Research of simulation of two-dimensional samples using polynomial splines

Abstract

Simulation is used to solve a variety of problems, for a certain set of reasons: imitation of "critical" modes that can be dangerous in real operation, saving time and material resources, the possibility of remote training, etc. In particular, in the case of research in multidimensional spaces, it is important not to model the operation of the system, but rather the data sequences of a certain type, could solve the problem of the lack of such data.

In [1] it is said that when simulating samples, the first thing to start from is the distribution model that needs to be obtained. The model can be defined by some analytical distribution law (normal, Weibull, uniform, etc.), and in this it depends on the parameters (parametric model). Usually, models are chosen such that their parameters carry some meaningful interpretation (a, b - the beginning and end of the interval in a uniform distribution, λ – intensity in an exponential distribution, etc.). Another class of models that reproduce distribution functions are nonparametric ones (kernel methods, histogram estimates of the empirical distribution function, spline approximation [4]). The main problem with parameter-based methods is their limitation, especially in two cases:

When modeling multivariate data - in this case, the work always leads to the transition to a multivariate normal distribution.

When modeling heterogeneous samples that are a mixture of several distributions (not necessarily from the same class), truncated or containing missing observations.

In this context, the use of parametric models is objectively impossible in its pure form. Therefore, having a tool that approximates heterogeneous data well is desirable for solving the problem of generating heterogeneous multivariate populations.