МЕТОДИЧНИЙ ПІДХІД ДО ВИЗНАЧЕННЯ СТАТИСТИЧНИХ ХАРАКТЕРИСТИК КОДОФАЗОМАНІПУЛЬОВАНОГО СИГНАЛУ В ІНФОРМАЦІЙНИХ СИСТЕМАХ

This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them. Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance. Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10. Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions. Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesaro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality. Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Levy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions. Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields. Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

In the statistical analysis of experimental results it is extremely important to know the distribution laws of the general population. ‎Because of all assumptions about the distribution laws are statistical hypotheses, they should be tested. ‎Testing hypotheses are carried out by using the statistical criteria that divided the multitude in two subsets: null and alternative. The ‎null hypothesis is accepted in subset null and is rejected in alternative subset. Knowledge of the distribution law is a prerequisite for the use of numerical mathematical methods. The hypothesis is accepted if the divergence between empirical and theoretical distributions will be random. The hypothesis is rejected if the divergence between empirical and theoretical distributions will be essential.
 There is a number of different agreement criteria for the statistical hypotheses testing. The paper continues ideas of the author’s works, devoted to advanced based tools of the mathematical statistics. This part of the paper is devoted to nonparametric agreement criteria.
 Nonparametric tests don’t allow us to include in calculations the parameters of the probability distribution and to operate with frequency only, as well as to assume directly that the experimental data have a specific distribution. Nonparametric criteria are widely used in analysis of the empirical data, in the testing of the simple and complex statistical hypotheses etc. They include the well known criteria of K. Pearson, A. Kolmogorov, N. H. Kuiper, G. S. Watson, T. W. Anderson, D. A. Darling, J. Zhang, Mann – Whitney U-test, Wilcoxon signed-rank test and so on. Pearson and Kolmogorov criteria are most frequently used in mathematical statistics.
 Pearson criterion (-criterion) is the universal statistical nonparametric criterion which has -distribution. It is used for the testing of the null hypothesis about subordination of the distribution of sample empirical to theory of general population at large amounts of sample (n>50). Pearson criterion is connected with calculation of theoretical frequency. Kolmogorov criterion is used for comparing empirical and theoretical distributions and permits to find the point in which the difference between these distributions is maximum and statistically reliable. Kolmogorov criterion is used at large amounts of sample too. It should be noted, that the results obtained by using Pearson criterion are more precise because practically all experimental data are used.
 The peculiarities of Pearson and Kolmogorov criteria are found out. The formulas for calculations are given and the typical tasks are suggested and solved. The typical tasks are suggested and solved that help us to understand more deeply the essence of Pearson and Kolmogorov criteria.

The subject matter of the article is the process of assessing the sustainable development indices of the European Union countries. The goal of the article is to develop a methodology for determining the number of periods for which it is necessary and sufficient to assess the sustainable development indices of states. The article results the following task: to develop a methodology for determining the law of distribution of random variables of sustainable development indices. Determination of the minimum number of periods for assessing the sustainable development indices of the European Union countries. Methods used: parametric and ordinal statistics. The following results are obtained: parametric and non-parametric methods of statistics, their advantages and disadvantages are considered. Various methods of estimating the distribution function for small samples, in particular the method of rectangular contributions and the method of uncertainty reduction are analysed. Particular attention is paid to the problem of changing the law of scattering of quality indicators when changing the conditions of technology. A graph-analytical method for identifying the law of distribution of random variables based on a small amount of statistical information is proposed. For this purpose, the theory of ordinal statistics was used. A step-by-step methodology for identifying the law of distribution of random variables using 10 ordered values has been developed. The mathematical expectations of ordinal statistics for three distribution laws are proposed. A methodology for determining the number of periods for assessing the indices of sustainable development of countries using ordinal statistics is developed. The study is based on the analysis of statistical data for the last ten years and their ordering in ascending order. The mathematical expectations of ordinal statistics are used to select appropriate distribution laws. Given the limited information available when working with small samples, the article proposes a methodology that allows obtaining the maximum amount of information from the available data. The developed approach makes it possible to take into account the uncertainty of the phenomenon under study and make informed decisions based on statistical analysis. Conclusions: based on the knowledge of the law of distribution, a methodology for determining the minimum number of periods for assessing the sustainable development indices of the European Union countries is proposed. Testing of the methodology on real numerical data has confirmed that the minimum number of periods is seven, provided that the distribution law follows the normal law.

A review of the tomographic-probability representation of classical and quantum states is presented. The tomographic entropies and entropic uncertainty relations are discussed in connection with ambiguities in the interpretation of the state tomograms which are considered either as a set of the probability distributions of random variables depending on extra parameters or as a single joint probability distribution of these random variables and random parameters with specific properties of the marginals. Examples of optical tomograms of photon states, symplectic tomograms, and unitary spin tomograms of qudits are given. A new universal integral inequality for generic wave function is obtained on the base of tomographic entropic uncertainty relations.

The five basic axioms of Kolmogorov define the probability in the real set R and do not take into cons ideration the imaginary part which takes place in the complex set C, a problem that we are facing in applied mathematics. Whatever the probability distribution of the random variable in R is, the corresponding probability in the whole set C equals always to one , so the outcome of the random experiment in C can be predicted totally. This is the consequence of the f act that the probability in C is got by subtracting the chaotic factor from the degree of our knowledge of the syst em. In this study, I will evaluate the complex rand om vectors and their resultant that represents the who le distribution and system in the complex space C. I will also define imaginary and complex expectations and variances and I will prove the law of large numbers usin g the concept of the resultant complex vector. In fact, a fter extending Kolmogorov’s system of axioms, the new axioms encompass the imaginary set of numbers and this by adding to the original five axioms of Kolmog orov an additional three axioms. Hence, the concept of c omplex random vector becomes clear, evident and it follows directly from the new axioms added. This re sult will be elaborated throughout this study using discrete probability distributions. Moreover, any experiment executed in the complex set C is the sum of the re al set R and the imaginary set M. Therefore, the whole probability distribution of random variables can be repr esented totally by the resultant complex random vector Z th at is used subsequently to prove the very well know n law of large numbers. In addition to my previous first paper, this second one elaborates the new field of “Complex Statistics” that considers random variables in the complex set C. Thus, the law of large numbers proves that this complex extension is successful and fruitful.

The limit measurement errors in the geodesy are determining for the normal distribution law, But the errors, exceeding the limits are throw aside and accidental errors are obey the cut law. Cut laws are not study enough. The article is leading the results of calculate the probabilities hit accidental measurements errors for intervals σ, 2σ, 2,5σ, 3σ for cut normal and cut logical distribution laws. The analysis was accomplished.The probabilities of the cut normal and normal laws is different immaterial. 0,043 - maximum of the different. The normal law replace to the cut normal law is not recommending. Probabilities of the cut law may to use, when it is allowing to avoid the repetition of the field measurements.

The paper analyzes the homogeneity of the stochastic structure of the samples of oil flax fibres of the Ukrainian and Polish selections. The distribution laws of fibres as mixtures of the normal and logarithmically normal distribution laws are identified. The general probability distribution law of fibre lengths of oil flax of the Ukrainian selection is a mixture of four logarithmically normal laws of distribution, and the general probability distribution law of fibre lengths of oil flax of the Polish selection is a mixture of a normal distribution law and three logarithmically normal laws of distribution. The parameters of the relevant components of the common distribution laws are found as a result of solving the problem of nonlinear optimization using the method of generalized lowering gradient that is implemented in Excel tabular processor. The adequacy of the built nonlinear models of the distribution laws is proved by estimating the value of the determination index. The comparative analysis of distribution laws of general totalities, from which the oil flax fibre samples of the Ukrainian and Polish selections are obtained, is conducted by the Kolmogorov-Smirnov statistical criterion. This criterion allows us to test and reject the hypothesis about belonging of two independent samples to the same general totality. The analysis of the general distribution laws shows that the oil flax fibres of the Ukrainian selection have a more pronounced tendency to stratification by a fibre length. Such a fibre is easily divided into factions. The oil flax fibres of the Polish selection have a larger part of short and extra-long fibres. Thus, it is clear that raw materials of the Ukrainian selection are more suitable for the production of non-woven materials of different target purposes, and the Polish selection - for the production of ropes, cords, etc. The obtained results can be used by specialists at processing enterprises for a reasonable choice of raw materials, optimization of processing modes and improving ready-made product quality.

ABSTRACTThis study proposes a copula-based chance-constrained waste management planning (CCWMP) method. The method can effectively reflect the interactions between random parameters of the waste management planning systems, and thus can help analyze the influences of their interactions on the entire systems. In particular, a joint distribution function is established using preestimated marginal distributions of random variables and an optimal copula selected from widely used Gaussian, Student’s t, Clayton, Frank, Gumbel, and Ali-Mikhail-Haq copulas. Then a set of joint probabilistic constraints in the chance-constrained programming problems is converted into individual probabilistic constraints using the joint distribution function. Further, this method is applied to residential solid waste management in the city of Regina in Canada for demonstrating its applicability. Nine scenarios based on different joint and marginal probability levels are considered within a multiperiod and multizone context to effectively reflect dynamic, uncertain, and interactive characteristics of the solid waste management systems in the city. The results provide many decision alternatives under these scenarios, including cost-effective and environmentally friendly decision schemes. Moreover, the results indicate that even though the effect of the joint probability levels on the system costs is more significant than that of the marginal probability levels, the effect of marginal probability levels is notable, and there exists a trade-off between the total system cost and the constraint-violation risk. Therefore, the results obtained from the present study would be useful to support the city’s long-term solid waste management planning and formulate local policies and regulation concerning the city’s waste generation and management.Implications: The CCWMP method not only can solve chance-constrained problems with unknown probability distributions of random variables in the right-hand sides of constraints, but also can effectively reflect the interactions between the random parameters and thus help analyze the influences of their interactions on the entire systems. The results obtained through applying this method to the city of Regina in Canada can provide many decision alternatives under different joint probability levels and marginal probability levels, and would be useful to support the city’s long-term solid waste management planning.

The aim of this study was to determine the best probability distributions for calculating the maximum annual daily precipitation with the specific probability of exceedance (Pmaxp%). The novelty of this study lies in using the peak-weighted root mean square error (PWRMSE), the root mean square error (RMSE), and the coefficient of determination (R2) for assessing the fit of empirical and theoretical distributions. The input data included maximum daily precipitation records collected in the years 1971–2014 at 51 rainfall stations from the Upper Vistula Basin, Southern Poland. The value of Pmaxp% was determined based on the following probability distributions of random variables: Pearson’s type III (PIII), Weibull’s (W), log-normal, generalized extreme value (GEV), and Gumbel’s (G). Our outcomes showed a lack of significant trends in the observation series of the investigated random variables for a majority of the rainfall stations in the Upper Vistula Basin. We found that the peak-weighted root mean square error (PWRMSE) method, a commonly used metric for quality assessment of rainfall-runoff models, is useful for identifying the statistical distributions of the best fit. In fact, our findings demonstrated the consistency of this approach with the RMSE goodness-of-fit metrics. We also identified the GEV distribution as recommended for calculating the maximum daily precipitation with the specific probability of exceedance in the catchments of the Upper Vistula Basin.

