Bayesian Formula Research Articles

Centralized Maintenance Time Prediction Algorithm for Freight Train Wheels Based on Remaining Useful Life Prediction

Many freight trains for special lines have in common the characteristics of a fixed group. Centralized Condition-Based Maintenance (CCBM) of key components, on the same freight train, can reduce maintenance costs and enhance transportation efficiency. To this end, an optimization algorithm based on the nonlinear Wiener process is proposed, for the prediction of the train wheels Remaining Useful Life (RUL) and the centralized maintenance timing. First, Hodrick–Prescott (HP) filtering algorithm is employed to process the raw monitoring data of wheel tread wear, extracting its trend components. Then, a nonlinear Wiener process model is constructed. Model parameters are calculated with a maximum likelihood estimation and the general deterioration parameters of wheel tread wear are obtained. Then, the updating algorithm for the drift coefficient is deduced using Bayesian formula. The online updating of the model is realized, based on individual wheel monitoring data, while a probability density function of individual wheel RUL is obtained. A prediction method of RUL for centralized maintenance is proposed, based on two set thresholds: “maintenance limit” and “the ratio of limit-arriving.” Meanwhile, a CCBM timing prediction algorithm is proposed, based on the expectation distribution of individual wheel RUL. Finally, the model is validated using a 500-day online monitoring data on a fixed group, consisting of 54 freight train cars. The validation result shows that the model can predict the wheels RUL of the train for CCBM. The proposed method can be used to predict the maintenance timing when there is a large number of components under the same working conditions and following the same path of degradation.

Predictions of nuclear charge radii and physical interpretations based on the naive Bayesian probability classifier

Background: The nuclear charge radii provide direct information for the nuclear structures. In recent years, many pioneering researches have been devoted to the nuclear charge radii based on the Bayesian neural networks (BNN) method. Purpose: The neural networks always have complex structure. To analyze the data relationships clearly, a statistical method is introduced to study the nuclear properties by combining the sophisticated nuclear models with the naive Bayesian probability (NBP) classifier. Method: In the framework of the NBP method, the predicted charge radii are interpreted as the most reasonable expectations. The prior probabilities and the condition probabilities are computed by the experimental data and the raw results of nuclear models. The posterior probabilities of expectations are updated by the Bayesian formula. The predicted charge radii are regarded as the expectations with maximum probability. Moreover, the abilities of global optimizations and extrapolations of the NBP method are analyzed to demonstrate the availability of the NBP method. Results: For the HFB model and the semiempirical formula, the accuracy of charge radii predictions improves 41% and 32% after NBP refinements, respectively. Calculations also illustrate that the NBP method has robust extrapolating abilities, and the charge radii of unknown regions in the nuclear chart can be predicted by the NBP method. Conclusions: The NBP method contains the advantages of local relations and global descriptions, which can provide fine-tuning for the theoretical results of sophisticated nuclear models. The method proposed in this paper can also be used for other research of nuclear properties.

Optimization Design of Groundwater Pollution Monitoring Scheme and Inverse Identification of Pollution Source Parameters Using Bayes’ Theorem

In the process of identifying groundwater pollution sources, in order to solve the problem that the monitoring data of monitoring wells was insufficient or the correlation between monitoring data and model parameters was weak, a monitoring well optimization method based on Bayesian formula and information entropy was proposed. Two-dimensional phreatic groundwater solute transport model was built and solved by using GMS software. To reduce the computational load of calling the numerical model repeatedly in the optimization design of the monitoring schemes and the identification process of the pollution sources, the Kriging method was used to establish the surrogate model of the numerical model. Under the condition of single well monitoring and determined monitoring frequency, with the target of optimization of monitoring position number D and monitoring time interval ∆t, both the single-objective monitoring scheme with the minimum information entropy of the model parameter posterior distribution and the multi-objective monitoring scheme with the minimum information entropy and the shortest monitoring time were optimized respectively. According to the above-optimized monitoring schemes, the delayed rejection adaptive Metropolis algorithm was used to identify the pollution source parameters. The case study results showed that under the condition of pre-set single well monitoring with monitoring frequency of 10 times, the single-objective optimized monitoring scheme was D = 37 and Δt = 20 days. Under this monitoring scheme, the mean errors of inversion pollution source parameters α = (XS, YS, T1, T2, QS) were 0.09%, 0.4%, 4.72%, 2.43%, and 9.29%, respectively. The multi-objective optimized monitoring scheme was D = 37 and Δt = 2 days. Under this monitoring scheme, the mean errors of the inversion parameters α = (XS, YS, T1, T2, QS) were 12.76%, 3.77%, 5.13%, 1.36%, and 7.68%, respectively. Compared with the monitoring scheme based on the single-objective optimization, although the inversion mean error of the five parameters based on the multi-objective optimized monitoring scheme increased by 2.75%, the monitoring time significantly reduced from 180 to 18 days.

Coupling model of fuzzy soft set and Bayesian method to forecast internal defects of ancient wooden structures based on nondestructive test

In order to improve the detection precision of internal defect in the ancient wooden structures, defect simulation tests on pine and elm commonly used in ancient buildings were performed by using stress wave detection and drilling resistance detection. Based on detection data, three typical evaluation criteria, which are the information entropy, the correlation coefficient, and residual sum of squares, were selected as a priori information. Combining with the expert’s fuzzy evaluation value, Bayesian formula was used to modify the prior information to determine the weight coefficients of the two detection methods, and a combined prediction model was established. The results show that the combination of subjectivity and objectivity enables the revised weights to more reasonably and accurately reflect the relative importance of each detection method in prediction evaluation, which reduces the forecasting error. Specifically speaking, the mean error of the combined model was reduced by 49.8% and 59.8%, respectively, compared with stress wave detection and drilling resistance detection. Moreover, the five error indicators of this combined forecasting model are the smallest in all methods, indicating the proposed method has a better forecasting effect. It provides an effective application tool for the practice of forecasting the internal defects of wooden components in ancient buildings.

Application of the Bayesian approach in an expert’s system for diagnostic fault in computing engineering

The problem of increasing the efficiency of maintenance and repair of computer equipment is solved by applying the method of probability theory in expert systems. An expert system is a set of programs that accumulates the knowledge of specialists in a specific subject area and replicates this empirical experience for consultations of less qualified specialists. A probabilistic analysis method of equipment operation is used in the expert system under development for diagnosing and preventing computer malfunctions, is used. This article provides detailed description of the expert system, diagnosing the causes of computer malfunctions, provides a structural diagram of the system, rules and facts of work are generated. Based on the well-known Bayesian formula, this article describes a method for using fault information to clarify initial probabilities.

Reliability Assessment for Very Few Failure Data and Weibull Distribution

Only very few failure data can be obtained for the time censored test of high-reliability and long-life products. For very few failure data, the current methods fail to obtain both the point estimation and confidence interval of reliability parameters. If the point estimation and confidence interval of reliability parameters are obtained based on different methods, the results tend to be unreliable. In this study, based on the existing research, a Bayesian reliability evaluation method for very few failure data under the Weibull distribution was proposed. First, the range of failure probability was limited based on the convexity and self-features of the Weibull distribution function. Second, based on the background of the sample with very few failure data, the pretest distribution function and parameters were set and solved. The point estimation and confidence interval model of failure probability based on the Bayesian formula was established. The improved match distribution curve method was used to compute both the point estimation and confidence interval of reliability parameters. Furthermore, by comparing the results of numerical examples, the calculation results obtained by the proposed method were verified as being very reasonable. Finally, taking wet friction plates as an example, the results showed the effectiveness of this method in engineering practice.

State estimation for nonlinear discrete–time fractional systems: A Bayesian perspective

In this paper, the state estimation issue for fractional dynamic systems is addressed. First, based on the Bayesian formula, a recursive Bayesian filter is derived from a probability density perspective. Then for a nonlinear fractional Gaussian system, a general fractional Gaussian filtering framework is investigated. Under this framework, the fractional particle filter is investigated systematically, and the working mechanism, time complexity and estimation accuracy of four suboptimal fractional filters are analyzed. Finally, several simulation examples verify the effectiveness of four suboptimal filters.

Vulnerability Assessment of a Multistate Component for IEMI Based on a Bayesian Method

A method based on the Bayesian approach is proposed in this paper to assess the vulnerability of a multistate component when the available experimental data are insufficient. According to the proposed definition of hierarchical vulnerability distributions, each state probability function of components is represented by adjacent level distributions, and the restriction of vulnerability distributions is analyzed. Depending on the credibility interval of the parameters of vulnerability distributions, the joint prior probability density functions (PDFs) of these parameters can be obtained in different scenarios. Therefore, the joint posterior PDF of these parameters can be obtained by combining the joint prior PDF and the likelihood function based on a Bayesian formula. Then, the posterior probability function of each state can be obtained by the proposed method. In realistic situations, the estimation of the state probability functions will be more practical than single estimation of the state probability under a specific threat level of intentional electromagnetic interference (IEMI). The specific process of the method is presented, while the hierarchical vulnerability distributions are represented by lognormal cumulative distribution functions. Finally, a case study demonstrates the effectiveness of the proposed method.

