uniform q series asymptotics for staircase polygons we present a uniform asymptotic expansion for the area perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q series 1 phi 1 0 y q x as q to 1 from a new integral representation this leads to a direct calculation of the scaling function for this model

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

shearless transport barriers in unsteady two dimensional flows and maps abstract we develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two dimensional flows and maps defined over a finite time interval this principle reveals that hyperbolic lagrangian coherent structures lcss and parabolic lcss or jet cores are the two main types of shearless barriers in unsteady flows based on the boundary conditions they satisfy parabolic barriers are found to be more observable and robust than hyperbolic barriers confirming widespread numerical observations both types of barriers are special null geodesics of an appropriate lorentzian metric derived from the cauchy green strain tensor using this fact we devise an algorithm for the automated computation of parabolic barriers we illustrate our detection method on steady and unsteady non twist maps and on the aperiodically forced bickley jet

sharp adaptation for inverse problems with random noise we consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems in particular multivariate ones we propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale that includes ellipoids with polynomially and exponentially decreasing axes and at the same time satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non increasing weights the construction of the estimator is based on a properly penalized blockwise stein s rule with weakly geometically increasing blocks as an application we construct sharp adaptive estimators in the problems of deconvolution and tomography

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

functional deconvolution in a periodic setting uniform case we extend deconvolution in a periodic setting to deal with functional data the resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation in the case when it is observed at a finite number of distinct points the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model we derive minimax lower bounds for the l 2 risk in the proposed functional deconvolution model when f is assumed to belong to a besov ball and the blurring function is assumed to possess some smoothness properties including both regular smooth and super smooth convolutions furthermore we propose an adaptive wavelet estimator of f that is asymptotically optimal in the minimax sense or near optimal within a logarithmic factor in a wide range of besov balls in addition we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points as an illustration we discuss particular examples for both continuous and discrete settings

statistical inverse estimation in hilbert scales the recovery of signals from indirect measurements blurred by random noise is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable for greater flexibility the general problem is embedded in an abstract hilbert scale in the applications sobolev scales are used for the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product where the operator is bounded but not necessarily compact a lower bound to certain minimax rates is included and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate examples include errors in variables deconvolution and indirect nonparametric regression special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

efficient computation of null geodesics with applications to coherent vortex detection recent results suggest that boundaries of coherent fluid vortices elliptic coherent structures can be identified as closed null geodesics of appropriate lorentzian metrics defined on the flow domain here we derive an automated method for computing such null geodesics based on the geometry of the underlying geodesic flow our approach simplifies and improves existing procedures for computing variationally defined eulerian and lagrangian vortex boundaries as an illustration we compute objective vortex boundaries from satellite inferred ocean velocity data a matlab implementation of our method is available at https github com mattiaserra closed null geodesics 2d

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

oracle inequalities for inverse problems we consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations let a finite set a of linear estimators be given our aim is to mimic the estimator in a that has the smallest risk on the true f under general conditions we show that this can be achieved by simple minimization of an unbiased risk estimator provided the singular values of the operator of the inverse problem decrease as a power law the main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact this inequality can also be used to obtain sharp minimax adaptive results in particular we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures

the principle of penalized empirical risk in severely ill posed problems we study a standard method of regularization by projections of the linear inverse problem y ax e where e is a white gaussian noise and a is a known compact operator it is assumed that the eigenvalues of aa converge to zero with exponential decay such behavior of the spectrum is typical for inverse problems related to elliptic differential operators as model example we consider recovering of unknown boundary conditions in the dirichlet problem for the laplace equation on the unit disk by using the singular value decomposition of a we construct a projection estimator of x the bandwidth of this estimator is chosen by a data driven procedure based on the principle of minimization of penalized empirical risk we provide non asymptotic upper bounds for the mean square risk of this method and we show in particular that this approach gives asymptotically minimax estimators in our model example

an infinite family of adsorption models and restricted lukasiewicz paths we define k l restricted lukasiewicz paths k l 0 and use these paths as models of polymer adsorption we write down a polynomial expression satisfied by the generating function for arbitrary values of k l the resulting polynomial is of degree l 1 and hence cannot be solved explicitly for sufficiently large l we provide two different approaches to obtain the phase diagram in addition to a more conventional analysis we also develop a new mathematical characterisation of the phase diagram in terms of the discriminant of the polynomial and a zero of its highest degree coefficient

