The transform likelihood ratio method for rare event simulation with heavy tails

We present a novel method, called the transform likelihood ratio (TLR) method, for estimation of rare event probabilities with heavy-tailed distributions. Via a simple transformation ( change of variables) technique the TLR method reduces the original rare event probability estimation with heavy tail distributions to an equivalent one with light tail distributions. Once this transformation has been established we estimate the rare event probability via importance sampling, using the classical exponential change of measure or the standard likelihood ratio change of measure. In the latter case the importance sampling distribution is chosen from the same parametric family as the transformed distribution. We estimate the optimal parameter vector of the importance sampling distribution using the cross-entropy method. We prove the polynomial complexity of the TLR method for certain heavy-tailed models and demonstrate numerically its high efficiency for various heavy-tailed models previously thought to be intractable. We also show that the TLR method can be viewed as a universal tool in the sense that not only it provides a unified view for heavy-tailed simulation but also can be efficiently used in simulation with light-tailed distributions. We present extensive simulation results which support the efficiency of the TLR method.

Improved algorithms for rare event simulation with heavy tails

The estimation of P(S-n > u) by simulation, where S, is the sum of independent. identically distributed random varibles Y-1,..., Y-n, is of importance in many applications. We propose two simulation estimators based upon the identity P(S-n > u) = nP(S, > u, M-n = Y-n), where M-n = max(Y-1,..., Y-n). One estimator uses importance sampling (for Y-n only), and the other uses conditional Monte Carlo conditioning upon Y1,..., Yn-1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

Heavy tails, importance sampling and cross-entropy

We consider the problem of estimating P(Yi + (...) + Y-n > x) by importance sampling when the Yi are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a toot for choosing good parameters in the importance sampling distribution; in doing so, we use the asymptotic description that given P(Y-1 + (...) + Y-n > x), n - 1 of the Yi have distribution F and one the conditional distribution of Y given Y > x. We show in some specific parametric examples (Pareto and Weibull) how this leads to precise answers which, as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/l and GI/G/l queues are also discussed.

A tutorial on the cross-entropy method

The cross-entropy (CE) method is a new generic approach to combinatorial and multi-extremal optimization and rare event simulation. The purpose of this tutorial is to give a gentle introduction to the CE method. We present the CE methodology, the basic algorithm and its modifications, and discuss applications in combinatorial optimization and machine learning. combinatorial optimization

Dapsone-induced agranulocytosis. The role of xenobiotic-metabolizing enzymes demonstrated by a case report [Dapson-induzierte agranulozytose. Die rolle fremdstoffmetabolisierender enzyme am beispiel einer kasuistik]

A 34-year-old female patient with a three year history of generalized granuloma annulare was treated systemically with dapsone (DADPS). Six weeks after the onset of treatment, the patient developed an extensive tonsillitis of the base of the tongue with fever and malaise. Routine laboratory work showed a leukocytopenia with agranulocytosis. Further investigation revealed a marked decrease of the enzyme activity of N-acetyltransferase 2, which plays an important role in dapsone metabolism. Treatment included the cessation of dapsone, antibiotic coverage, and G-CSF leading to the rapid improvement of symptoms and normalization of leukocyte counts. Dapsone-induced angina agranulocytotica is a rare event and is interpreted as an idiosyncratic reaction. Depending on genetic polymorphisms of various enzymes, dapsone can be metabolized to immunologically or toxicologically relevant intermediates. Because of the risk of severe hematologic reactions, dapsone should only be employed for solid indications and with appropriate monitoring.

A rare earth element study of complex zircons from early Archaean Amitsoq gneisses, Godthåbsfjord, south-west Greenland

We present high spatial resolution ion-microprobe rare earth element (REE) data for discrete growth phases of complex polyphase zircons from early Archaean Amitsoq gneisses, outer Godthabsfjord, SW Greenland. In Matsuda diagrams, the two major growth phases, >3.8 Ga cores and ca. 3.65 Ga rims, have steep positive slopes from La to Lu, prominent positive Ce anomalies and negative Eu anomalies that are consistent with growth in a melt. Exceptions to this are non-cathodolurnmescent zircon developed between the cores and rims, sometimes truncating zoning in the cores, and late Archaean prismatic tip overgrowths, both of which exhibit flatter light REE (LREE) patterns and have small or no Eu anomaly, which we interpret as the result of metamorphism and/or small-degree, isolated partial melting. Our data support previous interpretations that the ca. 3.65 Ga zircon phase was generated in a melt, with the >3.8 Ga phase representing either original protolith zircons in a large degree partial melt or inherited zircons in an introduced magma. Regardless which of these two interpretations is correct for these, and similar, rocks in the outer GodthAbsfjord, the 3.65 Ga event will have profoundly affected isotopic systems and obscured beyond recognition any earlier igneous features such as cross-cutting relationships, which may only be assigned a minimum 3.65 Ga age. (C) 2003 Elsevier Science B.V. All rights reserved.

Policies for saving a rare Australian glider: economics and ecology

This paper considers the economics of conserving a species with mainly non-use value, the endangered mahogany glider. Three serial surveys of Brisbane residents provide data on the knowledge of respondents about the mahogany glider. The results supply information about the attitudes of respondents to the mahogany glider, to its conservation and relevant public policies, and about variations in these factors as the knowledge of participants of the mahogany glider alters. Similarly, data are provided and analysed about the willingness to pay of respondents to conserve the mahogany glider and how it changes. Population viability analysis is applied to estimate the required habitat area for a minimum viable population of the mahogany glider to ensure at least a 95% probability of its survival for 100 years. Places are identified in Queensland where the requisite minimum area of critical habitat can be conserved. Using the survey results as a basis, the likely willingness of groups of Australians to pay for the conservation of the mahogany glider is estimated and consequently their willingness to pay for the minimum required area of its habitat. Methods for estimating the cost of protecting this habitat are outlined. Australia-wide benefits are estimated to exceed the costs. Establishing a national park containing the minimum viable population of the mahogany glider is an appealing management option. This would also be beneficial in conserving other endangered wildlife species and ecosystems. Therefore, additional economic benefits to those estimated on account of the mahogany glider itself can be obtained. (C) 2004 Elsevier Ltd. All rights reserved.

Introducing uncertainty into pattern discovery in temporal event sequences

Pattern discovery in temporal event sequences is of great importance in many application domains, such as telecommunication network fault analysis. In reality, not every type of event has an accurate timestamp. Some of them, defined as inaccurate events may only have an interval as possible time of occurrence. The existence of inaccurate events may cause uncertainty in event ordering. The traditional support model cannot deal with this uncertainty, which would cause some interesting patterns to be missing. A new concept, precise support, is introduced to evaluate the probability of a pattern contained in a sequence. Based on this new metric, we define the uncertainty model and present an algorithm to discover interesting patterns in the sequence database that has one type of inaccurate event. In our model, the number of types of inaccurate events can be extended to k readily, however, at a cost of increasing computational complexity.