Non-Dominated Immune-Endocrine Short Feedback Algorithm for Multi-Robot Maritime Patrolling

Multi-robot systems can be used to patrol a concerned part of ocean to ensure maritime safety under severe weather conditions. Seeking its optimal patrolling strategy to fulfill multi-optimization objectives is challenging. This paper presents a novel approach inspired by an immune-endocrine short feedback system to do so. Regulations produced by an endocrine system act on two phases of an artificial immune algorithm. First, a kind of hormone applied in a mutation process is proposed to decrease the number of undesirable solutions with the help of a Bayesian formula. Second, it is performed at a memory cell to suppress high-concentration antibodies and save high-quality individuals. To verify the feasibility of the proposed method, simulation experiments are conducted. The experimental results illustrate the desired effects of two regulation phases and the proposed method’s performance in solving a maritime patrolling problem. Its high search ability and convergence speed are shown via its comparison with the other well-known algorithms.

Optimization of Well Position and Sampling Frequency for Groundwater Monitoring and Inverse Identification of Contamination Source Conditions Using Bayes’ Theorem

Coupling Bayes’ Theorem with a two-dimensional (2D) groundwater solute advection‑diffusion transport equation allows an inverse model to be established to identify a set of contamination source parameters including source intensity (M), release location (X0,Y0) and release time (T0), based on monitoring well data. To address the issues of insufficient monitoring wells or weak correlation between monitoring data and model parameters, a monitoring well design optimization approach was developed based on the Bayesian formula and information entropy. To demonstrate how the model works, an exemplar problem with an instantaneous release of a contaminant in a confined groundwater aquifer was employed. The information entropy of the model parameters posterior distribution was used as a criterion to evaluate the monitoring data quantity index. The optimal monitoring well position and monitoring frequency were solved by the two-step Monte Carlo method and differential evolution algorithm given a known well monitoring locations and monitoring events. Based on the optimized monitoring well position and sampling frequency, the contamination source was identified by an improved Metropolis algorithm using the Latin hypercube sampling approach. The case study results show that the following parameters were obtained: 1) the optimal monitoring well position (D) is at (445, 200); and 2) the optimal monitoring frequency (∆t) is 7, providing that the monitoring events is set as 5 times. Employing the optimized monitoring well position and frequency, the mean errors of inverse modeling results in source parameters (M,X0,Y0,T0) were 9.20%, 0.25%, 0.0061%, and 0.33%, respectively. The optimized monitoring well position and sampling frequency can effectively safeguard the inverse modeling results in identifying the contamination source parameters. It was also learnt that the improved Metropolis‑Hastings algorithm (a Markov chain Monte Carlo method) can make the inverse modeling result independent of the initial sampling points and achieves an overall optimization, which significantly improved the accuracy and numerical stability of the inverse modeling results.

Long-term electricity consumption forecasting based on expert prediction and fuzzy Bayesian theory

Long-term electricity consumption (EC) forecasting is a very important part for the expansion planning of power system. Instead of point forecasting, based on fuzzy Bayesian theory and expert prediction, a novel long-term probability forecasting model is proposed to predict the Chinese per-capita electricity consumption (PEC) and its variation interval over the period 2010–2030. The special model structure can improve the reliability and accuracy of expert prediction through econometric methodology. It contains three components: fuzzy relation matrix, prior prediction, and fuzzy Bayesian formula. To contend with the long-term uncertainty, the prior prediction is implemented to combine the advantages of expert's experience with other time-based methods from the perspective of probability. With the utilization of fuzzy technique, the multiple effects of influencing factors (IFs) on PEC can be expressed as a fuzzy relation matrix. It can rule the results of prior prediction to obey the long-run equilibrium relationship of natural evolution thorough probability calibration. To demonstrate its efficiency and applicability, the result of this method is compared with that of other 6 approaches and 4 agencies. The case study shows that the proposed methodology has higher accuracy and adaptability.

Panoramic visual tracking based on adaptive mechanism

Panoramic visual tracking has high application value in many situations, but its visual distortion is likely to cause low tracking robustness or loss of target. This paper presents a panoramic visual tracking method based on adaptive feature fusion method, the size of the trapezoidal frame is calibrated with target moving in the method, and the linear model of trapezoidal frame parameters is fitted, then according to the model trapezoidal area of target extracted, and are modified to target trapezoidal region through the affine transformation; Based on filter tracking framework, the fusion of color and shape information is as the main characteristics of the target tracking, and Bayesian fusion recursive formula is used for calculating the particle weights. The experimental results show that the proposed algorithm is better than the existing methods in tracking precision and anti occlusion, and effectively improves the robustness of panoramic visual tracking.