geodesic theory of transport barriers in two dimensional flows abstract we introduce a new approach to locating key material transport barriers in two dimensional non autonomous dynamical systems such as unsteady planar fluid flows seeking transport barriers as minimally stretching material lines we obtain that such barriers must be shadowed by minimal geodesics under the riemannian metric induced by the cauchy green strain tensor as a result snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations using this approach we locate hyperbolic barriers generalized stable and unstable manifolds elliptic barriers generalized kam curves and parabolic barriers generalized shear jets in temporally aperiodic flows defined over a finite time interval our approach also yields a metric geodesic deviation that determines the minimal computational time scale needed for a robust numerical identification of generalized lagrangian coherent structures lcss as we show an extension of our transport barrier theory to non euclidean flow domains such as a sphere follows directly we illustrate our main results by computing key transport barriers in a chaotic advection map and in a geophysical model flow with chaotic time dependence

on convergence rates equivalency and sampling strategies in functional deconvolution models using the asymptotical minimax framework we examine convergence rates equivalency between a continuous functional deconvolution model and its real life discrete counterpart over a wide range of besov balls and for thel 2 risk for this purpose all possible models are divided into three groups for the models in the

robust and efficient recovery of a signal passed through a filter and then contaminated by non gaussian noise consider a channel where a continuous periodic input signal is passed through a linear filter and then is contaminated by an additive noise the problem is to recover this signal when we observe n repeated realizations of the output signal adaptive efficient procedures that are asymptotically minimax over all possible procedures are known for channels with gaussian noise and no filter the case of direct observation efficient procedures based on the smoothness of a recovered signal are known for the case of gaussian noise robust rate optimal procedures are known as well however there are no results on robust and efficient data driven procedures moreover the known results for the case of direct observation indicate that even a small deviation from gaussian noise may lead to a drastic change we show that for the considered case of indirect data and a particular class of so called supersmooth filters there exists a procedure of recovery of an input signal that possesses the desired properties namely it is adaptive to the smoothness of the input signal robust to the distribution of the noise globally and pointwise efficient that is its minimax global and pointwise risks converge with the best constant and rate over all possible estimators as n spl rarr spl infin and universal in the sense that for a wide class of linear not necessarily bounded operators the efficient estimator is a plug in one furthermore we explain how to employ the obtained asymptotic results for the practically important case of small n large noise

risk hull method and regularization by projections of ill posed inverse problems we study a standard method of regularization by projections of the linear inverse problem y af where is a white gaussian noise and a is a known compact operator with singular values converging to zero with polynomial decay the unknown function f is recovered by a projection method using the singular value decomposition of a the bandwidth choice of this projection regularization is governed by a data driven procedure which is based on the principle of risk hull minimization we provide nonasymptotic upper bounds for the mean square risk of this method and we show in particular that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation

ordered linear smoothers this paper deals with the following approach for estimating the mean mu of an n dimensional random vector y first a family s of n x n matrices is specified then an element s is an element of s is selected by mallows c l and mu s y the case is considered that s is an ordered linear smoother according to some easily interpretable qualitative conditions examples include linear smoothing procedures in nonparametric regression as e g smoothing splines minimax spline smoothers and kernel estimators stochastic probability bounds are given for the difference 1 n parallel to mu s y parallel to 2 2 1 n parallel to mu s mu y parallel to 2 2 where s mu denotes the minimizer of 1 n parallel to mu s y parallel to 2 2 for s is an element of s these probability bounds are generalized to the situation that s is the union of a moderate number of ordered linear smoothers the results complement work by li on the asymptotic optimality of c l implications for nonparametric regression are studied in detail it is shown that there exists a direct connection between james stein estimation and the use of smoothing procedures leading to a decision theoretic justification of the latter further conclusions concern the choice of the order of a smoothing spline or a minimax spline smoother and the rates of convergence of smoothing parameters

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

motzkin path models of long chain polymers in slits we consider motzkin path models for polymers confined to a slit the path interacts with each of the two confining lines and we define parameters a and b to characterize the strengths of the interactions with the two lines we consider the cases where i each vertex in a confining line contributes to the energy and ii where each edge in a confining line contributes to the energy for the vertex and edge versions of motzkin paths we can find the generating functions and give rigorous explicit expressions for the free energy at some special points in the a b plane and asymptotically i e for large slit widths elsewhere we find regions where the force between the lines is long range and repulsive short range and repulsive and short range and attractive our results indicate that the general form of the phase diagram is model independent and similar to previous results for a dyck path model although the details do depend on the underlying configurational model we also contrast the method used here to find the generating function with the transfer matrix and heap methods

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates