Finite-state Markov Chains obey Benford's Law

A sequence of real numbers (x_n) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_n) are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (P^n - P*) and (P^{n+1}-P^n) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (nonresonant) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.Comment: 31 pages, no figure

Application of a general risk management model to portfolio problems with elliptical distributions /

Focus is directed to a class of risk measures for portfolio optimization with two types of disutility functions, where the random return variables of financial instru¬ments are assumed to be distributed by multivariate elliptical distributions. Recent risk measures, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are consid¬ered under this setting. If the joint distribution function of the financial instruments is elliptical and disutility is taken as linear reflecting the behavior of risk neutral in¬vestors, then the optimal solution of the mathematical models with objective functions formed by VaR and CVaR measures, is equivalent to the solution of the corresponding Markowitz model. To solve the Markowitz model, a very fast, finite step algorithm proposed in the literature, has been modified and implemented . Finally, the CVaR model with convex increasing disutility functions reflecting the behavior of risk averse investors has been introduced. Although, a convex objective function exists in this case, an analytic form cannot be obtained. However, unlike generating scenarios from multivariate distributions as suggested in the literature, the objective function can be closely estimated by simulating realizations only from univariate distributions. The thesis concludes with a thorough computational study on a sample data collected from the Istanbul Stock Exchange for each different class introduced

Impact of pro-domain stability of matrix metalloproteinase-8 on the outcome of sepsis

Citation: Berx, B., M. Dickey-Collas, M.D. Skogen, Y.-H. De Roeck, H. Klein, R. Barciela, R.M. Forster, E. Dombrowsky, M. Huret, M. Payne, Y. Sagarminaga, and C. Schrum. 2011. Does operational oceanography address the needs of fisheries and applied environmental scientists? Oceanography 24(1):166–171, doi:10.5670/oceanog.2011.14.Although many oceanographic data products are now considered operational, continued dialogue between data producers and their user communities is still needed. The fisheries and environmental science communities have often been criticized for their lack of multidisciplinarity, and it is not clear whether recent developments in operational oceanographic products are addressing these needs. The International Council for the Exploration of the Sea (ICES) Working Group on Operational Oceanographic products for Fisheries and Environment (WGOOFE) identified a potential mismatch between user requirements and the perception of requirements by the providers. Through a questionnaire (98 respondents), WGOOFE identified some of these issues. Although products of physical variables were in higher demand, several biological parameters scored in the top 10 rankings. Users placed specific focus on historic time series products with monthly or annual resolution and updating on similar time scales. A significant percentage requested access to numerical data rather than graphical output. While the outcomes of this survey challenge our views of operational oceanography, several initiatives are already attempting to close the gap between user requirements and products available

Polatuzumab vedotin, rituximab, and bendamustine combination in relapsed or refractory diffuse large B-cell lymphoma: A real-world data from Turkey

Polatuzumab vedotin (Pola) with bendamustine and rituximab (BR) is a promising option for patients with relapsed/refractory (R/R) diffuse large B-cell lymphoma (DLBCL). We analyzed the data of 71 R/R DLBCL patients who had been treated with Pola-BR in the named patient program from March 2018 to April 2021 from 32 centers in Turkey. All patients received up to six cycles of Pola 1.8 mg/kg, rituximab 375 mg/m2 on day 1, and bendamustine 90 mg/m2 on days 1–2 of each cycle. Median age at Pola-BR initiation was 55 (19–84). The overall response rate was 47.9%, including 32.4% CR rate when a median of 3 cycles was applied. With a median follow-up of 5 months, the median OS was 5 months. Grade 3–4 neutropenia and thrombocytopenia were the most common hematological toxicities. The real-world data from our cohort showed the Pola-BR is an effective option with a manageable toxicity profile

The cross-entropy method with patching for rare-event simulation of large Markov chains

There are various importance sampling schemes to estimate rare event probabilities in Markovian systems such as Markovian reliability models and Jackson networks. In this work, we present a general state-dependent importance sampling method which partitions the state space and applies the cross-entropy method to each partition. We investigate two versions of our algorithm and apply them to several examples of reliability and queueing models. In all these examples we compare our method with other importance sampling schemes. The performance of the importance sampling schemes is measured by the relative error of the estimator and by the efficiency of the algorithm. The results from experiments show considerable improvements both in running time of the algorithm and the variance of the estimator.Cross-entropy Rare events Importance sampling Large-scale Markov chains

The Cross-Entropy Method With Patching For Rare-Event Simulation Of Large Markov Chains

There are various importance sampling schemes to estimate rare event probabilities in Markovian systems such as Markovian reliability models and Jackson networks. In this work, we present a general state dependent importance sampling method which partitions the state space and applies the cross-entropy method to each partition. We investigate two versions of our algorithm and apply them to several examples of reliability and queueing models. In all these examples we compare our method with other importance sampling schemes. The performance of the importance sampling schemes is measured by the relative error of the estimator and by the effciency of the algorithm. The results from experiments show considerable improvements both in running time of the algorithm and the variance of the estimator.Cross-Entropy, Rare Events, Importance Sampling, Large-Scale Markov Chains

Counting with Combined Splitting and Capture-Recapture Methods

We apply the splitting method to three well-known counting problems, namely 3-SAT, random graphs with prescribed degrees, and binary contingency tables. We present an enhanced version of the splitting method based on the capture-recapture technique, and show by experiments the superiority of this technique for SAT problems in terms of variance of the associated estimators, and speed of the algorithms.Counting, Gibbs Sampler, Capture-Recapture, Splitting