Probability Theory Research Articles

Not All Probability Density Functions Are Tomograms.

This paper delves into the significance of the tomographic probability density function (pdf) representation of quantum states, shedding light on the special classes of pdfs that can be tomograms. Instead of using wave functions or density operators on Hilbert spaces, tomograms, which are the true pdfs, are used to completely describe the states of quantum systems. Unlike quasi-pdfs, like the Wigner function, tomograms can be analysed using all the tools of classical probability theory for pdf estimation, which can allow a better quality of state reconstruction. This is particularly useful when dealing with non-Gaussian states where the pdfs are multi-mode. The knowledge of the family of distributions plays an important role in the application of both parametric and nonparametric density estimation methods. We show that not all pdfs can play the role of tomograms of quantum states and introduce the conditions that must be fulfilled by pdfs to be "quantum".

A semi-supervised cross-modal memory bank for cross-modal retrieval

The core of semi-supervised cross-modal retrieval tasks lies in leveraging limited supervised information to measure the similarity between cross-modal data. Current approaches assume an association between unlabelled data and pre-defined k-nearest neighbour data, relying on classifier performance for this selection. With diminishing labelled data, classifier performance weakens, resulting in erroneous associations among unlabelled instances. Moreover, the lack of interpretability in class probabilities of unlabelled data hinders classifier learning. Thus, this paper focuses on learning pseudo-labels for unlabelled data, providing pseudo-supervision to aid classifier learning. Specifically, a cross-modal memory bank is proposed, dynamically storing feature representations in a common space and class probability representations in a label space for each cross-modal data. Pseudo-labels are derived by computing feature representation similarity and adjusting class probabilities. During this process, imposing constraints on the classification loss between labelled data and contrastive losses between paired cross-modal data is a prerequisite for the successful learning of pseudo-labels. This procedure significantly contributes to enhancing the credibility of these pseudo-labels. Empirical findings demonstrate that using only 10% labelled data, compared to prevailing semi-supervised techniques, this method achieves improvements of 2.6%, 1.8%, and 4.9% in MAP@50 on the Wikipedia, NUS-WIDE, and MS-COCO datasets, respectively.

Research on the Setting and Control Method of Centrosymmetric Short Straight Lanes for Signalized Intersections

A new traffic organization method has been proposed to address the issue of traffic congestion caused by oversaturation through traffic during peak hours at signalized intersections. This method involves the installation of approximately centrosymmetric short straight lanes (CSSLs) to facilitate the passage of through vehicles, aiming to improve the traffic efficiency at signalized intersections. Firstly, two types of intersection designs with CSSLs were investigated, along with their operational principles. Secondly, a capacity model for the CSSL was established, taking into account the probability theory and mutual blockage considerations, as well as a delay model based on vehicle accumulation under oversaturated conditions. Finally, the intersection of Nanxu Road and Huangshan South Road in Zhenjiang City was chosen as an example. A simulation scheme was designed using VISSIM software to evaluate the effectiveness of the proposed method by implementing CSSLs before and after the intersection. The simulation results indicate that this method effectively increases the capacity of the oversaturated intersection approaches at the intersection and reduces the average delay. Under different lengths of short through lanes and utilization rates, the capacity of the eastbound short through lane and its adjacent lane increased by 20.7%, and the average delay decreased by 14.2% in all cases. Furthermore, a sensitivity analysis was conducted to examine the impact of short through lane capacity and the number of short through lanes on intersection capacity under different traffic volume levels. This analysis provides further insights into the influence of the CSSL on intersection capacity.

A graph neural network based fault diagnosis strategy for power communication networks

ABSTRACT The power communication network (PCN) is the backbone network of the power system. However, faulty nodes in the network may cause communication interruptions, seriously affecting the reliable operation of the power system. Therefore, accurate diagnosis of faulty nodes is crucial for timely detection, localization, and troubleshooting. Based on graph neural networks, we propose a Neighbour Selection and Merge Fault Diagnosis (NSMFD) strategy, which aims to identify faulty nodes capturing graph structure information and node feature information. First, we construct a graph representation of PCN, where nodes represent devices in the network and edges represent the connections between devices. Then, we use sampling and aggregation in node embedding to capture adjacent feature information of nodes, gradually fuse and update node representations through graph convolutional layers, which could be applied as the input layer of diagnostic process. Finally, we use softmax and cross entropy loss function to get a probabilistic representation and optimize it by backpropagation for prediction. We conduct experiments using real PCN datasets and compare it with other advanced diagnostic methods in terms of accuracy, false positive rate, and false negative rate. Compared with diagnostic methods, our method can accurately identify faulty nodes and achieve timely fault detection and recovery.

