Nonlinear Properties of the Rice Statistical Distribution: Theory and Applications in Stochastic Data Analysis

The paper considers the theoretical basics and the specific mathematical techniques having been developed for solving the tasks of the stochastic data analysis within the Rice statistical model in which the output signal’s amplitude is composed as a sum of the sought-for initial value and a random Gaussian noise. The Rician signal’s characteristics such as the average value and the noise dispersion have been shown to depend upon the Rice distribution’s parameters nonlinearly what has become a prerequisite for the development of a new approach to the stochastic Rician data analysis implying the joint signal and noise accurate evaluation. The joint computing of the Rice distribution’s parameters allows efficient reconstruction of the signal’s in-formative component against the noise background. A meaningful advantage of the proposed approach consists in the absence of restrictions connected with any a priori suppositions inherent to the traditional techniques. The results of the numerical experiments are provided confirming the efficiency of the elaborated approach to stochastic data analysis within the Rice statistical model.

Stein方法在求解随机变量线性组合概率分布中的应用

给出一种基于改进后的Stein方程求解随机变量线性组合概率分布的证明方法.基于随机变量Z_(r)和Z_(r)的线性组合H=m^(-1 )Z_r+n^(-1),利用伽马分布Γ(r,1)的Stein特征,求得关于H的Stein方程;加入二次可微函数,并采用改进后的Stein方程,求得H_(+)和H_(-)概率密度函数及显示公式.

利用概率分布函数法优化配网发电车调派的策略探究

鉴于发电车对配网发电的重要性,设计合理的调派方案,保证发电车调派合理性十分必要。概率分布函数是概率论中用来描述随机变量取值概率的函数,通过对任务的概率分布进行分析,可更合理地分配资源。文章将围绕如何利用概率分布函数法优化配网发电车调派进行分析,提出相应策略,以增强配网发电车调派方案的合理性。

Solution of the Fuzzy Equation <i>A + X = B</i>Using the Method of Superimposition

Fuzzy equations were solved by using different standard methods. One of the well-known methods is the method of α-cut. The method of superimposition of sets has been used to define arithmetic operations of fuzzy numbers. In this article, it has been shown that the fuzzy equation , where A, X, B are fuzzy numbers can be solved by using the method of superimposition of sets. It has also been shown that the method gives same result as the method of α-cut.

Construction of <i>k</i>-Variate Survival Functions with Emphasis on the Case <i>k</i>= 3

The purpose of this paper is to present a general universal formula for k-variate survival functions for arbitrary k = 2, 3, ..., given all the univariate marginal survival functions. This universal form of k-variate probability distributions was obtained by means of “dependence functions” named “joiners” in the text. These joiners determine all the involved stochastic dependencies between the underlying random variables. However, in order that the presented formula (the form) represents a legitimate survival function, some necessary and sufficient conditions for the joiners had to be found. Basically, finding those conditions is the main task of this paper. This task was successfully performed for the case k = 2 and the main results for the case k = 3 were formulated as Theorem 1 and Theorem 2 in Section 4. Nevertheless, the hypothetical conditions valid for the general k ≥ 4 case were also formulated in Section 3 as the (very convincing) Hypothesis. As for the sufficient conditions for both the k = 3 and k ≥ 4 cases, the full generality was not achieved since two restrictions were imposed. Firstly, we limited ourselves to the, defined in the text, “continuous cases” (when the corresponding joint density exists and is continuous), and secondly we consider positive stochastic dependencies only. Nevertheless, the class of the k-variate distributions which can be constructed is very wide. The presented method of construction by means of joiners can be considered competitive to the copula methodology. As it is suggested in the paper the possibility of building a common theory of both copulae and joiners is quite possible, and the joiners may play the role of tools within the theory of copulae, and vice versa copulae may, for example, be used for finding proper joiners. Another independent feature of the joiners methodology is the possibility of constructing many new stochastic processes including stationary and Markovian.

Dynamic Stability and Fuzzy Reliability Analysis of Toppling Perilous Rock Under Seismic Excitation

To predict the occurrence of the collapse disaster in toppling perilous rock under the action of bidirectional earthquakes,the dynamic stability and fuzzy reliability calculation method of toppling perilous rock under the action of bidirectional earthquakes is proposed.First,the mass viscoelasticity model is used to simulate two main control surfaces of toppling perilous rock,the seismic dynamic response model and motion equation of toppling perilous rock are established based on the D'Alembert principle,and the Newmark-β method is used to solve the dynamic motion equation.Then,the instability event of toppling perilous rock is considered a fuzzy event,the membership function expression of the stability coefficient of toppling perilous rock is determined based on the fuzzy failure criterion,the calculation equations of the toppling perilous rock dynamic stability coefficient and fuzzy reliability are established,and the fuzzy reliability evaluation method based on the probability distribution of reliability is proposed.Finally,the influence of different superposition modes of seismic excitation on the fuzzy reliability of toppling perilous rock is analyzed.The calculation results of toppling perilous rock in the engineering case show that the fuzzy reliability calculated after considering the fuzzy failure criterion is reduced by 10.73% to 25.66% compared with the classical reliability.Considering the bidirectional seismic excitation,the fuzzy reliability of toppling perilous rock is reduced by 5.46% to 14.89%.Compared with using the acceleration peak time encounter mode to superpose the seismic excitation,the fuzzy reliability of toppling perilous rock is reduced by 3.4% when the maximum action effect time encounter mode is adopted.

轨道不平顺小样本全信息表达的联合模拟方法

针对传统方法需要生成大量轨道不平顺样本才能涵盖所有随机特征,但样本数多又会导致计算效率低的问题,本文提出了一种自适应采样法(ASM)与改进的第二类随机谐和函数法(MSHF)结合的轨道不平顺模拟的联合方法。ASM自适应选取符合轨道不平顺功率谱密度函数(PSD)谱概率分布的高代表性样本点,从而减少样本数量;之后在SHF中添加一类符合谱概率分布的独立变量进行改进,通过将ASM选取的样本点代入新增变量,实现了轨道随机不平顺的时域小样本全信息表达。以车轨桥耦合系统(TTBS)随机动力分析为例,与传统一次性采样和SHF方法相比,本文方法在提高精度的同时可以减少41.67%的计算时间。

类比教学法在“概率论与数理统计”课程中的应用——以连续型随机变量为例

随机变量按取值形式可分为两类,离散型随机变量和非离散型随机变量。连续型随机变量作为非离散型随机变量中的一类,其相关内容是“概率论与数理统计”课程中的重难点。利用类比教学的方式,通过将连续型随机变量与学生熟悉的离散型随机变量对比,分别将连续型随机变量的概率密度、函数的分布、条件概率密度与离散型随机变量的分布律、函数的分布、条件分布律做类比,使学生更容易理解掌握连续型随机变量概率密度函数,函数的分布和条件概率密度。

Monte Carlo method for evaluation of surface emission rate measurement uncertainty

The aim of this study is to evaluate the uncertainty of 2πα and 2πβ surface emission rates using the windowless multiwire proportional counter method.This study used the Monte Carlo method (MCM) to validate the conventional Guide to the Expression of Uncertainty in Measurement (GUM) method.A dead time measurement model for the two-source method was established based on the characteristics of a single-channel measurement system,and the voltage threshold correction factor measurement function was indirectly obtained by fitting the threshold correction curve.The uncertainty in the surface emission rate was calculated using the GUM method and the law of propagation of uncertainty.The MCM provided clear definitions for each input quantity and its uncertainty distribution,and the simulation training was realized with a complete and complex mathematical model.The results of the surface emission rate uncertainty evaluation for four radioactive plane sources using both methods showed the uncertainty’s consistency E_(n)<0.070 for the comparison of each source,and the uncertainty results of the GUM were all lower than those of the MCM.However,the MCM has a more objective evaluation process and can serve as a validation tool for GUM results.

