semilinear copulas a family of copulas called semilinear is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square this family contains some other known families of copulas e g cuadras auge frechet and has a nice statistical interpretation several construction methods are provided especially concerning aggregation of semilinear copulas and a special form of ordinal sum construction is introduced some results about related families of quasi copulas and semicopulas are hence given

fast diffusion to self similarity complete spectrum long time asymptotics and numerology the complete spectrum is determined for the operator open image in new window on the sobolev space w1 2 rn formed by closing the smooth functions of compact support with respect to the norm open image in new window here the barenblatt profile is the stationary attractor of the rescaled diffusion equation open image in new window in the fast supercritical regime mopen image in new window the same diffusion dynamics represent the steepest descent down an entropy e u on probability measures with respect to the wasserstein distance d2 formally the operator h hess e is the hessian of this entropy at its minimum so the spectral gap h 2 n 1 m found below suggests the sharp rate of asymptotic convergence open image in new window from any centered initial data 0 u 0 x l1 rn with second moments this bound improves various results in the literature and suggests the conjecture that the self similar solution u t x r t n x r t is always slowest to converge the higher eigenfunctions which are polynomials with hypergeometric radial parts and the presence of continuous spectrum yield additional insight into the relations between symmetries of rn and the flow thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass preserving dilation or still better affine image the strange numerology of the spectrum is explained in terms of the number of moments of

open problems from the 12th international conference on fuzzy set theory and its applications eighteen open problems posed during fsta 2014 liptovsk jan slovakia are presented these problems concern fuzzy logics fuzzy partitions copulas triangular norms and related aggregation functions some open problems concerning effect and mv algebras are also included

on linear and quadratic constructions of aggregation functions abstract in the paper we introduce and discuss linear and quadratic constructions of aggregation functions based on an a priori given aggregation function we focus our attention on linear and quadratic constructions transforming any aggregation function from the considered class of aggregation functions into a new aggregation function belonging to the same class we study quadratic constructions of distinguished classes of aggregation functions with neutral element 0 1 in particular semi copulas quasi copulas copulas triangular norms and their duals in all cases the final results fully characterize quadratic functions which are universal for quadratic constructions of aggregation functions of the considered type

new constructions of diagonal patchwork copulas we present a method for constructing symmetric copulas which generalizes the diagonal patchwork construction of copulas procedure we also show how it is related to a new construction of a generalized farlie gumbel morgenstern distribution and to the copula transforms

conic aggregation functions inspired by the notion of conic t norms we introduce conic aggregation functions such aggregation functions are completely characterized by their zero set special classes of binary conic aggregation functions such as conic quasi copulas and conic copulas are considered we provide the necessary and sufficient conditions on the boundary curve of the zero set of a conic aggregation function to obtain a conic quasi copula and conclude that the class of conic copulas is a proper subclass of the class of conic quasi copulas moreover we characterize the class of singular conic copulas we investigate some aggregations of conic quasi copulas some examples are also provided

a class of bivariate copula mappings abstract we provide necessary and sufficient conditions on a function f 0 1 r so that h f c u v c u v f 1 u v c u v u v 0 1 2 is a copula for any bivariate copula c then we discuss several important bivariate copula properties preserved or not by the mapping c h f c the considered bivariate copula construction unifies many examples found in the literature and opens a qualitatively new way of obtaining new copulas by simply using a function f which has to satisfy a few easily verifiable properties

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

on a new construction of 1 lipschitz aggregation functions quasi copulas and copulas a new method of constructing 1 lipschitz aggregation functions is proposed and studied as a by product a new method of constructing quasi copulas from a given quasi copula c is obtained newly constructed copulas are negative quadrant dependent for any original copula c the only invariant copula under this construction is the frechet hoffding lower bound w the method preserves several important properties of copulas such as modularity absolute continuity or symmetry and commutes with some other constructions e g the construction of survival copulas

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

nonideal sampling and regularization theory shannon s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some shift invariant space which may or may not be bandlimited in this paper we present some further justification for this type of representation while addressing the issue of the specification of the best reconstruction space we consider a realistic setting where a multidimensional signal is prefiltered prior to sampling and the samples are corrupted by additive noise we adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a tikhonov like continuous domain l2 regularization to obtain the continuous space solution we present theoretical justification for the minimization of this cost functional and show that the globally minimal continuous space solution belongs to a shift invariant space generated by a function generalized b spline that is generally not bandlimited when the sampling is ideal we recover some of the classical smoothing spline estimators the optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a b spline like basis to make the scheme practical we specify the generating functions corresponding to the most popular families of regularization operators derivatives iterated laplacian as well as a new generalized one that leads to a new brand of matern splines we conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error mmse wiener solutions of the generalized sampling problem

a hitchhiker s guide to quasi copulas abstract this paper presents a review of the concept of an n quasi copula starting from the characterization of functions that can be derived from operations on random variables we recall several other characterizations and properties that have been proven in the literature highlighting the differences between the bivariate case and the higher dimensional case additionally we recall several applications of n quasi copulas such as their role in the study of bounds on sets of n copulas we also recall various classes of n quasi copulas that have been studied in the literature finally we present some questions that have not yet been answered in the literature

the nonlinear statistics of high contrast patches in natural images recently there has been a great deal of interest in modeling the non gaussian structures of natural images however despite the many advances in the direction of sparse coding and multi resolution analysis the full probability distribution of pixels values in a neighborhood has not yet been described in this study we explore the space of data points representing the values of 3 3 high contrast patches from optical and 3d range images we find that the distribution of data is extremely sparse with the majority of the data points concentrated in clusters and non linear low dimensional manifolds furthermore a detailed study of probability densities allows us to systematically distinguish between images of different modalities optical versus range which otherwise display similar marginal distributions our work indicates the importance of studying the full probability distribution of natural images not just marginals and the need to understand the intrinsic dimensionality and nature of the data we believe that object like structures in the world and the sensor properties of the probing device generate observations that are concentrated along predictable shapes in state space our study of natural image statistics accounts for local geometries such as edges in natural scenes but does not impose such strong assumptions on the data as independent components or sparse coding by linear change of bases

probabilistic analysis of regularization in order to use interpolated data wisely it is important to have reliability and confidence measures associated with it a method for computing the reliability at each point of any linear functional of a surface reconstructed using regularization is presented the proposed method is to define a probability structure on the class of possible objects and compute the variance of the corresponding random variable this variance is a natural measure for uncertainty and experiments have shown it to correlate well with reality the probability distribution used is based on the boltzmann distribution the theoretical part of the work utilizes tools from classical analysis functional analysis and measure theory on function spaces the theory was tested and applied to real depth images it was also applied to formalize a paradigm of optimal sampling which was successfully tested on real depth images

a probabilistic approach to pattern matching in the continuous domain the goal of this paper is to solve the following basic problem given discrete noisy samples from a continuous signal compute the probability distribution of its distance from a fixed template as opposed to the typical restoration problem which considers a single optimal signal the computation of the entire probability distribution necessitates integrating over the entire signal space to achieve this we apply path integration techniques the problem is studied in one and two dimensions and an accurate solution as well as an efficient approximation scheme are provided

the impact on the properties of the efgm copulas when extending this family abstract several extensions of the family of bivariate eyraud farlie gumbel morgenstern copulas efgm copulas are considered some of them are well known from the literature others have recently been suggested copulas based on quadratic constructions based on some forms of convexity and polynomial copulas for each of these extensions we analyze which properties of efgm copulas are preserved or even improved and which are partly lost such properties can be structural order theoretical or topological in nature or algebraic symmetry or being a polynomial or analytic absolute continuity other examples are forms of convexity quadrant dependence and symmetry with respect to copula transformations the last group of properties considered here is related to some dependence parameters

quadratic constructions of copulas in the paper we introduce and study quadratic constructions of copulas based on composition of a copula and some quadratic polynomial we characterize all quadratic polynomials whose composition with an arbitrary copula always results in a copula due to this result we can assign a two parametric class k c c d of copulas with parameters c d in a certain subset r 2 to each copula c moreover we also determine all copulas invariant with respect to the quadratic constructions presented this investigation brings two interesting parametric classes of copulas we show that the union of these two classes is equal to the so called plackett family of copulas we add some properties of these copulas and their statistical consequences

extremal lipschitz continuous aggregation functions with a given diagonal section abstract in this paper we study the smallest and the greatest m lipschitz continuous n ary aggregation functions with a given diagonal section we show that several properties that were studied for the smallest and the greatest 1 lipschitz continuous binary aggregation functions with a given diagonal section extend naturally to higher dimensions while considering different lipschitz constants just as in the binary case we show that the smallest n quasi copula with a given diagonal section coincides with the smallest 1 lipschitz n ary aggregation function with that diagonal section additionally we show that the smallest n quasi copula with a given diagonal section called the bertino n quasi copula is supermodular for any n 2

orthogonal grid constructions of copulas we establish a general framework for constructing copulas that can be regarded as a patchwork like assembly of arbitrary copulas with nonoverlapping rectangles as patches we derive a family of construction methods that require the choice of a single background copula when this background copula is the greatest copula we retrieve the well known ordinal sum construction while in the case of the smallest copula we obtain a construction method that is dual to the ordinal sum construction also nonsingular background copulas lead to suitable construction methods

a family of transformed copulas with a singular component abstract in this paper we present a family of bivariate copulas obtained by transforming a given copula function by means of two increasing functions named as transformed copula one distinctive characteristic of the transformed copula is its singular component along the main diagonal conditions guaranteeing the transformed function to be a copula function are provided and several classes of the transformed copulas are given the singular component along the main diagonal of the transformed copula is verified and the tail dependence coefficients of transformed copulas are obtained some properties of the transformed copula are discussed such as the total positivity of order 2 and the concordance order finally conditions for the proposed transformation being invariant are given sufficient conditions guaranteeing that the iterative transformation is a copula function are provided and the convergence of the iterative transformation is proved under some conditions