on degrees of freedom of projection estimators with applications to multivariate nonparametric regression abstract in this article we consider the nonparametric regression problem with multivariate predictors we provide a characterization of the degrees of freedom and divergence for estimators of the

total variation adaptive total variation and nonconvex smoothly clipped absolute deviation penalty for denoising blocky images the total variation based image denoising model has been generalized and extended in numerous ways improving its performance in different contexts we propose a new penalty function motivated by the recent progress in the statistical literature on high dimensional variable selection using a particular instantiation of the majorization minimization algorithm the optimization problem can be efficiently solved and the computational procedure realized is similar to the spatially adaptive total variation model our two pixel image model shows theoretically that the new penalty function solves the bias problem inherent in the total variation model the superior performance of the new penalty function is demonstrated through several experiments our investigation is limited to blocky images which have small total variation

edge preserving and noise reducing reconstruction for magnetic particle imaging magnetic particle imaging mpi is an emerging medical imaging modality which is based on the non linear response of magnetic nanoparticles to an applied magnetic field it is an important feature of mpi that even fast dynamic processes can be captured for 3d volumes the high temporal resolution in turn leads to large amounts of data which have to be handled efficiently but as the system matrix of mpi is non sparse the image reconstruction gets computationally demanding therefore currently only basic image reconstruction methods such as tikhonov regularization are used however tikhonov regularization is known to oversmooth edges in the reconstructed image and to have only a limited noise reducing effect in this work we develop an efficient edge preserving and noise reducing reconstruction method for mpi as regularization model we propose to use the nonnegative fused lasso model and we devise a discretization that is adapted to the acquisition geometry of the preclinical mpi scanner considered in this work we develop a customized solver based on a generalized forward backward scheme which is particularly suitable for the dense and not well structured system matrices in mpi already a non optimized prototype implementation processes a 3d volume within a few seconds so that processing several frames per second seems amenable we demonstrate the improvement in reconstruction quality over the state of the art method in an experimental medical setup for an in vitro angioplasty of a stenosis

the ii p method for estimating multivariate functions from noisy data the method for estimating an underlying smooth function of m variables x l xm using noisy data is based on approximating it by a sum of products of the form m m x m the problem is then reduced to estimating the univariate functions in the products a convergent algorithm is described the method keeps tight control on the degrees of freedom used in the fit many examples are given the quality of fit given by the method is excellent usually only a few products are enough to fit even fairly complicated functions the coding into products of univariate functions allows a relatively understandable interpretation of the multivariate fit

local extremes runs strings and multiresolution the paper considers the problem of nonparametric regression with emphasis on controlling the number of local extremes two methods the run method and the taut string multiresolution method are introduced and analyzed on standard test beds it is shown that the number and locations of local extreme values are consistently estimated rates of convergence are proved for both methods the run method converges slowly but can withstand blocks as well as a high proportion of isolated outliers the rate of convergence of the taut string multiresolution method is almost optimal the method is extremely sensitive and can detect very low power peaks section 1 contains an introduction with special reference to the number of local extreme values the run method is described in section 2 and the taut string multiresolution method in section 3 low power peaks are considered in section 4 section 5 contains a comparison with other methods and section 6 a short conclusion the proofs are given in section 7 and the taut string algorithm is described in the appendix

nonlinear ordinary differential equations resolvable with respect to an irregular singular point abstract we define a class of systems of nonlinear ordinary differential equations resolvable with respect to an irregular singular point say x it is proved that a resolvable system has a unique formal series solution in some fractional powers of x 1 moreover any system has a formal series solution if and only if it can be reduced to a resolvable one we present a method of constructing a proper solution in a suitable sectorial neighorhood of infinity which has the asymptotics in this region the second painleve equation is considered as an example

on the fly approximation of multivariate total variation minimization in the context of change point detection addressed by total variation minimization strategies an efficient on the fly algorithm has been designed leading to exact solutions for univariate data in this contribution an extension of such an on the fly strategy to multivariate data is investigated the proposed algorithm relies on the local validation of the karush kuhn tucker conditions on the dual problem showing that the non local nature of the multivariate setting precludes to obtain an exact on the fly solution we devise an on the fly algorithm delivering an approximate solution whose quality is controlled by a practitioner tunable parameter acting as a trade off between quality and computational cost performance assessment shows that high quality solutions are obtained on the fly while benefiting of computational costs several orders of magnitude lower than standard iterative procedures the proposed algorithm thus provides practitioners with an efficient multivariate change point detection on the fly procedure

