Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory

There has been considerable research conducted over the last 20 years focused on predicting motor vehicle crashes on transportation facilities. The range of statistical models commonly applied includes binomial, Poisson, Poisson-gamma (or negative binomial), zero-inflated Poisson and negative binomial models (ZIP and ZINB), and multinomial probability models. Given the range of possible modeling approaches and the host of assumptions with each modeling approach, making an intelligent choice for modeling motor vehicle crash data is difficult. There is little discussion in the literature comparing different statistical modeling approaches, identifying which statistical models are most appropriate for modeling crash data, and providing a strong justification from basic crash principles. In the recent literature, it has been suggested that the motor vehicle crash process can successfully be modeled by assuming a dual-state data-generating process, which implies that entities (e.g., intersections, road segments, pedestrian crossings, etc.) exist in one of two states—perfectly safe and unsafe. As a result, the ZIP and ZINB are two models that have been applied to account for the preponderance of “excess” zeros frequently observed in crash count data. The objective of this study is to provide defensible guidance on how to appropriate model crash data. We first examine the motor vehicle crash process using theoretical principles and a basic understanding of the crash process. It is shown that the fundamental crash process follows a Bernoulli trial with unequal probability of independent events, also known as Poisson trials. We examine the evolution of statistical models as they apply to the motor vehicle crash process, and indicate how well they statistically approximate the crash process. We also present the theory behind dual-state process count models, and note why they have become popular for modeling crash data. A simulation experiment is then conducted to demonstrate how crash data give rise to “excess” zeros frequently observed in crash data. It is shown that the Poisson and other mixed probabilistic structures are approximations assumed for modeling the motor vehicle crash process. Furthermore, it is demonstrated that under certain (fairly common) circumstances excess zeros are observed—and that these circumstances arise from low exposure and/or inappropriate selection of time/space scales and not an underlying dual state process. In conclusion, carefully selecting the time/space scales for analysis, including an improved set of explanatory variables and/or unobserved heterogeneity effects in count regression models, or applying small-area statistical methods (observations with low exposure) represent the most defensible modeling approaches for datasets with a preponderance of zeros

Experimental evaluation of hotspot identification methods

Identifying crash “hotspots”, “blackspots”, “sites with promise”, or “high risk” locations is standard practice in departments of transportation throughout the US. The literature is replete with the development and discussion of statistical methods for hotspot identification (HSID). Theoretical derivations and empirical studies have been used to weigh the benefits of various HSID methods; however, a small number of studies have used controlled experiments to systematically assess various methods. Using experimentally derived simulated data—which are argued to be superior to empirical data, three hot spot identification methods observed in practice are evaluated: simple ranking, confidence interval, and Empirical Bayes. Using simulated data, sites with promise are known a priori, in contrast to empirical data where high risk sites are not known for certain. To conduct the evaluation, properties of observed crash data are used to generate simulated crash frequency distributions at hypothetical sites. A variety of factors is manipulated to simulate a host of ‘real world’ conditions. Various levels of confidence are explored, and false positives (identifying a safe site as high risk) and false negatives (identifying a high risk site as safe) are compared across methods. Finally, the effects of crash history duration in the three HSID approaches are assessed. The results illustrate that the Empirical Bayes technique significantly outperforms ranking and confidence interval techniques (with certain caveats). As found by others, false positives and negatives are inversely related. Three years of crash history appears, in general, to provide an appropriate crash history duration.

Computed success rates of various carrier phase integer estimation solutions and their comparison with statistical success rates

The success rate of carrier phase ambiguity resolution (AR) is the probability that the ambiguities are successfully fixed to their correct integer values. In existing works, an exact success rate formula for integer bootstrapping estimator has been used as a sharp lower bound for the integer least squares (ILS) success rate. Rigorous computation of success rate for the more general ILS solutions has been considered difficult, because of complexity of the ILS ambiguity pull-in region and computational load of the integration of the multivariate probability density function. Contributions of this work are twofold. First, the pull-in region mathematically expressed as the vertices of a polyhedron is represented by a multi-dimensional grid, at which the cumulative probability can be integrated with the multivariate normal cumulative density function (mvncdf) available in Matlab. The bivariate case is studied where the pull-region is usually defined as a hexagon and the probability is easily obtained using mvncdf at all the grid points within the convex polygon. Second, the paper compares the computed integer rounding and integer bootstrapping success rates, lower and upper bounds of the ILS success rates to the actual ILS AR success rates obtained from a 24 h GPS data set for a 21 km baseline. The results demonstrate that the upper bound probability of the ILS AR probability given in the existing literatures agrees with the actual ILS success rate well, although the success rate computed with integer bootstrapping method is a quite sharp approximation to the actual ILS success rate. The results also show that variations or uncertainty of the unit–weight variance estimates from epoch to epoch will affect the computed success rates from different methods significantly, thus deserving more attentions in order to obtain useful success probability predictions.

