| 1. |
- Abramowicz, Konrad, et al.
(författare)
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Multivariate piecewise linear interpolation of a random field
- 2011
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 2. |
- Abramowicz, Konrad, 1983-
(författare)
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Numerical analysis for random processes and fields and related design problems
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D).In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed.In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided.In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity.In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.
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| 3. |
- Abramowicz, Konrad, 1983-, et al.
(författare)
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On the error of the Monte Carlo pricing method for Asian option
- 2008
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Ingår i: Journal of Numerical and Applied Mathematics. - 0868-6912. ; 96:1, s. 1-10
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Abramowicz, Konrad, et al.
(författare)
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Piecewise multilinear interpolation of a random field
- 2013
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 45:4, s. 945-959
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Abramowicz, Konrad, 1983-, et al.
(författare)
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Spline approximation of a random process with singularity
- 2011
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Ingår i: Journal of Statistical Planning and Inference. - : Elsevier. - 0378-3758 .- 1873-1171. ; 141:3, s. 1333-1342
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 6. |
- Abramowicz, Konrad, et al.
(författare)
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Stratified Monte Carlo quadrature for continuous random fields
- 2015
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Ingår i: Methodology and Computing in Applied Probability. - New York : Springer Science+Business Media B.V.. - 1387-5841 .- 1573-7713. ; 17:1, s. 59-72
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
- Ali-Eldin, Ahmed, 1985-, et al.
(författare)
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How will your workload look like in 6 years? : Analyzing Wikimedia's workload
- 2014
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Ingår i: Proceedings of the 2014 IEEE International Conference on Cloud Engineering (IC2E 2014). - : IEEE Computer Society. - 9781479937660 ; , s. 349-354
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Konferensbidrag (refereegranskat)abstract
- 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%.
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| 8. |
- Ali-Eldin, Ahmed, 1985-, et al.
(författare)
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Measuring cloud workload burstiness
- 2014
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Ingår i: 2014 IEEE/ACM 7th International Conference on Utility and Cloud Computing (UCC). - : IEEE conference proceedings. - 9781479978816 ; , s. 566-572
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 9. |
- Belyaev, Yuri K, et al.
(författare)
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Approaching in distribution with applications to resampling of stochastic processes
- 2000
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Ingår i: Scandinavian Journal of Statistics. - : Wiley-VCH Verlagsgesellschaft. - 0303-6898 .- 1467-9469. ; 27:2, s. 371-384
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 10. |
- Hashorva, Enkelejd, et al.
(författare)
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Approximation of a Random Process with Variable Smoothness
- 2015
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Ingår i: Mathematical Statistics and Limit Theorems. - Cham : Springer. - 9783319124414 - 9783319124421 ; , s. 189-208
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Bokkapitel (refereegranskat)abstract
- We consider the rate of piecewise constant approximation to a locally stationary process X(t),t ε [0,1] , having a variable smoothness index α(t). Assuming that α(⋅) attains its unique minimum at zero and satisfieswe propose a method for construction of observation points (composite dilated design) such that the integrated mean square errorwhere a piecewise constant approximation X n 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.
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