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

asymptotics beyond all orders and analytic properties of inverse laplace transforms of solutions a number of modern mathematical and physical problems require the study of delicate asymptotic properties lying beyond the power series asymptotics in this paper we suggest a link between these asymptotic problems and some analytic properties of inverse laplace transforms of the corresponding solutions the main result claims that these inverse transforms are holomorphic in an appropriately cut complex plane a direct consequence of this is the nonexistence of solutions to the class of asymptotics beyond all orders problems such as regular shocks of the kuramoto sivashinsky equaiton gr needle crystal solutions of the simple geometrical model of crystal growth ks solitary wave solutions to a class of the fifth order kortveg de vries equations ko sect 8 gj homoclinic orbits of some singularly perturbed mappings ec hm and others

estimation of a k monotone density limit distribution theory and the spline connection we study the asymptotic behavior of the maximum likelihood and least squares estimators of a kmonotone density g0 at a fixed point x0 when k 2 we find that the jth derivative of the estima tors at x0 converges at the rate n k j 2k 1 for j 0 k 1 the limiting distribution depends on an almost surely uniquely de fined stochastic process hk that stays above below the k fold in tegral of brownian motion plus a deterministic drift when k is even odd both the mle and lse are known to be splines of degree k 1 with simple knots establishing the order of the random gap n n where n denote two successive knots is a key ingredient of the proof of the main results we show that this gap problem can be solved if a conjecture about the upper bound on the error in a particular hermite interpolation via odd degree splines holds

stochastic models for sparse and piecewise smooth signals we introduce an extended family of continuous domain stochastic models for sparse piecewise smooth signals these are specified as solutions of stochastic differential equations or equivalently in terms of a suitable innovation model the latter is analogous conceptually to the classical interpretation of a gaussian stationary process as filtered white noise the two specific features of our approach are 1 signal generation is driven by a random stream of dirac impulses poisson noise instead of gaussian white noise and 2 the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes we provide a complete characterization of these finite rate of innovation signals within gelfand s framework of generalized stochastic processes we then focus on the class of scale invariant whitening operators which correspond to unstable systems we show that these can be solved by introducing proper boundary conditions which leads to the specification of random spline type signals that are piecewise smooth these processes are the poisson counterpart of fractional brownian motion they are nonstationary and have the same 1 type spectral signature we prove that the generalized poisson processes have a sparse representation in a wavelet like basis subject to some mild matching condition we also present a limit example of sparse process that yields a map signal estimator that is equivalent to the popular tv denoising algorithm

continuous domain solutions of linear inverse problems with tikhonov versus generalized tv regularization we consider one dimensional 1 d linear inverse problems that are formulated in the continuous domain the object of recovery is a function that is assumed to minimize a convex objective functional the solutions are constrained by imposing a continuous domain regularization we derive the parametric form of the solution representer theorems for tikhonov quadratic and generalized total variation gtv regularizations we show that in both cases the solutions are splines that are intimately related to the regularization operator in the tikhonov case the solution is smooth and constrained to live in a fixed subspace that depends on the measurement operator by contrast the gtv regularization results in a sparse solution composed of only a few dictionary elements that are upper bounded by the number of measurements and independent of the measurement operator our findings for the gtv regularization resonates with the minimization of the ell 1 norm which is its discrete counterpart and also produces sparse solutions finally we find the experimental solutions for some measurement models in one dimension we discuss the special case when the gtv regularization results in multiple solutions and devise an algorithm to find an extreme point of the solution set which is guaranteed to be sparse

adaptive nonparametric regression with the k nearest neighbour fused lasso the fused lasso also known as total variation denoising is a locally adaptive function estimator over a regular grid of design points in this article we extend the fused lasso to settings in which the points do not occur on a regular grid leading to a method for nonparametric regression this approach which we call the formula see text nearest neighbours fused lasso involves computing the formula see text nearest neighbours graph of the design points and then performing the fused lasso over this graph we show that this procedure has a number of theoretical advantages over competing methods specifically it inherits local adaptivity from its connection to the fused lasso and it inherits manifold adaptivity from its connection to the formula see text nearest neighbours approach in a simulation study and an application to flu data we show that excellent results are obtained for completeness we also study an estimator that makes use of an formula see text graph rather than a formula see text nearest neighbours graph and contrast it with the formula see text nearest neighbours fused lasso

a new look at q exponential distributions via excess statistics q exponential distributions play an important role in nonextensive statistics they appear as the canonical distributions i e the maximum generalized q entropy distributions under mean constraint their relevance is also independently justified by their appearance in the theory of superstatistics introduced by beck and cohen in this paper we provide a third and independent rationale for these distributions we indicate that q exponentials are stable by a statistical normalization operation and that pickands extreme values theorem plays the role of a clt like theorem in this context this suggests that q exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold moreover we give an asymptotic connection between excess distributions and maximum q entropy we also highlight the role of generalized pareto distributions in many applications and present several methods for the practical estimation of q exponential parameters

