on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

semiparametric estimation of spectral density with irregular observations we propose a semiparametric method for estimating spectral densities of isotropic gaussian processes with scattered data the spectral density function fourier transform of the covariance function is modeled as a linear combination of b splines up to a cutoff frequency and from this point a truncated algebraic tail we calculate an analytic expression for the covariance function and tackle several numerical issues that arise when calculating the likelihood the parameters are estimated by maximizing the likelihood using the simulated annealing method our method directly estimates the tail behavior of the spectral density which has the greatest impact on interpolation properties the use of the likelihood in parameter estimation takes the correlations between observations fully into account we compare our method with a kernel method proposed by hall et al and a parametric method using the matern model simulation results show that our method outperforms the other two by several criteria applicatio

abelianisation of orthogonal groups and the fundamental group of modular varieties abstract we study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms we show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry in particular the moduli of k3 surfaces we give applications to modular forms and to computing the fundamental groups of some moduli spaces

potential density of rational points for k3 surfaces over function fields we give examples of non isotrivial k3 surfaces over complex function fields with zariski dense rational points and n eron severi rank one

adaptive estimation of stationary gaussian fields we study the nonparametric covariance estimation of a stationary gaussian field x observed on a regular lattice in the time series setting some procedures like aic are proved to achieve optimal model selection among autoregressive models however there exists no such equivalent results of adaptivity in a spatial setting by considering collections of gaussian markov random fields gmrf as approximation sets for the distribution of x we introduce a novel model selection procedure for spatial fields for all neighborhoods m in a given collection mathcal m this procedure first amounts to computing a covariance estimator of x within the gmrfs of neighborhood m then it selects a neighborhood m by applying a penalization strategy the so defined method satisfies a nonasymptotic oracle type inequality if x is a gmrf the procedure is also minimax adaptive to the sparsity of its neighborhood more generally the procedure is adaptive to the rate of approximation of the true distribution by gmrfs with growing neighborhoods

moduli spaces of polarized symplectic o grady varieties and borcherds products we study moduli spaces of o grady s ten dimensional irreducible symplectic manifolds these moduli spaces are covers of modular varieties of dimension 21 namely quotients of hermitian symmetric domains by a suitable arithmetic group the interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group this makes it different from both the k3 and the k3 n case which are of dimension 19 and 20 respectively

on the nonparametric estimation of covariance functions we describe kernel methods for estimating the covariance function of a stationary stochastic process and show how to ensure that the estimator has the positive semidefiniteness property from a practical viewpoint our method is significant because it does not demand a parametric model for covariance from a technical angle our results exhibit a striking departure from those in more familiar cases of kernel estimation for example in the context of covariance estimation kernel estimators can have the same convergence rates as maximum likelihood estimators and can have exceptionally fast convergence rates when employed to estimate variance

parametric estimation for gaussian fields indexed by graphs in this paper using spectral theory of hilbertian operators we study a special class of gaussian processes indexed by graphs we extend whittle maximum likelihood estimation of the parameters for the corresponding spectral density and show their asymptotic optimality

on the kodaira dimension of orthogonal modular varieties we prove that up to scaling there are only finitely many integral lattices l of signature 2 n with n ge 21 or n 17 such that the modular variety defined by the orthogonal group of l is not of general type in particular when n ge 108 every modular variety defined by an arithmetic group for a rational quadratic form of signature 2 n is of general type we also obtain similar finiteness in n ge 9 for the stable orthogonal groups as a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order which proves a conjecture of gritsenko and nikulin

a three dimensional ball quotient in this paper we determine a very particular example of a picard modular variety of general type on its non singular models there exist many holomorphic differential forms in a forthcoming paper we will show that one can construct calabi yau manifolds by considering quotients of this variety and resolving singularities

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

stochastic processes with finite correlation time modeling and application to the generalized langevin equation the kangaroo process kp is characterized by various forms of covariance and can serve as a useful model of random noises we discuss properties of that process for the exponential stretched exponential and algebraic power law covariances then we apply the kp as a model of noise in the generalized langevin equation and simulate solutions by a monte carlo method some results appear to be incompatible with requirements of the fluctuation dissipation theorem because probability distributions change when the process is inserted into the equation we demonstrate how one can construct a model of noise free of that difficulty this form of the kp is especially suitable for physical applications

minimal penalties for gaussian model selection this paper is mainly devoted to a precise analysis of what kind of penalties should be used in order to perform model selection via the minimiza tion of a penalized least squares type criterion within some general gaussian framework including the classical ones as compared to our previous paper on this topic birge and massart in j eur math soc 3 203 268 2001 more elaborate forms of the penalties are given which are shown to be in some sense optimal we indeed provide more precise upper bounds for the risk of the penalized estimators and lower bounds for the penalty terms showing that the use of smaller penalties may lead to disastrous results these lower bounds may also be used to design a practical strategy that allows to estimate the penalty from the data when the amount of noise is unknown we provide an illustra tion of the method for the problem of estimating a piecewise constant signal in gaussian noise when neither the number nor the location of the change points are known

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

functional characterization of generalized langevin equations we present an exact functional formalism to deal with linear langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional no other hypothesis is assumed over the noise neither do we use the fluctuation dissipation theorem we find that the characteristic functional of the linear process can be expressed in terms of noise functional and the green function of the deterministic memory like dissipative dynamics this yields a procedure for calculating the full kolmogorov hierarchy of the non markov process as examples we have characterized through the 1 time probability a noise induced interplay between the dissipative dynamics and the structure of different noises conditions that lead to non gaussian statistics and distributions with long tails are analysed the introduction of arbitrary fluctuations in fractional langevin equations has also been pointed out

on complete intersections with trivial canonical class abstract we prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in some version of fano varieties our results imply in particular that batyrev borisov toric construction produces only a bounded set of hodge numbers in any given dimension even as the codimension is allowed to grow

bayesian estimation for homogeneous and inhomogeneous gaussian random fields this paper investigates bayesian estimation for gaussian markov random fields in particular a new class of compound model is proposed which describes the observed intensities using an inhomogeneous model and the degree of spatial variation described by a second random field the coupled markov random fields are used as prior distributions and combined with gaussian noise models to produce posterior distributions on which estimation is based all model parameters are estimated in a fully bayesian setting using the metropolis hasting algorithm the full posterior estimation procedures are illustrated and compared using various artificial examples for these examples the inhomogeneous model performs very favorably when compared to the homogeneous model allowing differential degrees of smoothing and varying local textures

the kodaira dimension of the moduli of k3 surfaces the global torelli theorem for projective k3 surfaces was first proved by piatetskii shapiro and shafarevich 35 years ago opening the way to treating moduli problems for k3 surfaces the moduli space of polarised k3 surfaces of degree 2d is a quasi projective variety of dimension 19 for general d very little has been known hitherto about the kodaira dimension of these varieties in this paper we present an almost complete solution to this problem our main result says that this moduli space is of general type for d 61 and for d 46 50 54 57 58 60

moment inequalities for functions of independent random variables a general method for obtaining moment inequalities for functions of independent random variables is presented it is a generalization of the entropy method which has been used to derive concentration inequalities for such functions boucheron lugosi and massart ann probab 31 2003 1583 1614 and is based on a generalized tensorization inequality due to latala and oleszkiewicz lecture notes in math 1745 2000 147 168 the new inequalities prove to be a versatile tool in a wide range of applications we illustrate the power of the method by showing how it can be used to effortlessly re derive classical inequalities including rosenthal and kahane khinchine type inequalities for sums of independent random variables moment inequalities for suprema of empirical processes and moment inequalities for rademacher chaos and u statistics some of these corollaries are apparently new in particular we generalize talagrand s exponential inequality for rademacher chaos of order 2 to any order we also discuss applications for other complex functions of independent random variables such as suprema of boolean polynomials which include as special cases subgraph counting problems in random graphs

mapping class group and a global torelli theorem for hyperkahler manifolds a mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies we compute a mapping class group of a hyperkahler manifold m showing that it is commensurable to an arithmetic lattice in so 3 b2 3 a teichmuller space of m is a space of complex structures on m up to isotopies we define a birational teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds we show that the period map gives the isomorphism between connected components of the birational teichmuller space and the corresponding period space so b2 3 3 so 2 so b2 3 1 we use this result to obtain a torelli theorem identifying each connected component of the birational moduli space with a quotient of a period space by an arithmetic group when m is a hilbert scheme of n points on a k3 surface with n 1 a prime power our torelli theorem implies the usual hodge theoretic birational torelli theorem for other examples of hyperkahler manifolds the hodge theoretic torelli theorem is known to be false

on the moduli space of pairs consisting of a cubic threefold and a hyperplane abstract we study the moduli space of pairs x h consisting of a cubic threefold x and a hyperplane h in p 4 the interest in this moduli comes from two sources the study of certain weighted hypersurfaces whose middle cohomology admit hodge structures of k3 type and on the other hand the study of the singularity o 16 the cone over a cubic surface in this paper we give a hodge theoretic construction of the moduli space of cubic pairs by relating x h to certain lattice polarized cubic fourfolds y a period map for the pairs x h is then defined using the periods of the cubic fourfolds y the main result is that the period map induces an isomorphism between a git model for the pairs x h and the baily borel compactification of some locally symmetric domain of type iv

functional characterization of linear delay langevin equations we present an exact functional characterization of linear delay langevin equations driven by any noise structure defined through its characteristic functional this method relies on the possibility of finding an explicitly analytical expression for each realization of the delayed stochastic process in terms of those of the driving noise general properties of the transient dissipative dynamics are analyzed the corresponding interplay with a color gaussian noise is presented as a full application of our functional method we study a model for population growth with non gaussian fluctuations the gompertz model driven by multiplicative white shot noise

edge detection spatial smoothing and image reconstruction with partially observed multivariate data situations with incomplete multivariate spatial data on a lattice are considered the goal is to impute the missing data in the presence of edges or boundaries and recover the image two methods based on bayesian hierarchical models that iterate between edge detection and spatial smoothing to impute the missing data within identified homogeneous regions are examined their performance is compared with another method that imputes the missing values using edge preserving spatial smoothers with locally varying weights the performances of the three methods are compared on artificial and real datasets it is seen that information from the multivariate data is critical in recovering the images an application with color images where only one of three primary colors red green or blue is observed at each pixel is used to illustrate the results

approximate bayesian inference for latent gaussian models by using integrated nested laplace approximations summary structured additive regression models are perhaps the most commonly used class of models in statistical applications it includes among others generalized linear models generalized additive models smoothing spline models state space models semiparametric regression spatial and spatiotemporal models log gaussian cox processes and geostatistical and geoadditive models we consider approximate bayesian inference in a popular subset of structured additive regression models latent gaussian models where the latent field is gaussian controlled by a few hyperparameters and with non gaussian response variables the posterior marginals are not available in closed form owing to the non gaussian response variables for such models markov chain monte carlo methods can be implemented but they are not without problems in terms of both convergence and computational time in some practical applications the extent of these problems is such that markov chain monte carlo sampling is simply not an appropriate tool for routine analysis we show that by using an integrated nested laplace approximation and its simplified version we can directly compute very accurate approximations to the posterior marginals the main benefit of these approximations is computational where markov chain monte carlo algorithms need hours or days to run our approximations provide more precise estimates in seconds or minutes another advantage with our approach is its generality which makes it possible to perform bayesian analysis in an automatic streamlined way and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged

valid parameter space of 2 d gaussian markov random fields the valid parameter spaces of infinite and finite lattice 2 d noncausal gaussian markov random fields gmrfs are investigated for the infinite lattice fields the valid parameter space is shown to admit an explicit description a procedure that yields the valid parameter space is presented this procedure is applied to the second order neighborhood 2 d gmrfs to obtain an explicit description of their valid parameter spaces for the finite lattice fields it is shown that the valid parameter space does not admit a simple description the conditions that ensure the positivity of the power spectrum are necessary sufficient and irreducible the set of conditions for the infinite lattice fields however serves as a good set of sufficient conditions for the finite lattice case the results readily extend to the class of d d real and complex gmrfs

wavelet based statistical signal processing using hidden markov models wavelet based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly gaussian these models are unrealistic for many real world signals we develop a new framework for statistical signal processing based on wavelet domain hidden markov models hmms that concisely models the statistical dependencies and non gaussian statistics encountered in real world signals wavelet domain hmms are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models efficient expectation maximization algorithms are developed for fitting the hmms to observational signal data the new framework is suitable for a wide range of applications including signal estimation detection classification prediction and even synthesis to demonstrate the utility of wavelet domain hmms we develop novel algorithms for signal denoising classification and detection

tautological classes on moduli spaces of hyper kahler manifolds we study algebraic cycles on moduli spaces f h of h polarized hyperkahler manifolds following previous work of marian oprea and pandharipande on the tautological conjecture on moduli spaces of k3 surfaces we first define the tautological ring on f h we then study the images of these tautological classes in the cohomology groups of f h and prove that most of them are linear combinations of noether lefschetz cycle classes in particular we prove the cohomological version of the tautological conjecture on moduli space of k3 n type hyperkahler man ifolds with n 2 secondly we prove the cohomological generalized franchetta conjecture on universal family of these hyperkahler mani folds

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

the unirationality of the moduli space of k 3 surfaces of genus 22 using the connection discovered by hassett between the noether lefschetz moduli space mathcal c 42 of special cubic fourfolds of discriminant 42 and the moduli space mathcal f 22 of polarized k3 surfaces of genus 22 we show that the universal k3 surface over mathcal f 22 is unirational