Joint PP and PS AVO inversion based on Bayes theorem

Based on the Aki-Richards approximate equations for reflection coefficients and Bayes theorem, we developed an inversion method to estimate P- and S-wave velocity contrasts and density contrast from combined PP and PS data. This method assumes that the parameters satisfy a normal distribution and introduces the covariance matrix to describe the degree of correlation between the parameters and thus to improve the inversion stability. Then, we suppose that the parameter sequence is subject to the Cauchy distribution and employs another matrix Q to describe the parameter sequence sparseness to improve the inversion result resolution. Tests on both synthetic and real multi-component data prove that this method is valid, efficient, more stable, and more accurate compared to methods using PP data only.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Whole Perfect Vectors and Fermat’s Last Theorem

A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.

Effective and Efficient Feature Selection for Large-scale Data Using Bayes' Theorem

This paper proposes one method of feature selection by using Bayes＇ theorem. The purpose of the proposed method is to reduce the computational complexity and increase the classification accuracy of the selected feature subsets. The dependence between two attributes （binary） is determined based on the probabilities of their joint values that contribute to positive and negative classification decisions. If opposing sets of attribute values do not lead to opposing classification decisions （zero probability）, then the two attributes are considered independent of each other, otherwise dependent, and one of them can be removed and thus the number of attributes is reduced. The process must be repeated on all combinations of attributes. The paper also evaluates the approach by comparing it with existing feature selection algorithms over 8 datasets from University of California, Irvine （UCI） machine learning databases. The proposed method shows better results in terms of number of selected features, classification accuracy, and running time than most existing 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(0 X,0 Y)and release time(0 T),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(Δt)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 canIt 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.

Bayes Theorem Based Virtual Machine Scheduling for Optimal Energy Consumption

This paper proposes an algorithm for scheduling Virtual Machines(VM)with energy saving strategies in the physical servers of cloud data centers.Energy saving strategy along with a solution for productive resource utilizationfor VM deployment in cloud data centers is modeled by a combination of“VirtualMachine Scheduling using Bayes Theorem”algorithm(VMSBT)and Virtual Machine Migration(VMMIG)algorithm.It is shown that the overall data center’sconsumption of energy is minimized with a combination of VMSBT algorithmand Virtual Machine Migration(VMMIG)algorithm.Virtual machine migrationbetween the active physical servers in the data center is carried out at periodicalintervals as and when a physical server is identified to be under-utilized.In VMscheduling,the optimal data centers are clustered using Bayes Theorem and VMsare scheduled to appropriate data center using the selection policy that identifiesthe cluster with lesser energy consumption.Clustering using Bayes rule minimizesthe number of server choices for the selection policy.Application of Bayestheorem in clustering has enabled the proposed VMSBT algorithm to schedule thevirtual machines on to the physical server with minimal execution time.The proposedalgorithm is compared with other energy aware VM allocations algorithmsviz.“Ant-Colony”optimization-based(ACO)allocation scheme and“min-min”scheduling algorithm.The experimental simulation results prove that the proposedcombination of‘VMSBT’and‘VMMIG’algorithm outperforms othertwo strategies and is highly effective in scheduling VMs with reduced energy consumptionby utilizing the existing resources productively and by minimizing thenumber of active servers at any given point of time.

复杂系统的Bayes可靠性评估

讨论了复杂系统的Bayes可靠性评估。首先 ,对各种不同分布类型的单元进行Bayes可靠性分析 ;其次利用最大熵方法把单元可靠性信息进行综合 ,得到系统可靠性的验前分布 ;最后根据Bayes定理 ,利用系统可靠性试验信息来更新系统可靠性的验前分布得到系统可靠性的验后分布 ,依据此验后分布对系统可靠性进行评估。

基于Bayes理论的计算机辅助诊断系统在孤立性肺结节CT诊断中的应用

目的初步探讨基于Bayes理论的计算机辅助诊断(computer-aided diagnosis,CAD)系统在孤立性肺结节(solitary pulmonary nodule,SPN)CT诊断中的价值。方法依据Bayes理论先从352例SPN训练集(恶性135例,良性217例)中求出恶性SPN的验前比及各临床和CT表现的似然比,再运用VC++语言编制基于Bayes理论的CAD系统,用它计算每个SPN的恶性概率,并前瞻性地检验该系统在132例SPN测试集(恶性61例,良性71例)中的诊断效能,与2位高年资和2位低年资放射科医师常规阅片的表现作比较。结果成功构建基于Bayes理论的CAD系统,它诊断训练集SPN的敏感度、特异度和符合率分别为88.9%、93.1%、91.5%,诊断测试集SPN的敏感度、特异度、符合率、阳性预测值及阴性预测值分别为88.5%、85.9%、87.1%、84.4%、89.7%,其诊断符合率与高年资甲、乙医师比较无统计学差异(P>0.05),但高于低年资丙、丁医师(P<0.05)。结论基于Bayes理论的CAD系统可帮助医师尤其是低年资医师提高鉴别SPN良恶性质的能力,并在指导SPN的临床决策中有一定的参考作用。

