Kartlos Kachiashvili

For solving the problems of study, analysis and quality management of the environment

there is necessary operatively to treat great amount of measuring information on physical,

chemical and biological parameters characteristic for them. To do it in a proper way, in

conformity to the modern requirements, is possible only by wide use of modern mathematical

methods and computers. For this purpose it is necessary to develop automated systems and

universal program packages with developed mathematical methods consisting of self-learning

algorithms requiring whenever it is possible minimum a prior information and having

capability of adaptation to the most unexpected changes of the character of the investigated

objects [1].

Among the most topical problems of monitoring of a natural water environment it is

necessary to single out the following issues: simulation of pollutants transferring in water

objects; methods of making decisions about condition of controlled objects and processes

taking place in them; identification of sources of emergency pollution to take measures for

their elimination. These problems are especially urgent in urban conditions because there

exist great number of sources of pollution. Their solution is of great ecological and

economical significance which makes possible to investigate the effect of different sources of

pollution on ecological object separately from each other, as well as jointly, to predict out-

comes of such an impact and consequences of the nature protection measures against the

sources of pollution. With their help, the minimization of technical means, in particular, those

of measurements, indispensable for the control and management of each source of pollution is

reached. They are also actual for large plants and factories having biochemical clearing of

sewages, on their design and ecological safe operation, as well as for detection of sites and

shops producing the excess of sewages pollution.

Algorithms were always an important part of many branches in the sciences. In many manuals and handbooks, algorithms of problems of computational mathematics are focused on the manual performance or by means of a calculator. In this book, descriptions of algorithms, their solutions and main characteristics are discussed. The present work is the outcome of many years of the authors’ work on solving different problems and tasks from domains of instruction making, metrology, system analysis, ecology, data analysis from ecology, agriculture, medicine and creation of corresponding universal computer packages and systems. (Imprint: Nova)

Algorithms were always an important part of many branches in the sciences. In many manuals and handbooks, algorithms of problems of computational mathematics are focused on the manual performance or by means of a calculator. In this book, descriptions of algorithms, their solutions and main characteristics are discussed.

The present work is the outcome of many years of the authors’ work on solving different problems and tasks from domains of instruction making, metrology, system analysis, ecology, data analysis from ecology, agriculture, medicine and creation of corresponding universal computer packages and systems.

The problems of one of the basic branches of mathematical statistics – statistical hypotheses testing – are considered in this book. The intensive development of these methods began at the beginning of the last century. The basic results of modern theory of statistical hypotheses testing belong to the cohort of famous statisticians of this period: Fisher, Neyman-Pearson, Jeffreys and Wald (Fisher, 1925; Neyman and Pearson, 1928, 1933; Jeffreys, 1939; Wald, 1947a,b). Many other bright scientists have brought their invaluable contributions to the development of this theory and practice. As a result of their efforts, many brilliant methods for different suppositions about the character of random phenomena are under study, as well as their applications for solving very complicated and diverse modern problems.

Since the mid-1970s, the author of this book has been engaged in the development of the methods of statistical hypotheses testing and their applications for solving practical problems from different spheres of human activity. As a result of this activity, a new approach to the solution of the considered problem has been developed, which was later named the Constrained Bayesian Methods (CBM) of statistical hypotheses testing. Decades were dedicated to the description, investigation and applications of these methods for solving different problems. The results obtained for the current century are collected in seven chapters and three appendices of this book. The short descriptions of existing basic methods of statistical hypotheses testing in relation to different CBM are examined in Chapter One. The formulations and solutions of conventional (unconstrained) and new (constrained) Bayesian problems of hypotheses testing are described in Chapter Two.

The investigation of singularities of hypotheses acceptance regions in CBM and new opportunities in hypotheses testing are presented in Chapter Three. Chapter Four is devoted to the investigations for normal distribution. Sequential analysis approaches developed on the basis of CBM for different kinds of hypotheses are described in Chapter Five. The special software developed by the author for statistical hypotheses testing with CBM (along with other known methods) is described in Chapter Six. The detailed experimental investigation of the statistical hypotheses testing methods developed on the basis of CBM and the results of their comparison with other known methods are given in Chapter Seven. The formalizations of absolutely different problems of human activity such as hypotheses testing problems in the solution – of which the author was engaged in different periods of his life – and some additional information about CBM are given in the appendices.

Finally, it should be noted that, for understanding the materials given in the book, the knowledge of the basics of the probability theory and mathematical statistics is necessary. I think that this book will be useful for undergraduate and postgraduate students in the field of mathematics, mathematical statistics, applied statistics and other subfields for studying the modern methods of statistics and their application in research. It will also be useful for researchers and practitioners in the areas of hypotheses testing, as well as the estimation theory who develop these new methods and apply them to the solutions of different problems.

Imitation models for computing the environmental water pollution level depending

on the intensity of pollution sources created by the author over the

years are presented. For this purpose, an additive model of a non-stationary

random process is considered. For the modeling of its components, models

that consider only dilution and self-purification processes are proposed for

waste water and three-dimensional turbulent diffusion equations for river

waters, and multidimensional Gaussian Markov series are proposed for modeling

the random component. The purpose, the capabilities and the peculiarities

of such imitation models are discussed taking into account the peculiarities

of the water objects. The modular principle of creating imitation models

is proposed to facilitate their development and use.

A short consideration of the existing approaches of statistical hypotheses

testing is given below. Among classical methods, comparatively new Constrained

Bayesian Method and its peculiarities are introduced. A brief description of the

essences of these methods is given. Recommendation for choosing a concrete

method for statistical hypotheses testing is given finally.

Since the mid-70s of the last century, the author of this paper has been engaged in

the development of themethods of statistical hypothesis testing and their applications

for solving practical problems from different spheres of human activity (Kachiashvili

1980, 2018; Kachiashvili et al. 2009). As a result of this activity, there developed

a new approach to the solution of the considered problem, which was later named

Constrained Bayesian Methods (CBM) of statistical hypotheses testing. The results

obtained in Kachiashvili (1989, 2003, 2011), Kachiashvili et al. (2012a, b), Kachiashvili

and Mueed (2013) and Kachiashvili (2019b, 2018) show that the Bayesian

approach to hypothesis testing, formulated as a constrained optimization problem,

allows us to combine the best features of both Bayesian and Neyman–Pearson

approaches. Moreover, it is the data-dependent method similar to the Fisher’s test

and gives a decision rule with new, more common properties than a usual decision

rule does. This approach opens new opportunities in inference theory and practice.

In particular, it allows us, on the basis of a common platform, to develop a set of

reliable, flexible, economic and universal methods for testing statistical hypotheses

of different types.

The present paper is devoted to an overview of the results obtained by us, using

this method, when examining different types of hypotheses under different criteria.

The rest of the article is organized as follows. The next section discusses CBM for

testing different types of hypotheses, comparative analysis of constrained Bayesian

and other methods are presented in Section 3 and conclusions are offered in Section 4.

Statistical hypotheses testing is one of the basic direction of mathematical statistics the methods of which are

widely used in theoretical research and practical applications. These methods are widely used in medical researches too. Scientists of different fields, among them of medical too, that are not experts in statistics, are often faced with the dilemma of which method to use for solving the problem they are interested. The article is devoted to helping the specialists in solving this problem and in finding the optimal resolution. For this purpose, here are very simple and clearly explained the essences of the existed approaches and are shown their positive and negative sides and are given the recommendations about their use depending on existed information and the aim that must be reached as a result of an investigation.

The original computer technologies for controlling and managing the ecological condition of the environmental water objects, developed

under the guidance and direct participation of the author, are described in the article. In particular, their purpose, capabilities

and peculiarities are briefly described. There is also given a short description of problems solved by using them.

By statistical processing of Georgian Cancer Registry data of 2015-2016, clustering (grouping) of Georgian populations was realized, according to the intensity of the cancer disease prevalence, for the purpose of priority distribution of existed resources and means in the country and for the reduction of the number of patients and improvement of the quality of treatment. Cluster analysis methods of mathematical statistics were used for the study, which was directly implemented using universal statistical software package SPSS. The concept of disease index was introduced for achieving the intruded purpose. Its several variants were determined. The study results using indexes showed that it is possible to group objectively populated areas and regions of the country by intensity of dissemination of cancer disease.

Constrained Bayesian methods (CBMs) and the concept of false discovery rates (FDRs) for testing directional hypotheses are considered in this article. It is shown that the direct application of CBM allows us to control FDR on the desired level for both one set of directional hypotheses and a multiple case when we consider m (m > 1) sets of directional hypotheses. When guaranteeing restriction on the desired level, a Bayesian sequential method can be applied, the stopping rules of which are proper and the sequential scheme for making a decision strongly controls the mixed directional FDR. Computational results of concrete examples confirm the correctness of the theoretical outcomes.

Application of CBM to the testing of the intersection of a sub-set of basic hypotheses against an alternative one is considered. Optimal decision rule allows us to restrict the Type-I and Type-II errors rates on the desired levels.

A statistical hypothesis is a formalized record of properties of the investigated phenomenon and relevant assumptions. The statistical hypotheses are set when random factors affect the investigated phenomena, i.e. when the observation results of the investigated phenomena are random. The properties of the investigated phenomenon are completely defined by its probability distribution law. Therefore, the statistical hypothesis is an assumption concerning this or that property of the probability distribution law of a random variable. Mathematical statistics is the set of the methods for studying the events caused by random variability and estimates the measures (the probabilities) of possibility of occurrence of these events. For this reason, it uses distribution laws as a rule. Practically all methods of mathematical statistics one way or another, in different doses, use hypotheses testing techniques. Therefore, it is very difficult to overestimate the meaning of the methods of statistical hypotheses testing in the theory and practice of mathematical statistics.

Constrained Bayesian method (CBM) and the concept of false discovery rates (FDR) for testing directional hypotheses is considered in the paper. Here is shown that the direct application of CBM allows us to control FDR on the desired level. Theoretically it is proved that mixed directional false discovery rates (mdFDR) are restricted on the desired levels at the suitable choice of restriction levels at different statements of CBM. The correctness of the obtained theoretical results is confirmed by computation results of concrete examples.

Constrained Bayesian method (CBM) and the concept of false discovery rates (FDR) for testing directional hypotheses is considered in the paper. Here is shown that the direct application of CBM allows us to control FDR on the desired level. Theoretically it is proved that mixed directional false discovery rates (mdFDR) are restricted on the desired levels at the suitable choice of restriction levels at different statements of CBM. The correctness of the obtained theoretical results is confirmed by computation results of concrete examples.

Constrained Bayesian method (CBM) and the concept of false discovery rates (FDR) for testing directional hypotheses is considered in the paper. Here is shown that the direct application of CBM allows us to control FDR on the desired level. Theoretically it is proved that mixed directional false discovery rates (mdFDR) are restricted on the desired levels at the suitable choice of restriction levels at different statements of CBM. The correctness of the obtained theoretical results is confirmed by computation results of concrete examples.

The paper discusses the generalization of constrained Bayesian method (CBM) for arbitrary loss functions and its application for testing the directional hypotheses. The problem is stated in terms of false and true discovery rates. One more criterion of estimation of directional hypotheses tests quality, the Type III errors rate, is considered. The ratio among discovery rates and the Type III errors rate in CBM is considered. The advantage of CBM in comparison with Bayes and frequentist methods is theoretically proved and demonstrated by an example.

The paper discusses the application of constrained Bayesian method (CBM) of testing the directional hypotheses. It is proved that decision rule of CBM restricts the mixed directional false discovery rate (mdFDR) and total Type III error rate as well.

The iteration algorithm of computation of effective estimators of the shape parameters of beta distributions using the unbiased estimators of the end point parameters of the random variable were obtained and investigated. For the cases when more accurate estimations of the parameters are required, one more step of computation, realized optimization of the obtained estimations, is necessary. The computation results, realized on the basis of the simulation of the appropriate random samples, demonstrate the correctness of the obtained theoretical outcomes.

