reductions of hidden information sources in all but special circumstances measurements of time dependent processes reflect internal structures and correlations only indirectly building predictive models of such hidden information sources requires discovering in some way the internal states and mechanisms unfortunately there are often many possible models that are observationally equivalent here we show that the situation is not as arbitrary as one would think we show that generators of hidden stochastic processes can be reduced to a minimal form and compare this reduced representation to that provided by computational mechanics the e machine on the way to developing deeper measure theoretic foundations for the latter we introduce a new two step reduction process the first step internal event reduction produces the smallest observationally equivalent algebra and the second internal state reduction removes algebra components that are redundant for optimal prediction for several classes of stochastic dynamical systems these reductions produce representations that are equivalent to e machines

combinatorial analogues of ad nilpotent ideals for untwisted affine lie algebras we study certain types of ideals in the standard borel subalgebra of an untwisted affine lie algebra we classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type a the formula involves various aspects of combinatorics of dyck paths and leads to a new interesting integral sequence

the complex cepstrum of higher order cumulants and nonminimum phase system identification a computationally efficient identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase system the method is based on the idea of computing the complex cepstrum of higher order cumulants of the system output in particular the differential cepstrum parameters of the nonminimum phase impulse response are estimated directly from higher order cumulants by least squares solution or two dimensional fft operations the method reconstructs the minimum phase and maximum phase impulse response components separately it is flexible enough to be applied on autoregressive ar moving average ma or arma system without a priori knowledge of the type of the system benchmark simulation examples demonstrate the effectiveness of the method even with short length data records

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

estimating the number of sinusoids in additive white noise the test proposed uses the eigenvector decomposition of the estimated autocorrelation matrix and is based on matrix perturbation analysis the estimator is shown to be able to resolve closely spaced sinusoids at lower signal to noise ratios than heuristic tests simulation results for two closely spaced sinusoids are detailed several unanswered questions are discussed

signal reconstruction from the phase of the bispectrum the authors present a simple procedure the bispectrum signal reconstruction bsr algorithm to recover the fourier phase of a signal from the phase of its bispectrum by simple analogy a procedure that recovers the fourier magnitude of a signal from the magnitude of its bispectrum is also presented in addition the authors propose an iterative scheme the bicepstrum iterative reconstruction algorithm bira for the reconstruction of a finite impulse response fir sequence from only the phase of its bispectrum and they demonstrate how some a priori information on the energy of the cepstra coefficients can improve significantly the convergence rate of the algorithm both schemes are based on the key observation that the differences of the bispectrum coefficients contain all the information concerning the fourier phase of the signal whereas their sums contain the fourier magnitude information

learning hidden markov models using nonnegative matrix factorization the baum welch algorithm together with its derivatives and variations has been the main technique for learning hidden markov models hmms from observational data we present an hmm learning algorithm based on the nonnegative matrix factorization nmf of higher order markovian statistics that is structurally different from the baum welch and its associated approaches the described algorithm supports estimation of the number of recurrent states of an hmm and iterates the nmf algorithm to improve the learned hmm parameters numerical examples are provided as well

characterization of ergodic hidden markov sources an algebraic criterion for the ergodicity of discrete random sources is presented for finite dimensional sources which contain hidden markov sources as a subclass the criterion can be effectively computed this result is obtained on the background of a novel elementary theory of discrete random sources which is based on linear spaces spanned by word functions and linear operators on these spaces an outline of basic elements of this theory is provided

efficient tests for equivalence of hidden markov processes and quantum random walks while two hidden markov process hmp resp quantum random walk qrw parametrizations can differ from one another the stochastic processes arising from them can be equivalent here a polynomial time algorithm is presented which can determine equivalence of two hmp parametrizations m1 m2 resp two qrw parametrizations q1 q2 in time o max n1 n2 4 where n1 n2 are the number of hidden states in m1 m2 resp the dimension of the state spaces associated with q1 q2 and is the set of output symbols previously avail able algorithms for testing equivalence of hmps were exponential in the number of hidden states in case of qrws algorithms for testing equivalence had not yet been presented the core subrou tines of this algorithm can also be used to efficiently test hidden markov processes and quantum random walks for ergodicity

