optimal eigen expansions and uniform bounds let x k k in mathbb z in mathbb l 2 mathcal t be a stationary process with associated lag operators varvec mathcal c h uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues spectral gap in addition the underlying dependence assumptions are optimal in a certain sense including both short and long memory processes this allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions among other things convergence to an extreme value distribution is shown we also discuss how the asymptotic expansions transfer to the long run covariance operator varvec mathcal g in a general framework

the problem of moments on polytopes and other bounded regions abstract the problem of moments is considered for polytopes in r d and other bounded regions specified piecewise by curves of the form c1x1 1 cdxd d c0 necessary and sufficient conditions are suggested by a simple geometric observation for some choices of c1 cd and 1 d these conditions reduce to the conditions of hausdorff for the unit interval of hildebrandt and schoenberg for the unit square and of dale for the unit simplex in r 2 whereas for other choices new conditions are required the proofs are obtained using probabilistic techniques

inductive robust principal component analysis in this paper we address the error correction problem that is to uncover the low dimensional subspace structure from high dimensional observations which are possibly corrupted by errors when the errors are of gaussian distribution principal component analysis pca can find the optimal in terms of least square error low rank approximation to high dimensional data however the canonical pca method is known to be extremely fragile to the presence of gross corruptions recently wright established a so called robust principal component analysis rpca method which can well handle the grossly corrupted data however rpca is a transductive method and does not handle well the new samples which are not involved in the training procedure given a new datum rpca essentially needs to recalculate over all the data resulting in high computational cost so rpca is inappropriate for the applications that require fast online computation to overcome this limitation in this paper we propose an inductive robust principal component analysis irpca method given a set of training data unlike rpca that targets on recovering the original data matrix irpca aims at learning the underlying projection matrix which can be used to efficiently remove the possible corruptions in any datum the learning is done by solving a nuclear norm regularized minimization problem which is convex and can be solved in polynomial time extensive experiments on a benchmark human face dataset and two video surveillance datasets show that irpca cannot only be robust to gross corruptions but also handle the new data well and in an efficient way

time average control of martingale problems a linear programming formulation one studies the average cost for controlled systems given as solutions of the martingale problem for their generator the control problem is reformulated as a linear programming problem and conditions are given for the existence of an optimal solution it is further shown that the optimal control can be taken to depend only on the history of the system and that this cost remains optimal for systems with different initial distributions

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

principal components and orthogonal regression based on robust scales both principal components analysis pca and orthogonal regression deal with finding a p dimensional linear manifold minimizing a scale of the orthogonal distances of the m dimensional data points to the manifold the main conceptual difference is that in pca p is estimated from the data to attain a small proportion of unexplained variability whereas in orthogonal regression p equals m 1 the two main approaches to robust pca are using the eigenvectors of a robust covariance matrix and searching for the projections that maximize or minimize a robust univariate dispersion measure this article is more akin to second approach but rather than finding the components one by one we directly undertake the problem of finding for a given p a p dimensional linear manifold minimizing a robust scale of the orthogonal distances of the data points to the manifold the scale may be either a smooth m scale or a trimmed scale an iterative algorithm is developed that is shown to converge to a local minimum a

online estimation of the geometric median in hilbert spaces nonasymptotic confidence balls estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with very large samples of high dimensional data the collected data may be contaminated by noise so that robust location indicators such as the geometric median may be preferred to the mean in this context an estimator of the geometric median based on a fast and efficient averaged nonlinear stochastic gradient algorithm has been developed by bernoulli19 2013 18 43 this work aims at studying more precisely the nonasymptotic behavior of this nonlinear algorithm by giving nonasymptotic confidence balls in general separable hilbert spaces this new result is based on the derivation of improved l 2 rates of convergence as well as an exponential inequality for the nearly martingale terms of the recursive nonlinear robbins monro algorithm

dispersion operators and resistant second order functional data analysis inferences related to the second order properties of functional data as expressed by covariance structure can become unreliable when the data are non gaussian or contain unusual observations in the functional setting it is often difficult to identify atypical observations as their distinguishing characteristics can be manifold but subtle in this paper we introduce the notion of a dispersion operator investigate its use in probing the second order structure of functional data and develop a test for comparing the second order characteristics of two functional samples that is resistant to atypical observations and departures from normality the proposed test is a regularized m test based on a spectrally truncated version of the hilbert schmidt norm of a score operator defined via the dispersion operator we derive the asymptotic distribution of the test statistic investigate the behaviour of the test in a simulation study and illustrate the method on a structural biology dataset

detecting and localizing differences in functional time series dynamics a case study in molecular biophysics abstractmotivated by the problem of inferring the molecular dynamics of dna in solution and linking them with its base pair composition we consider the problem of comparing the dynamics of functional time series fts and of localizing any inferred differences in frequency and along curvelength the approach we take is one of fourier analysis where the complete second order structure of the fts is encoded by its spectral density operator indexed by frequency and curvelength the comparison is broken down to a hierarchy of stages at a global level we compare the spectral density operators of the two fts across frequencies and curvelength based on a hilbert schmidt criterion then we localize any differences to specific frequencies and finally we further localize any differences along the length of the random curves that is in physical space a hierarchical multiple testing approach guarantees control of the averaged false discovery rate over the selected frequencies in this sense we are ab

computing moments of the exit time distribution for markov processes by linear programming we provide a new approach to the numerical computation of moments of the exit time distribution of markov processes the method relies on a linear programming formulation of a process exiting from a bounded domain the lp formulation characterizes the evolution of the process through the moments of the induced occupation measure and naturally provides upper and lower bounds for the exact values of the moments the conditions the moments have to satisfy are derived directly from the generator of the markov process and are not based on some approximation of the process excellent software is readily available because the computations involve finite dimensional linear programs

detection of change in the spatiotemporal mean function summary the paper develops inferential methodology for detecting a change in the annual pattern of an environmental variable measured at fixed locations in a spatial region using a framework built on functional data analysis we model observations as a collection of function valued time sequences available at many sites each sequence is modelled as an annual mean function which may change plus a sequence of error functions which are spatially correlated the tests statistics extend the cumulative sum paradigm to this more complex setting their asymptotic distributions are not parameter free because of the spatial dependence but can be effectively approximated by monte carlo simulations the new methodology is applied to precipitation data its finite sample performance is assessed by a simulation study

interpretable functional principal component analysis functional principal component analysis fpca is a popular approach to explore major sources of variation in a sample of random curves these major sources of variation are represented by functional principal components fpcs the intervals where the values of fpcs are significant are interpreted as where sample curves have major variations however these intervals are often hard for naive users to identify because of the vague definition of significant values in this article we develop a novel penalty based method to derive fpcs that are only nonzero precisely in the intervals where the values of fpcs are significant whence the derived fpcs possess better interpretability than the fpcs derived from existing methods to compute the proposed fpcs we devise an efficient algorithm based on projection deflation techniques we show that the proposed interpretable fpcs are strongly consistent and asymptotically normal under mild conditions simulation studies confirm that with a competitive performance in explaining variations of sample curves the proposed fpcs are more interpretable than the traditional counterparts this advantage is demonstrated by analyzing two real datasets namely electroencephalography data and canadian weather data

a wilcoxon mann whitney type test for infinite dimensional data the wilcoxon mann whitney test is a robust competitor of the t test in the univariate setting for finite dimensional multivariate non gaussian data several extensions of the wilcoxon mann whitney test have been shown to outperform hotelling s t 2 test in this paper we study a wilcoxon mann whitney type test based on spatial ranks in infinite dimensional spaces we investigate its asymptotic properties and compare it with several existing tests the proposed test is shown to be robust with respect to outliers and to have better power than some competitors for certain distributions with heavy tails we study its performance using real and simulated data

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

occupation measures for controlled markov processes characterization and optimality for controlled markov processes taking values in a polish space control problems with ergodic cost infinite horizon discounted cost and finite horizon cost are studied each is posed as a convex optimization problem wherein one tries to minimize a linear functional on a closed convex set of appropriately defined occupation measures for the problem these are characterized as solutions of a linear equation asssociated with the problem this characterization is used to establish the existence of optimal markov controls the dual convex optimization problem is also studied

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

bootstrap based inference on the difference in the means of two correlated functional processes we propose nonparametric inference methods on the mean difference between two correlated functional processes we compare methods that 1 incorporate different levels of smoothing of the mean and covariance 2 preserve the sampling design and 3 use parametric and nonparametric estimation of the mean functions we apply our method to estimating the mean difference between average normalized power of sleep electroencephalograms for 51 subjects with severe sleep apnea and 51 matched controls in the first 4 h after sleep onset we obtain data from the sleep heart health study the largest community cohort study of sleep although methods are applied to a single case study they can be applied to a large number of studies that have correlated functional data copyright 2012 john wiley sons ltd

robust orthogonal complement principal component analysis abstractrecently the robustification of principal component analysis pca has attracted lots of attention from statisticians engineers and computer scientists in this work we study the type of outliers that are not necessarily apparent in the original observation space but can seriously affect the principal subspace estimation based on a mathematical formulation of such transformed outliers a novel robust orthogonal complement principal component analysis roc pca is proposed the framework combines the popular sparsity enforcing and low rank regularization techniques to deal with row wise outliers as well as element wise outliers a nonasymptotic oracle inequality guarantees the accuracy and high breakdown performance of roc pca in finite samples to tackle the computational challenges an efficient algorithm is developed on the basis of stiefel manifold optimization and iterative thresholding furthermore a batch variant is proposed to significantly reduce the cost in ultra high dimensions th

white noise testing and model diagnostic checking for functional time series abstract this paper is concerned with white noise testing and model diagnostic checking for stationary functional time series to test for the functional white noise null hypothesis we propose a cramer von mises type test based on the functional periodogram introduced by panaretos and tavakoli 2013a using the hilbert space approach we derive the asymptotic distribution of the test statistic under suitable assumptions a new block bootstrap procedure is introduced to obtain the critical values from the non pivotal limiting distribution compared to existing methods our approach is robust to the dependence within white noise and it does not involve the choices of functional principal components and lag truncation number we employ the proposed method to check the adequacy of functional linear models and functional autoregressive models of order one by testing the uncorrelatedness of the residuals monte carlo simulations are provided to demonstrate the empirical advantages of the proposed method over existing alternatives our method is illustrated via an application to cumulative intradaily returns

testing for independence between functional time series frequently econometricians are interested in verifying a relationship between two or more time series such analysis is typically carried out by causality and or independence tests which have been well studied when the data is univariate or multivariate modern data though is increasingly of a high dimensional or functional nature for which finite dimensional methods are not suitable in the present paper we develop methodology to check the assumption that data obtained from two functional time series are independent our procedure is based on the norms of empirical cross covariance operators and is asymptotically validated when the underlying populations are assumed to be in a class of weakly dependent random functions which include the functional arma arch and garch processes

semantic preserving distance metric learning and applications how do we accurately browse a large set of images or efficiently annotate the images from an image library image clustering methods are invaluable tools for applications such as content based image retrieval and image annotation to perform these tasks it is critical to have proper features to describe the visual and semantic content of images and to define an accurate distance metric to measure the dissimilarity between any two images however existing methods which adopt the features of color histograms edge direction histograms and shape context lack the ability to describe semantic content to solve this problem we propose a new approach that utilizes user provided pairwise constraints to describe the semantic relationship between two images a semantic preserving distance metric learning sp dml algorithm is developed to explore the complementary characteristics of the visual features and pairwise constraints in a unified feature space in this space the learned distance metric can be used to measure the dissimilarity between two images specifically the manifold structure adopted in sp dml is revealed by the image s visual features to integrate semantic contents in distance metric learning sp dml utilizes pairwise constraints to build semantic patches and align these patches to obtain the optimal distance metric for the new feature space experimental results in image clustering demonstrate that the performance of sp dml is appealing

distribution of the residual autocorrelations in multivariate arma time series models the large sample distribution of the multivariate residual autocorrelations in the vector arma model is derived this result is somewhat less complicated for the vector autoregressive model a new multivariate portmanteau test for checking the adequacy of fitted vector arma models is developed a simulation study shows that a simple modification of the portmanteau test improves its accuracy in small samples

robpca a new approach to robust principal component analysis we introduce a new method for robust principal component analysis pca classical pca is based on the empirical covariance matrix of the data and hence is highly sensitive to outlying observations two robust approaches have been developed to date the first approach is based on the eigenvectors of a robust scatter matrix such as the minimum covariance determinant or an s estimator and is limited to relatively low dimensional data the second approach is based on projection pursuit and can handle high dimensional data here we propose the robpca approach which combines projection pursuit ideas with robust scatter matrix estimation robpca yields more accurate estimates at noncontaminated datasets and more robust estimates at contaminated data robpca can be computed rapidly and is able to detect exact fit situations as a by product robpca produces a diagnostic plot that displays and classifies the outliers we apply the algorithm to several datasets from chemometrics and engineering

bootstrap based testing of equality of mean functions or equality of covariance operators for functional data we investigate the properties of a simple bootstrap method for testing the equality of mean functions or of covariance operators in functional data theoretical size and power results are derived for certain test statistics whose limiting distributions depend on unknown infinite dimensional parameters simulations demonstrate good size and power of the bootstrap based functional tests

a geometrical characterization of multidimensional hausdorff polytopes with applications to exit time problems we present a formula for the corner points of the multidimensional hausdorff polytopes and show how this result can be used to improve linear programming models for computing e g moments of exit time distributions of diffusion processes specifically we compute the mean exit time of two dimensional brownian motion from the unit square as well as higher moments of the exit time of time space brownian motion i e the two dimensional process composed of a one dimensional wiener process and the time component from a rectangle the corner point formula is complemented by a convergence result which provides the analytical underpinning of the numerical method that we use

adaptive hypergraph learning and its application in image classification recent years have witnessed a surge of interest in graph based transductive image classification existing simple graph based transductive learning methods only model the pairwise relationship of images however and they are sensitive to the radius parameter used in similarity calculation hypergraph learning has been investigated to solve both difficulties it models the high order relationship of samples by using a hyperedge to link multiple samples nevertheless the existing hypergraph learning methods face two problems i e how to generate hyperedges and how to handle a large set of hyperedges this paper proposes an adaptive hypergraph learning method for transductive image classification in our method we generate hyperedges by linking images and their nearest neighbors by varying the size of the neighborhood we are able to generate a set of hyperedges for each image and its visual neighbors our method simultaneously learns the labels of unlabeled images and the weights of hyperedges in this way we can automatically modulate the effects of different hyperedges thorough empirical studies show the effectiveness of our approach when compared with representative baselines

properties of principal component methods for functional and longitudinal data analysis the use of principal component methods to analyze functional data is appropriate in a wide range of different settings in studies of functional data analysis it has often been assumed that a sample of random functions is observed precisely in the continuum and without noise while this has been the traditional setting for functional data analysis in the context of longitudinal data analysis a random function typically represents a patient or subject who is observed at only a small number of randomly distributed points with nonnegligible measurement error nevertheless essentially the same methods can be used in both these cases as well as in the vast number of settings that lie between them how is performance affected by the sampling plan in this paper we answer that question we show that if there is a sample of n functions or subjects then estimation of eigenvalues is a semiparametric problem with root n consistent estimators even if only a few observations are made of each function and if each observation is encumbered by noise however estimation of eigenfunctions becomes a nonparametric problem when observations are sparse the optimal convergence rates in this case are those which pertain to more familiar function estimation settings we also describe the effects of sampling at regularly spaced points as opposed to random points in particular it is shown that there are often advantages in sampling randomly however even in the case of noisy data there is a threshold sampling rate depending on the number of functions treated above which the rate of sampling either randomly or regularly has negligible impact on estimator performance no matter whether eigenfunctions or eigenvectors are being estimated

