Kriging Model with Fractional Euclidean Distance Matrices

Abstract

The multidimensional data model for kriging is developed using fractional Euclidean distance matrices (FEDM). The properties of FEDM are studied by means of the kernel matrix mehod. It has been shown that the factorization of kernel matrix enables us to create the embedded set being a nonsingular simplex. Using the properties of FEDM the Gaussian random field (GRF) is constructed doing it without positive definite correlation functions usually applied for such a purpose. Created GRF can be considered as a multidimensional analogue of the Wiener process, for instance, line realizations of this GRF are namely Wiener processes. Next, the kriging method is developed based on FEDM. The method is rather simple and depends on parameters that are simply estimated by the maximum likelihood method. Computer simulation of the developed kriging extrapolator has shown that it outperforms the well known Shepard inverse distance extrapolator. Practical application of the developed approach to surrogate modelling of wastewater treatment is discussed. Theoretical investigation, computer simulation, and a practical example demonstrate that the proposed kriging model, using FEDM, can be efficiently applied to multidimensional data modelling and processing.

Investigation of Filtration Capacity of Surface Wastewater Filters by Mathematical Modelling

“…In these cases, the Kriging method is proposed to increase the efficiency of solving modelling, prediction and optimization problems [25,26]. Since the Kriging method was developed as a consequence of its accurate predictions, it became one of the most promising numerical approaches in the various fields of engineering design, spatial statistics, experiment design, and so on [15,26]. In general, Kriging is a method of interpolation for which the interpolated values are modelled by a multivariate Gaussian process, governed by prior covariances.…”

“…Let us consider the approach to measurement data modelling by the Gaussian Random field, using geometrical properties of fractional Euclidean distance matrices [15]. The developed model allows us to represent the information, obtained from any number of measurements of the response function by a computational code or physical experiment.…”

“…Usually wastewater filters are complex systems, mathematical modelling of which is most often carried out using the difference differential models, which require complicated mathematical tools and expensive simulations. Therefore, the effectiveness of filters filled with construction waste and biochar is analysed using the Kriging method with distance matrices which requires only experimental data and enable us to avoid complicated mathematical modelling by difference-differential equations [15]. The developed method allows modelling filter characteristics with different filler ratios based on the previous experimental studies of filters.…”