Some new bounds for the blow-up time of solutions for certain nonlinear Volterra integral equations

In this paper, some new upper and lower bounds on the blow-up time of a certain class of nonlinear Volterra integral equations are presented and compared to previously proposed bounds. In most cases, the new bounds are a significant improvement on previously proposed bounds. We present many new bounds, with varying degrees of accuracy and computational complexity. The derivation of the upper bounds is made possible by a certain covariance inequality from advanced probability theory, which opens the door to many future investigations of blow-up time estimations in many other nonlinear integral equations.

Повышение выхода пиломатериалов при распиловке с брусовкой

The log cutting theory accepts that the volumetric yield of edged sawn timber from the maximum volume cant will be maximal. According to current standards, edged sawn timber must have specified thickness and width. Some cants are not used for the production of centre yield because the widths of their faces are aliquant of the centre yield thickness. The volume of such cant in centre yield production is not taken into account in the log cutting theory and the conclusion that their volumetric yield from the maximum volume cant will be maximal is not obvious. The 1st stage of cant sawing is obtaining a two-edged cant from a log. At that, due to the deviation of the roundwood axis from the centre line of the sawing pattern, a narrow and a wide face are obtained. We consider the dimensioning of the narrow face of a two-edged cant, as its size determines the volumetric yield of centre yield. Within the narrow face of a two-edged cant 2 zones are allocated: unconditional and probabilistic. In the unconditional zone, an integer number of edged boards is obtained. In the range of roundwood diameters from 17 to 29 cm, only the roundwood with the diameters of 21 and 25 cm have provided the maximum volume two-edged cants, but the volumetric yield of the centre yield from the roundwood of these diameters has not been maximal. It follows from this that the maximum volume cant does not guarantee the maximal volumetric centre yield. The probability zone includes a non-integer number of edged boards. It is impossible to determine their number in an analytical way, so the methods of probability theory have been used. The distribution function of the narrow face of a two-edged cant has been derived. In order to use the distribution function to obtain a non-integer number of edged boards, the width of the probability zone has been calculated, as well as the size of the part of the probability zone decisive in obtaining a non-integer number of edged boards and the confidence interval. Further, the “Distribution function of the width of the narrow face of a two-edged cant” table was used to determine the non-integer number of edged boards. Obtaining the noninteger number of the edged boards from a two-edged cant can be implemented in practice using changeable or adjacent sawing patterns. The presented results can be applied when determining the number of sorting groups of roundwood before its feeding to the sawmill and when changing the technology of centre yield production.

Coherent upper conditional expectations defined by fractal measures and the probabilistic representation of quantum states

Coherent upper conditional expectations defined by fractal measures and the probabilistic representation of quantum states

An appropriate approach of design and machining of mechanical part by means of probabilistic multi‐objective optimization

AbstractAppropriate design and machining of mechanical part are of great importance to guarantee the comprehensive quality of product. In this paper, a rational approach of design and machining of mechanical part with multiple objectives is formulated by combining the probability‐based method for multi‐objective optimization with sequential optimization. The probability‐based method for multi‐objective optimization is employed to transfer the problem of multiple objectives into a single objective one in the respect of probability theory, while the “sequential algorithm for optimization” is used to conduct successive deep optimization quantitatively. By performing the assessment of preferable probability of each scheme and the “sequential algorithm for optimization”, the optimal design of materials processing is thus completed rationally. Furthermore, the approach is used to deal with thermal design of spindle system for computerized numerical control grinder, optimal designs of large floor‐type milling and boring machine spindle box and bending performance of B‐pillar under condition of side impact as examples, respectively. The results indicate the rationality of the approach.

Quantum Theory in Finite Dimension Cannot Explain Every General Process with Finite Memory

Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory is a finite-dimensional quantum system, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of an explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite-dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.

Probability Representation of Nonclassical States of the Inverted Oscillator

Probability Representation of Nonclassical States of the Inverted Oscillator

The Philosophy of Probability Values Behaviour, through Fractions and Composite Probability Function in the Continuous Case

This paper deals with probability theory, and it is an extension to a published paper that has the same title, but for the discrete case. This present paper is aiming to study probability values behavior, in the case of continuous sample space, through fractions intervals and composite function. This aim tends to study the value behavior rather than finding the value itself. Also, this aim requires usage of some concepts of continuity, geometric probability, and measure theory, which also need a brief treatment. This paper is mainly using an experiment with a design that helps to study the probability fractions values in the form of intervals in the case of one direction movement and in the case of different directions. As a result, every case reflects some aspects of probability values behavior and can clarify many important characteristics of the probability theory. In addition to applying the composite function by some important theorems of conditional probability. These are besides a proven proposition that helps to design experiment, upon the understanding of the case nature. In addition to a corollary that allows to visualize negative probability values, as a particular case (trial), that upon the validity of the explanation of negativity which should be consistent with the probability axioms.

Updating information based on generalized credal sets. Part 1: Finite case

This paper considers updating information and statistical inference based on generalized credal sets. In the first part, we define possible ways of conditioning for generalized credal sets in a finite setting, and its realization based on generalized upper previsions with applications to decision making. We compare the obtained results with the generalized Bayes rule from the theory of imprecise probabilities, with Dempster's rule from the theory of belief functions, and with the models of imprecise probabilities based on the likelihood function proposed by M. Cattaneo.

Structural Reliability and Optimization Using Differential Geometric Approaches

This study investigates how differential geometry ideas can be used to effectively carry out structural optimization and reliability analysis. Strong mathematical representations and methods for examining intricate surfaces and forms are provided by differential geometry. The basic ideas of differential geometry, such as tensors, manifolds, and curvature, are initially introduced in the work. Then, to account for ambiguities in the geometry, the probability theory in tangent spaces is developed. As a result, structural reliability can be determined using propagating uncertainty. To enable reliability-based design optimization, differential geometry representations are linked with optimization techniques. The proposed differential geometry-based approach is applied in a number of case studies to trusses, airplane wings, car bodies, and ship hulls. The outcomes show a significant increase in productivity and scalability compared to conventional finite element methods. The article offers new tools for dealing with uncertainty.

General one-dimensional Clifford Fourier Transform and applications to probability theory

General one-dimensional Clifford Fourier Transform and applications to probability theory

Agreement and disagreement in a non-classical world.

The Agreement Theorem Aumann (1976 Ann. Stat. 4, 1236-1239. (doi:10.1214/aos/1176343654)) states that if two Bayesian agents start with a common prior, then they cannot have common knowledge that they hold different posterior probabilities of some underlying event of interest. In short, the two agents cannot 'agree to disagree'. This result applies in the classical domain where classical probability theory applies. But in non-classical domains, such as the quantum world, classical probability theory does not apply. Inspired principally by their use in quantum mechanics, we employ signed probabilities to investigate the epistemics of the non-classical world. We find that here, too, it cannot be common knowledge that two agents assign different probabilities to an event of interest. However, in a non-classical domain, unlike the classical case, it can be common certainty that two agents assign different probabilities to an event of interest. Finally, in a non-classical domain, it cannot be common certainty that two agents assign different probabilities, if communication of their common certainty is possible-even if communication does not take place. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.

Discrete cooperative coverage location models with alternative facility types in a probabilistic setting

AbstractIn this paper, we discuss discrete location problems where the objective is to locate a given number of facilities of different types in order to appropriately cover a given set of demand points. The coverage provided to each demand point is the result of the cooperation among the located facilities. The different types of facility refer to the coverage radius or quality of coverage that each type may provide. We present a non‐linear formulation of the problem where the objective is to maximize the percentage of demand points that are appropriately covered. We then show how the model can be linearized based on a representation of probabilities through a network structure. To address large instances of the problem, we introduce a genetic algorithm (GA) that is based on the representation of the solution by two chromosomes. Computational experiments over a wide range of randomly generated problems indicate the GA clearly outperforms CPLEX, especially in larger instances.

Probabilistic Hesitant Fuzzy Evidence Theory and Its Application in Capability Evaluation of a Satellite Communication System.

Evaluating the capabilities of a satellite communication system (SCS) is challenging due to its complexity and ambiguity. It is difficult to accurately analyze uncertain situations, making it difficult for experts to determine appropriate evaluation values. To address this problem, this paper proposes an innovative approach by extending the Dempster-Shafer evidence theory (DST) to the probabilistic hesitant fuzzy evidence theory (PHFET). The proposed approach introduces the concept of probabilistic hesitant fuzzy basic probability assignment (PHFBPA) to measure the degree of support for propositions, along with a combination rule and decision approach. Two methods are developed to generate PHFBPA based on multi-classifier and distance techniques, respectively. In order to improve the consistency of evidence, discounting factors are proposed using an entropy measure and the Jousselme distance of PHFBPA. In addition, a model for evaluating the degree of satisfaction of SCS capability requirements based on PHFET is presented. Experimental classification and evaluation of SCS capability requirements are performed to demonstrate the effectiveness and stability of the PHFET method. By employing the DST framework and probabilistic hesitant fuzzy sets, PHFET provides a compelling solution for handling ambiguous data in multi-source information fusion, thereby improving the evaluation of SCS capabilities.