Efficient numerical simulation methods for estimating fuzzy failure probability based importance measure indices

For the problem with a fuzzy failure state which commonly exists in degradation structures and systems, the fuzzy failure probability based importance measure indices can be used to measure the effect of the input variables on the fuzzy failure probability effectively. However, the computational cost is unaffordable for estimating the indices directly. For efficiently evaluating the fuzzy failure probability based importance measure indices, this paper proposed two numerical simulation methods, i.e., the direct Monte Carlo simulation method based on the Bayesian formula (B-DMCS) and the adaptive radial-based importance sampling method based on the Bayesian formula (B-ARBIS). The two proposed methods employ the Bayesian formula to eliminate the dependence of the computational cost on the dimensionality of the input variables. Compared with the B-DMCS method, the B-ARBIS method can enhance the computational efficiency significantly due to repeatedly utilizing the same group of samples of the input variables and the strategy to adaptively search the optimal hypersphere in the safety domain. After giving the principles and implementations of the two methods, three examples are employed to validate the effectiveness of the two proposed numerical simulation methods. The results of the examples demonstrate that the effectiveness of the two proposed methods is higher than the direct Monte Carlo method, and the B-ARBIS method can improve the efficiency obviously in contrast with the B-DMCS method.

Analysis and Comparison of Bayesian Methods for Measurement Uncertainty Evaluation

Based on the Bayesian principle, the modern uncertainty evaluation methods can fully integrate prior and current sample information, determine the prior distribution according to historical information, and deduce the posterior distribution by integrating prior distribution and the current sample data with the Bayesian model. As such, it is possible to evaluate uncertainty, updating in real time the uncertainty of the measuring instrument according to regular measurement, and timely reflect the latest information on the accuracy of the measurement system. Based on the Bayesian information fusion and statistical inference principle, the model of uncertainty evaluation is established. The maximum entropy principle and the hill-climbing search optimization algorithm are introduced to determine the prior distribution probability density function and the sample information likelihood function. The probability density function of posterior distribution is obtained by the Bayesian formula to achieve the optimization estimation of uncertainty. Three methods of measurement uncertainty evaluation based on Bayesian analysis are introduced: the noninformative prior, the conjugate prior, and the maximum entropy prior distribution. The advantages and limitations of each method are discussed.

APPLICATION OF BAYESIAN APPROACH OF LOGISTIC REGRESSION FORMATION WHILE PROCESSING THE TESTS RESULTS ON THE DURABILITY OF AIRCRAFTCONSTRUCTIONS ELEMENTS

When solving the problems of predicting the durability of aircraft structures elements in case of a bird strike, in a collision with large hail, foreign objects, it is necessary to know the dependence of the probability of the barrier penetration on the characteristics of the impactor and the impact speed. The results of the experiments have the form of a dichotomous variable having only two possible values (the barrier is punched or not), so the required dependence can be represented in the form of a logistic regression given by the integral function of the logistic distribution. To evaluate the parameters of logistic regression with a small number of experiments, it is advisable to use the Bayesian approach. One of the components of the Bayesian parameter estimation is the calculation of the likelihood function. The resulting expressions for the likelihood function and its logarithm were used to estimate the logistic regression parameters using the maximum likelihood method, which was realized by Newton method. The calculated iterative scheme of the maximum likelihood method is given. Based on the results of experimental data, estimates are obtained that are used to compare with the results obtained by the Bayesian approach. A special case is considered – the Bayesian estimation of the shift parameter of the logistic curve for a given value of the scale parameter, and also a general case – the estimation of the shift and scale parameters. A normal distribution is used as a priori distribution for the shift parameter, and a classical (two-parameter) gamma distribution is used for the scale parameter. Point estimation of logistic regression parameters was carried out by the modal value of a posteriori distribution. Using the modal rather than the mean value, made it possible to significantly reduce the amount of calculations due to the fact that in this case there is no need to calculate the integral probability in the denominator of the Bayesian formula.

The Impact of Automobile Recall on Consumers’ Purchase Decision Making Research

The automobile recall has been implemented in China for more than 10 years, the recall process has entered a mature stage and the recall rate also tends to be stable. However, the automobile recall, quite common in abroad, has a serious impact on the consumer’s purchase decision in the country. In this paper, firstly, analyze the data of automobile recall in the past 10 years; then using the Bayesian formula, to study the possibility change of consumer’s purchase decision under the condition of the automobile recall or not, and to study the impact of the recall on the final purchase decision making under consideration of performance and price.

