Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively. (C) 2000 Elsevier Science B.V. All rights reserved.

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.

Genome-wide search for markers associated with bovine spongiform encephalopathy

A genome-wide search for markers associated with BSE incidence was performed by using Transmission-Disequilibrium Tests (TDTs). Significant segregation distortion, i.e., unequal transmission probabilities of alleles within a locus, was found for three marker loci on Chromosomes (Chrs) 5, 10, and 20. Although TDTs are robust to false associations owing to hidden population substructures, it cannot distinguish segregation distortion caused by a true association between a marker and bovine spongiform encephalopathy (BSE) from a population-wide distortion. An interaction test and a segregation distortion analysis in half-sib controls were used to disentangle these two alternative hypotheses. None of the markers showed any significant interaction between allele transmission rates and disease status, and only the marker on Chr 10 showed a significant segregation distortion in control individuals. Nevertheless, the control group may have been a mixture of resistant and susceptible but unchallenged individuals. When new genotypes were generated in the vicinity of these three markers, evidence for an association with BSE was confirmed for the locus on Chr 5.

Integrating collaborative filtering and matching-based search for product recommendation

Currently, recommender systems (RS) have been widely applied in many commercial e-commerce sites to help users deal with the information overload problem. Recommender systems provide personalized recommendations to users and thus help them in making good decisions about which product to buy from the vast number of product choices available to them. Many of the current recommender systems are developed for simple and frequently purchased products like books and videos, by using collaborative-filtering and content-based recommender system approaches. These approaches are not suitable for recommending luxurious and infrequently purchased products as they rely on a large amount of ratings data that is not usually available for such products. This research aims to explore novel approaches for recommending infrequently purchased products by exploiting user generated content such as user reviews and product click streams data. From reviews on products given by the previous users, association rules between product attributes are extracted using an association rule mining technique. Furthermore, from product click streams data, user profiles are generated using the proposed user profiling approach. Two recommendation approaches are proposed based on the knowledge extracted from these resources. The first approach is developed by formulating a new query from the initial query given by the target user, by expanding the query with the suitable association rules. In the second approach, a collaborative-filtering recommender system and search-based approaches are integrated within a hybrid system. In this hybrid system, user profiles are used to find the target user’s neighbour and the subsequent products viewed by them are then used to search for other relevant products. Experiments have been conducted on a real world dataset collected from one of the online car sale companies in Australia to evaluate the effectiveness of the proposed recommendation approaches. The experiment results show that user profiles generated from user click stream data and association rules generated from user reviews can improve recommendation accuracy. In addition, the experiment results also prove that the proposed query expansion and the hybrid collaborative filtering and search-based approaches perform better than the baseline approaches. Integrating the collaborative-filtering and search-based approaches has been challenging as this strategy has not been widely explored so far especially for recommending infrequently purchased products. Therefore, this research will provide a theoretical contribution to the recommender system field as a new technique of combining collaborative-filtering and search-based approaches will be developed. This research also contributes to a development of a new query expansion technique for infrequently purchased products recommendation. This research will also provide a practical contribution to the development of a prototype system for recommending cars.

Improved estimates of percolation and anisotropic permeability from 3-D X-ray microtomography using stochastic analyses and visualization

X-ray microtomography (micro-CT) with micron resolution enables new ways of characterizing microstructures and opens pathways for forward calculations of multiscale rock properties. A quantitative characterization of the microstructure is the first step in this challenge. We developed a new approach to extract scale-dependent characteristics of porosity, percolation, and anisotropic permeability from 3-D microstructural models of rocks. The Hoshen-Kopelman algorithm of percolation theory is employed for a standard percolation analysis. The anisotropy of permeability is calculated by means of the star volume distribution approach. The local porosity distribution and local percolation probability are obtained by using the local porosity theory. Additionally, the local anisotropy distribution is defined and analyzed through two empirical probability density functions, the isotropy index and the elongation index. For such a high-resolution data set, the typical data sizes of the CT images are on the order of gigabytes to tens of gigabytes; thus an extremely large number of calculations are required. To resolve this large memory problem parallelization in OpenMP was used to optimally harness the shared memory infrastructure on cache coherent Non-Uniform Memory Access architecture machines such as the iVEC SGI Altix 3700Bx2 Supercomputer. We see adequate visualization of the results as an important element in this first pioneering study.

Linkages of development and environment : in search of an integrated approach through sustainable development

Least developed countries (LDCs) are the primary victims of environmental changes, including present and future impacts of climate change. Environmental degradation poses a serious threat to the conservation and sustainable use of natural resources, thus hindering development in LDCs. Simultaneously, poverty is itself both a major cause and effect of global environmental problems. Against this backdrop, this essay argues that without recognition and protection of a collective right to development, genuine environmental protection will remain unachievable. Further, this essay submits that, particularly in the context of LDCs, the right to environment and the right to development are inseparable. Finally, this essay argues that the relationship between the right to environment and the right to development must fall within the paradigm of sustainable development if the promotion and protection of those rights are to be justified.

Are consumers fooled by discounts? An experimental test in a consumer search environment

In this article, we investigate experimentally whether people search optimally and how price promotions influence search behaviour. We implement a sequential search task with exogenous price dispersion in a baseline treatment and introduce discounts in two experimental treatments. We find that search behaviour is roughly consistent with optimal search but also observe some discount biases. If subjects do not know in advance where discounts are offered, the purchase probability is increased by 19 percentage points in shops with discounts, even after controlling for the benefit of the discount and for risk preferences. If consumers know in advance where discounts are given, then the bias is only weakly significant and much smaller (7 percentage points).

The search for freedom in extreme sports : a phenomenological exploration

Participation in extreme sports is continuing to grow, yet there is still little understanding of participant motivations in such sports. The purpose of this paper is to report on one aspect of motivation in extreme sports, the search for freedom. The study utilized a hermeneutic phenomenological methodology. Fifteen international extreme sport participants who participated in sports such as BASE jumping, big wave surfing, extreme mountaineering, extreme skiing, rope free climbing and waterfall kayaking were interviewed about their experience of participating in an extreme sport. Results reveal six elements of freedom: freedom from constraints, freedom as movement, freedom as letting go of the need for control, freedom as the release of fear, freedom as being at one, and finally freedom as choice and responsibility. The findings reveal that motivations in extreme sport do not simply mirror traditional images of risk taking and adrenaline and that motivations in extreme sports also include an exploration of the ways in which humans seek fundamental human values.

The search for migraine genes : an overview of current knowledge

Migraine is a complex familial condition that imparts a significant burden on society. There is evidence for a role of genetic factors in migraine, and elucidating the genetic basis of this disabling condition remains the focus of much research. In this review we discuss results of genetic studies to date, from the discovery of the role of neural ion channel gene mutations in familial hemiplegic migraine (FHM) to linkage analyses and candidate gene studies in the more common forms of migraine. The success of FHM regarding discovery of genetic defects associated with the disorder remains elusive in common migraine, and causative genes have not yet been identified. Thus we suggest additional approaches for analysing the genetic basis of this disorder. The continuing search for migraine genes may aid in a greater understanding of the mechanisms that underlie the disorder and potentially lead to significant diagnostic and therapeutic applications.

Integrating collaborative filtering and matching-based search for product recommendations

Currently, recommender systems (RS) have been widely applied in many commercial e-commerce sites to help users deal with the information overload problem. Recommender systems provide personalized recommendations to users and, thus, help in making good decisions about which product to buy from the vast amount of product choices. Many of the current recommender systems are developed for simple and frequently purchased products like books and videos, by using collaborative-filtering and content-based approaches. These approaches are not directly applicable for recommending infrequently purchased products such as cars and houses as it is difficult to collect a large number of ratings data from users for such products. Many of the ecommerce sites for infrequently purchased products are still using basic search-based techniques whereby the products that match with the attributes given in the target user’s query are retrieved and recommended. However, search-based recommenders cannot provide personalized recommendations. For different users, the recommendations will be the same if they provide the same query regardless of any difference in their interest. In this article, a simple user profiling approach is proposed to generate user’s preferences to product attributes (i.e., user profiles) based on user product click stream data. The user profiles can be used to find similarminded users (i.e., neighbours) accurately. Two recommendation approaches are proposed, namely Round- Robin fusion algorithm (CFRRobin) and Collaborative Filtering-based Aggregated Query algorithm (CFAgQuery), to generate personalized recommendations based on the user profiles. Instead of using the target user’s query to search for products as normal search based systems do, the CFRRobin technique uses the attributes of the products in which the target user’s neighbours have shown interest as queries to retrieve relevant products, and then recommends to the target user a list of products by merging and ranking the returned products using the Round Robin method. The CFAgQuery technique uses the attributes of the products that the user’s neighbours have shown interest in to derive an aggregated query, which is then used to retrieve products to recommend to the target user. Experiments conducted on a real e-commerce dataset show that both the proposed techniques CFRRobin and CFAgQuery perform better than the standard Collaborative Filtering and the Basic Search approaches, which are widely applied by the current e-commerce applications.

A concept-based retrieval method for entity-oriented search

Entity-oriented retrieval aims to return a list of relevant entities rather than documents to provide exact answers for user queries. The nature of entity-oriented retrieval requires identifying the semantic intent of user queries, i.e., understanding the semantic role of query terms and determining the semantic categories which indicate the class of target entities. Existing methods are not able to exploit the semantic intent by capturing the semantic relationship between terms in a query and in a document that contains entity related information. To improve the understanding of the semantic intent of user queries, we propose concept-based retrieval method that not only automatically identifies the semantic intent of user queries, i.e., Intent Type and Intent Modifier but introduces concepts represented by Wikipedia articles to user queries. We evaluate our proposed method on entity profile documents annotated by concepts from Wikipedia category and list structure. Empirical analysis reveals that the proposed method outperforms several state-of-the-art approaches.

Distinguishing between mean-field, moment dynamics and stochastic descriptions of birth-death-movement processes

Mathematical descriptions of birth–death–movement processes are often calibrated to measurements from cell biology experiments to quantify tissue growth rates. Here we describe and analyze a discrete model of a birth–death-movement process applied to a typical two–dimensional cell biology experiment. We present three different descriptions of the system: (i) a standard mean–field description which neglects correlation effects and clustering; (ii) a moment dynamics description which approximately incorporates correlation and clustering effects, and; (iii) averaged data from repeated discrete simulations which directly incorporates correlation and clustering effects. Comparing these three descriptions indicates that the mean–field and moment dynamics approaches are valid only for certain parameter regimes, and that both these descriptions fail to make accurate predictions of the system for sufficiently fast birth and death rates where the effects of spatial correlations and clustering are sufficiently strong. Without any method to distinguish between the parameter regimes where these three descriptions are valid, it is possible that either the mean–field or moment dynamics model could be calibrated to experimental data under inappropriate conditions, leading to errors in parameter estimation. In this work we demonstrate that a simple measurement of agent clustering and correlation, based on coordination number data, provides an indirect measure of agent correlation and clustering effects, and can therefore be used to make a distinction between the validity of the different descriptions of the birth–death–movement process.