relative springer isomorphisms let g be a simple algebraic group over the algebraically closed field k a slightly strengthened version of a theorem of t a springer says that under some mild restrictions on g and k there exists a g equivariant isomorphism of varieties u n where u denotes the unipotent variety of g and n denotes the nilpotent variety of g lieg such is called a springer isomorphism let b be a borel subgroup of g u the unipotent radical of b and u the lie algebra of u in this note we show that a springer isomorphism induces a b equivariant isomorphism u m u m where m is any unipotent normal subgroup of b and m liem we call such a map a relative springer isomorphism we also use relative springer isomorphisms to describe the geometry of u orbits in u

nonminimum phase channel deconvolution using the complex cepstrum of the cyclic autocorrelation new discrete time blind deconvolution methods are proposed for nonminimum phase linear channels driven by cyclo stationary inputs the methods rely exclusively on second order statistics and do not impose any constraints on the distribution of the channel input as in the case of methods based on higher order statistics the output of the channel is fractionally sampled and then the complex cepstrum of the cyclic autocorrelation is obtained it is shown that this complex cepstrum preserves nonminimum phase information and thus the identification of nonminimum phase channels is possible practical constraints in the implementation of the methods and channel identifiability conditions are discussed the applicability of the methods to both channel identification and fractionally spaced linear and dfe equalization is described and verified by means of computer simulations

lumpable hidden markov models model reduction and reduced complexity filtering this paper is concerned with filtering of hidden markov processes hmp which possess or approximately possess the property of lumpability this property is a generalization of the property of lumpability of a markov chain which has been previously addressed by others in essence the property of lumpability means that there is a partition of the atomic states of the markov chain into aggregated sets which act in a similar manner as far as the state dynamics and observation statistics are concerned we prove necessary and sufficient conditions on the hmp for exact lumpability to hold for a particular class of hidden markov models hmm namely finite output alphabet models conditions for lumpability of all hmp representable by a specified hmm are given the corresponding optimal filter algorithms for the aggregated states are then derived the paper also describes an approach to efficient suboptimal filtering for hmp which are approximately lumpable by this we mean that the hmm generating the process may be approximated by a lumpable hmm this approach involves directly finding a lumped hmm which approximates the original hmm well in a matrix norm sense an alternative approach for model reduction based on approximating a given hmm by an exactly lumpable hmm is also derived this method is based on the alternating convex projections algorithm some simulation examples are presented which illustrate the performance of the suboptimal filtering algorithms

nonlinear adaptive blind whitening for mimo channels a nonlinear adaptive whitening method is proposed for blind deconvolution of mimo systems by whitening the received signals in both time and space with a highly nonlinear function of the past output data the whitened signals are isi free and can be viewed as outputs of a memoryless paraunitary mixing system the convergence of the proposed recursive algorithm is proved numerical simulation shows that the whitening method proposed in the paper works well even if the output signal is corrupted by additive noise

identification of linear stochastic systems via second and fourth order cumulant matching the identification problem for time invariant single input single output linear stochastic systems driven by non gaussian white noise is considered the system is not restricted to be minimum phase and it is allowed to contain all pass components a least squares criterion that involves matching the second and the fourth order cumulant functions of the noisy observations is proposed knowledge of the probability distribution of the driving noise is not required an order determination criterion that is a modification of the akaike information criterion is also proposed strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are presented to illustrate the method

spherical orbits and abelian ideals let g be a connected reductive complex algebraic group with lie algebra lie g g let b be a borel subgroup of g with unipotent radical bu we denote the lie algebra of b and bu by b and bu respectively the group b acts on any ideal of b by means of the adjoint representation after a preliminary section we study a relationship between spherical nilpotent orbits and abelian ideals a of b using the structure theory for these orbits from 10 the principal result of this section is that for an abelian ideal a of b any nilpotent orbit meeting a is a spherical g variety see theorem 2 3 as a consequence of this we obtain a short conceptual proof of a finiteness theorem from 14 namely for a parabolic subgroup p of g and an abelian ideal a of p lie p in the nilpotent radical pu lie pu the group p operates on a with finitely many orbits the proof of this fact in 14 involved long and tedious case by case considerations we also prove a partial converse to the result just mentioned following 4 we say that an ideal of b is ad nilpotent whenever it consists of nilpotent elements in case g is simply laced we show that an ad nilpotent ideal c of b is abelian provided any nilpotent orbit meeting c is spherical see proposition 2 7 in section 3 we consider some properties of ad nilpotent ideals of b in theorem 3 2 we give a description of the normaliser of such ideals this applies in particular to abelian ideals of b a remarkable theorem of doi 10 1006 aima 2000 1959 available online at http www idealibrary com on

