Quantification of Simultaneous-AND Gates in Temporal Fault Trees

Fault Tree Analysis has been a cornerstone of safety-critical systems for many years. It has seen various extensions to enable it to analyse dynamic behaviours exhibited by modern systems with redundant components. However, none of these extended FTA approaches provide much support for modelling situations where events have to be "nearly simultaneous", i.e., where events must occur within a certain interval to cause a failure. Although one such extension, Pandora, is unique in providing a "Simultaneous-AND" gate, it does not allow such intervals to be represented. In this work, we extend the Simultaneous-AND gate to include a parameterized interval - referred to as pSAND - such that the output event occurs if the input events occur within a defined period of time. This work then derives an expression for the exact quantification of pSAND for exponentially distributed events and provides an approximation using Monte Carlo simulation which can be used for other distributions

Improved Cross-Entropy Method for Estimation

The cross-entropy (CE) method is an adaptive importance sampling procedure that has been successfully applied to a diverse range of complicated simulation problems. However, recent research has shown that in some high-dimensional settings, the likelihood ratio degeneracy problem becomes severe and the importance sampling estimator obtained from the CE algorithm becomes unreliable. We consider a variation of the CE method whose performance does not deteriorate as the dimension of the problem increases. We then illustrate the algorithm via a high-dimensional estimation problem in risk management

Efficient simulation of ruin probabilities when claims are mixtures of heavy and light tails

We consider the classical Cram\'er-Lundberg risk model with claim sizes that are mixtures of phase-type and subexponential variables. Exploiting a specific geometric compound representation, we propose control variate techniques to efficiently simulate the ruin probability in this situation. The resulting estimators perform well for both small and large initial capital. We quantify the variance reduction as well as the efficiency gain of our method over another fast standard technique based on the classical Pollaczek-Khinchine formula. We provide a numerical example to illustrate the performance, and show that for more time-consuming conditional Monte Carlo techniques, the new series representation also does not compare unfavorably to the one based on the Pollaczek- Khinchine formula.Comment: 18 pages, 8 figure

Importance Sampling for Credit Risk Monte Carlo Simulations using the Cross Entropy method

For this thesis, we applied the Cross Entropy method on a credit risk model for the ING wholesale lending portfolio and some synthetically created realistic portfolios. The Cross Entropy method is found to be able to find appropriate Importance Sampling parameters within a relative modest resource budget. With the new parameters, the standard deviation of the estimate that the losses will exceed the available buffer can be decreased with more than 95%. A similar reduction with regular Monte Carlo would require the number of scenarios to increase four hundred times. Alternative methods provide similar reductions, but these use numerical methods that are more complex to implement and require more resources to calculate. Further tests show that the method is robust to the parameters used in the Cross Entropy method (within reasonable limits), it is not influenced significantly by the constitution of the portfolio and that none of the problems occur that the scientific literature warns about (in particular the “degeneracy of the likelihood ratio”)

Rare-event probability estimation with conditional Monte Carlo

Estimation of rare-event probabilities in high-dimensional settings via importance sampling is a difficult problem due to the degeneracy of the likelihood ratio. In fact, it is generally recommended that Monte Carlo estimators involving likelihood ratios should not be used in such settings. In view of this, we develop efficient algorithms based on conditional Monte Carlo to estimate rare-event probabilities in situations where the degeneracy problem is expected to be severe. By utilizing an asymptotic description of how the rare event occurs, we derive algorithms that involve generating random variables only from the nominal distributions, thus avoiding any likelihood ratio. We consider two settings that occur frequently in applied probability: systems involving bottleneck elements and models involving heavy-tailed random variables. We first consider the problem of estimating ℙ(X+· · ·+X>γ), where X,...,X are independent but not identically distributed (ind) heavy-tailed random variables. Guided by insights obtained from this model, we then study a variety of more general settings. Specifically, we consider a complex bridge network and a generalization of the widely popular normal copula model used in managing portfolio credit risk, both of which involve hundreds of random variables. We show that the same conditioning idea, guided by an asymptotic description of the way in which the rare event happens, can be used to derive estimators that outperform existing ones

