introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

preconditioned conjugate gradients for solving singular systems abstract in this paper the preconditioned conjugate gradient method is used to solve the system of linear equations ax b where a is a singular symmetric positive semi definite matrix the method diverges if b is not exactly in the range r a of a if the null space n a of a is explicitly known then this divergence can be avoided by subtracting from b its orthogonal projection onto n a as well as analysing this subtraction conditions necessary for the existence of a nonsingular incomplete cholesky decomposition are given finally the theory is applied to the discretized semi definite neumann problem

global structure and finite size effects in the f alpha of diffusion limited aggregates the multifractal spectrum f alpha characterizing the scaling properties of the growth probability on the boundary of radial diffusion limited aggregates is known to exhibit strong finite size effects we demonstrate that there exists a correlation between these finite size effects and those present in measurements of the angular width of the fjords which lie between the principal cluster arms we subsequently conclude that it is the evolution in the global structure of the clusters which is responsible for the slow convergence in f alpha and discuss how this global structure induces a phase transition in f alpha

comment on identification in the linear errors in variables model on considere un modele etudie par kapteyn et wansbeek 1983 et on presente des conditions necessaires et suffisantes d identification

generalized projection pursuit regression projection pursuit regression ppr can be used to estimate a smooth function of several variables from noisy and scattered data the estimate is a sum of smoothed one dimensional projections of the variables this paper discusses an extension of ppr to exponential family distributions called generalized projection pursuit regression gppr the proposed model allows multiple responses and nonlinear projections of the variables estimators are defined as minimizers of penalized cost functionals and estimation is related to the local scoring procedure for generalized additive models gams smooths are updated using a blockwise gauss seidel bgs method and convergence is shown for this procedure the use of generalized cross validation gcv to estimate smoothing parameters is discussed experimental results are shown for two types of data

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

iterative aggregation disaggregation techniques for nearly uncoupled markov chains iterative aggregation disaggregation methods provide an efficient approach for computing the stationary probability vector of nearly uncoupled also known as nearly completely decomposable markov chains three such methods that have appeared in the literature recently are considered and their similarities and differences are outlined specifically it is shown that the method of takahashi corresponds to a modified block gauss seidel step and aggregation whereas that of vantilborgh corresponds to a modified block jacobi step and aggregation the third method that of koury et al is equivalent to a standard block gauss seidel step and iteration for each of these methods a lemma is established which shows that the unique fixed point of the iterative scheme is the left eigenvector corresponding to the dominant unit eigenvalue of the stochastic transition probability matrix in addition conditions are established for the convergence of the first two of these methods convergence conditions for the third having already been established by stewart et al all three methods are shown to have the same asymptotic rate of convergence

self similarity of harmonic measure on dla abstract the right hand side of the curve of the harmonic measure on dla is undefined this does not necessarily imply that the harmonic measure and the dla geometry are not self similar we show for off lattice dla that the right hand tail satisfies a different rescaling rule this cauchy rescaling is compatible with self similarity the analysis is done on off off lattice dla in which both the brownian motion and the laplace equation are off lattice the cluster sizes range between 32 and 50 000 atoms the square lattice used to numerically estimate the laplacian potential introduces a lower cutoff on the spatial resolution of this potential we find a dependence of the right tail of the distribution of holders on this ultraviolet cutoff whereas the shape of the tail does depend on this ultraviolet lattice cutoff the applicability of the collapse rules do not

nonparametric identification and semiparametric estimation of classical measurement error models without side information virtually all methods aimed at correcting for covariate measurement error in regressions rely on some form of additional information e g validation data known error distributions repeated measurements or instruments in contrast we establish that the fully nonparametric classical errors in variables model is identifiable from data on the regressor and the dependent variable alone unless the model takes a very specific parametric form this parametric family includes but is not limited to the linear specification with normally distributed variables as a well known special case this result relies on standard primitive regularity conditions taking the form of smoothness constraints and nonvanishing characteristic functions assumptions our approach can handle both monotone and nonmonotone specifications provided the latter oscillate a finite number of times given that the very specific unidentified parametric functional form is arguably the exception rather than the rule this identification r

measurement error in multiple equations tobin s q and corporate investment saving and debt abstract we characterize the sharp identification regions for the coefficients in a system of linear equations that share an explanatory variable measured with classical error we demonstrate the identification gain from analyzing the equations jointly we derive the sharp identification regions under any configuration of three auxiliary assumptions these restrict the noise to signal ratio the coefficients of determination and the signs of the correlations among the cross equation disturbances for inference we implement results on intersection bounds the application studies the effects of cash flow on the investment saving and debt of firms when tobin s q serves as a proxy for marginal q