general subspace learning with corrupted training data via graph embedding we address the following subspace learning problem supposing we are given a set of labeled corrupted training data points how to learn the underlying subspace which contains three components an intrinsic subspace that captures certain desired properties of a data set a penalty subspace that fits the undesired properties of the data and an error container that models the gross corruptions possibly existing in the data given a set of data points these three components can be learned by solving a nuclear norm regularized optimization problem which is convex and can be efficiently solved in polynomial time using the method as a tool we propose a new discriminant analysis i e supervised subspace learning algorithm called corruptions tolerant discriminant analysis ctda in which the intrinsic subspace is used to capture the features with high within class similarity the penalty subspace takes the role of modeling the undesired features with high between class similarity and the error container takes charge of fitting the possible corruptions in the data we show that ctda can well handle the gross corruptions possibly existing in the training data whereas previous linear discriminant analysis algorithms arguably fail in such a setting extensive experiments conducted on two benchmark human face data sets and one object recognition data set show that ctda outperforms the related algorithms

estimation of the mean of functional time series and a two sample problem summary the paper is concerned with inference based on the mean function of a functional time series we develop a normal approximation for the functional sample mean and then focus on the estimation of the asymptotic variance kernel using these results we develop and asymptotically justify testing procedures for the equality of means in two functional samples exhibiting temporal dependence evaluated by means of a simulation study and application to a real data set these two sample procedures enjoy good size and power in finite samples

distance based tests for homogeneity of multivariate dispersions the traditional likelihood based test for differences in multivariate dispersions is known to be sensitive to nonnormality it is also impossible to use when the number of variables exceeds the number of observations many biological and ecological data sets have many variables are highly skewed and are zero inflated the traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non euclidean dissimilarity would be more appropriate distance based tests of homogeneity of multivariate dispersions which can be based on any dissimilarity measure of choice are proposed here they rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes the tests are straightforward multivariate extensions of levene s test with p values obtained either using the traditional f distribution or using permutation of either least squares or lad residuals examples illustrate the utility of the approach including the analysis of stabilizing selection in sparrows biodiversity of new zealand fish assemblages and the response of indonesian reef corals to an el nino monte carlo simulations from the real data sets show that the distance based tests are robust and powerful for relevant alternative hypotheses of real differences in spread

solving partially observable problems with inaccurate psr models modeling dynamical systems is a commonly used technique to solve partially observable problems in the artificial intelligence field predictive state representations psrs have been proposed as an alternative to partially observable markov decision processes pomdps to model dynamical systems although pomdps and psrs provide general frameworks to solve partially observable problems they rely heavily on a known and accurate model of the environment however in real world applications it is extremely difficult to build an accurate model in this paper we propose an algorithm to solve partially observable problems using an inaccurate psr model which is learned from samples the proposed algorithm can also improve the accuracy of the learned model given the inaccurate psr model the psr state is identified firstly then the traditional markov decision processes mdp techniques are used to solve the partially observable problem furthermore the learned model which may get off track as often happens when the model is learned from samples can be reset the effectiveness of our proposed algorithm is demonstrated based on a standard set of pomdp test problems

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

unified empirical likelihood ratio tests for functional concurrent linear models and the phase transition from sparse to dense functional data summary we consider the problem of testing functional constraints in a class of functional concurrent linear models where both the predictors and the response are functional data measured at discrete time points we propose test procedures based on the empirical likelihood with bias corrected estimating equations to conduct both pointwise and simultaneous inferences the asymptotic distributions of the test statistics are derived under the null and local alternative hypotheses where sparse and dense functional data are considered in a unified framework we find a phase transition in the asymptotic null distributions and the orders of detectable alternatives from sparse to dense functional data specifically the tests proposed can detect alternatives of n order when the number of repeated measurements per curve is of an order larger than n 0 with n being the number of curves the transition points 0 for pointwise and simultaneous tests are different and both are smaller than the transition point in the estimation problem simulation studies and real data analyses are conducted to demonstrate the methods proposed

linear programming formulation for optimal stopping problems optimal stopping problems for continuous time markov processes are shown to be equivalent to infinite dimensional linear programs over a space of pairs of measures under very general conditions the measures involved represent the joint distribution of the stopping time and stopping location and the occupation measure of the process until it is stopped these measures satisfy an identity for each function in the domain of the generator which is sufficient to characterize the stochastic process finite dimensional linear programs obtained using markov chain approximations are solved in two examples to illustrate the numerical accuracy of the linear programming formulation

average cost markov decision processes optimality conditions we are concerned in this paper with discrete time markov decision processes mdps with borel state and action spaces x and a respectively and the long run expected average cost criterion when x is a denumerable set man y necessary and or sufficient conditions for the existence of optimal control policies are known however when x is a borel space i e a borel subset of a complete separable metric space most of the available results impose on the mdp very restrictive topological conditions e g compactness and or strong recurrence assumptions such as doeblin s condition see e g 4 9 12 and their references another related work is 7 where we have studied mdps from the viewpoint of the recurrence or ergodicity properties of the state process ln the present paper however we are concerned with the existence of average optimal policies by looking at static optimization problems see condition cs in section 4 related in sorne cases equivalent to the existence of a bounded solution to the so called optimality equation see c4 in section 4 these optimization problems are dual in the sense that under appropriate conditions the existence of an optimal solution to one of the problems implies existence of an optimal

robust methods for testing the pattern of a single covariance matrix summary patterns for a single covariance matrix may be studied using likelihood ratio or quadratic form test statistics typically these tests have an asymptotic chi squared distribution when the data are normally distributed but the asymptotic distribution is sensitive to nonnormality this paper suggests two ways to robustify inference for patterned matrices i estimate the critical values of normal theory test statistics using a new bootstrap procedure which takes advantage of certain invariances under ho and ii modify the statistics so that an asymptotic chi squared distribution holds even under nonnormality

tests for error correlation in the functional linear model in this chapter we consider two tests for error correlation in the fully functional linear model which we call methods i and ii they complement the tools described in section 8 6 and the graphical goodness of fit checks used in chapter 9 to construct the test statistics finite dimensional residuals are computed in two different ways and then their autocorrelations are suitably defined from these autocorrelation matrices two quadratic forms are constructed whose limiting distribution are chi squared with known numbers of degrees of freedom different for the two forms the test statistics can be relatively easily computed using the r package fda

detecting changes in the mean of functional observations principal component analysis has become a fundamental tool of functional data analysis it represents tne functional data as x i t t 1 i l v l t wnere is the common mean v l are the eigenfunctions of the covariance operator and the i l are the scores inferential procedures assume that the mean function t is the same for all values of i if in fact the observations do not come from one population but rather their mean changes at some point s the results of principal component analysis are confounded by the change s it is therefore important to develop a methodology to test the assumption of a common functional mean we develop such a test using quantities which can be readily computed in the r package fda the null distribution of the test statistic is asymptotically pivotal with a well known asymptotic distribution the asymptotic test has excellent finite sample performance its application is illustrated on temperature data from england

asymptotic distributions of principal components based on robust dispersions algebraically principal components can be defined as the eigenvalues and eigenvectors of a covariance or correlation matrix but they are statistically meaningful as successive projections of the multivariate data in the direction of maximal variability an attractive alternative in robust principal component analysis is to replace the classical variability measure i e variance by a robust dispersion measure this projection pursuit approach was first proposed in li chen 1985 as a method of constructing a robust scatter matrix recent unpublished work of c croux and a ruiz gazen provided the influence functions of the resulting principal components the present paper focuses on the asymptotic distributions of robust principal components in particular we obtain the asymptotic normality of the principal components that maximise a robust dispersion measure we also explain the need to use a dispersion functional with a continuous influence function

penalized spline models for functional principal component analysis summary we propose an iterative estimation procedure for performing functional principal component analysis the procedure aims at functional or longitudinal data where the repeated measurements from the same subject are correlated an increasingly popular smoothing approach penalized spline regression is used to represent the mean function this allows straightforward incorporation of covariates and simple implementation of approximate inference procedures for coefficients for the handling of the within subject correlation we develop an iterative procedure which reduces the dependence between the repeated measurements that are made for the same subject the resulting data after iteration are theoretically shown to be asymptotically equivalent in probability to a set of independent data this suggests that the general theory of penalized spline regression that has been developed for independent data can also be applied to functional data the effectiveness of the proposed procedure is demonstrated via a simulation study and an application to yeast cell cycle gene expression data

robust functional estimation using the median and spherical principal components summary we present robust estimators for the mean and the principal components of a stochastic process in l2 r robustness and asymptotic properties of the estimators are studied theoretically by simulation and by example it is shown that the proposed estimators are generally more robust to outliers than the commonly used sample mean and principal components although their properties depend on the spacings of the eigenvalues of the covariance function

projection pursuit approach to robust dispersion matrices and principal components primary theory and monte carlo abstract this article proposes and discusses a type of new robust estimators for covariance correlation matrices and principal components via projection pursuit techniques the most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point besides they are qualitatively robust and consistent at elliptic underlying distributions the monte carlo study shows that the best of the new estimators compare favorably with other robust methods they provide as good a performance as m estimators and somewhat better empirical breakdown properties

second order comparison of gaussian random functions and the geometry of dna minicircles given two samples of continuous zero mean iid gaussian processes on 0 1 we consider the problem of testing whether they share the same covariance structure our study is motivated by the problem of determining whether the mechanical properties of short strands of dna are significantly affected by their base pair sequence though expected to be true had so far not been observed in three dimensional electron microscopy data the testing problem is seen to involve aspects of ill posed inverse problems and a test based on a karhunen loeve approximation of the hilbert schmidt distance of the empirical covariance operators is proposed and investigated when applied to a dataset of dna minicircles obtained through the electron microscope our test seems to suggest potential sequence effects on dna shape supplemental material available online

robust singular value decompositions a new approach to projection pursuit abstract robust location and covariance estimators are developed via general m estimation for covariance matrix eigenvectors and eigenvalues the solution to this gm estimation problem is obtained by transforming it into a series of robust regression problems based on a new algorithm for the singular value decomposition it is shown here that the singular value decomposition can be represented as an iteration of two steps a least squares regression fit of the data matrix followed by a rotation to the regression hyperplanes an algorithm to obtain the solution to this gm estimation problem is presented along with results of a monte carlo study and examples of its application in addition it is shown how the output of this algorithm can be used to numerically search for multivariate outliers which is especially useful in exploratory data analysis with high dimensional data and large sample sizes where standard graphical techniques are difficult to implement because the algorithm computes robust estima

robust low rank approximation of data matrices with elementwise contamination we propose a robust method to approximate an n p data matrix with one of rank q the method is based on yohai s regression mm estimates it is intended to be resistant against the existence of both atypical rows and of scattered atypical cells and also to be able to cope with missing values we propose an algorithm based on alternating m regressions and a starting estimate based on successive rank one fits which involves o npq operations simulations show that our estimate outperforms competing estimates in terms of both efficiency and resistance three high dimensional real data sets are analyzed the running time of our estimate for large data sets is shown to be less than that of its competitors

planning and acting in partially observable stochastic domains in this paper we bring techniques from operations research to bear on the problem of choosing optimal actions in partially observable stochastic domains we begin by introducing the theory of markov decision processes mdps and partially observable mdps pomdps we then outline a novel algorithm for solving pomdps off line and show how in some cases a finite memory controller can be extracted from the solution to a pomdp we conclude with a discussion of how our approach relates to previous work the complexity of finding exact solutions to pomdps and of some possibilities for finding approximate solutions

robust projection indices loosely speaking a robust projection index is one that prefers projections involving true clusters over projections consisting of a cluster and an outlier we introduce a mathematical definition of one dimensional index robustness and describe a numerical experiment to measure it we design five new indices based on measuring divergence from student s t distribution which are intended to be especially robust the experiment shows that they are more robust than several established indices the experiment also reveals more generally that the robustness of moment indices depends on the number of approximation terms providing additional practical guidance for existing projection pursuit implementations we investigate the theoretical properties of one new student t index and hall s index and show that the new index automatically adapts its robustness to the degree of outlier contamination we conclude by outlining the possibilities for extending our experiments to both higher dimensions and other new indices

percentile optimization for markov decision processes with parameter uncertainty markov decision processes are an effective tool in modeling decision making in uncertain dynamic environments because the parameters of these models typically are estimated from data or learned from experience it is not surprising that the actual performance of a chosen strategy often differs significantly from the designer s initial expectations due to unavoidable modeling ambiguity in this paper we present a set of percentile criteria that are conceptually natural and representative of the trade off between optimistic and pessimistic views of the question we study the use of these criteria under different forms of uncertainty for both the rewards and the transitions some forms are shown to be efficiently solvable and others highly intractable in each case we outline solution concepts that take parametric uncertainty into account in the process of decision making

existence of markov controls and characterization of optimal markov controls given a solution of a controlled martingale problem it is shown under general conditions that there exists a solution having markov controls which has the same cost as the original solution this result is then used to show that the original stochastic control problem is equivalent to a linear program over a space of measures under a variety of optimality criteria existence and characterization of optimal markov controls then follows an extension of echeverria s theorem characterizing stationary distributions for uncontrolled markov processes is obtained as a corollary in particular this extension covers diffusion processes with discontinuous drift and diffusion coefficients

low rank matrix fitting based on subspace perturbation analysis with applications to structure from motion the task of finding a low rank r matrix that best fits an original data matrix of higher rank is a recurring problem in science and engineering the problem becomes especially difficult when the original data matrix has some missing entries and contains an unknown additive noise term in the remaining elements the former problem can be solved by concatenating a set of r column matrices that share a common single r dimensional solution space unfortunately the number of possible submatrices is generally very large and hence the results obtained with one set of r column matrices will generally be different from that captured by a different set ideally we would like to find that solution that is least affected by noise this requires that we determine which of the r column matrices i e which of the original feature points are less influenced by the unknown noise term this paper presents a criterion to successfully carry out such a selection our key result is to formally prove that the more distinct the r vectors of the r column matrices are the less they are swayed by noise this key result is then combined with the use of a noise model to derive an upper bound for the effect that noise and occlusions have on each of the r column matrices it is shown how this criterion can be effectively used to recover the noise free matrix of rank r finally we derive the affine and projective structure from motion sfm algorithms using the proposed criterion extensive validation on synthetic and real data sets shows the superiority of the proposed approach over the state of the art

robust pca via outlier pursuit singular value decomposition svd and principal component analysis pca is one of the most widely used techniques for dimensionality reduction successful and efficiently computable it is nevertheless plagued by a well known well documented sensitivity to outliers recent work has considered the setting where each point has a few arbitrarily corrupted components yet in applications of svd or pca such as robust collaborative filtering or bioinformatics malicious agents defective genes or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted we present an efficient convex optimization based algorithm that we call outlier pursuit which under some mild assumptions on the uncorrupted points satisfied e g by the standard generative assumption in pca problems recovers the exact optimal low dimensional subspace and identifies the corrupted points such identification of corrupted points that do not conform to the low dimensional approximation is of paramount interest in bioinformatics financial applications and beyond our techniques involve matrix decomposition using nuclear norm minimization however our results setup and approach necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition since we develop an approach to recover the correct column space of the uncorrupted matrix rather than the exact matrix itself in any problem where one seeks to recover a structure rather than the exact initial matrices techniques developed thus far relying on certificates of optimality will fail we present an important extension of these methods which allows the treatment of such problems

principal components analysis based on multivariate mm estimators with fast and robust bootstrap we consider robust principal components analysis pca based on multivariate mm estimators we first study the robustness and efficiency of these estimators particularly in terms of eigenvalues and eigenvectors we then focus on inference procedures based on a fast and robust bootstrap for mm estimators this method is an alternative to the approach based on the asymptotic distribution of the estimators and can also be used to assess the stability of the principal components a formal consistency proof for the bootstrap method is given and its finite sample performance is investigated through simulations we illustrate the use of the robust pca and the bootstrap inference on a real dataset

robust sparse principal component analysis a method for principal component analysis is proposed that is sparse and robust at the same time the sparsity delivers principal components that have loadings on a small number of variables making them easier to interpret the robustness makes the analysis resistant to outlying observations the principal components correspond to directions that maximize a robust measure of the variance with an additional penalty term to take sparseness into account we propose an algorithm to compute the sparse and robust principal components the algorithm computes the components sequentially and thus it can handle datasets with more variables than observations the method is applied on several real data examples and diagnostic plots for detecting outliers and for selecting the degree of sparsity are provided a simulation experiment studies the effect on statistical efficiency by requiring both robustness and sparsity supplementary materials are available online on the journal web site

rank regularized estimation of approximate factor models abstract it is known that the common factors in a large panel of data can be consistently estimated by the method of principal components and principal components can be constructed by iterative least squares regressions replacing least squares with ridge regressions turns out to have the effect of removing the contribution of factors associated with small singular values from the common component the method has been used in the machine learning literature to recover low rank matrices we study the procedure from the perspective of estimating an approximate factor model under the rank constraint the common component is estimated by the space spanned by factors whose singular values exceed a threshold the desire for minimum rank and parsimony lead to a data dependent penalty for selecting the number of factors the new criterion is more conservative than the existing deterministic penalties and is appropriate when the nominal number of factors is inflated by the presence of weak factors or large measurement noise we provide asymptotic results that can be used to test economic hypotheses

the percentage bend correlation coefficient a well known result is that the usual correlation coefficient is highly nonrobust very slight changes in only one of the marginal distributions can alter by a substantial amount there are a variety of methods for correcting this problem this paper identifies one particular method which is useful in psychometrics and provides a simple test for independence it is not recommended that the new test replace the usual test ofh 0 0 but the new test has important advantages over the usual test in terms of both type i errors and power

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

robust procedures for testing equality of covariance matrices normal theory tests for common variance are very sensitive to departures from normality in both univariate and multivariate applications for mild departures from normality and moderate sample size nominal a 01 produced type i error rates in excess of 20 in simulation studies since a modification of levene s test has been shown to perform well for the univariate problem two multivariate generalizations of this test are proposed both procedures are robust to departures from normality and are simple computationally comparisons of the power of these two procedures and a procedure proposed previously by tiku and balakrishnan 1985 communications in statisticstheory and methods 14 3033 3051 indicate that the three methods should be viewed as complementary

free knot spline smoothing for functional data summary the paper introduces free knot regression spline estimators for the mean and the variance components of a sample of curves the asymptotic distribution of the mean estimator is derived and asymptotic confidence bands are constructed a comparative simulation study shows that free knot splines estimate salient features of the functions such as sharp peaks more accurately than smoothing splines this adaptive behaviour is also illustrated by an analysis of weather data

robust estimation of dispersion matrices and principal components abstract this paper uses monte carlo methods to compare the performances of several robust procedures for estimating a correlation matrix and its principal components the estimators are formed either from separate bivariate analyses or by simultaneous manipulation of all variables by using techniques such as multivariate trimming and m estimation the m estimators stand up exceptionally well they and the multivariate trimming procedure are especially effective at estimating the principal components including a near singularity however the m estimators can break down relatively easily when the dimensionality is large and the outliers are asymmetric with missing data the element wise approach becomes more attractive

robust estimation in the errors in variables model summary orthogonal regression analogues of m estimates of regression called here orthogonal regression m estimates are defined these estimates are shown to be consistent at elliptical errors in variables models and robust if the corresponding loss function is bounded the orthogonal regression analogues of regression scale estimates called here orthogonal regression s estimates are considered as well in particular they provide a robust estimate for the scale of the orthogonal residuals a crucial quantity in the computation of orthogonal regression m estimates finally we present an algorithm for computing orthogonal regression s and m estimates and the results of a small monte carlo experiment

portmanteau test of independence for functional observations in various fields observations are curves over some natural time interval these curves often arise from finely spaced measurements e g in physical sciences and finance or from smoothing unequally spaced sparse measurements recent years have seen the development of tools for analyzing such data in the growing field of functional data analysis to validate the assumptions underlying these tools it is important to verify that the functional observations form a simple random sample if the curves form a functional time series then model validation typically relies on checking whether model residuals are independent and identically distributed we propose a test for independence and identical distribution of functional observations to reduce dimension curves are projected on the most important functional principal components then a test statistic based on lagged cross covariances of the resulting vectors is constructed we show that this dimension reduction step introduces asymptotically negligib

approximation of infinite dimensional linear programming problems which arise in stochastic control we study a general approximation scheme for infinite dimensional linear programming lp problems which arise naturally in stochastic control we prove that the optimal value of the approximating problems converges to the value of the original lp problem for the controls we show that if the approximating optimal controls converge the limiting control is an optimal control for the original lp problem as an application of this theory we present numerical approximations to the lp formulation of stochastic control problems in continuous time we study long term average and discounted control problems for the example for which the theoretical solution is known our approximation results are very accurate

s estimators for functional principal component analysis principal component analysis is a widely used technique that provides an optimal lower dimensional approximation to multivariate or functional datasets these approximations can be very useful in identifying potential outliers among high dimensional or functional observations in this article we propose a new class of estimators for principal components based on robust scale estimators for a fixed dimension q we robustly estimate the q dimensional linear space that provides the best prediction for the data in the sense of minimizing the sum of robust scale estimators of the coordinates of the residuals we also study an extension to the infinite dimensional case our method is consistent for elliptical random vectors and is fisher consistent for elliptically distributed random elements on arbitrary hilbert spaces numerical experiments show that our proposal is highly competitive when compared with other methods we illustrate our approach on a real dataset where the robust estimator discovers atypi

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

on the rotational invariant l1 norm pca abstract principal component analysis pca is a powerful tool for dimensionality reduction unfortunately it is sensitive to outliers so that various robust pca variants were proposed in the literature one of the most frequently applied methods for high dimensional data reduction is the rotational invariant l 1 norm pca of ding and coworkers so far no convergence proof for this algorithm was available the main topic of this paper is to fill this gap we reinterpret this robust approach as a conditional gradient algorithm and show moreover that it coincides with a gradient descent algorithm on grassmann manifolds based on the latter point of view we prove global convergence of the whole series of iterates to a critical point using the kurdyka ojasiewicz property of the objective function where we have to pay special attention to so called anchor points where the function is not differentiable

components and completion of partially observed functional data functional data are traditionally assumed to be observed on the same domain motivated by a data set of heart rate temporal profiles we develop methodology for the analysis of incomplete functional samples where each curve may be observed on a subset of the domain and unobserved elsewhere we formalize this observation regime and develop the fundamental procedures of functional data analysis for this framework estimation of parameters mean and covariance operator and principal component analysis principal scores of a partially observed function cannot be computed directly and we solve this challenging issue by estimating their best predictions as linear functionals of the observed part of the trajectory next we propose a functional completion procedure that recovers the missing part by using the observed part of the curve we construct prediction intervals for principal scores and bands for missing parts of trajectories the prediction problems are seen to be ill posed inverse problems regularization techniques are used to obtain a stable solution a simulation study shows the good performance of our methods we illustrate the methods on the heart rate data and provide practical computational algorithms and theoretical arguments and proofs of all results

common functional principal components functional principal component analysis fpca based on the karhunen loeve decomposition has been successfully applied in many applications mainly for one sample problems in this paper we con sider common functional principal components for two sample prob lems our research is motivated not only by the theoretical challenge of this data situation but also by the actual question of dynamics of implied volatility iv functions for different maturities the log returns of ivs are samples of smooth random functions and the methods proposed here study the similarities of their stochastic be havior first we present a new method for estimation of functional principal components from discrete noisy data next we present the two sample inference for fpca and develop the two sample theory we propose bootstrap tests for testing the equality of eigenvalues eigenfunctions and mean functions of two functional samples illus trate the test properties by simulation study and apply the method to the iv analysis

semisupervised multiview distance metric learning for cartoon synthesis in image processing cartoon character classification retrieval and synthesis are critical so that cartoonists can effectively and efficiently make cartoons by reusing existing cartoon data to successfully achieve these tasks it is essential to extract visual features that comprehensively represent cartoon characters and to construct an accurate distance metric to precisely measure the dissimilarities between cartoon characters in this paper we introduce three visual features color histogram shape context and skeleton to characterize the color shape and action respectively of a cartoon character these three features are complementary to each other and each feature set is regarded as a single view however it is improper to concatenate these three features into a long vector because they have different physical properties and simply concatenating them into a high dimensional feature vector will suffer from the so called curse of dimensionality hence we propose a semisupervised multiview distance metric learning ssm dml ssm dml learns the multiview distance metrics from multiple feature sets and from the labels of unlabeled cartoon characters simultaneously under the umbrella of graph based semisupervised learning ssm dml discovers complementary characteristics of different feature sets through an alternating optimization based iterative algorithm therefore ssm dml can simultaneously accomplish cartoon character classification and dissimilarity measurement on the basis of ssm dml we develop a novel system that composes the modules of multiview cartoon character classification multiview graph based cartoon synthesis and multiview retrieval based cartoon synthesis experimental evaluations based on the three modules suggest the effectiveness of ssm dml in cartoon applications

robust functional principal components a projection pursuit approach in many situations data are recorded over a period of time and may be regarded as realizations of a stochastic process in this paper robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting our approach combines robust projection pursuit with different smoothing methods consistency of the estimators are shown under mild assumptions the performance of the classical and robust procedures are compared in a simulation study under different contamination schemes 1 introduction analogous to classical principal components analysis pca the projection pursuit approach to robust pca is based on finding projections of the data which have maximal dispersion instead of using the variance as a measure of dispersion a robust scale estimator sn is used for the maximization problem this approach was introduced by li and chen 1985 who proposed estimators based on maximizing or minimizing a robust scale in this way given a sample xi r d 1 i n the first robust principal component vector is defined as

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

outlier detection and robust normal curvature estimation in mobile laser scanning 3d point cloud data this paper proposes two robust statistical techniques for outlier detection and robust saliency features such as surface normal and curvature estimation in laser scanning 3d point cloud data one is based on a robust z score and the other uses a mahalanobis type robust distance the methods couple the ideas of point to plane orthogonal distance and local surface point consistency to get maximum consistency with minimum distance mcmd the methods estimate the best fit plane based on most probable outlier free and most consistent points set in a local neighbourhood then the normal and curvature from the best fit plane will be highly robust to noise and outliers experiments are performed to show the performance of the algorithms compared to several existing well known methods from computer vision data mining machine learning and statistics using synthetic and real laser scanning datasets of complex planar and non planar objects results for plane fitting denoising sharp feature preserving and segmentation are significantly improved the algorithms are demonstrated to be significantly faster more accurate and robust quantitatively for a sample size of 50 with 20 outliers the proposed mcmd z is approximately 5 15 and 98 times faster than the existing methods ulsif ransac and rpca respectively the proposed mcmd md method can tolerate 75 clustered outliers whereas rpca and ransac can only tolerate 47 and 64 outliers respectively in terms of outlier detection for the same dataset mcmd z has an accuracy of 99 72 0 4 false positive rate and 0 false negative rate for rpca ransac and ulsif the accuracies are 97 05 47 06 and 94 54 respectively and they have misclassification rates higher than the proposed methods the new methods have potential for local surface reconstruction fitting and other point cloud processing tasks highlightstwo statistical techniques are proposed for outlier detection in point cloud data the proposed methods can fit robust plane in laser scanning data the proposed methods produce robust normal and curvature in point cloud processing they are faster and robust than ransac robust pca and other existing efficient methods they have potential for point cloud denoising segmentation and reconstruction