# Deployment of Sensors According to Quasi-Random and Well Distributed Sequences or Nonparametric Estimation of Spatial Means of Random Fields

Ewa Skubalska-Rafajłowicz^{*} and Ewaryst Rafajłowicz

Spatial sampling is a crucial issue for proper estimation of parameters in spatio-temporal dynamical models \cite{Patan}, \cite{Ucinski} and for estimation of spatial fields (see \cite{P1}). Our aim is to discuss advantages in estimating the spatial mean by using quasi-random points (also known as uniformly distributed (UD) points \cite{Nieder}) and their sub-class recently proposed by the authors \cite{nD} that are well-distributed (WD). In opposite to most popular parameter estimation approaches, we consider a nonparametric estimator of the spatial mean, i.e., we do not assume a priori any parametric model in the same vain as it is done in nonparametric estimation problems. We shall prove the estimator convergence in the integrated mean square-error sense. UD and WD sequences have many interesting properties that are useful both for wireless sensors networks (coverage an and connectivity) and for large area networks such as radiological or environment pollution monitoring stations. For simplicity of exposition, we shall consider only independent errors in observations (see \cite{P2} and \cite{gr} for sensors' allocation tasks under more realistic covariance structures). \bibliographystyle{plain} \begin{thebibliography}{11} \bibitem{gr} Griffith D. A., {\em Statistical efficiency of model-informed geographic sampling designs}, 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences. Edited by M. Caetano and M. Painho, pp 91--98, 2006. \bibitem{Nieder} Kuipers, L., Niederreiter, H., {\em Uniform Distribution of Sequences}. Wiley, New York, 1974; Reprint, Dover, Mineola, 2006 \bibitem{Patan} Patan M., {\em Optimal sensor network scheduling in identification of distributed parameter systems}, Springer-Verlag, Lecture Notes in Control and Information Sciences, 2012. \bibitem{P2} Pilz J. and Sp\"ock G., {\em Spatial sampling design for prediction taking account of uncertain covariance structure}, 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences. Edited by M. Caetano and M. Painho, pp 109--119, 2006. \bibitem{P1} Rasch D., Pilz J., Verdooren L.R., Gebhardt A., {\em Optimal Experimental Design with R}, Francis \& Taylor, 2011 \bibitem{nD} Skubalska-Rafaj{\l}owicz E. and rafaj{\l}owicz E., {\em Sampling multidimensional signals by a new class of quasi-random sequences}, Multidim. Syst. Sign. Process., published on-line, June 2010, DOI 10.1007/s11045-010-0120-5 \bibitem{Ucinski} Ucinski D., {\em Optimal Measurement Methods for Distributed Parameter System Identification}, CRC Press, London, New York, 2005. \end{thebibliography}

Mathematics Subject Classification: 62G05 11K36

Keywords: sensors; allocation; nonparametric; estimation;

Minisymposion: Optimal Design of Spatio-Temporal Networks and Systems