Fluctuations in and out of equilibrium: Thermalization, quantum measurements and Coulomb disorder

This purely theoretical thesis covers aspects of two contemporary research fields: the non-equilibrium dynamics in quantum systems and the electronic properties of three-dimensional topological insulators. In the first part we investigate the non-equilibrium dynamics in closed quantum systems. Thanks to recent technologies, especially from the field of ultracold quantum gases, it is possible to realize such systems in the laboratory. The focus is on the influence of hydrodynamic slow modes on the thermalization process. Generic systems in equilibrium, either classical or quantum, in equilibrium are described by thermodynamics. This is characterized by an ensemble of maximal entropy, but constrained by macroscopically conserved quantities. We will show that these conservation laws slow down thermalization and the final equilibrium state can be approached only algebraically in time. When the conservation laws are violated thermalization takes place exponential in time. In a different study we calculate probability distributions of projective quantum measurements. Newly developed quantum microscopes provide the opportunity to realize new measurement protocols which go far beyond the conventional measurements of correlation functions. The second part of this thesis is dedicated to a new class of materials known as three-dimensional topological insulators. Also here new experimental techniques have made it possible to fabricate these materials to a high enough quality that their topological nature is revealed. However, their transport properties are not fully understood yet. Motivated by unusual experimental results in the optical conductivity we have investigated the formation and thermal destruction of spatially localized electron- and hole-doped regions. These are caused by charged impurities which are introduced into the material in order to make the bulk insulating. Our theoretical results are in agreement with the experiment and can explain the results semi-quantitatively. Furthermore, we study emergent lengthscales in the bulk as well as close to the conducting surface.

Optimal Budget-Constrained Sample Allocation for Selection Decisions with Multiple Uncertain Attributes

A decision-maker, when faced with a limited and fixed budget to collect data in support of a multiple attribute selection decision, must decide how many samples to observe from each alternative and attribute. This allocation decision is of particular importance when the information gained leads to uncertain estimates of the attribute values as with sample data collected from observations such as measurements, experimental evaluations, or simulation runs. For example, when the U.S. Department of Homeland Security must decide upon a radiation detection system to acquire, a number of performance attributes are of interest and must be measured in order to characterize each of the considered systems. We identified and evaluated several approaches to incorporate the uncertainty in the attribute value estimates into a normative model for a multiple attribute selection decision. Assuming an additive multiple attribute value model, we demonstrated the idea of propagating the attribute value uncertainty and describing the decision values for each alternative as probability distributions. These distributions were used to select an alternative. With the goal of maximizing the probability of correct selection we developed and evaluated, under several different sets of assumptions, procedures to allocate the fixed experimental budget across the multiple attributes and alternatives. Through a series of simulation studies, we compared the performance of these allocation procedures to the simple, but common, allocation procedure that distributed the sample budget equally across the alternatives and attributes. We found the allocation procedures that were developed based on the inclusion of decision-maker knowledge, such as knowledge of the decision model, outperformed those that neglected such information. Beginning with general knowledge of the attribute values provided by Bayesian prior distributions, and updating this knowledge with each observed sample, the sequential allocation procedure performed particularly well. These observations demonstrate that managing projects focused on a selection decision so that the decision modeling and the experimental planning are done jointly, rather than in isolation, can improve the overall selection results.

SEMIPARAMETRIC METHODS IN THE ESTIMATION OF TAIL PROBABILITIES AND EXTREME QUANTILES

In quantitative risk analysis, the problem of estimating small threshold exceedance probabilities and extreme quantiles arise ubiquitously in bio-surveillance, economics, natural disaster insurance actuary, quality control schemes, etc. A useful way to make an assessment of extreme events is to estimate the probabilities of exceeding large threshold values and extreme quantiles judged by interested authorities. Such information regarding extremes serves as essential guidance to interested authorities in decision making processes. However, in such a context, data are usually skewed in nature, and the rarity of exceedance of large threshold implies large fluctuations in the distribution's upper tail, precisely where the accuracy is desired mostly. Extreme Value Theory (EVT) is a branch of statistics that characterizes the behavior of upper or lower tails of probability distributions. However, existing methods in EVT for the estimation of small threshold exceedance probabilities and extreme quantiles often lead to poor predictive performance in cases where the underlying sample is not large enough or does not contain values in the distribution's tail. In this dissertation, we shall be concerned with an out of sample semiparametric (SP) method for the estimation of small threshold probabilities and extreme quantiles. The proposed SP method for interval estimation calls for the fusion or integration of a given data sample with external computer generated independent samples. Since more data are used, real as well as artificial, under certain conditions the method produces relatively short yet reliable confidence intervals for small exceedance probabilities and extreme quantiles.

Probabilistic methods for seasonal forecasting in a changing climate: Cox-type regression models.

For climate risk management, cumulative distribution functions (CDFs) are an important source of information. They are ideally suited to compare probabilistic forecasts of primary (e.g. rainfall) or secondary data (e.g. crop yields). Summarised as CDFs, such forecasts allow an easy quantitative assessment of possible, alternative actions. Although the degree of uncertainty associated with CDF estimation could influence decisions, such information is rarely provided. Hence, we propose Cox-type regression models (CRMs) as a statistical framework for making inferences on CDFs in climate science. CRMs were designed for modelling probability distributions rather than just mean or median values. This makes the approach appealing for risk assessments where probabilities of extremes are often more informative than central tendency measures. CRMs are semi-parametric approaches originally designed for modelling risks arising from time-to-event data. Here we extend this original concept beyond time-dependent measures to other variables of interest. We also provide tools for estimating CDFs and surrounding uncertainty envelopes from empirical data. These statistical techniques intrinsically account for non-stationarities in time series that might be the result of climate change. This feature makes CRMs attractive candidates to investigate the feasibility of developing rigorous global circulation model (GCM)-CRM interfaces for provision of user-relevant forecasts. To demonstrate the applicability of CRMs, we present two examples for El Ni ? no/Southern Oscillation (ENSO)-based forecasts: the onset date of the wet season (Cairns, Australia) and total wet season rainfall (Quixeramobim, Brazil). This study emphasises the methodological aspects of CRMs rather than discussing merits or limitations of the ENSO-based predictors.

BIAS-CORRECTED MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS OF THE WEIGHTED LINDLEY DISTRIBUTION

This report discusses the calculation of analytic second-order bias techniques for the maximum likelihood estimates (for short, MLEs) of the unknown parameters of the distribution in quality and reliability analysis. It is well-known that the MLEs are widely used to estimate the unknown parameters of the probability distributions due to their various desirable properties; for example, the MLEs are asymptotically unbiased, consistent, and asymptotically normal. However, many of these properties depend on an extremely large sample sizes. Those properties, such as unbiasedness, may not be valid for small or even moderate sample sizes, which are more practical in real data applications. Therefore, some bias-corrected techniques for the MLEs are desired in practice, especially when the sample size is small. Two commonly used popular techniques to reduce the bias of the MLEs, are ‘preventive’ and ‘corrective’ approaches. They both can reduce the bias of the MLEs to order O(n−2), whereas the ‘preventive’ approach does not have an explicit closed form expression. Consequently, we mainly focus on the ‘corrective’ approach in this report. To illustrate the importance of the bias-correction in practice, we apply the bias-corrected method to two popular lifetime distributions: the inverse Lindley distribution and the weighted Lindley distribution. Numerical studies based on the two distributions show that the considered bias-corrected technique is highly recommended over other commonly used estimators without bias-correction. Therefore, special attention should be paid when we estimate the unknown parameters of the probability distributions under the scenario in which the sample size is small or moderate.

Probabilistic Security Constrained Fuzzy Power Flow Models

In restructured power systems, generation and commercialization activities became market activities, while transmission and distribution activities continue as regulated monopolies. As a result, the adequacy of transmission network should be evaluated independent of generation system. After introducing the constrained fuzzy power flow (CFPF) as a suitable tool to quantify the adequacy of transmission network to satisfy 'reasonable demands for the transmission of electricity' (as stated, for instance, at European Directive 2009/72/EC), the aim is now showing how this approach can be used in conjunction with probabilistic criteria in security analysis. In classical security analysis models of power systems are considered the composite system (generation plus transmission). The state of system components is usually modeled with probabilities and loads (and generation) are modeled by crisp numbers, probability distributions or fuzzy numbers. In the case of CFPF the component’s failure of the transmission network have been investigated. In this framework, probabilistic methods are used for failures modeling of the transmission system components and possibility models are used to deal with 'reasonable demands'. The enhanced version of the CFPF model is applied to an illustrative case.

