Design of single and double acceptance sampling plans based on interval type-2 fuzzy sets

Defectiveness of items is generally considered as a certain value in acceptance sampling plans (ASPs). It is clear that, it may not be certainly known in some real-case problems. Uncertainties of the inspection process such as measurement errors, inspectors’ hesitancies or vagueness of the process etc. should be taken into account to obtain more reliable results. The fuzzy set theory (FST) is one of the best methods to overcome these problems. There are some studies in the literature formulating the ASPs with the help of FST. Deciding the right membership functions of the fuzzy sets (FSs) has a vital importance on the quality of the uncertainty modeling. Additionally, the fuzzy set extensions have been offered to model more complicated uncertainties to achieve better modeling. As one of these extensions, type-2 fuzzy sets (T2FSs) gives an ability to model uncertainty in situations where it is not possible to determine exact membership function parameters. In this study, single and double ASPs based on interval T2FSs (IT2FSs) have been designed for binomial and Poisson distributions. Thus, it becomes possible to make more flexible, sensitive and descriptive sensitivity analyzes. The main characteristic functions of ASPs have been derived and the suggested formulations have been illustrated on a comparative application from manufacturing process. Results allowing for more comprehensive analysis as against to the traditional and T1FSs based plans have been obtained.

Non-Debye Relaxations: The Ups and Downs of the Stretched Exponential vs. Mittag–Leffler’s Matchings

Experimental data collected to provide us with information on the course of dielectric relaxation phenomena are obtained according to two distinct schemes: one can measure either the time decay of depolarization current or use methods of the broadband dielectric spectroscopy. Both sets of data are usually fitted by time or frequency dependent functions which, in turn, may be analytically transformed among themselves using the Laplace transform. This leads to the question on comparability of results obtained using just mentioned experimental procedures. If we would like to do that in the time domain we have to go beyond widely accepted Kohlrausch–Williams–Watts approximation and become acquainted with description using the Mittag–Leffler functions. To convince the reader that the latter is not difficult to understand we propose to look at the problem from the point of view of objects which appear in the stochastic processes approach to relaxation. These are the characteristic exponents which are read out from the standard non-Debye frequency dependent patterns. Characteristic functions appear to be expressed in terms of elementary functions whose asymptotics is simple. This opens new possibility to compare behavior of functions used to describe non-Debye relaxations. It turnes out that the use of Mittag-Leffler function proves very convenient for such a comparison.

A new weighting scheme for arc based circle cone-beam CT reconstruction

In this paper, we present an arc based fan-beam computed tomography (CT) reconstruction algorithm by applying Katsevich’s helical CT image reconstruction formula to 2D fan-beam CT scanning data. Specifically, we propose a new weighting function to deal with the redundant data. Our weighting function ϖ ( x _ , λ ) is an average of two characteristic functions, where each characteristic function indicates whether the projection data of the scanning angle contributes to the intensity of the pixel x _ . In fact, for every pixel x _ , our method uses the projection data of two scanning angle intervals to reconstruct its intensity, where one interval contains the starting angle and another contains the end angle. Each interval corresponds to a characteristic function. By extending the fan-beam algorithm to the circle cone-beam geometry, we also obtain a new circle cone-beam CT reconstruction algorithm. To verify the effectiveness of our method, the simulated experiments are performed for 2D fan-beam geometry with straight line detectors and 3D circle cone-beam geometry with flat-plan detectors, where the simulated sinograms are generated by the open-source software “ASTRA toolbox.” We compare our method with the other existing algorithms. Our experimental results show that our new method yields the lowest root-mean-square-error (RMSE) and the highest structural-similarity (SSIM) for both reconstructed 2D and 3D fan-beam CT images.

Fuzzy logical conclusions and conclusions in expert systems of medical diagnostics

The main problems in making a correct diagnosis are: subjectivity and insufficient qualifications of the doctor, difficulties in correctly assessing the patient’s complaints, signs and symptoms of the disease observed in the patient, as well as individual manifestations of the symptoms of the disease. In publications on the use of expert systems for medical diagnostics using fuzzy logic, the main attention was paid to the medical features of the problem. In this work, for the first time, general methodological aspects of building such systems, creating databases, representing by fuzzy sets of real numbers, digital scales, linguistic and Boolean data of symptom values are formulated. The types of membership functions that are advisable to use to represent the symptoms of diseases are proposed. In fuzzy-logical conclusions, not only the values of the characteristic functions of the logical terms of individual symptoms, but also complex arithmetic functions of their values are used.

A Practical Method for the Automatic Recognition of Rock Structures in Panoramic Borehole Image during Deep-Hole Drilling Engineering

Digital panoramic borehole imaging technology has been widely used in the practice of drilling engineering. Based on many high-definition panoramic borehole images obtained by the borehole imaging system, this paper puts forward an automatic recognition method based on clustering and characteristic functions to perform intelligent analysis and automatic interpretation researches, and successfully applied to the analysis of the borehole images obtained at the Wudongde Hydropower Station in the south-west of China. The results show that the automatic recognition method can fully and quickly automatically identify most of the important structural planes and their position, dip, dip angle and gap width and other characteristic parameter information in the entire borehole image. The recognition rate of the main structural plane is about 90%. The accuracy rate is about 85%, the total time cost is about 3 h, and the accuracy deviation is less than 4% among the 12 boreholes with a depth of about 50 m. The application of automatic recognition technology to the panoramic borehole image can greatly improve work efficiency, reduce the time cost, and avoid the interference caused by humans, making it possible to automatically recognize the structural plane parameters of the full-hole image.

Weak Convergence of Distributions

This chapter introduces the fundamentals of weak convergence for real sequences. Definitions and examples are given. The Skorokhod representation theorem is proved and the chapter then considers the preservation of weak convergence under transformations. Next, the role of moments and characteristic functions is considered. In the leading case of random sums, the criteria for weak convergence and the concept of a stable distribution are studied.

Characteristic Functions

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.

Measures on Metric Spaces

This chapter lays the foundations for functional limit theory, considering the case of general metric spaces from a topological standpoint. The issues of separability and measurability and techniques for assigning measures in metric spaces are then discussed, developing tools to replace the methods of characteristic functions and the inversion theorem used for real sequences. The key cases of function spaces are studied and in particular the case C of continuous functions on the unit interval. Weiner measure (Brownian motion) is defined as the leading case of a measure on C.

Dimensionality Reduction of High Dimensional Time Series based on Artificial Neural Network

Molecular dynamics is a molecular simulation method which relies on Newtonian mechanics to simulate the motion of molecular system. In this method, some differential equations are integrated, and the results of integration are further processed to obtain the trajectory or momentum evolution process of some particles controlled by dynamic equations, and the technology of extracting the equilibrium state, motion process or related properties of classical particle system can be used. Through molecular dynamics simulation, we can obtain a series of properties of the system, which are widely used in experimental verification, theoretical derivation and other scenarios. Because it can obtain the dynamic state of macromolecules to make up for the limitations of these properties, it is widely used in the study of transmembrane proteins, polypeptide chains and other systems in life sciences. Through the kinetic path reduction of these systems, we can intuitively understand the characteristics of molecular folding, molecular motion and specific binding, which can play a very important role in the study of proteins and peptides. However, due to the characteristics of high-dimensional time series obtained by molecular dynamics simulation, it is difficult for us to pay attention to the collective state or characteristic process of the whole system in a non-equilibrium state or slow process. This is due to the difficulty in data processing and the difficulty in obtaining its characteristic function. This makes it very difficult to study the dynamic process of the whole system, especially the dynamic process at the intermediate non-equilibrium moment. It is difficult to solve this kind of problem by conventional methods, and only a few special simple systems can be solved by experience. Therefore, it is of great significance to find a method to obtain the characteristic function of the system through the trajectory obtained by molecular dynamics, and then reduce the molecular dynamics path. In order to solve this scientific problem, researchers focus on machine learning. In this study, machine learning method will be used to solve the overall non-equilibrium state of the system or the collective state of the slow process in molecular dynamics simulation. Firstly, we use this method to solve a simple one-dimensional four well model. By this method, we obtain a series of characteristic functions describing the motion process of the model. By sorting the eigenvalue contributions, we obtain some main characteristic functions describing the system. It includes the motion description of Markov smooth transition state and the motion description of four potential wells. At the same time, we use the traditional transition probability matrix to calculate. The difference between the characteristic function obtained by machine learning and the traditional method is very small, but the calculation method is simpler and more universal. After that, we apply the method to the actual scene. By solving the molecular dynamics simulation of alanine dipeptide structure in polymer protein molecule, the characteristic function of dihedral angle folding of alanine dipeptide structure was preliminarily calculated. The results were consistent with the traditional method.

Correlating Lévy processes with self-decomposability: applications to energy markets

Based on the concept of self-decomposability, we extend some recent multidimensional Lévy models built using multivariate subordination. Our aim is to construct multivariate Lévy processes that can model the propagation of the systematic risk in dependent markets with some stochastic delay instead of affecting all the markets at the same time. To this end, we extend some known approaches keeping their mathematical tractability, study the properties of the new processes, derive closed-form expressions for their characteristic functions and detail how Monte Carlo schemes can be implemented. We illustrate the applicability of our approach in the context of gas, power and emission markets focusing on the calibration and on the pricing of spread options written on different underlying commodities.