Two approaches to the entropy management of Gaussian stochastic system are considered. The first approach is scalar and implements the concept of "growth points". In this case the problem of maximizing (increasing) or minimizing (decreasing) the system entropy is solved. The second approach is the vector management, allowing to ensure effective changing of the entropy of two-dimensional vector, the components of which are randomness and self-organization entropies. For vector control an optimization problem on the conditional extremum is formulated. This problem can be solved using penalty methods. It is shown that the vector management of entropy for a number of cases has advantages compared to the scalar management. Examples of entropy models of real stochastic systems are provided.Keywords: differential entropy, multidimensional random variable, Gaussian stochastic system, covariance matrix, management, randomness, self-organization. IntroductionA number of real systems can be classified as complex multidimensional stochastic systems. The main important feature of these systems is having multiple elements that are intricately linked. In situations like this, the multidimensional stochastic system is often being modeled as a random vector.One of the perspectives of modeling complex stochastic systems is based on the application of entropy. It is known that entropy is a fundamental property of all systems with an ambiguous or probabilistic behavior [1] .The concept of entropy is rather flexible and it can be clearly interpreted in terms of that specific area, where it is applied. It is being widely used in modern science to describe the structural organization and disorganization, the degree of destruction of the connections between the elements of the system [2-8].Let's take a complex stochastic system S as a multidimensional continuous randomEach Y i element of the vector Y is a one-dimensional random variable which is characterizing the functioning of the particular element of the system under study. Those elements can be either interdependent or independent of each other.In [9] the differential entropy of random vector Y with p Y (x 1 , ..., x m ) joint probability density was introduced:Let's consider the approach to the management of multidimensional stochastic systems using differential entropy on the example of Gaussian systems. 38Journal of Computational and Engineering Mathematics COMPUTATIONAL MATHEMATICS The Entropy Model of a Gaussian Stochastic SystemWe take that the Y = (Y 1 , Y 2 , .., Y m ) random variable has a multivariate normal distribution with a Σ = {σ ij } m×m covariance matrix with its elements σ ij = cov (Y i , Y j ) is the covariance between Y i and Y j random variables, σ ii = σ 2 i is the variance of random variable Y i , i, j = 1, 2, ..., m.For a multidimensional normally distributed random variable Y the differential entropy (the entropy from now on) H(Y ) is equal to [10]:The entropy (1), being a functional from the set of probability densities of random variable Y , is a number...
The algorithms for solving the vector entropy control problem for Gaussian stochastic systems are considered in the article. To solve a nonlinear optimization problem for a conditional extremum, the method of penalty functions with unconditional optimization methods of various-orders is considered. A set of problem-oriented programs has been developed that implements the proposed algorithms. A comparative analysis of the computational efficiency of the proposed algorithms is performed based on Monte Carlo statistical simulation methods and simulation modeling.
SYSTEM-ENTROPY ANALYSIS OF THE EFFECTIVENESS HYPOLIPIDEMIC DRUGSThe system-entropy analysis of the effectiveness of hypolipidemicdrugs was carried out for the first time. The entropy of the lipid system was evaluated for the components "Total cholesterol", "Triglycerides", "HDLcholesterol" in the survey of 219 patients with ischemic heart disease. Entropy is a complex quantity that characterizes the measure of disorder (scattering of constituent elements) and the organization (mutual dependence of elements) of the system. Distribution of hypolipidemic drugs according to the degree of decrease in total entropy (in decreasing order): Atorvastatin>Fluvastatin> Ciprofibrate> Simvastatin> Lovastatin. The change in entropy was: -1,495; -1,124; -0,983; -0,528; -0,2 respectively. The decrease in the total entropy was due mainly to a decrease in the entropy of randomness. The entropy of self-organization in groups of patients taking Atorvastatin, Ciprofibrate has not changed, and in groups of patients taking other hypolipidemicdrugs increased. System-entropy analysis can be used for a comprehensive assessment of the effectiveness of medicines. Keywords: system-entropy analysis, entropy, entropy analysis, dyslipidemia treatment, hypolipidemic drugs, the effectiveness of statins, treatment of atherosclerosis, prevention of cardiovascular diseases.Применение лекарственных средств является важной составной частью лечения, профилактики заболеваний и укрепления здоровья населения. При этом врачи в клинической практике и исследователи при разработке новых лекарственных средств сталкиваются с проблемой оценки их эффективности. Применение медикаментозных препаратов сопровождается изменением многих параметров функционирования организма: одни увеличиваются, другие снижаются, третьи остаются на исходном уровне. Проблема
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