## Publications

92 results found

Percival D, Walden A, 2020, Spectral Analysis for Univariate Time Series, Publisher: Cambridge University Press, ISBN: 9781139235723

Percival D, Walden A, 2020, Combining Direct Spectral Estimators, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 351-444, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Harmonic Analysis, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 511-592, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Periodogram and Other Direct Spectral Estimators, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 163-244, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Introduction to Spectral Analysis, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 1-20, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Parametric Spectral Estimators, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 445-510, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Linear Time-Invariant Filters, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 132-162, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Lag Window Spectral Estimators, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 245-350, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Foundations for Stochastic Spectral Analysis, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 107-131, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Simulation of Time Series, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 593-641, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Deterministic Spectral Analysis, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 47-106, ISBN: 978-1-107-02814-2

Percival D, Walden A, 2020, Stationary Stochastic Processes, SPECTRAL ANALYSIS FOR UNIVARIATE TIME SERIES, Publisher: CAMBRIDGE UNIV PRESS, Pages: 21-46, ISBN: 978-1-107-02814-2

Lutzeyer JF, Walden AT, 2019, Comparing spectra of graph shift operator matrices, 8th International Conference on Complex Networks and Their Applications (COMPLEX NETWORKS), Publisher: Springer Cham, Pages: 191-202, ISSN: 1860-949X

Typically network structures are represented by one of three different graph shift operator matrices: the adjacency matrix and unnormalised and normalised Laplacian matrices. To enable a sensible comparison of their spectral (eigenvalue) properties, an affine transform is first applied to one of them, which preserves eigengaps. Bounds, which depend on the minimum and maximum degree of the network, are given on the resulting eigenvalue differences. The monotonicity of the bounds and the structure of networks are related. Bounds, which again depend on the minimum and maximum degree of the network, are also given for normalised eigengap differences, used in spectral clustering. Results are illustrated on the karate dataset and a stochastic block model. If the degree extreme difference is large, different choices of graph shift operator matrix may give rise to disparate inference drawn from network analysis; contrariwise, smaller degree extreme difference results in consistent inference.

Lutzeyer J, Walden A, 2019, Comparing Spectra of Graph Shift Operator Matrices, COMPLEX NETWORKS 2019: The 8th International Conference on Complex Networks and Their Applications VIII, Publisher: Springer Verlag, ISSN: 1860-949X

Lutzeyer JF, Walden AT, 2019, Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices I: Theory

The Davis-Kahan theorem can be used to bound the distance of the spacesspanned by the first $r$ eigenvectors of any two symmetric matrices. We extendthe Davis-Kahan theorem to apply to the comparison of the union of eigenspacesof any two symmetric matrices by making use of polynomial matrix transforms andin so doing, tighten the bound. The transform allows us to move requirementspresent in the original Davis-Kahan theorem, from the eigenvalues of thecompared matrices on to the transformation parameters, with the latter beingunder our control. We provide a proof of concept example, comparing the spacesspanned by the unnormalised and normalised graph Laplacian eigenvectors for$d$-regular graphs, in which the correct transform is automatically identified.

Lutzeyer JF, Walden AT, 2019, Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices II: Computation and Applications

The extended Davis-Kahan theorem makes use of polynomial matrixtransformations to produce bounds at least as tight as the standard Davis-Kahantheorem. The optimization problem of finding transformation parametersresulting in optimal bounds from the extended Davis-Kahan theorem is presentedfor affine transformations. It is demonstrated how globally optimal boundvalues can be computed automatically using fractional programming theory. Twodifferent solution approaches, the Charnes-Cooper transformation andDinkelbach's algorithm are reviewed. Our implementation of the extendedDavis--Kahan theorem is used to calculate bound values in three significantexamples. First, a pairwise comparison is made of the spaces spanned by theeigenvectors of the graph shift operator matrices corresponding to differentstochastic block model graphs. Second our bound is calculated on the distanceof the spaces spanned by eigenvectors of the graph shift operators and theircorresponding generating matrices in the stochastic blockmodel, and, third, onthe sample and population covariance matrices in a spiked covariance model. Ourextended bound values, using affine transformations, not only outperform thestandard Davis-Kahan bounds in all examples where both theorems apply, but alsodemonstrate good performance in several cases where the standard Davis-Kahantheorem cannot be used.

