replica treatment of non hermitian disordered hamiltonians we employ the fermionic and bosonic replicated nonlinear models to treat ginibre unitary symplectic and orthogonal ensembles of non hermitian random matrix hamiltonians using the saddle point approach combined with the borel resummation procedure we derive the exact large n results for microscopic density of states in all three ensembles we also obtain tails of the density of states as well as the two point function for the unitary ensemble

fourier inversion theorem in mesoscopic transport abstract the fourier inversion theorem on the graded hyperbolic plane a supermanifold closely analogous to the quotient spaces of efetov s nonlinear models is described an application to the calculation of the average conductance of quasi one dimensional disordered samples is given

discriminant analysis by gaussian mixtures fisher rao linear discriminant analysis lda is a valuable tool for multigroup classification lda is equivalent to maximum likelihood classification assuming gaussian distributions for each class in this paper we fit gaussian mixtures to each class to facilitate effective classification in non normal settings especially when the classes are clustered low dimensional views are an important by product of lda our new techniques inherit this feature we can control the within class spread of the subclass centres relative to the between class spread our technique for fitting these models permits a natural blend with nonparametric versions of lda

character expansion method for supergroups and extended superversions of the leutwyler smilga and berezin karpelevich integrals we introduce an extension of the character expansion method to the case of supergroups this method allows us to calculate a superversion of the leutwyler smilga integral which to the best of our knowledge has not been calculated before we also use the method to generalize a previously calculated superversion of the berezin karpelevich integral our character expansion method should also allow for the calculation of other supergroup integrals

negative moments of characteristic polynomials of random goe matrices and singularity dominated strong fluctuations we calculate the negative integer moments of the regularized characteristic polynomials of n n random matrices taken from the gaussian orthogonal ensemble goe in the limit as n the results agree nontrivially with a recent conjecture of berry and keating motivated by techniques developed in the theory of singularity dominated strong fluctuations this is the first example where nontrivial predictions obtained using these techniques have been proved

universality of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

on inconsistent bayes estimates of location on montre que dans certaines situations relativement naturelles les estimateurs de position bayesiens sont inconsistants

on the consistency of bayes estimates on etudie les proprietes de frequence des regles de bayes avec une attention particuliere a la consistence

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

toward a theory of the integer quantum hall transition continuum limit of the chalker coddington model an n channel generalization of the network model of chalker and coddington is considered the model for n 1 is known to describe the critical behavior at the plateau transition in systems exhibiting the integer quantum hall effect using a recently discovered equality of integrals the network model is transformed into a lattice field theory defined over efetov s model space with unitary symmetry the transformation is exact for all n no saddle point approximation is made and no massive modes have to be eliminated the naive continuum limit of the lattice theory is shown to be a supersymmetric version of pruisken s nonlinear model with couplings xx n 4 and xy n 2 at the symmetric point it follows that the model for n 2 which describes a spin degenerate landau level and the random flux problem is noncritical on the basis of symmetry considerations and inspection of the hamiltonian limit a modified network model is formulated which still lies in the quantum hall universality class the prospec

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

recursive construction for a class of radial functions ii superspace we extend the recursion formula for matrix bessel functions which we obtained previously to superspace it is sufficient to do this for the unitary orthosymplectic supergroup by direct computations we show that fairly explicit results can be obtained at least up to dimension 8 8 for the supermatrices since we introduce a new technique we discuss various of its aspects in some detail

on the density of states of sparse random matrices the supersymmetric method of calculation of the density of states of a sparse random matrix is shown to be absolutely equivalent to the replica trick a functional generalization of the hubbard stratonovich transformation was used in the course of the solution

derivation of determinantal structures for random matrix ensembles in a new way there are several methods to treat ensembles of random matrices in symmetric spaces circular matrices chiral matrices and others orthogonal polynomials and the supersymmetry method are particular powerful techniques here we present a new approach to calculate averages over ratios of characteristic polynomials at first sight paradoxically one can call our approach supersymmetry without supersymmetry because we use structures from supersymmetry without actually mapping onto superspaces we address two kinds of integrals which cover a wide range of applications for random matrix ensembles for probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way as a new application we derive an expression for the k point correlation function of an arbitrary rotation invariant probability density over the hermitian matrices in the presence of an external field

a comparison of the superbosonization formula and the generalized hubbard stratonovich transformation recently two different approaches were put forward to extend the supersymmetry method in random matrix theory from gaussian ensembles to general rotation invariant ensembles these approaches are the generalized hubbard stratonovich transformation and the superbosonization formula here we prove the equivalence of both approaches to this end we reduce integrals over functions of supersymmetric wishart matrices to integrals over quadratic supermatrices of certain symmetries

variable selection for model based clustering we consider the problem of variable or feature selection for model based clustering the problem of comparing two nested subsets of variables is recast as a model comparison problem and addressed using approximate bayes factors a greedy search algorithm is proposed for finding a local optimum in model space the resulting method selects variables or features the number of clusters and the clustering model simultaneously we applied the method to several simulated and real examples and found that removing irrelevant variables often improved performance compared with methods based on all of the variables our variable selection method consistently yielded more accurate estimates of the number of groups and lower classification error rates as well as more parsimonious clustering models and easier visualization of results

variable selection for clustering with gaussian mixture models this article is concerned with variable selection for cluster analysis the problem is regarded as a model selection problem in the model based cluster analysis context a model generalizing the model of raftery and dean 2006 journal of the american statistical association 101 168 178 is proposed to specify the role of each variable this model does not need any prior assumptions about the linear link between the selected and discarded variables models are compared with bayesian information criterion variable role is obtained through an algorithm embedding two backward stepwise algorithms for variable selection for clustering and linear regression the model identifiability is established and the consistency of the resulting criterion is proved under regularity conditions numerical experiments on simulated datasets and a genomic application highlight the interest of the procedure

asymptotic corrections to the wigner semicircular eigenvalue spectrum of a large real symmetric random matrix using the replica method the replica method has previously been used to calculate the semicircular averaged eigenvalue spectrum of the gaussian orthogonal ensemble of real symmetric n n random matrices in the limit where n to infinity the authors develop a perturbative scheme which within this same replica framework is used to calculate the corrections within this semicircular band of eigenvalues to order 1 n and 1 n2 a new and straightforward self consistency argument is presented and used to derive the shape of the averaged eigenvalue spectrum when n is large but finite and the scaling behaviour of this averaged eigenvalue spectrum near the band edges is demonstrated in a straightforward fashion some comments are made on the relation of the results to those of field theoretical calculations in zero dimensions

assessing a mixture model for clustering with the integrated completed likelihood we propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood for this purpose the observed data are assigned to unknown clusters using a maximum a posteriori operator then the integrated completed likelihood icl is approximated using the bayesian information criterion bic numerical experiments on simulated and real data of the resulting icl criterion show that it performs well both for choosing a mixture model and a relevant number of clusters in particular icl appears to be more robust than bic to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data

superbosonization formula and its application to random matrix theory starting from gaussian random matrix models we derive a new supermatrix field theory model in contrast to the conventional non linear sigma models the new model is applicable for any range of correlations of the elements of the random matrices we clarify the domain of integration for the supermatrices and give a demonstration of how the model works by calculating the density of states for an ensemble of almost diagonal matrices it is also shown how one can reduce the supermatrix model to the conventional sigma model

mixtures of shifted asymmetriclaplace distributions a mixture of shifted asymmetric laplace distributions is introduced and used for clustering and classification a variant of the em algorithm is developed for parameter estimation by exploiting the relationship with the generalized inverse gaussian distribution this approach is mathematically elegant and relatively computationally straightforward our novel mixture modelling approach is demonstrated on both simulated and real data to illustrate clustering and classification applications in these analyses our mixture of shifted asymmetric laplace distributions performs favourably when compared to the popular gaussian approach this work which marks an important step in the non gaussian model based clustering and classification direction concludes with discussion as well as suggestions for future work

a representation of exchangeable hierarchies by sampling from random real trees a hierarchy on a set s also called a total partition of s is a collection mathcal h of subsets of s such that s in mathcal h each singleton subset of s belongs to mathcal h and if a b in mathcal h then a cap b equals either a or b or varnothing every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h associated as follows with a random real tree mathcal t equipped with root element 0 and a random probability distribution p on the borel subsets of mathcal t given mathcal t p let t 1 t 2 ldots be independent and identically distributed according to p and let mathcal h comprise all singleton subsets of mathbb n and every subset of the form j t j in f x as x ranges over mathcal t where f x is the fringe subtree of mathcal t rooted at x there is also the alternative characterization every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h derived as follows from a random hierarchy mathscr h on 0 1 and a family u j of i i d uniform 0 1 random variables independent of mathscr h let mathcal h comprise all sets of the form j u j in b as b ranges over the members of mathscr h

optimum estimation via gradients of partition functions and information measures a statistical mechanical perspective in continuation to a recent work on the statistical mechanical analysis of minimum mean square error mmse estimation in gaussian noise via its relation to the mutual information the i mmse relation here we propose a simple and more direct relationship between optimum estimation and certain information measures e g the information density and the fisher information which can be viewed as partition functions and hence are amenable to analysis using statistical mechanical techniques the proposed approach has several advantages most notably its applicability to general sources and channels as opposed to the i mmse relation and its variants which hold only for certain classes of channels e g additive white gaussian noise channels we then demonstrate the derivation of the conditional mean estimator and the mmse in a few examples two of these examples turn out to be generalizable to a fairly wide class of sources and channels for this class the proposed approach is shown to yield an approximate conditional mean estimator and an mmse formula that has the flavor of a single letter expression we also show how our approach can easily be generalized to situations of mismatched estimation

critique of the replica trick it is shown that the replica trick fails to give the correct non perturbative result for the two point function s2 of the gaussian unitary ensemble of n n random matrices the failure arises from an incorrect description of the symmetries of the random matrix system in the limit n to infinity the correct description which involves integration over both non compact and compact degrees of freedom is obtained by using the method of superfields some implications for the localisation transition in disordered electronic systems and the theory of the quantised hall effect are suggested

random matrix theory by the supersymmetry method beyond the sigma model abstract the leading correction to the smoothed connected energy density density correlation function is obtained for the large energy difference within the context of the gaussian random matrix theory in order to achieve this result the supersymmetry method is extended beyond the sigma model to include small quadratic fluctuations around the saddle point special care is taken to avoid the potential divergence arising from the unbounded nature of the saddle point also in the small energy difference regime the leading correction to a two point correlation function is obtained

replica method for wide correlators in gaussian orthogonal unitary and symplectic random matrix ensembles we calculate connected correlators in gaussian orthogonal unitary and symplectic random matrix ensembles by the replica method in the 1 n expansion we obtain averaged one point green s functions up to the next to leading order wide two level correlators up to the first non trivial order and wide three level correlators up to the first non trivial order by carefully treating fluctuations in saddle point evaluation

eigenvalue correlations in non hermitean symplectic random matrices a correlation function of complex eigenvalues of n n random matrices drawn from non hermitean random matrix ensembles of symplectic symmetry is given in terms of a quaternion determinant spectral properties of gaussian ensembles are studied in detail in the regimes of weak and strong non hermiticity

integrability of zero dimensional replica field theories at β 1 building on insights from the theory of integrable lattices the integrability is claimed for nonlinear replica sigma models derived in the context of real symmetric random matrices specifically the fermionic and the bosonic replica partition functions are proven to form a single supersymmetric pfaff kp hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density density eigenvalue correlation function in the infinite dimensional gaussian orthogonal ensemble implications of the formalism outlined are briefly discussed

a robust method for cluster analysis let there be given a contaminated list of n r d valued observations coming from g different normally distributed populations with a common covariance matrix we compute the ml estimator with respect to a certain statistical model with n r outliers for the parameters of the g populations it detects outliers and simultaneously partitions their complement into g clusters it turns out that the estimator unites both the minimum covariance determinant rejection method and the well known pooled determinant criterion of cluster analysis we also propose an efficient algorithm for approximating this estimator and study its breakdown points for mean values and pooled ssp matrix

gaussian parsimonious clustering models gaussian clustering models are useful both for understanding and suggesting powerful criteria banfield and raftery 1993 have considered a parametrization of the variance matrix ek of a cluster pk in terms of its eigenvalue decomposition ek lkdkakd k lk where lk defines the volume of pk dk is an orthogonal matrix which defines its orientation and ak is a diagonal matrix with determinant 1 which defines its shape this parametrization allows us to propose many general clustering criteria from the simplest one spherical cluster with equal volumes which leads to the classical k means criterion to the most complex one unknown and different volumes orientations and shapes for all clusters methods of optimization to derive the maximum likelihood estimates as well as the practical usefulness of these models are discussed we especially analyze the influence of the volumes of clusters we report monte carlo simulations and an application on stellar data which dramatically illustrated the relevance of allowing clusters to have different volumes

wigner dyson statistics from the replica method we compute the correlation functions of the eigenvalues in the gaussian unitary ensemble using the fermionic replica method we show that non trivial saddle points which break replica symmetry must be included in the calculation in order to correctly reproduce the exact asymptotic behaviour of the correlation functions at a large distance

universal correlations of trapped one dimensional impenetrable bosons we calculate the asymptotic behaviour of the one body density matrix of one dimensional impenetrable bosons in finite size geometries our approach is based on a modification of the replica method from the theory of disordered systems we obtain explicit expressions for oscillating terms similar to fermionic friedel oscillations these terms are universal and originate from the strong short range correlations between bosons in one dimension

model based learning using a mixture of mixtures of gaussian and uniform distributions we introduce a mixture model whereby each mixture component is itself a mixture of a multivariate gaussian distribution and a multivariate uniform distribution although this model could be used for model based clustering model based unsupervised learning or model based classification model based semi supervised learning we focus on the more general model based classification framework in this setting we fit our mixture models to data where some of the observations have known group memberships and the goal is to predict the memberships of observations with unknown labels we also present a density estimation example a generalized expectation maximization algorithm is used to estimate the parameters and thereby give classifications in this mixture of mixtures model to simplify the model and the associated parameter estimation we suggest holding some parameters fixed this leads to the introduction of more parsimonious models a simulation study is performed to illustrate how the model allows for bursts of probability and locally higher tails two further simulation studies illustrate how the model performs on data simulated from multivariate gaussian distributions and on data from multivariate t distributions this novel approach is also applied to real data and the performance of our approach under the various restrictions is discussed

a new approach to derive pfaffian structures for random matrix ensembles correlation functions for matrix ensembles with orthogonal and unitary symplectic rotation symmetry are more complicated to calculate than in the unitary case the supersymmetry method and the orthogonal polynomials are two techniques to tackle this task recently we presented a new method to average ratios of characteristic polynomials over matrix ensembles invariant under the unitary group here we extend this approach to ensembles with orthogonal and unitary symplectic rotation symmetry we show that pfaffian structures can be derived for a wide class of orthogonal and unitary symplectic rotation invariant ensembles in a unifying way this also includes those for which this structure was not known previously as the real ginibre ensemble and the gaussian real chiral ensemble with two independent matrices as well