基于Bootstrap方法的岩土体参数联合分布模型识别

小样本容量岩土体参数最优联合概率分布模型的识别是一个富有挑战性的问题。基于Bootstrap提出了小样本容量岩土体参数最优边缘分布函数和最优Copula函数识别方法。简要介绍了岩土体参数联合概率分布函数构造的Copula方法,采用AIC准则识别最优的边缘分布函数和Copula函数。将识别结果表示为不同备选边缘分布函数和Copula函数为最优边缘分布和最优Copula的权重系数集合,以基桩荷载-位移双曲线参数试验数据为例证明了所提方法的有效性。结果表明:基于小样本容量岩土体参数试验数据估计的样本均值、标准差和相关系数具有较大的离散性,这种离散性进一步导致了统计量AIC值存在较大变异性。提出的基于Bootstrap的最优边缘分布函数和最优Copula函数识别方法不仅可以有效地考虑统计量AIC值的变异性,而且能够综合地反映不同备选边缘分布函数和Copula函数为最优边缘分布和最优Copula函数的概率,为小样本容量岩土体参数最优边缘分布函数和最优Copula函数的识别提供了一条有效的途径。

岩石力学参数概率分布的信息熵推断

基于信息论的最大熵原理,直接用牛顿迭代法,由样本矩来推断岩石力学参数的概率密度函数,并且用精度较高的K-S检验法,从理论上证明所求的密度函数的正确性。算例表明,该方法避免了复杂的数值计算,可以满足岩石工程随机可靠性分析的要求。直接根据试验样本信息和统计方法推断,而不是先假定成经典的理论概率分布,具有更充分的数学和物理意义。由于概率分布采用指数形式而不是幂函数系多项式,避免了计算中的震荡等不稳定现象。

最小二乘法的创立及其思想方法

目的探讨最小二乘法的历史发展过程及其创立者的思想与方法。方法历史考察与数理分析。结果勒让德在先驱者解线性方程组的基础上,以整体的思想方法创立了最小二乘法;高斯由寻找随机误差函数为突破,以独特的概率思想导出了正态分布,详尽地阐述了最小二乘法的理论依据。结论两位数学大师异曲同工地谱写了数理统计学的新篇。相比之下,高斯把最小二乘法推进得更远、更深刻,这极大地促进了数理统计学的发展。

有限数据条件下边坡可靠度分析的Bootstrap方法

基于Bootstrap抽样技术提出了有限数据条件下边坡可靠度分析方法。简要介绍了传统的边坡可靠度分析方法。采用Bootstrap方法模拟了抗剪强度参数概率分布函数的统计不确定性。以无限边坡为例研究了抗剪强度分布参数和分布类型不确定性对边坡可靠度的影响规律。结果表明:基于有限数据估计的样本均值、样本标准差和AIC值具有较大的变异性,这种变异性进一步导致了抗剪强度参数概率分布函数存在明显的统计不确定性。在考虑抗剪强度参数概率分布函数的统计不确定性时,边坡可靠度指标应为具有一定置信度水平的置信区间,而不是传统可靠度分析中的固定值。边坡可靠度指标的置信区间变化范围随安全系数的增加而增大,同时考虑分布参数和分布类型不确定性计算的可靠度指标具有更大的变异性和更宽的置信区间变化范围。Bootstrap方法为有限数据条件下抗剪强度参数概率分布函数统计不确定性的模拟以及边坡可靠度的评估提供了一条有效的途径。

万州区滑坡滑带土抗剪强度参数概率分布拟合及其优化

通过对三峡库区万州区几十处滑坡勘察、设计和施工资料的综合分析,进行了滑带土抗剪强度参数(粘聚力c 和内摩擦角?)的统计分析,利用χ2 检验法进行拟合,从而得出了c 和? 值的概率密度函数以及它们的概率分布形式:c 值服从对数正态分布,而? 值服从正态分布;以此为先验函数,利用Bayes 方法对万州区关塘口滑坡滑带土的小样本参数c 和? 值进行优化。计算结果表明,优化后方差和变异系数分别下降了37.6% 和32%,因而可以提高滑坡防治工程的可靠性,从而达到滑坡优化设计的目的。