Random vector representation of continuous functions and its applications in quantum mechanics

The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors where every random vector is defined on one of these probability spaces, such that the sequence of conditional mathematical expectations formed by the random vectors uniformly converges to the continuous function. This is a random vector representation of continuous functions, which is regarded as a bridge to be set up between real function theory and probability theory. By means of this conclusion, an interesting result about function approximation theory can be obtained. The random vector representation of continuous functions is of important applications in physics. Based on the conclusion, if a large proportion of certainty phenomena can be described by continuous functions and random phenomena can also be described by random variables or vectors, then any certainty phenomenon must be the limit state of a sequence of random phenomena. Then, in the approximation from a sequence of random vectors to a continuous function, the base functions are appropriately selected by us, and an important conclusion for quantum mechanics is deduced: classical mechanics and quantum mechanics are unified. Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.

Application of the Bayesian network theory in clinical trial data: Severity shift in spasticity numeric rating scale in patients with multiple sclerosis

BackgroundData digitization expands data collection opportunities, representing both a chance to understand interrelationships between variables and a challenge to identify the most appropriate clinical factors. Applications of causal inference techniques to clinical trial data is becoming very attractive, especially with the intent to provide insights into the relationships between baseline characteristics and outcomes. Graphical representations of model structures and conditional probabilities can be powerful tools to illustrate relationships in a high-dimensional data setting. MethodsWe review and apply Bayesian network theory to a clinical case study, presenting an analytical approach to investigating and visualizing causal relationships. We propose the use of the adherence score to compare data networks’ patterns based on different variables’ discretization. Data from adult patients with spasticity related to multiple sclerosis (MSS) from two randomized placebo-controlled clinical trials of nabiximols were used as analysis sets. The training and validation sets included 106 (53 treated, 53 placebo) and 155 (76 treated, 79 placebo) participants, respectively. The primary objective was to create a network and estimate the causal dependencies between participants’ characteristics, changes in MSS severity as reflected by shifts in the patient-reported numeric rating scale (NRS), and changes in symptoms, functional abilities, and quality of life factors. ResultsA causal network was identified between the key factors of assigned treatment, end of study spasticity NRS, and mental health/vitality subscales of the 36-Item Short Form Health Survey questionnaire (4 nodes and 3 edges; adherence score = 93%). In patients with mild spasticity, the impact of nabiximols on mental health or vitality subscales resulted in a probability ratio of 1.63. The decomposed mediation effect of spasticity NRS was observed through a mediation analysis between treatment and mental health (99.4%) or vitality (93.7%) subscales. ConclusionsThe use of innovative methods such as causal networks is highly encouraged to identify dependent relationships among key factors in clinical trial data and drive insights for additional research.

Disrupted brain structural networks associated with depression and cognitive dysfunction in cerebral small vessel disease with microbleeds

Emerging evidence highlights cerebral microbleeds (CMBs) as hallmarks of cerebral small vessel disease (CSVD) underlying depression and cognitive dysfunction. This study aimed to reveal how depression and cognition-related white matter (WM) abnormalities are topologically presented, and the network-level structural disruptions associated with CMBs in CSVD. We used probabilistic diffusion tractography and graph theory to investigate brain WM network topology in CSVD patients with (n = 64, CSVD-c) and without (n = 138, CSVD-n) CMBs and 90 healthy controls. Then we evaluated the Pearson's correlations between disrupted network metrics and neuropsychological parameters. For global topology, the CSVD-c group exhibited significantly decreased global (Eglob) and local (Eloc) efficiency and increased shortest path length compared with the controls, while no significant difference was found between the CSVD-c and CSVD-n groups. For regional topology, although all groups showed highly similar hub distributions, compare with control group, the CSVD-c group exhibited significantly decreased nodal efficiency mainly in the bilateral supplementary motor area (SMA), median cingulate gyrus (DCG) and right orbital middle frontal gyrus, while the CSVD-n group showed significantly decreased nodal efficiency only in the right SMA. Notably, Eglob, Eloc and nodal efficiency of the right anterior cingulate gyrus, DCG, middle temporal gyrus and left insula showed significantly negative correlations with depression score, significantly positive correlations with Rey auditory verbal learning test and symbol digit modalities test scores in CSVD-n group, as well as significantly negative correlations with Stroop color-word test scores in CSVD-c group. The WM networks of CSVD patients are characterized by decreased global integration and local specialization, and decreased nodal efficiency highly related to depression and cognitive dysfunction in the attention, default mode network and sensorimotor regions. These findings provide new insight into the neurobiological mechanisms of CSVD and concomitant affective and cognitive disorders.