adapt tools from sequential spatial point processes. For a range of models, the marginal and conditional distributions are derived. We discuss likelihood based inference as well as parameter estimation using the method of moments, conduct a simulation study for the important special case of renewal processes and analyse a data set collected by Diggle and Hawtin

Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets

We propose new summary statistics to quantify the association between the components in coverage-reweighted moment stationary multivariate random sets and measures. They are defined in terms of the coverage-reweighted cumulant densities and extend classic functional statistics for stationary random closed sets. We study the relations between these statistics and evaluate them explicitly for a range of models. Unbiased estimators are given for all statistics and applied to simulated examples.Comment: Added examples in version

A spectral mean for point sampled closed curves

We propose a spectral mean for closed curves described by sample points on its boundary subject to mis-alignment and noise. First, we ignore mis-alignment and derive maximum likelihood estimators of the model and noise parameters in the Fourier domain. We estimate the unknown curve by back-transformation and derive the distribution of the integrated squared error. Then, we model mis-alignment by means of a shifted parametric diffeomorphism and minimise a suitable objective function simultaneously over the unknown curve and the mis-alignment parameters. Finally, the method is illustrated on simulated data as well as on photographs of Lake Tana taken by astronauts during a Shuttle mission

A J-function for inhomogeneous spatio-temporal point processes

We propose a new summary statistic for inhomogeneous intensity-reweighted moment stationary spatio-temporal point processes. The statistic is defined through the n-point correlation functions of the point process and it generalises the J-function when stationarity is assumed. We show that our statistic can be represented in terms of the generating functional and that it is related to the inhomogeneous K-function. We further discuss its explicit form under some specific model assumptions and derive a ratio-unbiased estimator. We finally illustrate the use of our statistic on simulated data

Clustering methods based on variational analysis in the space of measures

We formulate clustering as a minimisation problem in the space of measures by modelling the cluster centres as a Poisson process with unknown intensity function.We derive a Ward-type clustering criterion which, under the Poisson assumption, can easily be evaluated explicitly in terms of the intensity function. We show that asymptotically, i.e. for increasing total intensity, the optimal intensity function is proportional to a dimension-dependent power of the density of the observations. For fixed finite total intensity, no explicit solution seems available. However, the Ward-type criterion to be minimised is convex in the intensity function, so that the steepest descent method of Molchanov and Zuyev (2001) can be used to approximate the global minimum. It turns out that the gradient is similar in form to the functional to be optimised. If we discretise over a grid, the steepest descent algorithm at each iteration step increases the current intensity function at those points where the gradient is minimal at the expense of regions with a large gradient value. The algorithm is applied to a toy one-dimensional example, a simulation from a popular spatial cluster model and a real-life dataset from Strauss (1975) concerning the positions of redwood seedlings. Finally, we discuss the relative merits of our approach compared to classical hierarchical and partition clustering techniques as well as to modern model based clustering methods using Markov point processes and mixture distributions

Summary statistics for inhomogeneous marked point processes

We propose new summary statistics for intensity-reweighted moment stationary marked point processes with particular emphasis on discrete marks. The new statistics are based on the n-point correlation functions and reduce to cross J- and D-functions when stationarity holds. We explore the relationships between the various functions and discuss their explicit forms under specific model assumptions. We derive ratio-unbiased minus sampling estimators for our statistics and illustrate their use on a data set of wildfires

Notes on the Markowitz portfolio selection method

Portfolio Investment;management science