Chance Theory Research Articles

Reliability analysis for the fractional-order circuit system subject to the uncertain random fractional-order model with Caputo type

IntroductionAccording to the competing failure theorem, the fractional-order Resistor Capacitance (RC) circuit system suffers not only from internal degradation but also from external shocks. However, due to the general differences of each failure type in the data availability and cognitive uncertainty, a better model is needed to describe the degradation process within the system. Also, a new reliability analysis method is needed for the circuit system under internal degradation and external shocks. ObjectivesTo demonstrate this problem, this paper proposes a novel class of Caputo-type uncertain random fractional-order model that focuses on the reliability analysis of a fractional-order RC circuit system. MethodsFirst, an uncertain Liu process is used to describe the internal degradation of soft faults and a stochastic process is used to describe the external random shocks of hard faults. Secondly, taking into account the correlation and competition among the fault types, an extreme shock model and a cumulative shock model are constructed, and chance theory is introduced to further explore the fault mechanisms, from which the corresponding reliability indices are derived. Finally, the predictor–corrector method is applied and numerical examples are given. ResultsThis paper presents a reliability analysis of a fractional-order RC circuit system with internal failure obeying an uncertain process and external failure obeying a stochastic process, and gives the calculation of the reliability indexes for different cases and the corresponding numerical simulations. ConclusionA new competing failure model for a fractional-order RC circuit system is presented and analyzed for reliability, which is proved to be of practical importance by numerical simulations.

John Haigh 1941–2021

John Haigh 1941–2021

Papillary and Medullary Thyroid Carcinomas Presenting as Collision Tumors: A Case Series of 21 Cases at a Tertiary Care Cancer Center.

Collision tumor is the occurrence of two histologically and morphologically distinct tumors within the same organ with no histological admixture. Collision tumors of the thyroid are extremely rare constituting < 1% of all thyroid tumors. Clinical profiles and pathological features of Medullary thyroid carcinoma (MTC) and Papillary thyroid carcinoma (PTC) presenting as Collision tumors of thyroid, diagnosed between 2009 and 2019, at a tertiary care cancer center were retrospectively analyzed. Collision tumors comprised 4.7% of all MTC cases diagnosed over 10years. A total of 21 cases (11males, 11 females, M:F = 1) were retrieved with the mean age of patients being 45.33years (range 26-77years). More than half of PTCs involved the right lobe of the thyroid (66.6%). About half (53.4%) of MTCs affected the left lobe. Imaging done pre-operatively failed to identify the smaller second tumor in 60% of the cases with both tumours in separate lobes. Pre-operative FNAC showed only MTC in all 8 cases in which it was done. Papillary microcarcinoma (m-PTC) was seen in 85.7% cases, with one case of multifocal m-PTC. MTC (mean size 3.12cm), on an average, was 3 times larger than the PTC (mean size 0.91cm). The histological variants of MTC included-oncocytic (1/21, 4.7%), spindle cell (1/21, 4.7%), epithelial (3/21, 14.2%) and classical (16/21, 76.2%) and of PTC included classic PTC (12/21, 57.14%), Hurthle cell (2/21, 9.52%), tall cell (1/21, 4.76%) and follicular variant of PTC (6/21, 28.57%). The microscopic extrathyroidal extension (ETE) due to MTC and PTC component was 42.8% and 9.5% respectively. Lymph node metastasis was seen in 16 (76.2%) cases; 87.5% (14/16) of which were contributed by MTC, 12.5% (2/16) by PTC alone, and 12.5% (2/16) cases showed metastasis from both MTC and PTC. MTC had a higher stage than PTC in 85.5% of cases. Collision tumors of the thyroid are exceedingly rare, and possibly underdiagnosed due to variation in sampling techniques, especially of the grossly "normal lobe". The low incidence in our cohort is in favor of the "Chance theory" of co-occurrence. This diagnosis is important due to its therapeutic and prognostic implications.

Chance Debugged

A ‘Big Bad Bug’ threatens Lewis’s Humean metaphysics of chance (Lewis 1986a, p. XIV); his Principal Principle provides an intuitive link between chance and credence. Yet on the one hand, certain future developments are incompatible with the true theory of chance, but on the other hand, such future developments have a positive chance to occur. The combination of these two claims with the Principal Principle leads to inconsistent credences. I present a Humean solution to the Bug: chances are relative to a limited perspective. The perspective comprises facts available as evidence to an ideal cognizer at a point in space-time. As a consequence, the same future event can have different chances of occurring provided the perspective is different. I show how this dissolves the Bug.

Uncertain Random Data Envelopment Analysis: Efficiency Estimation of Returns to Scale

Evaluating efficiency according to the different states of returns to scale (RTS) is crucial to resource allocation and scientific decision for decision-making units (DMUs), but this kind of evaluation will become very difficult when the DMUs are in an uncertain random environment. In this paper, we attempt to explore the uncertain random data envelopment analysis approach so as to solve the problem that the inputs and outputs of DMUs are uncertain random variables. Chance theory is applied to handling the uncertain random variables, and hence, two evaluating models, one for increasing returns to scale (IRS) and the other for decreasing returns to scale (DRS), are proposed, respectively. Along with converting the two uncertain random models into corresponding equivalent forms, we also provide a numerical example to illustrate the evaluation results of these models.

