Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed-form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This article provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox–Ingersoll–Ross and Ornstein–Uhlenbeck equations respectively.

The psychosocial factors influencing aggressive driving behaviour

Many drivers in highly motorised countries believe that aggressive driving is increasing. While the prevalence of the behaviour is difficult to reliably identify, the consequences of on-road aggression can be severe, with extreme cases resulting in property damage, injury and even death. This research program was undertaken to explore the nature of aggressive driving from within the framework of relevant psychological theory in order to enhance our understanding of the behaviour and to inform the development of relevant interventions. To guide the research a provisional ‘working’ definition of aggressive driving was proposed encapsulating the recurrent characteristics of the behaviour cited in the literature. The definition was: “aggressive driving is any on-road behaviour adopted by a driver that is intended to cause physical or psychological harm to another road user and is associated with feelings of frustration, anger or threat”. Two main theoretical perspectives informed the program of research. The first was Shinar’s (1998) frustration-aggression model, which identifies both the person-related and situational characteristics that contribute to aggressive driving, as well as proposing that aggressive behaviours can serve either an ‘instrumental’ or ‘hostile’ function. The second main perspective was Anderson and Bushman’s (2002) General Aggression Model. In contrast to Shinar’s model, the General Aggression Model reflects a broader perspective on human aggression that facilitates a more comprehensive examination of the emotional and cognitive aspects of aggressive behaviour. Study One (n = 48) examined aggressive driving behaviour from the perspective of young drivers as an at-risk group and involved conducting six focus groups, with eight participants in each. Qualitative analyses identified multiple situational and person-related factors that contribute to on-road aggression. Consistent with human aggression theory, examination of self-reported experiences of aggressive driving identified key psychological elements and processes that are experienced during on-road aggression. Participants cited several emotions experienced during an on-road incident: annoyance, frustration, anger, threat and excitement. Findings also suggest that off-road generated stress may transfer to the on-road environment, at times having severe consequences including crash involvement. Young drivers also appeared quick to experience negative attributions about the other driver, some having additional thoughts of taking action. Additionally, the results showed little difference between males and females in the severity of behavioural responses they were prepared to adopt, although females appeared more likely to displace their negative emotions. Following the self-reported on-road incident, evidence was also found of a post-event influence, with females being more likely to experience ongoing emotional effects after the event. This finding was evidenced by ruminating thoughts or distraction from tasks. However, the impact of such a post-event influence on later behaviours or interpersonal interactions appears to be minimal. Study Two involved the quantitative analysis of n = 926 surveys completed by a wide age range of drivers from across Queensland. The study aimed to explore the relationships between the theoretical components of aggressive driving that were identified in the literature review, and refined based on the findings of Study One. Regression analyses were used to examine participant emotional, cognitive and behavioural responses to two differing on-road scenarios whilst exploring the proposed theoretical framework. A number of socio-demographic, state and trait person-related variables such as age, pre-study emotions, trait aggression and problem-solving style were found to predict the likelihood of a negative emotional response such as frustration, anger, perceived threat, negative attributions and the likelihood of adopting either an instrumental or hostile behaviour in response to Scenarios One and Two. Complex relationships were found to exist between the variables, however, they were interpretable based on the literature review findings. Factor analysis revealed evidence supporting Shinar’s (1998) dichotomous description of on-road aggressive behaviours as being instrumental or hostile. The second stage of Study Two used logistic regression to examine the factors that predicted the potentially hostile aggressive drivers (n = 88) within the sample. These drivers were those who indicated a preparedness to engage in direct acts of interpersonal aggression on the road. Young, male drivers 17–24 years of age were more likely to be classified as potentially hostile aggressive drivers. Young drivers (17–24 years) also scored significantly higher than other drivers on all subscales of the Aggression Questionnaire (Buss & Perry, 1992) and on the ‘negative problem orientation’ and ‘impulsive careless style’ subscales of the Social Problem Solving Inventory – Revised (D’Zurilla, Nezu & Maydeu-Olivares, 2002). The potentially hostile aggressive drivers were also significantly more likely to engage in speeding and drink/drug driving behaviour. With regard to the emotional, cognitive and behavioural variables examined, the potentially hostile aggressive driver group also scored significantly higher than the ‘other driver’ group on most variables examined in the proposed theoretical framework. The variables contained in the framework of aggressive driving reliably distinguished potentially hostile aggressive drivers from other drivers (Nagalkerke R2 = .39). Study Three used a case study approach to conduct an in-depth examination of the psychosocial characteristics of n = 10 (9 males and 1 female) self-confessed hostile aggressive drivers. The self-confessed hostile aggressive drivers were aged 24–55 years of age. A large proportion of these drivers reported a Year 10 education or better and average–above average incomes. As a group, the drivers reported committing a number of speeding and unlicensed driving offences in the past three years and extensive histories of violations outside of this period. Considerable evidence was also found of exposure to a range of developmental risk factors for aggression that may have contributed to the driver’s on-road expression of aggression. These drivers scored significantly higher on the Aggression Questionnaire subscales and Social Problem Solving Inventory Revised subscales, ‘negative problem orientation’ and ‘impulsive/careless style’, than the general sample of drivers included in Study Two. The hostile aggressive driver also scored significantly higher on the Barrett Impulsivity Scale – 11 (Patton, Stanford & Barratt, 1995) measure of impulsivity than a male ‘inmate’, or female ‘general psychiatric’ comparison group. Using the Carlson Psychological Survey (Carlson, 1982), the self-confessed hostile aggressive drivers scored equal or higher scores than the comparison group of incarcerated individuals on the subscale measures of chemical abuse, thought disturbance, anti-social tendencies and self-depreciation. Using the Carlson Psychological Survey personality profiles, seven participants were profiled ‘markedly anti-social’, two were profiled ‘negative-explosive’ and one was profiled as ‘self-centred’. Qualitative analysis of the ten case study self-reports of on-road hostile aggression revealed a similar range of on-road situational factors to those identified in the literature review and Study One. Six of the case studies reported off-road generated stress that they believed contributed to the episodes of aggressive driving they recalled. Intense ‘anger’ or ‘rage’ were most frequently used to describe the emotions experienced in response to the perceived provocation. Less frequently ‘excitement’ and ‘fear’ were cited as relevant emotions. Notably, five of the case studies experienced difficulty articulating their emotions, suggesting emotional difficulties. Consistent with Study Two, these drivers reported negative attributions and most had thoughts of aggressive actions they would like to take. Similarly, these drivers adopted both instrumental and hostile aggressive behaviours during the self-reported incident. Nine participants showed little or no remorse for their behaviour and these drivers also appeared to exhibit low levels of personal insight. Interestingly, few incidents were brought to the attention of the authorities. Further, examination of the person-related characteristics of these drivers indicated that they may be more likely to have come from difficult or dysfunctional backgrounds and to have a history of anti-social behaviours on and off the road. The research program has several key theoretical implications. While many of the findings supported Shinar’s (1998) frustration-aggression model, two key areas of difference emerged. Firstly, aggressive driving behaviour does not always appear to be frustration driven, but can also be driven by feelings of excitation (consistent with the tenets of the General Aggression Model). Secondly, while the findings supported a distinction being made between instrumental and hostile aggressive behaviours, the characteristics of these two types of behaviours require more examination. For example, Shinar (1998) proposes that a driver will adopt an instrumental aggressive behaviour when their progress is impeded if it allows them to achieve their immediate goals (e.g. reaching their destination as quickly as possible); whereas they will engage in hostile aggressive behaviour if their path to their goal is blocked. However, the current results question this assertion, since many of the hostile aggressive drivers studied appeared prepared to engage in hostile acts irrespective of whether their goal was blocked or not. In fact, their behaviour appeared to be characterised by a preparedness to abandon their immediate goals (even if for a short period of time) in order to express their aggression. The use of the General Aggression Model enabled an examination of the three components of the ‘present internal state’ comprising emotions, cognitions and arousal and how these influence the likelihood of a person responding aggressively to an on-road situation. This provided a detailed insight into both the cognitive and emotional aspects of aggressive driving that have important implications for the design of relevant countermeasures. For example, the findings highlighted the potential value of utilising Cognitive Behavioural Therapy with aggressive drivers, particularly the more hostile offenders. Similarly, educational efforts need to be mindful of the way that person-related factors appear to influence one’s perception of another driver’s behaviour as aggressive or benign. Those drivers with a predisposition for aggression were more likely to perceive aggression or ‘wrong doing’ in an ambiguous on-road situation and respond with instrumental and/or hostile behaviour, highlighting the importance of perceptual processes in aggressive driving behaviour.