higher moment estimators for linear regression models with errors in the variables abstract this paper proposes consistent instrumental variable estimators for linear regression models with errors in the variables that require no extraneous information these estimators are based on sample moments of order higher than two while similar estimators proposed previously in the literature seem to be quite erratic our experimental f findings suggest that our estimators perform better than ordinary least squares estimators in terms of root mean squared errors and also in terms of size of type i errors of standard tests in many typical situations of economic analyses tests for the presence of errors in the variables are also described

an iterative aggregation disaggregation algorithm for solving linear equations a general procedure is given for solving large sets of linear equations by first rewriting them in a form suitable for aggregation of both the variables and equations followed by disaggregation a computational algorithm which iteratively aggregates and disaggregates is shown to converge geometrically to the exact solution provided the original problem has a structure suitable for such aggregation the algorithm exhibits fast computation times small main memory requirements and robustness to the starting point a rigorous foundation for aggregation and disaggregation is provided by the equations employed by this algorithm

triangular and skew symmetric splitting method for numerical solutions of markov chains in this paper a theorem is presented to indicate that there exists a nonnegative constant e 0 such that the matrix a q email protected is a positive definite matrix where email protected r n x n is an identity matrix and q email protected r n x n is a matrix with positive diagonal and nonpositive off diagonal elements then a class of triangular and skew symmetric splitting tss iteration method is applied to solve the positive definite linear system ax b for obtaining the stationary probability vector of an irreducible markov chain theoretical analyses show that the tss iteration method converges unconditionally to the unique solution of the linear system with the upper bound of its contraction factor dependent only on the spectrum of the triangular part and independent of the eigenvectors of the matrices involved moreover the inexact triangular and skew symmetric splitting itss iteration method which employs certain krylov subspace methods as the inner iteration processes at each step of the outer tss iteration method is proposed to accelerate the convergence of the tss iteration method numerical experiments are used to illustrate the effectiveness of the tss and itss iteration methods

aggregation disaggregation methods for computing the stationary distribution of a markov chain we implement and analyse aggregation disaggregation procedures constructed to accelerate the convergence of successive approximation methods suitable for computing the stationary distribution of a finite markov chain we define six of these methods and analyse them in detail in particular we show that some existing procedures lie in the aggregation disaggregation framework we set and hence can be considered as special cases also for all described methods we identify cases where they are promising numerical examples for the applications of some of the methods for nearly completely decomposable stochastic matrices are given as well

on a two level multigrid solution method for finite markov chains abstract we study a two level algebraic multigrid scheme for computing the stationary distribution of a homogeneous markov chain with a finite state space we reveal its relationship to an iterative aggregation disaggregation method provide an error analysis and prove its local convergence moreover we discuss the stochastic meaning of a block sor procedure that can be used as iterative smoother of the multigrid scheme

multilevel adaptive aggregation for markov chains with application to web ranking a multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described the method is a special case of the adaptive smoothed aggregation and adaptive algebraic multigrid methods for sparse linear systems and is also closely related to certain extensively studied iterative aggregation disaggregation methods for markov chains in contrast to most existing approaches our aggregation process does not employ any explicit advance knowledge of the topology of the markov chain instead adaptive agglomeration is proposed that is based on the strength of connection in a scaled problem matrix in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level the strength of connection is determined as in the algebraic multigrid method and the aggregation process is fully adaptive with optimized aggregates chosen in each step of the iteration and at all recursive levels the multilevel method is applied to a set of stochastic matrices that provide models for web page ranking numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods the tests also provide more insight into why google s pagerank model is a successful model for determining a ranking of web pages

the smoothed laplacian model a mean field approximation to laplacian growth abstract we introduce a new model for irreversible growth inspired by laplacian growth processes the growth probability is defined using an approximation for the laplacian field which takes into account the screening effect only at the largest scale for this reason the resulting cluster is not a fractal as those generated with diffusion limited aggregation but this makes the model fully soluble in a mean field approximation the results are non trivial and give a clear description of the stationary regime of growth which corresponds to the dynamical equilibrium between the growth probability and the evolving cluster this model then provides a new perspective in the study of laplacian growth phenomena from which it is possible to make predictions on the behaviour of more complex systems for example it is possible to recover in a simple way the correct value of the finger width in the saffman taylor problem as the only one consistent with this equilibrium situation it is also possible to understand the physical origin of the statistical properties which give rise to the breakdown of the multifractal spectrum for negative moments applied to the probability distribution of the dielectric breakdown model

mean field analysis of the equilibrium of patterns growing in a laplacian field the properties of the shape of patterns generated by random growth in a laplacian field are investigated using a simple model the smoothed laplacian model the relation between this model and more complex systems such as saffman taylor fingers or diffusion limited aggregates stands in the recently recognized universality of patterns whose evolution is controlled by the laplace equation and in an analytical approximation for the laplacian field itself the smoothed laplacian model allows one to treat analytically the equilibrium of the structure in the smoothed laplacian field in a mean field approximation the results on this model are extended to radial geometry thus providing theoretical insight in the dependence of the shape on the geometry of boundary conditions in which the process takes place and on the parameter that tunes the strength of the laplacian field in the same way as the parameter eta does for the dielectric breakdown model this new picture is compared to numerical simulation results of dla and dbm and proves to be consistent with them

smoothed aggregation multigrid for markov chains a smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse markov chain it is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for markov chains that have been proposed in the literature the proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems supplemented with a new lumping technique that assures well posedness of the coarse level problems the coarse level operators are singular m matrices on all levels resulting in strictly positive coarse level corrections on all levels numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems both when geometric and algebraic aggregation strategies are used

numerical methods in markov chain modeling this paper describes and compares several methods for computing stationary probability distributions of markov chains the main linear algebra problem consists of computing an eigenvector of a sparse nonsymmetric matrix associated with a known eigenvalue it can also be cast as a problem of solving a homogeneous singular linear system we present several methods based on combinations of krylov subspace techniques single vector power iteration relaxation procedures and acceleration techniques we compare the performance of these methods on some realistic problems

an arnoldi extrapolation algorithm for computing pagerank the arnoldi type algorithm proposed by golub and greif g golub c greif an arnoldi type algorithm for computing pagerank bit 46 2006 759 771 is a restarted krylov subspace method for computing pagerank however this algorithm may not be efficient when the damping factor is high and the dimension of the search subspace is small in this paper we first develop an extrapolation method based on ritz values we then consider how to periodically knit this extrapolation method together with the arnoldi type algorithm the resulting algorithm is the arnoldi extrapolation algorithm the convergence of the new algorithm is analyzed numerical experiments demonstrate the numerical behavior of this algorithm

fixed scale transformation approach to the multifractcal properties of the growth probabilities in the dielectric breakdown model the separation of the properties of the growth probability distribution in two different contributions as discussed in the previous paper corresponds naturally to the approximation scheme of the fixed scale transformation fst method the growth probabilities used to compute the fst matrix elements represent the essential elements of the multiplicative process that gives rise to the regular part the only one relevant to the growth process of the multifractal spectrum the fst uses these probabilities directly without the need of introducing a multifractal spectrum explicitly this however can be obtained as a by product of the fst method we present here analytical calculations for the regular part of the multifractal spectrum of the dielectric breakdown model with different values of the parameter the results are good for 1 and less accurate for 1 in fact for small values and in order to recover the eden limit it is necessary to go to higher order and possibly to include self affine properties explicitly

constructing instruments for regressions with measurement error when no additional data are available with an application to patents and r d constructing instruments for regressions with measurement error when no additional data are available with an application to patents and r d

an algebraic multigrid preconditioner for a class of singular m matrices we apply algebraic multigrid amg as a preconditioner for solving large singular linear systems of the type i t t x 0 with gmres here t is assumed to be the transition matrix of a markov process although amg and gmres were originally designed for the solution of regular systems with adequate adaptation their applicability can be extended to problems as described above

the numerical solution of stochastic automata networks abstract stochastic automata networks san s have recently received attention in the literature as an efficient means of modelling parallel systems such as communicating processes concurrent processors shared memory etc the advantage that the san approach has over generalized stochastic petri nets and indeed over any markovian analysis that requires the generation of a transition matrix is that its representation remains compact even as the number of states in the underlying markov chain begins to explode our concern in this paper is with the numerical issues that are involved in solving san networks we introduce stochastic automata and consider the numerical difficulties that result from their interaction we examine how the product of a vector with a compact san descriptor may be formed for this operation is the basis of all iterative solution methods we describe possible solution methods including the power method the method of arnoldi and gmres and show that the two latter methods greatly out perform the power method the method usually used in conjunction with stochastic automata networks finally we consider one possible means of preconditioning but conclude that further research is needed