The Development and Application of Bayesian Networks Used in Data Mining Under Big Data

Big data era has arrived. With the rapid development of big data, the machine learning based on probability statistics has drawn great attention in recent years from both industry and academia. It also has great applications in the fields of the Internet, finance, natural language, biology, among which the Bayesian network has experienced significant development last year and has become a kind of important machine learning method. This paper summarizes the data mining concept principle and basic method of data mining under the background of big data. Since data mining has strong statistic characteristic, it can be integrated with the Bayesian networks naturally. Therefore, this paper starts with the Bayesian networks, stretching out the Bayesian formula, Bayesian network technology, as well as the advantages of the combination of data mining and the Bayesian networks. Finally, with the biomedicine as an example, it summarizes its development and applications.

Temporal Parameters of Spontaneous Speech in Forensic Speaker Identification in Case of Language Mismatch: Serbian as L1 and English as L2

The purpose of the research is to examine the possibility of forensic speaker identification if question and suspect sample are in different languages using temporal parameters (articulation rate, speaking rate, degree of hesitancy, percentage of pauses, average pause duration). The corpus includes 10 female native speakers of Serbian who are proficient in English. The parameters are tested using Bayesian likelihood ratio formula in 40 same-speaker and 360 different-speaker pairs, including estimation of error rates, equal error rates and Overall Likelihood Ratio. One-way ANOVA is performed to determine whether inter-speaker variability is higher than intra- speaker variability across languages. The most successful discriminant is degree of hesitancy with ER of 42.5%/28%, (EER: 33%), followed by average pause duration with ER 35%/45.56%, (EER: 40%). Although the research features a closed-set comparison, which is not very common in forensic reality, the results are still relevant for forensic phoneticians working on criminal cases or as expert witnesses. This study pioneers in forensically comparing Serbian and English as well as in forensically testing temporal parameters on bilingual speakers. Further research should focus on comparing two stress-timed or two syllable-timed languages to test whether they will be more comparable in terms of temporal aspects of speech.

Bayesian seismic multi-scale inversion in complex Laplace mixed domains

Seismic inversion performed in the time or frequency domain cannot always recover the long-wavelength background of subsurface parameters due to the lack of low-frequency seismic records. Since the low-frequency response becomes much richer in the Laplace mixed domains, one novel Bayesian impedance inversion approach in the complex Laplace mixed domains is established in this study to solve the model dependency problem. The derivation of a Laplace mixed-domain formula of the Robinson convolution is the first step in our work. With this formula, the Laplace seismic spectrum, the wavelet spectrum and time-domain reflectivity are joined together. Next, to improve inversion stability, the object inversion function accompanied by the initial constraint of the linear increment model is launched under a Bayesian framework. The likelihood function and prior probability distribution can be combined together by Bayesian formula to calculate the posterior probability distribution of subsurface parameters. By achieving the optimal solution corresponding to maximum posterior probability distribution, the low-frequency background of subsurface parameters can be obtained successfully. Then, with the regularization constraint of estimated low frequency in the Laplace mixed domains, multi-scale Bayesian inversion in the pure frequency domain is exploited to obtain the absolute model parameters. The effectiveness, anti-noise capability and lateral continuity of Laplace mixed-domain inversion are illustrated by synthetic tests. Furthermore, one field case in the east of China is discussed carefully with different input frequency components and different inversion algorithms. This provides adequate proof to illustrate the reliability improvement in low-frequency estimation and resolution enhancement of subsurface parameters, in comparison with conventional Bayesian inversion in the frequency domain.

An improved CSMA/CA algorithm based on WSNs of the drug control system

A new improved CSMA/CA algorithm for wireless sensor networks (WSNs) is proposed in this paper to save energy and prolong the life cycle of WSNs. This algorithm is combined with the artificial neural network and Bayesian algorithm according to the practical applications of the drug control system of the Internet of Things. The algorithm is divided into two parts: first, the artificial neural network algorithm is used to estimate the data of WSNs, the results are the reference for the conversion of routing node frequency; second, by using the Bayesian formula, valuation method, and the CSMA/CA’s collision detection mechanism, the algorithm adjusts the frequency of the routing node and the relevant node frequency to establish the normal communication of packets sent by nodes and the aggregation node packets. In this way, it will reduce the collision detection and the back off time and avoid data packet duplication. The simulation tool-NS2 is used to configure an appropriate simulation scene for the experiment, which analyses and compares the received packet rate, the overall energy consumption of the network, and so on. The results demonstrate that the proposed algorithm ensures high energy efficiency and balanced energy consumption. Therefore the results show that the improved algorithm increases the efficiency, so that the network has the function of intelligent learning.