the estimation of the model order in exponential families estimators are sought that achieve high exponential rate of decrease in the underestimation probability while keeping the overestimation probability exponent at a certain prescribed level it is assumed that a given integer is known to upper bound the true order some examples are given of possible applications of the proposed method in specific order estimation and hypothesis testing problems

learning high dimensional markov forest distributions analysis of error rates the problem of learning forest structured discrete graphical models from i i d samples is considered an algorithm based on pruning of the chow liu tree through adaptive thresholding is proposed it is shown that this algorithm is both structurally consistent and risk consistent and the error probability of structure learning decays faster than any polynomial in the number of samples under fixed model size for the high dimensional scenario where the size of the model d and the number of edges k scale with the number of samples n sufficient conditions on n d k are given for the algorithm to satisfy structural and risk consistencies in addition the extremal structures for learning are identified we prove that the independent resp tree model is the hardest resp easiest to learn using the proposed algorithm in terms of error rates for structure learning

darboux transforms deep reductions and solitons explicit formalisms for deep reductions of matrix differential equations and darboux covariance properties are presented the matrix 2 2 spectral problem of the second order is considered this problem with the reduction constraints being imposed on the potentials is the first equation of the lax representation of the hirota satsuma system 1981 the two reduction cases are treated with the help of the bilinear delta forms the covariance of these forms with respect to the darboux transforms under restrictions gives rise to explicit formulas of n soliton solutions in particular the two parameter soliton solutions of the hirota satsuma system are obtained the specific feature of such solutions evolution is that the singularity appears in some parameter region the yajima oikawa system 1976 is given as an example of the technique application to a 3 3 spectral problem

multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system compared with most of the existing tanh methods the jacobi elliptic function method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters the solutions obtained in this paper include a kink shaped and bell shaped soliton solutions b rational solutions c triangular periodic solutions and d jacobi and weierstrass doubly periodic wave solutions among them the jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition the efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper kdv mkdv ito s fifth mkdv hirota nizhnik novikov veselov broer kaup generalized coupled hirota satsuma coupled schrodinger kdv 2 1 dimensional dispersive long wave 2 1 dimensional davey stewartson equations in addition as an illustrative sample the properties of the soliton solutions and jacobi doubly periodic solutions for the hirota equation are shown by some figures the links among our proposed method the tanh method extended tanh method and the jacobi elliptic function method are clarified generally

high dimensional structure estimation in ising models local separation criterion we consider the problem of high dimensional ising graphical model selection we propose a simple algorithm for structure estimation based on the thresholding of the empirical conditional variation distances we introduce a novel criterion for tractable graph families where this method is efficient based on the presence of sparse local separators between node pairs in the underlying graph for such graphs the proposed algorithm has a sample complexity of n omega j min 2 log p where p is the number of variables and j min is the minimum absolute edge potential in the model we also establish nonasymptotic necessary and sufficient conditions for structure estimation

forest density estimation we study graph estimation and density estimation in high dimensions using a family of density estimators based on forest structured undirected graphical models for density estimation we do not assume the true distribution corresponds to a forest rather we form kernel density estimates of the bivariate and univariate marginals and apply kruskal s algorithm to estimate the optimal forest on held out data we prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest for graph estimation we consider the problem of estimating forests with restricted tree sizes we prove that finding a maximum weight spanning forest with restricted tree size is np hard and develop an approximation algorithm for this problem viewing the tree size as a complexity parameter we then select a forest using data splitting and prove bounds on excess risk and structure selection consistency of the procedure experiments with simulated data and microarray data indicate that the methods are a practical alternative to gaussian graphical models

interactions among periodic waves and solitary waves for a higher dimensional system the general variable separated approach is successfully extended to the 2 1 dimensional physical model and we obtain a universal formula involving arbitrary number of variable separated functions because of the existence of the arbitrary functions in the universal formula many new types of structures of periodic waves for example the periodic periodic interaction waves periodic kink interaction waves periodic peakon interaction waves periodic compacton interaction waves periodic foldon interaction waves etc are investigated both analytically and graphically some novel features or interesting behaviours are revealed

solitary wave and periodic wave solutions for the thermally forced gravity waves in atmosphere by introducing a new transformation a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system which extends fan s direct algebraic method to the case when r 4 the solutions of a first order nonlinear ordinary differential equation with a higher degree nonlinear term and fan s direct algebraic method of obtaining exact solutions to nonlinear partial differential equations are applied to the combined kdv mkdv gkdv equation which is derived from a simple incompressible non hydrostatic boussinesq equation with the influence of thermal forcing and is applied to investigate internal gravity waves in the atmosphere as a result by taking advantage of the new first order nonlinear ordinary differential equation with a fifth degree nonlinear term and an eighth degree nonlinear term periodic wave solutions associated with the jacobin elliptic function and the bell and kink profile solitary wave solutions are obtained under the effect of thermal forcing most importantly the mechanism of propagation and generation of the periodic waves and the solitary waves is analysed in detail according to the values of the heating parameter which show that the effect of heating in atmosphere helps to excite westerly or easterly propagating periodic internal gravity waves and internal solitary waves in atmosphere which are affected by the local excitation structures in atmosphere in addition as an illustrative sample the properties of the solitary wave solution and jacobin periodic solution are shown by some figures under the consideration of heating interaction

high dimensional gaussian graphical model selection walk summability and local separation criterion we consider the problem of high dimensional gaussian graphical model selection we identify a set of graphs for which an efficient estimation algorithm exists and this algorithm is based on thresholding of empirical conditional covariances under a set of transparent conditions we establish structural consistency or sparsistency for the proposed algorithm when the number of samples n jmin 2 log p where p is the number of variables and jmin is the minimum absolute edge potential of the graphical model the sufficient conditions for sparsistency are based on the notion of walk summability of the model and the presence of sparse local vertex separators in the underlying graph we also derive novel non asymptotic necessary conditions on the number of samples required for sparsistency

learning factor graphs in polynomial time and sample complexity we study the computational and sample complexity of parameter and structure learning in graphical models our main result shows that the class of factor graphs with bounded degree can be learned in polynomial time and from a polynomial number of training examples assuming that the data is generated by a network in this class this result covers both parameter estimation for a known network structure and structure learning it implies as a corollary that we can learn factor graphs for both bayesian networks and markov networks of bounded degree in polynomial time and sample complexity importantly unlike standard maximum likelihood estimation algorithms our method does not require inference in the underlying network and so applies to networks where inference is intractable we also show that the error of our learned model degrades gracefully when the generating distribution is not a member of the target class of networks in addition to our main result we show that the sample complexity of parameter learning in graphical models has an o 1 dependence on the number of variables in the model when using the kl divergence normalized by the number of variables as the performance criterion

on the estimation of the order of a markov chain and universal data compression the authors estimate the order of a finite markov source based on empirically observed statistics the performance criterion adopted is to minimize the probability of underestimating the model order while keeping the overestimation probability exponent at a prescribed level a universal asymptotically optimal test in the sense just defined is proposed for the case where a given integer is known to be the upper bound of the true order for the case where such a bound is unavailable an alternative rule based on the lempel ziv data compression algorithm is shown to be asymptotically optimal also and computationally more efficient

numerical integration of a coupled korteweg de vries system abstract we introduce a numerical method for general coupled korteweg de vries systems the scheme is valid for solving cauchy problems for an arbitrary number of equations with arbitrary constant coefficients the numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes the method is applied to the hirota satsuma hs system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy we also illustrate the method to show the effects of constants with a transition to nonintegrable cases

robust estimation of latent tree graphical models inferring hidden states with inexact parameters latent tree graphical models are widely used in computational biology signal and image processing and network tomography here we design a new efficient estimation procedure for latent tree models including gaussian and discrete reversible models that significantly improves on previous sample requirement bounds our techniques are based on a new hidden state estimator that is robust to inaccuracies in estimated parameters more precisely we prove that latent tree models can be estimated with high probability in the so called kesten stigum regime with o log2n samples where n is the number of nodes

a generalized hirota satsuma coupled kdv system darboux transformations and reductions a darboux transformation is constructed for the generalized hirota satsuma coupled kdv system and the result is compared with the recent work of geng and his collaborators x g geng et al phys rev e 79 056602 2009 and x g geng and g l he j math phys 51 033514 2010 it is shown that our darboux transformation may be applied to three interesting reductions of the general system in addition the iteration of this darboux transformation is worked out and some solutions to the associated systems are obtained

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

algebraic construction of the darboux matrix revisited we present algebraic construction of darboux matrices for 1 1 dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case we discuss different approaches to the darboux backlund transformation based on different dependences of the darboux matrix polynomial sum of partial fractions or the transfer matrix form we derive symmetric n soliton formulae in the general case the matrix spectral parameter and dressing actions in loop groups are also discussed we describe reductions to twisted loop groups unitary reductions the matrix lax pair for the kdv equation and reductions of chiral models harmonic maps to su n and to grassmann spaces we show that in the kdv case the nilpotent darboux matrix generates the binary darboux transformation the paper is intended as a review of known results usually presented in a novel context but some new results are included as well e g general compact formulae for n soliton surfaces and linear and bilinear constraints on the nonisospectral lax pair matrices which are preserved by darboux transformations

a large deviation analysis of the maximum likelihood learning of markov tree structures the problem of maximum likelihood ml estimation of discrete tree structured distributions is considered chow and liu established that ml estimation reduces to the construction of a maximum weight spanning tree using the empirical mutual information quantities as the edge weights using the theory of large deviations we analyze the exponent associated with the error probability of the event that the ml estimate of the markov tree structure differs from the true tree structure given a set of independently drawn samples by exploiting the fact that the output of ml estimation is a tree we establish that the error exponent is equal to the exponential rate of decay of a single dominant crossover event we prove that in this dominant crossover event a non neighbor node pair replaces a true edge of the distribution that is along the path of edges in the true tree graph connecting the nodes in the non neighbor pair using ideas from euclidean information theory we then analyze the scenario of ml estimation in the very noisy learning regime and show that the error exponent can be approximated as a ratio which is interpreted as the signal to noise ratio snr for learning tree distributions we show via numerical experiments that in this regime our snr approximation is accurate

the consistency of the bic markov order estimator the bayesian information criterion bic estimates the order of a markov chain with finite alphabet a from observation of a sample path x 1 x 2 x n as that value k k that minimizes the sum of the negative logarithm of the kth order maximum likelihood and the penalty term a k a 1 2 log n we show that k equals the correct order of the chain eventually almost surely as n oo thereby strengthening earlier consistency results that assumed an apriori bound on the order a key tool is a strong ratio typicality result for markov sample paths we also show that the bayesian estimator or minimum description length estimator of which the bic estimator is regarded as an approximation fails to be consistent for the uniformly distributed i i d process

constructing of new exact solutions to the gkdv mkdv equation with any order nonlinear terms by g g expansion method abstract the g g expansion method is introduced to construct more general exact travelling wave solutions to the generalized kdv mkdv equation with any order nonlinear terms as a result hyperbolic function solution trigonometric function solution and rational solution with parameters are obtained when the parameters take up special values the solitary waves are also derived from the travelling wave solutions the g g expansion method presents a wider applicability for handling nonlinear wave equations

learning gaussian tree models analysis of error exponents and extremal structures the problem of learning tree structured gaussian graphical models from independent and identically distributed i i d samples is considered the influence of the tree structure and the parameters of the gaussian distribution on the learning rate as the number of samples increases is discussed specifically the error exponent corresponding to the event that the estimated tree structure differs from the actual unknown tree structure of the distribution is analyzed finding the error exponent reduces to a least squares problem in the very noisy learning regime in this regime it is shown that the extremal tree structure that minimizes the error exponent is the star for any fixed set of correlation coefficients on the edges of the tree if the magnitudes of all the correlation coefficients are less than 0 63 it is also shown that the tree structure that maximizes the error exponent is the markov chain in other words the star and the chain graphs represent the hardest and the easiest structures to learn in the class of tree structured gaussian graphical models this result can also be intuitively explained by correlation decay pairs of nodes which are far apart in terms of graph distance are unlikely to be mistaken as edges by the maximum likelihood estimator in the asymptotic regime

classification of polynomial integrable systems of mixed scalar and vector evolution equations i we perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries we consider polynomial systems that are homogeneous under a suitable weighting of variables this paper deals with the kdv weighting the burgers or potential kdv or modified kdv weighting the ibragimov shabat weighting and two unfamiliar weightings the case of other weightings will be studied in a subsequent paper making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems we give the complete lists of second order systems with a third order or a fourth order symmetry and third order systems with a fifth order symmetry for all but a few systems in the lists we show that the system or at least a subsystem of it admits either a lax representation or a linearizing transformation a thorough comparison with recent work of foursov and olver is made