properties of the growth probability for the dielectric breakdown model in cylinder geometry abstract we study the properties of the growth probabilities for diffusion limited aggregation and the dielectric breakdown model in the steady state regime of the cylinder geometry the results show a rather unambiguous picture with the following properties the projection of the growth probability along the growth direction is exponential contrary to the gaussian behavior of the radial case one can distinguish two regions one with simple multifractal properties corresponding to the growing zone and a second one which accounts for the exponential decay of the growth probability this situation can be explained in terms of a first order transition in the multifractal spectrum at q 1 this corresponds to a different picture with respect to those that have been proposed in the literature the properties of the growing interface could be universal while the small probability part is non universal and it depends on the particular geometry however we can show that this part is irrelevant with respect to the growth process even though it is determined by it

theory of laplacian fractals diffusion limited aggregation and dielectric breakdown model abstract we describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the laplace equation and a stochastic field this new theory provides a systematic method for the calculation of the fractal dimension d and of the multifractal spectrum of the growth probability a detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented our approach exploits the scale invariance of the laplace equation that implies that the structure is self similar both under growth and scale transformation this allows one to introduce a fixed scale transformation instead of coarse graining as in the renormalization group theory that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process for the calculation of the matrix elements of this transformation one has to consider an infinite but rapidly convergent number of processes that occurs outside a considered diagram

product type skew hermitian triangular splitting iteration methods for strongly non hermitian positive definite linear systems by further generalizing the modified skew hermitian triangular splitting iteration methods studied in l wang z z bai skew hermitian triangular splitting iteration methods for non hermitian positive definite linear systems of strong skew hermitian parts bit numer math 44 2004 363 386 in this paper we present a new iteration scheme called the product type skew hermitian triangular splitting iteration method for solving the strongly non hermitian systems of linear equations with positive definite coefficient matrices we discuss the convergence property and the optimal parameters of this method moreover when it is applied to precondition the krylov subspace methods the preconditioning property of the product type skew hermitian triangular splitting iteration is analyzed in detail numerical results show that the product type skew hermitian triangular splitting iteration method can produce high quality preconditioners for the krylov subspace methods for solving large sparse positive definite systems of linear equations of strong skew hermitian parts

minimum distance estimation of the errors in variables model using linear cumulant equations abstract we consider a multiple mismeasured regressor errors in variables model we develop closed form minimum distance estimators from any number of estimating equations which are linear in the third and higher cumulants of the observable variables using the cumulant estimators alters qualitative inference relative to ordinary least squares in two applications related to investment and leverage regressions the estimators perform well in monte carlos calibrated to resemble the data from our applications although the cumulant estimators are asymptotically equivalent to the moment estimators from erickson and whited 2002 the finite sample performance of the cumulant estimators exceeds that of the moment estimators

circulant preconditioners for stochastic automata networks stochastic automata networks sans are widely used in modeling communication systems manufacturing systems and computer systems the san approach gives a more compact and efficient representation of the network when compared to the stochastic petri nets approach to find the steady state distribution of sans it requires solutions of linear systems involving the generator matrices of the sans very often direct methods such as the lu decomposition are inefficient because of the huge size of the generator matrices an efficient algorithm should make use of the structure of the matrices iterative methods such as the conjugate gradient methods are possible choices however their convergence rates are slow in general and preconditioning is required we note that the milu and minv based preconditioners are not appropriate because of their expensive construction cost in this paper we consider preconditioners obtained by circulant approximations of sans they have low construction cost and can be inverted efficiently we prove that if only one of the automata is large in size compared to the others then the preconditioned system of the normal equations will converge very fast numerical results for three different sans solved by cgs are given to illustrate the fast convergence of our method

estimation of regression coefficients with the help of characteristic functions abstract in this paper some easily applicable estimators of the regression coefficient in a simple regression model with errors in variables are derived usually one uses sample moments to estimate the regression coefficient however we use empirical characteristic functions in several points besides theoretical derivations a small study with generated data is discussed this study compares our method with estimating methods which use higher order moments

algebraic schwarz methods for the numerical solution of markov chains the convergence of additive and multiplicative schwarz methods for computing certain characteristics of markov chains such as stationary probability vectors and mean first passage matrices is studied the main result is a convergence theorem for multiplicative schwarz iterations when applied to singular systems as a byproduct a convergence result for alternating iterations is also obtained it is also shown that when the markov chain is ergodic additive and multiplicative schwarz methods can be applied to the nonsingular systems that result from reducing the equations the so called coarse grid corrections are also studied