the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

multiple testing for pattern identification with applications to microarray time course experiments in time course experiments it is often desirable to identify genes that exhibit a specific pattern of differential expression over time and thus gain insights into the mechanisms of the underlying biological processes two challenging issues in the pattern identification problem are i how to combine the simultaneous inferences across multiple time points and ii how to control the multiplicity while accounting for the strong dependence we formulate a compound decision theoretic framework for set wise multiple testing and propose a data driven procedure that aims to minimize the missed set rate subject to a constraint on the false set rate the hidden markov model proposed in yuan and kendziorski 2006 is generalized to capture the temporal correlation in the gene expression data both theoretical and numerical results are presented to show that our data driven procedure controls the multiplicity provides an optimal way of combining simultaneous inferences across multiple time points and greatly i

nearest neighbor classification with dependent training sequences the asymptotic classification risk for nearest neighbor procedures is well understood in the case of i i d training sequences in this article we generalize these results to a class of dependent models including hidden markov models in the case where the observed patterns have lebesgue densities the asymptotic risk takes the same expression as in the i i d case for discrete distributions we show that the asymptotic risk depends on the rule used for breaking ties of equal distances

stepwise signal extraction via marginal likelihood this article studies the estimation of a stepwise signal to determine the number and locations of change points of the stepwise signal we formulate a maximum marginal likelihood estimator which can be computed with a quadratic cost using dynamic programming we carry out an extensive investigation on the choice of the prior distribution and study the asymptotic properties of the maximum marginal likelihood estimator we propose to treat each possible set of change points equally and adopt an empirical bayes approach to specify the prior distribution of segment parameters a detailed simulation study is performed to compare the effectiveness of this method with other existing methods we demonstrate our method on single molecule enzyme reaction data and on dna array comparative genomic hybridization cgh data our study shows that this method is applicable to a wide range of models and offers appealing results in practice supplementary materials for this article are available online

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

a two state markov mixture model for a time series of epileptic seizure counts this paper discusses a model for a time series of epileptic seizure counts in which the mean of a poisson distribution changes according to an underlying two state markov chain the em algorithm dempster laird and rubin 1977 journal of the royal statistical society series b 39 1 38 is used to compute maximum likelihood estimators for the parameters of this two state mixture model and extensions are made allowing for nonstationarity the model is illustrated using daily seizure counts for patients with intractable epilepsy and results are compared with a simple poisson distribution and poisson regressions some simulation results are also presented to demonstrate the feasibility of this model

convergence of the maximum a posteriori path estimator in hidden markov models in a hidden markov model hmm the underlying finite state markov chain cannot be observed directly but only by an additional process we are interested in estimating the unknown path of the markov chain the most widely used estimator is the maximum a posteriori path estimator map path estimator it can be calculated effectively by the viterbi 1967 algorithm as is e g frequently done in the field of coding theory correction of intersymbol interference and speech recognition we investigate component wise convergence of the map path estimator convergence is shown under the condition of unbounded likelihood ratios this condition is satisfied in the important case of hmms with additive white gaussian noise we also prove convergence if the markov chain has two states the so called viterbi paths are an important tool for obtaining these results

a nonlinear renewal theory soit t le premier instant auquel une marche aleatoire perturbee traverse une frontiere non lineaire on considere les approximations de la distribution de l exces au dela de la frontiere de l esperance du temps d arret et de la variance du temps d arret

exact likelihood evaluation in a markov mixture model for time series of seizure counts this paper provides an alternative to albert s 1991 biometrics 47 1371 1381 approximation to the e step when using the em algorithm for parameter estimation in markov mixture models use of a recursive algorithm of baum et al 1970 annals of mathematical statistics 41 164 171 results in exact evaluation of the likelihood optimal parameter estimates and very efficient computation applications to time series of seizure counts and fetal movements clearly show the advantages of this exact approach

asymptotical statistics of misspecified hidden markov models this paper deals with the problem of modeling data generated by an ergodic stochastic process as the output of a hidden markov model hmm more specifically we consider the problem of fitting a parametric family of hmm with continuous output to an ergodic stochastic process with continuous values which does not necessarily belong to the family in this context we derive the main asymptotic results almost sure consistency of the maximum likelihood estimator asymptotic normality of the estimation error and the exact rates of almost sure convergence

