Browsing by Subject "Stochastic models"

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Open Access
Analyzing large sparse Markov chains of Kronecker products
(IEEE, 2009) Dayar, Tuğrul
Kronecker products are used to define the underlying Markov chain (MC) in various modeling formalisms, including compositional Markovian models, hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker structured representation rather than a flat one is to alleviate the storage requirements associated with the MC. With this approach, systems that are an order of magnitude larger can be analyzed on the same platform. In the Kronecker based approach, the generator matrix underlying the MC is represented using Kronecker products [6] of smaller matrices and is never explicitly generated. The implementation of transient and steady-state solvers rests on this compact Kronecker representation, thanks to the existence of an efficient vector-Kronecker product multiplication algorithm known as the shuffle algorithm [6]. The transient distribution can be computed through uniformization using vector-Kronecker product multiplications. The steady-state distribution also needs to be computed using vector-Kronecker product multiplications, since direct methods based on complete factorizations, such as Gaussian elimination, normally introduce new nonzeros which cannot be accommodated. The two papers [2], [10] provide good overviews of iterative solution techniques for the analysis of MCs based on Kronecker products. Issues related to reachability analysis, vector-Kronecker product multiplication, hierarchical state space generation in Kronecker based matrix representations for large Markov models are surveyed in [5]. Throughout our discussion, we make the assumption that the MC at hand does not have unreachable states, meaning it is irreducible. And we take an algebraic view [7] to discuss recent results related to the analysis of MCs based on Kronecker products independently from modeling formalisms. We provide background material on the Kronecker representation of the generator matrix underlying a CTMC, show that it has a rich structure which is nested and recursive, and introduce a small CTMC whose generator matrix is expressed as a sum of Kronecker products; this CTMC is used as a running example throughout the discussion. We also consider preprocessing of the Kronecker representation so as to expedite numerical analysis. We discuss permuting the nonzero structure of the underlying CTMC symmetrically by reordering, changing the orders of the nested blocks by grouping, and reducing the size of the state space by lumping. The steady-state analysis of CTMCs based on Kronecker products is discussed for block iterative methods, multilevel methods, and preconditioned projection methods, respectively. The results can be extended to DTMCs based on Kronecker products with minor modifications. Areas that need further research are mentioned as they are discussed. Our contribution to this area over the years corresponds to work along iterative methods based on splittings and their block versions [11], associated preconditioners to be used with projection methods [4], near complete decomposability [8], a method based on iterative disaggregation for a class of lumpable MCs [9], a class of multilevel methods [3], and a recent method based on decomposition for weakly interacting subsystems [1]. © 2009 IEEE.
Open Access
An integrated approach to inventory and flexible capacity management subject to fixed costs and non-stationary stochastic demand
(Springer, 2009) Tan, T.; Alp O.
In a manufacturing system with flexible capacity, inventory management can be coupled with capacity management in order to handle fluctuations in demand more effectively. Typical examples include the effective use of temporary workforce and overtime production. In this paper, we discuss an integrated model for inventory and flexible capacity management under non-stationary stochastic demand with the possibility of positive fixed costs, both for initiating production and for using contingent capacity. We analyze the characteristics of the optimal policies for the integrated problem. We also evaluate the value of utilizing flexible capacity under different settings, which enable us to develop managerial insights. © 2008 The Author(s).
Open Access
On compact solution vectors in Kronecker-based Markovian analysis
(Elsevier, 2017) Buchholz, P.; Dayar T.; Kriege, J.; Orhan, M. C.
State based analysis of stochastic models for performance and dependability often requires the computation of the stationary distribution of a multidimensional continuous-time Markov chain (CTMC). The infinitesimal generator underlying a multidimensional CTMC with a large reachable state space can be represented compactly in the form of a block matrix in which each nonzero block is expressed as a sum of Kronecker products of smaller matrices. However, solution vectors used in the analysis of such Kronecker-based Markovian representations require memory proportional to the size of the reachable state space. This implies that memory allocated to solution vectors becomes a bottleneck as the size of the reachable state space increases. Here, it is shown that the hierarchical Tucker decomposition (HTD) can be used with adaptive truncation strategies to store the solution vectors during Kronecker-based Markovian analysis compactly and still carry out the basic operations including vector–matrix multiplication in Kronecker form within Power, Jacobi, and Generalized Minimal Residual methods. Numerical experiments on multidimensional problems of varying sizes indicate that larger memory savings are obtained with the HTD approach as the number of dimensions increases. © 2017 Elsevier B.V.
Open Access
On the modeling of CO2 EUA and CER prices of EU-ETS for the 2008–2012 period
(John Wiley and Sons, 2016) Gürler, Ü.; Yenigün, D.; Çağlar, M.; Berk, E.
Increased consumption of fossil fuels in industrial production has led to a significant elevation in the emission of greenhouse gases and to global warming. The most effective international action against global warming is the Kyoto Protocol, which aims to reduce carbon emissions to desired levels in a certain time span. Carbon trading is one of the mechanisms used to achieve the desired reductions. One of the most important implications of carbon trading for industrial systems is the risk of uncertainty about the prices of carbon allowance permits traded in the carbon markets. In this paper, we consider stochastic and time series modeling of carbon market prices and provide estimates of the model parameters involved, based on the European Union emissions trading scheme carbon allowances data obtained for 2008–2012 period. In particular, we consider fractional Brownian motion and autoregressive moving average–generalized autoregressive conditional heteroskedastic modeling of the European Union emissions trading scheme data and provide comparisons with benchmark models. Our analysis reveals evidence for structural changes in the underlying models in the span of the years 2008–2012. Data-driven methods for identifying possible change-points in the underlying models are employed, and a detailed analysis is provided. Our analysis indicated change-points in the European Union Allowance (EUA) prices in the first half of 2009 and in the second half of 2011, whereas in the Certified Emissions Reduction (CER) prices three change-points have appeared, in the first half of 2009, the middle of 2011, and in the second half of 2012. These change-points seem to parallel the global economic indicators as well.
Open Access
On the number of clusters in channel model
(IEEE, 2006-08) Akkaya, Keziban; Tunç, Celal Alp; Aktaş, Defne; Altıntaş, Ayhan
Typically, scatterers in an environment are not distributed uniformly but rather observed to occur in clusters. Identification of clusters is an issue under discussion. To this end, we study the effect of number of clusters on channel model through computer simulations. We focus on a geometric stochastic directional channel model based on COST259. Fixing a scatterer scenario, we calculate root mean square delay and angular spreads when scatterers are grouped into varying numbers of clusters and study how sensitive these parameters are to the number of clusters used in this channel model. © 2006 IEEE.
Open Access
On the numerical solution of Kronecker-based infinite level-dependent QBD processes
(2013) Baumann, H.; Dayar, T.; Orhan, M. C.; Sandmann, W.
Infinite level-dependent quasi-birth-and-death (LDQBD) processes can be used to model Markovian systems with countably infinite multidimensional state spaces. Recently it has been shown that sums of Kronecker products can be used to represent the nonzero blocks of the transition rate matrix underlying an LDQBD process for models from stochastic chemical kinetics. This paper extends the form of the transition rates used recently so that a larger class of models including those of call centers can be analyzed for their steady-state. The challenge in the matrix analytic solution then is to compute conditional expected sojourn time matrices of the LDQBD model under low memory and time requirements after truncating its countably infinite state space judiciously. Results of numerical experiments are presented using a Kronecker-based matrix-analytic solution on models with two or more countably infinite dimensions and rules of thumb regarding better implementations are derived. In doing this, a more recent approach that reduces memory requirements further by enabling the computation of steady-state expectations without having to obtain the steady-state distribution is also considered. © 2013 Elsevier B.V. All rights reserved.
Open Access
A simulation model for military deployment
(IEEE, 2007) Yıldırım, Uğur Z.; Sabuncuoğlu, İhsan; Tansel, Barbaros
The Deployment Planning Problem (DPP) for military units may in general be defined as the problem of planning the movement of geographically dispersed military units from their home bases to their final destinations using different transportation assets and a multimodal transportation network while obeying the constraints of a time-phased force deployment data describing the movement requirements for troops and equipment. Our main contribution is to develop a GISbased, object-oriented, loosely-coupled, modular, platformindependent, multi-modal and medium-resolution discrete event simulation model to test the feasibility of deployment scenarios. While our simulation model is not a panacea for all, it allows creation and testing the feasibility of a given scenario under stochastic conditions and can provide insights into potential outcomes in a matter of a few hours.
Open Access
Simultaneous 3-D motion estimation and wire-frame model adaptation including photometric effects for knowledge-based video coding
(IEEE, 1994) Bozdağı, Gözde; Tekalp, A. M.; Onural, Levent
We address the problem of 3-D motion estimation in the context of knowledge-based coding of facial image sequences. The proposed method handles the global and local motion estimation and the adaptation of a generic wire-frame to a particular speaker simultaneously within an optical flow based framework including the photometric effects of motion. We use a flexible wire-frame model whose local structure is characterized by the normal vectors of the patches which are related to the coordinates of the nodes. Geometrical constraints that describe the propagation of the movement of the nodes are introduced, which are then efficiently utilized to reduce the number of independent structure parameters. A stochastic relaxation algorithm has been used to determine optimum global motion estimates and the parameters describing the structure of the wire-frame model. For the initialization of the motion and structure parameters, a modified feature based algorithm is used. Experimental results with simulated facial image sequences are given.
Open Access
Space weather activities of IONOLAB group using TNPGN GPS Network
(IEEE, 2011) Aktug, B.; Lenk O.; Kurt, M.; Parmaksiz, E.; Ozdemir, S.; Arikan F.; Sezen, U.; Toker, C.; Arıkan, Orhan
Characterization and constant monitoring of variability of the ionosphere is of utmost importance for the performance improvement of HF communication, Satellite communication, navigation and guidance systems, Low Earth Orbit (LEO) satellite systems, Space Craft exit and entry into the atmosphere and space weather. Turkish National Permanent GPS Network (TNPGN) is the Reference Station Network of 146 continuously-operating GNSS stations of which are distributed uniformly across Turkey and North Cyprus Turkish Republic since May 2009. IONOLAB group is currently investigating new techniques for space-time interpolation, and automatic mapping of TEC through a TUBITAK research grant. It is utmost importance to develop regional stochastic models for correction of ionospheric delay in geodetic systems and also form a scientific basis for communication link characterization. This study is a brief summary of the efforts of IONOLAB group in monitoring of space weather, and correction of geodetic positioning errors due to ionosphere using TNPGN. © 2011 IEEE.
Open Access
Unsupervised classification of remotely sensed images using Gaussian mixture models and particle swarm optimization
(IEEE, 2010) Arı, Çağlar; Aksoy, Selim
Gaussian mixture models (GMM) are widely used for un-supervised classification applications in remote sensing. Expectation-Maximization (EM) is the standard algorithm employed to estimate the parameters of these models. However, such iterative optimization methods can easily get trapped into local maxima. Researchers use population-based stochastic search algorithms to obtain better estimates. We present a novel particle swarm optimization-based algorithm for maximum likelihood estimation of Gaussian mixture models. The proposed approach provides solutions for important problems in effective application of population-based algorithms to the clustering problem. We present a new parametrization for arbitrary covariance matrices that allows independent updating of individual parameters during the search process. We also describe an optimization formulation for identifying the correspondence relations between different parameter orderings of candidate solutions. Experiments on a hyperspectral image show better clustering results compared to the commonly used EM algorithm for estimating GMMs. © 2010 IEEE.