Analytical and computational methods for the study of rare event probabilities in dispersive and dissipative waves

The main focus of this dissertation is the application of importance sampling (IS) to calculate the probabilities associated with rare events in nonlinear, large-dimensional lightwave systems that are driven by noise, including models for fiber-based optical communication system and mode-locked lasers. Throughout the last decade, IS has emerged as a valuable tool for improving the efficiency of simulating rare events in such systems. In particular, it has shown great success in simulating various sources of transmission impairments found in optical communication systems, with examples ranging from large polarization fluctuations resulting from randomly varying fiber birefringence to large pulse-width fluctuations resulting from imperfections in the optical fiber. In many cases, the application of IS is guided by a low-dimensional reduction of the system dynamics. Combining the low-dimensional reduction with Monte Carlo simulations of the original system has been shown to be an extremely effective scheme for computing, for example, the probability with which a pulse deviates significantly from its initial form due to a random forcing. In the context of nonlinear optics, this might represent a transmission error where the propagation model is the nonlinear Schr¨odinger equation (NLSE) with additive or multiplicative noise. A shortcoming of this method is that the efficiency of the IS technique depends strongly on the accuracy of the low-dimensional reduction used to guide the simulations. These low-dimensional reductions are often derived from a formal perturbation theory, referred to as soliton perturbation theory (SPT) for the case of soliton propagation under the forced NLSE. As demonstrated here, such reduction methodsare often inadequate in their description of the pulse\u27s dynamics. In particular, the interaction between a propagating pulse and dispersive radiation leads to a radiation-induced drift in a pulse\u27s phase, which is largely unaccounted for in the reduced systems currently in use. The first part of this dissertation is devoted to understanding the interaction between a pulse and dispersive radiation, leading to the derivation of an improved reduced system based on a variational approach. Once this system is derived and verified numerically, it serves as the basis for an improved IS method that incorporates the dynamics of the radiation, which is subsequently extended to more realistic propagation models. Of particular interest is the case of the NLSE with a periodic modulation of the dispersion constant, referred to as dispersion management (DM), and a related model where this modulation is averaged to give an autonomous, nonlocal equation. Following the nomenclature commonly use in literature, the former (nonautonomous) equation will be referred to as the NLSE+DM and the latter (autonomous) equation as the DMNLSE. A complicating aspect of these more realistic models is that, unlike the NLSE, exact solutions only exist as numerical objects rather than as closed-form solutions, which introduces an addition source of error in the derivation of a reduced system for the pulse dynamics. In the second part of this dissertation, the IS method is extended to the calculation of phase-slip probabilities in mode-locked lasers (MLL). Realistic models for pulse propagation in MLL include the dissipative effects of gain and loss, in addition to nonlocal saturation effects. As a result most of the reduced systems derived for pulse dynamics are extremely complicated, which diminishes their applicability as guides for IS simulations. Therefore, a MLL operating in the soliton propagation regime is considered, where the effects of gain, loss and saturation are treated perturbatively. A simple reduced system for the pulse dynamics is derived for this MLL model, allowing the IS technique to be effectively applied

Statistical and Computational Methods for Differential Expression Analysis in High-throughput Gene Expression Data

In this dissertation, we develop novel statistical and computational methods for differential expression analysis in high-throughput gene expression data. In the first part, we develop statistical models for differential expression with a variety of study designs. In project one, we present an efficient algorithm for the detection of differential expression and splicing of genes in RNA-Seq data. Our approach considers three cases for each gene: no differential expression, differential expression without differential splicing, and differential splicing. We use a Poisson regression framework to model the read counts and a hierarchical likelihood ratio test for model selection. In project two, we present a non-parametric approach for the joint detection of differential expression and splicing of genes by introducing a new statistic named gene-level differential score and using a permutation test to assess the statistical significance. The method can be applied to a variety of experimental designs, including those with two (unpaired or paired) or multiple biological conditions, and those with quantitative or survival outcomes. In project three, we model the single-cell gene expression data using a two-part mixed model, which not only adequately accounts for the distinct features of single cell expression data, including extra zero expression values, high variability and clustered design, but also provides the flexibility of adjusting for covariates. Comparisons with existing methods, our approach achieves improved power for detecting differential expressed genes. In the second part, we propose novel methods to improve the computational efficiency of resampling-based test methods in genomics. In project four, we present a fast algorithm for evaluating small p-values from permutation tests based on the cross-entropy method. In chapter five, we develop an efficient algorithm for estimating small p-values in parametric bootstrap tests using the improved cross-entropy method to approximate the optimal proposal density and the Hamiltonian Monte Carlo method to efficiently sample from the optimal proposal density. These methods together address a critical challenge for resampling-based tests in genomics since an enormous number of resamples is needed for estimating very small p-values. Simulations and applications to real data demonstrate that our methods achieve significant gains in computational efficiency comparing with existing methods.PHDBiostatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135864/1/shyboy_1.pd