maximum likelihood versus likelihood free quantum system identification in the atom maser we consider the problem of estimating a dynamical parameter of a markovian quantum open system the atom maser by performing continuous time measurements in the system s output outgoing atoms two estimation methods are investigated and compared firstly the maximum likelihood estimator mle takes into account the full measurement data and is asymptotically optimal in terms of its mean square error secondly the likelihood free method of approximate bayesian computation abc produces an approximation of the posterior distribution for a given set of summary statistics by sampling trajectories at different parameter values and comparing them with the measurement data via chosen statistics building on previous results which showed that atom counts are poor statistics for certain values of the rabi angle we apply mle to the full measurement data and estimate its fisher information we then select several correlation statistics such as waiting times distribution of successive identical detections and use them as input of the abc algorithm the resulting posterior distribution follows closely the data likelihood showing that the selected statistics capture most statistical information about the rabi angle

equivalence classes and local asymptotic normality in system identification for quantum markov chains we consider the problem of identifying and estimating dynamical parameters of an ergodic quantum markov chain when only the stationary output is accessible for measurements the starting point of the analysis is the fact that the knowledge of the output state completely fixes the dynamics up to an equivalence class of coordinate transformation consisting of a multiplication by a phase and a unitary conjugation of the kraus operators

asymptotic normality of the maximum likelihood estimator for general hidden markov models hidden markov models hmms have during the last decade become a widespread tool for modeling sequences of dependent random variables inference for such models is usually based on the maximum likelihood estimator mle and consistency of the mle for general hmms was recently proved by leroux in this paper we show that under mild conditions the mle is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the fisher information

limit theorems in hidden markov models in this paper under mild assumptions we derive a law of large numbers a central limit theorem with an error estimate an almost sure invariance principle and a variant of the chernoff bound in finite state hidden markov models these limit theorems are of interest in certain areas of information theory and statistics particularly we apply the limit theorems to derive the rate of convergence of the maximum likelihood estimator in finite state hidden markov models

an efficient numerical scheme for precise time integration of a diffusion dissolution precipitation chemical system a numerical scheme based on an operator splitting method and a dense output event location algorithm is proposed to integrate a diffusion dissolution precipitation chemical initial boundary value problem with jumping nonlinearities the numerical analysis of the scheme is carried out and it is proved to be of order 2 in time this global order estimate is illustrated numerically on a test case

diffusion and dissolution precipitation in an open porous reactive medium in this article a mathematical model is proposed for dissolution precipitation with diffusion in a reactive open porous medium when many chemical species in solid phase can dissolve for this purpose the concept of saturation concentration for a chemical species in liquid phase is introduced and the mathematical properties of the functions representing such saturation concentrations are investigated then the equations of dissolution precipitation are stated and investigated in order to be reformulated as an obstacle problem finally an original mathematical model is derived and this model can be used for predictive numerical simulations

consistency and asymptotic normality of the quasi maximum likelihood estimator in igarch 1 1 and covariance stationary garch 1 1 models this paper provides a proof of the consistency and asymptotic normality of the quasi maximum likelihood estimator in garch 1 1 and igarch 1 1 models in contrast to the case of a unit root in the conditional mean the presence of a unit root in the conditional variance does not affect the limiting distribution of the estimators in both models estimators are normally distributed in addition a consistent estimator of the covariance matrix is available enabling the use of standard test statistics for inference

asymptotic normality of the maximum likelihood estimator in state space models state space models is a very general class of time series models capable of modelling dependent observations in a natural and interpretable way inference in such models has been studied by bickel ritov and ryden who consider hidden markov models which are special kinds of state space models and prove that the maximum likelihood estimator is asymptotically normal under mild regularity conditions in this paper we generalize the results of bickel ritov and ryden to state space models where the latent process is a continuous state markov chain satisfying regularity conditions which are fulfilled if the latent process takes values in a compact space

information bounds and quick detection of parameter changes in stochastic systems by using information theoretic bounds and sequential hypothesis testing theory this paper provides a new approach to optimal detection of abrupt changes in stochastic systems this approach not only generalizes previous work in the literature on optimal detection far beyond the relatively simple models treated but also suggests alternative performance criteria which are more tractable and more appropriate for general stochastic systems in addition it leads to detection rules which have manageable computational complexity for on line implementation and yet are nearly optimal under the different performance criteria considered

asymptotic expansions for sums of weakly dependent random vectors it is shown that formal edgeworth expansions are valid for sums of weakly dependent random vectors the error of approximation has ordero n s 2 2 if the strong mixing coefficients in iii are decreasing at an exponential rate the above conditions can easily be checked and are often satisfied when the sequence of random vectors is a gaussian or a markov or an autoregressive process explicit formulas are given for the distribution of finite fourier transforms of a strictly stationary time series

decision theoretic optimality of the cusum procedure suppose x 1 x 2 are independent random variables such that for some unknown each of x 1 x 1 is distributed according to f 0 while x x 1 are all distributed according to f 1 we prove a result of moustakides that claims that the cusum procedures are optimal in the sense of lorden we do that by proving that the procedures are bayes for some stochastic mechanism of generating

asymptotic minimax robust quickest change detection for dependent stochastic processes with parametric uncertainty in this paper we consider the problem of quickly detecting an unknown change in the conditional densities of a dependent stochastic process in contrast to the existing quickest change detection approaches for dependent stochastic processes we propose minimax robust versions of the popular lorden pollak and bayesian criteria for when there is uncertainty about the parameter of the post change conditional densities under an information theoretic pythagorean inequality condition on the uncertainty set of possible post change parameters we identify asymptotic minimax robust solutions to our lorden pollak and bayesian problems finally through simulation examples we illustrate that asymptotically minimax robust rules can provide detection performance comparable to the popular but more computationally expensive generalized likelihood ratio rule

heisenberg algebra and hilbert schemes of points on projective surfaces the purpose of this paper is to throw a bridge between two seemingly unrelated subjects one is the hilbert scheme of points on projective surfaces which has been intensively studied by various people see e g i br es g61 g62 the other is the infinite dimensional heisenberg algebra which is closely related to affine lie algebras see e g k we shall construct a representation of the heisenberg algebra on the homology group of the hilbert scheme in other words the homology group will become a fock space the basic idea is to introduce certain correspondences in the product of the hilbert scheme then they define operators on the homology group by a well known procedure they give generators of the heisenberg algebra and the only thing we must check is that they satisfy the defining relation here we remark that the components of the hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components our correspondences live in the product of the different components thus it is quite essential to study all components together our construction has the same spirit as the author s construction nal na4 of representations of affine lie algebras on homology groups of moduli spaces of instantons 1 on ale spaces which are minimal resolutions of simple singularities certain correspondences called hecke correspondences were used to define operators these twist instantons along curves irreducible components of the exceptional set while ours twist ideals around points in fact the hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles or more precisely torsion free sheaves our construction should be considered as a first step to extend nal na4 to general 4 manifolds the same program was also proposed by ginzburg kapranov and vasserot gkv

properties of the maximum a posteriori path estimator in hidden markov models a hidden markov model hmm consists of a nonobservable markov chain x x0 x1 and a measured process y y 0 y1 whose distribution is determined by x to estimate the hidden path of x up to time n x0 x1 xn by the observations y 0 y1 yn usually the maximum a posteriori probability path estimator map path estimator is applied an effective means for calculating this estimator is the viterbi algorithm which is widely employed in the fields of coding theory correction of intersymbol interference and text recognition here properties of the map estimator are derived under a certain condition c it is shown that the limiting process u u0 u1 is a regenerative process particularly this means that u has an asymptotic distribution satisfies the central limit theorem and possesses a mean error furthermore condition c is satisfied for a broad class of hmms including the most important case for applications the hmm with additive white gaussian noise

optimal stopping times for detecting changes in distributions it is shown that page s stopping time is optimal for the detection of changes in distributions in a well defined sense this work is a generalization of an existing result where it was shown that page s stopping time is optimal asymptotically

joint universal lossy coding and identification of stationary mixing sources with general alphabets in this paper we consider the problem of joint universal variable rate lossy coding and identification for parametric classes of stationary beta mixing sources with general polish alphabets compression performance is measured in terms of lagrangians while identification performance is measured by the variational distance between the true source and the estimated source provided that the sources are mixing at a sufficiently fast rate and satisfy certain smoothness and vapnik chervonenkis vc learnability conditions it is shown that for bounded metric distortions there exist universal schemes for joint lossy compression and identification whose lagrangian redundancies converge to zero as radic vn logn n as the block length n tends to infinity where vn is the vc dimension of a certain class of decision regions defined by the n dimensional marginal distributions of the sources furthermore for each n the decoder can identify n dimensional marginal of the active source up to a ball of radius o radic vnlogn n in variational distance eventually with probability one the results are supplemented by several examples of parametric sources satisfying the regularity conditions

bayesian inference in hidden markov models through the reversible jump markov chain monte carlo method hidden markov models form an extension of mixture models which provides a flexible class of models exhibiting dependence and a possibly large degree of variability we show how reversible jump markov chain monte carlo techniques can be used to estimate the parameters as well as the number of components of a hidden markov model in a bayesian framework we employ a mixture of zero mean normal distributions as our main example and apply this model to three sets of data from finance meteorology and geomagnetism

input output properties of the page hinkley detector we consider the stochastic input output properties of a simple non linear dynamical system the so called page hinkley detector playing a key role in change detection and also in queuing theory we show that for l mixing inputs with negative expectation the output process of this system is l mixing the result is applied to get an upper bound for the false alarm rate the proof is then adapted to get a similar result for the case of random i i d inputs possible extensions and open problems are given in the discussion

refinements in asymptotic expansions for sums of weakly dependent random vectors let s0 denote the nth normalized partial sum of a sequence of mean zero weakly dependent random vectors this paper gives asymptotic expansions for ef sn under weaker moment conditions than those of g6tze and hipp 1983 it is also shown that an expansion for ef sn with an error term o n s 2 2 is valid without any cramer type condition if f has partial derivatives of order s 1 only this settles a conjecture of g6tze and hipp in their 1983 paper

asymptotic properties of the maximum likelihood estimator in autoregressive models with markov regime an autoregressive process with markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable markov chain in this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden markov chain conditional on the observations

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

sequential changepoint detection in quality control and dynamical systems after a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection this theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on line implementation and yet are nearly optimal under several performance criteria

optimal error exponents in hidden markov models order estimation we consider the estimation of the number of hidden states the order of a discrete time finite alphabet hidden markov model hmm the estimators we investigate are related to code based order estimators penalized maximum likelihood ml estimators and penalized versions of the mixture estimator introduced by liu and narayan 1994 we prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works we prove a version of stein s lemma for hmm order estimation and derive an upper bound on underestimation exponents then we prove that this upper bound can be achieved by the penalized ml estimator and by the penalized mixture estimator the proof of the latter result gets around the elusive nature of the ml in hmm by resorting to large deviation techniques for empirical processes finally we prove that for any consistent hmm order estimator for most hmm the overestimation exponent is null

bayesian predictive information criterion for the evaluation of hierarchical bayesian and empirical bayes models summary the problem of evaluating the goodness of the predictive distributions of hierarchical bayesian and empirical bayes models is investigated a bayesian predictive information criterion is proposed as an estimator of the posterior mean of the expected loglikelihood of the predictive distribution when the specified family of probability distributions does not contain the true distribution the proposed criterion is developed by correcting the asymptotic bias of the posterior mean of the loglikelihood as an estimator of its expected loglikelihood in the evaluation of hierarchical bayesian models with random effects regardless of our parametric focus the proposed criterion considers the bias correction of the posterior mean of the marginal loglikelihood because it requires a consistent parameter estimator the use of the bootstrap in model evaluation is also discussed

analyticity convergence and convergence rate of recursive maximum likelihood estimation in hidden markov models this paper considers the asymptotic properties of the recursive maximum likelihood estimator for hidden markov models the paper is focused on the analytic properties of the asymptotic log likelihood and on the point convergence and convergence rate of the recursive maximum likelihood estimator using the principle of analytic continuation the analyticity of the asymptotic log likelihood is shown for analytically parameterized hidden markov models relying on this fact and some results from differential geometry lojasiewicz inequality the almost sure point convergence of the recursive maximum likelihood algorithm is demonstrated and relatively tight bounds on the convergence rate are derived as opposed to the existing result on the asymptotic behavior of maximum likelihood estimation in hidden markov models the results of this paper are obtained without assuming that the log likelihood function has an isolated maximum at which the hessian is strictly negative definite

optimal detection of a change in distribution abstract document discusses a stopping rule that is a limit of bayes rules derived then an almost minimax rule is presented

large scale multiple testing under dependence summary the paper considers the problem of multiple testing under dependence in a compound decision theoretic framework the observed data are assumed to be generated from an underlying two state hidden markov model we propose oracle and asymptotically optimal datadriven procedures that aim to minimize the false non discovery rate fnr subject to a constraint on the false discovery rate fdr it is shown that the performance of a multiple testing procedure can be substantially improved by adaptively exploiting the dependence structure among hypotheses and hence conventional fdr procedures that ignore this structural information are inefficient both theoretical properties and numerical performances of the procedures proposed are investigated it is shown that the procedures proposed control fdr at the desired level enjoy certain optimality properties and are especially powerful in identifying clustered non null cases the new procedure is applied to an influenza like illness surveillance study for detecting the timing of epidemic periods

quickest change detection and kullback leibler divergence for two state hidden markov models in this paper the quickest change detection problem is studied in two state hidden markov models hmm where the vector parameter theta of the hmm changes from theta 0 to theta 1 at some unknown time and one wants to detect the true change as quickly as possible while controlling the false alarm rate it turns out that the generalized likelihood ratio glr scheme while theoretically straightforward is generally computationally infeasible for the hmm to develop efficient but computationally simple schemes for the hmm we first discuss a subtlety in the recursive form of the generalized likelihood ratio glr scheme for the hmm then we show that the recursive cusum scheme proposed in fuh ann statist 2003 can be regarded as a quasi glr scheme for pseudo post change hypotheses with certain dependence structure between pre and postchange observations next we extend the quasi glr idea to propose recursive score schemes in the scenario when the postchange parameter theta 1 of the hmm involves a real valued nuisance parameter finally the kullback leibler kl divergence plays an essential role in the quickest change detection problem and many other fields however it is rather challenging to numerically compute it in hmms here we develop a non monte carlo method that computes the kl divergence of two state hmms via the underlying invariant probability measure which is characterized by the fredholm integral equation numerical study demonstrates an unusual property of the kl divergence for hmm that implies the severe effects of misspecifying the postchange parameter for the hmm

subspace estimation and prediction methods for hidden markov models hidden markov models hmms are probabilistic functions of finite markov chains or put in other words state space models with finite state space in this paper we examine subspace estimation methods for hmms whose output lies a finite set as well in particular we study the geometric structure arising from the nonminimality of the linear state space representation of hmms and consistency of a subspace algorithm arising from a certain factorization of the singular value decomposition of the estimated linear prediction matrix for this algorithm we show that the estimates of the transition and emission probability matrices are consistent up to a similarity transformation and that the m step linear predictor computed from the estimated system matrices is consistent i e converges to the true optimal linear m step predictor

universal filtering via hidden markov modeling the problem of discrete universal filtering in which the components of a discrete signal emitted by an unknown source and corrupted by a known discrete memoryless channel dmc are to be causally estimated is considered a family of filters are derived and are shown to be universally asymptotically optimal in the sense of achieving the optimum filtering performance when the clean signal is stationary ergodic and satisfies an additional mild positivity condition our schemes are comprised of approximating the noisy signal using a hidden markov process hmp via maximum likelihood ml estimation followed by the use of the forward recursions for hmp state estimation it is shown that as the data length increases and as the number of states in the hmp approximation increases our family of filters attains the performance of the optimal distribution dependent filter an extension to the case of channels with memory is also established

inference in arch and garch models with heavy tailed errors arch and garch models directly address the dependency of conditional second moments and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present these include financial time series which can be particularly heavy tailed however little is known about properties of arch or garch models in the heavy tailed setting and no methods are available for approximating the distributions of parameter estimators there in this paper we show that for heavy tailed errors the asymptotic distributions of quasi maximum likelihood parameter estimators in arch and garch models are nonnormal and are particularly difficult to estimate directly using standard parametric methods standard bootstrap methods also fail to produce consistent estimators to overcome these problems we develop percentile t subsample bootstrap approximations to estimator distributions studentizing is employed to approximate scale and the subsample bootstrap is used to estimate shape the good performance of this approach is demonstrated both theoretically and numerically

an algorithm for detecting a change in a stochastic process the problem of detecting a change from one given stationary and ergodic stochastic process to another such process is considered it is assumed that both stochastic processes are processes with memory and that they are mutually independent a sequential test is proposed and analyzed it is proved that the proposed test is asymptotically optimal in a mathematically precise sense

acceleration of the em algorithm by using quasi newton methods the em algorithm is a popular method for maximum likelihood estimation its simplicity in many applications and desirable convergence properties make it very attractive its sometimes slow convergence however has prompted researchers to propose methods to accelerate it we review these methods classifying them into three groups pure hybrid and em type accelerators we propose a new pure and a new hybrid accelerator both based on quasi newton methods and numerically compare these and two other quasi newton accelerators for this we use examples in each of three areas poisson mixtures the estimation of covariance from incomplete data and multivariate normal mixtures in these comparisons the new hybrid accelerator was fastest on most of the examples and often dramatically so in some cases it accelerated the em algorithm by factors of over 100 the new pure accelerator is very simple to implement and competed well with the other accelerators it accelerated the em algorithm in some cases by factors of over 50 to obtain standard errors we propose to approximate the inverse of the observed information matrix by using auxiliary output from the new hybrid accelerator a numerical evaluation of these approximations indicates that they may be useful at least for exploratory purposes

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

average run lengths of an optimal method of detecting a change in distribution abstract suppose one accumulates independent observations from a certain process initially the process is at state 0 at some unknown point in time something occurs e g a breakdown which puts the process in state 1 and consequently the stochastic behavior of the observations changes it is of interest to declare that a change took place to raise an alarm as soon as possible after its occurrence subject to a restriction on the rate of false detections it is assumed that the aforementioned observations are the only information one has about the process and the problem is to construct a good detection scheme practical examples of this problem arise in areas such as health quality control ecological monitoring etc for instance consider surveillance for congenital malformations in newborn infants under normal circumstances the percentage of babies born with a certain type of malformation has a known value should something occur such as an environmental change the introduction of a new drug to the market etc the percentage may increase one would want to raise an alarm as quickly as possible after a change would have taken place subject to an acceptable rate of false alarms

a note on optimal detection of a change in distribution suppose x 1 x 2 dots x nu 1 are iid random variables with distribution f 0 and x nu x nu 1 dots are are iid with distributed f 1 the change point nu is unknown the problem is to raise an alarm as soon as possible after the distribution changes from f 0 to f 1 detect the change but to avoid false alarms pollak found a version of the shiryayev roberts procedure to be asymptotically optimal for the problem of minimizing the average run length to detection over all stopping times which satisfy a given constraint on the rate of false alarms here we find that this procedure is strictly optimal for a slight reformulation of the problem he considered explicit formulas are developed for the calculation of the average run length both before and after the change for the optimal stopping time

a numerical scheme for the pore scale simulation of crystal dissolution and precipitation in porous media in this paper we analyze a numerical scheme for a dissolution and precipitation model in porous media we focus here on the chemistry which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion defined on the boundary we use a regularization approach for constructing a semi implicit scheme that is stable and convergent for dealing with the emerging time discrete nonlinear problems we propose a simple fixed point iterative procedure the paper is concluded by numerical results

likelihood ratio tests for hidden markov models summary we consider hidden markov models as a versatile class of models for weakly dependent random phenomena the topic of the present paper is likelihood ratio testing for hidden markov models and we show that under appropriate conditions the standard asymptotic theory of likelihood ratio tests is valid such tests are crucial in the specification of multivariate gaussian hidden markov models which we use to illustrate the applicability of our general results finally the methodology is illustrated by means of a real data set

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

practical filtering with sequential parameter learning summary the paper develops a simulation based approach to sequential parameter learning and filtering in general state space models our approach is based on approximating the target posterior by a mixture of fixed lag smoothing distributions parameter inference exploits a sufficient statistic structure and the methodology can be easily implemented by modifying state space smoothing algorithms we avoid reweighting particles and hence sample degeneracy problems that plague particle filters that use sequential importance sampling the method is illustrated by using two examples a benchmark auto regressive model with observation error and a high dimensional dynamic spatiotemporal model we show that the method provides accurate inference in the presence of outliers model misspecification and high dimensionality

sprt and cusum in hidden markov models in this paper we study the problems of sequential probability ratio tests for parameterized hidden markov models we investigate in some detail the performance of the tests and derive corrected brownian approximations for error probabilities and expected sample sizes asymptotic optimality of the sequential probability ratio test for testing simple hypotheses based on hidden markov chain data is established next we consider the cumulative sum cusum procedure for change point detection in this model based on the renewal property of the stopping rule cusum can be regarded as a repeated one sided sequential probability ratio test asymptotic optimality of the cusum procedure is proved in the sense of lorden 1971 motivated by the sequential analysis in hidden markov models wald s likelihood ratio identity and wald s equation for products of markov random matrices are also given we apply these results to several types of hidden markov models i i d hidden markov models switch gaussian regression and switch gaussian autoregression which are commonly used in digital communications speech recognition bioinformatics and economics

incidence relations among the schubert cells of equivariant punctual hilbert schemes let mathbb h ab h be the equivariant hilbert scheme parametrizing the zero dimensional subschemes of the affine plane k 2 fixed under the one dimensional torus t ab t b t a t in k and whose hilbert function is h this hilbert scheme admits a natural stratification in schubert cells which extends the notion of schubert cells on grassmannians however the incidence relations between the cells become more complicated than in the case of grassmannians in this paper we give a necessary condition for the closure of a cell to meet another cell in the particular case of grassmannians it coincides with the well known necessary and sufficient incidence condition there is no known example showing that the condition wouldn t be sufficient

identifiability of hidden markov information sources and their minimum degrees of freedom if only a function of the state in a finite state markov chain is observed then the stochastic process is no longer markovian in general this type of information source is found widely and the basic problem of its identifiability remains open that is the problem of showing when two different markov chains generate the same stochastic process the identifiability problem is completely solved by linear algebra where a block structure of a markov transition matrix plays a fundamental role and from which the minimum degree of freedom for a source is revealed

a constructive proof of the existence of viterbi processes since the early days of digital communication hidden markov models hmms have now been also routinely used in speech recognition processing of natural languages images and in bioinformatics in an hmm x t y t t 1 observations x 1 x 2 are assumed to be conditionally independent given a markov process y 1 y 2 which itself is not observed moreover the conditional distribution of x t depends solely on y t central to the theory and applications of hmm is the viterbi algorithm to find a maximum a posteriori probability map estimate v x 1 t v 1 v 2 vt of y 1 t given observed data x 1 t maximum a posteriori paths are also known as the viterbi paths or alignments recently attempts have been made to study behavior of the viterbi alignments when t thus it has been shown that in some cases a well defined limiting viterbi alignment exists while innovative these attempts have relied on rather strong assumptions and involved proofs which are existential this work proves the existence of infinite viterbi alignments in a more constructive manner and for a very general class of hmms

causal recursive parameter estimation for discrete time hidden bivariate markov chains an algorithm for causal recursive parameter estimation of a discrete time hidden bivariate markov chain is developed in this model a discrete time bivariate markov chain is observed through a discrete time memoryless channel the algorithm relies on the em based recursive approach developed by stiller and radons for hidden markov models a distinct advantage of the discrete time hidden bivariate markov chain model is that the sojourn time distribution of its observable process in each state is phase type rather than geometric as in the hidden markov model phase type distributions can approximate any desired sojourn time distribution particular phase type distributions include mixtures and convolutions of geometric distributions the parameter estimation algorithm requires causal recursive estimation of the relevant statistics in each em step these statistics include the number of jumps of the bivariate markov chain from one state to another including self transitions in the given observation interval and first and second order statistical averages of the observable process in each state when the memoryless channel is gaussian we develop the explicit recursions and demonstrate the performance of the algorithm in estimating the model s parameter and its sojourn time distribution in a numerical example

consistency of the maximum likelihood estimator for general hidden markov models consider a parametrized family of general hidden markov models where both the observed and unobserved components take values in a complete separable metric space we prove that the maximum likelihood estimator mle of the parameter is strongly consistent under a rather minimal set of assumptions as special cases of our main result we obtain consistency in a large class of nonlinear state space models as well as general results on linear gaussian state space models and finite state models a novel aspect of our approach is an information theoretic technique for proving identifiability which does not require an explicit representation for the relative entropy rate our method of proof could therefore form a foundation for the investigation of mle consistency in more general dependent and non markovian time series also of independent interest is a general concentration inequality for v uniformly ergodic markov chains

hidden markov models with applications in cell adhesion experiments estimation of the number of hidden states is challenging in hidden markov models motivated by the analysis of a specific type of cell adhesion experiments a new framework based on a hidden markov model and double penalized order selection is proposed the order selection procedure is shown to be consistent in estimating the number of states a modified expectation maximization algorithm is introduced to efficiently estimate parameters in the model simulations show that the proposed framework outperforms existing methods applications of the proposed methodology to real data demonstrate the accuracy of estimating receptor ligand bond lifetimes and waiting times which are essential in kinetic parameter estimation supplementary materials for this article are available online

markov switching model selection using kullback leibler divergence abstract in markov switching regression models we use kullback leibler kl divergence between the true and candidate models to select the number of states and variables simultaneously specifically we derive a new information criterion markov switching criterion msc which is an estimate of kl divergence msc imposes an appropriate penalty to mitigate the over retention of states in the markov chain and it performs well in monte carlo studies with single and multiple states small and large samples and low and high noise we illustrate the usefulness of msc via applications to the u s business cycle and to media advertising

asymptotic properties of the maximum likelihood estimator in regime switching econometric models abstract markov regime switching models have been widely used in numerous empirical applications in economics and finance however the asymptotic distribution of the maximum likelihood estimator mle has not been proven for some empirically popular markov regime switching models in particular the asymptotic distribution of the mle has been unknown for models in which some elements of the transition probability matrix have the value of zero as is commonly assumed in empirical applications with models with more than two regimes this also includes models in which the regime specific density depends on both the current and the lagged regimes such as the seminal model of hamilton 1989 and switching arch model of hamilton and susmel 1994 this paper shows the asymptotic normality of the mle and consistency of the asymptotic covariance matrix estimate of these models

asymptotic operating characteristics of an optimal change point detection in hidden markov models let 0 1 1 be observations from the hidden markov model with probability distribution p 0 and let 1 be observations from the hidden markov model with probability distribution p 1 the parameters 0 and 1 are given while the change point is unknown the problem is to raise an alarm as soon as possible after the distribution changes from p 0 to p 1 but to avoid false alarms specifically we seek a stopping rule n which allows us to observe the s sequentially such that e n is large and subject to this constraint sup k e k n k n k is as small as possible here e k denotes expectation under the change point k and e denotes expectation under the hypothesis of no change whatever in this paper we investigate the performance of the shiryayev roberts pollak srp rule for change point detection in the dynamic system of hidden markov models by making use of markov chain representation for the likelihood function the structure of asymptotically minimax policy and of the bayes rule and sequential hypothesis testing theory for markov random walks we show that the srp procedure is asymptotically minimax in the sense of pollak ann statist 13 1985 206 227 next we present a second order asymptotic approximation for the expected stopping time of such a stopping scheme when w 1 motivated by the sequential analysis in hidden markov models a nonlinear renewal theory for markov random walks is also given

chern classes of tautological sheaves on hilbert schemes of points on surfaces we give an algorithmic description of the action of the chern classes of tautological bundles on the cohomology of hilbert schemes of points on a smooth surface within the framework of nakajima s oscillator algebra this leads to an identification of the cohomology ring of hilb n a2 with a ring of explicitly given differential operators on a fock space we end with the computation of the top segre classes of tautological bundles associated to line bundles on hilb n up to n 7 extending computations of severi lebarz tikhomirov and troshina and give a conjecture for the generating series

irreducible components of the equivariant punctual hilbert schemes abstract let h ab be the equivariant hilbert scheme parameterizing the zero dimensional subschemes of the affine plane invariant under the natural action of the one dimensional torus t ab t b t a t k we compute the irreducible components of h ab they are in one to one correspondence with the set of possible hilbert functions as a by product of the proof we give new proofs of results by ellingsrud and stromme namely the main lemma of the computation of the betti numbers of the hilbert scheme h l parametrizing the zero dimensional subschemes of the affine plane of length l invent math 87 1988 343 and a description of bialynicki birula cells on h l by means of explicit flat families invent math 91 1988 365 in particular we give precise conditions when this last description applies

a syzygetic approach to the smoothability of zero dimensional schemes abstract we consider the question of which zero dimensional schemes deform to a collection of distinct points equivalently we ask which artinian k algebras deform to a product of fields we introduce a syzygetic invariant which sheds light on this question for zero dimensional schemes of regularity two this invariant imposes obstructions for smoothability in general and it completely answers the question of smoothability for certain zero dimensional schemes of low degree the tools of this paper also lead to other results about hilbert schemes of points including a characterization of nonsmoothable zero dimensional schemes of minimal degree in every embedding dimension d 4

efficient likelihood estimation in state space models motivated by studying asymptotic properties of the maximum likelihood estimator mle in stochastic volatility sv models in this paper we investigate likelihood estimation in state space models we first prove under some regularity conditions there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the fisher information as its variance with an extra assumption that the likelihood equation has a unique root for each n then there is a consistent sequence of estimators of the unknown parameters if in addition the supremum of the log likelihood function is integrable the mle exists and is strongly consistent edge worth expansion of the approximate solution of likelihood equation is also established several examples including markov switching models arma models g arch models and stochastic volatility sv models are given for illustration

maximum penalized likelihood estimation for independent and markov dependent mixture models summary this paper concerns the use and implementation of maximum penalized likelihood procedures for choosing the number of mixing components and estimating the parameters in independent and markov dependent mixture models computation of the estimates is achieved via algorithms for the automatic generation of starting values for the em algorithm computation of the information matrix is also discussed poisson mixture models are applied to a sequence of counts of movements by a fetal lamb in utero obtained by ultrasound the resulting estimates are seen to provide plausible mechanisms for the physiological process the analysis of count data that are overdispersed relative to the poisson distribution i e variance mean has received considerable recent attention such data might arise in a clinical study in which overdispersion is caused by unexplained or random subject effects alternatively we might observe a time series of counts in which temporal patterns in the data suggest that a poisson model and its implied randomness are inappropriate this paper is motivated by analysis of a time series of overdispersed count data generated in a study of central nervous system development in fetal lambs our data set consists of observed movement counts in 240 consecutive 5 second intervals obtained from a single animal in analysing these data we focus on the use of poisson mixture models assuming independent observations and also markov dependent mixture models or hidden markov models these models assume that the counts follow independent poisson distributions conditional on the rates which are generated from a mixing distribution either independently or with markov dependence we believe finite mixture models are particularly attractive because they provide plausible explanations for variation in the data this paper will emphasize the following issues concerning estimation inference and application of mixture models i choosing the number of model components ii applying the em algorithm to obtain parameter estimates iii generating sufficiently many starting values to identify a global maximum of the likelihood iv avoiding numerical instability

smooth and irreducible multigraded hilbert schemes abstract the multigraded hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed hilbert function we prove that any multigraded hilbert scheme is smooth and irreducible when the polynomial ring is z x y which establishes a conjecture of haiman and sturmfels