jump and sharp cusp detection by wavelets summary a method is proposed to detect jumps and sharp cusps in a function which is observed with noise by checking if the wavelet transformation of the data has significantly large absolute values across fine scale levels asymptotic theory is established and practical implementation is discussed the method is tested on simulated examples and applied to stock market return data the analysis of change points which describe sudden localised changes has recently found increasing interest change points can be used to model practical problems arising in fields such as quality control economics medicine signal and image processing and physical sciences for example in electroencephalogram signals sharp cusps exhibit the accelerations and decelerations in the beating of the hearts many practical problems like this involve functions which have jumps and sharp cusps the recently developed theory of wavelets has drawn much attention from both math ematicians statisticians and engineers in the seminal work of donoho 1993 donoho johnstone 1994 1995a b and donoho johnstone et al 1995 orthonormal bases of compactly supported wavelets have been used to estimate functions the theory of wavelets permits decomposition of functions into localised oscillating components this is an ideal tool to study localised changes such as jumps and sharp cusps in one dimension as well as several dimensions unlike traditional smoothing methods based on a fixed spatial scale the wavelet method is a multiresolution approach and has local adaptivity in this

mice multiple peak identification characterization and estimation summary mice multiple peak identification characterization and estimation is a procedure for estimating a lower bound of the number of frequency peaks and for estimating the frequency peak parameters the leading application is protein structure determination using nuclear magnetic resonance nmr experiments nmr frequency data are multiple peak data where each frequency peak corresponds to two connected atoms in the three dimensional protein structure we analyze the nmr frequency data through a series of steps a preliminary step for separating the signal from the background followed by identification of local maxima up to a noise level dependent threshold estimation of the frequency peak parameters using an iterative algorithm and detection of mixtures of peaks using hypothesis testing

on the local well posedness of quasilinear schrodinger equations in arbitrary space dimension abstract a class of quasilinear schrodinger equations is studied which contain strongly singular nonlinearities including derivatives of second order such equations have been derived as models of several physical phenomena the results of this paper apply to the superfluid film equation in fluid mechanics the local well posedness for smooth solutions of the cauchy problem is proved in arbitrary space dimension without any smallness assumption on the initial value this improves results in the literature which cover the one dimensional case only almost global well posedness results are obtained as well the proof combines nash moser techniques on implicit function theorems in frechet spaces with hyperbolic semigroup theory including evolution systems using the intersection of all sobolev spaces as basic function space the nondissipativity is overcome by a transformation procedure yielding the evolution operator and a priori estimates of higher order sobolev norms for the linearized problem here a certain trace has to vanish in view of apparent integrability conditions the transformations and the regularity theory are substantially different from the easier one dimensional case in particular continuity estimates are proved for the operator norms of the linear evolution operator in negative sobolev spaces

quantum correlations and distinguishability of quantum states a survey of various concepts in quantum information is given with a main emphasis on the distinguishability of quantum states and quantum correlations covered topics include generalized and least square measurements state discrimination quantum relative entropies the bures distance on the set of quantum states the quantum fisher information the quantum chernoff bound bipartite entanglement the quantum discord and geometrical measures of quantum correlations the article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them and for mathematicians looking for an up to date introductory course on these subjects which are mainly developed in the physics literature

multiscale maximum likelihood analysis of a semiparametric model with applications a special semiparametric model for a univariate density is introduced that allows analyzing a number of problems via appropriate transformations two problems treated in some detail are testing for the presence of a mixture and detecting a wear out trend in a failure rate the analysis of the semiparametric model leads to an approach that advances the maximum likelihood theory of the grenander estimator to a multiscale analysis the construction of the corresponding test statistic rests on an extension of a result on a two sided brownian motion with quadratic drift to the simultaneous control of excursions under parabolas at various scales of a brownian bridge the resulting test is shown to be asymptotically optimal in the minimax sense regarding both rate and constant and adaptive with respect to the unknown parameter in the semiparametric model the performance of the method is illustrated with a simulation study for the failure rate problem and with data from a flow cytometry experiment for the mixture analysis

simultaneous confidence bands for linear regression and smoothing suppose we observe y i f x i varepsilon i i 1 ldots n we wish to find approximate 1 alpha simultaneous confidence regions for f x x in mathscr x our regions will be centered around linear estimates hat f x of nonparametric or nonparametric f x there is a large amount of previous work on this subject substantial restrictions have been usually placed on some or all of the dimensionality of x the class of functions f that can be considered the class of linear estimates hat f and the region mathscr x the method we present is an approximation to the tube formula dn can be used for multidimensional x and a wide class of linear estimates by considering the effect of bias we are able to relax assumptions on the class of functions f which are considered simultaneous and numerical computations are used to illustrate the performance

false discovery control for random fields this article extends false discovery rates to random fields for which there are uncountably many hypothesis tests we develop a method for finding regions in the field s domain where there is a significant signal while controlling either the proportion of area or the proportion of clusters in which false rejections occur the method produces confidence envelopes for the proportion of false discoveries as a function of the rejection threshold from the confidence envelopes we derive threshold procedures to control either the mean or the specified tail probabilities of the false discovery proportion an essential ingredient of this construnction is a new algorithm to compute a confidence superset for the set of all true null locations we demonstrate our method with applications to scan statistics and functional neuroimaging

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

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

estimation of parameters of exponentially damped sinusoids using fast maximum likelihood estimation with application to nmr spectroscopy data we present fast maximum likelihood fml estimation of parameters of multiple exponentially damped sinusoids the fml algorithm was motivated by the desire to analyze data that have many closely spaced components such as the nmr spectroscopy data of human blood plasma the computational efficiency of fml lies in reducing the multidimensional search involved in ml estimation into multiple 1 d searches this is achieved by using our knowledge of the shape of the compressed likelihood function clf in the parameter space the proposed fml algorithm is an iterative method that decomposes the original data into its constituent signal components and estimates the parameters of the individual components efficiently using our knowledge of the shape of the clf the other striking features of the proposed algorithm are that it provides procedures for initialization has a fast converging iteration stage and makes use of the information extracted in preliminary iterations to segment the data suitably to increase the effective signal to noise ratio snr the computational complexity and the performance of the proposed algorithm are compared with other existing methods such as those based on linear prediction kiss iqml alternating projections ap and expectation maximization em

a note on the aitkin rubin approach to hypothesis testing in mixture models on montre que les conditions de regularite habituellement utilisees pour obtenir des resultats asymptotiques standards ne conviennent pas meme en utilisant une distribution a priori sur le vecteur des proportions de melange

existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in rn abstract mountain pass in a suitable orlicz space is employed to prove the existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in r n these equations contain strongly singular nonlinearities which include derivatives of the second order such equations have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

bayes cost reduction algorithm in quantum hypothesis testing corresp an iterative procedure is described for reducing the bayes cost in decisions among m 2 quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses the resulting decision strategy is a projection valued measure and yields an upper bound to the minimum attainable bayes cost from it is derived an algorithm for finding the optimum measurement states for choosing among m linearly independent pure states with minimum probability of error the method is also applied to decisions among m unimodal coherent quantum signals in thermal noise

discord of response the presence of quantum correlations in a quantum state is related to the state s response to local unitary perturbations such a response is quantified by the distance between the unperturbed and perturbed states minimized with respect to suitably identified sets of local unitary operations in order to be a bona fide measure of quantum correlations the distance function must be chosen among those that are contractive under completely positive and trace preserving cptp maps the most relevant instances of such physically well behaved metrics include the trace the bures and the hellinger distance to each of these metrics one can associate the corresponding discord of response namely the trace or hellinger or bures minimum distance from the set of unitarily perturbed states all these three discords of response satisfy the basic axioms for a proper measure of quantum correlations in the present work we focus in particular on the bures distance which enjoys the unique property of being both riemannian and contractive under cptp maps and admits important operational interpretations in terms of state distinguishability we compute analytically the bures discord of response for two qubit states with maximally mixed marginals and we compare it with the corresponding bures geometric discord namely the geometric measure of quantum correlations defined as the bures distance from the set of classical quantum states finally we investigate and identify the maximally quantum correlated two qubit states according to the bures discord of response these states exhibit a remarkable nonlinear dependence on the global state purity

geometric quantum discord with bures distance the qubit case the minimal bures distance of a quantum state of a bipartite system ab to the set of classical states for subsystem a defines a geometric measure of quantum discord when a is a qubit we show that this geometric quantum discord is given in terms of the eigenvalues of a 2nb 2nb hermitian matrix nb being the hilbert space dimension of the other subsystem b as a first application we calculate the geometric discord for the output state of the dqc1 algorithm we find that it takes its highest value when the unitary matrix from which the algorithm computes the trace has its eigenvalues uniformly distributed on the unit circle modulo a symmetry with respect to the origin as a second application we derive an explicit formula for the geometric discord of a two qubit state with maximally mixed marginals and compare it with other measures of quantum correlations we also determine the closest classical states to

quantum discord a measure of the quantumness of correlations two classically identical expressions for the mutual information generally differ when the systems involved are quantum this difference defines the quantum discord it can be used as a measure of the quantumness of correlations separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord thus showing that absence of entanglement does not imply classicality we relate this to the quantum superposition principle and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system i e the pointer states

multiscale testing of qualitative hypotheses suppose that one observes a process y on the unit interval where dy t n1 2f t dt dw t with an unknown function parameter f given scale parameter n 1 sample size and standard brownian motion w we propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity these tests are asymptotically optimal and adaptive in a certain sense they are constructed via a new class of multiscale statistics and an extension of levy s modulus of continuity of brownian motion

unifying approach to the quantification of bipartite correlations by bures distance the notion of distance defined on the set of states of a composite quantum system can be used to quantify total quantum and classical correlations in a unifying way we provide new closed formulae for classical and total correlations of two qubit bell diagonal states by considering the bures distance complementing the known corresponding expressions for entanglement and more general quantum correlations we thus complete the quantitative hierarchy of bures correlations for bell diagonal states we then explicitly calculate bures correlations for two relevant families of states werner states and rank 2 bell diagonal states highlighting the subadditivity which holds for total correlations with respect to the sum of classical and quantum ones when using bures distance finally we analyse a dynamical model of two independent qubits locally exposed to non dissipative decoherence channels where both quantum and classical correlations measured by bures distance exhibit freezing phenomena in analogy with other known quantifiers of correlations

the dip test of unimodality the dip test measures multimodality in a sample by the maximum difference over all sample points between the empirical distribution function and the unimodal distribution function that minimizes that maximum difference the uniform distribution is the asymptotically least favorable unimodal distribution and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform

time dependent dissipation in nonlinear schrodinger systems a coupled nonlinear schrodinger poisson equation is considered which contains a time dependent dissipation function as a specific model of dissipation effects in nonlinear quantum transport theory and other areas the wigner poisson equation associated with this system is derived using conservation and quasiconservation laws and certain growth assumptions for the nonlinearities and the dissipation function global existence of solutions to the cauchy problem of the time dependent schrodinger poisson system is shown both for small attractive case or arbitrary data repulsive case

testing monotonicity of a regression function we consider the problem of testing monotonicity of the regression function in a nonparametric regression model we introduce test statistics that are functionals of a certain natural u process we study the limiting distribution of these test statistics through strong approximation methods and the extreme value theory for gaussian processes we show that the tests are consistent against general alternatives

a modified likelihood ratio test for homogeneity in finite mixture models summary testing for homogeneity in finite mixture models has been investigated by many researchers the asymptotic null distribution of the likelihood ratio test lrt is very complex and difficult to use in practice we propose a modified lrt for homogeneity in finite mixture models with a general parametric kernel distribution family the modified lrt has a x2 type of null limiting distribution and is asymptotically most powerful under local alternatives simulations show that it performs better than competing tests they also reveal that the limiting distribution with some adjustment can satisfactorily approximate the quantiles of the test statistic even for moderate sample sizes

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

unified view of quantum and classical correlations we discuss the problem of the separation of total correlations in a given quantum state into entanglement dissonance and classical correlations using the concept of relative entropy as a distance measure of correlations this allows us to put all correlations on an equal footing entanglement and dissonance whose definition is introduced here jointly belong to what is known as quantum discord our methods are completely applicable for multipartite systems of arbitrary dimensions we investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

estimation and hypothesis testing in finite mixture models summary finite mixture models are a useful class of models for application to data when sample sizes are not large and the number of underlying densities is in question likelihood ratio tests based on joint maximum likelihood estimation of the mixing parameter x and the parameter of the underlying densities 0 are problematical our approach places a prior distribution on x and estimates 0 by maximizing the likelihood of the data given 0 with x integrated out advantages of this approach computational issues using the em algorithm and directions for further work are discussed the technique is applied to two examples

excess mass estimates and tests for multimodality abstract we propose and study a method for analyzing the modality of a distribution the method is based on the excess mass functional this functional measures excessive empirical mass in comparison with multiples of uniform distribution by the excess mass approach we separate the investigation about the number of modes from questions concerning their location for distributions on the line the excess mass functional can be estimated at a square root rate the asymptotic behavior of estimators is studied and tests for multimodality based on the excess mass are derived

optimal detection of symmetric mixed quantum states we develop a sufficient condition for the least squares measurement lsm or the square root measurement to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states using this condition we derive the optimal measurement for state sets with a broad class of symmetries we first consider geometrically uniform gu state sets with a possibly non abelian generating group and show that if the generator satisfies a weighted norm constraint then the lsm is optimal in particular for pure state gu ensembles the lsm is shown to be optimal for arbitrary gu state sets we show that the optimal measurement operators are gu with generator that can be computed very efficiently in polynomial time within any desired accuracy we then consider compound gu cgu state sets which consist of subsets that are gu when the generators satisfy a certain constraint the lsm is again optimal for arbitrary cgu state sets the optimal measurement operators are shown to be cgu with generators that can be computed efficiently in polynomial time

geometric coherence and quantum state discrimination the operational meaning of coherence measure lies at the very heart of the coherence theory in this paper we provide an operational interpretation for geometric coherence by proving that the geometric coherence of a quantum state is equal to the minimum error probability to discriminate a set of pure states with von neumann measurement on the other hand we also show that a task to ambiguously discriminate a set of linearly independent pure states can be regarded as a problem of calculating geometric coherence that is we reveal an equivalence relation between ambiguous quantum state discrimination and geometric coherence based on this equivalence moreover we improve the upper bound for geometric coherence and give the explicit expression of geometric coherence for a class of states besides we establish a complementarity relation of geometric coherence and path distinguishability with which the relationship between l 1 norm of coherence and geometric coherence is obtained finally with geometric coherence we study multiple copies of quantum state discrimination and give an example to show how to discriminate two pure states

estimating the number of peaks in a random field using the hadwiger characteristic of excursion sets with applications to medical images excursion sets has been studied by adler who defined the ig integral geometry characteristic of excursion sets as a multidimensional analogue of the number of upcrossings of the threshold by a unidimensional process the ig characteristic equals the euler characteristic of an excursion set provided that the set does not touch the boundary of the volume and adler found the expected ig characteristic for a stationary random field inside a fixed volume worsley evans marrett and neelin used the ig characteristic as an estimator of the number of regions of activation of positron emission tomography pet images of blood flow in the brain unfortunately the ig characteristic is only defined on intervals it is not invariant under rotations and it can fail to count connected regions that touch the boundary this is important since activation often occurs in the cortical regions near the boundary of the brain in this paper we study the hadwiger characteristic which is defined on arbitrary sets is invariant under rotations and does count connected regions whether they touch the boundary or not our main result is a simple expression for the expected hadwiger characteristic for an isotropic stationary random field in two and three dimensions and on a smooth surface embedded in three dimensions results are applied to pet studies of pain perception and word recognition

a graphical technique for determining the number of components in a mixture of normals abstract when a population is assumed to be composed of a finite number of subpopulations a natural model to choose is the finite mixture model it will often be the case however that the number of component distributions is unknown and must be estimated this problem can be difficult for instance the density of two mixed normals is not bimodal unless the means are separated by at least 2 standard deviations hence modality of the data per se can be an insensitive approach to component estimation we demonstrate that a mixture of two normals divided by a normal density having the same mean and variance as the mixed density is always bimodal this analytic result and other related results form the basis for a diagnostic and a test for the number of components in a mixture of normals the density is estimated using a kernel density estimator under the null hypothesis the proposed diagnostic can be approximated by a stationary gaussian process under the alternative hypothesis components in the mixtu