Statistical analyses of weigh-in-motion data for bridge live load development

This paper discusses the statistical analyses used to derive bridge live loads models for Hong Kong from a 10-year weigh-in-motion (WIM) data. The statistical concepts required and the terminologies adopted in the development of bridge live load models are introduced. This paper includes studies for representative vehicles from the large amount of WIM data in Hong Kong. Different load affecting parameters such as gross vehicle weights, axle weights, axle spacings, average daily number of trucks etc are first analyzed by various stochastic processes in order to obtain the mathematical distributions of these parameters. As a prerequisite to determine accurate bridge design loadings in Hong Kong, this study not only takes advantages of code formulation methods used internationally but also presents a new method for modelling collected WIM data using a statistical approach.

On the consistency of multiclass classification methods

Binary classification is a well studied special case of the classification problem. Statistical properties of binary classifiers, such as consistency, have been investigated in a variety of settings. Binary classification methods can be generalized in many ways to handle multiple classes. It turns out that one can lose consistency in generalizing a binary classification method to deal with multiple classes. We study a rich family of multiclass methods and provide a necessary and sufficient condition for their consistency. We illustrate our approach by applying it to some multiclass methods proposed in the literature.

New variational Bayesian approaches for statistical data mining : with applications to profiling and differentiating habitual consumption behaviour of customers in the wireless telecommunication industry

This thesis investigates profiling and differentiating customers through the use of statistical data mining techniques. The business application of our work centres on examining individuals’ seldomly studied yet critical consumption behaviour over an extensive time period within the context of the wireless telecommunication industry; consumption behaviour (as oppose to purchasing behaviour) is behaviour that has been performed so frequently that it become habitual and involves minimal intentions or decision making. Key variables investigated are the activity initialised timestamp and cell tower location as well as the activity type and usage quantity (e.g., voice call with duration in seconds); and the research focuses are on customers’ spatial and temporal usage behaviour. The main methodological emphasis is on the development of clustering models based on Gaussian mixture models (GMMs) which are fitted with the use of the recently developed variational Bayesian (VB) method. VB is an efficient deterministic alternative to the popular but computationally demandingMarkov chainMonte Carlo (MCMC) methods. The standard VBGMMalgorithm is extended by allowing component splitting such that it is robust to initial parameter choices and can automatically and efficiently determine the number of components. The new algorithm we propose allows more effective modelling of individuals’ highly heterogeneous and spiky spatial usage behaviour, or more generally human mobility patterns; the term spiky describes data patterns with large areas of low probability mixed with small areas of high probability. Customers are then characterised and segmented based on the fitted GMM which corresponds to how each of them uses the products/services spatially in their daily lives; this is essentially their likely lifestyle and occupational traits. Other significant research contributions include fitting GMMs using VB to circular data i.e., the temporal usage behaviour, and developing clustering algorithms suitable for high dimensional data based on the use of VB-GMM.

Statistical eye model for normal eyes

Purpose. To create a binocular statistical eye model based on previously measured ocular biometric data. Methods. Thirty-nine parameters were determined for a group of 127 healthy subjects (37 male, 90 female; 96.8% Caucasian) with an average age of 39.9 ± 12.2 years and spherical equivalent refraction of −0.98 ± 1.77 D. These parameters described the biometry of both eyes and the subjects' age. Missing parameters were complemented by data from a previously published study. After confirmation of the Gaussian shape of their distributions, these parameters were used to calculate their mean and covariance matrices. These matrices were then used to calculate a multivariate Gaussian distribution. From this, an amount of random biometric data could be generated, which were then randomly selected to create a realistic population of random eyes. Results. All parameters had Gaussian distributions, with the exception of the parameters that describe total refraction (i.e., three parameters per eye). After these non-Gaussian parameters were omitted from the model, the generated data were found to be statistically indistinguishable from the original data for the remaining 33 parameters (TOST [two one-sided t tests]; P < 0.01). Parameters derived from the generated data were also significantly indistinguishable from those calculated with the original data (P > 0.05). The only exception to this was the lens refractive index, for which the generated data had a significantly larger SD. Conclusions. A statistical eye model can describe the biometric variations found in a population and is a useful addition to the classic eye models.