a parsimonious mixture of gaussian trees model for oversampling in imbalanced and multimodal time series classification we propose a novel framework of using a parsimonious statistical model known as mixture of gaussian trees for modeling the possibly multimodal minority class to solve the problem of imbalanced time series classification by exploiting the fact that close by time points are highly correlated due to smoothness of the time series our model significantly reduces the number of covariance parameters to be estimated from o d 2 to o ld where l is the number of mixture components and d is the dimensionality thus our model is particularly effective for modeling high dimensional time series with limited number of instances in the minority positive class in addition the computational complexity for learning the model is only of the order o ln d 2 where n is the number of positively labeled samples we conduct extensive classification experiments based on several well known time series data sets both single and multimodal by first randomly generating synthetic instances from our learned mixture model to correct the imbalance we then compare our results with several state of the art oversampling techniques and the results demonstrate that when our proposed model is used in oversampling the same support vector machines classifier achieves much better classification accuracy across the range of data sets in fact the proposed method achieves the best average performance 30 times out of 36 multimodal data sets according to the f value metric our results are also highly competitive compared with nonoversampling based classifiers for dealing with imbalanced time series data sets

hamilton jacobi isaacs equations for differential games with asymmetric information on probabilistic initial condition abstract we investigate hamilton jacobi isaacs equations associated to a two players zero sum differential game with incomplete information the first player has complete information on the initial state of the game while the second player has only information of a possibly uncountable probabilistic nature he knows a probability measure on the initial state such differential games with finite type incomplete information can be viewed as a generalization of the famous aumann maschler theory for repeated games the main goal and novelty of the present work consists in obtaining and investigating a hamilton jacobi isaacs equation satisfied by the upper and the lower values of the game since we obtain a uniqueness result for such hamilton jacobi equation as a byproduct this gives an alternative proof of the existence of a value of the differential game which has been already obtained in the literature by different technics since the hamilton jacobi equation is naturally stated in the space of probability measures we use the wasserstein distance and some tools of optimal transport theory

new rational solitary wave solutions of 2 1 dimensional burgers equation abstract based on the computer symbolic system maple and the rational solitary wave solutions and rational periodic solutions obtained by others previously we present a new direct ansatz in terms of the hyperbolic functions and periodic function to solve the nonlinear evolution equations that is to say we assume the equations have the solution in the following form u i a i 0 j 1 n i g j 1 a i j f b i j g c i j h d i j r where f tanh sech a b tanh c sech 2 g sech a b tanh c sech 2 h sech 2 a b tanh c sech 2 r tanh a b tanh c sech 2 and a 2 b 2 c 2 0 compared with the solutions obtained by the existing tanh methods riccati equation method and all kinds of their improved methods the solutions obtained by the proposed direct method not only include some obtained solutions but also include some new and general rational solutions and we choose the 2 1 dimensional burgers equation to illustrate the key step as a result we obtain abundant families of rational solitary wave solutions and rational periodic solutions furthermore the direct method is proved to be efficient

a rational generalization of fan s method and its application to generalized shallow water wave equations in this paper we employ a rational expansion to generalize fan s method for exact travelling wave solutions for nonlinear partial differential equations pdes to verify the reliability of the proposed method the generalized shallow water wave gsww equation has been investigated as an example kinds of new exact travelling wave solutions of a rational form have been obtained this indicates that the proposed method provides a more general result for exact solution of nonlinear equations

beyond independent components trees and clusters we present a generalization of independent component analysis ica where instead of looking for a linear transform that makes the data components independent we look for a transform that makes the data components well fit by a tree structured graphical model this tree dependent component analysis tca provides a tractable and flexible approach to weakening the assumption of independence in ica in particular tca allows the underlying graph to have multiple connected components and thus the method is able to find clusters of components such that components are dependent within a cluster and independent between clusters finally we make use of a notion of graphical models for time series due to brillinger 1996 to extend these ideas to the temporal setting in particular we are able to fit models that incorporate tree structured dependencies among multiple time series

phase transition in the sample complexity of likelihood based phylogeny inference reconstructing evolutionary trees from molecular sequence data is a fundamental problem in computational biology stochastic models of sequence evolution are closely related to spin systems that have been extensively studied in statistical physics and that connection has led to important insights on the theoretical properties of phylogenetic reconstruction algorithms as well as the development of new inference methods here we study maximum likelihood a classical statistical technique which is perhaps the most widely used in phylogenetic practice because of its superior empirical accuracy at the theoretical level except for its consistency that is the guarantee of eventual correct reconstruction as the size of the input data grows much remains to be understood about the statistical properties of maximum likelihood in this context in particular the best bounds on the sample complexity or sequence length requirement of maximum likelihood that is the amount of data required for correct reconstruction are exponential in the number n of tips far from known lower bounds based on information theoretic arguments here we close the gap by proving a new upper bound on the sequence length requirement of maximum likelihood that matches up to constants the known lower bound for some standard models of evolution more specifically for the r state symmetric model of sequence evolution on a binary phylogeny with bounded edge lengths we show that the sequence length requirement behaves logarithmically in n when the expected amount of mutation per edge is below what is known as the kesten stigum threshold in general the sequence length requirement is polynomial in n our results imply moreover that the maximum likelihood estimator can be computed efficiently on randomly generated data provided sequences are as above our main technical contribution which may be of independent interest relates the total variation distance between the leaf state distributions of two trees with a notion of combinatorial distance between the trees in words we show in a precise quantitative manner that the more different two evolutionary trees are the easier it is to distinguish their output

nonlocal symmetries of the hirota satsuma coupled korteweg de vries system and their applications exact interaction solutions and integrable hierarchy the nonlocal symmetry is derived from the known darboux transformation dt of the hirota satsuma coupled korteweg de vries hs ckdv system and infinitely many nonlocal symmetries are given by introducing the internal parameters by extending the hs ckdv system to an auxiliary system with five dependent variables the prolongation is found to localize the so called seed nonlocal symmetry related to the dt by applying the general lie point symmetry method to this enlarged system we obtain two main results a new type of finite symmetry transformation is derived which is different from the initial dt and can generate new solutions from old ones some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and painleve waves are computed through similarity reductions in addition two kinds of new integrable models are proposed from the obtained nonlocal symmetry the negative hs ckdv hierarchy by introducing the internal parameters the integrable models both in lower and higher dimensions by restricting the symmetry constraints

constructing of exact solutions to the kdv and burgers equations with power law nonlinearity by the extended g g expansion method abstract the extended g g expansion method is introduced to construct more general exact traveling wave solutions to the kdv and burgers equations with any order nonlinear terms the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions when the parameters take up special values new solitary periodic and complex solutions are also derived from the traveling wave solutions

forest learning from data and its universal coding this paper considers structure learning from data with n samples of p variables assuming that the structure is a forest using the chow liu algorithm specifically for incomplete data we construct two model selection algorithms that complete in o p 2 steps one obtains a forest with the maximum posterior probability given the data and the other obtains a forest that converges to the true one as n increases we show that the two forests are generally different when some values are missing in addition we present estimations for benchmark data sets to demonstrate that both algorithms work in realistic situations moreover we derive the conditional entropy provided that no value is missing and we evaluate the per sample expected redundancy for the universal coding of incomplete data in terms of the number of non missing samples

context tree estimation for not necessarily finite memory processes via bic and mdl the concept of context tree usually defined for finite memory processes is extended to arbitrary stationary ergodic processes with finite alphabet these context trees are not necessarily complete and may be of infinite depth the familiar bayesian information criterion bic and minimum description length mdl principles are shown to provide strongly consistent estimators of the context tree via optimization of a criterion for hypothetical context trees of finite depth allowed to grow with the sample size n as o logn algorithms are provided to compute these estimators in o n time and to compute them on line for all i les n in o nlogn time

elementary and binary darboux transformations at rings for a given idempotent p and some element from a differential associative ring we introduce gauge transformation p with the spectral parameter that leaves some linear operators form invariant the explicit form of the is derived for the generalized zakharov shabat problem the maps that factorize darboux transformations are referred to as elementary ones binary transformations that correspond to the iteration of elementary maps with the special choice of solutions of the direct and conjugate problems are introduced and widen the potential reductions set we show an infinitesimal version of iterated transforms the spectral operator and soliton theories applications are outlined the nonabelian n wave interaction equation modification with additional linear terms its zero curvature representation soliton solutions and their stability with respect to infinitesimal deformations are studied

some new integrable nonlinear evolution equations and darboux transformation two classes of new nonlinear evolution equations are derived with the aid of a 4 4 matrix spectral problem with three potentials a darboux transformation for the first nontrivial nonlinear evolution equation is given based on the gauge transformation between the corresponding 4 4 matrix spectral problems as an application of the darboux transformation some explicit solutions of the nontrivially nonlinear evolution equation are obtained

new analytical positon negaton and complexiton solutions of a coupled kdv mkdv system new positon negaton and complexiton solutions of a coupled kdv mkdv system are constructed directly from zero seed solution by means of darboux transformation these new exact solutions are all analytical with different spectral parameters and constants selection