Løsninger på utfordringer knyttet til snø for økt utbredelse av solcelleanlegg

Photovoltaic (PV) systems are becoming more competitive due to a cost reduction of the technology and increased electricity prices. As the technology extends to cold climates with lower irradiance, a knowledge gap in how PV systems are affected by the environment arises, which can limit PV system deployment. This thesis focuses on the impact of snow, which is perhaps the most distinguished environmental impact compared to the high irradiance climates where PV systems traditionally have been deployed. An interdisciplinary perspective is used to investigate different snow challenges connected to the deployment of PV in cold climates, and how they can be resolved. One of the challenges explored in the thesis is the development of snowdrifts in ground mounted PV plants. This challenge is relevant for PV systems installed in exposed snowdrift climates. To document the challenge itself, field measurements of snowdrift development in a small-scale PV plant in a polar climate were performed. The study concludes that PV plants designed with established principles commonly used at lower latitudes are susceptible to snowdrift accumulation. To achieve a snowdrift resilient plant, the design of the plant itself can be adapted. This strategy is further investigated in a numerical study using Computational Fluid Dynamics and energy yield simulations to quantify the impact of changing the design parameters on the snowdrift accumulation conditions and the energy yield. It is found that all the design parameters can be adjusted to improve the snowdrift conditions, with variable effect on the yield. Based on these results, adaptions to local climate conditions can be made to increase the snowdrift resiliency of the PV plant while minimizing an adverse impact on the yield, enabling the use of ground mounted PV plants in exposed snowdrift climates. Another of the investigated snow challenges is the use of active snow mitigation with PV systems on existing building roofs. Such systems reduce heavy snow loads so that roofs which lack structural capacity can be utilised for PV power production. In the thesis, PV snow mitigation systems are analysed in two separate studies focusing on the influence of active snow mitigation with PV systems on (i) the structural safety of building roofs, and (ii) the energy consumption and production compared to ordinary PV systems. The results provide a foundation for estimating which structures and climates PV snow mitigation systems are suitable. The research address former knowledge gaps for the use of PV snow mitigation systems and can contribute to increased utilisation of roof area for PV power production in the built environment. Snow contributes to an uncertainty in the yield of PV systems as it is difficult to predict snow shedding from the PV modules. There are several models for estimating yield losses in PV systems based on empirical data of snow shedding, but due to being developed based on single systems, the applicability to different configurations in different snow climates are limited. With the intent of achieving a model with wider applicability, an existing snow loss model is improved by considering the influence of snow depth on the snow shedding. By applying the model to seven different PV systems in different snow climates, the error in estimation of snow loss is reduced by 23 percentage points compared to the original model. The model contributes to reducing the uncertainty in PV yield estimations without the need for system specific empirical data of snow shedding. The overall contribution of the work is to resolve specific snow challenges which limit the deployment of PV systems in cold climates. Additional snow challenges have been identified during the work with the thesis, and recommendations for paths for future work are suggested. With ongoing research on this topic, the limitations for PV deployment in cold climates can be resolved and PV systems can contribute to increased renewable energy production in cold climates.Reduserte produksjonskostnader og økte strømpriser øker konkurransedyktigheten til solcelleanlegg. Solcelleanlegg har vært mest utbredt i, og delvis blitt utviklet for, varme klima med mye stråling, men når solcelleanlegg sprer seg til kaldere klima begrenses bruken av teknologien av manglende kunnskap om klimapåkjenninger. En av de største forandringene i klimapåkjenninger i kalde klima kontra varme klima er påvirkningen fra snø. Denne avhandlingen omhandler hvordan snø begrenser bruk av solcelleanlegg og hvordan slike utfordringer kan løses. En av utfordringene som undersøkes er snøfonndannelse i bakkemonterte solcelleanlegg. For å undersøke hvor utsatt solcelleanlegg er for snøfonndannelse er det gjennomført feltforsøk på et bakkemontert solkraftverk i et polart klima. Studien viser at solcelleanlegg som er designet ut ifra samme prinsipper som på lavere breddegrader gir en utforming som er svært utsatt for snøfonndannelse. En måte å redusere risikoen for snøfonndannelse på er å tilpasse designet av anlegget. For å undersøke denne tilpasningsstrategien er det gjennomført en numerisk studie som anvender fluidmekanikk- og energiytelsessimuleringer til å kvantifisere hvilken påvirkning det gir å endre utformingen av solkraftverket. Resultatene viser at alle de undersøkte designparameterne i solcelleanlegg kan tilpasses for å redusere risikoen for snøfonndannelse, men at de forskjellige designparameterne gir forskjellig påvirkning på energiytelsen. Resultatene fra disse studiene gir et grunnlag for å tilpasse utformingen av solkraftverk til klima med betydelig snødriv samtidig som ytelsen ivaretas. En annen utfordring som undersøkes er hvordan solcelleanlegg med snøsmeltefunksjon kan benyttes på eksisterende takkonstruksjoner som ikke tåler den totale vekten av snølasten og solcelleanlegget. I avhandlingen undersøkes det hvordan slike solcelleanlegg påvirker konstruksjonssikkerheten til bygg ved å benytte statistiske metoder. Resultatene tydeliggjør påvirkningen styringen og designet av slike anlegg har på konstruksjonssikkerheten til bygg, samt hvordan forskjellige kapasitets- og lastforutsetninger påvirker utbyttet av slike anlegg. I tillegg til påvirkningen på konstruksjonssikkerhet undersøkes energibehovet og hvilken potensiell produksjonsøkning det medfører å aktivt redusere snølasten på tak i en studie som benytter en kombinasjon av numeriske verktøy. Resultatene viser hvilke type klimatiske forhold som gir lavest energibruk og høyest økning i produksjon. En sammenstilling av resultatene fra de to studiene danner et grunnlag for å vurdere hvilke konstruksjoner og klima som egner seg for å benytte solcelleanlegg med snøsmeltefunksjon. Forskningen reduserer kunnskapshull for bruken av solcelleanlegg på tak med begrenset bæreevne og kan bidra til økt utnyttelse av eksisterende takflater til solstrømproduksjon. Den siste undersøkte utfordringen omhandler modellering av påvirkningen snø har på solcelleanleggs ytelse. En begrensing med mange eksisterende modeller for ytelsestap fra snø er at de er utviklet med empiriske data fra ett type snøklima og ikke nødvendigvis gir gode resultater når de anvendes i andre klimaforhold. Dette forsøkes å forbedres ved å videreutvikle en eksisterende snøtapsmodell til å ta hensyn til snødybde i avsklidningen av snø fra solcellepanelene. Sammenlignet med den opprinnelige snøtapsmodellen reduseres nøyaktigheten til modellen med 23 prosentpoeng når den anvendes til syv forskjellige solcelleanlegg. Modellen kan bidra til å redusere usikkerheten til ytelsen av solcelleanlegg i forskjellige type snøklima. Det overordnede bidraget til avhandlingen er å løse utfordringer snø gir for bruk av solcelleanlegg. Gjennom arbeidet har det blitt oppdaget ytterligere utfordringer. På bakgrunn av dette foreslås det hva som er aktuelt å fokusere på i fremtidig forskning på solcelleanlegg i klima med snø. Videre forskning på temaet kan føre til at bruken av solcelleanlegg i mindre grad hindres av snø og til å redusere klimautslipp i kalde klima