estimating time evolving partial coherence between signals via multivariate locally stationary wavelet processes we consider the problem of estimating time localized cross dependence in a collection of nonstationary signals to this end we develop the multivariate locally stationary wavelet framework which provides a time scale decomposition of the signals and thus naturally captures the time evolving scale specific cross dependence between components of the signals under the proposed model we rigorously define and estimate two forms of cross dependence measures wavelet coherence and wavelet partial coherence these dependence measures differ in a subtle but important way the former is a broad measure of dependence which may include indirect associations i e dependence between a pair of signals that is driven by another signal conversely wavelet partial coherence measures direct linear association between a pair of signals i e it removes the linear effect of other observed signals our time scale wavelet partial coherence estimation scheme thus provides a mechanism for identifying hidden dynamic relationships within a network of nonstationary signals as we demonstrate on electroencephalograms recorded in a visual motor experiment

most recent changepoint detection in panel data detecting recent changepoints in time series can be important for short term prediction as we can then base predictions just on the data since the changepoint in many applications we have panel

spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

reformulation of the covering and quantizer problems as ground states of interacting particles it is known that the sphere packing problem and the number variance problem closely related to an optimization problem in number theory can be posed as energy minimizations associated with an infinite number of point particles in d dimensional euclidean space r d interacting via certain repulsive pair potentials we reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in r d that generally involve single body two body three body and higher body interactions this is done by linking the covering and quantizer problems to certain optimization problems involving the void nearest neighbor functions that arise in the theory of random media and statistical mechanics these reformulations which again exemplify the deep interplay between geometry and physics allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems the covering and quantizer problems have relevance in numerous applications including wireless communication network layouts the search of high dimensional data parameter spaces stereotactic radiation therapy data compression digital communications meshing of space for numerical analysis and coding and cryptography among other examples in the first three space dimensions the best known solutions of the sphere packing and number variance problems or their dual solutions are directly related to those of the covering and quantizer problems but such relationships may or may not exist for d 4 depending on the peculiarities of the dimensions involved our reformulation sheds light on the reasons for these similarities and differences we also show that disordered saturated sphere packings provide relatively thin economical coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions in the case of the quantizer problem we derive improved upper bounds on the quantizer error using sphere packing solutions which are generally substantially sharper than an existing upper bound in low to moderately large dimensions we also demonstrate that disordered saturated sphere packings yield relatively good quantizers finally we remark on possible applications of our results for the detection of gravitational waves

changepoint estimation another look at multiple testing problems summary we consider large scale multiple testing for data that have locally clustered signals with this structure we apply techniques from change point analysis and propose a boundary detection algorithm so that the clustering information can be utilised consequently the precision of the multiple testing procedure is substantially improved we study tests with independent as well as dependent p values monte carlo simulations suggest that the methods perform well with realistic sample sizes and show improved detection ability compared with competing methods our procedure is applied to a genome wide association dataset of blood lipids

wild binary segmentation for multiple change point detection we propose a new technique called wild binary segmentation wbs for consistent estimation of the number and locations of multiple change points in data we assume that the number of change points can increase to infinity with the sample size due to a certain random localisation mechanism wbs works even for very short spacings between the change points and or very small jump magnitudes unlike standard binary segmentation on the other hand despite its use of localisation wbs does not require the choice of a window or span parameter and does not lead to a significant increase in computational complexity wbs is also easy to code we propose two stopping criteria for wbs one based on thresholding and the other based on what we term the strengthened schwarz information criterion we provide default recommended values of the parameters of the procedure and show that it offers very good practical performance in comparison with the state of the art the wbs methodology is implemented in the r package wbs available on cran in addition we provide a new proof of consistency of binary segmentation with improved rates of convergence as well as a corresponding result for wbs

slex analysis of multivariate nonstationary time series we develop a procedure for analyzing multivariate nonstationary time series using the slex library smooth localized complex exponentials which is a collection of bases each basis consisting of waveforms that are orthogonal and time localized versions of the fourier complex exponentials under the slex framework we build a family of multivariate models that can explicitly characterize the time varying spectral and coherence properties every model has a spectral representation in terms of a unique slex basis before selecting a model we first decompose the multivariate time series into nonstationary components with uncorrelated nonredundant spectral information the best slex model is selected using the penalized log energy criterion which we derive in this article to be the kullback leibler distance between a model and the slex principal components of the multivariate time series the model selection criterion takes into account all of the pairwise cross correlation simultaneously in the multivar

on almost regular tournament matrices abstract spectral and determinantal properties of a special class m n of 2n 2n almost regular tournament matrices are studied in particular the maximum perron value of the matrices in this class is determined and shown to be achieved by the brualdi li matrix which has been conjectured to have the largest perron value among all tournament matrices of even order we also establish some determinantal inequalities for matrices in m n and describe the structure of their associated walk spaces

detection of changes in multivariate time series with application to eeg data the primary contributions of this article are rigorously developed novel statistical methods for detecting change points in multivariate time series we extend the class of score type change point statistics considered in 2007 by huskova praskova and steinebach to the vector autoregressive var case and the epidemic change alternative our proposed procedures do not require the observed time series to actually follow the var model instead following the strategy implicitly employed by practitioners our approach takes model misspecification into account so that our detection procedure uses the model background merely for feature extraction we derive the asymptotic distributions of our test statistics and show that our procedure has asymptotic power of 1 the proposed test statistics require the estimation of the inverse of the long run covariance matrix which is particularly difficult in higher dimensional settings i e where the dimension of the time series and the dimension of the parameter vecto

on tournament matrices abstract let t be a tournament of order n with adjacency matrix m we find several conditions that are equivalent to m being singular a correlation between the number of 3 cycles in t and the rank of m is established it is shown that asymptotically at least 1 2 of the tournament matrices are nonsingular we also derive bounds on the spectral radius of tournament matrices with a given row sum vector

spectral radii of tournament matrices whose graphs are related by an arc reversal abstract given an irreducible tournament matrix t and a pair of distinct indices i and j let t i j be the matrix obtained from t by transposing its principal submatrix on rows and columns i and j we establish one condition on rows i and j of t under which the spectral radius of t i j is no smaller than that of t and another condition on the i th and j th entries of the left and right perron vectors of t under which the spectral radius of t i j must be strictly smaller than that of t these conditions are used to compare the spectral radii of a class of toeplitz tournament matrices and the resulting comparison sheds light on some conjectures of brualdi and li further if t yields equality in a certain lower bound on the spectral radius of a tournament matrix then for any i and j we provide simple necessary and sufficient conditions for the spectral radius of t i j to be larger than that of t to be smaller than that of t and to be equal to that of t

optimal detection of changepoints with a linear computational cost in this article we consider the problem of detecting multiple changepoints in large datasets our focus is on applications where the number of changepoints will increase as we collect more data for example in genetics as we analyze larger regions of the genome or in finance as we observe time series over longer periods we consider the common approach of detecting changepoints through minimizing a cost function over possible numbers and locations of changepoints this includes several established procedures for detecting changing points such as penalized likelihood and minimum description length we introduce a new method for finding the minimum of such cost functions and hence the optimal number and location of changepoints that has a computational cost which under mild conditions is linear in the number of observations this compares favorably with existing methods for the same problem whose computational cost can be quadratic or even cubic in simulation studies we show that our new method can

some results on h hypertournament matrices abstract in this work we find a necessary and sufficient condition for the normality of an h hypertournament matrix moreover we give a sufficient condition for n 1 2 to be the spectral radius of a normal h hypertournament matrix of order n also we answer an open question suggested by kirkland