Stochastic models for regulatory networks of the genetic toggle switch

Bistability arises within a wide range of biological systems from the λ phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existence of bistability properties under various system conditions, these models cannot realize cell-to-cell fluctuations in genetic switching. However, there is a lag in the development of stochastic models for studying the impact of noise in bistable systems because of the lack of detailed knowledge of biochemical reactions, kinetic rates, and molecular numbers. In this work, we develop a previously undescribed general technique for developing quantitative stochastic models for large-scale genetic regulatory networks by introducing Poisson random variables into deterministic models described by ordinary differential equations. Two stochastic models have been proposed for the genetic toggle switch interfaced with either the SOS signaling pathway or a quorum-sensing signaling pathway, and we have successfully realized experimental results showing bimodal population distributions. Because the introduced stochastic models are based on widely used ordinary differential equation models, the success of this work suggests that this approach is a very promising one for studying noise in large-scale genetic regulatory networks.

Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Stochastic simulation of chemical reactions in spatially complex media

Discrete stochastic simulations, via techniques such as the Stochastic Simulation Algorithm (SSA) are a powerful tool for understanding the dynamics of chemical kinetics when there are low numbers of certain molecular species. However, an important constraint is the assumption of well-mixedness and homogeneity. In this paper, we show how to use Monte Carlo simulations to estimate an anomalous diffusion parameter that encapsulates the crowdedness of the spatial environment. We then use this parameter to replace the rate constants of bimolecular reactions by a time-dependent power law to produce an SSA valid in cases where anomalous diffusion occurs or the system is not well-mixed (ASSA). Simulations then show that ASSA can successfully predict the temporal dynamics of chemical kinetics in a spatially constrained environment.

Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.