Fuzzy Probability Theory Research Articles

The Intersections Based on Joint Observables In Fuzzy Probability

“Fuzzy probability theory” appeared as a smooth extension of classical probability theory in 1995. It was expected that it will be of great importance in quantum mechanics, but the theory doesn’t keep its development as it was expected. This necessitates revising some of its fundamental basic concepts. We argue that if quantum probability theory should have less constrained than classical probability theory as can be seen in the case of joint random variables, we surely need to weaken the definition of the intersection operation. In this paper, discuss the definition validity in quantum probability theory and to discuss the consistency of the given definitions with the whole theory and the possibility to have a more suitable definition.

Numerical Calculation Method of Target Damage Effectiveness Evaluation under Uncertain Information of Warhead Fragments

In order to scientifically evaluate the target damage effectiveness when warhead fragments formed by projectile explosion attack targets, this paper proposes a new method by using the multi-layer warhead fragments distribution division mechanism to study an evaluation and calculation model of target damage effectiveness under uncertain information of warhead fragments. On the basis of the spatial relationship between projectile explosion position and target, we first analyze the target damage rules based on the target’s own vulnerability characteristics. Then we use the fuzzy prior probability theory to establish the membership function of the target damage factor and an evaluation model of the target damage grade, and use the stochastic multi-criteria acceptability analysis method to obtain the evaluation index value of the target damage level. Finally, considering the ratio between the effective area of the warhead fragment group attacking target and the area of target itself and the effective number and area ratio of warhead fragments penetrating target, we give a numerical calculation method of target damage effectiveness. Through simulation and calculation, the results show that when the distance between the projectile explosion position and the target is less than 7 m and the coordinate interval of the projectile flight direction is −2.5 m to 2.5 m, the damage probability of warhead fragments to the target is greater than 75%. When the weight of the damage factor of the target changes, the damage effectiveness of the target also changes, and this result is in accordance with the actual experiment, which verifies the rationality of the proposed mathematical model.

Measures of uncertainty for a fuzzy probabilistic approximation space

An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

Efficient reliability analysis of complex systems in consideration of imprecision

In this work, the reliability of complex systems under consideration of imprecision is addressed. By joining two methods coming from different fields, namely, structural reliability and system reliability, a novel methodology is derived. The concepts of survival signature, fuzzy probability theory and the two versions of non-intrusive stochastic simulation (NISS) methods are adapted and merged, providing an efficient approach to quantify the reliability of complex systems taking into account the whole uncertainty spectrum. The new approach combines both of the advantageous characteristics of its two original components: 1. a significant reduction of the computational effort due to the separation property of the survival signature, i.e., once the system structure has been computed, any possible characterization of the probabilistic part can be tested with no need to recompute the structure and 2. a dramatically reduced sample size due to the adapted NISS methods, for which only a single stochastic simulation is required, avoiding the double loop simulations traditionally employed.Beyond the merging of the theoretical aspects, the approach is employed to analyze a functional model of an axial compressor and an arbitrary complex system, providing accurate results and demonstrating efficiency and broad applicability.

Two-parameter KNN algorithm and its application in recognition of brand rice

In response to the frequent counterfeiting of Wuchang rice in the market, an effective method to identify brand rice is proposed. Taking the near-infrared spectroscopy data of a total of 373 grains of rice from the four origins (Wuchang, Shangzhi, Yanshou, and Fangzheng) as the observations, kernel principal component analysis(KPCA) was employed to reduce the dimensionality, and Fisher discriminant analysis(FDA) and k-nearest neighbor algorithm (KNN) were used to identify brand rice respectively. The effects of the two recognition methods are very good, and that of KNN is relatively better. Howerver the shortcomings of KNN are obvious. For instance, it has only one test dimension and its test of samples is not delicate enough. In order to further improve the recognition accuracy, fuzzy k-nearest neighbor set is defined and fuzzy probability theory is employed to get a new recognition method –Two-Parameter KNN discrimination method. Compared with KNN algorithm, this method increases the examination dimension. It not only examines the proportion of the number of samples in each pattern class in the k-nearest neighbor set, but also examines the degree of similarity between the center of each pattern class and the sample to be identified. Therefore, the recognition process is more delicate and the recognition accuracy is higher. In the identification of brand rice, the discriminant accuracy of Two-Parameter KNN algorithm is significantly higher than that of FDA and that of KNN algorithm.