基于Bayes法的C-12蛋白芯片联合检测判别函数的建立及临床意义分析

目的结合C-12蛋白芯片检测结果,采用Bayes法建立多肿瘤标志物判别函数,以明确肿瘤类型,提高肿瘤诊断准确率;比较多肿瘤标志物判别函数与C-12多肿瘤标志物蛋白芯片检测系统的差异及对肿瘤诊断的临床意义。方法用C-12蛋白芯片法检测2 177例恶性肿瘤患者(肺癌836例,肝癌318例,胰腺癌84例,胃癌123例,肠癌338例,乳腺癌206例,卵巢癌47例,子宫内膜癌28例,食管癌197例)和2 111例正常及患良性疾病的非肿瘤患者的12项常见肿瘤标志物,并结合临床资料进行回顾性研究。应用Bayes法建立肿瘤判别诊断函数,比较12种肿瘤标志物和判别诊断函数对肿瘤诊断的准确率差异。结果成功建立了多肿瘤标志物的三级诊断判别函数。一级判别函数诊断的特异度为82.11%,灵敏度为71.28%,准确率为83.97%,而C-12蛋白芯片法检测12项肿瘤标志物的特异度为70.11%,灵敏度为66.10%,准确率为68.07%。二级判别函数能显著提高10种常规肿瘤诊断的灵敏度、特异度和准确率。三级诊断判别函数对确定部分肿瘤的类型有意义。结论基于Bayes法建立的C-12蛋白芯片联合检测判别函数具有较高的灵敏度、特异度和准确率,对肿瘤的诊断具有重要临床价值。

二项分布Bayes鉴定试验次数的选择与分析

讨论了二项分布的可靠性Bayes鉴定试验方案问题。在试验样本量一定的条件下,充分利用验前信息弥补现场试验信息的不足,从平均后验风险角度制定了成败型系统可靠性鉴定试验的Bayes方案。密切结合工程试验,从生产和使用双方的实际风险的角度,研究分析了Bayes可靠性鉴定试验与评估的联系与差异。最后,与传统的鉴定试验方案进行了分析比较,明确了Bayes方法的使用范围,对于工程试验方案的选择具有实际的指导意义。

Bayes理论在河流水质模型参数识别中的应用

参数识别是水环境数学模型建模的重要步骤.在实际模拟过程中,往往难以获得理想的数据进行模型参数识别.充分利用研究者已有的经验,可在一定程度上减少模拟过程的风险.Bayes理论为把研究者的经验或前验信息纳入到水质模拟提供了一个定量手段.采用离散Bayes理论的基本方法,以国内某河段实际监测数据为基础,完成了模型的参数识别过程,并对识别结果进行了分析.验证结果表明,采用Bayes理论获得的参数识别结果能够达到模型验证的要求.

基于Bayes定理的分类规则研究

研究了利用Bayes定理发现分类规则的方法,用Bayes定理可以发现分类规则,然后用分类规则进行数据分类。结合实例针对概念性数据集及包含数值性属性和概念性属性的数据集两种情况进行讨论。通过实例说明Bayes定理是数据挖掘中一种有效的数据分类方法。

测量数据处理中的Bayes理论与最大熵方法

以Bayes理论和Jaynes的最大熵方法为基础,分析了测量数据概率空间的特点,提出了将Bayes理论与最大熵方法结合,利用最大熵方法求取Bayes理论中的先验分布和似然函数的测量数据PDF反演方法。实验结果证明此方法的反演结果优于单独运用Bayes或最大熵方法计算的反演结果。

用Bayes定理多指标筛查Wilson病家系

利用临床诊断Ｗｉｌｓｏｎ病（ＷＤ）的常用生化指标：血浆铜蓝蛋白、血清铜和尿铜，结合酯酶Ｄ多态性、ＷＤ发病风险与年龄分布的关系，应用微机化的Ｂａｙｅｓ定理对８个家系的１２例先证者同胞进行了三种状态──症状前患者、杂合子和正常个体的综合判别分析。家系１、家系６和家系８的筛查结果与已经完成的ＤＮＡ诊断结果相吻合。本研究可用于临床诊断、治疗ＷＤ以及遗传咨询。

估计构件疲劳极限的Bayes极小样本方法

从材料的S—N曲线与构件的S—N曲线形状相似原理出发,结合Bayes估计理论和加速寿命试验理论,提出一个估计构件疲劳极限的极小样本方法。在疲劳寿命服从对数正态函数分布的假设下,研究常幅和变幅载荷作用下由构件的疲劳寿命计算其疲劳极限的问题。两个算例表明,文中提出的方法利用以往积累的大量疲劳试验数据和现场试验数据,估计构件的疲劳极限,计算方便。计算结果与传统方法确定的疲劳极限的结果符合较好。

基于Bayes方法的诊断模型分析

文中证明了在Balyes框架下,当漂移参数γ服从有信息先验时,在相当广泛的统计模型中,数据删除模型(CDM)和均值漂移模型(MSOM)的参数估计不相等,几个数值例子验证了相应的结论.

基于Bayes理论的测试性数据融合评估技术

针对目前高可靠性产品测试性评估验证面临的样本量不足的问题,提出一种基于Bayes理论的内外场数据融合评估技术,并给出详细的内外场测试性数据收集方法及原则,利用某产品的试验数据对该方法进行了验证。该方法工程应用性强,可作为高可靠性产品测试性评估的依据。

极值分布参数、可靠寿命、可靠性的 Bayes 精确区间估计

在无信息先验分布下，根据定数及随机截尾样本数据，给出了极小值分布参数，可靠寿命、可靠性的Ｂａｙｅｓ精确区间估计，然后，用数值例说明了这些方法，并将它与基于ＢＬＵＥ，ＢＬＩＥ的方法进行了比较，指出了文中方法的优越性。

Bayes可靠性增长模型先验参数的确定

Gamma分布Gamma(a ,b) (其中a >0 ,b>0 ) ,通常用来作指数分布Bayes可靠性增长模型的先验分布 ,参数a和b的不同 ,将使可靠性增长试验的评估结果有很大的差异。根据可靠性增长和“弱”先验分布原理 ,通过对数值例的分析、比较 ,得出了先验参数的三种取法对可靠性评估结果的影响 ,并给出了指数分布Bayes可靠性增长模型的先验参数a和b的一种确定方法。