multivariable self tuning control a model following approach abstract a multivariable self tuning control algorithm based on the model following approach is proposed in this paper reference is made to systems described by arm ax models with an a apriori known interactor matrix the cost function is defined in terms of variance of the norm of a suitably defined generalized error a detuned version of the model following procedure is also given so as to cope with non minimum phase systems the proposed algorithm is finally applied to the self tuning control problem of a simulated binary distillation column

relative negligibility of linear automorphisms let u and v be finite dimensional vector spaces over a field k gl u gl v and i be the identity transformation on v denote by and i the induced linear automorphisms on u v and i can also be regarded as k automorphisms on the function field k u v it is elementary to check whether and i are conjugate within gl u v by examining their rational canonical forms in this paper we shall give necessary and sufficient conditions for and i to be conjugate within autk k u v for this characterization we introduce the concept of the generalized order through this invariant we also settle the question of when two different polynomials are minimal polynomials of the same linear automorphism of a rational function field

on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

spherical regression models using projective linear transformations this article studies the problem of modeling relationship between two spherical or directional random variables in a regression setup here the predictor and the response variables are constrained to be on a unit sphere and due to this nonlinear condition the standard euclidean regression models do not apply several past papers have studied this problem termed spherical regression by modeling the response variable with a von mises fisher vmf density with the mean given by a rotation of the predictor variable the few papers that go beyond rigid rotations are limited to one or two dimensional spheres this article extends the mean transformations to a larger group the projective linear group of transformations on unit spheres of arbitrary dimensions while keeping the vmf density to model the noise it develops a newton raphson algorithm on the special linear group for estimating the mle of regression parameter and establishes its asymptotic properties when the sample size becomes large through

a note on algebraic automorphisms abstract let v be a vector space over the field k of a possibly infinite dimension and let t be an automorphism of v of a possibly infinite order assume that t is algebraic with minimal polynomial p t k t let k v be the field of quotients of the symmetric k algebra of v and let s be the k automorphism on k v induced by t we show that s is completely determined by p t and dimk v

improved pivotal methods for constructing confidence regions with directional data abstract the importance of pivoting is well established in the context of nonparametric confidence regions it ensures enhanced coverage accuracy however pivoting for directional data cannot be achieved simply by rescaling a somewhat cumbersome pivotal method which involves passing first into a space of higher dimension has been developed by fisher and hall for samples of unit vectors although that method has some advantages over nonpivotal techniques it does suffer from certain drawbacks in particular the operation of passing to a higher dimension here we suggest alternative pivotal approaches the implementation of which does not require us to increase the intrinsic dimension of the data and which in practice seem to achieve greater coverage accuracy these methods are of two types new pivotal bootstrap techniques and techniques that exploit the implicit pivotalness of the empirical likelihood algorithm unlike the method proposed by fisher and hall these methods are also applicable to axia

large sample theory of intrinsic and extrinsic sample means on manifolds ii this article develops nonparametric inference procedures for estimation and testing problems for means on manifolds a central limit theorem for frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on riemannian manifolds central limit theorems are also obtained for extrinsic sample means w r t an arbitrary embedding of a differentiable manifold in a euclidean space bootstrap methods particularly suitable for these problems are presented applications are given to distributions on the sphere s d directional spaces real projective space rp n 1 axial spaces complex projective space cp k 2 planar shape spaces w r t veronese whitney embeddings and a three dimensional shape space 4 3

generalized minimum variance control of mimo systems a stability result abstract this note describes a generalized minimum variance gmv control algorithm for square systems the controller embeds an integral action on the error vector it is shown how an appropriate yet simple choice of some design parameters guarantees overall asymptotic stability for open loop asymptotically stable and possibly nonminimum phase plants