blind equalization of simo fir channels driven by colored signals with unknown statistics we consider the blind equalization of single input multiple output finite impulse response channel driven by colored signals the statistics of the input colored signals are unknown a priori by exploiting the inherent structural relationship between the source autocorrelation matrices of different delay lags the closed form zero forcing and minimum mean square error equalizers of desired delays are estimated from the second order statistics of the received data the blind equalizability conditions of the proposed method are investigated numerical simulation results are presented to illustrate the performance of the proposed algorithm

blind identification of lti zmnl lti nonlinear channel models a simple method is proposed for blind identification of discrete time nonlinear models consisting of two linear time invariant lti subsystems separated by a polynomial type zero memory nonlinearity zmnl of order n the lti zmnl lti model the linear subsystems are allowed to be of nonminimum phase nmp though the first lti can be completely identified only if it is of minimum phase with a circularly symmetric gaussian input the linear subsystems can be identified using simple cepstral operations on a single 2 d slice of the n 1 th order polyspectrum of the output signal the linear subsystem of an lti zmnl model can be identified using only a 1 d moment or polyspectral slice if it is of minimum phase the zmnl coefficients are not identified and need not be known the order n of the nonlinearity can in principle be estimated from the output signal the methods are analytically simple computationally efficient and possess noise suppression characteristics computer simulations are presented to support the theory

blind equalization using a tricepstrum based algorithm an adaptive blind equalization method is introduced for nonminimum phase communication channels the method estimates the inverse channel impulse response by using the complex cepstrum of the fourth order cumulants tricepstrum of the synchronously sampled received signal as such the proposed adaptive method depends only on the statistics of the received sequence and is capable of reconstructing separately both the minimum and maximum phase response of the channel it is demonstrated by means of extensive simulations that the proposed tricepstrum based equalization scheme performs well and outperforms other existing blind equalizers at the expense of higher computational complexity

the signature of a toric variety we identify a combinatorial quantity the alternating sum of the h vector defined for any simple polytope as the signature of a toric variety this quantity was introduced by r charney and m davis in their work which in particular showed that its nonnegativity is closely related to a conjecture of h hopf on the euler characteristic of a nonpositively curved manifold we prove positive or nonnegative lower bounds for this quantity under geometric hypotheses on the polytope and in particular resolve a special case of their conjecture these hypotheses lead to ampleness or weaker conditions for certain line bundles on toric divisors and then the lower bounds follow from calculations using the hirzebruch signature formula moreover we show that under these hypotheses on the polytope the ith l class of the corresponding toric variety is 1 i times an effective class for any i

observable operator models for discrete stochastic time series a widely used class of models for stochastic systems is hidden markov models systems that can be modeled by hidden markov models are a proper subclass of linearly dependent processes a class of s

why the characteristic polynomial factors we survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently the first technique uses geometric ideas and is is based on zaslavsky s theory of signed graphs the second approach is algebraic and employs results of saito and terao about free hyperplane arrangements finally we consider a purely combinatorial theorem of stanley about supersolvable lattices and its generalizations

fourier series based nonminimum phase model for statistical signal processing in this paper a parametric fourier series based model fsbm for or as an approximation to an arbitrary nonminimum phase linear time invariant lti system is proposed for statistical signal processing applications where a model for lti systems is needed based on the fsbm a minimum phase linear prediction error lpe filter for amplitude estimation of the unknown lti system together with the cramer rao cr bounds is presented then an iterative algorithm for obtaining the optimum lpe filter with finite data is presented that is also an approximate maximum likelihood algorithm when data are gaussian then three iterative algorithms using higher order statistics hos with finite non gaussian data are presented to estimate parameters of the fsbm followed by some simulation results as well as some experimental results with real speech data to support the efficacy of the proposed algorithms using the fsbm finally we draw some conclusions

cross spectrum based blind channel identification a novel cross correlation based framework is proposed for the problem of blind equalization in communications we assume that we have access to two observations obtained either by sampling at the symbol rate the outputs of two sensors or by oversampling by a factor of two the output of a single sensor in either case the two observations correspond to the outputs of two channels excited by the same input the channels are estimated using the theory of signal reconstruction from phase only the phase used is the phase of the cross spectrum of the observations filtered through their minimum phase equivalent filters we provide an analytical study of the propagation of noise effects in the phase estimate comparisons with existing methods indicate that the proposed approach is robust to noise and at low signal to noise ratio snr leads to significantly smaller channel estimation errors besides robustness to noise the proposed method does not require knowledge of channel lengths which are determined via an iterative procedure

