on variable bandwidth selection in local polynomial regression the performances of data driven bandwidth selection procedures in local polynomial regression are investigated by using asymptotic methods and simulation the bandwidth selection procedures considered are based on minimizing prelimit approximations to the conditional mean squared error mse when the mse is considered as a function of the bandwidth h we first consider approximations to the mse that are based on taylor expansions around h 0 of the bias part of the mse these approximations lead to estimators of the mse that are accurate only for small bandwidths h we also consider a bias estimator which instead of using small h approximations to bias naively estimates bias as the difference of two local polynomial estimators of different order and we show that this estimator performs well only for moderate to large h we next define a hybrid bias estimator which equals the taylor expansion based estimator for small h and the difference estimator for moderate to large h we find that the mse estimator based on this hybrid bias estimator leads to a bandwidth selection procedure with good asymptotic and for our monte carlo examples finite sample properties

seismic inversion based on proximal objective function optimization algorithm abstractseismic impedance inversion has become a common approach in reservoir prediction at present the critical issue in the application of seismic inversion is its low computational efficiency especially in 3d to improve the computational efficiency we have developed an inversion method derived from the proximal objective function optimization algorithm our inversion method calculates each unknown parameter in the model vector one by one during iteration compared with routine gradient dependent inversion algorithms such as the iteratively reweighted least squares irls algorithm our inversion method has lower computational complexity as well as higher efficiency in addition to obtain a sparse reflectivity series a long tailed cauchy distribution is used as the a priori constraint the weak nonlinear problem owing to the introduction of cauchy sparse constraint is addressed by taking advantage of reweighting strategy results of synthetic and real data tests illustrate that the proposed inv

dynamics of one and two dimensional fronts in a bistable equation with time delayed global feedback propagation failure and control mechanisms we study the evolution of fronts in a bistable equation with time delayed global feedback in the fast reaction and slow diffusion regime this equation generalizes the hodgkin grafstein and allen cahn equations we derive a nonlinear equation governing the motion of fronts which includes a term with delay in the one dimensional case this equation is linear we study the motion of one and two dimensional fronts finding a much richer dynamics than for the previously studied cases without time delayed global feedback we explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a hopf bifurcation as the delay time increases we show that for certain delay times the prevailing phase is different from that corresponding to the system in the absence of global coupling numerical simulations of the partial differential equation are in agreement with the analytical predictions

wave propagation in a fitzhugh nagumo type model with modified excitability we examine a generalized fitzhugh nagumo fhn type model with modified excitability derived from the diffusive morris lecar equations for neuronal activity we obtain exact analytic solutions in the form of traveling waves using a piecewise linear approximation for the activator and inhibitor reaction terms we study the existence and stability of waves and find that the inhibitor species exhibits different types of wave forms fronts and pulses while the activator wave maintains the usual kink front shape the nonequilibrium ising bloch bifurcation for the wave speed that occurs in the fhn model where the control parameter is the ratio of inhibitor to activator time scales persists when the strength of the inhibitor nonlinearity is taken as the bifurcation parameter

properties of synchronization in the systems of non identical coupled van der pol and van der pol duffing oscillators broadband synchronization the particular properties of dynamics are discussed for dissipatively coupled van der pol oscillators non identical in values of parameters controlling the hopf bifurcation possibility of a special synchronization regime in an infinitively long band between oscillator death and quasiperiodic areas is shown for such system features of the bifurcation picture are discussed for different values of the control parameters and for the case of additional duffing type nonlinearity analysis of the slow flow equations is presented

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

the multivariate l1 median and associated data depth this paper gives three related results i a new simple fast monotonically converging algorithm for deriving the l1 median of a data cloud in d a problem that can be traced to fermat and has fascinated applied mathematicians for over three centuries ii a new general definition for depth functions as functions of multivariate medians so that different definitions of medians will correspondingly give rise to different dept functions and iii a simple closed form formula of the l1 depth function for a given data cloud in d

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

families of non linear subdivision schemes for scattered data fitting and their non tensor product extensions abstract in this article families of non linear subdivision schemes are presented that are based on univariate polynomials up to degree three these families of schemes are constructed by using dynamic iterative re weighed least squares method these schemes are suitable for fitting noisy data with outliers the codes are designed in a python environment to numerically fit the given data points although these schemes are non interpolatory but have the ability to preserve the shape of the initial polygon in case of non noisy initial data the numerical examples illustrate that the schemes constructed by non linear polynomials give better performance than the schemes that are constructed by linear polynomials mustafa et al 2015 moreover the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data furthermore families of non linear bivariate subdivision schemes are also presented that are based on linear and non linear bivariate polynomials

a modified weiszfeld algorithm for the fermat weber location problem this paper gives a new simple monotonically convergent algorithm for the fermat weber location problem with extensions covering more general cost functions

a canard mechanism for localization in systems of globally coupled oscillators localization in a discrete system of oscillators refers to the partition of the population into a subset that oscillates at high amplitudes and anotherthat oscillates at much loweramplitudes motivated by experimental results on the belousov zhabotinsky reaction which oscillates in the relaxation regime we study a mechanism of localization in a discrete system of relaxation oscillators globally coupled via inhibition the mechanism is based on the canard phenomenon for a single relaxation oscillator a rapid explosion in the amplitude of the limit cycle as a parameter governing the relative position of the nullclines is varied starting from a parameter regime in which each uncoupled oscillatorhas a lar ge amplitude and no otherper iodic orotherstable solutions we show that the canard phenomenon can be induced by increasing a global negative feedback parameter with the network then partitioned into low and high amplitude oscillators for the case in which the oscillators are synchronous within each of the two such populations we can assign a canard inducing critical value of separately to each of the two clusters localization occurs when the value for the system is between the critical values of the two clusters we show that the larger the cluster size the smaller is the corresponding critical value of implying that it is the smallerclusterthat oscillates at large amplitude the theory shows that the above results come from a kind of self inhibition of each cluster induced by the local feedback in the full system there are also effects of interactions between the clusters and we present simulations showing that these nonlocal interactions do not destroy the localization created by the self inhibition

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

classification by pairwise coupling we discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes and then coupling the estimates together the coupling model is similar to the bradley terry method for paired comparisons we study the nature of the class probability estimates that arise and examine the performance of the procedure in real and simulated data sets classifiers used include linear discriminants nearest neighbors adaptive nonlinear methods and the support vector machine

projection pursuit density estimation abstract the projection pursuit methodology is applied to the multivariate density estimation problem the resulting nonparametric procedure is often less biased than the kernel and near neighbor methods in addition graphical information is produced that can be used to help gain geometric insight into the multivariate data distribution

estimation and inference in two stage semi parametric models of production processes many papers have regressed non parametric estimates of productive efficiency on environmental variables in two stage procedures to account for exogenous factors that might affect firms performance none of these have described a coherent data generating process dgp moreover conventional approaches to inference employed in these papers are invalid due to complicated unknown serial correlation among the estimated efficiencies we first describe a sensible dgp for such models we propose single and double bootstrap procedures both permit valid inference and the double bootstrap procedure improves statistical efficiency in the second stage regression we examine the statistical performance of our estimators using monte carlo experiments

spatial median and directional data summary we introduce the normalized spatial median as an estimator of location for rotationally symmetric distributions on the hypersphere we investigate some of its asymptotic properties and use them to obtain confidence regions for the modal direction of a distribution on the hypersphere these results are then applied to the von mises fisher distribution and to a contamination model it is seen that the normalized spatial median can perform more efficiently than the normalized mean in presence of outliers

synchronization of weakly coupled canard oscillators synchronization has been studied extensively in the context of weakly coupled oscillators using the so called phase response curve prc which measures how a change of the phase of an oscillator is affected by a small perturbation this approach was based upon the work of malkin and it has been extended to relaxation oscillators namely synchronization conditions were established under the weak coupling assumption leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points where canard solutions can arise in the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles in particular we highlight a new special role of the maximal canard in separating two distinct synchronization regimes the hopf regime and the relaxation regime phase plane analysis of slow fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

bump hunting with non gaussian kernels it is well known that the number of modes of a kernel density estimator is monotone nonincreasing in the bandwidth if the kernel is a gaussian density there is numerical evidence of nonmonotonicity in the case of some non gaussian kernels but little additional information is available the present paper provides theoretical and numerical descriptions of the extent to which the number of modes is a nonmonotone function of bandwidth in the case of general compactly supported densities our results address popular kernels used in practice for example the epanechnikov biweight and triweight kernels and show that in such cases nonmonotonicity is present with strictly positive probability for all sample sizes n 3 in the epanechnikov and biweight cases the probability of nonmonotonicity equals 1 for all n 2 nevertheless in spite of the prevalence of lack of monotonicity revealed by these results it is shown that the notion of a critical bandwidth the smallest bandwidth above which the number of modes is guaranteed to be monotone is still well defined moreover just as in the gaussian case the critical bandwidth is of the same size as the bandwidth that minimises mean squared error of the density estimator these theoretical results and new numerical evidence show that the main effects of nonmonotonicity occur for relatively small bandwidths and have negligible impact on many aspects of bump hunting jj

estimating a unimodal distribution from interval censored data in this article we consider three nonparametric maximum likelihood estimators based on mixed case interval censored data apart from the unrestricted estimator we consider estimators under the assumption that the underlying distribution function of event times is concave or unimodal characterizations of the estimates are derived and algorithms are proposed for their computation the estimators are shown to be asymptotically consistent and the benefits of additional constraints are illustrated through simulations finally the estimators are used as an ingredient in a nonparametric comparison of two samples

statistical uses of the spatial median the spatial median called the mediancentre in an earlier paper by j c gower is defined as the bivariate location measure to minimize the sum of absolute distances to observations its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2 ir 0 637 its estimating equations have an angular aspect and are used to develop angle tests which are analogues of sign tests in one dimension

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

comparison of data driven bandwidth selectors abstract this article compares several promising data driven methods for selecting the bandwidth of a kernel density estimator the methods compared are least squares cross validation biased cross validation and a plug in rule the comparison is done by asymptotic rate of convergence to the optimum and a simulation study it is seen that the plug in bandwidth is usually most efficient when the underlying density is sufficiently smooth but is less robust when there is not enough smoothness present we believe the plug in rule is the best of those currently available but there is still room for improvement

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

on the convergence of alternating minimization for convex programming with applications to iteratively reweighted least squares and decomposition schemes this paper is concerned with the alternating minimization am method for solving convex minimization problems where the decision variables vector is split into two blocks the objective function is a sum of a differentiable convex function and a separable possibly nonsmooth extended real valued convex function and consequently constraints can be incorporated we analyze the convergence rate of the method and establish a nonasymptotic sublinear rate of convergence where the multiplicative constant depends on the minimal block lipschitz constant we then analyze the iteratively reweighted least squares irls method for solving convex problems involving sums of norms based on the results derived for the am method we establish a nonasymptotic sublinear rate of convergence of the irls method in addition we show an asymptotic rate of convergence whose efficiency estimate does not depend on the data of the problem finally we study the convergence properties of a decomposition based approach designed t

the discrimination power of projection pursuit with different density estimators summary we explore the properties of projection pursuit discriminant analysis this discriminant method is very powerful but relies heavily on a univariate density estimate we show that the procedure based on wavelets maintains the same rate of convergence as with univariate wavelet density estimation we also show the bayes risk strong consistency of both the kernel and wavelet based methods simulated data and real data concerning character recognition show that the method is effective and robust against the curse of dimensionality the wavelet alternative seems more likely than the kernel counterpart to find an interesting projection wavelets are often criticised for giving too wiggly an estimate and for being too localised to give good global properties in the above context these potential drawbacks do not weaken the method but the use of wavelets seems to enhance it a multiple projection generalisation is also considered

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

localized and asynchronous patterns via canards in coupled calcium oscillators in this paper we consider the mechanism for localized behavior in coupled calcium oscillators described by the canonical two pool model localization occurs when the individual cells oscillate with amplitudes of different orders of magnitude our analysis and computations show that a combination of diffusive coupling heterogeneity and the underlying canard structure of the oscillators all contribute to the localized behavior two key quantities characterize the different states of the system by representing the effects of both the autonomous and the non autonomous terms which are due to the coupling by highlighting the influence of the canard phenomenon these quantities identify stabilizing and destabilizing effects of the coupling on the localized behavior we compare our analysis with computations describing multi mode states and asynchronized large oscillations in addition to the localized states

applied smoothing techniques for data analysis the kernel approach with s plus illustrations 1 density estimation for exploring data 2 density estimation for inference 3 nonparametric regression for exploring data 4 inference with nonparametric regression 5 checking parametric regression models 6 comparing regression curves and surfaces 7 time series data 8 an introduction to semiparametric and additive models references

structural source identification using a generalized tikhonov regularization abstract this paper addresses the problem of identifying mechanical exciting forces from vibration measurements the proposed approach is based on a generalized tikhonov regularization that allows taking into account prior information on the measurement noise as well as on the main characteristics of sources to identify like its sparsity or regularity to solve such a regularization problem efficiently a generalized iteratively reweighted least squares girls algorithm is introduced proposed numerical and experimental validations reveal the crucial role of prior information in the quality of the source identification and the performance of the girls algorithm

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

sungular hopf bifurcation to relaxation oscillations this paper is a continuation of our study of singular hopf bifurcations to relaxation oscillations siam journal on applied mathematics 46 1986 pp 721 739 in our previous analysis a system of two first order nonlinear equatins is investigated and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first order equations in this paper a method based on analyzing nearly conservative quantities was developed to find the bifurcation diagram of the periodic solutions emerging from singular hopf bifurcation points specifically the subcritical hopf bifurcation of the fitzhugh nagumo equations is analyzed the extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading order nonlinear oscillator at a critical value of a parameter the first integral exists up to an additional order in the perturbed equation this condition permits the resolution of features determined by h