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

paper the role of the interactor matrix in multivariable stochastic adaptive control this paper is concerned with the extension of the well known minimum variance control strategy to the multi input multi output case it is shown that this extension is straightforward when the system interactor matrix is diagonal but presents some unexpected difficulties in the general case we develop a suitable stochastic controller for the general case as a logical extension of the single input algorithm we also explore the properties of the algorithm in detail we also address the question of adaptive control of multivariable stochastic systems and investigate one possible strategy for overcoming the requirement of knowing the system interactor matrix a priori

gait based human recognition by classification of cyclostationary processes on nonlinear shape manifolds we study the problem of analyzing and classifying human gait by modeling it as a stochastic process on a shape space we consider gait as a evolution of human silhouettes as seen in video sequences and focus on their shapes more specifically we define a shape space of planar closed curves and model a human gait as a stochastic process on this space due to the periodic nature of human walk this process is naturally constrained to be cyclostationary that is its mean path is assumed to be cyclic we compare two subjects using a metric that quantifies differences between average gait cycles of each subject this computation uses several tools from differential geometry of the shape space including computation of geodesics estimation of means of observed shapes interpolation between observed shapes and temporal registration of two gait cycles finally we apply a nearest neighbor classifier using the gait metric to perform human recognition and present results from an experiment involving 26 sub

saddlepoint approximations for the bingham and fisher bingham normalising constants the fisher bingham distribution is obtained when a multivariate normal random vector is conditioned to have unit length its normalising constant can be expressed as an elementary function multiplied by the density evaluated at 1 of a linear combination of independent noncentral chi 1 2 random variables hence we may approximate the normalising constant by applying a saddlepoint approximation to this density three such approximations implementation of each of which is straightforward are investigated the first order saddlepoint density approximation the second order saddlepoint density approximation and a variant of the second order approximation which has proved slightly more accurate than the other two the numerical and theoretical results we present show that this approach provides highly accurate approximations in a broad spectrum of cases

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

a probabilistic framework for 3d visual object representation we present an object representation framework that encodes probabilistic spatial relations between 3d features and organizes these features in a hierarchy features at the bottom of the hierarchy are bound to local 3d descriptors higher level features recursively encode probabilistic spatial configurations of more elementary features the hierarchy is implemented in a markov network detection is carried out by a belief propagation algorithm which infers the pose of high level features from local evidence and reinforces local evidence from globally consistent knowledge effectively producing a likelihood for the pose of the object in the detection scene we also present a simple learning algorithm that autonomously builds hierarchies from local object descriptors we explain how to use our framework to estimate the pose of a known object in an unknown scene experiments demonstrate the robustness of hierarchies to input noise viewpoint changes and occlusions

shape procrustes tangent projections and bilateral symmetry summary in shape analysis with concentrated data the procrustes tangent projection of shapes plays a fundamental role since this projection can be used to convert shape analysis to standard multivariate analysis in view of its importance a systematic and rigorous treatment of the procrustes tangent projection is developed here to facilitate its use in applications one important application involves an analytical description of bilateral symmetry of objects together with some related distributional results an explicit expression for the tangent projection is also derived

the complex watson distribution and shape analysis the complex watson distribution is an important simple distribution on the complex sphere which is used in statistical shape analysis we describe the density obtain the integrating constant and provide large sample approximations maximum likelihood estimation and hypothesis testing procedures for one and two samples are described the particular connection with shape analysis is discussed and we consider an application examining shape differences between normal and schizophrenic brains we make some observations about bayesian shape inference and finally we describe a more general rotationally symmetric family of distributions

intrinsic manova for riemannian manifolds with an application to kendall s space of planar shapes we propose an intrinsic multifactorial model for data on riemannian manifolds that typically occur in the statistical analysis of shape due to the lack of a linear structure linear models cannot be defined in general to date only one way manova is available for a general multifactorial model we assume that variation not explained by the model is concentrated near elements defining the effects by determining the asymptotic distributions of respective sample covariances under parallel transport we show that they can be compared by standard manova often in applications manifolds are only implicitly given as quotients where the bottom space parallel transport can be expressed through a differential equation for kendall s space of planar shapes we provide an explicit solution we illustrate our method by an intrinsic two way manova for a set of leaf shapes while biologists can identify genotype effects by sight we can detect height effects that are otherwise not identifiable