Modelling frequency distributions of 5 minute-averaged solar radiation indexes using Beta probability functions

Five minute-averaged values of sky clearness, direct and diffuse indices, were used to model the frequency distributions of these variables in terms of optical air mass. From more than four years of solar radiation observations it was found that variations in the frequency distributions of the three indices of optical air mass for Botucatu, Brazil, are similar to those in other places, as published in the literature. The proposed models were obtained by linear combination of normalized Beta probability functions, using the observed distributions derived from three years of data. The versatility of these functions allows modelling of all three irradiance indexes to similar levels of accuracy. A comparison with the observed distributions obtained from one year of observations indicate that the models are able to reproduce the observed frequency distributions of all three indices at the 95% confidence level.

Probability distributed time delays: integrating spatial effects into temporal models

Background In order to provide insights into the complex biochemical processes inside a cell, modelling approaches must find a balance between achieving an adequate representation of the physical phenomena and keeping the associated computational cost within reasonable limits. This issue is particularly stressed when spatial inhomogeneities have a significant effect on system's behaviour. In such cases, a spatially-resolved stochastic method can better portray the biological reality, but the corresponding computer simulations can in turn be prohibitively expensive. Results We present a method that incorporates spatial information by means of tailored, probability distributed time-delays. These distributions can be directly obtained by single in silico or a suitable set of in vitro experiments and are subsequently fed into a delay stochastic simulation algorithm (DSSA), achieving a good compromise between computational costs and a much more accurate representation of spatial processes such as molecular diffusion and translocation between cell compartments. Additionally, we present a novel alternative approach based on delay differential equations (DDE) that can be used in scenarios of high molecular concentrations and low noise propagation. Conclusions Our proposed methodologies accurately capture and incorporate certain spatial processes into temporal stochastic and deterministic simulations, increasing their accuracy at low computational costs. This is of particular importance given that time spans of cellular processes are generally larger (possibly by several orders of magnitude) than those achievable by current spatially-resolved stochastic simulators. Hence, our methodology allows users to explore cellular scenarios under the effects of diffusion and stochasticity in time spans that were, until now, simply unfeasible. Our methodologies are supported by theoretical considerations on the different modelling regimes, i.e. spatial vs. delay-temporal, as indicated by the corresponding Master Equations and presented elsewhere.

Sampling stored-product insect pests : a comparison of four statistical sampling models for probability of pest detection

Background: Developing sampling strategies to target biological pests such as insects in stored grain is inherently difficult owing to species biology and behavioural characteristics. The design of robust sampling programmes should be based on an underlying statistical distribution that is sufficiently flexible to capture variations in the spatial distribution of the target species. Results: Comparisons are made of the accuracy of four probability-of-detection sampling models - the negative binomial model,1 the Poisson model,1 the double logarithmic model2 and the compound model3 - for detection of insects over a broad range of insect densities. Although the double log and negative binomial models performed well under specific conditions, it is shown that, of the four models examined, the compound model performed the best over a broad range of insect spatial distributions and densities. In particular, this model predicted well the number of samples required when insect density was high and clumped within experimental storages. Conclusions: This paper reinforces the need for effective sampling programs designed to detect insects over a broad range of spatial distributions. The compound model is robust over a broad range of insect densities and leads to substantial improvement in detection probabilities within highly variable systems such as grain storage.

Visualization of predictive distributions for discrete spatial-temporal log Cox processes approximated with MCMC

An important aspect of decision support systems involves applying sophisticated and flexible statistical models to real datasets and communicating these results to decision makers in interpretable ways. An important class of problem is the modelling of incidence such as fire, disease etc. Models of incidence known as point processes or Cox processes are particularly challenging as they are ‘doubly stochastic’ i.e. obtaining the probability mass function of incidents requires two integrals to be evaluated. Existing approaches to the problem either use simple models that obtain predictions using plug-in point estimates and do not distinguish between Cox processes and density estimation but do use sophisticated 3D visualization for interpretation. Alternatively other work employs sophisticated non-parametric Bayesian Cox process models, but do not use visualization to render interpretable complex spatial temporal forecasts. The contribution here is to fill this gap by inferring predictive distributions of Gaussian-log Cox processes and rendering them using state of the art 3D visualization techniques. This requires performing inference on an approximation of the model on a discretized grid of large scale and adapting an existing spatial-diurnal kernel to the log Gaussian Cox process context.

