Principles of choosing an approximation basis and a kernel function in a computational algorithm for approximating the probability density from a given sample
0009-0004-4417-0842 This paper formulates two important principles – multiplicativity of the form (in the multivariate case) and finiteness (concentration of supports near the nodes of the approximation grid) – when choosing the approximation basis and the kernel function for constructing a cost-effective computational (computer) algorithm for approximating a probability density fr om a given sample. Taking into account the features of the considered grid-based computational scheme, new criteria for the optimal choice of the kernel function are proposed. These include appropriate combinations of the second moments and the integrals of the squares of these functions, which simultaneously determine the bias components and the stochastic components of the root-mean-square error of the considered kernel algorithm. A particularly important special case is highlighted – the multivariate analog of the frequency polygon (wh ere the chosen kernel function is piecewise constant) – for which it is possible to find parameters that ensure minimal computational cost (for a given error level). A test example demonstrates that the choice of known types of kernel functions, different from piecewise constant ones, does not allow for algorithm optimization and increases computation time (for a given error level).
УДК 519.245
Keywords: computational (computer) functional kernel algorithm, choice of approximation basis, choice of kernel function, multiplicative form of chosen functions, finiteness of chosen functions, root-mean-square error, multivariate analog of the frequency polygon.