Statistical aspects of carbon fiber risk assessment modeling

The probabilistic and statistical aspects of the carbon fiber risk assessment modeling of fire accidents involving commercial aircraft are examined. Three major sources of uncertainty in the modeling effort are identified. These are: (1) imprecise knowledge in establishing the model; (2) parameter estimation; and (3)Monte Carlo sampling error. All three sources of uncertainty are treated and statistical procedures are utilized and/or developed to control them wherever possible

Concentration fluctuations in growing and dividing cells:Insights into the emergence of concentration homeostasis

Intracellular reaction rates depend on concentrations and hence their levels are often regulated. However classical models of stochastic gene expression lack a cell size description and cannot be used to predict noise in concentrations. Here, we construct a model of gene product dynamics that includes a description of cell growth, cell division, size-dependent gene expression, gene dosage compensation, and size control mechanisms that can vary with the cell cycle phase. We obtain expressions for the approximate distributions and power spectra of concentration fluctuations which lead to insight into the emergence of concentration homeostasis. We find that (i) the conditions necessary to suppress cell division-induced concentration oscillations are difficult to achieve; (ii) mRNA concentration and number distributions can have different number of modes; (iii) two-layer size control strategies such as sizer-timer or adder-timer are ideal because they maintain constant mean concentrations whilst minimising concentration noise; (iv) accurate concentration homeostasis requires a fine tuning of dosage compensation, replication timing, and size-dependent gene expression; (v) deviations from perfect concentration homeostasis show up as deviations of the concentration distribution from a gamma distribution. Some of these predictions are confirmed using data for E. coli, fission yeast, and budding yeast

Quantile Approximation of the Erlang Distribution using Differential Evolution Algorithm

Erlang distribution is a particular case of the gamma distribution and is often used in modeling queues, traffic congestion in wireless sensor networks, cell residence duration and finding the optimal queueing model to reduce the probability of blocking. The application is limited because of the unavailability of closed-form expression for the quantile (inverse cumulative distribution) function of the distribution. The problem is primarily tackled using approximation since the inversion method cannot be applied. This paper extended a six parameter quantile model earlier proposed to the Nakagami distribution to the Erlang distributions. Consequently, the established relationship between the two distributions is now extended to their quantile functions. The quantile model was used to fit the machine (R software) values with their corresponding quartiles in two ways. Firstly, artificial neural network (ANN) was used to establish that a curve fitting can be achieved. Lastly, differential evolution (DE) algorithm was used to minimize the errors obtained from the curve fitting and hence estimate the values of the six parameters of the quantile model that will ensure the best possible fit, for different values of the parameters that characterize Erlang distribution. Hence, the problem is constrained optimization in nature and the DE algorithm was able to find the different values of the parameters of the quantile model. The simulation result corroborates theoretical findings. The work is a welcome result for the quest for a universal quantile model that can be applied to different distributions

IP Traffic Statistics - A Markovian Approach

Data originating from non-voice sources is expected to play an increasingly important role in the next generation mobile communication services. To plan these networks, a detailed understanding of their traffic load is essential. Recent experimental studies have shown that network traffic originating from data applications can be self-similar, leading to a different queueing behavior than predicted by conventional traffic models. Heavy tailed probability distributions are appropriate for capturing this property, but including those random processes in a performance analysis makes it difficult and often impossible to find numerical results. In this thesis three related topics are addressed: It is shown that Markovian models with a large state space can be used to describe traffic which is self-similar over a large time scale, a Maximum Likelihood approach to fit parallel Erlang-k distributions directly to time series is developed, and the performance of a channel assignment procedure in a wireless communication network is evaluated using the above mentioned techniques to set up a Markovian model. Outcomes of the performance analysis are blocking probabilities and latency due to restrictions of the channel assignment procedure as well as estimations of the overall bandwidth that the system is required to offer in order to support a given number of users

Methods for generating variates from probability distributions

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Diverse probabilistic results are used in the design of random univariate generators. General methods based on these are classified and relevant theoretical properties derived. This is followed by a comparative review of specific algorithms currently available for continuous and discrete univariate distributions. A need for a Zeta generator is established, and two new methods, based on inversion and rejection with a truncated Pareto envelope respectively are developed and compared. The paucity of algorithms for multivariate generation motivates a classification of general methods, and in particular, a new method involving envelope rejection with a novel target distribution is proposed. A new method for generating first passage times in a Wiener Process is constructed. This is based on the ratio of two random numbers, and its performance is compared to an existing method for generating inverse Gaussian variates. New "hybrid" algorithms for Poisson and Negative Binomial distributions are constructed, using an Alias implementation, together with a Geometric tail procedure. These are shown to be robust, exact and fast for a wide range of parameter values. Significant modifications are made to Atkinson's Poisson generator (PA), and the resulting algorithm shown to be complementary to the hybrid method. A new method for Von Mises generation via a comparison of random numbers follows, and its performance compared to that of Best and Fisher's Wrapped Cauchy rejection method. Finally new methods are proposed for sampling from distribution tails, using optimally designed Exponential envelopes. Timings are given for Gamma and Normal tails, and in the latter case the performance is shown to be significantly better than Marsaglia's tail generation procedure.Governors of Dundee College of Technolog

Multivariate phase type distributions - Applications and parameter estimation

Den bedst kendte univariate sandsynlighedsfordeling er normalfordelingen. Den er grundigt beskrevet i litteraturen inden for et bredt felt af anvendelsesområder. I de tilfælde, hvor det ikke er meningsfuldt at anvende normalfordelingen, findes alternative sandsynlighedsfordelinger som alle er godt beskrevet; mange af disse tilhører klassen af fasetypefordelinger. Fasetypefordelinger har adskillige fordele. De er alsidige forstået på den måde, at de kan benyttes til at tilnærme en vilkårlig sandsynlighedsfordeling defineret på den positive reelle akse. Der eksisterer generelle probabilistiske resultater for hele klassen af fasetypefordelinger, hvilket bidrager til anvendelsen af forskellige estimeringsmetoder på enten klassen af fasetypefordelinger eller dens delklasser. Disse egenskaber gør klassen af fasetypefordelinger til et interessant alternativ til normalfordelingen.Når det kommer til multivariate problemer, så er den multivariate normalfordeling den eneste generelle fordeling, der tillader parameterestimering og statistisk inferens. Desværre er kendskabet til egenskaberne af den multivariate fasetypefordeling stærk begrænset. Resultaterne for parameterestimering og inferensteori for den univariate fasetypefordeling indikerer et potentiale for lignende gode resultater for klassen af multivariate fasetypefordelinger. Mit ph.d.-studium var en del afWork Package 3 i UNITE-projektet. UNITEprojektet arbejder mod det overordnede mål at forbedre kvaliteten af beslutningsgrundlaget for projekter. Dette gøres ved at reducere systematisk model bias og ved at beskrive og reducere model usikkerheder generelt. Forskning har vist, at afvigelsen fra omkostningsestimater for infrastrukturprojekter tydeligvis ikke er normaltfordelt men i stedet hælder mod budgetoverskridelser. Denne skævhed kan beskrives med fasetypefordelinger. Cost-benefit-analyser bruges til at evaluere potentielle fremtidige projekter og til at udvikle pålidelige omkostningsvurderinger. Successiv Princippet er en gruppebaseret analysemetode, der primært bruges til at prædiktere omkostninger og varighed af mellem til store projekter. Vi mener, at den matematiske modellering, der ligger til grund for Successiv Princippet, kan forbedres. Vi foreslår derfor en ny tilgang til modellering af den samlede varighed af et projekt ved hjælp af univariate fasetypefordelinger. Den matematiske model er dernæst udvidet til også at beskrive korrelationen mellem projektvarighed og omkostninger nu baseret på bivariate fasetypefordelinger. Vores model kan anvendes til at forbedre estimater for varighed og omkostninger, og derved hjælpe projekters beslutningstagere til at træffe en optimal beslutning.Det arbejde, jeg har udført som en del af mit ph.d.-studium, sigtede efter at belyse klassen af multivariate fasetypefordelinger. Denne afhandling indeholder analytiske og numeriske resultater for parameterestimering og inferensteori for en gruppe af multivariate fasetypefordelinger. Resultaterne kan betragtes som et første skridt i retning af en mere tilbundsgående forståelse af multivariate fasetypefordelinger. Vi er imidlertid langt fra at have afdækket det fulde potentiale af generelle fasetypefordelinger. En dybere forståelse af multivariate fasetypefordelinger vil åbne op for et bredt felt af anvendelsesområder.Afhandlingen består af en opsummerende rapport og to videnskabelige artikler. Det bagvedliggende arbejde var udført i perioden 2010 til 2014.The best known univariate probability distribution is the normal distribution. It is used throughout the literature in a broad field of applications. In cases where it is not sensible to use the normal distribution alternative distributions are at hand and well understood, many of these belonging to the class of phase type distributions. Phase type distributions have several advantages. They are versatile in the sense that they can be used to approximate any given probability distribution on the positive reals. There exist general probabilistic results for the entire class of phase type distributions, allowing for different estimation methods for the whole class or subclasses of phase type distributions. These attributes make this class of distributions an interesting alternative to the normal distribution. When facing multivariate problems, the only general distribution that allows for estimation and statistical inference, is the multivariate normal distribution. Unfortunately only little is known about the general class of multivariate phase type distribution. Considering the results concerning parameter estimation and inference theory of univariate phase type distributions, the class of multivariate phase type distributions shows potential for similar great results.My PhD studies were part of the the work package 3 of the UNITE project. The overall goal of the UNITE project is to improve the decision support prior to deciding on a project by reducing systematic model bias and by quantifying and reducing model uncertainties.Research has shown that the errors on cost estimates for infrastructure projects clearly do not follow a normal distribution but is skewed towards cost overruns. This skewness can be described using phase type distributions. Cost benefit analysis assesses potential future projects and depend on reliable cost estimates. The Successive Principle is a group analysis method primarily used for analyzing medium to large projects in relation to cost or duration. We believe that the mathematical modeling used in the Successive Principle can be improved. We suggested a novel approach for modeling the total duration of a project using a univariate phase type distribution. The model is then extended to catch the correlation between duration and cost estimates using a bivariate phase type distribution. The use of our model can improve estimates for duration and costs and therefore help project management to make the optimal decisions. The work conducted during my PhD studies aimed at shedding light on the class of multivariate phase type distributions. This thesis contains analytical and numerical results for parameter estimations and inference theory for a family of multivariate phase type distributions. The results can be used as a stepping stone towards understanding multivariate phase type distributions better. However, we are far from uncovering the full potential of general multivariate phase type distributions. Deeper understanding of multivariate phase type distributions will open up a broad field of research areas they can be applied to.This thesis consists of a summary report and two research papers. The work was carried out in the period 2010 - 2014

Approximating the randomized hitting time distribution of a non-stationary gamma process

The non-stationary gamma process is a non-decreasing stochasticprocess with independent increments. By this monotonic behavior thisstochastic process serves as a natural candidate for modellingtime-dependent phenomena such as degradation. In condition-basedmaintenance the first time such a process exceeds a random thresholdis used as a model for the lifetime of a device or for the randomtime between two successive imperfect maintenance actions. Thereforethere is a need to investigate in detail the cumulative distributionfunction (cdf) of this so-called randomized hitting time. We firstrelate the cdf of the (randomized) hitting time of a non-stationarygamma process to the cdf of a related hitting time of a stationarygamma process. Even for a stationary gamma process this cdf has ingeneral no elementary formula and its evaluation is time-consuming.Hence two approximations are proposed in this paper and both have aclear probabilistic interpretation. Numerical experiments show thatthese approximations are easy to evaluate and their accuracy dependson the scale parameter of the non-stationary gamma process. Finally,we also consider some special cases of randomized hitting times forwhich it is possible to give an elementary formula for its cdf.approximation;condition based maintencance;first hitting time;non-stationary gamma process;random threshold