asymptotics for spherical needlets we investigate invariant random fields on the sphere using a new type of spherical wavelets called needlets these are compactly supported in frequency and enjoy excellent localization properties in real space with quasi exponentially decaying tails we show that for random fields on the sphere the needlet coefficients are asymptotically uncorrelated for any fixed angular distance this property is used to derive clt and functional clt convergence results for polynomial functionals of the needlet coefficients here the asymptotictheory is considered in the high frequency sense our proposals emerge from strong empirical motivations especially in connection with the analysis of cosmological data sets 1 introduction over the last two decades wavelets have emerged as one of the most interesting tools of statistical investigation in this paper we give an application to the statistical analysis of data sets indexed by the unit sphere s 2 this is motivated mostly by the analysis of the cosmic

statistical shape analysis clustering learning and testing using a differential geometric treatment of planar shapes we present tools for 1 hierarchical clustering of imaged objects according to the shapes of their boundaries 2 learning of probability models for clusters of shapes and 3 testing of newly observed shapes under competing probability models clustering at any level of hierarchy is performed using a minimum variance type criterion and a markov process statistical means of clusters provide shapes to be clustered at the next higher level thus building a hierarchy of shapes using finite dimensional approximations of spaces tangent to the shape space at sample means we implicitly impose probability models on the shape space and results are illustrated via random sampling and classification hypothesis testing together hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval examples are presented using shapes and images from eth surrey and amcom databases

the strong consistency of the stochastic gradient algorithm of adaptive control by use of a technique due to chen sufficient conditions are established for the strong consistency of the stochastic gradient sg algorithm for mimo armax stochastic systems without monitoring this result is then used in conjunction with the method of disturbed adaptive controls introduced in 1 2 hence it is shown that the sg algorithm generates strongly consistent parameter estimates while it is operating as a part of the sg algorithm of adaptive control of goodwin ramadge and caines

efficient generation of the ring of invariants we shall use the binet minc formula in the theory of permanents to prove david richman s theorem letgbe a finite group acting ona r a1 ar whereris any commutative ring with 1 g r then the ring of invariantsagis generated overrby g a 11a 22 a rr where 1 r g applications of permanents to other problems related to invariants are given also

negligibility of projective linear automorphisms abstract letkbe any field s aij 1 i j n 1 gln 1 k k k x1 xn 1 xn xn 1 y1 yn 1 yn yn 1 a field extension of transcendence degree 2noverk the diagonal action ofsonkis defined as equation equation theorem there existz1 zn ksuch thatk k x1 xn 1 xn xn 1 z1 zn ands zi zifor 1 i n in particular ifk x1 xn 1 xn xn 1 s is rational purely transcendental overk so isk s overk

some actions of symmetrical groups abstract let k be any field v the n dimensional vector space over k on which the symmetric group sn acts by permuting a basis then the fixed fields k skv sn and k k v sn are rational over k where skv and k v are the kth symmetric power and exterior power of v respectively the rationality of some other fixed fields are established also

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

detecting the direction of a signal on high dimensional spheres non null and le cam optimality results we consider one of the most important problems in directional statistics namely the problem of testing the null hypothesis that the spike direction pmb theta of a fisher von mises langevin distribution on the p dimensional unit hypersphere is equal to a given direction pmb theta 0 after a reduction through invariance arguments we derive local asymptotic normality lan results in a general high dimensional framework where the dimension p n goes to infinity at an arbitrary rate with the sample size n and where the concentration kappa n behaves in a completely free way with n which offers a spectrum of problems ranging from arbitrarily easy to arbitrarily challenging ones we identify various asymptotic regimes depending on the convergence divergence properties of kappa n that yield different contiguity rates and different limiting experiments in each regime we derive le cam optimal tests under specified kappa n and we compute from the le cam third lemma asymptotic powers of the classical watson test under contiguous alternatives we further establish lan results with respect to both spike direction and concentration which allows us to discuss optimality also under unspecified kappa n to investigate the non null behavior of the watson test outside the parametric framework above we derive its local asymptotic powers through martingale clts in the broader semiparametric model of rotationally symmetric distributions a monte carlo study shows that the finite sample behaviors of the various tests remarkably agree with our asymptotic results

convergence of a multivariable self tuning controller with integral action abstract in this note global convergence is established for the multivariable self tuning control algorithm presented in 11 the plant is assumed to be described by a carima model the control algorithm relies on the generalized minimum variance approach the convergence proof is carried out by resorting to an interlaced stochastic gradient identification algorithm

analysis of planar shapes using geodesic paths on shape spaces for analyzing shapes of planar closed curves we propose differential geometric representations of curves using their direction functions and curvature functions shapes are represented as elements of infinite dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces we use a fourier basis to represent tangents to the shape spaces and then use a gradient based shooting method to solve for the tangent that connects any two shapes via a geodesic using the surrey fish database we demonstrate some applications of this approach 1 interpolation and extrapolations of shape changes 2 clustering of objects according to their shapes 3 statistics on shape spaces and 4 bayesian extraction of shapes in low quality images

recursive system identification and adaptive control by use of the modified least squares algorithm in this paper we first use a weaker than persistent excitation condition to establish the strong consistency of mls for mimo stochastic systems without monitoring and give a comparison between various conditions for consistency then we show the global convergence for adaptive tracking based on mls and finally the adaptive tracking and strong consistency results are combined

a multivariable self tuning controller with integral action abstract in this paper a novel stochastic self tuning control algorithm for multivariable linear systems described by controlled autoregressive integrating moving average carima models is proposed in particular the presence of an integral action on each component of the error vector ensures the robust offset rejection for any constant load disturbance acting on the plant the leading assumption is made that the system interactor matrix is known a priori the algorithm is derived by resorting to the generalized minimum variance approach two simulation examples illustrate the main features of the method

a multi dimensional scaling approach to shape analysis summary we propose an alternative to kendall s shape space for reflection shapes of configurations in rm with k labelled vertices where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations the proposed approach embeds the space of such shapes into the space p k 1 of k 1 x k 1 real symmetric positive semidefinite matrices which is the closure of an open subset of a euclidean space and defines mean shape as the natural projection of euclidean means in p k 1 on to the embedded copy of the shape space this approach has strong connections with multi dimensional scaling and the mean shape so defined gives good approximations to other commonly used definitions of mean shape we also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions

the complex bingham distribution and shape analysis summary a complex version of the bingham distribution is defined on the unit complex sphere in ck various statistical properties including asymptotic normality under high concentration are derived symmetries in the distribution make it a natural tool for the analysis of the shape of landmark data in two dimensions strengths and weaknesses of this approach are investigated links and contrasts with bookstein co ordinates are discussed a hybrid approach to principal component analysis based on complex and real co ordinates is suggested

rationality of some projective linear actions let k be any field which may not be algebraically closed v a finite dimensional vector space over k gl v where the order of can be finite or infinite theorem if dimk v 3 then both k v and k p v are rational purely transcendental over k similar results hold for a cyclic affine action

directions and projective shapes this paper deals with projective shape analysis which is a study of finite configurations of points modulo projective transformations the topic has various applications in machine vision we introduce a convenient projective shape space as well as an appropriate coordinate system for this shape space for generic configurations of k points in m dimensions the resulting projective shape space is identified as a product of k m 1 copies of axial spaces rp m this identification leads to the need for developing multivariate directional and multivariate axial analysis and we propose parametric models as well as nonparametric methods for these areas in particular we investigate the frechet extrinsic mean for the multivariate axial case asymptotic distributions of the appropriate parametric and nonparametric tests are derived we illustrate our methodology with examples from machine vision

statistical analysis on high dimensional spheres and shape spaces we consider the statistical analysis of data on high dimensional spheres and shape spaces the work is of particular relevance to applications where high dimensional data are available a commonly encountered situation in many disciplines first the uniform measure on the infinite dimensional sphere is reviewed together with connections with wiener measure we then discuss densities of gaussian measures with respect to wiener measure some nonuniform distributions on infinite dimensional spheres and shape spaces are introduced and special cases which have important practical consequences are considered we focus on the high dimensional real and complex bingham uniform von mises fisher fisher bingham and the real and complex watson distributions asymptotic distributions in the cases where dimension and sample size are large are discussed approximations for practical maximum likelihood based inference are considered and in particular we discuss an application to brain shape modeling