Fuzzy probability based person recognition in smart environments

Biometric features are used to verify the people identity in the living places like smart apartments. To increase the chance of classification and recognition rate, the recognizing procedure contains various steps such as detection of silhouette from the gait profile, silhouette segmentation, reading features from the silhouette, classification of features and finally recognition of person using its probability value. Person recognition accuracy will be oscillated and declined due to blockage, radiance and posture variance problems. In the proposed work, the gait profile will be formed by capturing the gait of a targeted person in stipulated time to reach the destination. From the profile the silhouettes are detected using frame difference and segmented from the background using immediate thresholding and features are extracted from the silhouette using gray-level covariance matrix and optimized feature set is formed using PSO. These optimized features are fused, trained and classified using nearest neighbor support vectors. The fuzzy probability method is used for recognizing the person based on the probability value of the authentic and imposter scores. The relationship between the CMS, TPR, TNR and F-rate are calculated for 1 : 1 matcher from the gallery set. The performance of the classifiers are found to be perfect by plotting the DET graph and ROC curve. The proposed fuzzy probability theory is mingled with GLCMPSO and NSFV method for human recognition purpose. The performance of the proposed is proved to be acceptable for recognition with the optimal parameters (Entropy, SSIM, PSNR, CQM) calculation From the work, it is clear that, the rank probability is proportional to the match score value of the silhouette stored in the gallery.

Research on Probability Mean-Lower Semivariance-Entropy Portfolio Model with Background Risk

In the financial market, investors must deal with uncertain risk, and they also face background risk and many uncertain factors caused by their own characteristics. Considering the fuzzy nature of these factors as well as investors’ risk preferences, transaction costs, and so on, in order to reduce investment risk, an improved probability entropy measure is introduced, and a probability mean-lower semivariance-entropy model with different risk attitudes is established by using fuzzy sets and probability theory. To solve the portfolio model, an improved differential evolution algorithm is proposed and a numerical example is given. The numerical results show that the proposed algorithm is effective and that the model can disperse the financial risk to a certain extent and reasonably solve the portfolio problem under many different conditions.

Reliability analysis of corrosion initiation in reinforced concrete structures subjected to chlorides in presence of epistemic uncertainties

This paper presents a methodology for assessing the corrosion initiation in reinforced concrete (RC) structures subjected to chlorides, with a focus on the influence of epistemic uncertainties inherent to input parameters. Epistemic uncertain parameters were modeled using fuzzy probability theory. The chloride penetration was modeled using a finite element solution of the coupled heat transfer, moisture and chloride transport in concrete. The environmental conditions were modeled using a stochastic process. The reliability analysis was undertaken using a hybrid fuzzy Monte Carlo simulation. As an illustration, the durability of concrete basins of a newly constructed waste water treatment plant, with respect to reinforcement corrosion, was examined. The time to corrosion initiation was found to approximately follow a generalized extreme value distribution with fuzzy parameters. The stochastic modeling of exposure conditions accelerated the corrosion initiation. The time to corrosion initiation is highly sensitive to the epistemic uncertainty in the threshold chloride level in concrete. Reducing this uncertainty will result in an important improvement in accuracy. The proposed method is suitable for durability assessment of recent RC structures with little or no available performance data.

Risk Assessment and Decision-Making of a Listed Enterprise’s L/C Settlement Based on Fuzzy Probability and Bayesian Game Theory

Letter of Credit (L/C) is currently a very popular international settlement method frequently used in international trade processes amongst countries around the globe. Compared with other international settlement methods, however, L/C has some obvious shortcomings. Firstly, it is not easy to use due to the sophisticated processes its usage involves. Secondly, it is sometimes accompanied by a few risks and some uncertainty. Thus, highly efficient methods need to be used to assess and control these risks. To begin with, FAHP and KMV methods are used to resolve the problem of incomplete information associated with L/C and then, on this basis, Bayesian game theory is used in order to make more scientific and reasonable decisions with respect to international trade.

Assessment and modelling of uncertainty in precipitation forecasts from TIGGE using fuzzy probability and Bayesian theory

The precipitation forecasts from numerical weather prediction have a variety of potential uses in flood forecasting and reservoir operation, but suffer from relatively poor performance due to the uncertainty of the hydrometeorological system. In this study, the control forecasts of four global weather centres were selected and assessed against the measured data using several verification metrics during the flood season (May–September) over the Shihe River catchment in the Huaihe River basin of China. The results show that the daily rainfall forecasts have low prediction ability and cannot meet the demand of reservoir regulation. To describe the uncertainty of precipitation forecasts for the safety of flood control, a new model was proposed using fuzzy probability and Bayesian theory on the basis of the generalized probability density function (GPDF). The performance of the new model is examined by using various probability measures and compared with the ensemble forecasts generated by the weather centres. It is proved that the fuzzy Bayesian model can generate the conditional probability distribution of actual precipitation from a single predicted value based on the historical observation and forecast data, and has strong generalization ability. Compared with the ensemble forecasts, although the fuzzy Bayesian model shows a slightly improvement in accuracy, it has better performance in sharpness and reliability. Meanwhile, the new model is easy to update with new samples by modifying its likelihood function, which is favourable for real-time reservoir regulation. In addition, the uncertainty of the precipitation forecast was analysed with the model in different lead times. Generally, the uncertainty increases with the growth of lead time, and the probability distribution of the rainfall forecast within 3 days is an acceptable result for a risk and benefit analysis of the flood control system. To be more specific, a fuzzy Bayesian model based on the GPDF is an efficient way to generate the probability distribution of the precipitation with forecast data for the uncertainty analysis, and makes it possible to provide a reference for reservoir managers to plan regulation strategies with a lead time of at least 3 days.

Simulated Hesitant Fuzzy Linguistic Term Sets-Based Approach for Modeling Uncertainty in AHP Method

One of the important points in decision-making is representing the uncertainty. There have been developed approaches which are based on either fuzzy sets theory or probability theory to model the inherent uncertainty and imprecision in the pairwise comparison process of analytic hierarchy process (AHP). This study proposes a hybrid approach that integrates fuzzy sets theory and simulation technique into AHP method to better represent the uncertainty in decision-making process. The aim of the proposed approach is to benefit the advantages of both probability theory and fuzzy sets theory which may provide a new paradigm for modeling uncertainty in multi-criteria decision-making (MCDM). An illustrative example is also presented to show the applicability of the proposed approach.

Multiple Deferred State Sampling Plan with Fuzzy Parameter

Traditional multiple deferred state sampling plan by attribute is an inspection with crisp parameter. In this plan, in addition to using the current lot information, the future successive lots information can also be utilized on sentencing the current lot. In this study, a fuzzy multiple deferred state (FMDS) sampling plan by attribute is proposed when the proportion of defective items (p) is uncertain. So, the fuzzy probability theory is applied to construct the operating characteristic curve of mentioned plan. Then, we concentrate more on a special feature of FMDS(0,1,2) plan. We show that the proposed plan is well-defined because it converts to classic case when p is not imprecise. In order to be fully informed, some examples will also be discussed. The comparisons between the FMDS(0,1,2) plan and the existing fuzzy single sampling plan (FSSP) with zero and one acceptance number show that the proposed plan is more powerful than the FSSP in distinguishing good and bad quality of the lot. It also enjoys the least sum of the consumer and producer risks.

Factorization of Observables

Categorical approach to probability leads to better understanding of basic notions and constructions in generalized (fuzzy, operational, quantum) probability, where observables—dual notions to generalized random variables (statistical maps)—play a major role. First, to avoid inconsistencies, we introduce three categories $\mathbb {L}$ , $\mathbb {S}$ , and $\mathbb {P}$ , the objects and morphisms of which correspond to basic notions of fuzzy probability theory and operational probability theory, and describe their relationships. To illustrate the advantages of categorical approach, we show that two categorical constructions involving observables (related to the representation of generalized random variables via products, or smearing of sharp observables, respectively) can be described as factorizing a morphism into composition of two morphisms having desired properties. We close with a remark concerning products.

Risk identification and assessment for engineering procurement construction management projects using fuzzy set theory

Considerable work has been carried out on risk qualitative and quantitative assessment but far less on risk identification. This paper introduces a newly developed method for risk identification, based on micro risk breakdown structure and newly introduced identification procedure called preventive root cause and effective remedial. It also introduces a risk responsibility matrix that distributes the responsibilities associated with each risk among project stakeholders and introduces a newly developed method for qualitative and quantitative assessment of each item using fuzzy set and fuzzy probability theories. Output of the proposed assessment method is pre-mitigation contingency of each risk which represents a quantitative indicator for decision making whether to mitigate or not the risk being considered. Two case studies and one numerical example are presented to demonstrate the applicability and illustrate the essential features of proposed identification, allocation, and assessment methods.