sequential multi sensor change point detection we develop a mixture procedure to monitor parallel streams of data for a change point that affects only a subset of them without assuming a spatial structure relating the data streams to one another observations are assumed initially to be independent standard normal random variables after a change point the observations in a subset of the streams of data have nonzero mean values the subset and the post change means are unknown the procedure we study uses stream specific generalized likelihood ratio statistics which are combined to form an overall detection statistic in a mixture model that hypothesizes an assumed fraction p 0 of affected data streams an analytic expression is obtained for the average run length arl when there is no change and is shown by simulations to be very accurate similarly an approximation for the expected detection delay edd after a change point is also obtained numerical examples are given to compare the suggested procedure to other procedures for unstructured problems and in one case where the problem is assumed to have a well defined geometric structure finally we discuss sensitivity of the procedure to the assumed value of p 0 and suggest a generalization

detection of multiple structural breaks in multivariate time series we propose a new nonparametric procedure referred to as mubred for the detection and estimation of multiple structural breaks in the autocovariance function of a multivariate second order piecewise stationary process which also identifies the components of the series where the breaks occur mubred is based on a comparison of the estimated spectral distribution on different segments of the observed time series and consists of three steps it starts with a consistent test which allows us to prove the existence of structural breaks at a controlled type i error second it estimates sets containing possible break points and finally these sets are reduced to identify the relevant structural breaks and corresponding components which are responsible for the changes in the autocovariance structure in contrast to all other methods proposed in the literature our approach does not make any parametric assumptions is not especially designed for detecting one single change point and addresses the problem of m

on the location of the spectrum of hypertournament matrices inclusion regions for the spectrum of a hypertournament matrix a are obtained based on a complex curve that relates the real and imaginary parts of the eigenvalues these results generalize and in certain cases improve the work of s kirkland linear and multilinear algebra 30 1991 261 the bounds obtained depend on the variance of the score vector their tightness is investigated using the notion of numerical range 2001 elsevier science inc all rights reserved

algebraic multiplicity of the eigenvalues of a tournament matrix let f denote the set of irreducible n x n tournament matrices here arc our main results 1 for all n 3 every matrix in k has at least three distinct eigenvalues such a matrix has exactly three distinct eigenvalues if and only if it is a hadamard tournament matrix 2 for all n 2 3 there is a matrix in y having n distinct eigenvalues 3 if cr denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z then ln 21 2 6 4 if r is the minimum perron value i e spectral radius of all matrices in r then 2 8

structural break estimation for nonstationary time series models this article considers the problem of modeling a class of nonstationary time series using piecewise autoregressive ar processes the number and locations of the piecewise ar segments as well as the orders of the respective ar processes are assumed unknown the minimum description length principle is applied to compare various segmented ar fits to the data the goal is to find the best combination of the number of segments the lengths of the segments and the orders of the piecewise ar processes such a best combination is implicitly defined as the optimizer of an objective function and a genetic algorithm is implemented to solve this difficult optimization problem numerical results from simulation experiments and real data analyses show that the procedure has excellent empirical properties the segmentation of multivariate time series is also considered assuming that the true underlying model is a segmented autoregression this procedure is shown to be consistent for estimating the location of

multiple change point detection for high dimensional time series via sparsified binary segmentation time series segmentation a k a multiple change point detection is a well established problem however few solutions are designed specifically for high dimensional situations in this paper our interest is in segmenting the second order structure of a high dimensional time series in a generic step of a binary segmentation algorithm for multivariate time series one natural solution is to combine cusum statistics obtained from local periodograms and cross periodograms of the components of the input time series however the standard maximum and average methods for doing so often fail in high dimensions when for example the change points are sparse across the panel or the cusum statistics are spuriously large in this paper we propose the sparsified binary segmentation sbs algorithm which aggregates the cusum statistics by adding only those that pass a certain threshold this sparsifying step reduces the impact of irrelevant noisy contributions which is particularly beneficial in high dimensions in order to show the consistency of sbs we introduce the multivariate locally stationary wavelet model for time series which is a separate contribution of this work

normal matrices and their principal submatrices of co order one abstract let a be a normal matrix v be any of its indices a v be the matrix obtained from a by deleting the v th row and column and be an eigenvalue of a v in our paper we construct the eigenspace of a associated with from that of a v in particular it is shown that if there is a unique jordan block of size strictly greater than one in the part of the jordan form of a v corresponding to then the geometric multiplicity of decreases by one under the transition from a v to a in other words the typical change of the spectral properties holds for the results obtained are applied to circulant matrices moreover in appendix to our paper we consider almost regular tournament matrices as principal submatrices of co order one of regular tournament matrices in particular it is observed that the brualdi li tournament matrix b 2 n of order 2n is permutationally similar to a principal submatrix of co order one of the circulant matrix of order 2n 1 with the first row 0 1 1 n 0 0 n as a consequence of this fact the weak brualdi li conjecture is formulated for principal submatrices of co order one of the adjacency matrices of cayley tournaments

a nonparametric approach for multiple change point analysis of multivariate data change point analysis has applications in a wide variety of fields the general problem concerns the inference of a change in distribution for a set of time ordered observations sequential detection is an online version in which new data are continually arriving and are analyzed adaptively we are concerned with the related but distinct offline version in which retrospective analysis of an entire sequence is performed for a set of multivariate observations of arbitrary dimension we consider nonparametric estimation of both the number of change points and the positions at which they occur we do not make any assumptions regarding the nature of the change in distribution or any distribution assumptions beyond the existence of the th absolute moment for some 0 2 estimation is based on hierarchical clustering and we propose both divisive and agglomerative algorithms the divisive method is shown to provide consistent estimates of both the number and the location of change points under standard

tournament matrices with extremal spectral properties abstract for a tournament matrix m of order n we define its walk space wm to be span m j 1 j 0 n 1 where 1 is the all ones vector we show that the dimension of wm equals the number of eigenvalues of m whose real parts are greater than 1 2 we then focus on tournament matrices whose walk space has particularly simple structure and characterize them in terms of their spectra specifically we characterize those tournament matrices such that mj1 is an eigenvector of m for some j 0 we also characterize the tournament matrices m such that jn 2m is a skew hadamard matrix throughout we illustrate our results with examples

estimates of the optimal density of sphere packings in high dimensions the problem of finding the asymptotic behavior of the maximal density max of sphere packings in high euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry one century ago minkowski obtained a rigorous lower bound on max that is controlled asymptotically by 1 2d where d is the euclidean space dimension an indication of the difficulty of the problem can be garnered from the fact that exponential improvement of minkowski s bound has proved to be elusive even though existing upper bounds suggest that such improvement should be possible using a statistical mechanical procedure to optimize the density associated with a test pair correlation function and a conjecture concerning the existence of disordered sphere packings s torquato and f h stillinger exp math 15 307 2006 the putative exponential improvement on max was found with an asymptotic behavior controlled by 1 2 0 77865 d using the same methods we investigate whether this exponential im

pick s inequality and tournaments abstract let t be a tournament on n nodes and let a be its adjacency matrix a brauer and i gentry observed that an inequality due to g pick implies that im 1 2 cot 2n for all eigenvalues of a we say that t is a pick or p tournament if equality holds for some we determine when equality holds in pick s inequality for arbitrary real matrices and use this to show that the p tournaments can all be constructed from the transitive tournament m by reversing the arcs between the sets of certain node partitions or cuts u the cuts are specified by 1 n vectors u such that utw 0 where w 1 n 1 t e i n this links the cuts to cyclotomic polynomials there is at least one p tournament on n nodes if and only if n 2k k 1 up to isomorphism there is precisely one p tournament on n nodes if and only if n p or n 2p for some odd prime p for odd n up toisomorphism the only regular p tournament on n nodes has matrix zn circ 0 1 1 0 0 a composition rule is used on the matrices zp to form all p tournaments on n 2kpl nodes l 1

a pairwise likelihood based approach for changepoint detection in multivariate time series models this paper develops a composite likelihood based approach for multiple changepoint estimation in multivariate time series we derive a criterion based on pairwise likelihood and minimum description length for estimating the number and locations of changepoints and for performing model selection in each segment the number and locations of the changepoints can be consistently estimated under mild conditions and the computation can be conducted efficiently with a pruned dynamic programming algorithm simulation studies and real data examples demonstrate the statistical and computational efficiency of the proposed method