Expected value analysis of multiparameter systems

This paper proposes a new method, called EVA (expected value analysis), to analyze the effect of the uncertainties of the parameters on the output function of systems described as differential equation initial value problems. Starting from the model equations, the associated EVA differential equations are shown to be readily handled through available computer codes. A quantitative analysis of the EVA approximation is presented along with a comparison to local SSA (statistical sensitivity analysis) and global FAST (Fourier amplitude sensitivity test).     

Expected value analysis: Experimental considerations and convergence properties

In a companion paper, we developed a method, EVA (expected value analysis), to analyze the effect of the uncertainty of parameters in multiparameter systems described by ordinary differential equations. In this paper, we first interpret the output mean of EVA from an experimental point of view. Secondly, we discuss the convergence properties of EVA. The only approximation is the truncation of higher order correlation effects. A quantitative analysis of the approximations is presented, justifying the methodology.     

Expected Conditioning

A perturbation analysis based on probabilistic arguments isdeveloped for a range of problems in numerical linear algebra,including well-determined and over-determined linear systems.Condition matrices and condition numbers are determined forthe expected value of the actual condition number of a problem.These enable attainable lower and upper bounds on the expectedcondition properties of a matrix to be given, independent ofany particular linear system. These estimates are much morereliable than those derived from conventional norm conditionnumbers, and are shown to reveal features which the latter cannot. The expected condition analysis has desirable properties underscaling transformations which is not the case for the norm conditionanalysis. It is shown that an optimal (or natural) scaling canbe associated with any matrix which moreover is readily computed.This enables the equilibration of a matrix to be carried out.Because this process is a uniquely defined projection it alwaysenables the best conditioned of all the possible equilibratedmatrices to be determined.     

Alternative formulations of a combined trip generation,trip distribution,modal split,and trip assignment model

The traditional four-step model has been widely used in travel demand forecasting by considering trip generation, trip distribution, modal split and traffic assignment sequentially in a fixed order. However, this sequential approach suffers from the inconsistency among the level-of-service and flow values in each step of the procedure. In the last two decades, this problem has been addressed by many researchers who have sought to develop combined (or integrated) models that can consider travelers’ choice on different stages simultaneously and give consistent results. In this paper, alternative formulations, including mathematical programming (MP) formulation and variational inequality (VI) formulations, are provided for a combined travel demand model that integrates trip generation, trip distribution, modal split, and traffic assignment using the random utility theory framework. Thus, the proposed alternative formulations not only allow a systematic and consistent treatment of travel choice over different dimensions but also have behavioral richness. Qualitative properties of the formulations are also given to ensure the existence and uniqueness of the solution. Particularly, the model is analyzed for a special but useful case where the probabilistic travel choices are assumed to be a hierarchical logit model. Furthermore, a self-adaptive Goldstein–Levitin–Polyak (GLP) projection algorithm is adopted for solving this special case.     

Expected Frobenius numbers

Given a primitive positive integer vector a, the Frobenius number F(a) is the largest integer that cannot be represented as a non-negative integral combination of the coordinates of a. We show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bound.     

Expected present value of total dividends in a delayed claims risk model under stochastic interest rates

In this paper, a compound binomial risk model with a constant dividend barrier under stochastic interest rates is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. A system of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is derived and solved. Explicit results are obtained when the claim sizes are K_n distributed or the claim size distributions have finite support. Numerical results are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends.     

Expected heights in heaps

In this paper several recurrences and formulas are presented leading to upper and lower bounds, both logarithmic, for the expected height of a node in a heap. These bounds are of interest for algorithms that select thekth smallest element in a heap.     

最大期望收益与最小期望损失决策一致性条件

本首先建立了条件收益值与条件损失值之间的关系，其次在理想期望概念的基础上建立了期望收益与期望损失之间的关系。然后给出了依据最大期望收益准则与最小期望损失准则决策一致性条件。     

Expected utility for nonstochastic risk

Stochastic random phenomena considered in von Neumann–Morgenstern utility theory constitute only a part of all possible random phenomena (Kolmogorov, 1986). We show that any sequence of observed consequences generates a corresponding sequence of frequency distributions, which in general does not have a single limit point but a non-empty closed limit set in the space of finitely additive probabilities. This approach to randomness allows to generalize the expected utility theory in order to cover decision problems under nonstochastic random events. We derive the maxmin expected utility representation for preferences over closed sets of probability measures. The derivation is based on the axiom of preference for stochastic risk, i.e. the decision maker wishes to reduce a set of probability distributions to a single one. This complements Gilboa and Schmeidler’s (1989) consideration of the maxmin expected utility rule with objective treatment of multiple priors.     

Expected reliability of communication protocols

We consider the problem of sending a message from a sender $s$ to a receiver $r$ through an unreliable network by specifying in a protocol what each vertex is supposed to do if it receives the message from one of its neighbors. A protocol for routing a message in such a graph is finite if it never floods $r$ with an infinite number of copies of the message. The expected reliability of a given protocol is the probability that a message sent from $s$ reaches $r$ when the edges of the network fail independently with probability $1?p$.We discuss, for given networks, the properties of finite protocols with maximum expected reliability in the case when $p$ is close to 0 or 1, and we describe networks for which no one protocol is optimal for all values of $p$. In general, finding an optimal protocol for a given network and fixed probability is challenging and many open problems remain.     

Expected Distances in Distribution Problems

In many problems in distribution management it is necessary to take account of the expected distances that result from dispatching vehicles to meet customer demand. For example, in mathematical models for determining the optimal location of depots, the sum of radial distances (between customers and the depot), or the sum of the weighted distances, is used as a measure of the delivery "costs". Since actual delivery operations from the depot usually consist of truck-routes with each truck delivering to more than one customer at a time, it is important to know to what extent the above simplification is valid, namely to find a relationship between the actual route-distances and the sum of the radial distances.This paper makes use of an algorithm which plans optimal or near optimal routes to estimate this relationship by solving a large number of randomly generated problems. The discrepancies between the two methods are shown to be significant under certain circumstances.     

Asymptotics of the Expected Posterior

Let (X₁,...,X,...,X_n) be independently and identically distributed observations from an exponential family p equipped with a smooth prior density w on a real d-dimensional parameter . We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, ₀, admits an asymptotic expansion in terms of the Fisher information I(₀), the prior w, and their first two derivatives. The leading term of the expansion is of the form n^d/2c₁(d, 0) and the second order term is of the form n^4/2-1c₂(d, ₀>, w), with an error term that is o(n^d/2-1). We identify the functions c₁ and c₂ explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at ₀. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.     

Expected utility with nonlinear threshold

This paper axiomatizes a nonlinear threshold representation for expected utility, which generalizes a linear threshold representation established by Nakamura [10]. The advantage of a nonlinear threshold lies in its applicability to the monetary context. When the consequence space is the real line, interpreted as wealth, the nonlinear threshold can account for the intransitivity of the indifference relation over gambles, whereas the threshold vanishes in the linear threshold representation.     

Expected numbers at hitting times

We determine exactly the expected number of hamilton cycles in the random graph obtained by starting with n isolated vertices and adding edges at random until each vertex degree is at least two. This complements recent work of Cooper and Frieze. There are similar results concerning expected numbers, for example, of perfect matchings, spanning trees, hamilton paths, and directed hamilton cycles.     

差分—等级依赖效用模型研究

等级依赖效用模型在决策权重设置上存在理论缺陷;差分-等级依赖效用模型在有限理性假设下,考虑结果差分变化对决策权重的影响,提出基于差分的决策权重与概率权重的组合权重控制机制,可克服等级依赖效用模型的缺陷;针对差分-等级依赖效用模型的检验表明模型可很好解释拆分效应;差分-等级依赖效用模型保持了等级依赖效用模型的理论框架,可对人们的风险决策行为做出更好的解释和预测.     

Expected utility approximation and portfolio optimisation

Classical portfolio selection problems that optimise expected utility can usually not be solved in closed form. It is natural to approximate the utility function, and we investigate the accuracy of this approximation when using Taylor polynomials. In the important case of a Merton market and power utility we show analytically that increasing the order of the polynomial does not necessarily improve the approximation of the expected utility. The proofs use methods from the theory of parabolic second-order partial differential equations. All results are illustrated by numerical examples.     

Expected Sums of General Parking Functions

A (u₁, u₂, . . . )-parking function of length n is a sequence (x₁, x₂, . . . , x_n) whose order statistics (the sequence (x₍₁₎, x₍₂₎, . . . , x_(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x_(i) u_(i). The Gonarov polynomials g _n (x; a₀, a ₁, . . . , a _n-1) are polynomials biorthogonal to the linear functionals (a _i) Dⁱ, where (a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gonarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u _i=a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u_i+1 - u_i. Finally, we give analogues of our results for real-valued parking functions.AMS Subject Classification: 05A15, 05A19, 05A20, 05E35.