Uncertainty propagation in structural reliability with implicit limit state functions under aleatory and epistemic uncertainties

Uncertainty propagation plays a pivotal role in structural reliability assessment. This paper introduces a novel uncertainty propagation method for structural reliability under different knowledge stages based on probability theory, uncertainty theory and chance theory. Firstly, a surrogate model combining the uniform design and least-squares method is presented to simulate the implicit limit state function with random and uncertain variables. Then, a novel quantification method based on chance theory is derived herein, to calculate the structural reliability under mixed aleatory and epistemic uncertainties. The concepts of chance reliability and chance reliability index (CRI) are defined to show the reliable degree of structure. Besides, the selection principles of uncertainty propagation types and the corresponding reliability estimation methods are given according to the different knowledge stages. The proposed methods are finally applied in a practical structural reliability problem, which illustrates the effectiveness and advantages of the techniques presented in this work.

Опыт проведения дистанционной олимпиады по теории вероятностей в техническом вузе

The article analyzed the experience of distance Olympiad about theory of chances. It was in technical university. In this article demonstrates the tasks that were proposed for solving. This was a qualitative assessment of the results was carried out. The solution of several tasks is analyzed.

Uncertain random linear quadratic control with multiplicative and additive noises

AbstractThis paper studies two types of linear quadratic (LQ) optimal control models for multistage uncertain random systems. The first model is an LQ model with additive noises, while the second model is an LQ model with both multiplicative noises and additive noises. Chance theory is a useful tool to deal with the analysis of indeterminacy including both uncertainty and randomness. Based on Bellman's principle and chance theory, the recurrence equations are presented for settling such optimal control problems. Then, by applying recurrence equations, the analytical expressions are derived for solving both the optimal objective function and optimal control strategies of the LQ models. Meanwhile, an application and a numerical example are provided to show the effectiveness of our results.

Multistage Uncertain Random Linear Quadratic Optimal Control

In this paper, linear quadratic (LQ) optimal control problems are investigated for two types of uncertain random systems which consider the coefficient of the perturbed term as a constant vector or a vector-valued function of state vector and control vector. First, the uncertain random optimal control model is established under expected value criterion. Second, based on Bellman’s principle, recurrence equations are presented for settling such problem. Then by applying the recurrence equations and chance theory, the analytical expressions of the optimal results for the LQ problems are derived. Furthermore, some examples and an application are given to show the effectiveness of our results.

Is Evolution a Chance Process?

It is commonly thought that evolution is a chance process, an idea found in popular writings on evolution, but also in academic writing in a broad range of scientific disciplines: scientific, philosophical and theological. One problem is that words such as ‘chance’ and ‘random’ are used with a range of different meanings according to context, and in evolutionary biology the word ‘chance’ is sometimes used in a way that is different from its use in mathematics and philosophy. The present article aims to clarify the range of meanings and to argue the case that the evolutionary process is far from being a ‘theory of chance’ from biological, mathematical, or indeed philosophical and theological perspectives.

Elliptic entropy of uncertain random variables with application to portfolio selection

This paper investigates an elliptic entropy of uncertain random variables and its application in the area of portfolio selection. We first define the elliptic entropy to characterize the uncertainty of uncertain random variables and give some mathematical properties of the elliptic entropy. Then we derive a computational formula to calculate the elliptic entropy of function of uncertain random variables. Furthermore, we use the elliptic entropy to characterize the risk of investment and establish a mean-entropy portfolio selection model, in which the future security returns are described by uncertain random variables. Based on the chance theory, the equivalent form of the mean–entropy model is derived. To show the performance of the mean–entropy portfolio selection model, several numerical experiments are presented. We also numerically compare the mean–entropy model with the mean–variance model, the equi-weighted portfolio model, and the most diversified portfolio model by using three kinds of diversification indices. Numerical results show that the mean-entropy model outperforms the mean–variance model in selecting diversified portfolios no matter of using which diversification index.

Color distribution of three drawn balls from Ellsberg urn

Ellsberg urn is a complicated system with uncertainty (the unknown numbers of the colored balls) and randomness (the randomly drawn balls). By supposing that two numbers of colored balls are unknown in an Ellsberg urn, this paper applies uncertainty theory, probability theory and chance theory as rigorous mathematical tools to formulating the color distribution of the drawn balls when three balls are randomly drawn from the urn. Furthermore, an intuitive example is given to illustrate the results obtained by the mathematical method.

Partial similarity measure of uncertain random variables and its application to portfolio selection

A similarity measure determines the similarity between two objects. As important roles of similarity measure in chance theory, this paper introduces the concept of partial similarity measure for two uncertain random variables. Based on maximum simila

A SOLUTION ALGORITHM FOR p-MEDIAN LOCATION PROBLEM ON UNCERTAIN RANDOM NETWORKS

This paper investigatesthe classical $p$-median location problem in a network in which some of the vertex weights and the distances between vertices are uncertain and while others are random. For solving the $p$-median problem in an uncertain random network, an optimization model based on the chance theory is proposed first and then an algorithm is presented to find the $p$-median. Finally, a numerical example is given to illustrate the efficiency of the proposed method

ASSESSMENT OF THE RISK OF SKIN LESIONS ASSOCIATED WITH DEMODEX MITES IN STUDENTS WITH ACNE VULGARIS

The study of comorbid pathologies that influence the severity of the disease and impair the effectiveness of treatment is carried out to optimize the treatment of the main disease. Materials and methods. The total of 132 medical students was divided into two groups: the 1st group: 56 students who didn`t have acne elements, the 2nd group: 76 students with mild acne vulgaris. At the baseline and after 12 months epithelial tape-test was performed to define Demodex mites presence. The theory of chances was used to assess the impact of acne on the occurrence of Demodex mites. Results. The presence of Demodex mites was observed in 3.57% of students in the 1st group and in 21.05% of students in the 2nd group on repeated tape-test. The probability value (P) among students of the 1st group was P1 = 0.037; among students of the 2nd group – P2 = 0.266. The odds (O) indicator in the 1st group of students without acne elements was O1 = 0.04, in the 2nd group diagnosed with acne – O2 = 0.21. The odds ratio (OR) value in the case/control groups was 7.20. Confidence intervals (95% CI) – (1.582, 32.765): the lower limit of the confidence interval (2.019, + ∞); the upper limit of the confidence interval (-∞, 25680). The significance test p-value was 0.00533. Conclusion. Acne vulgaris is one of the aggravating factor that increases the risk of Demodex mites.

Reliability Modeling for Repairable Systems With Stochastic Lifetimes and Uncertain Repair Times

In actual industrial systems, there are often situations where the fault data are sufficient, whereas the repair data are insufficient. It is not appropriate to treat the system as a stochastic system or an uncertain system in this case. In this paper, the lifetimes of components are assumed to be random variables and the repair times of components are assumed to be uncertain variables. The entire system can be regarded as an uncertain random repairable system accordingly. Based on chance theory, some common reliability indexes of repairable system, such as availability, steady-state failure frequency, mean up time, and mean down time, are redefined and some properties are also presented. Then, the reliability mathematical models of repairable series systems, repairable parallel systems, repairable series-parallel systems, and repairable parallel-series systems are established, respectively. Furthermore, the formulas of reliability indexes of each repairable system are obtained. Finally, some numerical examples are presented to illustrate the calculation of the reliability indexes.

Postulating the Theory of Experience and Chance

The aim of the paper is the axiomatic justification of the theory of experience and chance, one of the dual halves of which is the Kolmogorov probability theory. The author’s main idea was the natural inclusion of Kolmogorov’s axiomatics of probability theory in a number of general concepts of the theory of experience and chance. The main result of this work is the axiom of co~event, intended for the sake of constructing a theory formed by dual theories of believabilities and probabilities, each of which itself is postulated by its own Kolmogorov system of axioms. Of course, other systems of postulating the theory of experience and chance can be imagined, however, in this work a preference is given to a system of postulates that is able to describe in the most simple manner the results of what I call an experienced-random experiment

Two-degree-of-freedom Ellsberg urn problem

Traditional model assumes there is only randomness existing in the urn problem. However, if the numbers of the colored balls are unknown, then they should be regarded as uncertain variable. Since a ball is drawn randomly, Ellsberg urn problem is essentially a complicated system with randomness and uncertainty. Instead of psychological experiment, this paper applies uncertainty theory and chance theory to provide a rigorous mathematical method for formulating the general case of a one-degree-of-freedom Ellsberg urn problem. Furthermore, a two-degree-of-freedom Ellsberg urn problem is proposed, and the formulation for the problem is given to deal with three unknown numbers of colored balls.

Reliability analysis for devices subject to competing failure processes based on chance theory

This paper studies the reliability for devices subject to independent competing failure processes of degradation and shocks in an uncertain random environment. The continuous degradation is governed by an uncertain process, and external shocks arrive according to an uncertain random renewal reward process, in which the inter-arrival times of shocks and the shock sizes are assumed to be random variables and uncertain variables, respectively. The device reliability is defined as the chance measure that the uncertain degradation signals do not exceed a soft failure threshold L, and the uncertain random shocks do not cause the device failure. The device reliability is obtained by employing chance theory under four different shock patterns. Finally, a case study on a gas insulated transmission line is carried out to show the implementation of the proposed model.

Propensities and Possibilities

Abstract This paper is a reply to a recent Metaphysica paper advocating an ‘unrestricted actualism’ which lets the actual world include unrealised possible outcomes of propensities. I argue that the actual world can accommodate propensity theories of chance without including unrealised possibilities.