characteristic polynomials of subspace arrangements and finite fields let a be any subspace arrangement in rndefined over the integers and let fqdenote the finite field withqelements letqbe a large prime we prove that the characteristic polynomial a q of a counts the number of points in fnqthat do not lie in any of the subspaces ofa viewed as subsets of fnq this observation which generalizes a theorem of blass and sagan about subarrangements of the bnarrangement reduces the computation of a q to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years the basic idea has its origins in the work of crapo and rota 1970 we find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers

markov property for a function of a markov chain a linear algebra approach abstract in this paper we address whether a probabilistic function of a finite homogeneous markov chain still enjoys a markov type property we propose a complete answer to this question using a linear algebra approach at the core of our approach is the concept of invariance of a set under a matrix in that sense the framework of this paper is related to the so called geometric approach in control theory for linear dynamical systems this allows us to derive a collection of new results under generic assumptions on the original markov chain in particular we obtain a new criterion for a function of a markov chain to be a homogeneous markov chain we provide a deterministic polynomial time algorithm for checking this criterion moreover a non standard notion of observability for a linear system will be used this allows one to show that the set of all stochastic matrices for which our criterion holds is nowhere dense in the set of stochastic matrices

consistent and asymptotically normal parameter estimates for hidden markov models hidden markov models are today widespread for modeling of various phenomena it has recently been shown by leroux that the maximum likelihood estimate mle of the parameters of a such a model is consistent and local asymptotic normality has been proved by bickel and ritov in this paper we propose a new clans of estimates which are consistent asymptoti cally normal and almost as good an the mle

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

ad nilpotent ideals of minimal dimension abstract we prove the conjecture in 5 10 about the lower bounds of ad nilpotent ideals with the same associated orbit more precisely for each nilpotent orbit of classical type we construct explicit ad nilpotent ideals of minimal dimension for type a n 1 an explicit dimension formula for minimal ideals is provided based on partitions of n

an interval amplitude algorithm for deinterleaving stochastic pulse train sources we consider the deinterleaving of pulse trains transmitted by n independent sources the deinterleaving problem considered has applications in spectral estimation where n known number stochastic parameterized sources are sampled using a fast sensor recording the sign of the signal from each source due to communication constraints the recorded signals pulse trains or sequences of zeros and ones are superimposed and transmitted through a single gaussian communication channel the aim of this paper is to estimate the parameters that characterize the sources and identify those sources that are responsible for the observed noisy pulses our proposed algorithm subject to modeling assumptions optimally combines hidden markov model and binary time series estimation techniques and yields maximum likelihood parameter estimates of the sources

on a refinement of the generalized catalan numbers for weyl groups let be an irreducible crystallographic root system with weyl group w coroot lattice q and coxeter number h spanning a euclidean space v and let m be a positive integer it is known that the set of regions into which the fundamental chamber of w is dissected by the hyperplanes in v of the form x k for a and k 1 2 m is equinumerous to the set of orbits of the action of w on the quotient q mh 1 q a bijection between these two sets as well as a bijection to the set of certain chains of order ideals in the root poset of are described and are shown to preserve certain natural statistics on these sets the number of elements of these sets and their corresponding refinements generalize the classical catalan and narayana numbers which occur in the special case m 1 and a n 1

on the existence of a positive realization the positive realization problem for linear systems is to find conditions for a given transfer function with nonnegative impulse response to have a realization such that the resulting system is a positive system recently it has been shown that under a mild assumption on the long term behaviour of the impulse response this problem is related to the maximum modulus poles only in this paper necessary and sufficient conditions for positive realizability of discrete time systems are given they show that also nondominant poles play a role in the most general case positive realizability conditions for the continuous time case are also given

