RECONSTRUCTING DISTRIBUTION DENSITY OF PARTICLES FOR DISPERSE MATERIALS BY THE PARZEN-ROSENBLATT WINDOW METHOD

Subject: the article contains the description and the possibility of using the Parzen-Rosenblatt window method which belongs to the methods of non-parametric statistics for estimation of empirical density of distribution of particles for disperse materials - compacted microsilica (silica fume). As a dispersed material, the compacted silica fume is considered. Microsilica is a byproduct of the metallurgical industry and is used as a pozzolanic additive for the manufacture of various types of concrete. The compacted silica fume consists of spherical particle-clusters formed of individual silica fume particles. Research objectives: description and implementation of the Parzen-Rosenblatt window method for obtaining empirical density distribution function for the diameter of particles of compacted silica fume; comparison of the histogram method with the Parzen-Rosenblatt window method in estimating the distribution of the diameter of particles of compacted silica fume. Materials and methods: the Gaussian weight (kernel) functions are used to implement the Parzen-Rosenblatt window method. The Sheather-Jones plug-in method is used to find the optimal bandwidth of the kernel functions. In the Sheather-Jones method, the non-linear equation for finding the optimal bandwidth is solved numerically using the Newton’s method. Implementation of the methods is performed in the programming language of the numerical computing environment Matlab. Results: the Parzen-Rosenblatt window method was described and implemented, and by implementing this method, the estimate of empirical density of distribution of the diameter of particles for compacted silica fume was obtained. A comparison of the Parzen-Rosenblatt window method and the histogram method is also given, for example, by reconstructing the density distribution of the diameter of particles of compacted silica fume. Conclusions: application of the Parzen-Rosenblatt window method allows us to solve the problems that arise when using the methods of parametric statistics and the histogram method in estimating the empirical distribution density of particles of disperse materials. In particular, there is no need to assign unknown statistics for the methods of parametric statistics and determine the number of intervals for the histogram method. Density distributions obtained by the Parzen-Rosenblatt window method can be used for statistical modeling of physical and mechanical properties of building materials.