László Márkus

Spatial prediction of the intensity of latent effects governing hydrogeological phenomena

Fluctuation of the groundwater level in karstic areas results from different cumulative effects, such as water infiltration from precipitation, human interference on the water resources of a certain area, the Moons tide effect, air pressure fluctuation, etc. It is practically impossible to measure these effects directly, generating the need for estimations by indirect methods, applied to the registered water level in monitoring wells. The conventional tool to determine latent or background effects governing variability or fluctuations is factor analysis. Since ordinary factor analysis has been elaborated for independent observations and the observed karstwater levels at a certain location represent realizations of time series, dynamic factor analysis has to be used instead. The obtained dynamic factors can then be identified as the above mentioned effects. Once the intensity of an effect has been determined at every observation site, its spatial structure and prediction is aimed to be given. To achieve this goal, universal kriging is used and analyzed for the spatial process of estimated loadings of dynamic factors. The present paper concentrates on the effect of infiltration. The obtained map of the loadings of the dynamic factor corresponding to infiltration provides valuable information on the intensity of this effect. Its importance from the environmental protection point of view is immense, because it helps to determine those potentially dangerous areas where infiltrating contaminated water can quickly, and in vast quantity, reach underground water resources. Copyright

A light-tailed conditionally heteroscedastic model with applications to river flows

A conditionally heteroscedastic model, different from the more commonly used autoregressive moving average-generalized autoregressive conditionally heteroscedastic (ARMA-GARCH) processes, is established and analysed here. The time-dependent variance of innovations passing through an ARMA filter is conditioned on the lagged values of the generated process, rather than on the lagged innovations, and is defined to be asymptotically proportional to those past values. Designed this way, the model incorporates certain feedback from the modelled process, the innovation is no longer of GARCH type, and all moments of the modelled process are finite provided the same is true for the generating noise. The article gives the condition of stationarity, and proves consistency and asymptotic normality of the Gaussian quasi-maximum likelihood estimator of the variance parameters, even though the estimated parameters of the linear filter contain an error. Copyright 2007 The Authors Journal compilation 2007 Blackwell Publishing Ltd.

A Hierarchical Bayesian model to predict belatedly reported claims in insurances

Latent, that is Incurred But Not Reported (IBNR) claims influence heavily the calculation of the reserves of an insurer, necessitating an accurate estimation of such claims. The highly diverse estimations of the latent claim amount produced by the traditional estimation methods (chain-ladder, etc.) underline the need for more sophisticated modelling. We are aimed at predicting the number of latent claims, not yet reported. This means the continuation the so called run-off triangle by filling in the lower triangle of the delayed claims matrix. In order to do this the dynamics of claims occurrence and reporting tendency is specified in a hierarchical Bayesian model. The complexity of the model building requires an algorithmic estimation method, that we carry out along the lines of the Bayesian paradigm using the MCMC technique. The predictive strength of the model against the future disclosed claims is analysed by cross validation. Simulations serve to check model stability. Bootstrap methods are also available as we have full record of the individual claims at our disposal. Those methods are used for assessing the variability of the estimated structural parameters.

Exploring Potentially Hazardous Areas for Water Quality Using Dynamic Factor Analysis

Jozsef Kovacs1, Laszlo Markus2, Jozsef Szalai3, Marton Barcza6, Gyorgy Bernath1, Ilona Kovacsne Szekely4 and Gabor Halupka5 1Eotvos Lorand University, Department of Physical and Applied Geology, 2Eotvos Lorand University, Department of Probability Theory and Statistics, 3VITUKI Environmental and Water Menagement Research Institute Non-Profit Ltd., 4Budapest Business School, Institute of Methodology, 5Repet Ltd., Environmental Consulting Office, 6University of Szeged, Department of Mineralogy, Geochemistry and Petrology, Hungary

Analysis of spatial structure of latent effects governing hydrogeological phenomena

In the present study we investigate the data provided by the karstwater level monitoring system set up in the Transdanubian Mountains, more precisely in the Bakony, the Keszthelyi Mountains and the Balaton-Highland. (Here, like in the sequel, the term karstwater is used for groundwater in karstic areas.) The detailed description of the monitoring system itself and the geological and hydrogeological situation in which the system was planned to function and collect data about the water level can be found in Markus et al (1997) as well as the results of our previous study in determining the underlying (called also latent or background) effects driving the karstwater fluctuations.

Hierarchical Bayesian Modelling of Geographic Dependence of Risk in Household Insurance

We analyse here the spatial dependence structure for the counts of certain type of claims occurring in household insurance in Hungary. We build up a Bayesian hierarchical model for the Gaussian Markov random field driven nonhomogeneous spatial Poisson process of claims counts. Model estimation is carried out by the MCMC technique, by applying sparse matrices, and a novel approach for updates by radial and angular components to obtain better mixing in over 3000 dimensions. We design a procedure that tailors the acceptance rate automatically during burn-in to a prescribed (optimal) value while updating the chain.

The Variation of the Forecast of Lévy’s Brownian Motion as the Observation Domain Undergoes Deformation

While one observes a random phenomenon evolving in an area or in space it may well happen, that the supposed observation domain undergoes a deformation under the influence of heat, pressure, etc. Therefore, in this paper we do not regard the observation domain as usual, something known, immovable and unchangeable.On the contrary, we wish to investigate the dependence of the forecast on the observation domain. In a simple case we wish to give the variation of the forecast of Levy’s Brownian Motion (LBM) for infinitesimal deformations of the observation domain.

On the solution of a generalized Schrödinger-type equation

AbstractРассматривается лин ейное дифференциаль ное уравнение с обобщенн ым потенциаломLu(t)+(u,F)g(t)=f(t), t∈S. Исследуются обобщен ные граничные услови я, при которых существует е динственное решение (в зависимостиF, g). Анало гично исследуется и стохастическое урав нение.