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

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

estimation of primary user parameters in cognitive radio systems via hidden markov model for a cognitive radio cr system we investigate the estimation problem in which a secondary user su estimates the channel parameters such as the sojourn times on the active and the inactive states of the primary user pu as well as the pu signal strength on the basis of the sequence of the sensing results by modeling the cr system as a hidden markov model hmm the channel parameters are estimated by the standard expectation maximization em algorithm we focus on mathematically analyzing the condition under which the em algorithm can find the true channel parameters for this we apply the theory of the equivalence and the identifiability to the proposed hmm model for a cr system based on the identifiability analysis we propose a parameter estimation algorithm for our problem by extending the em algorithm the numerical results show that the proposed algorithm successfully estimates the true channel parameters as long as the condition for finding the channel parameters is satisfied

ad nilpotent ideals of a borel subalgebra ii we provide an explicit bijection between the ad nilpotent ideals of a borel subalgebra of a simple lie algebra g and the orbits of q h 1 q under the weyl group q being the coroot lattice and h the coxeter number of g from this result we deduce in a uniform way a counting formula for the ad nilpotent ideals

asymptotic mean stationarity of sources with finite evolution dimension the notion of the evolution of a discrete random source with finite alphabet is introduced and its behavior under the action of an associated linear evolution operator is studied viewing these sources as possibly stable dynamical systems it is proved that all random sources with finite evolution dimension are asymptotically mean stationary which implies that such random sources have ergodic properties and a well defined entropy rate it is shown that the class of random sources with finite evolution dimension properly generalizes the well studied class of finitary stochastic processes which includes hidden markov sources as special cases

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

anomaly preserving ell _ scriptscriptstyle 2 infty optimal dimensionality reduction over a grassmann manifold in this paper we address the problem of redundancy reduction of high dimensional noisy signals that may contain anomaly rare vectors which we wish to preserve since anomaly data vectors contribute weakly to the l2 norm of the signal as compared to the noise l2 based criteria are unsatisfactory for obtaining a good representation of these vectors as a remedy a new approach named min max svd mx svd was recently proposed for signal subspace estimation by attempting to minimize the maximum of data residual l2 norms denoted as l2 l and designed to represent well both abundant and anomaly measurements however the mx svd algorithm is greedy and only approximately minimizes the proposed l2 l norm of the residuals in this paper we develop an optimal algorithm for the minization of the l2 l norm of data misrepresentation residuals which we call maximum orthogonal complements optimal subspace estimation moose the optimization is performed via a natural conjugate gradient learning approach carried out on the set of n dimensional subspaces in irm m n which is a grassmann manifold the results of applying moose mx svd and l2 based approaches are demonstrated both on simulated and real hyperspectral data

prediction error method for second order blind identification blind channel identification methods based on the oversampled channel output are a problem of current theoretical and practical interest in this paper we introduce a second order blind identification technique based on a linear prediction approach in contrast to eigenstructure based methods it will be shown that the linear prediction error method is robust to order overdetermination an asymptotic performance analysis of the proposed estimation method is carried out consistency and asymptotic normality of the estimates is established a closed form expression for the asymptotic covariance of the estimates is given numerical simulations and investigations are finally presented to demonstrate the potential and the robustness of the proposed method

blind fractionally spaced equalization of noisy fir channels direct and adaptive solutions blind fractionally spaced equalizers reduce intersymbol interference using second order statistics without the need for training sequences methods for finding fir zero forcing blind equalizers directly from the observations are described and adaptive versions are developed in contrast most current methods require channel estimation as a first step to estimating the equalizer the direct methods can be zero forcing minimum mean square error or even minimum mean square error mmse within the class of zero forcing equalizers performance of the proposed methods and comparisons with existing approaches are shown for a variety of channels including an empirically measured digital microwave channel

persistency of excitation sufficient richness and parameter convergence in discrete time adaptive control abstract the main result of this paper is to give new techniques for deriving explicit conditions on the exogenously specified reference trajectory to guarantee parameter convergence for a class of either single input single output or multi input multi output discrete time adaptive control schemes other interesting results are relations between persistently exciting signals and sufficiently rich signals

blind identification of linear subsystems of lti zmnl lti models with cyclostationary inputs discrete time nonlinear models consisting of two linear time invariant lti filters separated by a finite order zero memory nonlinearity zmnl of the polynomial type the lti zmnl lti model are appropriate in a large number of practical applications we discuss some approaches to the problem of blind identification of such nonlinear models it is shown that for an nth order nonlinearity the possibly non minimum phase finite memory linear subsystems of lti zmnl and lti zmnl lti models can be identified using the n 1th order cyclic statistics of the output sequence alone provided the input is cyclostationary and satisfies certain conditions the coefficients of the zmnl are not needed for identification of the linear subsystems and are not estimated it is shown that the theory presented leads to analytically simple identification algorithms that possess several noise and interference suppression characteristics

