
 

Course given at 25th Finnish Summer School on Probability Theory, Nagu/Nauvo, The Archipelago of Finland, 26 of June, 2003 Measures EverywhereTo the paper: A. Lindo, S. Zuyev and S. Sagitov Nonparametric estimation of infinitely divisible distributions based on variational analysis on measuresIn this paper we develop algorithms of steepest descent type to estimate the compounding measure of a sample distributed as compound Poisson. They are realised in our Rlibrary mesop (the source code). It realises the steepest descent method to find extremum for a strongly differentiable functional of a measure in the class of positive measures without any additional constraints. To see example of its use, see the example provided in man(GoSteep).To the paper: M. Zolghadr and S. Zuyev Optimal design of dilution experiments under volume constraintsThe paper aims at finding optimal designs of dilution experiments under the total volume constraint typical for stem cells research. Here is the Rcode providing the designs described in the paper. It also uses medea Rlibrary available for download at the bottom of this page.To the paper: X. Li, D. K. Hunter and S. Zuyev Coverage Properties of the Target Area in Wireless Sensor NetworksThe paper studies tricovered areas with respect to a twodimensional Poisson process representing wireless sensors. Two scripts in freely available Statistical Programming language R are provided. The code triangsimul.R estimates the density of tricovered area by MonteCarlo simulations, the code triangestim.R provides a lower analytical bound, see the paper for details. Both scripts are extensively commented and contain examples of the usage.To the paper: K. Tchoumatchenko and S. Zuyev Aggregate and fractal tessellationsThese tessellations are defined as follows. Let C^{0}(X^{0}_{i}) be tessellation's cells of level 0 with the nuclei X^{0}_{i }, i=1...N_{0}. To construct an aggregate tessellation of level 1 consider a 1st level tessellation {C^{1}(X^{1}_{j})} with nuclei {X^{1}_{j}}, j=1...N_{1}. Now the aggregate cells of level 1 are the sets C^{1}_{0}(X^{0}_{i}) associated with the 0level nuclei X^{0}_{i} that are the union of those C^{1}(X^{1}_{j}) whose nuclei X^{1}_{j} lie inside C^{0}(X^{0}_{i}). Given a 2ndlevel tessellation {C^{2}(X^{2}_{k})}, the second level aggregate cells C^{2}_{0}(X^{0}_{i}) are the union of those C^{2}(X^{2}_{k}) having X^{2}_{k }belonging to C^{1}_{0}(X^{0}_{i}), etc., etc... This construction allows to obtain very wierd tessellations with the cells that are generally not convex, not connected and do not necessarily contain the nucleus. You can download my program called pvat that draws PoissonVoronoi aggregate tessellations. Here is a screenshot of how they look in pvat.To the paper: I. Molchanov and S. Zuyev Steepest descent algorithms in space of measuresIn this paper propose a "true" steepest descent method for optimisation of functionals defined on nonnegative measures. The corresponding algorithms are encoded in Splus and R languages and available as source Rlibraries mefista (for finding an optimal measures with a fixed total mass) and and medea (for optimisation under many constraints). Windows binaries (compiled under R version 3) are: mefista.zip and medea.zip.To the paper: I. Molchanov and S. Zuyev Optimisation in space of measures and optimal designIn this paper we propose a unified approach to finding an optimal design measure. In a common case when an analytical solution is impossible, the steepest descent algorithms described in the previous paper allow for a numerical solution. This could be done with the help of our Rlibraries mefista (for finding an optimal measures with a fixed total mass) and and medea (for optimisation under many constraints). Windows binaries (compiled under R version 3) are: mefista.zip and medea.zip. Here is a screenshot from an Rwindow. 