岩土参数概率分布推断的模糊BAYES方法探讨

探讨了如何在有限的少量样本条件下，利用已有的经验和试验资料确定岩土参数的概率分布。用模糊综合评判方法与ＢＡＹＥＳ理论相结合，给出由小样本试验数据确定岩土参数的概率分布。

中国大陆地区年最大平均风速的概率密度函数

根据中国大陆地区1951-2002年的实测风速数据资料,采用密度演化方法进行了各站点年最大平均风速的概率密度函数估计,给出了大陆地区年最大平均风速均值和标准差的等值线分布图。与常用的基于拟合优度检验的概率密度函数估计和统计量分析不同,所提方法不需要进行先验分布类型的假定,可以通过直接计算给出与基本数据经验分布函数符合得较好的概率密度函数与概率分布函数。对比研究表明,该方法给出的统计结果是可信的,具有工程实用参考价值。

极不充分开采地表移动和变形预计的概率密度函数法

通过理论分析和计算机模拟,在极不充分开采条件下,离散介质中碎块体的移动概率逐渐趋近于正态分布的概率密度函数.提出了采用概率密度函数法对极不充分开采条件下地表移动和变形预计的方法,预计结果与实测资料进行了对比,在预计参数选取比较准确的情况下,预计的相对中误差可以控制在±10%以内.解决了概率积分法预计极不充分开采地表移动和变形误差偏大的问题.

混响室的概率统计分析方法及其蒙特卡罗模拟

提出了一种研究混响室问题的理论分析方法,从理想混响室的离散概率模型出发,采用蒙特卡罗模拟来获得室内电磁场量及其相关物理量的概率统计特性。运用该方法获得了混响室内若干场量(均为归一化值)的概率密度函数、室内一段均匀无耗单导体传输线的负载平均耦合截面及负载电流响应模值的概率分布,所得结果与已有文献给出的结果(包括由解析公式计算或测量所得的结果)吻合良好。由于蒙特卡罗方法的普适性,采用该方法可以获得任意待分析参量的全部概率统计特性(只要样本数足够大)。同时该方法无需针对具体混响室进行,只要在其测试区域能较好地与理想混响室的基本概率模型相吻合,就能给出较精确的计算结果,且其计算量远远小于基于全波分析的蒙特卡罗方法。

基于集合预报的四川夏季强降水订正试验

四川历来多暴雨洪涝,然而其发生具有很多不确定的因素,预报难度很大。从观测与模式预报的累积概率密度函数角度出发,利用2007—2012年6—8月中国降水观测格点资料和2012—2013年6—8月ECMWF模式集合预报资料,探索了一种提高四川地区强降水预报准确率的方法——概率阈值订正法,并运用该方法对2012年6—8月盆地东部的降水过程进行批量试验。试验结果表明:订正后的模式预报相比订正前的预报而言,不仅强降水落区更接近实况,而且较大程度地延长了预报时效,能提前6~7天给出强降水过程的警示信息,经过订正后显著提升了ECMWF模式的降水预报水平。

概率法储量计算在CLFS项目中的应用

采用蒙特卡洛模型进行储量计算,承认储量参数具有不确定性。鉴于各地质参数本身的随机性特点,将其视为以不同概率在实数域上随机取值的随机变量,较好地掌握参数的取值范围。同时,结合CLFS项目储量计算实例,按照SPE标准,详细分析了具有较大不确定性参数的取值方法,包括含油面积、孔隙度、有效厚度、饱和度和原油体积系数等多个储量参数。确定了各个参数的取值范围,求出了各个参数的P10,P50和P90值。最终的储量结果是一个储量范围,涵盖了从保守到乐观的估值范围,从而更准确地把握了油田储量状况,为油田的勘探开发决策提供了更可靠的依据,降低了油气勘探开发过程中决策的风险。