an investigation into the characteristics of non linear frequency response functions part 1 understanding the higher dimensional frequency spaces the characteristics of generalized frequency response functions gfrfs of non linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques it is shown how a systematic analysis can be achieved for a wide class of non linear systems in the frequency domain using the proposed methods the paper is divided into two parts in part 1 the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi dimensional frequency spaces the visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time in part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non linear systems described by the narmax class of models case studies are included in part 2 to illustrate the application of the new methods

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

density estimation in the l infty norm for dependent data with applications to the gibbs sampler this paper investigates the density estimation problem in the lx norm for dependent data it is shown that the iid optimal minimax rates are also optimal for smooth classes of stationary sequences satisfying certain 8 mixing or absolutely regular conditions moreover for given 3 mixing coefficients bounds on uniform convergence rates of kernel estimators are computed in terms of the mixing coefficients the rates and the bounds obtained are not only for estimating the density but also for its derivatives the results are then applied to give uniform convergence rates in problems associated with the gibbs sampler

stability of stochastic approximation under verifiable conditions in this paper we address the problem of the stability and convergence of the stochastic approximation procedure theta n 1 theta n gamma n 1 h theta n xi n 1 the stability of such sequences theta n is known to heavily rely on the behavior of the mean field h at the boundary of the parameter set and the magnitude of the stepsizes used the conditions typically required to ensure convergence and in particular the boundedness or stability of theta n are either too difficult to check in practice or not satisfied at all this is the case even for very simple models the most popular technique for circumventing the stability problem consists of constraining theta n to a compact subset mathcal k in the parameter space this is obviously not a satisfactory solution as the choice of mathcal k is a delicate one in this paper we first prove a deterministic stability result which relies on simple conditions on the sequences xi n and gamma n we then propose and analyze an algorithm based on projections on adaptive truncation sets which ensures that the aforementioned conditions required for stability are satisfied we focus in particular on the case where xi n is a so called markov state dependent noise we establish both the stability and convergence with probability 1 w p 1 of the algorithm under a set of simple and verifiable assumptions we illustrate our results with an example related to adaptive markov chain monte carlo algorithms

noise contrastive estimation of unnormalized statistical models with applications to natural image statistics we consider the task of estimating from observed data a probabilistic model that is parameterized by a finite number of parameters in particular we are considering the situation where the model probability density function is unnormalized that is the model is only specified up to the partition function the partition function normalizes a model so that it integrates to one for any choice of the parameters however it is often impossible to obtain it in closed form gibbs distributions markov and multi layer networks are examples of models where analytical normalization is often impossible maximum likelihood estimation can then not be used without resorting to numerical approximations which are often computationally expensive we propose here a new objective function for the estimation of both normalized and unnormalized models the basic idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise with this approach the normalizing partition function can be estimated like any other parameter we prove that the new estimation method leads to a consistent convergent estimator of the parameters for large noise sample sizes the new estimator is furthermore shown to behave like the maximum likelihood estimator in the estimation of unnormalized models there is a trade off between statistical and computational performance we show that the new method strikes a competitive trade off in comparison to other estimation methods for unnormalized models as an application to real data we estimate novel two layer models of natural image statistics with spline nonlinearities

the design of energy transfer filters for energy focus filtering this paper is concerned with extending the energy transfer filters etfs designs to a more general case where the design requirement is to focus signal energy from different input frequency bands into a desired output frequency range etfs are a class of non linear filters recently proposed by the authors to employ non linear effects to transfer signal energy from one frequency band to a different frequency location the new etf designs considered in the present study involve the development of an algorithm for determining output frequencies of non linear systems subject to an input the frequency components of which are located in two separate frequency intervals and a new procedure for the implementation of the filter design the results achieved provide an effective approach to the solution to a more general etf design problem

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

on variance reduction for stochastic smooth convex optimization with multiplicative noise we propose dynamic sampled stochastic approximation sa methods for stochastic optimization with a heavy tailed distribution with finite 2nd moment the objective is the sum of a smooth convex function with a convex regularizer typically it is assumed an oracle with an upper bound sigma 2 on its variance oubv differently we assume an oracle with multiplicative noise this rarely addressed setup is more aggressive but realistic where the variance may not be uniformly bounded our methods achieve optimal iteration complexity and near optimal oracle complexity for the smooth convex class we use an accelerated sa method a la fista which achieves given tolerance varepsilon 0 the optimal iteration complexity of mathscr o varepsilon frac 1 2 with a near optimal oracle complexity of mathscr o varepsilon 2 ln varepsilon frac 1 2 2 this improves upon ghadimi and lan math program 156 59 99 2016 where it is assumed an oubv for the strongly convex class our method achieves optimal iteration complexity of mathscr o ln varepsilon 1 and optimal oracle complexity of mathscr o varepsilon 1 this improves upon byrd et al math program 134 127 155 2012 where it is assumed an oubv in terms of variance our bounds are local they depend on variances sigma x 2 at solutions x and the per unit distance multiplicative variance sigma 2 l for the smooth convex class there exist policies such that our bounds resemble up to absolute constants those obtained in the mentioned papers if it was assumed an oubv with sigma 2 sigma x 2 for the strongly convex class such property is obtained exactly if the condition number is estimated or in the limit for better conditioned problems or for larger initial batch sizes in any case if it is assumed an oubv our bounds are thus sharper since typically max sigma x 2 sigma l 2 ll sigma 2

morphologically constrained grfs applications to texture synthesis and analysis a new class of gibbs random fields grfs is proposed capable of modeling geometrical constraints in images by means of mathematical morphology the proposed approach known as morphologically constrained grfs models images by means of their size density since the size density is a multiresolution statistical summary morphologically constrained grfs explicitly incorporate multiresolution information into image modeling important properties are studied and their implication to texture synthesis and analysis is discussed statistical inference can be easily implemented by means of mathematical morphology this allows the design of a computationally simple morphological bayes classifier which produces excellent results in classifying natural textures

a monte carlo implementation of the em algorithm and the poor man s data augmentation algorithms abstract the first part of this article presents the monte carlo implementation of the e step of the em algorithm given the current guess to the maximizer of the posterior distribution latent data patterns are generated from the conditional predictive distribution the expected value of the augmented log posterior is then updated as a mixture of augmented log posteriors mixed over the generated latent data patterns multiple imputations in the m step of the algorithm this mixture is maximized to obtain the update to the maximizer of the observed posterior the gradient and hessian of the observed log posterior are also expressed as mixtures mixed over the multiple imputations the relation between the monte carlo em mcem algorithm and the data augmentation algorithm is noted two modifications to the mcem algorithm the poor man s data augmentation algorithms which allow for the calculation of the entire posterior are then presented these approximations serve as diagnostics for the validity o

spatiotemporal prediction for log gaussian cox processes space time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems we describe a flexible class of space time point processes our models are cox processes whose stochastic intensity is a space time ornstein uhlenbeck process we develop moment based methods of parameter estimation show how to predict the underlying intensity by using a markov chain monte carlo approach and illustrate the performance of our methods on a synthetic data set

non euclidean restricted memory level method for large scale convex optimization we propose a new subgradient type method for minimizing extremely large scale nonsmooth convex functions over simple domains the characteristic features of the method are a the possibility to adjust the scheme to the geometry of the feasible set thus allowing to get nearly dimension independent and nearly optimal in the large scale case rate of convergence results for minimization of a convex lipschitz continuous function over a euclidean ball a standard simplex and a spectahedron the set of positive semidefinite symmetric matrices of given size with unit trace b flexible handling of accumulated information allowing for tradeoff between the level of utilizing this information and iteration s complexity we present extensions of the scheme for the cases of minimizing non lipschitzian convex objectives finding saddle points of convex concave functions and solving variational inequalities with monotone operators finally we report on encouraging numerical results of experiments with test problems of dimensions up to 66 000

analysis of output frequencies of nonlinear systems in this paper an algorithm is derived for the determination of the output frequency ranges of nonlinear systems which extends previous results on the output frequencies of nonlinear systems to a more general situation the new results are significant for the analysis of the output frequency response of a wide class of nonlinear systems

output frequency response characteristics of nonlinear systems part ii overlapping effects and commensurate multi tone excitations the output response of nonlinear systems is strongly dependent on the extent to which different input frequency combinations intermodulations overlap one another and contribute to a common output frequency component this makes it hard to determine the actual number of output frequencies or to determine which of the many possible combinations contribute to a specified output frequency of interest systematic algorithms are presented for resolving these issues in a manner which is both efficient and straightforward to implement on a computer three broad classes of input are considered multi tone but nominally incommensurate sinusoids multi tone harmonically related sinusoids and multi tone excitations that are equi spaced in frequency the considered inputs also admit the possibility of a dc bias component since this can significantly affect the system response the algorithms are therefore applicable to a broad range of practical inputs and their use and benefits are illustrated with several examples

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

splitting of exact sequences of frechet spaces in the absence of continuous norms abstract in the present paper we study the splitting of a short exact sequence 0 g e s f 0 of nuclear frechet spaces where f need not admit a continuous norm i e e or f may be spaces of continuous functions on an open subset of r n or on a compact c manifold if e is such a space then we give a necessary and sufficient condition for the splitting of the sequence in terms of a condition on s and of linear topological invariants loc and dnloc this is used to give a structure theory of the space s n a space which is isomorphic to many spaces of smooth functions and to study the splitting of differential complexes the paper offers a different approach to problems which have been studied in domanski vogt j funct anal 153 1998 203

output frequency response characteristics of nonlinear systems part i general multi tone inputs a systematic algorithm for determining the output frequencies generated by a nonlinear system subject to a general multi tone input including a bias input is presented the algorithm is very efficient and circumvents many of the difficulties of duplicate frequencies that are often generated using previous methods the inclusion of the dc bias component in the analysis is also important because of the significant effect that non zero mean inputs can have on nonlinear system response the algorithm is straightforward to implement on a computer and the results are illustrated by means of an example

bayesian inference for logistic models using polya gamma latent variables we propose a new data augmentation strategy for fully bayesian inference in models with binomial likelihoods the approach appeals to a new class of polya gamma distributions which are constructed in detail a variety of examples are presented to show the versatility of the method including logistic regression negative binomial regression nonlinear mixed effect models and spatial models for count data in each case our data augmentation strategy leads to simple effective methods for posterior inference that 1 circumvent the need for analytic approximations numerical integration or metropolis hastings and 2 outperform other known data augmentation strategies both in ease of use and in computational efficiency all methods including an efficient sampler for the polya gamma distribution are implemented in the r package bayeslogit supplementary materials for this article are available online

ascent based monte carlo expectation maximization summary the expectation maximization em algorithm is a popular tool for maximizing likelihood functions in the presence of missing data unfortunately em often requires the evaluation of analytically intractable and high dimensional integrals the monte carlo em mcem algorithm is the natural extension of em that employs monte carlo methods to estimate the relevant integrals typically a very large monte carlo sample size is required to estimate these integrals within an acceptable tolerance when the algorithm is near convergence even if this sample size were known at the onset of implementation of mcem its use throughout all iterations is wasteful especially when accurate starting values are not available we propose a data driven strategy for controlling monte carlo resources in mcem the algorithm proposed improves on similar existing methods by recovering em s ascent i e likelihood increasing property with high probability being more robust to the effect of user defined inputs and handling classical monte carlo and markov chain monte carlo methods within a common framework because of the first of these properties we refer to the algorithm as ascent based mcem we apply ascent based mcem to a variety of examples including one where it is used to accelerate the convergence of deterministic em dramatically

on the convergence of monte carlo maximum likelihood calculations summary monte carlo maximum likelihood for normalized families of distributions can be used for an extremely broad class of models given any family he 0 e 0 of non negative integrable functions maximum likelihood estimates in the family obtained by normalizing the functions to integrate to 1 can be approximated by monte carlo simulation the only regularity conditions being a compactification of the parameter space such that the evaluation maps 0 h0 x remain continuous then with probability 1 the monte carlo approximant to the log likelihood hypoconverges to the exact log likelihood its maximizer converges to the exact maximum likelihood estimate approximations to profile likelihoods hypoconverge to the exact profile and level sets of the approximate likelihood support regions converge to the exact sets in painleve kuratowski set convergence the same results hold when there are missing data if a wald type integrability condition is satisfied asymptotic normality of the monte carlo error and convergence of the monte carlo approximation to the observed fisher information are also shown

stochastic approximation algorithms for partition function estimation of gibbs random fields we present an analysis of previously proposed monte carlo algorithms for estimating the partition function of a gibbs random field we show that this problem reduces to estimating one or more expectations of suitable functionals of the gibbs states with respect to properly chosen gibbs distributions as expected the resulting estimators are consistent certain generalizations are also provided we study computational complexity with respect to grid size and show that monte carlo partition function estimation algorithms can be classified into two categories e type algorithms that are of exponential complexity and p type algorithms that are of polynomial complexity turing reducible to the problem of sampling from the gibbs distribution e type algorithms require estimating a single expectation whereas p type algorithms require estimating a number of expectations with respect to gibbs distributions which are chosen to be sufficiently close to each other in the latter case the required number of expectations is of polynomial order with respect to grid size we compare computational complexity by using both theoretical results and simulation experiments we determine the most efficient e type and p type algorithms and conclude that p type algorithms are more appropriate for partition function estimation we finally suggest a practical and efficient p type algorithm for this task

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

baseband volterra filters for implementing carrier based nonlinearities a diagonal coordinate representation for volterra filters is developed and exploited to derive efficient volterra filter implementations for processing carrier based input signals in the diagonal coordinate representation the output is expressed as a sum of linear filters applied to modified input signals hence linear filtering methods are employed to implement the nonlinear filter on a baseband version of the input downsampling is then used to reduce computational complexity the same approach is employed to develop efficient implementations for processing continuous time carrier based signals pulse amplitude modulated signals and frequency division multiplexed input signals

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

segmentation of macular fluorescein angiographies a statistical approach this paper is concerned with the use of bayesian methods in the segmentation of macular uorescein angiographies fluorescein angiography is used in ophthalmic practice to evaluate vascular retinopathies and choroidopathies sodium uorescein is injected in the arm s cubital vein of the patient and its distribution is observed along retinal vessels at certain times in this task a previous and essential step is the segmentation of the image into its relevant components in order to obtain this segmentation bayesian methods can be used because a previous knowledge about the spatial structure of the scene to be segmented is available in this kind of images the stochastic model assumed for the observed intensities is a simple model with a gaussian noise process which is statiscally independent between pixels the process of labels x is modelled as a markov random eld with a space dependent external eld expressing the anatomy of the ocular fundus and higher order interactions encouraging blood vessels to be thin and large this procedure is applied to di erent cases of diabetic retinophaty and vein occlusions two algorithms have been used to estimate x simulated annealing and iterated conditional modes in order to evaluate the accuracy of the estimation several error measures have been calculated 2001 pattern recognition society published by elsevier science ltd all rights reserved

mapping nonlinear integro differential equations to a generalized describing function form an algebraic relationship is derived which enables the generalized amplitude dependent describing function to be expressed in terms of the parameters of a general nonlinear integro differential equation the algorithm is not restricted to a specific input but treats the whole class of harmonically related phase independent sinusoidal inputs and is illustrated by a number of examples

practical frequency response analysis of non linear time delayed differential or difference equation models a systematic algorithm for generating the polyharmonic balance equations for any system within a broad class of time delayed differential or non linear difference equation models is presented the method which is readily automated enables the balance equations to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of a general harmonic waveform the system frequency response or amplitude dependent describing function is then readily computed the method is illustrated by means of examples including both a time delayed differential system example and a discrete time narx model application the results are validated against detailed numeric simulation which confirms the accuracy and efficacy of the approach

convergence of the wang landau algorithm we analyze the convergence properties of the wang landau algorithm this sampling method belongs to the general class of adaptive importance sampling strategies which use the free energy along a chosen reaction coordinate as a bias such algorithms are very helpful to enhance the sampling properties of markov chain monte carlo algorithms when the dynamics is metastable we prove the convergence of the wang landau algorithm and an associated central limit theorem

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

mapping from parametric characteristics to generalized frequency response functions of non linear systems based on the parametric characteristic of the nth order generalized frequency response function gfrf for non linear systems described by a non linear differential equation nde model a mapping function from the parametric characteristics to the gfrfs is established by which the nth order gfrf can be directly written into a more straightforward and meaningful form in terms of the first order gfrf i e an n degree polynomial function of the first order gfrf the new expression has no recursive relationship between different order gfrfs and demonstrates some new properties of the gfrfs which can explicitly unveil the linear and non linear factors included in the gfrfs and reveal clearly the relationship between the nth order gfrf and its parametric characteristic as well as the relationship between the nth order gfrf and the first order gfrf the new results provide a novel and useful insight into the frequency domain analysis and design of non linear systems based on the gfrfs several examples are given to illustrate the theoretical results

learning in gibbsian fields how accurate and how fast can it be gibbsian fields or markov random fields are widely used in bayesian image analysis but learning gibbs models is computationally expensive the computational complexity is pronounced by the recent minimax entropy frame models which use large neighborhoods and hundreds of parameters in this paper we present a common framework for learning gibbs models we identify two key factors that determine the accuracy and speed of learning gibbs models the efficiency of likelihood functions and the variance in approximating partition functions using monte carlo integration we propose three new algorithms in particular we are interested in a maximum satellite likelihood estimator which makes use of a set of precomputed gibbs models called satellites to approximate likelihood functions this algorithm can approximately estimate the minimax entropy model for textures in seconds in a hp workstation the performances of various learning algorithms are compared in our experiments

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

improved contact prediction in proteins using pseudolikelihoods to infer potts models spatially proximate amino acids in a protein tend to coevolve a protein s three dimensional 3d structure hence leaves an echo of correlations in the evolutionary record reverse engineering 3d structures from such correlations is an open problem in structural biology pursued with increasing vigor as more and more protein sequences continue to fill the data banks within this task lies a statistical inference problem rooted in the following correlation between two sites in a protein sequence can arise from firsthand interaction but can also be network propagated via intermediate sites observed correlation is not enough to guarantee proximity to separate direct from indirect interactions is an instance of the general problem of inverse statistical mechanics where the task is to learn model parameters fields couplings from observables magnetizations correlations samples in large systems in the context of protein sequences the approach has been referred to as direct coupling analysis here we show that the pseudolikelihood method applied to 21 state potts models describing the statistical properties of families of evolutionarily related proteins significantly outperforms existing approaches to the direct coupling analysis the latter being based on standard mean field techniques this improved performance also relies on a modified score for the coupling strength the results are verified using known crystal structures of specific sequence instances of various protein families code implementing the new method can be found at http plmdca csc kth se

high dimensional ising model selection using l1 regularized logistic regression we consider the problem of estimating the graph associated with a binary ising markov random field we describe a method based on l1 regularized logistic regression in which the neighborhood of any given node is estimated by performing logistic regression subject to an l1 constraint the method is analyzed under high dimensional scaling in which both the number of nodes p and maximum neighborhood size d are allowed to grow as a function of the number of observations n our main results provide sufficient conditions on the triple n p d and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously with coherence conditions imposed on the population fisher information matrix we prove that consistent neighborhood selection can be obtained for sample sizes n d3log p with exponentially decaying error when these same conditions are imposed directly on the sample matrices we show that a reduced sample size of n d2log p suffices for the method to estimate neighborhoods consistently although this paper focuses on the binary graphical models we indicate how a generalization of the method of the paper would apply to general discrete markov random fields

asymptotic behavior of the gibbs sampler mild sufficient conditions for the ergodicity of the gibbs sampler are obtained the geometric ergodicity of the gibbs sampler is then discussed where the main tool is the drift criterion this criterion requires the construction of a generalized energy function so that the chain is dissipating energy at a suitable rate whenever it lies outside the center of the state space for the two variable case a useful recipe is provided for constructing appropriate energy functions these results are illustrated with four examples

using em to obtain asymptotic variance covariance matrices the sem algorithm abstract the expectation maximization em algorithm is a popular and often remarkably simple method for maximum likelihood estimation in incomplete data problems one criticism of em in practice is that asymptotic variance covariance matrices for parameters e g standard errors are not automatic byproducts as they are when using some other methods such as newton raphson in this article we define and illustrate a procedure that obtains numerically stable asymptotic variance covariance matrices using only the code for computing the complete data variance covariance matrix the code for em itself and code for standard matrix operations the basic idea is to use the fact that the rate of convergence of em is governed by the fractions of missing information to find the increased variability due to missing information to add to the complete data variance covariance matrix we call this supplemented em algorithm the sem algorithm theory and particular examples reinforce the conclusion that the sem alg

a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series abstract high order frequency response functions based on the volterra series are employed to represent the input output characteristics of the duffing oscillator subject to sinusoidal excitation from these a series representation of a first order frequency response function of the non linear system is obtained and used to derive a simple criterion which provides an estimate of the magnitude of the input sine wave above which the predicted response begins to diverge the criterion results in a small overestimate of the magnitude of the input force excitation at the undamped natural frequency condition although the derived criterion strictly applies only to the duffing oscillator the procedure can be applied generally and is extended to a two degree of freedom system with a non linear cubic stiffness producing a similar criterion to that of the single degree of freedom system

convergence of adaptive and interacting markov chain monte carlo algorithms adaptive and interacting markov chain monte carlo algorithms mcmc have been recently introduced in the literature these novel simulation algorithms are designed to increase the simulation efficiency to sample complex distributions motivated by some recently introduced algorithms such as the adaptive metropolis algorithm and the interacting tempering algorithm we develop a general methodological and theoretical framework to establish both the convergence of the marginal distribution and a strong law of large numbers this framework weakens the conditions introduced in the pioneering paper by roberts and rosenthal j appl probab 44 2007 458 475 it also covers the case when the target distribution is sampled by using markov transition kernels with a stationary distribution that differs from

markov chain monte carlo for autologistic regression models with application to the distribution of plant species summary in this paper we explore using autologistic regression models for spatial binary data with covariates autologistic regression models can handle binary responses exhibiting both spatial correlation and dependence on covariates we use markov chain monte carlo mcmc to estimate the parameters in these models the distributional behavior of the mcmc maximum likelihood estimates mcmc mles is studied via simulation we find that the mcmc mles are approximately normally distributed and that the mcmc estimates of fisher information may be used to estimate the variance of the mcmc mles and to construct confidence intervals finally we illustrate by example how our studies may be applied to model the distribution of plant species

on perturbed proximal gradient algorithms we study a version of the proximal gradient algorithm for which the gradient is intractable and is approximated by monte carlo methods and in particular markov chain monte carlo we derive conditions on the step size and the monte carlo batch size under which convergence is guaranteed both increasing batch size and constant batch size are considered we also derive non asymptotic bounds for an averaged version our results cover both the cases of biased and unbiased monte carlo approximation to support our findings we discuss the inference of a sparse generalized linear model with random effect and the problem of learning the edge structure and parameters of sparse undirected graphical models