total bounded variation regularization as a bilaterally constrained optimization problem it is demonstrated that the predual for problems with total bounded variation regularization terms can be expressed as a bilaterally constrained optimization problem existence of a lagrange multiplier and an optimality system are established this allows us to utilize efficient optimization methods developed for problems with box constraints in the context of bounded variation formulations here in particular the primal dual active set method considered as a semismooth newton method is analyzed and superlinear convergence is proved as a by product we obtain that the lagrange multiplier associated with the box constraints acts as an edge detector numerical results for image denoising and zooming resizing show the efficiency of the new approach

locally adaptive regression splines in this paper least squares penalized regression estimates withtotal variation penalities are considered it is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints algorithms and asymptotic properties are discussed

on the equivalence of soft wavelet shrinkage total variation diffusion total variation regularization and sides soft wavelet shrinkage total variation tv diffusion tv regularization and a dynamical system called sides are four useful techniques for discontinuity preserving denoising of signals and images in this paper we investigate under which circumstances these methods are equivalent in the one dimensional case first we prove that haar wavelet shrinkage on a single scale is equivalent to a single step of space discrete tv diffusion or regularization of two pixel pairs in the translationally invariant case we show that applying cycle spinning to haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of tv diffusion we prove that space discrete tv diffusion and tv regularization are identical and that they are also equivalent to the sides system when a specific force function is chosen afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the laplacian pyramid of the signal we analyze possibilities to avoid gibbs like artifacts for multiscale haar wavelet shrinkage by scaling the thresholds finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and pde variational approaches these methods are based on iterated shift invariant wavelet shrinkage at multiple scales with scaled thresholds

consistencies and rates of convergence of jump penalized least squares estimators we study the asymptotics for jump penalized least squares regression aiming at approximating a regression function by piecewise constant functions besides conventional consistency and convergence rates of the estimates in l 2 0 1 our results cover other metrics like skorokhod metric on the space of cadlag functions and uniform metrics on c 0 1 we will show that these estimators are in an adaptive sense rate optimal over certain classes of approximation spaces special cases are the class of functions of bounded variation piecewise holder continuous functions of order 0 1 and the class of step functions with a finite but arbitrary number of jumps in the latter setting we will also deduce the rates known from change point analysis for detecting the jumps finally the issue of fully automatic selection of the smoothing parameter is addressed

sharp oracle bounds for monotone and convex regression through aggregation we derive oracle inequalities for the problems of isotonic and convex regression using the combination of q aggregation procedure and sparsity pattern aggregation this improves upon the previous results including the oracle inequalities for the constrained least squares estimator one of the improvements is that our oracle inequalities are sharp i e with leading constant 1 it allows us to obtain bounds for the minimax regret thus accounting for model misspecification which was not possible based on the previous results another improvement is that we obtain oracle inequalities both with high probability and in expectation

empirical regularities of opening call auction in chinese stock market abstract we study the statistical regularities of an opening call auction using the ultra high frequency data of 22 liquid stocks traded on the shenzhen stock exchange in 2003 the distribution of the relative price defined as the relative difference between the order price in the opening call auction and the closing price on the last trading day is asymmetric and that the distribution displays a sharp peak at the zero relative price and a relatively wide peak at the negative relative price the detrended fluctuation analysis dfa method is adopted to investigate the long term memory of relative order prices we further study the statistical regularities of order sizes in the opening call auction and observe a phenomenon of number preference known as order size clustering the probability density function pdf of order sizes could be well fitted by a q gamma function and the long term memory also exists in order sizes in addition both the average volume and the average number of orders decrease exponentially with the price level away from the best bid or ask price level in the limit order book lob established immediately after the opening call auction and a price clustering phenomenon is observed

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

statistical mechanics of topological phase transitions in networks we provide a phenomenological theory for topological transitions in restructuring networks in this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges the associated microscopic dynamics satisfies the detailed balance condition and is equivalent to a lattice gas model on the edge dual graph of a fully connected network in our studies based on an exact enumeration method monte carlo simulations and theoretical considerations we find a rich variety of topological phase transitions when the temperature is varied these transitions signal singular changes in the essential features of the global structure of the network depending on the energy function chosen the observed transitions can be best monitored using the order parameters f s5s max m i e the size of the largest connected component divided by the number of edges or f k5k max m the largest degree in the network divided by the number of edges if for example the energy is chosen to be e52s max the observed transition is analogous to the percolation phase transition of random graphs for this choice of the energy the phase diagram in the k t plane is constructed single vertex energies of the form e5 i f k i where k i is the degree of vertex i are also studied depending on the form of f ki first order and continuous phase transitions can be observed in case of f k i 52 k i1a ln ki the transition is continuous and at the critical temperature scale free graphs can be recovered finally by abruptly decreasing the temperature nonequilibrium processes e g nucleation and growth of particular topological phases can also be interpreted by the present approach

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

monotonically equivalent entropies and solution of additivity equation generalized entropies are studied as lyapunov functions for the master equation markov chains three basic properties of these lyapunov functions are taken into consideration universality independence of the kinetic coefficients trace form the form of sum over the states and additivity for composition of independent subsystems all the entropies which have all three properties simultaneously and are defined for positive probabilities are found they form a one parametric family we consider also pairs of entropies s1 s2 which are connected by the monotonous transformation s2 f s1 equivalent entropies all classes of pairs of universal equivalent entropies one of which has a trace form and another is additive these entropies can be different one from another were found these classes consist of two one parametric families the family of entropies which are equivalent to the additive trace form entropies and the family of renyi tsallis entropies

entropy and complexity analysis of dirac delta like quantum potentials abstract the dirac delta like quantum mechanical potentials are frequently used to describe and interpret numerous phenomena in many scientific fields including atomic and molecular physics condensed matter and quantum computation the entropy and complexity properties of potentials with one and two dirac delta functions are here analytically calculated and numerically discussed in both position and momentum spaces we have studied the information theoretic lengths of fisher renyi and shannon types as well as the cramer rao fisher shannon and lmc shape complexities of the lowest lying stationary states of one delta and twin delta they allow us to grasp and quantify different facets of the spreading of the charge and momentum of the system far beyond the celebrated standard deviation

fractal surfaces from simple arithmetic operations fractal surfaces patchwork quilts are shown to arise under most general circumstances involving simple bitwise operations between real numbers a theory is presented for all deterministic bitwise operations on a finite alphabet it is shown that these models give rise to a roughness exponent h that shapes the resulting spatial patterns larger values of the exponent leading to coarser surfaces

generalized kolmogorov entropy in the dynamics of multifractal generation abstract we point out that applying a maximization principle on a tsallis like generalized form of the kolmogorov entropy for iterated function systems naturally provides a canonical statistical frame for the description of the multifractal measures generated by such dynamical processes multifractal spectra can then be characterized by usual statistical parameters in particular the temperature we show that in the limit of zero temperature the multifractal measure collapses to a homogeneous distribution over a fractal support for finite temperatures multifractal spectra are studied numerically in an illustrative example

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

local asymptotics for quantile smoothing splines quantile smoothing splines were introduced by koenker ng and portnoy as natural and appealing estimates of conditional quantiles of response variables the natural setting for the problem considers minimization of a weighted combination of a fit penalty and a roughness penalty over the space of functions whose derivatives have bounded variation although this space is not traditional shen has shown recently that the quantile smoothing splines do indeed converge at the usual optimal rate n 2 5 in various norms here local asymptotic results are obtained by establishing bahadur representations for local parameters of the splines these are used to obtain local rates of convergence to establish uniform convergence rates to provide local distribution theory for quantile b splines and to expand the fit measure in order to analyze an information criterion for determining the smoothing parameter examples of using derivatives of the smoothing splines for estimating jump functions are also presented

densities spectral densities and modality this paper considers the problem of specifying a simple approximating density function for a given data set x 1 x n simplicity is measured by the number of modes but several different definitions of approximation are introduced the taut string method is used to control the numbers of modes and to produce candidate approximating densities refinements are introduced that improve the local adaptivity of the procedures and the method is extended to spectral densities

disequilibrium complexity the schottky effect and q entropies in paramagnetism abstract we investigate connections between statistical quantifiers and paramagnetism more concretely we apply the notions of i disequilibrium and ii statistical complexity to a paramagnetic system of non coupled dipoles interesting insights are thereby obtained in particular we encounter a kind of criticality not associated to the temperature but to the disequilibrium

inference in high dimensional generalized linear models based on soft thresholding we further develop and analyse penalized likelihood estimators for generalized linear models with a large number of coefficients the methodology proposed leads to an adaptive selection of model terms without substantial variance inflation our proposal extends the soft thresholding strategy of donoho and johnstone and the lasso of tibshirani to generalized linear models and multiple predictor variables in addition we develop an estimator for the covariance matrix of the estimated coefficients which can even be used for terms dropped from the model used in connection with basis functions the methodology proposed provides an alternative to other generalized function estimators it leads to an adaptive economical description of the results in terms of basis functions specifically it is shown how adaptive regression splines and qualitative restrictions can be incorporated our approach is demonstrated by applications to a prognosis of solvency and rental guides

on the stability of generalized entropies by the use of a probabilistic coupling technique and generalizing a recent result of zhang 2007 lett math phys 80 171 we derive several sharp inequalities concerning the uniform continuity of some generalized entropies these inequalities capture certain aspects of the interplay between the entropies and the uniform distance variational distance and establish the stability of the involved entropies

piecewise linear regularized solution paths we consider the generic regularized optimization problem argmin l y x j efron hastie johnstone and tibshirani ann statist 32 2004 407 499 have shown that for the lasso that is if l is squared error loss and j 1 is the l 1 norm of the optimal coefficient path is piecewise linear that is is piecewise constant we derive a general characterization of the properties of loss l penalty j pairs which give piecewise linear coefficient paths such pairs allow for efficient generation of the full regularized coefficient paths we investigate the nature of efficient path following algorithms which arise we use our results to suggest robust versions of the lasso for regression and classification and to develop new efficient algorithms for existing problems in the literature including mammen and van de geer s locally adaptive regression splines

indices for testing neural codes one of the most critical challenges in systems neuroscience is determining the neural code a principled framework for addressing this can be found in information theory with this approach one can determine whether a proposed code can account for the stimulus response relationship specifically one can compare the transmitted information between the stimulus and the hypothesized neural code with the transmitted information between the stimulus and the behavioral response if the former is smaller than the latter i e if the code cannot account for the behavior the code can be ruled out the information theoretic index most widely used in this context is shannon s mutual information the shannon test however is not ideal for this purpose while the codes it will rule out are truly nonviable there will be some nonviable codes that it will fail to rule out here we describe a wide range of alternative indices that can be used for ruling codes out the range includes a continuum from shannon information to measures of the performance of a bayesian decoder we analyze the relationship of these indices to each other and their complementary strengths and weaknesses for addressing this problem

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

convergence in reaction diffusion systems an information theory approach abstract we have applied an information theory approach in order to study the problem of convergence to point like or extended attractors in reaction diffusion systems a distance between two states based on the kullback leibler relative information was defined different forms of the probability distribution some of them based on the knowledge of the nonequilibrium potential when accesible give the possibility to look for a faster and or more accurate convergence this approach offers the chance to estimate the attraction basins of the different attractors as well as detecting limit circles together with an easy evaluation of their periods

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

asymptotic integration of linear differential equations abstract we establish a new result on the asymptotic integration of linear differential systems of the form y r y where is diagonal and r is small more specifically we allow for different scales of asymptotic behaviour of the individual matrix elements of and r we also discuss an application to a self adjoint fourth order equation

diffusion on fractal phase spaces and entropy production abstract complex systems may show strongly non exponential relaxation the origins of these behavior seem to be related to the fractal structure of the phase space in this article we consider the diffusion on a dilute hypercube this stochastic process is a coarse grained model for the time evolution of a glassy system where both the dynamics and the structure of the phase space was considered as simple as possible we characterize this process using two quantities a memory function and the tsallis entropy the first one is a measure of mixing while the second is related to non extensivity we characterize quantitatively their relationship finally we discuss the implications for other glassy and complex systems

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

two statistical mechanics aspects of complex networks abstract by adopting an ensemble interpretation of non growing rewiring networks network theory can be reduced to a counting problem of possible network states and an identification of their associated probabilities we present two scenarios of how different rewirement schemes can be used to control the state probabilities of the system in particular we review how by generalizing the linking rules of random graphs in combination with superstatistics and quantum mechanical concepts one can establish an exact relation between the degree distribution of any given network and the nodes linking probability distributions in a second approach we control state probabilities by a network hamiltonian whose characteristics are motivated by biological and socio economical statistical systems we demonstrate that a thermodynamics of networks becomes a fully consistent concept allowing to study e g phase transitions and computing entropies through thermodynamic relations

student s t and r distributions unified derivation from an entropic variational principle abstract the optimization of the recently generalized entropy of the form s q 1 dx p x q q 1 with the constraints dx p x 1 and x 2 q dx x 2 p x q 1 yields the student s t distribution for q 1 and the r distribution for q

properties and refinements of the fused lasso we consider estimating an unknown signal both blocky and sparse which is corrupted by additive noise we study three interrelated least squares procedures and their asymptotic properties the first procedure is the fused lasso put forward by friedman et al ann appl statist 1 2007 302 332 which we modify into a different estimator called the fused adaptive lasso with better properties the other two estimators we discuss solve least squares problems on sieves one constrains the maximal l 1 norm and the maximal total variation seminorm and the other restricts the number of blocks and the number of nonzero coordinates of the signal we derive conditions for the recovery of the true block partition and the true sparsity patterns by the fused lasso and the fused adaptive lasso and we derive convergence rates for the sieve estimators explicitly in terms of the constraining parameters

multiscale change point inference summary we introduce a new estimator the simultaneous multiscale change point estimator smuce for the change point problem in exponential family regression an unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level the probability of overestimating the true number of change points k is controlled by the asymptotic null distribution of the multiscale test statistic further we derive exponential bounds for the probability of underestimating k by balancing these quantities will be chosen such that the probability of correctly estimating k is maximized all results are even non asymptotic for the normal case on the basis of these bounds we construct asymptotically honest confidence sets for the unknown step function and its change points at the same time we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate up to a log term finally the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n even for an unbounded number of change points we illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions the performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

on the statistical analysis of smoothing by maximizing dirty markov random field posterior distributions we consider bayesian nonparametric function estimation using a markov random field prior based on the laplace distribution we describe efficient methods for finding the exact maximum a posteriori estimate which handle constraints naturally and avoid the problems posed by nondifferentiability of the posterior distribution the methods also make links to spline and wavelet smoothers and to a dual posterior distribution three automatic smoothing parameter selection procedures are described empirical bayes two fold cross validation and a universal rule for the laplace prior monte carlo simulation with gaussian and poisson responses demonstrates that the new estimator can give better estimates of nonsmooth functions than can a similar prior based on the gaussian distribution or wavelet based competitors applications are given to spectral density estimation and to poisson image denoising

zeroth law compatibility of nonadditive thermodynamics nonextensive thermodynamics is criticized by the statement that the zeroth law cannot be satisfied with nonadditive composition rules corrected in this paper we determine the general functional form of those nonadditive composition rules that are compatible with the zeroth law of thermodynamics we find that this general form is additive for the formal logarithms of the original quantities and the familiar relations of thermodynamics apply to these our result offers a possible solution to the long standing questions about equilibrium between extensive and nonextensive systems or systems with different nonextensivity parameters

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

splines in higher order tv regularization splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces in this paper we show in a strictly discrete setting that splines of degree m 1 solve also a minimization problem with quadratic data term and m th order total variation tv regularization term in contrast to problems with quadratic regularization terms involving m th order derivatives the spline knots are not known in advance but depend on the input data and the regularization parameter more precisely the spline knots are determined by the contact points of the m th discrete antiderivative of the solution with the tube of width 2 around the m th discrete antiderivative of the input data we point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the sobolev space w 2 0 m from this point of view the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines

multiwavelet packet entropy and its application in transmission line fault recognition and classification multiwavelets possess better properties than traditional wavelets multiwavelet packet transformation has more high frequency information spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal this paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults and uses a radial basis function rbf neural network to recognize and classify 10 fault types of power transmission lines first the preprocessing and postprocessing problems of multiwavelets are presented shannon entropy and tsallis entropy are introduced and their difference is discussed second multiwavelet packet energy entropy time entropy shannon singular entropy and tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals third the plan of transmission line fault recognition using multiwavelet packet entropies and an rbf neural network is proposed finally the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition the performance with sa4 symmetric antisymmetric multiwavelet packet tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

non existence of elliptic travelling wave solutions of the complex ginzburg landau equation abstract we give a simple proof that for generic parameter values the cubic complex one dimensional ginzburg landau equation has no elliptic travelling wave solutions this is contrary to the expectations of musette and conte in physica d 181 2003 70 79 that elliptic solutions of zero codimension should exist the method of proof based on the residue theorem is very general and can be applied to determine necessary conditions for the existence of elliptic travelling waves for any autonomous partial differential equation as another application we prove that kudryashov s codimension one elliptic solution of the generalized kuramoto sivashinsky equation is the only one possible