a least squares approach to blind channel identification conventional blind channel identification algorithms are based on channel outputs and knowledge of the probabilistic model of channel input in some practical applications however the input statistical model may not be known or there may not be sufficient data to obtain accurate enough estimates of certain statistics in this paper we consider the system input to be an unknown deterministic signal and study the problem of blind identification of multichannel fir systems without requiring the knowledge of the input statistical model a new blind identification algorithm based solely on the system outputs is proposed necessary and sufficient identifiability conditions in terms of the multichannel systems and the deterministic input signal are also presented

zero forcing blind equalization based on subspace estimation for multiuser systems input signal recovery from frequency selective fading channels is a problem of great theoretical and practical importance we present several new blind algorithms that utilize second order statistics for direct multichannel equalization the algorithms are based on the subspace extraction of a preselected block column of the channel convolution matrix for a multiuser system user signal separation can be achieved based on partial information of the composite channel response these equalization algorithms do not rely on the precise separation of signal and noise subspaces and therefore tend to be less sensitive to channel order or column rank estimation errors equalization is directly achieved without channel identification furthermore the equalizability conditions of these algorithms are discussed

deterministic blind identification of iir systems with output switching operations in this paper a deterministic blind identification approach is proposed for linear output switching systems which are modeled by multiple infinite impulse response iir dynamic functions by adopting a new over sampling strategy the concerned single input single output siso output switching system is equivalently transformed into a time invariant multi input multi output mimo system further by exploring the mutual relations among the multiple inputs the time invariant mimo system model and subsequently the output switching system model are identified uniquely up to a scalar constant using the proposed identification approach sufficient identifiability conditions are provided for output switching systems and numerical simulations are carried out to validate the proposed approach

fast subspace decomposition many recent signal processing techniques in various areas e g array signal processing system identification and spectrum estimation require a step of extracting the low dimensional subspace from a large space this step is usually called subspace decomposition which is conventionally accomplished by an eigendecomposition ed since an ed requires o m sup 3 flops for an m spl times m matrix it may represent a barrier to the real time implementation of these algorithms the authors present a fast algorithm for signal subspace decomposition which is termed fsd that exploits the special matrix structure low rank plus identity associated with signal subspace algorithms and requires only o m sup 2 d flops where d often spl lt m denotes the signal subspace dimension unlike most of alternatives the dimension of the signal subspace is not assumed known apriori and is estimated as part of the procedure moreover theoretical analysis has been conducted and its results show the strong consistency of the new detection scheme and the asymptotic equivalence between fsd and ed in estimating the signal subspace their new approach can also exploit other matrix structure common in signal processing areas e g toeplitz or hankel and thus achieve another order of computational reduction moreover it can be easily implemented in parallel to reduce further the computation time to as little as o md using o m simple processors

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

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

nilpotent ideals in having a fixed class of nilpotence combinatorics and enumeration we study the combinatorics of ad nilpotent ideals of a borel subalgebra of sl n 1 c we provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence we study the relationships between these results and the combinatorics of dyck paths based upon a remarkable bijection between ad nilpotent ideals and dyck paths finally we propose a q t analogue of the catalan number c n these q t catalan numbers count on the one hand ad nilpotent ideals with respect to dimension and class of nilpotence and on the other hand admit interpretations in terms of natural statistics on dyck paths

ill convergence of godard blind equalizers in data communication systems the existence of stable undesirable equilibria for the godard algorithm is demonstrated through a simple autoregressive ar channel model these undesirable equilibria correspond to local but nonglobal minima of the underlying mean cost function and thus permit the ill convergence of the godard algorithms which are stochastic gradient descent in nature simulation results confirm predicted misbehavior for channel input of constant modulus it is shown that attaining the global minimum of the mean cost necessarily implies correct equalization a criterion is also presented for allowing a decision at the equalizer as to whether a global or nonglobal minimum has been reached

norm observable operator models hidden markov models hmms are one of the most popular and successful statistical models for time series observable operator models ooms are generalizations of hmms that exhibit several attractive advantages in particular a variety of highly efficient constructive and asymptotically correct learning algorithms are available for ooms however the oom theory suffers from the negative probability problem npp a given learned oom may sometimes predict negative probabilities for certain events it was recently shown that it is undecidable whether a given oom will eventually produce such negative values we propose a novel variant of ooms called norm observable operator models nooms which avoid the npp by design like ooms nooms use a set of linear operators to update system states but differing from ooms they represent probabilities by the square of the norm of system states thus precluding negative probability values while being free of the npp nooms retain most advantages of ooms for example nooms also capture some processes that cannot be modeled by hmms more importantly in principle nooms can be learned from data in a constructive way and the learned models are asymptotically correct we also prove that nooms capture all markov chain mc describable processes this letter presents the mathematical foundations of nooms discusses the expressiveness of the model class and explains how a noom can be estimated from data constructively

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

evam an eigenvector based algorithm for multichannel blind deconvolution of input colored signals a new algorithm is proposed for the deconvolution of an unknown possibly colored gaussian or nonstationary signal that is observed through two or more unknown channels described by rational system transfer functions more specifically not only the root pole and zero locations but also the orders of the channel transfer functions are unknown it is assumed that the channel orders may be overestimated the proposed algorithm estimates the orders and root locations of the channel transfer functions therefore it can also be used in multichannel system identification problems the input signal is allowed to be nonstationary and the channel transfer functions may be a nonminimum phase as well as noncausal hence the proposed algorithm is particularly suitable for applications such as dereverberation of speech signals recorded through multiple microphones several experimental results indicate improvement compared to the existing methods in the literature

ad nilpotent ideals and equivalence relations abstract in this paper we study ad nilpotent ideals of a complex simple lie algebra g and their connections with affine weyl groups and nilpotent orbits we define a left equivalence relation for ad nilpotent ideals based on their normalizer and generators and prove that the equivalence relation corresponds to lusztig s star operation for simply laced algebraic group g

on analytic properties of entropy rate entropy rate is a real valued functional on the space of discrete random sources for which it exists however it lacks existence proofs and or closed formulas even for classes of random sources which have intuitive parameterizations a good way to overcome this problem is to examine its analytic properties relative to some reasonable topology a canonical choice of a topology is that of the norm of total variation as it immediately arises with the idea of a discrete random source as a probability measure on sequence space it is shown that both upper and lower entropy rate hence entropy rate itself if it exists are lipschitzian relative to this topology which by well known facts is close to differentiability an application of this theorem leads to a simple and elementary proof of the existence of entropy rate of random sources with finite evolution dimension this class of sources encompasses arbitrary hidden markov sources and quantum random walks

new criteria for blind deconvolution of nonminimum phase systems channels a necessary and sufficient condition for blind deconvolution without observing the input of nonminimum phase linear time invariant systems channels is derived based on this condition several optimization criteria are proposed and their solution is shown to correspond to the desired response these criteria involve the computation only of second and fourth order moments implying a simple tap update procedure the proposed methods are universal in the sense that they do not impose any restrictions on the probability distribution of the unobserved input sequence it is shown that in several important cases e g when the additive noise is gaussian the proposed criteria are essentially unaffected

minimal realization problems for hidden markov models this paper addresses two fundamental problems in the context of hidden markov models hmms the first problem is concerned with the characterization and computation of a minimal order hmm that realizes the exact joint densities of an output process based on only finite strings of such densities known as hmm partial realization problem the second problem is concerned with learning a hmm from finite output observations of a stochastic process we review and connect two fields of studies realization theory of hmms and the recent development in spectral methods for learning latent variable models our main results in this paper focus on generic situations namely statements that will be true for almost all hmms excluding a measure zero set in the parameter space in the main theorem we show that both the minimal quasi hmm realization and the minimal hmm realization can be efficiently computed based on the joint probabilities of length n strings for n in the order of cal o log d k in other words learning a quasi hmm and an hmm have comparable complexity for almost all hmms

ad nilpotent ideals of a parabolic subalgebra abstract we extend the results of cellini and papi p cellini p papi ad nilpotent ideals of a borel subalgebra j algebra 225 2000 130 140 p cellini p papi ad nilpotent ideals of a borel subalgebra ii j algebra 258 2002 112 121 on the characterizations of ad nilpotent and abelian ideals of a borel subalgebra to parabolic subalgebras of a simple lie algebra these characterizations are given in terms of elements of the affine weyl group and faces of alcoves in the case of a parabolic subalgebra of a classical simple lie algebra we give formulas for the number of these ideals