Delineating environmental envelopes to improve mapping of species distributions, via a hurdle model with CART &/or MaxEnT

Species distribution modelling (SDM) typically analyses species’ presence together with some form of absence information. Ideally absences comprise observations or are inferred from comprehensive sampling. When such information is not available, then pseudo-absences are often generated from the background locations within the study region of interest containing the presences, or else absence is implied through the comparison of presences to the whole study region, e.g. as is the case in Maximum Entropy (MaxEnt) or Poisson point process modelling. However, the choice of which absence information to include can be both challenging and highly influential on SDM predictions (e.g. Oksanen and Minchin, 2002). In practice, the use of pseudo- or implied absences often leads to an imbalance where absences far outnumber presences. This leaves analysis highly susceptible to ‘naughty-noughts’: absences that occur beyond the envelope of the species, which can exert strong influence on the model and its predictions (Austin and Meyers, 1996). Also known as ‘excess zeros’, naughty noughts can be estimated via an overall proportion in simple hurdle or mixture models (Martin et al., 2005). However, absences, especially those that occur beyond the species envelope, can often be more diverse than presences. Here we consider an extension to excess zero models. The two-staged approach first exploits the compartmentalisation provided by classification trees (CTs) (as in O’Leary, 2008) to identify multiple sources of naughty noughts and simultaneously delineate several species envelopes. Then SDMs can be fit separately within each envelope, and for this stage, we examine both CTs (as in Falk et al., 2014) and the popular MaxEnt (Elith et al., 2006). We introduce a wider range of model performance measures to improve treatment of naughty noughts in SDM. We retain an overall measure of model performance, the area under the curve (AUC) of the Receiver-Operating Curve (ROC), but focus on its constituent measures of false negative rate (FNR) and false positive rate (FPR), and how these relate to the threshold in the predicted probability of presence that delimits predicted presence from absence. We also propose error rates more relevant to users of predictions: false omission rate (FOR), the chance that a predicted absence corresponds to (and hence wastes) an observed presence, and the false discovery rate (FDR), reflecting those predicted (or potential) presences that correspond to absence. A high FDR may be desirable since it could help target future search efforts, whereas zero or low FOR is desirable since it indicates none of the (often valuable) presences have been ignored in the SDM. For illustration, we chose Bradypus variegatus, a species that has previously been published as an exemplar species for MaxEnt, proposed by Phillips et al. (2006). We used CTs to increasingly refine the species envelope, starting with the whole study region (E0), eliminating more and more potential naughty noughts (E1–E3). When combined with an SDM fit within the species envelope, the best CT SDM had similar AUC and FPR to the best MaxEnt SDM, but otherwise performed better. The FNR and FOR were greatly reduced, suggesting that CTs handle absences better. Interestingly, MaxEnt predictions showed low discriminatory performance, with the most common predicted probability of presence being in the same range (0.00-0.20) for both true absences and presences. In summary, this example shows that SDMs can be improved by introducing an initial hurdle to identify naughty noughts and partition the envelope before applying SDMs. This improvement was barely detectable via AUC and FPR yet visible in FOR, FNR, and the comparison of predicted probability of presence distribution for pres/absence.

A new family of multivariate heavy-tailed distributions with variable marginal amounts of tailweight: Application to robust clustering

We propose a family of multivariate heavy-tailed distributions that allow variable marginal amounts of tailweight. The originality comes from introducing multidimensional instead of univariate scale variables for the mixture of scaled Gaussian family of distributions. In contrast to most existing approaches, the derived distributions can account for a variety of shapes and have a simple tractable form with a closed-form probability density function whatever the dimension. We examine a number of properties of these distributions and illustrate them in the particular case of Pearson type VII and t tails. For these latter cases, we provide maximum likelihood estimation of the parameters and illustrate their modelling flexibility on simulated and real data clustering examples.