A non-linear fuzzy degradation model for estimating reliability of a polymeric coating

Oftentimes, material components follow non-linear degradation because the material loss trough time is non-linear; the components present different levels of resistance during a degradation process. Constructing predictions of the component life span and performing reliability analyses regarding degradation is possible by non-linear regression. Generally, this method ignores the prediction error in the extrapolated time and it does not account measurement error in the observed sample paths. In this paper, it is proposed new means for predicting time to failure of the components, using a calibration regression method for measuring the error prediction in the extrapolation process. In order to consider the uncertainty, we proposed to perform the estimated times by a trapezoidal fuzzy numbers, providing a more reliable prediction without loss of information. Since the times to failure are shaped by fuzzy numbers, it was adapted the fuzzy probability theory to the classical reliability analysis for estimating the fuzzy reliability of a component. An accelerated test of a polymeric coating process was performed and also an application for a preventive maintenance frequency was proposed.

Fuzzy Probability Algebra (FPA): A Theory of Fuzzy Probability for Fuzzy Inference and Computational Intelligence

Fuzzy Probability Algebra (FPA): A Theory of Fuzzy Probability for Fuzzy Inference and Computational Intelligence

Real functions and the extension of generalized probability measures II

Abstract We continue our study of the extensions of generalized probability measures. First, we describe some extensions of generalized random events (represented by classes of functions with values in [0,1]) to which generalized probability measures can be extended. Second, we study products of domains of probability and describe states on such products. Third, we show that the events in IF-probability, introduced by B. Riečan, form a suitable category isomorphic to a subcategory of the category of fuzzy random events. Consequently, IF-probability can be interpreted within fuzzy probability theory. We put forward some problems related to the extensions of probability domains and hint some applications.

Learning parameters of fuzzy Bayesian Network based on imprecise observations

In recent years, Bayesian Network has become an important modeling method for decision making problems of real-world applications. In this paper learning parameters of a fuzzy Bayesian Network (BN) based on imprecise/fuzzy observations is considered, where imprecise observations particularly refers to triangular fuzzy numbers. To achieve this, an extension to fuzzy probability theory based on imprecise observations is proposed which employs both the truth concept of Yager and the Extension Principle in fuzzy set theory. In addition, some examples are given to demonstrate the concepts of the proposed idea. The aim of our suggestion is to be able to estimate joint fuzzy probability and the conditional probability tables (CPTs) of Bayesian Network based on imprecise observations. Two real-world datasets, Car Evaluation Database (CED) and Extending Credibility (EC), are employed where some of attributes have crisp (exact) and some of them have fuzzy observations. Estimated parameters of the CED's corresponding network, using our extension, are shown in tables. Then, using Kullback-Leibler divergence, two scenarios are considered to show that fuzzy parameters preserve more knowledge than that of crisp parameters. This phenomenon is also true in cases where there are a small number of observations. Finally, to examine a network with fuzzy parameters versus the network with crisp parameters, accuracy result of predictions is provided which shows improvements in the predictions.

On D-posets of fuzzy sets

Abstract D-posets of fuzzy sets constitute a natural simple mathematical structure in which relevant notions of generalized probability theory can be formalized. We present a classification of D-posets leading to a hierarchy of distinguished subcategories of D-posets related to probability and study their relationships. This contributes to a better understanding of the transition from classical probability theory to fuzzy probability theory. In particular, we describe the transition from the Boolean cogenerator {0, 1} to the fuzzy cogenerator [0, 1] and prove that the generated Łukasiewicz tribes form an epireflective subcategory of the bold algebras.

Fuzzy reliability analysis with only censored data

It is expected that samples in reliability analysis contain both censored and complete failure data; thus, the maximum likelihood method is used to estimate the parameters of the related distribution. Nonetheless, samples may contain only censored data; therefore introducing a high degree of uncertainty which does result in non-viability for either the likelihood method or for statistical inference. This paper proposes the use of fuzzy probability theory to account for the uncertainty and the prior knowledge of the process in the parameters׳ estimation, for censored data. The proposed method was applied to risk based inspection. Results demonstrate that our method represents a reliable option for using the expert knowledge about the component and the physics of the failure mode. Additionally, an inspection time was estimated based on target risk; the results confirm that the methodology could be used to develop maintenance plans.