blind convolution using signal reconstruction from partial higher order cepstral information a blind deconvolution scheme for the reconstruction of a signal that propagates in the presence of reverberation and additive noise is presented the reconstruction employs data collected by two receivers placed well apart from each other each received sequence consists of the convolution of the transmitted signal with a channel applying the bicepstrum iterative reconstruction algorithm on the differences of the cepstra coefficients of the two received sequences the cepstra coefficients of the transmission channels can be computed and used for the reconstruction of the transmitted signal the deconvolution can be performed even if the data sequences are corrupted by additive noise the computation of the cepstra coefficients is based on the cross bispectrum if the noise processes present in the observation sequences are zero mean and uncorrelated while the bicepstrum of each observation is used if the noise processes are zero mean correlated with a symmetric probability density function

application of fast subspace decomposition to signal processing and communication problems the authors previously described a class of fast subspace decomposition fsd algorithms though these algorithms can be applied to solve a variety of signal processing and communication problems with significant computational reduction they focus their discussion on two typical applications i e sensor array processing and time series analysis in many cases replacing the usual eigenvalue decomposition evd or singular value decomposition svd by the fsd is quite straightforward however the fsd approach can exploit more structure of some special problems to further simplify the implementation they first discuss the implementation details of the fsd such as how to choose an optimal starting vector how to handle correlated noise and how to exploit additional matrix structure for further computational reduction then they describe an fsd approach targeted at data matrices rectangular n spl times m n spl ges m which requires only o nmd flops where d denotes the signal subspace dimension versus a regular o nm sup 2 m sup 3 svd the computational reduction is substantial in typical scenarios i e d spl lt m spl les n in the spectrum estimation problems the data matrix has additional structure such as toeplitz or hankel they finally show how the fsd can exploit such structure for further computational reduction

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

eigenanalysis based blind methods for identification equalization and inversion of linear time invariant channels some blind algorithms are presented for estimation of equalizers for linear time invariant lti channels from the eigenvectors of certain rank one matrices constructed from the second order statistics of the oversampled received signal it is shown that the channel can also be identified from the same matrices it can be shown that in multipath dominated environments equalizers with symbol spread of only one referred to as inverters can be used when sufficient diversity is available because of the manner in which structure in the channel distortion is exploited the proposed identification and equalization algorithms are also applicable to this case for the same reason the proposed algorithms do not require estimation of the channel memory only an upper bound is required equalizers of desired delay are estimated directly independent of others

fractionally spaced blind equalization using polyperiodic linear filtering performance of fractionally spaced equalizers using polyperiodic linear filtering is depicted the fractionally sampled version of an arbitrary stable channel s outputs with stationary random inputs exhibit cyclostationarity with cycle frequencies being integer multiples of the reciprocal of the oversampling factor by optimally polyperiodic linear filtering the oversampled outputs we can fully utilize the cyclostationary property with some traditional bussgang techniques such as stop and go decision directed method to adaptively update the filter coefficients the convergence rate of the fractionally spaced equalizers proposed is improved while the complexity retains in the same order as original bussgang ones some computer simulation results are shown to demonstrate the improvement we can achieve

observability and nonlinear filtering this paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability we consider a general class of hidden markov models in continuous time with compact signal state space and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process we demonstrate that observability implies stability of the filter i e the filtered estimates become insensitive to the initial measure at large times for the special case where the signal is a finite state markov process and the observations are of the white noise type a complete necessary and sufficient characterization of filter stability is obtained in terms of a slightly weaker detectability condition in addition to observability the role of controllability is explored finally the results are partially extended to non compact signal state spaces

links between multiplicity automata observable operator models and predictive state representations a unified learning framework stochastic multiplicity automata sma are weighted finite automata that generalize probabilistic automata they have been used in the context of probabilistic grammatical inference observable operator models ooms are a generalization of hidden markov models which in turn are models for discrete valued stochastic processes and are used ubiquitously in the context of speech recognition and bio sequence modeling predictive state representations psrs extend ooms to stochastic input output systems and are employed in the context of agent modeling and planning we present sma ooms and psrs under the common framework of sequential systems which are an algebraic characterization of multiplicity automata and examine the precise relationships between them furthermore we establish a unified approach to learning such models from data many of the learning algorithms that have been proposed can be understood as variations of this basic learning scheme and several turn out to be closely related to each other or even equivalent