generalizations of chebyshev polynomials and polynomial mappings in this paper we show how polynomial mappings of degree from a union of disjoint intervals onto generate a countable number of special cases of generalizations of chebyshev polynomials we also derive a new expression for these generalized chebyshev polynomials for any genus from which the coefficients of can be found explicitly in terms of the branch points and the recurrence coefficients we find that this representation is useful for specializing to polynomial mapping cases for small where we will have explicit expressions for the recurrence coefficients in terms of the branch points we study in detail certain special cases of the polynomials for small degree mappings and prove a theorem concerning the location of the zeroes of the polynomials we also derive an explicit expression for the discriminant for the genus 1 case of our chebyshev polynomials that is valid for any configuration of the branch point

some comments on tridiagonal p r toeplitz matrices abstract the aim of this note is to extend the notion of a tridiagonal p r toeplitz matrix recently introduced and to provide procedures for short proofs to results on the invertibility of such matrices

adaptive cost dynamic time warping distance in time series analysis for classification dynamic time warping dtw distance is commonly used in measuring similarity between time series for classification in order to obtain the minimum cumulative distance however dtw distance may map multiple points on one time series to one point on another and this makes time series over stretched and compressed resulting in missing important feature information thus influence the classification accuracy in this paper we propose a method called adaptive cost dynamic time warping distance ac dtw which adjusts the number of points on one time series mapped to the points on another ac dtw records the trajectories of all points and then adaptively allocates the cost rate to each point by calculating cost function at the next step the results of the experiments implemented on 17 ucr datasets by using nearest neighbor classifier demonstrate that ac dtw prevails in criterion of higher accuracy rate in comparison with some existing methods adaptive cost is introduced to dynamic time warping for better classification cost function is proposed for dynamic time warping experiments on ucr datasets prove that ac dtw perform better than other methods

orthogonal and extremal polynomials on several intervals abstract let a1 1 where is positive on el are characterised by orthogonality conditions and some new results are proved then the connection with polynomials orthogonal with respect to h where has an odd number of zeros in each interval a2j a2j 1 j 1 l 1 is shown and a full description of orthogonal polynomials having periodic respectively asymptotic periodic recurrence coefficients is given finally the asymptotic behaviour of polynomials orthogonal on several intervals is discussed

possibilistic nonlinear dynamical analysis for pattern recognition a nonlinear dynamical system can be defined as a study of any system that implies motion change or evolution in time where a change in one variable is not proportional to a change in a related variable the mathematical operations underlying such a system are very useful for pattern recognition with time series data one of the most recent developments in nonlinear dynamical analysis is the so called approximate entropy family however its algorithms are deterministic and do not consider uncertainty where the modeling of possibility can be appropriate and advantageous in many practical situations thus possibilistic entropy algorithms are proposed in this paper as a new methodology for nonlinear dynamical analysis the proposed approach is based on the notions of the approximate entropy family geostatistics and the theory of fuzzy sets furthermore for the first time nonlinear dynamical analysis of mass spectrometry data is presented for computer based recognition of potential protein biomarkers and classification which can be utilized for early disease prediction experimental results using proteomic and genetic data have shown the potential application of the proposed possibilistic nonlinear dynamical analysis to the study of complex biosignals

some comments on k tridiagonal matrices recently several papers emerged in the literature discussing the determinant the spectrum and the inverse of the so called k tridiagonal matrices the aim of this comment is to provide a general unified and attractive approach to them recalling some less known results

garch based robust clustering of time series in this paper we propose different robust fuzzy clustering models for classifying heteroskedastic volatility time series following the so called model based approach to time series clustering and using a partitioning around medoids procedure the proposed models are based on a garch parametric modeling of the time series i e the unconditional volatility and the time varying volatility garch representation of the time series we first suggest a timid robustification of the fuzzy clustering then we propose three robust fuzzy clustering models belonging to the so called metric noise and trimmed approaches respectively each model neutralizes the negative effects of the outliers in the clustering process in a different manner in particular the first robust model based on the metric approach achieves its robustness with respect to outliers by taking into account a robust distance measure the second based on the noise approach achieves its robustness by introducing a noise cluster represented by a noise prototype the third based on the trimmed approach achieves its robustness by trimming away a certain fraction of outlying time series the usefulness and effectiveness of the proposed clustering models is illustrated by means of a simulation study and two applications in finance and economics

the eigenvalues of a family of persymmetric anti tridiagonal 2 hankel matrices in this note we provide the general expression for the eigenvalues of a type of anti tridiagonal matrices

on the solution of a second order linear homogeneous difference equation with variable coefficients abstract a closed form solution of a second order linear homogeneous difference equation with variable coefficients is presented as an application of this solution we obtain expressions for cos n and sin n 1 sin as polynomials in cos

a recurrent probabilistic neural network with dimensionality reduction based on time series discriminant component analysis this paper proposes a probabilistic neural network nn developed on the basis of time series discriminant component analysis tsdca that can be used to classify high dimensional time series patterns tsdca involves the compression of high dimensional time series into a lower dimensional space using a set of orthogonal transformations and the calculation of posterior probabilities based on a continuous density hidden markov model with a gaussian mixture model expressed in the reduced dimensional space the analysis can be incorporated into an nn which is named a time series discriminant component network tsdcn so that parameters of dimensionality reduction and classification can be obtained simultaneously as network coefficients according to a backpropagation through time based learning algorithm with the lagrange multiplier method the tsdcn is considered to enable high accuracy classification of high dimensional time series patterns and to reduce the computation time taken for network training the validity of the tsdcn is demonstrated for high dimensional artificial data and electroencephalogram signals in the experiments conducted during the study

inversion of a tridiagonal jacobi matrix abstract a formula for the inverse of a general tridiagonal matrix is given in terms of the principal minors

inversion of tridiagonal matrices this paper presents a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist ukita s theorem is extended to characterize the class of matrices that have tridiagonal inverses

spectra of certain jacobi operators we discuss the spectra and in particular the essential spectra of some bounded self adjoint jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings

orthogonal and l q extremal polynomials on inverse images of polynomial mappings abstract let t be a polynomial of degree n and let k be a compact set with c first it is shown if zero is a best approximation to f from p n on k with respect to the lq norm q 1 then zero is also a best approximation to f t on t 1 k with respect to the l q t norm where t arises from by the transformation t in particular t is the equilibrium measure on t 1 k if is the equilibrium measure on k for q i e the sup norm a corresponding result is presented in this way polynomials minimal on several intervals on lemniscates on equipotential lines of compact sets etc are obtained special attention is given to lq minimal polynomials on julia sets next based on asymptotic results of widom we show that the minimum deviation of polynomials orthogonal with respect to a positive measure on t 1 k behaves asymptotically periodic and that the orthogonal polynomials have an asymptotically periodic behaviour too some open problems are also given

analytical inversion of general periodic tridiagonal matrices in this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed toeplitz for both periodic and non periodic tridiagonal matrices an efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given it is suited for implementation using computer algebra systems such as maple matlab macsyma and mathematica an example is also given to illustrate the algorithm

dynamic time warping under limited warping path length dynamic time warping dtw is probably the most popular distance measure for time series data because it captures flexible similarities under time distortions however dtw has long been suffering from the pathological alignment problem and most existing solutions which essentially impose rigid constraints on the warping path are likely to miss the correct alignments a crucial observation on pathological alignment is that it always leads to an abnormally large number of links between two sequences based on this new observation we propose a novel variant of dtw called ldtw which limits the total number of links during the optimization process of dtw ldtw not only oppresses the pathological alignment effectively but also allows more flexibilities when measuring similarities it is a softer constraint because we still let the optimization process of dtw decide how many links to allocate to each data point and where to put these links in this paper we introduce the motivation and algorithm of ldtw and we conduct a nearest neighbor classification experiment on ucr time series archive to show its performance

dynamic time warping under pointwise shape context similarity measures are of fundamental importance in time series data mining dynamic time warping dtw is a quite popular measure because it handles time distortions well however dtw has an inherent shortcoming in that dtw can lead to pathological alignments between time series where a single point maps onto a large subsection of another time series to overcome this problem we propose a novel variant of dtw named sc dtw sc dtw employs shape context a rich local shape descriptor to replace the raw observed values considered by conventional dtw the main novelties of sc dtw are 1 it deeply explores both the numerical nature and shape nature of time series and 2 neighborhood information for each point is taken into account sc dtw can generate a more feature to feature alignment between time series and thus serves as a robust similarity measure we test the performance of sc dtw on ucr time series datasets using the one nearest neighbor 1nn classifier compared with other well established methods sc dtw provides better accuracy on 24 of 34 datasets

eigenproblems for tridiagonal 2 toeplitz matrices and quadratic polynomial mappings abstract given a system of monic orthogonal polynomials mops p n x n 0 we characterize all the sequences of monic orthogonal polynomials q n x n 0 such that q 1 x x b q 2 n x p n 2 x n 0 1 2 where 2 is a fixed polynomial of degree exactly 2 and b is a fixed complex number with an appropriate choice of the mops p n x n 0 our results enables us to solve the eigenproblem of a tridiagonal 2 toeplitz matrix giving an alternative proof to a recent result by m j c gover we also find the relations between the jacobi matrices corresponding to the mops p n x n 0 and q n x n 0 finally we show that if p n x n 0 is a semiclassical orthogonal polynomial sequence then so is q n x n 0 and in particular we analyze the classical case in detail

inversion of jacobi s tridiagonal matrix abstract this paper gives a simple algorithm for finding the explicit inverse of a general jacobi tridiagonal matrix a few illustrations are also provided to demonstrate the applicability of our algorithm

tridiagonal matrices invertibility and conditioning abstract tridiagonal matrices arise in a large variety of applications most of the time they are diagonally dominant and this is indeed the case most extensively studied in this paper we study in a unified approach the invertibility and the conditioning of such matrices the results presented provide practical criteria for a tridiagonal and irreducible matrix to be both invertible and well conditioned an application to a singular perturbation boundary value problem is then presented

explicit inverse of a tridiagonal k toeplitz matrix we obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k toeplitz matrix a the proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of chebyshev polynomials of the second kind we also compute the characteristic polynomial of a which enables us to state some conditions for the existence of a 1 our results also extend known results for the case when the residue mod k of the order of a is equal to 0 or k 1 numer math 10 1967 pp 153 161

an inversion algorithm for a banded matrix in this paper an inversion algorithm for a banded matrix is presented the n twisted decompositions of a banded matrix are given first then the inverse of the matrix is obtained one column at time the method is about two times faster than the standard method based on the lu decomposition as is shown with the analysis of computing complexity and the numerical experiments

eigenvalues of a certain class of tridiagonal matrices we make a short comment on a paper involving the minimal eigenvalues of a specialized version of 2 toeplitz tridiagonal matrices

orthogonal polynomials and cubic transformations abstract starting from a sequence pn of orthogonal polynomials with respect to a quasi definite or regular linear functional we construct a new sequence qn of orthogonal polynomials such that q3n x pn x3 the basic tool is the three term recurrence relation which such a family satisfies as an application we present some examples concerning chebyshev and hermite polynomials

on the eigenvalues of some tridiagonal matrices a solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by william trench the method presented can be generalizable to other problems

inverses of k toeplitz matrices with applications to resonator arrays with multiple receivers abstract we find closed form algebraic formulas for the elements of the inverses of tridiagonal 2 and 3 toeplitz matrices which are symmetric and have constant upper and lower diagonals these matrices appear respectively as the impedance matrices of resonator arrays in which a receiver is placed over every 2 or 3 resonators consequently our formulas allow to compute the currents of a wireless power transfer system in closed form allowing for a simple exact and symbolic analysis thereof small numbers are chosen for illustrative purposes but the elementary linear algebra techniques used can be extended to k toeplitz matrices of this special form with k arbitrary hence resonator arrays with a receiver placed over every k resonators can be analysed in the same way

faster retrieval with a two pass dynamic time warping lower bound the dynamic time warping dtw is a popular similarity measure between time series the dtw fails to satisfy the triangle inequality and its computation requires quadratic time hence to find closest neighbors quickly we use bounding techniques we can avoid most dtw computations with an inexpensive lower bound lb keogh we compare lb keogh with a tighter lower bound lb improved we find that lb improved based search is faster as an example our approach is 2 3 times faster over random walk and shape time series

explicit inverse of a tridiagonal p r toeplitz matrix abstract we have named tridiagonal p r toeplitz matrix to those tridiagonal matrices in which each diagonal is a quasi periodic sequence d p j r d j so with period p n but multiplied by a real number r we present here the necessary and sufficient conditions for the invertibility of this kind of matrices and explicitly compute their inverse the techniques we use are related with the solution of boundary value problems associated to second order linear difference equations these boundary value problems can be expressed throughout the discrete schrodinger operator and their solutions can be computed using recent advances in the study of linear difference equations with quasi periodic coefficients the conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix whereas the solutions of the boundary value problems provides the entries of the inverse matrix

the inverse of a tridiagonal matrix in this paper explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case and then applying these extended results to a boundary value problem a formula for the characteristic polynomial is obtained in the process we also establish a connection between the matrix inverse and orthogonal polynomials in addition the case of a cyclic tridiagonal system is discussed

characterizations of generalized hermite and sieved ultraspherical polynomials a new characterization of the generalized hermite polynomials and of the orthogonal polynomials with respect to the measure ixy 1 x2 1 2dx is derived which is based on a reversing property of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of laguerre and jacobi polynomials a similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind these results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

iterative solution of indefinite symmetric linear systems by methods using orthogonal polynomials over two disjoint intervals in the classical chebyshev algorithm for solving symmetric positive definite linear systems ax b the approximate solution is of the form x k x 0 s k a r 0 where r 0 b ax 0 is the initial residual and s k is a polynomial of degree k 1 which minimizes 1 lambda s k lambda infty where cdot infty is the uniform norm on some interval s containing the spectrum of a but not the origin if a is not positive definite the set s can be taken to be the union of two disjoint intervals a b and c d with b 0 c and a generalization of the chebyshev algorithm based on the minimax polynomial has been considered by several authors in this paper we propose a new technique based upon the least squares polynomial in the set s i e the polynomial t k which minimizes 1 lambda t k lambda w where cdot w is an l 2 norm with respect to a weight function w lambda defined on s with a suitable weight function w lambda the po

sieved ultraspherical polynomials the continuous q ultraspherical polynomials contain a number of important examples as limiting or special cases one of these arose in allaway s ph d thesis in a previous paper we solved a characterization problem essentially equivalent to allaway s and showed that these polynomials arose from the q ultraspherical polynomials when q approached a root of unity a second class of such polynomials is found and the recurrence relation and orthogonality relation are found for each of these polynomials the orthogonality is interesting because the weight function has a finite number of zeros in 1 1 generating functions and other formulas are also found

on the inverses of general tridiagonal matrices in this work the sign distribution for all inverse elements of general tridiagonal h matrices is presented in addition some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained based on the sign distribution these bounds greatly improve some well known results due to ostrowski 1952 23 shivakumar and ji 1996 26 nabben 1999 21 22 and recently given by peluso and politi 2001 24 peluso and popolizio 2008 25 and so forth it is also stated that the inverse of a general tridiagonal matrix may be described by 2n 2 parameters k k 2n and k k 1n 1 instead of 2n 2 ones as given by el mikkawy 2004 3 el mikkawy and karawia 2006 4 and huang and mccoll 1997 10 according to these results a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented which improves some recent results finally several applications to the preconditioning technology the numerical solution of differential equations and the birth death processes together with numerical tests are given

modelling ordered categorical data recent advances and future challenges this article summarizes recent advances in the modelling of ordered categorical ordinal response variables we begin by reviewing some models for ordinal data introduced in the literature in the past 25 years we then survey recent extensions of these models and related methodology for special types of applications such as for repeated measurement and other forms of clustering we also survey other aspects of ordinal modelling such as small sample analyses power and sample size considerations and availability of software throughout we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data

the score test for independence in r x c contingency tables with missing data in this paper the score test statistic for testing independence in r x c contingency tables with missing data is proposed under the null hypothesis of independence the statistic has an approximate chi squared distribution with r 1 c 1 degrees of freedom the proposed test statistic is quite similar to the pearson chi squared statistic with complete data and unlike the likelihood ratio statistic for testing independence its computation is simple and noniterative in addition a score test statistic is proposed for testing independence when the rows and columns of the r x c table are ordinal finally extensions of the score statistics to test for conditional independence in a set of r x c contingency tables with missing data are described this yields score test statistics that are natural extensions of the mantel haenszel statistic an example using a subset of data from the six cities study is presented to illustrate the methods

comments on the spectrum of a tridiagonal k toeplitz matrix abstract in this note we will discuss some errors presented in a recent paper on r periodic tridiagonal matrices in addition we present several results that had been previously reported in mathematical literature

orthogonality properties of iterated polynomial mappings we consider a measure defined on a complex contour and its associated orthogonal polynomials the action of a polynomial transformation on the measure and the transformation laws of the corresponding orthogonal polynomials are given iterating the transformation provides an invariant measure whose support is the julia set corresponding to the polynomial transformation it appears that up to a constant the iterated polynomials generated by the initial mapping form a subset of the invariant set of orthogonal polynomials which fulfill a three term recursion relation an algorithm is given to compute the coefficients of this recursion relation which can be interpreted as a linear extension of the iterative procedure

spectral properties of certain tridiagonal matrices abstract we study spectral properties of irreducible tridiagonal k toeplitz matrices and certain matrices which arise as perturbations of them

analytical inversion of general tridiagonal matrices in this paper we give a complete analysis for general tridiagonal matrix inversion for both non block and block cases and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or toeplitz tridiagonal matrices

on the number of p vertices of some graphs abstract we provide positive answers to some open questions presented recently by kim and shader on a continuity like property of the p vertices of nonsingular matrices whose graph is a path a criterion for matrices associated with more general trees to have at most n 1 p vertices is established the cases of the cycles and stars are also analyzed several algorithms for generating matrices with a given number of p vertices are proposed

orthogonal polynomials of equilibrium measures supported on cantor sets the equilibrium measure of a compact set is a fundamental object in logarithmic potential theory we compute numerically this measure and its orthogonal polynomials when the compact set is a cantor set defined by an iterated function system we first construct sequences of discrete measures via the solution of large systems of non linear equations that converge weakly to the equilibrium measure successively we compute their jacobi matrices via standard procedures enhanced for the scope numerical estimates of the convergence rate to the limit jacobi matrix are provided which show stability and efficiency of the whole procedure as a companion result we also compute jacobi matrices in two other cases equilibrium measures on finite sets of intervals and balanced measures of iterated function systems these algorithms can reach large polynomial orders therefore we study the asymptotic behavior of the orthogonal polynomials and by a natural extension of the concept of regular root asymptotics we derive an efficient scheme for the computation of complex green s functions and of related conformal mappings

generalized fibonacci sequences via orthogonal polynomials abstract using tools from the theory of orthogonal polynomials we obtain and extend some recent results on generalized fibonacci sequences by stating a binet s type formula for a sequence of real or complex numbers q n n 0 defined by q 0 0 q 1 1 q m a j q m 1 b j q m 2 m j mod k where k 3 is a fix integer number and a 0 a 1 a k 1 and b 0 b 1 b k 1 are 2 k given real or complex numbers with b j 0 for 0 j k 1

solutions of linear difference equations with variable coefficients the explicit solution of a linear difference equation of unbounded order with variable coefficients is presented as special cases the solutions of nonhomogeneous and homogeneous linear difference equations of order n with variable coefficients are obtained from these solutions we also get expressions for the product of companion matrices and the power of a companion matrix

multi modal and multi scale temporal metric learning for a robust time series nearest neighbors classification abstract the definition of a metric between time series is inherent to several data analysis and mining tasks including clustering classification or forecasting time series data present naturally several modalities covering their amplitude behavior or frequential spectrum that may be expressed with varying delays and at multiple temporal scales exhibited globally or locally combining several modalities at multiple temporal scales to learn a holistic metric is a key challenge for many real temporal data applications this paper proposes a multi modal and multi scale temporal metric learning m 2 tml approach for a robust time series nearest neighbors classification the solution lies in embedding time series into a dissimilarity space where a pairwise svm is used to learn both linear and non linear combined metric a sparse and interpretable variant of the solution shows the ability of the learned temporal metric to localize accurately discriminative modalities as well as their temporal scales a wide range of 30 public and challenging datasets encompassing images traces and ecg data are used to show the efficiency and the potential of m 2 tml for an effective time series nearest neighbors classification

inversion of k tridiagonal matrices with toeplitz structure in this paper we consider an inverse problem with the k tridiagonal toeplitz matrices a theoretical result is obtained that under certain assumptions the explicit inverse of a k tridiagonal toeplitz matrix can be derived immediately two numerical examples are given to demonstrate the validity of our results

explicit inverses of some tridiagonal matrices abstract we give explicit inverses of tridiagonal 2 toeplitz and 3 toeplitz matrices which generalize some well known results concerning the inverse of a tridiagonal toeplitz matrix

a limit q 1 for the big q jacobi polynomials we study a new family of classical orthogonal polynomials here called big 1 jacobi polynomials which satisfy apart from a 3 term recur rence relation an eigenvalue problem with differential operators of dunkl type these polynomials can be obtained from the big q jacobi polynomials in the limit q 1 an explicit expression of these polynomials in terms of gauss hypergeometric functions is found the big 1 jacobi polynomials are orthogo nal on the union of two symmetric intervals of the real axis we show that the big 1 jacobi polynomials can be obtained from the bannai ito polynomials when the orthogonality support is extended to an infinite number of points we further indicate that these polynomials provide a nontrivial realization of the askey wilson algebra for q 1 the novelty lies in the fact that l is a differential difference operator of special type namely l is a linear operator which is of first order in the derivative operator x and contains also the reflection operator r which acts as rf x f x roughly speaking one can say that l belongs to the class of dunkl operators 8 which contain both the operators x and r nevertheless the operator l differs from the standard dunkl operators in a fundamental way indeed l preserves the linear space of polynomials of any given maximal degree this basic property allows to construct a complete system of polynomials pn x n 0 1 2 as eigenfunctions of the operator l guided by the q 1 limit of the little q jacobi polynomials we derived in 20 an explicit expression of the polynomials pn x in terms of gauss hypergeometric functions we also found explicitly the recurrence coefficients and s that the polynomials pn x are orthogonal on the interval 1 1 with a weight function related to the weight function of the generalized jacobi polynomials 7 we also proved that they admit the dunkl classical property 5 and further demonstrated that the operator l together with the multiplication operator x form a special case of the askey wilson algebra aw 3 21 corresponding to the parameter q 1

weighted dynamic time warping for time series classification dynamic time warping dtw which finds the minimum path by providing non linear alignments between two time series has been widely used as a distance measure for time series classification and clustering however dtw does not account for the relative importance regarding the phase difference between a reference point and a testing point this may lead to misclassification especially in applications where the shape similarity between two sequences is a major consideration for an accurate recognition therefore we propose a novel distance measure called a weighted dtw wdtw which is a penalty based dtw our approach penalizes points with higher phase difference between a reference point and a testing point in order to prevent minimum distance distortion caused by outliers the rationale underlying the proposed distance measure is demonstrated with some illustrative examples a new weight function called the modified logistic weight function mlwf is also proposed to systematically assign weights as a function of the phase difference between a reference point and a testing point by applying different weights to adjacent points the proposed algorithm can enhance the detection of similarity between two time series we show that some popular distance measures such as dtw and euclidean distance are special cases of our proposed wdtw measure we extend the proposed idea to other variants of dtw such as derivative dynamic time warping ddtw and propose the weighted version of ddtw we have compared the performances of our proposed procedures with other popular approaches using public data sets available through the ucr time series data mining archive for both time series classification and clustering problems the experimental results indicate that the proposed approaches can achieve improved accuracy for time series classification and clustering problems

on sieved orthogonal polynomials iii orthogonality on several intervals on introduit deux generalisations des polynomes de chelryshev dont le spectre continu est x 2 c 1 c t k x 2 c 1 c ou c est un parametre positif on definit les fonctions poids on obtient les fonctions generatrices et les formules explicites

on semiclassical orthogonal polynomials via polynomial mappings abstract we consider orthogonal polynomials via polynomial mappings in the framework of the semiclassical class we prove that this class is stable under polynomial transformations several consequences of this fact are deduced as an application we analyze in detail cubic transformations for semiclassical orthogonal polynomials of class at most 2 recovering and extending some results proved recently for class 1 and producing new examples of semiclassical orthogonal polynomials of class 2 in particular we show how to obtain integral representations for the regular functionals with respect to which these new semiclassical families are orthogonal

on some n 1 symmetric linear functionals abstract many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle particular aspects are treated for quadratic or cubic decompositions in this paper we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality this technique is based on the decomposition with respect to the cyclic group of order n the main theorem is applied to the orthogonality on n 1 symmetric curves some particular cases are singled out these results may be useful in studying hermite pade approximations vector continued fractions and dynamical systems

the eigenproblem of a tridiagonal 2 toeplitz matrix abstract the characteristic polynomial of a tridiagonal 2 toeplitz matrix is shown to be closely connected to polynomials which satisfy the three point chebyshev recurrence relationship this is an extension of the well known result for a tridiagonal toeplitz matrix when the order of the matrix is odd the eigenvalues are found explicitly in terms of the chebyshev zeros the eigenvectors are found in terms of the polynomials satisfying the three point recurrence relationship

on the powers and the inverse of a tridiagonal matrix in this paper we present an eigendecomposition of a tridiagonal matrix tridiagonal matrix powers and inverse are derived as consequence we get some relations verified by the coefficients of the inverse and the powers of a tridiagonal matrix

shapedtw shape dynamic time warping abstract dynamic time warping dtw is an algorithm to align temporal sequences with possible local non linear distortions and has been widely applied to audio video and graphics data alignments dtw is essentially a point to point matching method under some boundary and temporal consistency constraints although dtw obtains a global optimal solution it does not necessarily achieve locally sensible matchings concretely two temporal points with entirely dissimilar local structures may be matched by dtw to address this problem we propose an improved alignment algorithm named shape dynamic time warping shapedtw which enhances dtw by taking point wise local structural information into consideration shapedtw is inherently a dtw algorithm but additionally attempts to pair locally similar structures and to avoid matching points with distinct neighborhood structures we apply shapedtw to align audio signal pairs having ground truth alignments as well as artificially simulated pairs of aligned sequences and obtain quantitatively much lower alignment errors than dtw and its two variants when shapedtw is used as a distance measure in a nearest neighbor classifier nn shapedtw to classify time series it beats dtw on 64 out of 84 ucr time series datasets with significantly improved classification accuracies by using a properly designed local structure descriptor shapedtw improves accuracies by more than 10 on 18 datasets to the best of our knowledge shapedtw is the first distance measure under the nearest neighbor classifier scheme to significantly outperform dtw which had been widely recognized as the best distance measure to date our code is publicly accessible at https github com jiapingz shapedtw

pearson s chi square test and rank correlation inferences for clustered data pearson s chi square test has been widely used in testing for association between two categorical responses spearman rank correlation and kendall s tau are often used for measuring and testing association between two continuous or ordered categorical responses however the established statistical properties of these tests are only valid when each pair of responses are independent where each sampling unit has only one pair of responses when each sampling unit consists of a cluster of paired responses the assumption of independent pairs is violated in this article we apply the within cluster resampling technique to u statistics to form new tests and rank based correlation estimators for possibly tied clustered data we develop large sample properties of the new proposed tests and estimators and evaluate their performance by simulations the proposed methods are applied to a data set collected from a pet ct imaging study for illustration

a new recursive algorithm for inverting general tridiagonal and anti tridiagonal matrices in the current article the authors present a new recursive symbolic computational algorithm for inverting general tridiagonal and anti tridiagonal matrices an illustrative example is given

universality limits for generalized jacobi measures abstract in this paper universality limits are studied in connection with measures which exhibit power type singular behavior somewhere in their support we extend the results of lubinsky for jacobi measures supported on 1 1 to generalized jacobi measures supported on a compact subset of the real line where the singularity can be located in the interior or at an endpoint of the support the analysis is based upon the riemann hilbert method christoffel functions the polynomial inverse image method of totik and the normal family approach of lubinsky

a time series representation model for accurate and fast similarity detection similarity search and detection is a central problem in time series data processing and management most approaches to this problem have been developed around the notion of dynamic time warping whereas several dimensionality reduction techniques have been proposed to improve the efficiency of similarity searches due to the continuous increasing of sources of time series data and the cruciality of real world applications that use such data we believe there is a challenging demand for supporting similarity detection in time series in a both accurate and fast way our proposal is to define a concise yet feature rich representation of time series on which the dynamic time warping can be applied for effective and efficient similarity detection of time series we present the derivative time series segment approximation dsa representation model which originally features derivative estimation segmentation and segment approximation to provide both high sensitivity in capturing the main trends of time series and data compression we extensively compare dsa with state of the art similarity methods and dimensionality reduction techniques in clustering and classification frameworks experimental evidence from effectiveness and efficiency tests on various datasets shows that dsa is well suited to support both accurate and fast similarity detection

sieved orthogonal polynomials vii generalized polynomial mappings systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second order difference equations are studied in detail applications are given to the theory of the recently discovered sieved orthogonal polynomials the connection with polynomial mappings is examined an example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included

a time series forest for classification and feature extraction a tree ensemble method referred to as time series forest tsf is proposed for time series classification tsf employs a combination of entropy gain and a distance measure referred to as the entrance entropy and distance gain for evaluating the splits experimental studies show that the entrance gain improves the accuracy of tsf tsf randomly samples features at each tree node and has computational complexity linear in the length of time series and can be built using parallel computing techniques the temporal importance curve is proposed to capture the temporal characteristics useful for classification experimental studies show that tsf using simple features such as mean standard deviation and slope is computationally efficient and outperforms strong competitors such as one nearest neighbor classifiers with dynamic time warping

functional echo state network for time series classification abstract echo state networks esns are a new approach to recurrent neural networks rnns that have been successfully applied in many domains nevertheless an esn is a predictive model rather than a classifier and methods to employ esns in time series classification tsc tasks have not yet been fully explored in this paper we propose a novel esn approach named functional echo state network fesn for time series classification the basic idea behind fesn is to replace the numeric variable output weights of an esn with time varying output weight functions and introduce a temporal aggregation operator to the output layer that can project temporal signals into discrete class labels thereby transforming the esn from a predictive model into a true classifier subsequently to learn the output weight functions a spatio temporal aggregation learning algorithm is proposed based on orthogonal function basis expansion by leveraging the nonlinear mapping capacity of a reservoir and the accumulation of temporal information in the time domain fesn can not only enhance the separability of different classes in a high dimensional functional space but can also consider the relative importance of temporal data at different time steps according to dynamic output weight functions theoretical analyses and experiments on an extensive set of ucr data were conducted on fesn the results show that fesn yields better performance than single algorithm methods has comparable accuracy with ensemble based methods and exhibits acceptable computational complexity interestingly for some time series datasets we visualized some interpretable features extracted by fesn via specific patterns within the output weight functions

error in the upwind differencing of the convection diffusion equation abstract in convection diffusion problems a first order upwind difference approximation is usually used on a locally refined mesh to get a wiggle free solution which is then corrected using some form of deferred correction to get second order accuracy pereyra 1966 has shown that improving accuracy of a low order scheme using deferred correction requires the existence of an asymptotic expansion for the discretization error of the scheme concerned in this paper bounds on the discrezazation error are obtained when the first order upwind difference scheme is applied to the convection diffusion equation on a locally refined mesh both in one and two dimensions existence of an asymptotic expansion for the discretization error is established different formulae at the nodes separating subregions of differing mesh sizes are analysed and numerical tests performed to illustrate validity of the resulting expressions

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

on some tridiagonal k toeplitz matrices algebraic and analytical aspects applications in this paper we use the analytic theory for 2 and 3 toeplitz matrices to obtain the explicit expressions for the eigenvalues eigenvectors and the spectral measure associated to the corresponding infinite matrices as an application we consider two solvable models related with the so called chain model some numerical experiments are also included

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials