Variance Reduction Techniques for Rare Events Simulations

In many real-life applications, one often encounters the problem of estimating events with very low probabilities, but their occurrences are critical and can result in severe consequences (major earthquakes, floods, etc.). In the field of wireless communication systems, the rare event can, for instance, be the event that the system is in an outage, and hence it does not operate properly. For sake of illustration, error probabilities of the order of 10-9 need to be estimated with high precision for ultra-reliable 5G and 6G wireless systems. Naive Monte Carlo methods can be used to estimate these probabilities. However, it is well-acknowledged that these methods are computationally expensive when dealing with rare events, that is a substantial amount of computational effort is required in order to achieve a good accuracy requirement. Variance reduction techniques are good alternatives that can be used to overcome the failure of naive Monte Carlo methods. Importance sampling, conditional Monte Carlo, and splitting are among the most common variance reduction techniques that were extensively used to develop efficient estimators for rare event probabilities [1]. The main objective of the Ph.D. project is to develop efficient importance sampling techniques to estimate critical rare event quantities. The focus will be on estimating tail probabilities of the form P(S(X) > b), where X is a random vector, S is a given real-valued function, and b is a threshold value. The tail probabilities of this form are motivated by several engineering applications. In financial engineering, the problem of estimating the value-at-risk can be related to that computing the left tail of sums of random variables [3]. The right tail of sums of random variables can model the ruin probability of an insurance company receiving a large number of claims [2]. Finally, tail probabilities of the form P(S(X) > b) can serve to compute error probabilities in the field of wireless communication systems [4,5]. Technically, the focus of the project will be to develop state-dependent importance sampling techniques to efficiently estimate the quantity of interest. By state-dependent schemes, we mean that the importance sampling parameters are dynamically chosen as a function of the system’s current time and space. Techniques from dynamic programming will be employed to obtain the optimal importance sampling parameters. The efficiency of the proposed importance sampling estimator will be then compared to state-of-the-art approaches in terms of the amount of variance reduction as well as the total computational effort.

<p style="margin-bottom:11px">References</p> <p>[1] D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo methods, <a href="https://onlinelibrary.wiley.com/series/1345">Wiley Series in Probability and Statistics</a>, 2011.</p> <p>[2] S. Asmussen and P. W. Glynn, Stochastic simulation: algorithms and analysis, Springer Science & Business Media, 2007.</p> <p>[3] S. Asmussen, J. L. Jensen, and L. Rojas-Nandayapa, Exponential family techniques for the Lognormal left tail, Scandinavian Journal of Statistics, Vol. 43, No. 3, 2016.</p> <p>[4] N. Ben Rached, A. Kammoun, M.-S. Alouini, and R. Tempone, Unified importance sampling Schemes for efficient simulation of outage capacity over generalized fading channels, IEEE Journal of Selected Topics in Signal Processing (Special Issue on: Stochastic Simulation and Optimization in Signal Processing), Vol. 10, issue 2, Mar. 2016.</p> <p>[5] E. Ben Amar, N. Ben Rached, A.-L. Haji-Ali, and R. Tempone, State-dependent importance sampling for estimating expectations of functionals of sums of independent random variables, arXiv preprint arXiv:2201.01340, 2022.</p>

<p>To apply for this project you will need to make a formal application for research degree study through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University website</a>. You will need to create a login ID with a username and PIN. </p> <p>•    For <strong>Application type</strong> please select <strong>Research Degrees – Research Postgraduate</strong>. <br /> •    The admission year for this project is <strong>2027/28</strong> Academic Year. <br /> •    You will need to select your <strong>Planned Course of Study</strong> from a drop-down menu. For this project, scroll down and select <strong>PHP Statistics Full Time.</strong> <br /> •    The project start date for this project is <strong>1 October 2027</strong>, please use this as your <strong>Proposed Start Date of Research</strong>. <br /> •    Please state clearly in the<strong> Research Information Section</strong> that the research degree you wish to be considered for is <strong>Variance Reduction Technique for Rare Events Simulations<em> </em></strong>as well as <a href="https://eps.leeds.ac.uk/faculty-engineering-physical-sciences/staff/12206/dr-nadhir-ben-rached">Dr Nadhir Ben Rached</a> as your proposed supervisor.</p> <p><strong>Please state clearly in the Finance section</strong>, <strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong>.</p> <p>More information on how to apply is available on our website <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">here</a>. You will be required to provide a personal statement which outlines your interest in the project you are applying for, why you have chosen it and how your skills map onto the requirements of the project.</p> <p>We will assess applications continuously as we receive them. We welcome and strongly encourage any potential applicants to contact the supervisor(s) for an informal discussion, prior to applying, and recommend submitting your application early.</p> <p><strong>If you are applying with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<p>This is a non funded PhD project. Applicants are expected to be self funded or to secure external funding.</p> <p>You can find out more about our Funding Opportunities on our <a href="https://phd.leeds.ac.uk/">Postgraduate Research Opportunities</a> website.</p> <p><strong>Important: </strong>Please note that all costs associated with your arrival at Leeds (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source.</p>

<p>For general enquiries about your application, please contact PGR Admissions by email to <a href="mailto:epspgradmissions@leeds.ac.uk">epspgradmissions@leeds.ac.uk</a></p> <p>For questions about this research project, please contact Dr Nadhir Ben Rached by email to <a href="mailto:n.benrached@leeds.ac.uk">n.benrached@leeds.ac.uk</a></p>