Two local dissimilarity measures for weighted graphs with application to protein interaction networks

We extend the Czekanowski-Dice dissimilarity measure, classically used to cluster the vertices of unweighted graphs, to weighted ones. The first proposed formula corresponds to edges weighted by a probability of existence. The second one is adapted to edges weighted by intensity or strength. We show on simulated graphs that the class identification process is improved by computing weighted compared to unweighted edges. Finally, an application to a drosophila protein network illustrates the fact that using these new formulas improves the ’biological accuracy’ of partitioning.     

Boole''s conditions of possible experience and reasoning under uncertainty

Consider a set of logical sentences together with probabilities that they are true. These probabilities must satisfy certain conditions for this system to be consistent. It is shown that an analytical form of these conditions can be obtained by enumerating the extreme rays of a polyhedron. We also consider the cases when (i) intervals of probabilities are given, instead of single values; and (ii) best lower and upper bounds on the probability of an additional logical sentence to be true are sought. Enumeration of vertices and extreme rays is used. Each vertex defines a finear expression and the maximum (minimum) of these defines a best possible lower (upper) bound on the probability of the additional logical sentence to be true. Each extreme ray leads to a constraint on the probabilities assigned to the initial set of logical sentences. Redundancy in these expressions is studied. Illustrations are provided in the domain of reasoning under uncertainty.     

随机利率下双分红的变保费复合帕斯卡模型

利润最大化风险最小化是保险公司运营所追求的目标,破产概率为公司进行风险决策提供了依据。本文基于随机利率环境下,保费随公司盈余水平调整的双分红复合帕斯卡模型,研究了股份制保险公司的有限时间破产概率。我们证明了公司盈余过程的齐次马氏性,得到了有限时间破产概率的计算方法,最后给出了具体算例。     

具有相依结构离散时间模型破产概率的上界

研究两类具有相依结构的离散时间风险模型的破产概率问题.其中,索赔和利率过程假设为2个不同的自回归移动平均模型.利用更新递归技巧,首先得到了该模型下破产概率所满足的递归方程.然后,根据该递归方程得到了破产概率的上界估计.最后对两类风险模型的破产概率的上界进行了比较.     

A note on the sum of uniform random variables

An inductive procedure is used to obtain distributions and probability densities for the sum S_n of independent, non-equally uniform random variables. Some known results are then shown to follow immediately as special cases. Under the assumption of equally uniform random variables some new formulas are obtained for probabilities and means related to S_n. Finally, some new recursive formulas involving distributions are derived.     

Threshold probability of non-terminal type in finite horizon Markov decision processes

We consider a class of problems concerned with maximizing probabilities, given stage-wise targets, which generalizes the standard threshold probability problem in Markov decision processes. The objective function is the probability that, at all stages, the associatively combined accumulation of rewards earned up to that point takes its value in a specified stage-wise interval. It is shown that this class reduces to the case of the nonnegative-valued multiplicative criterion through an invariant imbedding technique. We derive a recursive formula for the optimal value function and an effective method for obtaining the optimal policies.     

On the steady-state queue size distribution of the discrete-timeGeo/G/1 queue with repeated customers

In this paper, we study the steady-state queue size distribution of the discrete-timeGeo/G/1 retrial queue. We derive analytic formulas for the probability generating function of the number of customers in the system in steady-state. It is shown that the stochastic decomposition law holds for theGeo/G/1 retrial queue. Recursive formulas for the steady-state probabilities are developed. Computations based on these recursive formulas are numerically stable because the recursions involve only nonnegative terms. Since the regularGeo/G/1 queue is a special case of theGeo/G/1 retrial queue, the recursive formulas can also be used to compute the steady-state queue size distribution of the regularGeo/G/1 queue. Furthermore, it is shown that a continuous-timeM/G/1 retrial queue can be approximated by a discrete-timeGeo/G/1 retrial queue by dividing the time into small intervals of equal length and the approximation approaches the exact when the length of the interval tends to zero. This relationship allows us to apply the recursive formulas derived in this paper to compute the approximate steady-state queue size distribution of the continuous-timeM/G/1 retrial queue and the regularM/G/1 queue.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0046415.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0105828.     

Convergence of multi-level iterative aggregation-disaggregation methods

This paper introduces an error propagation formula of a certain class of multi-level iterative aggregation-disaggregation (IAD) methods for numerical solutions of stationary probability vectors of discrete finite Markov chains. The formula can be used to investigate convergence by computing the spectral radius of the error propagation matrix for specific Markov chains. Numerical experiments indicate that the same type of the formula could be used for a wider class of the multi-level IAD methods. Using the formula we show that for given data there is no relation between convergence of two-level and of multi-level IAD methods.     

Exact waiting time and queue size distributions for equilibrium <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 queues with Pareto service

This paper solves the problem of finding exact formulas for the waiting time cdf and queue length distribution of first-in-first-out M/G/1 queues in equilibrium with Pareto service. The formulas derived are new and are obtained by directly inverting the relevant Pollaczek-Khinchin formula and involve single integrals of non-oscillating real valued functions along the positive real line. Tables of waiting time and queue length probabilities are provided for certain parameter values under heavy traffic conditions.      

Cardy's formula for some dependent percolation models

We prove Cardy's formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice  (with alternating updates of two sublattices) [7]; it may also be realized on the triangular lattice 𝕋 with flips when a site disagrees with six, five and sometimes four of its six neighbors. Received: 24 December 2001     

Applications of some formulas for finite Markov chains

We present some explicit matrix formulas for a finite state Markov chain. The first gives sums of probabilities along some general subsets of paths. Another formula yields the probability mass function (pmf) of the random variable which adds costs along subsets of paths. We then discuss how these formulas can be used to efficiently compute expected values of a function of the sum of costs along paths, as well as related applications. We conclude by describing a procedure allowing us to avoid using Monte Carlo simulation in stochastic approaches to solving some general boundary value problems. Instead, we show how to evaluate the relevant expected values exactly for discretizations of the original continuous problem.     

Convergence of multi-level iterative aggregation–disaggregation methods

This paper introduces an error propagation formula of a certain class of multi-level iterative aggregation–disaggregation (IAD) methods for numerical solutions of stationary probability vectors of discrete finite Markov chains. The formula can be used to investigate convergence by computing the spectral radius of the error propagation matrix for specific Markov chains. Numerical experiments indicate that the same type of the formula could be used for a wider class of the multi-level IAD methods. Using the formula we show that for given data there is no relation between convergence of two-level and of multi-level IAD methods.     

The Goodman–Nguyen relation within imprecise probability theory

The Goodman–Nguyen relation is a partial order generalising the implication (inclusion) relation to conditional events. As such, with precise probabilities it both induces an agreeing probability ordering and is a key tool in a certain common extension problem. Most previous work involving this relation is concerned with either conditional event algebras or precise probabilities. We investigate here its role within imprecise probability theory, first in the framework of conditional events and then proposing a generalisation of the Goodman–Nguyen relation to conditional gambles. It turns out that this relation induces an agreeing ordering on coherent or C-convex conditional imprecise previsions. In a standard inferential problem with conditional events, it lets us determine the natural extension, as well as an upper extension. With conditional gambles, it is useful in deriving a number of inferential inequalities.     

二维风险模型中Copula 相依下的破产概率

在本文中, 我们把Copula 连结函数用到二维的风险模型中, 考虑两个模型索赔额之间基于Copula 的相依关系. 首先对二维复合Poisson 模型给出了最早破产时刻定义下的生存概率满足的偏微分方程; 然后对二维的复合二项模型, 分别在连续型索赔额分布和离散型索赔额分布下给出了不同定义的生存概率和破产概率的递归公式, 并且特别选择了FGM Copula 连结函数, 给出了相应的结果; 另外在离散型分布下, 对于其Copula 函数的不唯一性进行了说明.     

Recursive methods for a multi-dimensional risk process with common shocks

In this paper, a multi-dimensional risk model with common shocks is studied. Using a simple probabilistic approach via observing the risk processes at claim instants, recursive integral formulas are developed for the survival probabilities as well as for a class of Gerber-Shiu expected discounted penalty functions that include the surplus levels at ruin. Under the assumption of exponential or mixed Erlang claims, the recursive integrals can be simplified to give recursive sums which are computationally more tractable. Numerical examples including an optimal capital allocation problem are also given towards the end.     

Intrinsic ordering,combinatorial numbers and reliability engineering

A new algorithm for evaluating the top event probability of large fault trees (FTs) is presented. This algorithm does not require any previous qualitative analysis of the FT. Indeed, its efficiency is independent of the FT logic, and it only depends on the number n of basic system components and on their failure probabilities. Our method provides exact lower and upper bounds on the top event probability by using new properties of the intrinsic order relation between binary strings. The intrinsic order enables one to select binary n  -tuples with large occurrence probabilities without necessity to evaluate them. This drastically reduces the complexity of the problem from exponential (2ⁿ$2^n$ binary n-tuples) to linear (n Boolean variables). Our algorithm is mainly based on a recursive formula for rapidly computing the sum of the occurrence probabilities of all binary n-tuples with weight m whose 1s are placed among the k right-most positions. This formula, as well as the balance between accuracy and computational cost, is closely related to the famous Pascal’s triangle.     

Expansion formulae for characteristics of cumulative cost in finite horizon production models

We consider the expected value and the tail probability of cumulative shortage and holding cost (i.e. the probability that cumulative cost is more than a certain value) in finite horizon production models. An exact expression is provided for the expected value of the cumulative cost for general production functions. This expression is then used to compute the optimal production rate when the production function is linear. An expansion formula (whose coefficients can be obtained recursively) is provided for the tail probability. Examples are given to illustrate the results.     

The recursive dimension formula for graded lie algebras and its applications

In this paper, we obtain a recursive dimension formula for all γ- graded Lie algebras 𝔏 = +_α∈γ𝔩_α with finite dimensional homogeneous sub-spaces, where T is a countable abelian semigroup satisfying a certain finiteness condition. We apply this dimension formula to several Lie algebras to obtain explicit and simple dimension formulas.     

Markovian start-up demonstration tests with rejection of units upon observing d failures

By conditioning on the time of the first failure, several results are derived for demonstration tests of the start-up reliability of equipment. The start-ups are assumed to follow a Markov chain. The equipment is accepted if k consecutive successful start-ups occur before d failures. We compute the probability of the demonstration test ending with acceptance of the unit and the expectation and variance of the number of start-ups until termination of the test. We also give recursive formulas for computing the probability distribution of the number of start-ups and the probability of acceptance or rejection of the equipment in a specified number of trials. The conditional distribution of the number of start-ups in the test given that the unit is accepted or rejected is obtained as a by-product. Examples are given to illustrate the usefulness of these results.     

Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties.