Towards the nonparametric estimation of measures of complex dependencies in dynamic systems

Nonparametric methods of estimation in stochastic systems are certainly more general than parametric estimation methods. The latter assume the presence of a priori information, which ultimately assumes the knowledge of the number of estimated parameters and the structure of the system itself. In nonparametric methods, such information is not required. The paper objective: to develop methods for nonparametric estimation of dependence measures for complexly organized random processes. The article introduces a measure of dependence that links k pairs of random processes. Such a measure, based on the use of conditional mathematical expectations of processes, can be considered as a further generalization of dispersion functions. Convergence with probability 1 of nonparametric estimates of such a measure is derived using sample data. These estimates are used to construct sample analogues of some nonlinear measures of stochastic dependence of random processes, in particular, to obtain a consistent measure of dependence in the sense of Kolmogorov, i.e., a measure that vanishes if and only if the given random processes are stochastically independent. As a direct consequence, the consistency of the measure of dependence in the sense of Rényi, i.e., a measure that satisfies the corresponding Rényi axioms, will immediately follow from the obtained results. The developed estimation algorithms converging with probability 1 do not require any a priori information about the system and can be used to construct input-output mappings of nonlinear systems without any special requirements for the system.