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Researcher profile

Anatoly Zhigljavsky

Anatoly Zhigljavsky contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.STStatistics TheoryPopulations and EvolutionApplicationsMachine Learningmath.COmath.NAmath.PRNumerical Analysisphysics.soc-ph

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Published work

10 published item(s)

preprint2026arXiv

Non-asymptotic quantisation of spherically symmetric distributions

Zador's celebrated theorem is a cornerstone of optimal quantisation, establishing both the weak limit of the empirical distribution of an $n$-point optimal quantiser in $R^d$ and the decay rate of the associated $L_s$-mean quantisation error. However, for large dimensions $d$, observing this asymptotic behaviour demands an astronomically large sample size $n$, which grows super-exponentially with $d$. Through a detailed analysis of the quantisation problem for spherically symmetric distributions, we demonstrate that for moderate $n$ random quantisers uniformly distributed on a sphere of suitable radius $r$ achieve exceptional performance. The expected distortion, expressed as a triple integral, can be computed with arbitrary precision, and the optimal radius $r$ can be efficiently determined numerically. Leveraging results from extreme-value theory, we derive approximations for $r$, particularly in scenarios where $n$ scales with $d$. Depending on the growth rate of $n$, $r$ may either converge to zero or approach a limiting value that is independent of $s$.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Efficient quantization and weak covering of high dimensional cubes

Let $\mathbb{Z}_n = \{Z_1, \ldots, Z_n\}$ be a design; that is, a collection of $n$ points $Z_j \in [-1,1]^d$. We study the quality of quantization of $[-1,1]^d$ by the points of $\mathbb{Z}_n$ and the problem of quality of coverage of $[-1,1]^d$ by ${\cal B}_d(\mathbb{Z}_n,r)$, the union of balls centred at $Z_j \in \mathbb{Z}_n$. We concentrate on the cases where the dimension $d$ is not small ($d\geq 5$) and $n$ is not too large, $n \leq 2^d$. We define the design ${\mathbb{D}_{n,δ}}$ as a $2^{d-1}$ design defined on vertices of the cube $[-δ,δ]^d$, $0\leq δ\leq 1$. For this design, we derive a closed-form expression for the quantization error and very accurate approximations for {the coverage area} vol$([-1,1]^d \cap {\cal B}_d(\mathbb{Z}_n,r))$. We provide results of a large-scale numerical investigation confirming the accuracy of the developed approximations and the efficiency of the designs ${\mathbb{D}_{n,δ}}$.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Random and quasi-random designs in group testing

For large classes of group testing problems, we derive lower bounds for the probability that all significant items are uniquely identified using specially constructed random designs. These bounds allow us to optimize parameters of the randomization schemes. We also suggest and numerically justify a procedure of constructing designs with better separability properties than pure random designs. We illustrate theoretical considerations with a large simulation-based study. This study indicates, in particular, that in the case of the common binary group testing, the suggested families of designs have better separability than the popular designs constructed from disjunct matrices. We also derive several asymptotic expansions and discuss the situations when the resulting approximations achieve high accuracy.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Approximations for the boundary crossing probabilities of moving sums of random variables

In this paper we study approximations for the boundary crossing probabilities of moving sums of i.i.d. normal r.v. We approximate a discrete time problem with a continuous time problem allowing us to apply established theory for stationary Gaussian processes. By then subsequently correcting approximations for discrete time, we show that the developed approximations are very accurate even for small window length. Also, they have high accuracy when the original r.v. are not exactly normal and when the weights in the moving window are not all equal. We then provide accurate and simple approximations for ARL, the average run length until crossing the boundary.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Blind deconvolution of covariance matrix inverses for autoregressive processes

Matrix $\mathbf{C}$ can be blindly deconvoluted if there exist matrices $\mathbf{A}$ and $\mathbf{B}$ such that $\mathbf{C}= \mathbf{A} \ast \mathbf{B}$, where $\ast$ denotes the operation of matrix convolution. We study the problem of matrix deconvolution in the case where matrix $\mathbf{C}$ is proportional to the inverse of the autocovariance matrix of an autoregressive process. We show that the deconvolution of such matrices is important in problems of Hankel structured low-rank approximation (HSLRA). In the cases of autoregressive models of orders one and two, we fully characterize the range of parameters where such deconvolution can be performed and provide construction schemes for performing deconvolutions. We also consider general autoregressive models of order $p$, where we prove that the deconvolution $\mathbf{C}= \mathbf{A} \ast \mathbf{B}$ does not exist if the matrix $\mathbf{B}$ is diagonal and its size is larger than $p$.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Comparison of different exit scenarios from the lock-down for COVID-19 epidemic in the UK and assessing uncertainty of the predictions

We model further development of the COVID-19 epidemic in the UK given the current data and assuming different scenarios of handling the epidemic. In this research, we further extend the stochastic model suggested in \cite{us} and incorporate in it all available to us knowledge about parameters characterising the behaviour of the virus and the illness induced by it. The models we use are flexible, comprehensive, fast to run and allow us to incorporate the following: -time-dependent strategies of handling the epidemic; -spatial heterogeneity of the population and heterogeneity of development of epidemic in different areas; -special characteristics of particular groups of people, especially people with specific medical pre-histories and elderly. Standard epidemiological models such as SIR and many of its modifications are not flexible enough and hence are not precise enough in the studies that requires the use of the features above. Decision-makers get serious benefits from using better and more flexible models as they can avoid of nuanced lock-downs, better plan the exit strategy based on local population data, different stages of the epidemic in different areas, making specific recommendations to specific groups of people; all this resulting in a lesser impact on economy, improved forecasts of regional demand upon NHS allowing for intelligent resource allocation.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Covering of high-dimensional cubes and quantization

As the main problem, we consider covering of a $d$-dimensional cube by $n$ balls with reasonably large $d$ (10 or more) and reasonably small $n$, like $n=100$ or $n=1000$. We do not require the full coverage but only 90\% or 95\% coverage. We establish that efficient covering schemes have several important properties which are not seen in small dimensions and in asymptotical considerations, for very large $n$. One of these properties can be termed `do not try to cover the vertices' as the vertices of the cube and their close neighbourhoods are very hard to cover and for large $d$ there are far too many of them. We clearly demonstrate that, contrary to a common belief, placing balls at points which form a low-discrepancy sequence in the cube, makes for a very inefficient covering scheme. For a family of random coverings, we are able to provide very accurate approximations to the coverage probability. We then extend our results to the problems of coverage of a cube by smaller cubes and quantization, the latter being also referred to as facility location. Along with theoretical considerations and derivation of approximations, we discuss results of a large-scale numerical investigation.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Generic probabilistic modelling and non-homogeneity issues for the UK epidemic of COVID-19

Coronavirus COVID-19 spreads through the population mostly based on social contact. To gauge the potential for widespread contagion, to cope with associated uncertainty and to inform its mitigation, more accurate and robust modelling is centrally important for policy making. We provide a flexible modelling approach that increases the accuracy with which insights can be made. We use this to analyse different scenarios relevant to the COVID-19 situation in the UK. We present a stochastic model that captures the inherently probabilistic nature of contagion between population members. The computational nature of our model means that spatial constraints (e.g., communities and regions), the susceptibility of different age groups and other factors such as medical pre-histories can be incorporated with ease. We analyse different possible scenarios of the COVID-19 situation in the UK. Our model is robust to small changes in the parameters and is flexible in being able to deal with different scenarios. This approach goes beyond the convention of representing the spread of an epidemic through a fixed cycle of susceptibility, infection and recovery (SIR). It is important to emphasise that standard SIR-type models, unlike our model, are not flexible enough and are also not stochastic and hence should be used with extreme caution. Our model allows both heterogeneity and inherent uncertainty to be incorporated. Due to the scarcity of verified data, we draw insights by calibrating our model using parameters from other relevant sources, including agreement on average (mean field) with parameters in SIR-based models.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Non-lattice covering and quanitization of high dimensional sets

The main problem considered in this paper is construction and theoretical study of efficient $n$-point coverings of a $d$-dimensional cube $[-1,1]^d$. Targeted values of $d$ are between 5 and 50; $n$ can be in hundreds or thousands and the designs (collections of points) are nested. This paper is a continuation of our paper \cite{us}, where we have theoretically investigated several simple schemes and numerically studied many more. In this paper, we extend the theoretical constructions of \cite{us} for studying the designs which were found to be superior to the ones theoretically investigated in \cite{us}. We also extend our constructions for new construction schemes which provide even better coverings (in the class of nested designs) than the ones numerically found in \cite{us}. In view of a close connection of the problem of quantization to the problem of covering, we extend our theoretical approximations and practical recommendations to the problem of construction of efficient quantization designs in a cube $[-1,1]^d$. In the last section, we discuss the problems of covering and quantization in a $d$-dimensional simplex; practical significance of this problem has been communicated to the authors by Professor Michael Vrahatis, a co-editor of the present volume.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Asymptotic optimal designs under long-range dependence error structure

We discuss the optimal design problem in regression models with long-range dependence error structure. Asymptotic optimal designs are derived and it is demonstrated that these designs depend only indirectly on the correlation function. Several examples are investigated to illustrate the theory. Finally, the optimal designs are compared with asymptotic optimal designs which were derived by Bickel and Herzberg [Ann. Statist. 7 (1979) 77--95] for regression models with short-range dependent error.

0 citations0 reviewsNo code linkedOpen access