The paper deals with the problem of electronic verification of people on the basis of measurement information of a fingerprint reader and new approaches to its solution. The offered method guaranties the restriction of error probabilities of both type at the desired level at making a decision about permitting or rejecting the request on service in the system. On the basis of investigation of real data obtained in the real biometrical system, the choice of distribution laws is substantiated and the proper estimations of their parameters are

obtained. Using chosen distribution laws, the normal distribution for measurement results of characteristics of the people having access to the system and the beta distribution for the people having no such access, the optimal rule based on the Constrained Bayesian Method (CBM) of making a decision about giving a permission of access to the users of the system is justified. The CBM, the Neyman–Pearson and classical Bayes methods are investigated and their good and negative points are examined. Computation results obtained by direct computation, by simulation and using real data completely confirm the suppositions made and the high quality of verification results obtained on their basis.

For solving the problems of study, analysis and quality management of the environment there is necessary operatively to treat great amount of measuring information on physical, chemical and biological parameters characteristic for them. To do it in a proper way is possible only by wide use of modern mathematical methods and computers. For this purpose, it is necessary to develop automated systems and universal program packages with modern mathematical methods consisting of self-learning algorithms requiring minimal a prior information and having capability of adaptation to the most unexpected changes of the character of the investigated objects. Among the most topical problems of monitoring of a natural water environment it is

necessary to develop: the automated water quality control systems for operative control and management of water pollution level; simulation of pollutants transferring in water objects; methods of making decisions about condition of controlled objects and processes taking place in them; identification of sources of emergency pollution. These problems are especially urgent in urban conditions because their great number of sources of pollution exist. Their solution is of great ecological and economical significance which makes possible to investigate the effect of different sources of pollution on ecological object separately from each other, as well as jointly, to predict outcomes of such an impact and consequences of the nature protection measures against the sources of pollution. They are also actual for large plants and factories having biochemical clearing of sewages, on their design and ecological safe operation.

Technological progress brought the civilization to the emergence of numerous artificial factors, the influence of which on the environment becomes in all perceptible. This stipulates the quantitative and qualitative change in the environment. In this connection, there arises an actual problem of studying and analyzing the current situation with the purpose of elaboration of the principles and facilities of saving the environment in the suitable condition of living. It is necessary to have adequate information about the quality of the environment for its study, analysis and management. The environment is characterized by an enormous number of physical, chemical and biological parameters. A lot of measurements are to be carried out for permanent control of these parameters. Therefore, the solution of the problems of control and management of the quality of the environment can be realized only by using automated, continuously operating pollution analyzers and automated environment control systems.

The profitability and efficiency of farming economies achieved by introduction of advanced agricultural methods during two years, in particular, the increase in the productivity of soil with the use of advanced manure, treated in biogas facilities, the increase in the productivity of degraded and low-productive agricultural lands (exchange of seeds, introduction of new cultures, drainage etc.) and soil erosion prevention techniques were investigated by application of the methods of mathematical statistics to the obtained results. High profitability, efficiency and economic justification of these methods are shown as a result of this investigation.

The program packages of realization of mathematical models of pollutants transport in rivers and for identification of river water excessive pollution sources located between two controlled cross-sections of the river will be considered and demonstrated. The software has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers and statistical hypotheses testing methods. The identification algorithms were elaborated with the supposition that the pollution sources discharge different compositions of pollutants or (at the identical composition) different proportions of pollutants into the rivers. One-, two-, and three-dimensional advection-diffusion mathematical models of river water quality formation both under classical and new, original boundary conditions are realized in the package. New finite-difference schemes of calculation have been developed and the known ones have been improved for these mathematical models. At the same time, a number of important problems which provide practical realization, high accuracy and short time of obtaining the solution by computer have been solved. Classical and new constrained Bayesian methods of hypotheses testing for identification of river water excessive pollution sources are realized in the appropriate software. The packages are designed as a up-to-date convenient, reliable tools for specialists of various areas of knowledge such as ecology, hydrology, building, agriculture, biology, ichthyology and so on. They allow us to calculate pollutant concentrations at any point of the river depending on the quantity and the conditions of discharging from several pollution sources and to identify river water excessive pollution sources when such necessity arise.

The paper deals with the constrained Bayesian Method (CBM) for testing composite hypotheses. It is shown

that, similarly to the cases when CBM is optimal for testing simple and multiple hypotheses in parallel and sequential experiments, it keeps the optimal properties at testing composite hypotheses. In particular, it easily, without special efforts, overcomes the Lindley’s paradox arising when testing a simple hypothesis versus a composite one. The CBM is compared with Bayesian test in the classical case and when the a priori probabilities are chosen in a special manner for overcoming the Lindley’s paradox. Superiority of CBM against these tests is demonstrated by simulation. The justice of the theoretical judgment is supported by many computation results of different characteristics of the considered methods.

Statistical methods all are more widely used in all spheres of human activity. Their importance in medicine and biology especially intensively is developing and increasing since the latest decade of the previous century. The reason of this circumstance consists in especial complexity of the problems of these domains caused by complexity of their character, by the great number of the parameters included in them and of the factors influencing their. Many of the factors affecting the observation results used for investigation of the problems under study are random by their nature and, hence, the observation results are random. Therefore the study and solution of these problems require the application of the modern methods of probability and mathematical statistics.

We obtained and investigated consistent, unbiased and efficient estimators of the parameters of irregular right-angled triangular distribution on the basis of maximum likelihood estimators. Some computation results realized on the basis of simulation of the appropriate random samples demonstrate theoretical outcomes.

We have developed an original, simple and convenient software for testing statistical hypotheses concerning the parameters of probability distribution laws. It is intended for the users who are not professionals in the field of applied statistics and computer science because it is very simple and convenient for use, and the results of

application of the methods realized in the package are given as simple for understanding texts in outcomes of

the programs. The problems and the algorithms (of only original) realized in the package, as well as the features and the opportunities of their application are briefly described. Those features that distinguish favorably this package from other similar products are emphasized. Some examples showing singularities and

efficiency of the algorithms realized in the package are cited.

A sequential method based on constrained Bayesian methods is developed for testing multiple hypotheses. It controls the family-wise error rate and the family-wise power in a more accurate form than the Bonferroni or intersection scheme using the ideas of stepup and step-down methods for multiple comparisons of sequential designs. The new method surpasses the existing testing methods proposed earlier in a substantial reduction of the expected sample size.

The article focuses on the discussion of basic approaches to hypotheses testing, which are Fisher, Jeffreys,

Neyman, Berger approaches and a new one proposed by the author of this paper and called the constrained Bayesian method (CBM). Wald and Berger sequential tests and the test based on CBM are presented also. The positive and negative aspects of these approaches are considered on the basis of computed examples. Namely, it is shown that CBM has all positive characteristics of the above-listed methods. It is a data-dependent measure like Fisher’s test for making a decision, uses a posteriori probabilities like the Jeffreys test and computes error probabilities Type I and Type II like the Neyman-Pearson’s approach does. Combination of these properties assigns new properties to the decision regions of the offered method. In CBM the observation space contains regions for making the decision and regions for no-making the decision. The regions for no-making the decision are separated into the regions of impossibility of making a decision and the regions of impossibility of making a unique decision. These properties bring the statistical hypotheses testing rule in CBM much closer to the everyday decision-making rule when, at shortage of necessary information, the acceptance of one of made suppositions is not compulsory. Computed practical examples clearly demonstrate high quality and reliability of CBM. In critical situations, when other tests give opposite decisions, it gives the most logical

decision. Moreover, for any information on the basis of which the decision is made, the set of error robabilities is defined for which the decision with given reliability is possible.

New sequential methods of multiple testing problems based on special properties of hypotheses acceptance regions in the constrained Bayesian tasks of testing hypotheses are offered. Results of an investigation on the properties of one of these methods are given. They show the consistency, simplicity, and optimality of the results obtained in the sense of the chosen criterion. The essence of the criterion is to restrict from above the probability of the error of one type and to minimize the probability of the error of the second type. The facts of the validity of the suitable properties of the method are proved. Examples of testing of hypotheses for the sequentially obtained independent samples from the multivariate normal distribution with correlated components are cited. They show the high quality of the proffered methods. The results of the Wald sequential method are given for the examples with two hypotheses and compared with the results obtained by the proffered method.

The results of investigation of the properties of new sequential methods of testing many hypotheses based on special properties of hypotheses acceptance regions in the constrained Bayesian tasks of testing many hypotheses are offered. In particular, some relations between the errors of the first and the second kinds in constrained Bayesian task and in sequential method of Bayesian type depending on the divergence between the tested hypotheses are given. Also dependences of the Lagrange multiplier and the risk function on the probability of incorrectly accepted hypotheses are presented. Theses results are necessary for computation of errors of made decisions at testing multiple hypotheses using offered new sequential methods of testing hypotheses. Computation results of some examples confirm the rightness of theoretical researches.

Formulae for computation of probability of errors in sequential method of Bayesian type are offered. In particular, some relations between the errors of the first and the second kinds in constrained Bayesian task and in sequential method of Bayesian type depending on the divergence between the tested hypotheses are given. Dependencies of the Lagrange multiplier and the risk function on the probability of incorrectly accepted hypotheses are also presented. Theses results are necessary for computation of errors of made decisions at

testing multiple hypotheses using the offered new sequential methods of testing hypotheses. Computation results of some examples confirm the rightness of theoretical analysis.

Conditional Bayesian task of testing many hypotheses is stated and solved. The concept of conditionality is used for the designation of the fact that the Bayesian task is stated as a conditional optimization problem where the probability of one-type error is restricted and, under such a condition, the probability of second-type error is minimized. The offered statement gives the decision rule which allows us not to accept any hypothesis if, on the basis of the available information, it is impossible to make a decision with the set significance level. In such a case, it is necessary to ensure the additional information in the form of additional observation results or a change in the significant level of hypotheses testing. These properties make our statement more general than the usual statement of the Bayesian problem which is a special case of the one offered and improve the reliability of the made decision. The calculation results completely confirm the results of theoretical investigations.

Quasi-optimal procedures of testing many hypotheses are described in this paper. They significantly simplify

the Bayesian algorithms of hypothesis testing and computation of the risk function. The relations allowing for obtaining the estimations for the values of average risks in optimum tasks are given. The obtained general solutions are reduced to concrete formulae for a multivariate normal distribution of probabilities. The methods of approximate computation of the risk functions in Bayesian tasks of testing many hypotheses are offered. The properties and interrelations of the developed methods and algorithms are investigated. On the basis of a simulation, the validity of the obtained results and conclusions drawn is presented.

Specific features of the regions of acceptance of hypotheses in conditional Bayesian problems of statistical hypotheses testing are discussed. It is shown that the classical Bayesian statement of the problem of statistical hypotheses testing in the form of an unconditional optimizing problem is a special case of conditional Bayesian problems of hypotheses testing set in the form of conditional optimizing problems. It is also shown that, at acceptance of hypotheses in conditional problems of hypotheses testing, the situation is similar to the sequential analysis. It is possible an occurrence of the situation when the acceptance of a hypothesis with specified validity on the basis of the available information is impossible. In such a situation, the actions are

similar to the sequential analysis, i.e. it is necessary to obtain additional information in the form of new observation results or to change the significance level of a test.

The problem of choosing the loss function in the Bayesian problem of many hypotheses testing is considered. It is shown that linear and quadratic loss functions are the most-used ones. For any kind of loss function, the risk function in the Bayesian problem of many hypotheses testing contains the errors of two kinds. The Bayesian decision rule minimizes the total effect of these errors. The share of each of them in the optimal value of risk function is unknown. When solving many important problems, the results caused by different errors significantly differ from each other. Therefore, it is necessary to guarantee the limitation on the most undesirable kind of these errors and to minimize the errors of the second kind. For solving these problems, this article are states and solves conditional Bayesian tasks of testing many hypotheses. The results of sensitivity analysis of the classical and conditional Bayesian problems are given and their advantages and drawbacks are considered.

The computation of the multivariate normal integral over a Complex Subspace is a challenge, especially when the inte-gration region is of a complex nature. Such integrals are met with, for example, in the generalized Neyman-Pearson criterion, conditional Bayesian problems of testing many hypotheses and so on. The Monte-Carlo methods could be used for their computation, but at increasing dimensionality of the integral the computation time increases unjustifiedly. Therefore a method of computation of such integrals by series after reduction of dimensionality to one without informa-tion loss is offered below. The calculation results are given.

In the work the problem of sustainable development of production, i.e., an optimum choice of parameter values of technological process with the purpose of minimization of risk of obtaining production of not planed quality also incorrect making decision about quality of production and maximization of profit of production at the guaranteed social and economic effects is formalized. Different statements of the problem depending on the put ultimate purpose are considered. The general method of solution of the put task using Bayesian approach of testing many hypotheses is offered.

We offer an original, simple and convenient software package for testing statistical hypotheses concerning the parameters of probability distribution laws. The methods of statistical hypotheses testing allow us to solve problems from many spheres of human activity. Applications include engineering, physics, chemistry, medicine, biology, economics, defense, ecology, sociology and so on [see, for example, 1-6]. The problems arising in these areas include the fact that as a rule, the applications contain numerous parameters, i.e. the tasks are multivariate. The dimensionality of the tasks very often reaches up to several tens, even several hundreds. It is practically impossible to solve these tasks without special software. Despite the variety of statistical packages in which the methods of statistical hypotheses testing are realized, there is not known package realizing the methods similar to the ones realized in the offered package. In this package, in addition to well-known, classical methods of hypotheses testing, such as the Bayesian method with general and step loss functions; Sign Test; Mann-Whitney Test; Wilcoxon Test; Wilcoxon Signed-Rank Sum Test and the Wald sequential method [see for example, 7,8], entirely new, original methods, are realized. Among these are

constrained Bayesian methods with restrictions on the probabilities of errors of the first or second types, quasi-optimum methods, and sequential analysis methods of Bayesian type for testing any number of hypotheses [9-13]. Simple, convenient and reliable methods of statistical hypotheses testing, based on

different information distances (Euclidian and Makhalanobis) between them are also realized in the package.

The developed package is completely original and has no analogue. All algorithms, programs, texts, schedules, drawings, tables, registration etc. realized in the package belong to the authors.

In Bayesian statement of hypotheses testing, instead of unconditional problem of minimization of average risk caused by the errors of the first and the second types, there is offered to solve the conditional optimization problem when restrictions are imposed on the errors of one type and, under such conditions, the errors of the second type are minimized. Depending on the type of restrictions, there are considered different conditional optimization problems. Properties of hypotheses acceptance regions for the stated problems are investigated and, finally, comparison of the properties of unconditional and conditional methods is realized. The results of the computed example confirm the validities of the theoretical judgments.

There are considered three original software packages developed by authors: the first is for statistical processing of the experimental information; the second is a software package of realization of mathematical models of pollutants transport in rivers and the third is for identification of river water excessive pollution sources located between two controlled cross-sections of the river. They are designated on identical methodological basis and are intended for the users who are not professionals in the field of applied mathematics and computer science. The packages are universal, simple and convenient for understanding and application.

In Bayesian statement of hypotheses testing, instead of unconditional problem of minimization of average

risk caused by the errors of the first and the second types, there is offered to solve the conditional optimization problem when restrictions are imposed on the errors of one type and, under such conditions, the errors of the second type are minimized. Depending on the type of restrictions, there are considered different conditional optimization problems. Properties of hypotheses acceptance regions for the stated problems are investigated and, finally, comparison of the properties of unconditional and conditional methods is realized. The results of the computed example confirm the validities of the theoretical judgments.

There is an original software package offered for statistical processing of experimental information. It is designated for users who are not professionals in the field of applied statistics and computer science. The package is universal, simple and convenient for understanding and application. The problems and the algorithms realized in the package, the features and the opportunities of their application are described. Those features which distinguish favorably this package from other similar products are emphasized.

The examples showing the efficiency of the algorithms realized in the package are cited. Serviceability of the suggested package was tested in various modes at solving the problems from different fields of knowledge. The obtained results justify the stability and the reliability of algorithms and the high accuracy of the calculated values.

New sequential method of testing many hypotheses based on special properties of decision-making areas in the conditional Bayesian task of testing many hypotheses is offered. The results of research of the properties of this method are given. They show the consistency, simplicity and optimality of the obtained results in the sense of the chosen criterion, which consists in the upper restriction of the probability of the error of one kind

and the minimization of the probability of the error of the second kind. The examples of testing of hypotheses for the case of the sequential independent sample from the multidimensional normal law of probability distribution with correlated components are cited. They show the high quality of the offered methods.

An original software package for determination of biological age has been offered. The package is simple for

understanding and convenient in application. It is designed for the users who are not professionals in the fields of applied statistics or computer science. The problems and the algorithms realized in the package, the features and the possibilities of their application are described in brief.

The package can be used both for fundamental theoretical research in which various logical-mathematical methods of determination of biological age are compared with each other and for applied work in a geriatric clinic.

A software package of realization of mathematical models of pollutants transport in rivers is offered. This

package is designed as a up-to-date convenient, reliable tool for specialists of various areas of knowledge

such as ecology, hydrology, building, agriculture, biology, ichthyology and so on. It allows us to calculate

pollutant concentrations at any point of the river depending on the quantity and the conditions of discharging

from several pollution sources. One-, two-, and three-dimensional advection–diffusion mathematical models of river water quality formation both under classical and new, original boundary conditions are realized in the package. New finite-difference schemes of calculation have been developed and the known ones have been improved for these mathematical models. At the same time, a number of important problems which provide practical realization, high accuracy and short time of obtaining the solution by computer have been solved. In particular: (a) the analytical description of plane or spatial region for which the diffusion equations and boundary conditions are investigated, i.e. the analytical description of the bank line and the bottom of the river; (b) the analytical description of dependence of coefficients of the equation on the spatial coordinates; (c) the analytical description of dependence of non-homogeneous parts of the diffusion equation (i.e. the capacities of pollution sources) on the spatial coordinates and on the time; (d) the correct choice of ratios between spatial steps of the grid, and also between them and the step of digitations of time in the difference scheme.

The methods of mathematical statistics by their nature are universal in the sense that the same methods can be used for solving the problems of absolutely different nature. The same mathematical methods successfully solve a great diversity of problems from different areas of knowledge. For illustration of this fact, in this work, the formalization of three absolutely different problems from different areas of knowledge is given (air defense, the environment monitoring, sustainable development of production). They show that, despite their absolutely different nature and character at first sight, the formalization reduces to identical mathematical tasks which could be solved by using the same methods of mathematical statistics. For solving of these tasks, unconditional and conditional Bayesian methods of testing of many hypotheses are used, which gives the opportunities of decision-making with certain significance level of criterion.

The methods of mathematical statistics by their nature are universal in the sense that the same methods can be used for solving the problems of absolutely different nature. The same mathematical methods successfully solve a great diversity of problems from different areas of knowledge. For illustration of this fact, in this work, the formalization of three absolutely different problems from different areas of knowledge is given (air defense, the environment monitoring, sustainable development of production). They show that, despite their absolutely different nature and character at first sight, the formalization reduces to identical mathematical tasks which could be solved by using the same methods of mathematical statistics. For solving of these tasks, unconditional and conditional Bayesian methods of testing of many hypotheses are used, which gives the opportunities of decision-making with certain significance level of criterion.

The methods of mathematical statistics by their nature are universal in the sense that the same methods can be used for solving the problems of absolutely different nature. The same mathematical methods successfully solve a great diversity of problems from different areas of knowledge. For illustration of this fact, in this work, the formalization of three absolutely different problems from different areas of knowledge is given (air defense, the environment monitoring, sustainable development of production). They show that, despite their absolutely different nature and character at first sight, the formalization reduces to identical mathematical tasks which could be solved by using the same methods of mathematical statistics. For solving of these tasks, unconditional and conditional Bayesian methods of testing of many hypotheses are used, which gives the opportunities of decision-making with certain significance level of criterion.

The methods of mathematical statistics by their nature are universal in the sense that the same methods can be used for solving the problems of absolutely different nature. The same mathematical methods successfully solve a great diversity of problems from different areas of knowledge. For illustration of this fact, in this work, the formalization of three absolutely different problems from different areas of knowledge is given (air defense, the environment monitoring, sustainable development of production). They show that, despite their absolutely different nature and character at first sight, the formalization reduces to identical mathematical tasks which could be solved by using the same methods of mathematical statistics. For solving of these tasks, unconditional and conditional Bayesian methods of testing of many hypotheses are used, which gives the opportunities of decision-making with certain significance level of criterion.

The methods of mathematical statistics by their nature are universal in the sense that the same methods can be used for solving the problems of absolutely different nature. The same mathematical methods successfully solve a great diversity of problems from different areas of knowledge. For illustration of this fact, in this work, the formalization of three absolutely different problems from different areas of knowledge is given (air defense, the environment monitoring, sustainable development of production). They show that, despite their absolutely different nature and character at first sight, the formalization reduces to identical mathematical tasks which could be solved by using the same methods of mathematical statistics. For solving of these tasks, unconditional and conditional Bayesian methods of testing of many hypotheses are used, which gives the opportunities of decision-making with certain significance level of criterion.

The methods of mathematical statistics by their nature are universal in the sense that the same methods can be used for solving the problems of absolutely different nature. The same mathematical methods successfully solve a great diversity of problems from different areas of knowledge. For illustration of this fact, in this work, the formalization of three absolutely different problems from different areas of knowledge is given (air defense, the environment monitoring, sustainable development of production). They show that, despite their absolutely different nature and character at first sight, the formalization reduces to identical mathematical tasks which could be solved by using the same methods of mathematical statistics. For solving of these tasks, unconditional and conditional Bayesian methods of testing of many hypotheses are used, which gives the opportunities of decision-making with certain significance level of criterion.

The program package for identification of river water excessive pollution sources located between two controlled cross-sections of the river is described in this paper. The software has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers and statistical hypotheses checking methods. The identification algorithms were elaborated with the supposition that the pollution sources discharge different compositions of pollutants or (at the identical composition) different proportions of pollutants into the rivers.

The program package for identification of river water excessive pollution sources located between two controlled cross-sections of the river is described in this paper. The software has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers and statistical hypotheses checking methods. The identification algorithms were elaborated with the supposition that the pollution sources discharge different compositions of pollutants or (at the identical composition) different proportions of pollutants into the rivers.

The program package for identification of river water excessive pollution sources located between two controlled cross-sections of the river is described in this paper. The software has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers and statistical hypotheses checking methods. The identification algorithms were elaborated with the supposition that the pollution sources discharge different compositions of pollutants or (at the identical composition) different proportions of pollutants into the rivers.

The program package for identification of river water excessive pollution sources located between two controlled cross-sections of the river is described in this paper. The software has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers and statistical hypotheses checking methods. The identification algorithms were elaborated with the supposition that the pollution sources discharge different compositions of pollutants or (at the identical composition) different proportions of pollutants into the rivers.

The aim of the present study was to examine the distribution of pollutants in two coastal systems in Georgia: (1) Kubitskali river which flows into the Black sea through the city of Batumi and is polluted mainly from the effluents of an oil refinery; (2) Paliastomi lake, which is a shallow water body at the south-east of the city of Poti. During 2000–2001, two samplings took place in each system, one in the low-flow period and one in the high-flow period. During these samplings, pH, temperature, dissolved oxygen, and salinity were measured in situ, whereas water samples were collected for the analysis of trace metals, nutrients, and organic pollutants with standard methods. The results of the measurements indicate the significant pollution of both systems by ammonia and in the case of Kubitskali River also by oil products. The need for a sustainable management plan of the activities taking place in the river basin is urgent.

The aim of the present study was to examine the distribution of pollutants in two coastal systems in Georgia: (1) Kubitskali river which flows into the Black sea through the city of Batumi and is polluted mainly from the effluents of an oil refinery; (2) Paliastomi lake, which is a shallow water body at the south-east of the city of Poti. During 2000–2001, two samplings took place in each system, one in the low-flow period and one in the high-flow period. During these samplings, pH, temperature, dissolved oxygen, and salinity were measured in situ, whereas water samples were collected for the analysis of trace metals, nutrients, and organic pollutants with standard methods. The results of the measurements indicate the significant pollution of both systems by ammonia and in the case of Kubitskali River also by oil products. The need for a sustainable management plan of the activities taking place in the river basin is urgent.

The aim of the present study was to examine the distribution of pollutants in two coastal systems in Georgia: (1) Kubitskali river which flows into the Black sea through the city of Batumi and is polluted mainly from the effluents of an oil refinery; (2) Paliastomi lake, which is a shallow water body at the south-east of the city of Poti. During 2000–2001, two samplings took place in each system, one in the low-flow period and one in the high-flow period. During these samplings, pH, temperature, dissolved oxygen, and salinity were measured in situ, whereas water samples were collected for the analysis of trace metals, nutrients, and organic pollutants with standard methods. The results of the measurements indicate the significant pollution of both systems by ammonia and in the case of Kubitskali River also by oil products. The need for a sustainable management plan of the activities taking place in the river basin is urgent.

Developed by the authors general methods of identification of nonlinear regressions for the

certain class of functional dependences which is determined on the basis of expert estimations

of the conducting experts of a number of institutes as most frequently meeting in researches

is realized in the work. The essence general methods consists in minimization of the modified

method of the least squares, which is reduced to the solving of the nonlinear equations (in

considered cases one or two) by means of an iterative algorithm, for which the initial range

of definition of parameters finds with elaborated by the authors a modified method of tests

using algorithms of interpolation [1, 2, 3]. Besides the properties of the considered nonlinear

functions depending on the values of parameters included in them are investigated in the work

and the appropriate diagrams are given. The knowledge of the latter is very important at

identification of functional dependences for a correct choice of an analytical kind of function

appropriate to experimental data.

In the paper are considered one-parameter families of functions from polynomials and

a set of non-linear functions of real variables depended from many parameters. Is offered

a general method of definition of unknown parameters values both for equal-distanced and

non equal-distanced values of argument. The method allows transfer of interpolation task to

the solving of non linear equations system (consisted from one or two equations), to find the

initial approximations for roots of these equations for which is guaranteed the monotonous

convergence of the iteration consistencies to the unknown root of the system.

In the paper are considered one-parameter families of functions from polynomials and

a set of non-linear functions of real variables depended from many parameters. Is offered

a general method of definition of unknown parameters values both for equal-distanced and

non equal-distanced values of argument. The method allows transfer of interpolation task to

the solving of non linear equations system (consisted from one or two equations), to find the

initial approximations for roots of these equations for which is guaranteed the monotonous

convergence of the iteration consistencies to the unknown root of the system.

In the paper are considered one-parameter families of functions from polynomials and

a set of non-linear functions of real variables depended from many parameters. Is offered

a general method of definition of unknown parameters values both for equal-distanced and

non equal-distanced values of argument. The method allows transfer of interpolation task to

the solving of non linear equations system (consisted from one or two equations), to find the

initial approximations for roots of these equations for which is guaranteed the monotonous

convergence of the iteration consistencies to the unknown root of the system.

In the paper are considered one-parameter families of functions from polynomials and

a set of non-linear functions of real variables depended from many parameters. Is offered

a general method of definition of unknown parameters values both for equal-distanced and

non equal-distanced values of argument. The method allows transfer of interpolation task to

the solving of non linear equations system (consisted from one or two equations), to find the

initial approximations for roots of these equations for which is guaranteed the monotonous

convergence of the iteration consistencies to the unknown root of the system.

There are different methods of statistical hypotheses testing.1 – 4 Among them, a special place

has Bayesian approach. A generalization of Bayesian rule of many hypotheses testing is given below.

It consists in increasing of decision rule dimensionality with respect to the number of tested

hypoteses, which allows to make decisions more differentiated than in the classical case and to

state, instead of unconstrained optimozation problem, constrained one that enables to make

guaranteed decisions concerning errors of true decisions rejection, which is the key point when

solving a number of practical problems. These generalizations are given both for a set of simple

hypotheses, each containing one space point, and hypotheses containing a finite set of separated

space points.

At the solution of many theoretical and applied problems the broad application are found

the confidence intervals for parameters of probabilities distribution laws of studied

phenomena. The quality of a confidence interval is determined by its width for a given

confidence coefficient. There are three basic methods of finding of confidence intervals

[3.44], which one are based: 1) on frequency probability theory; 2) on fiducial distributions;

3) on the theorem of Bayesian. The first method use an asymptotic normality of the first

derivative of a logarithm of likelyhood function. According to the theorem Wilks, for large

sampling this method gives shortest on the average intervals for the definite class of

distributions [3.44] (hereinafter we shall call this method classic). The second method use

fiducial distributions conforming to considered distribution. In the third method of

confidence limits are established on the basis of an a posteriori probability distribution of

considered parameter.

Among the most urgent problems of environmental monitoring, note should be

taken of simulation of pollutants transfer in water objects. This problem is especially

urgent for urban conditions as there exist lost of pollution sources there. It solution is

of great ecological and economical significance allowing to investigate the influence

of different pollution sources on an ecological object, both separately from each

other and jointly, to predict the results of such influence and the consequences of

nature protective decision made against the pollution sources. By using mathematical

models, minimization of technical facilities, in particular, measurement equipment

indispensable for control and management of each pollution source is achieved. They

also are needed for large plants and factories with biochemical purification of waste

water, for their designing and ecologically secure exploitation, and also for of those

sites, shops that are guilty of waste water pollution over the norm.

For effective study and analysis of a condition of quality of a water environment, acceptance

of the conforming solutions on its improvement the adequate information is indispensable, that

is connected to huge number of measurement of different parameters which are carried out

with the help of automatic, permanent systems.

The environment is fast varying dynamic object, the control behind a condition which one by

non computerized methods is hindered and is economically unprofitable. At realization of the

analyses more than 3-4 times per day economically are expedient to use the automated systems

for status monitoring of environment. In these systems the cost of the information in 2-6 times

is less, than at usage of laboratory methods.

General technique of identification of non-linear regression relations below is offered,

which one is designed with the purpose of overcoming two basic difficulties not only

regression analysis, but also all modern mathematics: non-linearity and multidimensionality

of a problem [3.36]. The universal algorithm of optimum definition of areas of finding of

unknown values of parameters of regression models is designed where are contained these

unknown parameters with probability close to unit. On successful finding of these areas

depend quality of activity and outcomes of iteration search algorithms of extremum of

criterion of identification. The given methodology is suitable for the rather broad class of

non-linear regression models at classic regression and passive experiment and at its qualified

application, as against a customary non-linear parameter estimation, considerably reduces

time indispensable for the solution of a problem for identification and provides given

veracity. At some hardening of imposed limitations on nature of noise the obtained

outcomes are just also at active experiment.

The problems related to practical realization (as a computer program) of difference schemes of the solution of diffusion equations that describe pollutants transport in the river water are considered in this paper. In particular, the problems of optimum choosing of the algorithm parameters on which depend the accuracy, the time and the possibility of practical realization of the equation solution are considered. The demand to reduce as much as possible the time and the errors of calculations, and also the simplicity of the functions of certain classes used in mathematical models and their maximum accordance to real physical conditions are considered as criteria of optimality.

The problems related to practical realization (as a computer program) of difference schemes of the solution of diffusion equations that describe pollutants transport in the river water are considered in this paper. In particular, the problems of optimum choosing of the algorithm parameters on which depend the accuracy, the time and the possibility of practical realization of the equation solution are considered. The demand to reduce as much as possible the time and the errors of calculations, and also the simplicity of the functions of certain classes used in mathematical models and their maximum accordance to real physical conditions are considered as criteria of optimality.

The problems related to practical realization (as a computer program) of difference schemes of the solution of diffusion equations that describe pollutants transport in the river water are considered in this paper. In particular, the problems of optimum choosing of the algorithm parameters on which depend the accuracy, the time and the possibility of practical realization of the equation solution are considered. The demand to reduce as much as possible the time and the errors of calculations, and also the simplicity of the functions of certain classes used in mathematical models and their maximum accordance to real physical conditions are considered as criteria of optimality.

By support of World Bank the Georgian government has prepared "The Project of agricultural research, introduction - consultation and training", which is financed by the help both of the international development association (IDA) and global environment fund (GEF). Within the framework of the given project in the basin of the river Khobistskali, which runs into the black sea within the limits of Georgia, on farms introduce the biological methods of struggle against the agricultural pests, carry out the measures of reduction of erosion fields and increase of agricultural cultures productivity, build bio-installations for processing of stock-raising wastes and manufacture of high-quality organic fertilizers. The purpose of realization of these measures is the increase of socially-economic condition of the population and improvement of ecological condition as the environment, as growing agricultural products by the farmers. For estimation of the achievement of socially-economic and ecological effects by introduced the new technologies here realize monitoring of pollution: river waters, waters which are washed off from farmer's fields and stock-raising farms, soils of agricultural fields on the different depths and underground waters. By the help of the computer processing of monitoring results by the modern methods of the data analysis investigated the pollution process of the rivers from farms. Are restored dependences between of soil pollution levels, superficial and underground waters, contents in ground of nitrates and phosphates and volumes of received harvests.

At the solution of many theoretical and applied problems the broad application are found

the confidence intervals for parameters of probabilities distribution laws of studied

phenomena. The quality of a confidence interval is determined by its width for a given

confidence coefficient. There are three basic methods of finding of confidence intervals

[3.44], which one are based: 1) on frequency probability theory; 2) on fiducial distributions;

3) on the theorem of Bayesian. The first method use an asymptotic normality of the first

derivative of a logarithm of likelyhood function. According to the theorem Wilks, for large

sampling this method gives shortest on the average intervals for the definite class of

distributions [3.44] (hereinafter we shall call this method classic). The second method use

fiducial distributions conforming to considered distribution. In the third method of

confidence limits are established on the basis of an a posteriori probability distribution of

considered parameter.

For effective study and analysis of a condition of quality of a water environment, acceptance

of the conforming solutions on its improvement the adequate information is indispensable, that

is connected to huge number of measurement of different parameters which are carried out

with the help of automatic, permanent systems.

transfer in water objects. This problem is especially urgent for urban conditions as there exist lost of pollution sources there. It solution is of great ecological and economical significance allowing to investigate the influence of different pollution sources on an ecological object, both separately from each other and jointly, to predict the results of such influence and the consequences of nature protective decision made against the pollution sources. By using mathematical models, minimization of technical facilities, in particular, measurement equipment indispensable for control and management of each pollution source is achieved. They

also are needed for large plants and factories with biochemical purification of waste water, for their designing and ecologically secure exploitation, and also for of those sites, shops that are guilty of waste water pollution over the norm.

In the paper are considered one-parameter families of functions from polynomials and a set of non-linear functions of real variables depended from many parameters. Is offered a general method of definition of unknown parameters values both for equal-distanced and non equal-distanced values of argument. The method allows transfer of interpolation task to the solving of non linear equations system (consisted from one or two equations), to find the initial approximations for roots of these equations for which is guaranteed the monotonous convergence of the iteration consistencies to the unknown root of the system.

General procedures of identification of non-linear regression relations is offered below, which one is developed with the purpose of overcoming two basic difficulties not only regression analysis but also all modern mathematics: non-linearity and multidimensionality of a problem [73-75]. The universal algorithm of optimum definition of regions of finding of unknown values of parameters of regression models, in which these unknown parameters with probability close to unit are contained, is developed. The quality of working and

obtained results of iteration algorithms of search of extremum of criterion of identification depend on successful finding of these regions. Given methods is suitable for the rather wide class of non-linear regression models at classical regression and passive experiment and at its qualified application, in despite of usual non-linear estimation of parameters, considerably reduces the time necessary for solving identification problem and provides the given reliability. At some hardening of imposed restrictions on the nature of noises, the obtained results are also correct at active experiment.