A review of current statistical methodologies for in storage sampling and surveillance in the grains industry

Effective, statistically robust sampling and surveillance strategies form an integral component of large agricultural industries such as the grains industry. Intensive in-storage sampling is essential for pest detection, Integrated Pest Management (IPM), to determine grain quality and to satisfy importing nation’s biosecurity concerns, while surveillance over broad geographic regions ensures that biosecurity risks can be excluded, monitored, eradicated or contained within an area. In the grains industry, a number of qualitative and quantitative methodologies for surveillance and in-storage sampling have been considered. Primarily, research has focussed on developing statistical methodologies for in storage sampling strategies concentrating on detection of pest insects within a grain bulk, however, the need for effective and statistically defensible surveillance strategies has also been recognised. Interestingly, although surveillance and in storage sampling have typically been considered independently, many techniques and concepts are common between the two fields of research. This review aims to consider the development of statistically based in storage sampling and surveillance strategies and to identify methods that may be useful for both surveillance and in storage sampling. We discuss the utility of new quantitative and qualitative approaches, such as Bayesian statistics, fault trees and more traditional probabilistic methods and show how these methods may be used in both surveillance and in storage sampling systems.

Development and use of risk adjusted statistical process control tools for the monitoring and improvement of clinical outcomes in interventional cardiology

This thesis explored the development of statistical methods to support the monitoring and improvement in quality of treatment delivered to patients undergoing coronary angioplasty procedures. To achieve this goal, a suite of outcome measures was identified to characterise performance of the service, statistical tools were developed to monitor the various indicators and measures to strengthen governance processes were implemented and validated. Although this work focused on pursuit of these aims in the context of a an angioplasty service located at a single clinical site, development of the tools and techniques was undertaken mindful of the potential application to other clinical specialties and a wider, potentially national, scope.

Humanities and social scientific research methods in porn studies

Porn studies researchers in the humanities have tended to use different research methods from those in social sciences. There has been surprisingly little conversation between the groups about methodology. This article presents a basic introduction to textual analysis and statistical analysis, aiming to provide for all porn studies researchers a familiarity with these two quite distinct traditions of data analysis. Comparing these two approaches, the article suggests that social science approaches are often strongly reliable – but can sacrifice validity to this end. Textual analysis is much less reliable, but has the capacity to be strongly valid. Statistical methods tend to produce a picture of human beings as groups, in terms of what they have in common, whereas humanities approaches often seek out uniqueness. Social science approaches have asked a more limited range of questions than have the humanities. The article ends with a call to mix up the kinds of research methods that are applied to various objects of study.

Controlled Monte Carlo data generation for statistical damage identification employing Mahalanobis squared distance

The use of Mahalanobis squared distance–based novelty detection in statistical damage identification has become increasingly popular in recent years. The merit of the Mahalanobis squared distance–based method is that it is simple and requires low computational effort to enable the use of a higher dimensional damage-sensitive feature, which is generally more sensitive to structural changes. Mahalanobis squared distance–based damage identification is also believed to be one of the most suitable methods for modern sensing systems such as wireless sensors. Although possessing such advantages, this method is rather strict with the input requirement as it assumes the training data to be multivariate normal, which is not always available particularly at an early monitoring stage. As a consequence, it may result in an ill-conditioned training model with erroneous novelty detection and damage identification outcomes. To date, there appears to be no study on how to systematically cope with such practical issues especially in the context of a statistical damage identification problem. To address this need, this article proposes a controlled data generation scheme, which is based upon the Monte Carlo simulation methodology with the addition of several controlling and evaluation tools to assess the condition of output data. By evaluating the convergence of the data condition indices, the proposed scheme is able to determine the optimal setups for the data generation process and subsequently avoid unnecessarily excessive data. The efficacy of this scheme is demonstrated via applications to a benchmark structure data in the field.

Comparing methods for modelling spreading cell fronts

Spreading cell fronts play an essential role in many physiological processes. Classically, models of this process are based on the Fisher-Kolmogorov equation; however, such continuum representations are not always suitable as they do not explicitly represent behaviour at the level of individual cells. Additionally, many models examine only the large time asymptotic behaviour, where a travelling wave front with a constant speed has been established. Many experiments, such as a scratch assay, never display this asymptotic behaviour, and in these cases the transient behaviour must be taken into account. We examine the transient and asymptotic behaviour of moving cell fronts using techniques that go beyond the continuum approximation via a volume-excluding birth-migration process on a regular one-dimensional lattice. We approximate the averaged discrete results using three methods: (i) mean-field, (ii) pair-wise, and (iii) one-hole approximations. We discuss the performace of these methods, in comparison to the averaged discrete results, for a range of parameter space, examining both the transient and asymptotic behaviours. The one-hole approximation, based on techniques from statistical physics, is not capable of predicting transient behaviour but provides excellent agreement with the asymptotic behaviour of the averaged discrete results, provided that cells are proliferating fast enough relative to their rate of migration. The mean-field and pair-wise approximations give indistinguishable asymptotic results, which agree with the averaged discrete results when cells are migrating much more rapidly than they are proliferating. The pair-wise approximation performs better in the transient region than does the mean-field, despite having the same asymptotic behaviour. Our results show that each approximation only works in specific situations, thus we must be careful to use a suitable approximation for a given system, otherwise inaccurate predictions could be made.