optimal spatial adaptation to inhomogeneous smoothness an approach based on kernel estimates with variable bandwidth selectors a new v ria e 1 s e 0r estimation is proposed the application of this bandwidth selector leads to kernel estimates that achieve optimal rates of convergence over b 3 sses this implies that the procedure adapts to spatially inhomogeneous s noothness in particu lar the estimates share optimality properties with wavelet estimates based on thresholding of empirical wavelet coefficients 1 introduction in nonparametric curve estimation the statistical anal ysis may focus on the inference of the qualitative structure of the analyzed curve often interesting features of the curve are connected with spatially inhomogeneous smoothness in this case curve estimates that are spatially adaptlve ar e appropriaie a variety of such procedures have been proposed in the literature in breiman friedman olshen and stone 1983 piecewise constant least squares estimates are considered with a data adaptive choice of the pieces cart more generally friedman and silverman 1989 friedman 1991 and luo and wahba 1995 use variable knot splines mars knot points are added removed and allocated recursively using cross validation techniques these methods have shown good performance in simulations and real data examples however no asymptotic theory is available mammen and van de geer 1997 discuss penalized least squares curve estimation for spatially inhomogeneous curves they propose penalty terms which allow more spatial inhomogeneity than the usual l2 norms of deriva tives of the curve the estimates turn out to be variable knot splines see also mammen 1991 results on rates of convergence and a pointwise asymptotic distribution theory are given miiller and stadtmiiller 1987 staniswalis 1989 and brockmann gasser and hermann 1993 propose kernel estimation with locally variable band width selectors the calculation of local bandwidths is based on pilot estima tion of local smoothness characteristics an asymptotic analysis is available

tsallis like information measures and the analysis of complex signals abstract a tsallis like information measure that elaborates upon those of the tsallis type is shown to be a rather powerful tool for the analysis of complex non stationary signals as exemplified by the henon map and eegs the ensuing results could be of importance concerning the automatic detection of epileptic spikes

unified model for network dynamics exhibiting nonextensive statistics we introduce a dynamical network model which unifies a number of network families which are individually known to exhibit q exponential degree distributions the present model dynamics incorporates static nongrowing self organizing networks preferentially growing networks and preferentially rewiring networks further it exhibits a natural random graph limit the proposed model generalizes network dynamics to rewiring and growth modes which depend on internal topology as well as on a metric imposed by the space they are embedded in in all of the networks emerging from the presented model we find q exponential degree distributions over a large parameter space we comment on the parameter dependence of the corresponding entropic index q for the degree distributions and on the behavior of the clustering coefficients and neighboring connectivity distributions

the dfs fused lasso linear time denoising over general graphs the fused lasso also known as anisotropic total variation denoising is widely used for piecewise constant signal estimation with respect to a given undirected graph the fused lasso estimate is highly nontrivial to compute when the underlying graph is large and has an arbitrary structure but for a special graph structure namely the chain graph the fused lasso or simply 1d fused lasso can be computed in linear time in this paper we revisit a result recently established in the online classification literature herbster et al 2009 cesa bianchi et al 2013 and show that it has important implications for signal denoising on graphs the result can be translated to our setting as follows given a general graph if we run the standard depth first search dfs traversal algorithm then the total variation of any signal over the chain graph induced by dfs is no more than twice its total variation over the original graph this result leads to several interesting theoretical and computational conclusions letting m and n denote the number of edges and nodes respectively of the graph in consideration it implies that for an underlying signal with total variation t over the graph the fused lasso properly tuned achieves a mean squared error rate of t2 3n 2 3 moreover precisely the same mean squared error rate is achieved by running the 1d fused lasso on the dfs induced chain graph importantly the latter estimator is simple and computationally cheap requiring o m operations to construct the dfs induced chain and o n operations to compute the 1d fused lasso solution over this chain further for trees that have bounded maximum degree the error rate of t2 3n 2 3 cannot be improved in the sense that it is the minimax rate for signals that have total variation t over the tree finally several related results also hold for example the analogous result holds for a roughness measure defined by the l0 norm of differences across edges in place of the total variation metric

a shape based cutting and clustering algorithm for multiple change point detection abstract the detection of multiple change points without prior knowledge of the number and location of the change points is investigated in this paper by making full use of the geometric information of local statistics a new change point detection algorithm called the shape based cutting and clustering scc algorithm is established there are three key techniques in the proposed scc procedure data driven threshold adaptive bandwidth and single peak recognition our simulation results show that the proposed method is highly competitive in terms of computational speed and effectiveness in order to validate the feasibility of the proposed algorithm we apply the methodology to an operational problem in renewable integrated electrical distribution networks the results of the real data analysis illustrate the effectiveness of the proposed algorithm

conditional growth charts growth charts are often more informative when they are customized per subject taking into account prior measurements and possibly other covariates of the subject we study a global semiparametric quantile regression model that has the ability to estimate conditional quantiles without the usual distributional assumptions the model can be estimated from longitudinal reference data with irregular measurement times and with some level of robustness against outliers and it is also flexible for including covariate information we propose a rank score test for large sample inference on covariates and develop a new model assessment tool for longitudinal growth data our research indicates that the global model has the potential to be a very useful tool in conditional growth chart analysis

scale free networks from a hamiltonian dynamics contrary to many recent models of growing networks we present a model with fixed number of nodes and links where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is introduced by applying a local rewiring move the network reaches equilibrium states assuming broad degree distributions which have a power law form in an intermediate range of the parameters used interestingly in the same range we find nontrivial hierarchical clustering

complex networks emerging from fluctuating random graphs analytic formula for the hidden variable distribution in analogy to superstatistics which connects boltzmann gibbs statistical mechanics to its generalizations through temperature fluctuations complex networks are constructed from fluctuating erd os r enyi random graphs using a quantum mechanical method the exact analytic formula for the hidden variable distribution is presented which describes the nature of the fluctuations and generates a generic degree distribution through the poisson transformation as an example a static scale free network is discussed and the corresponding hidden variable distribution is found to decay as a power law and to diverge at the origin

trend filtering on graphs we introduce a family of adaptive estimators on graphs based on penalizing the l1 norm of discrete graph differences this generalizes the idea of trend filtering kim et al 2009 tibshirani 2014 used for univariate nonparametric regression to graphs analogous to the univariate case graph trend filtering exhibits a level of local adaptivity unmatched by the usual l2 based graph smoothers it is also defined by a convex minimization problem that is readily solved e g by fast admm or newton algorithms we demonstrate the merits of graph trend filtering through both examples and theory

on the derivation of power law distributions within classical statistical mechanics far from the thermodynamic limit we show that within classical statistical mechanics without taking the thermodynamic limit the most general boltzmann factor for the canonical ensemble is a q exponential function the only assumption here is that microcanonical distributions have to be separated from the total system energy which is the prerequisite for any sensible measurement we derive that all separable distributions are parametrized by a mathematical separation constant q which can be related to the non extensivity q parameter in tsallis distributions we further demonstrate that nature fixes the separation constant q to 1 for large dimensionality of gibbs phase space our results will be relevant for systems with a low dimensional space for example nanosystems comprised of a small number of particles or for systems with a dimensionally collapsed phase space which might be the case for a large class of complex systems

extracting features of gaussian self similar stochastic processes via the bandt pompe approach by recourse to appropriate information theory quantifiers normalized shannon entropy and martin plastino rosso intensive statistical complexity measure we revisit the characterization of gaussian self similar stochastic processes from a bandt pompe viewpoint we show that the ensuing approach exhibits considerable advantages with respect to other treatments in particular clear quantifiers gaps are found in the transition between the continuous processes and their associated noises

convergence analysis of generalized iteratively reweighted least squares algorithms on convex function spaces the computation of robust regression estimates often relies on minimization of a convex functional on a convex set in this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization minimization algorithms our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case we prove convergence on convex function spaces for general coercive and convex functionals f and derive geometric convergence in certain unconstrained settings the algorithm is applied to total variation tv penalized quantile regression and is compared with a step size corrected newton raphson algorithm it is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better whereas the newton type method outpaces the former only after many iterations finally in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows one to utilize highly efficient algorithms for special quadratic programs in more complex settings

estimating a density under order restrictions nonasymptotic minimax risk on considere la classe de toutes les densites unimodales definies sur un intervalle de longueur l et borne par h on etudie le risque minimax sur cette classe lorsqu on estime en utilisant n observations i i d la perte etant mesuree par la distance l 1 entre l estimateur et la densite reelle

nonparametric regression with adaptive truncation via a convex hierarchical penalty we consider the problem of nonparametric regression with a potentially large number of covariates we propose a convex penalized estimation framework that is particularly well suited to high dimensional sparse additive models and combines the appealing features of finite basis representation and smoothing penalties in the case of additive models a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions possess different levels of complexity in contrast a smoothing spline type penalty on the component functions is adaptive but does not provide a parsimonious representation our proposal simultaneously achieves parsimony and adaptivity in a computationally efficient way we demonstrate these properties through empirical studies and show that our estimator converges at the minimax rate for functions within a hierarchical class we further establish minimax rates for a large class of sparse additive models we also develop an efficient algorithm that scales similarly to the lasso with the number of covariates and sample size

quantile smoothing splines summary although nonparametric regression has traditionally focused on the estimation of conditional mean functions nonparametric estimation of conditional quantile functions is often of substantial practical interest we explore a class of quantile smoothing splines defined as solutions to