Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations

The pioneering work of Runge and Kutta a hundred years ago has ultimately led to suites of sophisticated numerical methods suitable for solving complex systems of deterministic ordinary differential equations. However, in many modelling situations, the appropriate representation is a stochastic differential equation and here numerical methods are much less sophisticated. In this paper a very general class of stochastic Runge-Kutta methods is presented and much more efficient classes of explicit methods than previous extant methods are constructed. In particular, a method of strong order 2 with a deterministic component based on the classical Runge-Kutta method is constructed and some numerical results are presented to demonstrate the efficacy of this approach.

Synthesis and characterisation of organo-beidellite

Today, there are growing concerns about the presence of environmental pollutants in many parts of the world. In particular, a lot of attention has been drawn to the levels of water and soil contaminants (de Paiva et al., 2008). The majority of these contaminants consist of NOCs (non-ionic organic compounds) and can enter our waterways through industrial activities, mining operations, crop and animal production, waste disposal and accidental leakage (de Paiva et al., 2008; Park et al., 2011). Therefore, there is an increased interest in the synthesis of new materials that can be used to remove potentially carcinogenic and toxic water contaminants. Smectite type organoclays are widely used in numerous applications, such as sorbent agents for environmental remediation, due to their unique properties (Jiunn-Fwu et al., 1990; Sheng et al., 1996; Zhou et al., 2007; Bektas et al., 2011; Park et al., 2011). This investigation focuses on beidellite (SBId-1), which belongs to the smectite clay family. Their properties include high cation exchange capacity (CEC), swelling properties, porous, high surface area and consequential strong adsorption/absorption capacity (Xi et al., 2007). However, swelling clays in general are not an effective sorbent agent in nature due to their hydrophilic properties. The hydrophilic properties of the clay can be changed to organophilic by intercalating a cationic surfactant. Many applications of organoclays are strongly dependent on their structural properties and hence, a better understanding of the configuration and structural change of organoclay is crucial. Organoclays were synthesised through ion exchange of 21CODTMA (MW: 392.5 g mol-1) and characterised using XRD and FTIR spectroscopy. This study investigates the structural and conformational changes of beidellite intercalated with octadecyltrimethylammonium bromide.

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.

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.

Thai family support (TFS) : working with families of young substance users in primary health care

Living with substance users negatively impacts upon family members in many ways, and distress is common. Despite these deep and wide-ranging impacts, supportive interventions for family members in their own right are rarely available. Thailand has substantial and growing problems with substance use, and there is very little support or family members of drug users, especially in community setting. The Thai Family Support (TFS) program was designed for implementation in primary health care units (PCUs) in Thailand. TFS was based on two approaches with existing empirical support in Western contexts—the 5-step method and CRAFT—with adaptations to a Thai setting that included integration with Buddhist practices. Its aims were to increase well-being of family members, reduce mental distress, improve family relationships between family members, and engage substance users in behaviour change. A small-scale randomised controlled trial on TFS with a Delayed Treatment control was conducted, with assessments at 8 weeks (Post 1) and 20-24 weeks (Post 2). Structured interviews with participants and PCU staff and an examination of five case studies augmented the quantitative results. Mixed Model Analyses were applied to quantitative outcomes, and thematic analysis was used for qualitative data. Thirty-six participants (18 in each of Immediate and Delayed Conditions) were recruited. A significant difference at Baseline between the two conditions was observed on the Thai GHQ-28 and Gender, but it was not possible to statistically control for these effects. There was a significant Time by Condition interaction on the Thai GHQ-28, WHOQOL-BREF-THAI and FAS, reflecting greater improvements in the Immediate condition by Post 1, but with the Delayed condition meeting or exceeding that effect by Post 2. On FES Cohesion and Conflict, there were falls across conditions at Post 2, but only Cohesion also showed a Time by Condition interaction, and that effect was consistent with a delayed impact of treatment. Overall, TFS by PCU staff in the Delayed Condition gave similar results to TFS conducted by the researcher, supporting the viability of its dissemination to standard health services. Qualitative data also confirmed the quantitative results. Most participants reported physiological and psychological improvements even though their substance-using relative did not change their drug use behaviour. After completing TFS, participants reported increased knowledge, group support and sharing feeling, having positive patient-professional relationship, having greater knowledge of substance abuse and social support. In particular, they changed their behaviour towards the substance user, resulting in improvements to family relationships. PCU staff gave similar responses on the efficacy of TFS, and saw it as feasible for routine use, although some implementation challenges were identified. The cultural adaptation and in particular the religious activities, were recognised by participants and PCU staff as an important component of TFS to support psychological health and well-being. Findings from this study showed the impact of substance use on family members and difficulties that they experienced when living with the substance users, resulting distresses and burden that may develop severe mental health disease. Drug use policies should be modified to support family members and response to their needs effectively for early prevention. This study also gave preliminary support for application of the TFS program in rural primary care settings and identified some policies that will be required for it to be disseminated more broadly.

Family-based programmes for preventing smoking by children and adolescents (Protocol)

This is the protocol for a review and there is no abstract. The objectives are as follows: To assess the effectiveness of interventions to help family members strengthen non-smoking attitudes and promote non-smoking by children and other family members by identifying and assessing RCT's that provide training, skills and support to family members to prevent smoking initiation. Hypothesis: This is an exploratory review, and only one hypothesis based on the literature review will be tested: "Interventions to help family members strengthen non-smoking attitudes and promote non-smoking by children and other family members are more effective in preventing children starting smoking than no intervention."

Effect of family relatedness on characteristics of estimated IBD probabilities in relation to precision of QTL estimates

Background: A random QTL effects model uses a function of probabilities that two alleles in the same or in different animals at a particular genomic position are identical by descent (IBD). Estimates of such IBD probabilities and therefore, modeling and estimating QTL variances, depend on marker polymorphism, strength of linkage and linkage disequilibrium of markers and QTL, and the relatedness of animals in the pedigree. The effect of relatedness of animals in a pedigree on IBD probabilities and their characteristics was examined in a simulation study. Results: The study based on nine multi-generational family structures, similar to a pedigree structure of a real dairy population, distinguished by an increased level of inbreeding from zero to 28 % across the studied population. Highest inbreeding level in the pedigree, connected with highest relatedness, was accompanied by highest IBD probabilities of two alleles at the same locus, and by lower relative variation coefficients. Profiles of correlation coefficients of IBD probabilities along the marked chromosomal segment with those at the true QTL position were steepest when the inbreeding coefficient in the pedigree was highest. Precision of estimated QTL location increased with increasing inbreeding and pedigree relatedness. A method to assess the optimum level of inbreeding for QTL detection is proposed, depending on population parameters. Conclusions: An increased overall relationship in a QTL mapping design has positive effects on precision of QTL position estimates. But the relationship of inbreeding level and the capacity for QTL detection depending on the recombination rate of QTL and adjacent informative marker is not linear. © 2010 Freyer et al., licensee BioMed Central Ltd.

A strong quantitative trait locus for wing length on chromosome 2 in a wild population of great reed warblers

Wing length is a key character for essential behaviours related to bird flight such as migration and foraging. In the present study, we initiate the search for the genes underlying wing length in birds by studying a long-distance migrant, the great reed warbler (Acrocephalus arundinaceus). In this species wing length is an evolutionary interesting trait with pronounced latitudinal gradient and sex-specific selection regimes in local populations. We performed a quantitative trait locus (QTL) scan for wing length in great reed warblers using phenotypic, genotypic, pedigree and linkage map data from our long-term study population in Sweden. We applied the linkage analysis mapping method implemented in GRIDQTL (a new web-based software) and detected a genome-wide significant QTL for wing length on chromosome 2, to our knowledge, the first detected QTL in wild birds. The QTL extended over 25 cM and accounted for a substantial part (37%) of the phenotypic variance of the trait. A genome scan for tarsus length (a bodysize-related trait) did not show any signal, implying that the wing-length QTL on chromosome 2 was not associated with body size. Our results provide a first important step into understanding the genetic architecture of avian wing length, and give opportunities to study the evolutionary dynamics of wing length at the locus level. This journal is© 2010 The Royal Society.

Effects of family environment on negative symptoms and quality of life of psychotic patients

This study tested the hypothesis that negative symptoms and quality of life for patients with functional psychoses are associated with family environment. Fifty-seven first-admission patients with functional psychoses were assessed at hospital admission for severity of psychopathology and premorbid adjustment. Relatives residing with patients rated the family environment at admission and one month after discharge on the Family Environment Scale. Patients made the same ratings after discharge. Six months later, patients were reassessed on severity of psychopathology, negative symptoms, and quality of life. Multiple regression analyses showed that higher levels of positive emotional expressiveness in the family predicted milder and fewer negative symptoms and better quality of life at follow-up. The prediction was statistically independent of the initial severity of psychopathology or premorbid adjustment

Power of QTL detection using association tests with family controls

The power of testing for a population-wide association between a biallelic quantitative trait locus and a linked biallelic marker locus is predicted both empirically and deterministically for several tests. The tests were based on the analysis of variance (ANOVA) and on a number of transmission disequilibrium tests (TDT). Deterministic power predictions made use of family information, and were functions of population parameters including linkage disequilibrium, allele frequencies, and recombination rate. Deterministic power predictions were very close to the empirical power from simulations in all scenarios considered in this study. The different TDTs had very similar power, intermediate between one-way and nested ANOVAs. One-way ANOVA was the only test that was not robust against spurious disequilibrium. Our general framework for predicting power deterministically can be used to predict power in other association tests. Deterministic power calculations are a powerful tool for researchers to plan and evaluate experiments and obviate the need for elaborate simulation studies.

Candidate gene analysis for quantitative traits using the transmission disequilibrium test : the example of the melanocortin 4-receptor in pigs

Population-wide associations between loci due to linkage disequilibrium can be used to map quantitative trait loci (QTL) with high resolution. However, spurious associations between markers and QTL can also arise as a consequence of population stratification. Statistical methods that cannot differentiate between loci associations due to linkage disequilibria from those caused in other ways can render false-positive results. The transmission-disequilibrium test (TDT) is a robust test for detecting QTL. The TDT exploits within-family associations that are not affected by population stratification. However, some TDTs are formulated in a rigid-form, with reduced potential applications. In this study we generalize TDT using mixed linear models to allow greater statistical flexibility. Allelic effects are estimated with two independent parameters: one exploiting the robust within-family information and the other the potentially biased between-family information. A significant difference between these two parameters can be used as evidence for spurious association. This methodology was then used to test the effects of the fourth melanocortin receptor (MC4R) on production traits in the pig. The new analyses supported the previously reported results; i.e., the studied polymorphism is either causal of in very strong linkage disequilibrium with the causal mutation, and provided no evidence for spurious association.

Alcoholism and family interaction

In i6 families, half of which had an alcoholic parent, both parents and an adolescent were videotaped interacting with each other. Mothers, fathers and the adolescent in each family viewed the videotaped interaction and completed ratings of themselves and the other two family members on levels of anxiety, involvement, dominance and friendliness. In families with an alcoholic parent, adolescents and their mothers rated family members as less anxious than did adolescents and mothers in families without a drinking problem. Also mothers in the alcoholic families rated family members as being more involved, and their ratings were higher than mothers in other families. Alcoholic families rated parent-adolescent interactions as more dominant and friendlier. At least in these videotaped interactions where alcohol was not being consumed, mothers in alcoholic families adopted a more positive view of family members than mothers in other families. In addition, possibly due to the efforts of fathers not to drink and memories of interactions when he was drunk, alcoholic families perceived their family interactions as more dominant and friendlier than families without an alcohol-related problem. [Schweitzer R, Wilks j, Callan vJ. Alcoholism and family interaction.

Use of concept maps in reforming teaching Family and Consumer Sciences in Kuwait

This paper addresses the professional development of Kuwaiti teachers in the use of concept maps to teach Family and Consumer Science. A key aim of the study was to evaluate the degree to which the use of concept maps would influence the way Kuwaiti teachers approach and teach Family and Consumer Studies (FCS) subjects and the degree to which concept maps empower students to critically identify and express their knowledge of the subject being taught. A case study methodology was adopted to follow the implementation of lessons using concept maps by four teachers of middle years. An analysis of the data revealed the positive impact that student-centred teaching tools can have on the reformation of traditional teaching environments. For all teachers, the primary strengths of using concept maps were the ability to generate student interest, to motivate student participation and to enhance student understanding of content. Although a case study design may limit the generalisation and comparative value of the study, the findings of this study remain important to the planning of future professional development programs and the use of concept maps within Kuwait’s FCS curriculum area.

Teaching and learning family and consumer sciences through K-W-L charts

While much of the control and many of the activities found in today’s classrooms have been placed in the hands of the learners and learning has become inquiry-based, there remains a need for teachers to use teaching tools that would facilitate this student-centered teaching process. This article identifies the K-W-L Chart as one such tool and follows a case study of four Kuwaiti ‘Family and Consumer Sciences’ teaching / learning events to evaluate their ability to enhance the learning outcomes of eight students. The research was designed from a qualitative, multi-tiered design approach and was assessed through a constant comparative method of data analysis of interview responses, classroom observations and worksheet-assessments. The results showed that the use of K-W-L Charts influenced the teachers and learners toward a more inquiry-based approach and facilitated a more student-centered and collaborative learning environment, raising the level of interest and the amount of personal input given by the students.