convergence of contrastive divergence algorithm in exponential family the contrastive divergence cd algorithm has achieved notable success in training energy based models including restricted boltzmann machines and played a key role in the emergence of deep learning the idea of this algorithm is to approximate the intractable term in the exact gradient of the log likelihood function by using short markov chain monte carlo mcmc runs the approximate gradient is computationally cheap but biased whether and why the cd algorithm provides an asymptotically consistent estimate are still open questions this paper studies the asymptotic properties of the cd algorithm in canonical exponential families which are special cases of the energy based model suppose the cd algorithm runs m mcmc transition steps at each iteration t and iteratively generates a sequence of parameter estimates theta t t ge 0 given an i i d data sample x i i 1 n sim p theta star under conditions which are commonly obeyed by the cd algorithm in practice we prove the existence of some bounded m such that any limit point of the time average sum s 0 t 1 theta s t as t to infty is a consistent estimate for the true parameter theta star our proof is based on the fact that theta t t ge 0 is a homogenous markov chain conditional on the data sample x i i 1 n this chain meets the foster lyapunov drift criterion and converges to a random walk around the maximum likelihood estimate the range of the random walk shrinks to zero at rate mathcal o 1 sqrt 3 n as the sample size n to infty

annealing markov chain monte carlo with applications to ancestral inference abstract markov chain monte carlo mcmc the metropolis hastings algorithm has been used for many statistical problems including bayesian inference likelihood inference and tests of significance though the method generally works well doubts about convergence often remain here we propose mcmc methods distantly related to simulated annealing our samplers mix rapidly enough to be usable for problems in which other methods would require eons of computing time they simulate realizations from a sequence of distributions allowing the distribution being simulated to vary randomly over time if the sequence of distributions is well chosen then the sampler will mix well and produce accurate answers for all the distributions even when there is only one distribution of interest these annealing like samplers may be the only known way to get a rapidly mixing sampler these methods are essential for attacking very hard problems which arise in areas such as statistical genetics we illustrate the methods wi

on the epiconvergence of stochastic optimization problems the problem of strong consistency of sequences of optimal solutions to stochastic optimization problems is considered this problem is related to a large number of applications including bayesian decision problems and monte carlo simulations as well as a number of statistical methodologies such as maximum likelihood estimation the theory of epiconvergence being a framework within which such results can be established the epiconvergence of the performance criteria of a sequence of stochastic optimization problems is proved under simple weak assumptions

automatic computation of polyharmonic balance equations for non linear differential systems an automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross product non linear differential system to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of a general harmonic waveform the system frequency response in the form of a multi input amplitude dependent describing function is therefore readily computed the method is illustrated by means of an example and the results validated against detailed numeric simulation

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

adaptive subgradient methods for online learning and stochastic optimization we present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient based learning metaphorically the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features our paradigm stems from recent advances in stochastic optimization and online learning which employ proximal functions to control the gradient steps of the algorithm we describe and analyze an apparatus for adaptively modifying the proximal function which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight we give several efficient algorithms for empirical risk minimization problems with common and important regularization functions and domain constraints we experimentally study our theoretical analysis and show that adaptive subgradient methods outperform state of the art yet non adaptive subgradient algorithms

stochastic approximation in monte carlo computation the wang landau wl algorithm is an adaptive markov chain monte carlo algorithm used to calculate the spectral density for a physical system a remarkable feature of the wl algorithm is that it is not trapped by local energy minima which is very important for systems with rugged energy landscapes this feature has led to many successful applications of the algorithm in statistical physics and biophysics however there does not exist rigorous theory to support its convergence and the estimates produced by the algorithm can reach only a limited statistical accuracy in this article we propose the stochastic approximation monte carlo samc algorithm which overcomes the shortcomings of the wl algorithm we establish a theorem concerning its convergence the estimates produced by samc can be improved continuously as the simulation proceeds samc also extends applications of the wl algorithm to continuum systems the potential uses of samc in statistics are discussed through two classes of applications i

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data

monte carlo likelihood inference for missing data models we describe a monte carlo method to approximate the maximum likelihood estimate mle when there are missing data and the observed data likelihood is not available in closed form this method uses simulated missing data that are independent and identically distributed and independent of the observed data our monte carlo approximation to the mle is a consistent and asymptotically normal estimate of the minimizer of the kullback leibler information as both monte carlo and observed data sample sizes go to infinity simultaneously plug in estimates of the asymptotic variance are provided for constructing confidence regions for we give logit normal generalized linear mixed model examples calculated using an r package

describing functions volterra series and the analysis of non linear systems in the frequency domain a relationship between the higher order frequency response functions of the volterra series model and the well known describing function representation for non linear systems is derived it is shown that a large class of generalized describing functions can be derived from the volterra model and that this approach removes the restriction to specific inputs the concept of a worst case transfer function is introduced and examples are included to illustrate the results

convergence of the monte carlo expectation maximization for curved exponential families the monte carlo expectation maximization mcem algorithm is a versatile tool for inference in incomplete data models especially when used in combination with markov chain monte carlo simulation methods in this contribution the almost sure convergence of the mcem algorithm is established it is shown using uniform versions of ergodic theorems for markov chains that mcem converges under weak conditions on the simulation kernel practical illustrations are presented using a hybrid random walk metropolis hastings sampler and an independence sampler the rate of convergence is studied showing the impact of the simulation schedule on the fluctuation of the parameter estimate at the convergence a novel averaging procedure is then proposed to reduce the simulation variance and increase the rate of convergence

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

an investigation into the characteristics of non linear frequency response functions part 2 new analysis methods based on symbolic expansions and graphical techniques in part 1 of this paper the concepts of input and output frequency subdomains were introduced to give insight into the relation between one dimensional and multi dimensional frequency spaces the visualisation of both magnitude and phase responses of third order generalized frequency response functions was also presented in this the second part symbolic expansion techniques are introduced and together with the results achieved in part 1 yield new methods for analysing the properties of generalized frequency response functions case studies are included to illustrate the application of the new methods

interpretation of non linear frequency response functions analytical and graphical methods of interpreting generalised frequency response functions for non linear systems are derived it is shown that non linear phenomena can be classified into intra kernel and inter kernel interference and that worst case responses can be computed the results are illustrated using several discrete and continuous time non linear systems

robust feedback synthesis for nonlinear integro differential equation models using generalized describing functions we propose a technique for synthesizing robust feedback systems for plants described by nonlinear integro differential equations nide with considerable coefficient uncertainties the technique is based on the concept of generalized frequency response functions gfrf and exploits recent results that enable ready derivation of these from the nide description of the plant a powerful synthesis procedure that combines the gfrf and quantitative feedback theory qft technique is given the proposed procedure has several advantages over the existing nonlinear qft methods a nonlinear chemical reactor example is solved using the procedure and is found to yield satisfactory results

a functional version of the birkhoff ergodic theorem for a normal integrand a variational approach in this paper we prove a new version of the birkhoff ergodic theorem bet for random variables depending on a parameter alias integrands this involves variational convergences namely epigraphical hypographical and uniform convergence and requires a suitable definition of the conditional expectation of integrands we also have to establish the measurability of the epigraphical lower and upper limits with respect to the field of invariant subsets from the main result applications to uniform versions of the bet to sequences of random sets and to the strong consistency of estimators are briefly derived

a splitting theorem for the space of smooth functions we will show using purely linear functional analytic methods that each exact complex0 f c s0 p0 c s1 p1 c s2 p2 wherepiare matrices of convolution operators in particular linear differential operators with constant coefficients splits fromp1on in the category of topological vector spaces moreover we characterize when these complexes split completely the obtained result covers all known particular cases as well as some more general cases independently of the analytic nature ofpi the result is a consequence of studying splitting of short exact sequences0 f jx qg 0 wheref x gare graded frechet spaces i e frechet spaces equipped with fixed reduced spectra representing them andj qare consistent with the graded structures we characterize splitting of in case i forgare spacesc where is an open subset in rn or ii gis the space of all sequences

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

regeneration in markov chain samplers abstract markov chain sampling has recently received considerable attention in particular in the context of bayesian computation and maximum likelihood estimation this article discusses the use of markov chain splitting originally developed for the theoretical analysis of general state space markov chains to introduce regeneration into markov chain samplers this allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance the approach is applied to several samplers including certain metropolis samplers that can be used on their own or in hybrid samplers and is illustrated in several examples

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples