Assessing significance in a Markov chain without mixing

We present a new statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of the Markov chain, we would like to demonstrate rigorously that the presented state is an outlier with respect to the values, by establishing a $p$-value for observations we make about the state under the null hypothesis that it was chosen uniformly at random. A simple heuristic used in practice is to sample ranks of states from long random trajectories on the Markov chain, and compare these to the rank of the presented state; if the presented state is a $0.1\%$-outlier compared to the sampled ranks (i.e., its rank is in the bottom $0.1\%$ of sampled ranks) then this should correspond to a $p$-value of $0.001$. This test is not rigorous, however, without good bounds on the mixing time of the Markov chain, as one must argue that the observed states on the trajectory approximate the stationary distribution. Our test is the following: given the presented state in the Markov chain, take a random walk from the presented state for any number of steps. We prove that observing that the presented state is an $\varepsilon$-outlier on the walk is significant at $p=\sqrt {2\varepsilon}$, under the null hypothesis that the state was chosen from a stationary distribution. Our result assumes nothing about the structure of the Markov chain beyond reversibility, and we construct examples to show that significance at $p\approx\sqrt \varepsilon$ is essentially best possible in general. We illustrate the use of our test with a potential application to the rigorous detection of gerrymandering in Congressional districtings.Comment: 15 pages, 3 figure

Предпрогнозний аналіз часових рядів методами фрактального аналізу та фазових траєкторій

The procedure of the qualitative analysis of time series, for which the hypothesis of trend existence isn’t confirmed, with application of the methods of nonlinear dynamics and the theory of chaos, is presented. The real time series characterizing prevalence of various skin diseases in Ukraine are considered. The basis for similar researches is Takens’s theorem. The randomness of the studied dynamical system given by time realizations is established by means of Lyapunov’s indicator. The state stability is estimated by Hausdorf’s fractal dimension and the fractality index. Visual evaluation of the time series was carried out by means of the phase trajectory restoration procedure. As a result of the analysis of phase pointsin the phase space the split attractor is indicated, which gives the chance to speak about its bifurcation.Запропоновано процедуру якісного аналізу часових рядів, для яких не підтверджується гіпотеза про наявність тренда, із застосуванням методів нелінійної динаміки, теорії хаосу. Розглянуто реальні часові ряди, що характеризують поширення різних шкірних захворювань в Україні. Обґрунтуванням для подібних досліджень є теорема Такенса. Хаотичність досліджуваної динамічної системи, що задана часовими реалізаціями, встановлена за допомогоюпоказника Ляпунова. Оцінка стійкості стану оцінювалась фрактальною розмірністю Хаусдорфа і індексом фрактальності. Візуальна оцінка часового ряду проводилась за допомогоюпроцедури відновлення фазових траєкторій. В результаті аналізу фазових точок фазового простору виявлено розщеплений атрактор, що дає можливість говорити про його біфуркацію

Prophylaxis of Acute Arthritis at Initiation of Urate-Lowering Therapy in Gout Patients

During the first months after the initiation of urate-lowering therapy in gout patients, the risk of exacerbation of arthritis considerably rises, which often results in discontinuation of the prescribed therapy by patients. The main way to avoid this risk is preventive prescription of colchicine, NSAIDs or glucocorticoids. Such prophylaxis of acute arthritis has been specified in a large number of the latest editions of various national and international guidelines; however, this tactics is rarely used in practice. The chapter includes the most significant studies on this problem

Effect of boundary conditions on the character of ambipolar diffusion in electrolytes

PREInternational audienceWe discuss the details of ambipolar relaxation of the electric field in liquid asymmetric electrolytes to its stationary value. It is demonstrated that the account for finite boundary conditions modifies the existing concepts of this diffusion process. In particular, we succeeded to suggest a qualitatively correct explanation of the observed distribution of the electric fields over the bulk of the cuvette and its nonmonotonic behavior in measurements on the finite-size cuvette. We analyze the conditions of such an anomaly at the intermediate stages of the relaxation proces