algebraic reduction of the ising model we consider the ising model on a cylindrical lattice of l columns with fixed spin boundary conditions on the top and bottom rows the spontaneous magnetization can be written in terms of partition functions on this lattice we show how we can use the clifford algebra of kaufman to write these partition functions in terms of l by l determinants and then further reduce them to m by m determinants where m is approximately l 2 in this form the results can be compared with those of the ising case of the superintegrable chiral potts model they point to a way of calculating the spontaneous magnetization of that more general model algebraically

variational bayesian image restoration based on a product of t distributions image prior image priors based on products have been recognized to offer many advantages because they allow simultaneous enforcement of multiple constraints however they are inconvenient for bayesian inference because it is hard to find their normalization constant in closed form in this paper a new bayesian algorithm is proposed for the image restoration problem that bypasses this difficulty an image prior is defined by imposing student t densities on the outputs of local convolutional filters a variational methodology with a constrained expectation step is used to infer the restored image numerical experiments are shown that compare this methodology to previous ones and demonstrate its advantages

variational bayesian sparse kernel based blind image deconvolution with student s t priors in this paper we present a new bayesian model for the blind image deconvolution bid problem the main novelty of this model is the use of a sparse kernel based model for the point spread function psf that allows estimation of both psf shape and support in the herein proposed approach a robust model of the bid errors and an image prior that preserves edges of the reconstructed image are also used sparseness robustness and preservation of edges are achieved by using priors that are based on the student s t probability density function pdf this pdf in addition to having heavy tails is closely related to the gaussian and thus yields tractable inference algorithms the approximate variational inference methodology is used to solve the corresponding bayesian model numerical experiments are presented that compare this bid methodology to previous ones using both simulated and real data

neumann series and lattice sums we consider sums over the square lattice which depend only on radial distance and provide formulas which enable sums of functions with neumann series to be reexpressed as combinations of hypergeometric series we illustrate the procedure using trigonometric sums previously studied by borwein and borwein sums combining logarithms bessel functions j and powers of distance and sums of neumann functions we also exhibit sums which may be evaluated analytically and recurrence formulas linking sums

fuzzy membership descriptors for images the fuzzy membership function based local image descriptors are introduced as a competing alternative to widely accepted histogram based image descriptors the fuzzy membership descriptors are highly distinctive and thus facilitate an accurate image matching this study utilizes fuzzy membership descriptors to design a method meant for image matching the method finds the correspondence between the two images the study also introduces a gamma mixture fuzzy model to detect geometrically consistent correspondence between the two images the gamma mixture fuzzy model combines a finite number of gamma distributions through a fuzzy model the parameters of the gamma mixture fuzzy model are inferred by a method similar to the variational bayes the experimental studies support the claim of fuzzy membership descriptors being highly distinctive the method was also applied to 2 d ear images for an automated human identification the experimental results achieved the rank 1 recognition accuracy of text 97 5659 on a database of 125 subjects containing 493 ear images the motivation of this study is derived from the application potential of fuzzy membership functions in characterizing the local image features

superintegrable chiral potts model thermodynamic properties an inverse model and a simple associated hamiltonian the partition function of then state superintegrable chiral potts model is obtained exactly and explicitly if not completely rigorously for a finite lattice with particular boundary conditions this yields the bulk and surface free energies and horizontal and vertical correlation lengths and interfacial tensions the critical exponents are 1 2 n hor hor 2 n and vert vert 1 and the finite size corrections are obtained at criticality the eigenvalue spectrum of the column to column transfer matrix is that of a direct product ofn byn matrices inverting this matrix gives a related solvable model which is a generalization of the free fermion model the associated hamiltobian has a very simple form suggesting there may be a more direct algebraic method perhaps a generalized clifford algebra for obtaining its eigenvalues

fast algorithms for helmholtz green s functions the formal representation of the quasi periodic helmholtz green s function obtained by the method of images is only conditionally convergent and thus requires an appropriate summation convention for its evaluation instead of using this formal sum we derive a candidate green s function as a sum of two rapidly convergent series one to be applied in the spatial domain and the other in the fourier domain as in ewald s method we prove that this representation of green s function satisfies the helmholtz equation with the quasi periodic condition and furthermore leads to a fast algorithm for its application as an operator we approximate the spatial series by a short sum of separable functions given by gaussians in each variable for the series in the fourier domain we exploit the exponential decay of its terms to truncate it we use fast and accurate algorithms for convolving functions with this approximation of the quasi periodic green s function the resulting method yields a fast solver for the helmholtz equation with the quasi periodic boundary condition the algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when green s function is applied to discontinuous functions or potentials with singularities we also construct helmholtz green s functions with dirichlet neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi periodic green s function to apply them the complexity in dimension d 2 of these algorithms is graphic 1 d log c log 1 d where is the desired accuracy is proportional to the number of wavelengths contained in the computational domain and c is a constant we illustrate our approach with examples 1 embed inline graphic 1 gif

efficient computation of the 2d periodic green s function using the ewald method an efficient computation of the periodic helmholtz green s function for a 2d array of point sources using the ewald method is presented limitations on the numerical accuracy when using the optimum e parameter which gives optimum asymptotic convergence at high frequency are discussed a best e parameter is then derived to overcome these limitations which allows for the fastest convergence while maintaining a specific level of accuracy loss of significant figures in the final result the actual loss of significant figures has been verified through numerical simulations formulas for the number of terms needed for convergence have also been derived for both the spectral and the spatial series that appear in the ewald method and are found to be accurate in almost all cases

bayesian spectral analysis with student t noise we introduce a bayesian spectral analysis model for one dimensional signals where the observation noise is assumed to be student t distributed for robustness to outliers and we estimate the posterior distributions of the student t hyperparameters as well as the amplitudes and phases of the component sinusoids the integrals required for exact bayesian inference are intractable so we use variational approximation we show that the approximate phase posteriors are generalised von mises distributions of order 2 and that their spread increases as the signal to noise ratio decreases the model is demonstrated against synthetic data and real gps and wolf s sunspot data

robust statistical modeling using the t distribution abstract the t distribution provides a useful extension of the normal for statistical modeling of data sets involving errors with longer than normal tails an analytical strategy based on maximum likelihood for a general model with multivariate t errors is suggested and applied to a variety of problems including linear and nonlinear regression robust estimation of the mean and covariance matrix with missing data unbalanced multivariate repeated measures data multivariate modeling of pedigree data and multivariate nonlinear regression the degrees of freedom parameter of the t distribution provides a convenient dimension for achieving robust statistical inference with moderate increases in computational complexity for many models estimation of precision from asymptotic theory and the bootstrap is discussed and graphical methods for checking the appropriateness of the t distribution are presented

inexact bayesian point pattern matching for linear transformations we introduce a novel bayesian inexact point pattern matching model that assumes that a linear transformation relates the two sets of points the matching problem is inexact due to the lack of one to one correspondence between the point sets and the presence of noise the algorithm is itself inexact we use variational bayesian approximation to estimate the posterior distributions in the face of a problematic evidence term the method turns out to be similar in structure to the iterative closest point algorithm

a conjecture for the superintegrable chiral potts model we adapt our previous results for the partition function of the superintegrable chiral potts model with open boundaries to obtain the corresponding matrix elements of e h where h is the associated hamiltonian the spontaneous magnetization r can be expressed in terms of particular matrix elements of e hs1re h where s1 is a diagonal matrix we present a conjecture for these matrix elements as an m by m determinant where m is proportional to the width of the lattice the author has previously derived the spontaneous magnetization of the chiral potts model by analytic means but hopes that this work will facilitate a more algebraic derivation similar to that of yang for the ising model

spontaneous magnetization of the superintegrable chiral potts model calculation of the determinant d pq for the ising model the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant yang did this by calculating the eigenvalues in the large lattice limit montroll potts and ward expressed it as a toeplitz determinant and used szeg s theorem this is almost certainly the route originally travelled by onsager for the corresponding problem in the superintegrable chiral potts model neither approach appears to work here we show that the determinant dpq can be expressed as that of a product of two cauchy like matrices one can then use the elementary exact formula for the cauchy determinant one of course regains the known result originally conjectured in 1989

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

the order parameter of the chiral potts model an outstanding problem in statistical mechanics is the order parameter of the chiral potts model an elegant conjecture for this was made in 1983 it has since been successfully tested against series expansions but there is as yet no proof of the conjecture here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy then one can indeed verify the conjecture the method is based on the broken rapidity line approach pioneered by jimbo et al j phys a 26 2199 2210 1993

robust autoregression student t innovations using variational bayes autoregression ar is a tool commonly used to understand and predict time series data traditionally the excitation noise is modelled as a gaussian however real world data may not be gaussian in nature and it is known that gaussian models are adversely affected by the presence of outliers we introduce a bayesian ar model in which the excitation noise is assumed to be student t distributed variational bayesian approximations to the posterior distributions of the model parameters are used to overcome the intractable integrations inherent in the bayesian model independent automatic relevance determination ard priors over each of the ar coefficients are used to estimate the model order using synthetic data we show that the student t model performs well against both gaussian and leptokurtic data in terms of parameter estimation including the model order and is much more robust to outliers than either gaussian or finite mixtures of gaussian models we apply the model to strongly leptokurtic eeg signals and show that the student t model makes more accurate one step ahead predictions than the gaussian model and provides more consistent estimates of the ar coefficients over simultaneously recorded eeg channels

regularized lti system identification in the presence of outliers a variational em approach abstract regularized system identification of linear time invariant systems in the presence of outliers is investigated the finite impulse response fir model and the gaussian scale mixture are chosen to be the system model and the noise model respectively two special cases of the noise model are considered the well known student s t distribution and a proposed g confluent distribution both the fir model parameter and the latent variables in the noise model are treated as parameters of our statistical model and moreover the scale of the noise variance is treated as a hyper parameter besides the hyper parameters used to parameterize the priors of the impulse response and the latent variables then a variational expectation maximization algorithm is proposed for inference of the parameters and hyper parameters of the statistical model and the algorithm is guaranteed to converge to a stationary point monte carlo numerical simulations show that when the relative size of outliers is small the proposed approach performs comparably to a state of the art method and when the relative size of outliers and or the occurrence probability of outliers is large the proposed approach outperforms the state of the art method

blind deconvolution with model discrepancies blind deconvolution is a strongly ill posed problem comprising of simultaneous blur and image estimation recent advances in prior modeling and or inference methodology led to methods that started to perform reasonably well in real cases however as we show here they tend to fail if the convolution model is violated even in a small part of the image methods based on variational bayesian inference play a prominent role in this paper we use this inference in combination with the same prior for noise image and blur that belongs to the family of independent non identical gaussian distributions known as the automatic relevance determination prior we identify several important properties of this prior useful in blind deconvolution namely enforcing non negativity of the blur kernel favoring sharp images over blurred ones and most importantly handling non gaussian noise which as we demonstrate is common in real scenarios the presented method handles discrepancies in the convolution model and thus extends applicability of blind deconvolution to real scenarios such as photos blurred by camera motion and incorrect focus

bayesian estimation of an autoregressive model using markov chain monte carlo abstract we present a complete bayesian treatment of autoregressive model estimation incorporating choice of autoregressive order enforcement of stationarity treatment of outliers and allowance for missing values and multiplicative seasonality the paper makes three distinct contributions first we enforce the stationarity conditions using a very efficient metropolis within gibbs algorithm to generate the partial autocorrelations second we show how to carry out the gibbs sampler when the autoregressive order is unknown third we show how to combine the various aspects of fitting an autoregressive model giving a more comprehensive and efficient treatment than previous work we illustrate our methodology with a real example

lattice sums and the two dimensional periodic green s function for the helmholtz equation many algorithms that are currently used for the solution of the helmholtz equation in periodic domains require the evaluation of the green s function g x x0 the fact that the natural representat

short note the type 3 nonuniform fft and its applications the nonequispaced or nonuniform fast fourier transform nufft arises in a variety of application areas including imaging processing and the numerical solution of partial differential equations in its most general form it takes as input an irregular sampling of a function and seeks to compute its fourier transform at a nonuniform sampling of frequency locations this is sometimes referred to as the nufft of type 3 like the fast fourier transform the amount of work required is of the order o nlogn where n denotes the number of sampling points in both the physical and spectral domains in this short note we present the essential ideas underlying the algorithm in simple terms we also illustrate its utility with application to problems in magnetic resonance imagin and heat flow

robust bayesian model selection for autoregressive processes with additive outliers abstract autoregressive ar models of order k are often used for forecasting and control of time series as well as for the estimation of functionals such as the spectrum here we propose a method that consists of calculating the posterior probabilities of the competing ar k models in a way that is robust to outliers and then obtaining the predictive distributions of quantities of interest such as future observations and the spectrum as a weighted average of the predictive distributions conditional on each model this method is based on the idea of robust bayes factors calculated by replacing the likelihood for the nominal model by a robust likelihood it draws on and synthesizes several recent research advances namely robust filtering and the laplace method for integrals modified to take account of the finite range of the parameters the method performs well in simulation experiments and on real and artificial data software is available from statlib

quasi periodic green s functions of the helmholtz and laplace equations a classical problem of free space green s function g0 representations of the helmholtz equation is studied in various quasi periodic cases i e when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space exponentially convergent series for the free space quasi periodic g0 and for the expansion coefficients dl of g0 in the basis of regular cylindrical in two dimensions and spherical in three dimension 3d waves or lattice sums are reviewed and new results for the case of a one dimensional 1d periodicity in 3d are derived from a mathematical point of view a derivation of exponentially convergent representations for schlomilch series of cylindrical and spherical hankel functions of any integer order is accomplished exponentially convergent series for g0 and lattice sums dl hold for any value of the bloch momentum and allow g0 to be efficiently evaluated also in the periodicity plane the quasi periodic green s functions of the laplace equation are obtained from the corresponding representations of g0 of the helmholtz equation by taking the limit of the wave vector magnitude going to zero the derivation of relevant results in the case of a 1d periodicity in 3d highlights the common part which is universally applicable to any of remaining quasi periodic cases the results obtained can be useful for the numerical solution of boundary integral equations for potential flows in fluid mechanics remote sensing of periodic surfaces periodic gratings and infinite arrays of resonators coupled to a waveguide in many contexts of simulating systems of charged particles in molecular dynamics for the description of quasi periodic arrays of point interactions in quantum mechanics and in various ab initio first principle multiple scattering theories for the analysis of diffraction of classical and quantum waves

efficient calculation of lattice sums for free space periodic green s function an efficient method to calculate the lattice sums is presented for a one dimensional 1 d periodic array of line sources the method is based on the recurrence relations for hankel functions and the fourier integral representation of the zeroth order hankel function the lattice sums of arbitrary high order are then expressed by an integral of elementary functions which is easily computed using a simple scheme of numerical integration the calculated lattice sums are used to evaluate the free space periodic green s function the numerical results show that the proposed method provides a highly accurate evaluation of the green s function with far less computation time even when the observation point is located near the plane of the array

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

leptokurtic and platykurtic class of robust symmetrical and asymmetrical time series models abstract in this study we examined the well known autoregressive time series model in which innovations follow the flexible class of two piece distributions based on the scale mixtures of normal tp smn family the mentioned class of distributions is a rich class of distributions family that covers the robust symmetric asymmetric light heavy tailed distributions a key feature of this study is using a new suitable hierarchical representation of the family to obtain the maximum likelihood estimates of the model parameters via an expectation maximization em type algorithm performances and usefulness of the proposed model and estimates are shown via simulation studies and a real dataset

a variational bayesian approach to robust sensor fusion based on student t distribution in this paper a robust sensor fusion method is proposed where the measurement noise is modeled by a student t distribution the student t distribution has a heavy tail compared to the gaussian distribution and is robust to outliers we formulate sensor fusion as a state space estimation problem in the bayesian framework both batch and recursive variational bayesian vb algorithms are developed to perform this non gaussian state space estimation problem to obtain the fusion results computer simulations show that the proposed approach has an improved fusion performance and a lower computation cost compared to methods based on gaussian and finite gaussian mixture models

variational bayes for generalized autoregressive models we describe a variational bayes vb learning algorithm for generalized autoregressive gar models the noise is modeled as a mixture of gaussians rather than the usual single gaussian this allows different data points to be associated with different noise levels and effectively provides robust estimation of ar coefficients the vb framework is used to prevent overfitting and provides model order selection criteria both for ar order and noise model order we show that for the special case of gaussian noise and uninformative priors on the noise and weight precisions the vb framework reduces to the bayesian evidence framework the algorithm is applied to synthetic and real data with encouraging results