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  • 1.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Multivariate piecewise linear interpolation of a random field2011Manuscript (preprint) (Other academic)
    Abstract [en]

    We consider a multivariate piecewise linear interpolation of a continuous random field on a-dimensional cube. The approximation performance is measured by the integrated mean square error. Multivariate piecewise linear interpolator is defined by N field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field in mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields we provide the upper bound for the approximation accuracy in the uniform mean square norm.

  • 2.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On the error of the Monte Carlo pricing method for Asian option2008In: Journal of Numerical and Applied Mathematics, ISSN 0868-6912, Vol. 96, no 1, p. 1-10Article in journal (Refereed)
    Abstract [en]

    We consider a Monte Carlo method to price a continuous arithmetic Asian option with a given precision. Piecewise constant approximation and plain simulation are used for a wide class of models based on L\'{e}vy processes. We give bounds of the possible discretization and simulation errors. The sufficient numbers of discretization points and simulations to obtain requested accuracy are derived. To demonstrate the general approach, the Black-Scholes model is studied in more detail. We undertake the case of continuous averaging and starting time zero,  but the obtained results can be applied to the discrete case  and generalized for any time before an execution date. Some numerical experiments and comparison to the PDE based method are also presented.

  • 3.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Piecewise multilinear interpolation of a random field2013In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 45, no 4, p. 945-959Article in journal (Refereed)
    Abstract [en]

    We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

  • 4.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Spline approximation of a random process with singularity2011In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 141, no 3, p. 1333-1342Article in journal (Refereed)
    Abstract [en]

    Let a continuous random process X defined on [0,1] be (m+β)-smooth, 0m, 0<β1, in quadratic mean for all t>0 and have an isolated singularity point at t=0. In addition, let X be locally like a m-fold integrated β-fractional Brownian motion for all nonsingular points. We consider approximation of X by piecewise Hermite interpolation splines with n free knots (i.e., a sampling design, a mesh). The approximation performance is measured by mean errors (e.g., integrated or maximal quadratic mean errors). We construct a sequence of sampling designs with asymptotic approximation rate n^(m+β) for the whole interval.

  • 5.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stratified Monte Carlo quadrature for continuous random fields2015In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 17, no 1, p. 59-72Article in journal (Refereed)
    Abstract [en]

    We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behavior is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the H¨older class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.

  • 6.
    Ali-Eldin, Ahmed
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Rezaie, Ali
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Mehta, Amardeep
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Razroev, Stanislav
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sjöstedt-de Luna, Sara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Tordsson, Johan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Elmroth, Erik
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    How will your workload look like in 6 years?: Analyzing Wikimedia's workload2014In: Proceedings of the 2014 IEEE International Conference on Cloud Engineering (IC2E 2014) / [ed] Lisa O’Conner, IEEE Computer Society, 2014, p. 349-354Conference paper (Refereed)
    Abstract [en]

    Accurate understanding of workloads is key to efficient cloud resource management as well as to the design of large-scale applications. We analyze and model the workload of Wikipedia, one of the world's largest web sites. With descriptive statistics, time-series analysis, and polynomial splines, we study the trend and seasonality of the workload, its evolution over the years, and also investigate patterns in page popularity. Our results indicate that the workload is highly predictable with a strong seasonality. Our short term prediction algorithm is able to predict the workload with a Mean Absolute Percentage Error of around 2%.

  • 7.
    Ali-Eldin, Ahmed
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sjöstedt-de Luna, Sara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Tordsson, Johan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Elmroth, Erik
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Measuring cloud workload burstiness2014In: 2014 IEEE/ACM 7th International Conference on Utility and Cloud Computing (UCC), IEEE conference proceedings, 2014, p. 566-572Conference paper (Refereed)
    Abstract [en]

    Workload burstiness and spikes are among the main reasons for service disruptions and decrease in the Quality-of-Service (QoS) of online services. They are hurdles that complicate autonomic resource management of datacenters. In this paper, we review the state-of-the-art in online identification of workload spikes and quantifying burstiness. The applicability of some of the proposed techniques is examined for Cloud systems where various workloads are co-hosted on the same platform. We discuss Sample Entropy (SampEn), a measure used in biomedical signal analysis, as a potential measure for burstiness. A modification to the original measure is introduced to make it more suitable for Cloud workloads.

  • 8.
    Belyaev, Yuri K
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Moscow state university.
    Approaching in distribution with applications to resampling of stochastic processes2000In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 27, no 2, p. 371-384Article in journal (Refereed)
    Abstract [en]

    We introduce the notion of weak approaching and conditionally weak approaching sequences of random processes. This notion generalizes the conventional weak convergence, and has been proposed for real valued random variables in Belyaev (1995). Some of the standard tools for an investigation of the behaviour of weak approaching sequences of random elements in metric spaces are developed. The spaces of smoothed and right-continuous functions with left-hand limits are considered. This technique allows us to use the resampling approach for an evaluation of distributions of continuous functionals on realizations of sum of an increasing number of independent random processes. Two numerical examples are presented for such functionals as supremum and number of level crossings.

  • 9. Hashorva, Enkelejd
    et al.
    Lifshits, Mikhail
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Approximation of a Random Process with Variable Smoothness2015In: Mathematical Statistics and Limit Theorems: Festschrift in Honour of Paul Deheuvels / [ed] Hallin, Marc; Mason, David M.; Pfeifer, Dietmar; Steinebach, Josef G., Springer International Publishing , 2015, p. 189-208Chapter in book (Other academic)
    Abstract [en]

    We consider the rate of piecewise constant approximation to a locally stationary process X(t),t∈[0,1] , having a variable smoothness index α(t) . Assuming that α(⋅) attains its unique minimum at zero and satisfies α(t)=α0+btγ+o(tγ) as t→0, we propose a method for construction of observation points (composite dilated design) such that the integrated mean square error ∫10E{(X(t)−Xn(t))2}dt∼Knα0(logn)(α0+1)/γ as n→∞, where a piecewise constant approximation Xn is based on N(n)∼n observations of X . Further, we prove that the suggested approximation rate is optimal, and then show how to find an optimal constant K .

  • 10. Hashorva, Enkelejd
    et al.
    Mishura, Yuliya
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Boundary non-crossing probabilities for fractional Brownian motion with trend2015In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 87, no 6, p. 946-965Article in journal (Refereed)
    Abstract [en]

    In this paper, we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product, we solve a minimization problem related to the norm of the trend function.

  • 11. Hashorva, Enkelejd
    et al.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Tan, Zhongquan
    Approximation of maximum of Gaussian random fields2018In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 457, no 1, p. 841-867Article in journal (Refereed)
    Abstract [en]

    This contribution is concerned with Gumbel limiting results for supremum M-n = sup(t epsilon[0,Tn])X(n)(t)vertical bar with X (n) ,n epsilon N-2 centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for M-n as n -> infinity and show a second-order approximation for E{M-n(p)}(1/p) for any p >= 1.

  • 12. Kubilius, Kęstutis
    et al.
    Mishura, Yuliya
    Ralchenko, Kostiantyn
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index H ∈ (0, 12)2015In: Electronic Journal of Statistics, ISSN 1935-7524, E-ISSN 1935-7524, Vol. 9, p. 1799-1825Article in journal (Refereed)
    Abstract [en]

    parameter θ and where the noise is modeled as fractional Brownian motionwith Hurst index H ∈ (0, 12 ). The solution corresponds to the fractionalOrnstein–Uhlenbeck process. We construct an estimator, based on discreteobservations in time, of the unknown drift parameter, that is similar in formto the maximum likelihood estimator for the drift parameter in Langevinequation with standard Brownian motion. It is assumed that the intervalbetween observations is n−1, i.e. tends to zero (high-frequency data) andthe number of observations increases to infinity as nm with m > 1. It isproved that for strictly positive θ the estimator is strongly consistent forany m > 1, while for θ ≤ 0 it is consistent when m > 12H .

  • 13.
    Källberg, David
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Leonenko, Nikolaj
    School of Mathematics, Cardiff University, Cardiff, UK.
    Oleg, Seleznjev
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Statistical inference for Rényi entropy functionals2012In: Conceptual modelling and its theoretical foundations / [ed] Antje Düsterhöft, Meike Klettke, Klaus-Dieter Schewe, Springer Berlin/Heidelberg, 2012, p. 36-51Chapter in book (Refereed)
    Abstract [en]

    Numerous entropy-type characteristics (functionals) generalizing Rényi entropy are widely used in mathematical statistics, physics, information theory, and signal processing for characterizing uncertainty in probability distributions and distribution identification problems. We consider estimators of some entropy (integral) functionals for discrete and continuous distributions based on the number of epsilon-close vector records in the corresponding independent and identically distributed samples from two distributions. The proposed estimators are generalized U-statistics. We show the asymptotic properties of these estimators (e.g., consistency and asymptotic normality). The results can be applied in various problems in computer science and mathematical statistics (e.g., approximate matching for random databases, record linkage, image matching).

  • 14.
    Källberg, David
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Leonenko, Nikolaj
    Cardiff University, School of Mathematics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Statistical estimation of quadratic Rényi entropy for a stationary m-dependent sequence2014In: Journal of nonparametric statistics (Print), ISSN 1048-5252, E-ISSN 1029-0311, Vol. 26, no 2, p. 385-411Article in journal (Refereed)
    Abstract [en]

    The Rényi entropy is a generalization of the Shannon entropy and is widely used in mathematical statistics and applied sciences for quantifying the uncertainty in a probability distribution. We consider estimation of the quadratic Rényi entropy and related functionals for the marginal distribution of a stationary m-dependent sequence. The U-statistic estimators under study are based on the number of ε-close vector observations in the corresponding sample. A variety of asymptotic properties for these estimators are obtained (e.g., consistency, asymptotic normality, Poisson convergence). The results can be used in diverse statistical and computer science problems whenever the conventional independence assumption is too strong (e.g., ε-keys in time series databases, distribution identication problems for dependent samples).

  • 15.
    Källberg, David
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Estimation of entropy-type integral functionals2016In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 45, no 4, p. 887-905Article in journal (Other academic)
    Abstract [en]

    Entropy-type integral functionals of densities are widely used in mathematical statistics, information theory, and computer science. Examples include measures of closeness between distributions (e.g., density power divergence) and uncertainty characteristics for a random variable (e.g., Renyi entropy). In this paper, we study U-statistic estimators for a class of such functionals. The estimators are based on ε-close vector observations in the corresponding independent and identically distributed samples. We prove asymptotic properties of the estimators (consistency and asymptotic normality) under mild integrability and smoothness conditions for the densities. The results can be applied in diverse problems in mathematical statistics and computer science (e.g., distribution identication problems, approximate matching for random databases, two-sample problems).

  • 16.
    Källberg, David
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Estimation of quadratic density functionals under m-dependenceManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper, we study estimation of certain integral functionals of one or two densities with samples from stationary m-dependent sequences. We consider two types of U-statistic estimators for these functionals that are functions of the number of epsilon-close vector observations in the samples. We show that the estimators are consistent and obtain their rates of convergence under weak distributional assumptions. In particular, we propose estimators based on incomplete U-statistics which have favorable consistency properties even when m-dependence is the only dependence condition that can be imposed on the stationary sequences. The results can be used for divergence and entropy estimation, and thus find many applications in statistics and applied sciences.

  • 17.
    Leonenko, Nikolaj
    et al.
    School of Mathematics, Cardi® University, Senghennydd Road, Cardi® CF24 4AG, UK.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Statistical inference for the epsilon-entropy and the quadratic Renyi entropy2010In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 101, no 9, p. 1981-1994Article in journal (Refereed)
    Abstract [en]

    Entropy and its various generalizations are widely used in mathematical statistics, communication theory, physical and computer sciences for characterizing the amount of information in a probability distribution. We consider estimators of the quadratic Rényi entropy and some related characteristics of discrete and continuous probability distributions based on the number of coincident (or-close) vector observations in the corresponding independent and identically distributed sample. We show some asymptotic properties of these estimators (e.g., consistency and asymptotic normality). These estimators can be used in various problems in mathematical statistics and computer science (e.g., distribution identi¯cation problems, average case analysis for random databases, approximate pattern matching in bioinformatics, cryptography).

  • 18.
    Mishura, Yuliya
    et al.
    Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine .
    Ral’chenko, Kostiantyn
    Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Shevchenko, Georgiy
    Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine .
    Asymptotic properties of drift parameter estimator based on discrete observations of stochastic differential equation driven by fractional brownian motion2014In: Modern stochastics and applications / [ed] Volodymyr Korolyuk, Nikolaos Limnios, Yuliya Mishura, Lyudmyla Sakhno, Georgiy Shevchenko, Springer, 2014, p. 303-318Chapter in book (Refereed)
    Abstract [en]

    In this chapter, we consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic differential equations are constructed. It is proved that the estimators converge almost surely to the parameter value, as the observation interval expands and the distance between observations vanishes. A bound for the rate of convergence is given and numerical simulations are presented. As an auxilliary result of independent interest we establish global estimates for fractional derivative of fractional Brownian motion.

  • 19.
    Sedaghat, Mina
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wilkes, John
    Sjöstedt de Luna, Sara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Elmroth, Erik
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    DieHard: Reliable Scheduling to Survive Correlated failures in Cloud Data Centers2016In: 2016 16th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing (CCGrid), IEEE, 2016Conference paper (Refereed)
    Abstract [en]

    In large scale data centers, a single fault can lead to correlated failures of several physical machines and the tasks running on them, simultaneously. Such correlated failures can severely damage the reliability of a service or a job running on the failed hardware. This paper models the impact of stochastic and correlated failures on job reliability in a data center. We focus on correlated failures caused by power outages or failures of network components, on jobs running multiple replicas of identical tasks. We present a statistical reliability model and an approximation technique for computing a job’s reliability in the presence of correlated failures. In addition, we address the problem of scheduling a job with reliability constraints.We formulate the scheduling problem as an optimization problem, with the aim being to maintain the desired reliability with the minimum number of extra tasks to resist failures.We present a scheduling algorithm that approximates the minimum number of required tasks and a placement to achieve a desired job reliability. We study the efficiency of our algorithm using an analytical approach and by simulating a cluster with different failure sources and reliabilities. The results show that the algorithm can effectively approximate the minimum number of extra tasks required to achieve the job’s reliability.

  • 20.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Asymptotic behavior of mean uniform norms for sequences of Gaussian processes and fields2004Report (Other academic)
  • 21.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Asymptotic behavior of mean uniform norms for sequences of Gaussian processes and fields2006In: Extremes.: Statistical Theory and Applications in Science, Engineering and Economics, ISSN 1386-1999, Vol. 8, p. 161-169Article in journal (Refereed)
  • 22.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Random databases with approximate record mathching and e-entropy2004Report (Other academic)
  • 23.
    Seleznjev, Oleg
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Shykula, Mykola
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Quantization of random sequences and related statistical problems2006Report (Other academic)
  • 24.
    Seleznjev, Oleg
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Shykula, Mykola
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Run-length compression of quantized Gaussian stationary signals2012In: Random Operators and Stochastic Equations, ISSN 0926-6364, E-ISSN 1569-397X, Vol. 20, no 4, p. 311-328Article in journal (Refereed)
    Abstract [en]

    We consider quantization of random continuous-valued signals. In practice, analogue signals are quantized at sampling points with further compression. We study probabilistic models for run-length encoding (RLE) algorithm applied to quantized sampled random signals (Gaussian processes). This compression technique is widely used in digital signal and image processing. The mean (inverse) RLE compression ratio (or data rate savings) and its statistical inference are considered. In particular, the asymptotic normality for some estimators of this characteristic is shown. Numerical experiments for synthetic and real data are presented.

  • 25.
    Seleznjev, Oleg
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Shykula, Mykola
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Uniform and non-uniform quantization of Gaussian processes2012In: Mathematical Communications, ISSN 1331-0623, E-ISSN 1848-8013, Vol. 17, no 2, p. 447-460Article in journal (Refereed)
    Abstract [en]

    Quantization of a continuous-value signal into a discrete form (or discretization of amplitude) is a standard task in all analog/digital devices. We consider quantization of a signal (or random process) in a probabilistic framework. The quantization method presented in this paper can be applied to signal coding and storage capacity problems. In order to demonstrate a general approach, both uniform and non-uniform quantization of a Gaussian process are studied in more detail and compared with a conventional piecewise constant approximation. We investigate asymptotic properties of some accuracy characteristics, such as a random quantization rate, in terms of the correlation structure of the original random process when quantization cellwidth tends to zero. Some examples and numerical experiments are presented.

  • 26.
    Seleznjev, Oleg
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Thalheim, Bernard
    Joining multiple random tables2005Report (Other academic)
  • 27.
    Seleznjev, Oleg
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Moscow MV Lomonosov State Univ, Fac Math & Mech, Moscow 119992, Russia.
    Thalheim, Bernhard
    Kiel University, Institute of Informatics.
    Random databases with approximate record matching2010In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 12, no 1, p. 63-89Article in journal (Refereed)
    Abstract [en]

    In many database applications in   telecommunication, environmental and health sciences,  bioinformatics, physics, and econometrics, real-world data are uncertain  and subjected to errors. These data are processed, transmitted and stored in large databases. We consider stochastic modelling for databases with uncertain data and for some basic database operations (for example, join, selection) with exact and approximate matching.  Approximate join is used for merging or data deduplication in large databases. Distribution and mean of the join sizes are studied for random databases.  A random database is treated as a table with independent random records with a  common distribution (or a set of random tables). These results can be used for  integration of information from different databases, multiple join optimization, and various probabilistic algorithms for structured random data.

  • 28.
    Shykula, Mykola
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stochastic structure of asymptotic quantization errors2006In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 76, no 5, p. 453-464Article in journal (Refereed)
    Abstract [en]

    We consider quantization of continuous-valued random variables and processes in a probabilistic framework. Stochastic structure for non-uniform quantization errors is studied for a wide class of random variables. Asymptotic properties of the additive quantization noise model for a random process are derived for uniform and non-uniform quantizers. Some examples and numerical experiments demonstrating the rate of convergence in the obtained asymptotic results are presented.

  • 29.
    Shykula, Mykola
    et al.
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Stochastic structure of asymptotic quantization errors2004Report (Other academic)
  • 30.
    Shykula, Mykola
    et al.
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Uniform quantization of random processes2004Report (Other academic)
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  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf