on local smoothing of nonparametric curve estimators abstract we develop new local versions of familiar smoothing methods such as cross validation and smoothed cross validation in the contexts of density estimation and regression these new methods are locally adaptive in the sense that they capture smooth local fluctuations in the curve by using smoothly varying bandwidths that change as the character of the curve changes moreover the new methods are accurate easy to apply and computationally expedient

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

a unified formulation and fast accelerated proximal gradient method for classification binary classification is the problem of predicting the class a given sample belongs to to achieve a good prediction performance it is important to find a suitable model for a given dataset however it is often time consuming and impractical for practitioners to try various classification models because each model employs a different formulation and algorithm the difficulty can be mitigated if we have a unified formulation and an efficient universal algorithmic framework for various classification models to expedite the comparison of performance of different models for a given dataset in this paper we present a unified formulation of various classification models including c svm l2 svm svm mm fda mm mpm logistic regression distance weighted discrimination and develop a general optimization algorithm based on an accelerated proximal gradient apg method for the formulation we design various techniques such as backtracking line search and adaptive restarting strategy in order to speed up the practical convergence of our method we also give a theoretical convergence guarantee for the proposed fast apg algorithm numerical experiments show that our algorithm is stable and highly competitive to specialized algorithms designed for specific models e g sequential minimal optimization smo for svm

smoothed cross validation for bandwidth selection of a kernel density estimator a generalization of the widely studied least squares cross validation method is considered the essential idea is to do a particular type of presmoothing of the data this is seen to be essentially the same as using the smoothed bootstrap estimate of the mean integrated squared error analysis reveals that a rather large amount of presmoothing yields excellent asymptotic performance the rate of convergence to the optimum is known to be best possible under a wide range of smoothness conditions the method is more appealing than other selectors with this property because its motivation is not heavily dependent on precise asymptotic analysis and because its form is simple and intuitive theory is also given for choice of the amount of presmoothing and this is used to derive a data based method for this choice

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

stochastic models for closed boundary analysis representation and reconstruction the analysis of closed boundaries of arbitrary shapes on a plane is discussed specifically the problems of representation and reconstruction are considered a one to one correspondence between the given closed boundary and a univariate or multivariate sequence of real numbers is set up univariate or multivariate circular autoregressive models are suggested for the representation of the sequence of numbers derived from the closed boundary the stochastic model representing the closed boundary is invariant to transformations like sealing translation choice of starting point and rotation over angles that are multiples of 2 pi n where n is the number of observations methods for estimating the unknown parameters of the model are given and a decision rule for choosing the appropriate number of coefficients is included constraints on the estimates are derived so that the estimates are invariant to the transformations of the boundaries the stochastic model enables the reconstruction of a dosed boundary using fft algorithms results of simulations are included and the application to contour coding is discussed

local bandwidth selection for kernel estimation of population densities with line transect sampling summary seber 1986 biometrics42 267 292 suggested an approach to biological population density estimation using kernel estimates of the probability density of detection distances in line transect sampling chen 1996a applied statistics45 135 150 and others have employed cross validation to choose a global bandwidth for the kernel estimator or have suggested adaptive kernel estimation chen 1996b biometrics52 1283 1294 because estimation of the density is required at only a single point we investigate a local bandwidth selection procedure that is a modification of the method of schucany 1995 journal of the american statistical association90 535 540 for nonparametric regression we report on simulation results comparing the proposed method and a local normal scale rule with cross validation and adaptive estimation the local bandwidths and normal scale rule produce estimates with mean squares that are half the size of the others in most cases consistency results are also provided

multicategory ψ learning in binary classification margin based techniques usually deliver high performance as a result a multicategory problem is often treated as a sequence of binary classifications in the absence of a dominating class this treatment may be suboptimal and may yield poor performance such as for support vector machines svms we propose a novel multicategory generalization of learning that treats all classes simultaneously the new generalization eliminates this potential problem while at the same time retaining the desirable properties of its binary counterpart we develop a statistical learning theory for the proposed methodology and obtain fast convergence rates for both linear and nonlinear learning examples we demonstrate the operational characteristics of this method through a simulation our results indicate that the proposed methodology can deliver accurate class prediction and is more robust against extreme observations than its svm counterpart

another look at distance weighted discrimination summary distance weighted discrimination dwd is a modern margin based classifier with an interesting geometric motivation it was proposed as a competitor to the support vector machine svm despite many recent references on dwd dwd is far less popular than the svm mainly because of computational and theoretical reasons we greatly advance the current dwd methodology and its learning theory we propose a novel thrifty algorithm for solving standard dwd and generalized dwd and our algorithm can be several hundred times faster than the existing state of the art algorithm based on second order cone programming in addition we exploit the new algorithm to design an efficient scheme to tune generalized dwd furthermore we formulate a natural kernel dwd approach in a reproducing kernel hilbert space and then establish the bayes risk consistency of the kernel dwd by using a universal kernel such as the gaussian kernel this result solves an open theoretical problem in the dwd literature a comparison study on 16 benchmark data sets shows that data driven generalized dwd consistently delivers higher classification accuracy with less computation time than the svm

indirect cross validation for density estimation a new method of bandwidth selection for kernel density estimators is proposed the method termed indirect cross validation icv makes use of so called selection kernels least squares cross validation lscv is used to select the bandwidth of a selection kernel estimator and this bandwidth is appropriately rescaled for use in a gaussian kernel estimator the proposed selection kernels are linear combinations of two gaussian kernels and need not be unimodal or positive a theory is developed showing that the relative error of icv bandwidths can converge to 0 at a rate of n 1 4 which is substantially better than the n 1 10 rate of lscv interestingly the selection kernels that are best for purposes of bandwidth selection are very poor if used to actually estimate the density function this property appears to be part of the larger and well documented paradox to the effect that the harder the estimation problem the better cross validation performs the icv method uniformly outperforms lscv in a simul

solution path for pin svm classifiers with positive and negative tau values applying the pinball loss in a support vector machine svm classifier results in pin svm the pinball loss is characterized by a parameter tau its value is related to the quantile level and different tau values are suitable for different problems in this paper we establish an algorithm to find the entire solution path for pin svm with different tau values this algorithm is based on the fact that the optimal solution to pin svm is continuous and piecewise linear with respect to tau we also show that the nonnegativity constraint on tau is not necessary i e tau can be extended to negative values first in some applications a negative tau leads to better accuracy second tau 1 corresponds to a simple solution that links svm and the classical kernel rule the solution for tau 1 can be obtained directly and then be used as a starting point of the solution path the proposed method efficiently traverses tau values through the solution path and then achieves good performance by a suitable tau in particular tau 0 corresponds to c svm meaning that the traversal algorithm can output a result at least as good as c svm with respect to validation error

generalized self concordant functions a recipe for newton type methods we study the smooth structure of convex functions by generalizing a powerful concept so called self concordance introduced by nesterov and nemirovskii in the early 1990s to a broader class of convex functions which we call generalized self concordant functions this notion allows us to develop a unified framework for designing newton type methods to solve convex optimization problems the proposed theory provides a mathematical tool to analyze both local and global convergence of newton type methods without imposing unverifiable assumptions as long as the underlying functionals fall into our class of generalized self concordant functions first we introduce the class of generalized self concordant functions which covers the class of standard self concordant functions as a special case next we establish several properties and key estimates of this function class which can be used to design numerical methods then we apply this theory to develop several newton type methods for solving a class of smooth convex optimization problems involving generalized self concordant functions we provide an explicit step size for a damped step newton type scheme which can guarantee a global convergence without performing any globalization strategy we also prove a local quadratic convergence of this method and its full step variant without requiring the lipschitz continuity of the objective hessian mapping then we extend our result to develop proximal newton type methods for a class of composite convex minimization problems involving generalized self concordant functions we also achieve both global and local convergence without additional assumptions finally we verify our theoretical results via several numerical examples and compare them with existing methods

incorporating prior probabilities into high dimensional classifiers in standard parametric classifiers or classifiers based on nonparametric methods but where there is an opportunity for estimating population densities the prior probabilities of the respective populations play a key role however those probabilities are largely ignored in the construction of high dimensional classifiers partly because there are no likelihoods to be constructed or bayes risks to be estimated nevertheless including information about prior probabilities can reduce the overall error rate particularly in cases where doing so is most important i e when the classification problem is particularly challenging and error rates are not close to zero in this paper we suggest a general approach to reducing error rate in this way by using a method derived from breiman s bagging idea the potential improvements in performance are identified in theoretical and numerical work the latter involving both applications to real data and simulations the method is simple and explicit to apply and does not involve choice of any tuning parameters

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

confidence intervals for nonparametric curve estimates numerous nonparametric regression methods exist that yield consistent estimators of function curves often one is also interested in constructing confidence intervals for the unknown function when a function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 on average but because bias in nonparametric estimation is not uniform they do not hold the desired level uniformly at all design points most research in this area has focused on mean squared error properties of the estimator for example mise itself a global measure in addition measures like mise are one step removed from the practical issue of coverage probability recent work that focuses on coverage probability has considered only coverage in an average sense ignoring the important issue of uniformity of coverage across the design space to deal with this problem a new estimator is developed which uses a local cross validation criterion lcv to determi

machine learning with data dependent hypothesis classes we extend the vc theory of statistical learning to data dependent spaces of classifiers this theory can be viewed as a decomposition of classifier design into two components the first component is a restriction to a data dependent hypothesis class and the second is empirical risk minimization within that class we define a measure of complexity for data dependent hypothesis classes and provide data dependent versions of bounds on error deviance and estimation error we also provide a structural risk minimization procedure over data dependent hierarchies and prove consistency we use this theory to provide a framework for studying the trade offs between performance and computational complexity in classifier design as a consequence we obtain a new family of classifiers with dimension independent performance bounds and efficient learning procedures

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

transformations in density estimation abstract for the density estimation problem the global window width kernel density estimator does not perform well when the underlying density has features that require different amounts of smoothing at different locations in this article we propose to transform the data with the intention that a global window width is more appropriate for the density of the transformed data the density estimate of the original data is the back transform by change of variables of the global window width estimate of the transformed data s density we explore choosing the transformation from suitable parametric families data based selection rules for the choice of transformations and the window width are discussed application to real and simulated data demonstrates the usefulness of our proposals

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

wavelet transforms versus fourier transforms this note is a very basic introduction to wavelets it starts with an orthogonal basis of piecewise constant functions constructed by dilation and translation the wavelet transform maps each f x to its coefficients with respect to this basis the mathematics is simple and the transform is fast faster than the fast fourier transform which we briefly explain but approximation by piecewise constants is poor to improve this first wavelet we are led to dilation equations and their unusual solutions higher order wavelets are constructed and it is surprisingly quick to compute with them always indirectly and recursively

scale adjustments for classifiers in high dimensional low sample size settings distance based classifiers are generally considered to be effective at discriminating between populations that differ in location indeed nearest neighbour methods and the support vector machine are frequently used in very high dimensional problems involving gene expression data where it is believed that elevated levels of expression convey much of the information for classification however one problem inherent to distance based classifiers is that scale differences can mask location differences in consequence such classifiers can have poor performance if the information for classification accumulates through a large number of relatively small location differences in data components rather than via large differences in this paper we show that a simple adjustment for scale applicable to a variety of distance based classifiers can remedy the problem for some classifiers such as those based on the support vector machine or the centroid method scale corrections are important primarily in the case of small training sample sizes however for other classifiers including those based on nearest neighbour and average distance methods scale adjustments are helpful more generally

asymptotic behavior of support vector machine for spiked population model for spiked population model we investigate the large dimension n and large sample size m asymptotic behavior of the support vector machine svm classification method in the limit of n m at fixed m n we focus on the generalization performance by analytically evaluating the angle between the normal direction vectors of svm separating hyperplane and corresponding bayes optimal separating hyperplane this is an analogous result to the one shown in paul 2007 and nadler 2008 for the angle between the sample eigenvector and the population eigenvector in random matrix theorem we provide not just bound but sharp prediction of the asymptotic behavior of svm that can be determined by a set of nonlinear equations based on the analytical results we propose a new method of selecting tuning parameter which significantly reduces the computational cost a surprising finding is that svm achieves its best performance at small value of the tuning parameter under spiked population model these results are confirmed to be correct by comparing with those of numerical simulations on finite size systems we also apply our formulas to an actual dataset of breast cancer and find agreement between analytical derivations and numerical computations based on cross validation

distance weighted discrimination high dimension low sample size statistical analysis is becoming increasingly important in a wide range of applied contexts in such situations the popular support vector machine suffers from data piling at the margin which can diminish generalizability this leads naturally to the development of distance weighted discrimination which is based on second order cone programming a modern computationally intensive optimization method

analysis of principal nested spheres a general framework for a novel non geodesic decomposition of high dimensional spheres or high dimensional shape spaces for planar landmarks is discussed the decomposition principal nested spheres leads to a sequence of submanifolds with decreasing intrinsic dimensions which can be interpreted as an analogue of principal component analysis in a number of real datasets an apparent one dimensional mode of variation curving through more than one geodesic component is captured in the one dimensional component of principal nested spheres while analysis of principal nested spheres provides an intuitive and flexible decomposition of the high dimensional sphere an interesting special case of the analysis results in finding principal geodesics similar to those from previous approaches to manifold principal component analysis an adaptation of our method to kendall s shape space is discussed and a computational algorithm for fitting principal nested spheres is proposed the result provides a coordinate system to visualize the data structure and an intuitive summary of principal modes of variation as exemplified by several datasets

a bivariate autoregressive technique for analysis and classification of planar shapes a bivariate autoregressive model is introduced for the analysis and classification of closed planar shapes the boundary coordinate sequence of a digitized binary image is sampled to produce a polygonal approximation to an object s shape this circular sample sequence is then represented by a vector autoregressive difference equation which models the individual cartesian coordinate sequences as well as coordinate interdependencies several classification features which are functions or transformations of the estimated coefficient matrices and the associated residual error covariance matrices are developed these features are shown to be invariant to object transformations such as translation rotation and scaling laboratory experiments involving object sets representative of industrial shapes are presented superior classification results are demonstrated

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

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

on locally adaptive density estimation abstract theoretical and practical aspects of the sample point adaptive positive kernel density estimator are examined a closed form expression for the mean integrated squared error is obtained through the device of preprocessing the data by binning with this expression the exact behavior of the optimally adaptive smoothing parameter function is studied for the first time the approach differs from most earlier techniques in that bias of the adaptive estimator remains o h 2 and is not improved to the rate o h 4 a practical algorithm is constructed using a modification of least squares cross validation simulated and real examples are presented including comparisons with a fixed bandwidth estimator and a fully automatic version of abramson s adaptive estimator the results are very promising

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

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

flexible high dimensional classification machines and their asymptotic properties classification is an important topic in statistics and machine learning with great potential in many real applications in this paper we investigate two popular large margin classification methods support vector machine svm and distance weighted discrimination dwd under two contexts the high dimensional low sample size data and the imbalanced data a unified family of classification machines the flexible assortment machine flame is proposed within which dwd and svm are special cases the flame family helps to identify the similarities and differences between svm and dwd it is well known that many classifiers overfit the data in the high dimensional setting and others are sensitive to the imbalanced data that is the class with a larger sample size overly influences the classifier and pushes the decision boundary towards the minority class svm is resistant to the imbalanced data issue but it overfits high dimensional data sets by showing the undesired data piling phenomenon the dwd method was proposed to improve svm in the high dimensional setting but its decision boundary is sensitive to the imbalanced ratio of sample sizes our flame family helps to understand an intrinsic connection between svm and dwd and provides a trade off between sensitivity to the imbalanced data and overfitting the high dimensional data several asymptotic properties of the flame classifiers are studied simulations and real data applications are investigated to illustrate theoretical findings

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

geometric representation of high dimension low sample size data summary high dimension low sample size data are emerging in various areas of science we find a common structure underlying many such data sets by using a non standard type of asymptotics the dimension tends to while the sample size is fixed our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex essentially all the randomness in the data appears only as a random rotation of this simplex this geometric representation is used to obtain several new statistical insights

object oriented data analysis of cell images this article discusses a study of cell images in cell culture biology from an object oriented point of view the motivation of this research is to develop a statistical approach to cell image analysis that better supports the automated development of stem cell growth media a major hurdle in this process is the need for human expertise based on studying cells under the microscope to make decisions about the next step of the cell culture process we aim to use digital imaging technology coupled with statistical analysis to tackle this important problem the discussion in this article highlights a common critical issue choice of data objects instead of conventionally treating either the individual cells or the wells a container in which the cells are grown as data objects a new type of data object is proposed that is the union of a well with its corresponding set of cells the image data analysis suggests that the cell well unions can be a better choice of data objects than the cells or the wells al

formulae for mean integrated squared error of nonlinear wavelet based density estimators we provide an asymptotic formula for the mean integrated squared error mise of nonlinear wavelet based density estimators we show that unlike the analogous situation for kernel density estimators this mise formula is relatively unaffected by assumptions of continuity in particular it is available for densities which are smooth in only a piecewise sense another difference is that in the wavelet case the classical mise formula is valid only for sufficiently small values of the bandwidth for larger bandwidths mise assumes a very different form and hardly varies at all with changing bandwidth this remarkable property guarantees a high level of robustness against oversmoothing not encountered in the context of kernel methods we also use the mise formula to describe an asymptotically optimal empirical bandwidth selection rule

complex autoregressive model for shape recognition a complex autoregressive model for invariant feature extraction to recognize arbitrary shapes on a plane is presented a fast algorithm to calculate complex autoregressive coefficients and complex parcor coefficients of the model is also shown the coefficients are invariant to rotation around the origin and to choice of the starting point in tracing a boundary it is possible to make them invariant to scale and translation experimental results that the complicated shapes like nonconvex boundaries can be recognized in high accuracy even in the low order model it is seen that the complex parcor coefficients tend to provide more accurate classification than the complex ar coefficients

the high dimension low sample size geometric representation holds under mild conditions summary high dimension low small sample size datasets have different geometrical properties from those of traditional low dimensional data in their asymptotic study regarding increasing dimensionality with a fixed sample size hall et al 2005 showed that each data vector is approximately located on the vertices of a regular simplex in a high dimensional space a perhaps unappealing aspect of their result is the underlying assumption which requires the variables viewed as a time series to be almost independent we establish an equivalent geometric representation under much milder conditions using asymptotic properties of sample covariance matrices we discuss implications of the results such as the use of principal component analysis in a high dimensional space extension to the case of nonindependent samples and also the binary classification problem

invariant distance measures for planar shapes based on complex autoregressive model abstract several distance measures euclidean distances between complex autoregressive coefficients or complex partial correlation coefficients log likelihood distance and complex power cepstrum distance between planar shapes are presented on the basis of a complex autoregressive model the measures are invariant to translation rotation and scale of patterns since the concept of shape is essentially by independent of such transformations of patterns these measures are suitable for classification identification or clustering of planar shapes like unsupervised applications the properties of the proposed distance measures are shown with their experimental comparisons

hyperdisk based large margin classifier we introduce a large margin linear binary classification framework that approximates each class with a hyperdisk the intersection of the affine support and the bounding hypersphere of its training samples in feature space and then finds the linear classifier that maximizes the margin separating the two hyperdisks we contrast this with support vector machines svms which find the maximum margin separator of the pointwise convex hulls of the training samples arguing that replacing convex hulls with looser convex class models such as hyperdisks provides safer margin estimates that improve the accuracy on some problems both the hyperdisks and their separators are found by solving simple quadratic programs the method is extended to nonlinear feature spaces using the kernel trick and multi class problems are dealt with by combining binary classifiers in the same ways as for svms experiments on a range of data sets show that the method compares favourably with other popular large margin classifiers