Notions of separation in graphs of dynamical systems

Another title could have been "A probabilistic factorization algorithm in GL(2,p)". However, the problem is to calculate a fast and short signature associated with a plaintext inscribed on an erasable support. The signature should be written down in a book accompanying the record in order that it could be check ed anytime that the latter has not been changed. J. BOSSET [1] suggest such a scheme together with an algorithm for computing a signature. The 64 characters needed for the plaintext are identified with a subset of GL(2,p), p=997. The signature is the product of the matrices corresponding to the plaintext characters taken in the order where they appear. Such a scheme could be broken if it is possible to factorize an element of GL(2,p) into t=16 r factors, each one in a subset Ui of GL(2,p) of size 64 , i=1,...,t. We here assume one hypothesis only on uniform probability distributions of random variables defined on product sets Vj=Ujr+1×...×U(j+1)r, j=0,...,15. In consideration on which, a probabilistic factorization algorithm in GL(2,p) is introduced.It is shown that for p=10,007, drawing according to a uniform probability distribution a sequence of 11,952 elements in each Vj provides the whole needed material to factorizing with a probability of success of at least 97%. The most expensive operation in the algorithm is sorting each of the sequences.

In this work, we focus on nonparametric kernel methods for estimating the probability density function (pdf). The convergence of a kernel estimator depends crucially on the choice of the smoothing parameter. We present in this paper, a new method for optimizing the bandwidth of an estimator of the probability density function: the adaptive kernel estimator. This optimized estimator is used to construct the Bayes classifier. In this sense, we have proposed a new approach to optimize the pdf based on the statistical properties of the probability distributions of random variables. We adopt the maximum entropy principle (MEP) in order to determine the optimal value of the smoothing parameter used in the estimator. In the proposed criterion, the estimated probability density function is called optimal in the sense of having a minimum error rate of classifying data. Finally, we illustrate the robustness of our optimization process of the kernel estimation methods by using a set of DNA microarray data showing that our approach effectively improves the performance of the classification process.

We propose a minirnax model with a quadratic recourse. In stochastic linear programming models, a decision maker has been assumed to know the probability distribution of random variables. Here we consider the case that the parameters of distribution are unknown. We impose the restrictions on the unknown parameters from the view point of a confidence region, and then seek a minimax solution that minimizes the worst case of the parameters. This model reflects the situation minimizing the maximal possible damage. Especially, the independent normal distribution model is discussed in detail. The analysis for a sufficiently large sample size and a numerical result are given.

Compared to traditional resources, user-side resources are of various types and have more significant uncertainty about their regulatory capacity, leading to difficulties in coordinating decisions about their simultaneous participation in the electric energy and peaking ancillary services markets. This paper proposes a joint bidding decision-making method for the day-ahead electricity energy and peak shaving auxiliary service market based on distributed robust opportunity constraints, which addresses the problem of difficulty in using an accurate probability density distribution to represent the uncertainty process of user-side resources. Firstly, a data-driven method for characterizing the uncertainty of load regulation capacity is investigated, and fuzzy sets are constructed without assuming specific probability distributions of random variables. Then, to minimize the risk expectation of the joint bidding cost on the customer side, a bidding strategy that considers the uncertainty is proposed. Finally, an example simulation verifies the reasonableness and effectiveness of the proposed joint bidding method, and the results show that the constructed model overcomes the problem of over-conservatism of the robust model, and the computational adaptability is better than that of the stochastic model, which achieves a better balance between robustness and economy.

This paper proposes a novel approach to multiple input and multiple output (MIMO) simulation metamodeling using Bayesian networks (BNs). A BN is a probabilistic model that represents the joint probability distribution of a set of random variables and enables the efficient calculation of their marginal and conditional distributions. A BN metamodel gives a non-parametric description for the joint probability distribution of random variables representing simulation inputs and outputs by combining MIMO data provided by stochastic simulation with available background knowledge about the system under consideration. The BN metamodel allows various what-if analyses that are used for studying the marginal probability distributions of the outputs, the input uncertainty, the dependence between the inputs and the outputs, and the dependence between the outputs as well as for inverse reasoning. The construction and utilization of BN metamodels in simulation studies are illustrated with an example involving a queueing model.