Walden A, Zhuang L, 2019, Constructing brain connectivity group graphs from EEG time series, *Journal of Applied Statistics*, Vol: 46, Pages: 1107-1128, ISSN: 0266-4763

Graphical analysis of complex brain networks is a fundamental area of modern neuroscience. Functional connectivity is important since many neurological and psychiatric disorders, including schizophrenia, are described as ‘dys-connectivity’ syndromes. Using electroencephalogram time series collected on each of a group of 15 individuals with a common medical diagnosis of positive syndrome schizophrenia we seek to build a single, representative, brain functional connectivity group graph. Disparity/distance measures between spectral matrices are identified and used to define the normalized graph Laplacian enabling clustering of the spectral matrices for detecting ‘outlying’ individuals. Two such individuals are identified. For each remaining individual, we derive a test for each edge in the connectivity graph based on average estimated partial coherence over frequencies, and associated p-values are found. For each edge these are used in a multiple hypothesis test across individuals and the proportion rejecting the hypothesis of no edge is used to construct a connectivity group graph. This study provides a framework for integrating results on multiple individuals into a single overall connectivity structure.

Walden AT, Leong ZZ, 2018, Tapering promotes propriety for Fourier transforms of real-valued time series, *IEEE Transactions on Signal Processing*, Vol: 66, Pages: 4585-4597, ISSN: 1053-587X

We examine Fourier transforms of real-valued stationary time series from the point of view of the statistical propriety. Processes with a large dynamic range spectrum have transforms that are very significantly improper for some frequencies; the real and imaginary parts can be highly correlated, and the periodogram will not have the standard chi-square distribution at these frequencies, nor have two degrees of freedom. Use of a taper reduces impropriety to just frequencies close to zero and Nyquist only, and frequency ranges where the propriety breaks down can be quite accurately and easily predicted by half the autocorrelation width of |H * H(2f)|, denoted by c, where H(f) is the Fourier transform of the taper and * denotes convolution. For vector time series we derive an improved distributional approximation for minus twice the log of the generalized likelihood ratio test statistic for testing for propriety of the Fourier transform at any frequency, and compare frequency range cutoffs for propriety determined by the hypothesis test with those determined by c.

Zhuang L, Walden AT, 2017, Sample mean versus sample Frechet mean for combining complex Wishart matrices: a statistical study, *IEEE Transactions on Signal Processing*, Vol: 65, Pages: 4551-4561, ISSN: 1941-0476

The space of covariance matrices is a non-Euclidean space. The matrices form a manifold which if equipped with a Riemannian metric becomes a Riemannian manifold, and recently this idea has been used for comparison and clustering of complex valued spectral matrices, which at a given frequency are typically modelled as complex Wishart-distributed random matrices. Identically distributed sample complex Wishart matrices can be combined via a standard sample mean to derive a more stable overall estimator. However, using the Riemannian geometry their so-called sample Fr´echet mean can also be found. We derive the expected value of the determinant of the sample Fr´echet mean and the expected value of the sample Fr´echet mean itself. The population Fr´echet mean is shown to be a scaled version of the true covariance matrix. The risk under convex loss functions for the standard sample mean is never larger than for the Fr´echet mean. In simulations the sample mean also performs better for the estimation of an important functional derived from the estimated covariance matrix, namely partial coherence.

Chandna S, Walden AT, 2016, A frequency domain test for propriety of complex-valued vector time series, *IEEE Transactions on Signal Processing*, Vol: 65, Pages: 1425-1436, ISSN: 1941-0476

This paper proposes a frequency domain approachto test the hypothesis that a stationary complex-valued vectortime series is proper, i.e., for testing whether the vector time seriesis uncorrelated with its complex conjugate. If the hypothesis isrejected, frequency bands causing the rejection will be identifiedand might usefully be related to known properties of the physicalprocesses. The test needs the associated spectral matrix whichcan be estimated by multitaper methods using, say,Ktapers.Standard asymptotic distributions for the test statistic are of nouse since they would requireK→∞,but, asKincreases so doesresolution bandwidth which causes spectral blurring. In manyanalysesKis necessarily kept small, and hence our efforts aredirected at practical and accurate methodology for hypothesistesting for smallK.Our generalized likelihood ratio statisticcombined with exact cumulant matching gives very accuraterejection percentages. We also prove that the statistic on whichthe test is based is comprised of canonical coherencies arisingfrom our complex-valued vector time series. Frequency specifictests are combined using multiple hypothesis testing to give anoverall test. Our methodology is demonstrated on ocean currentdata collected at different depths in the Labrador Sea. Overallthis work extends results on propriety testing for complex-valuedvectors to the complex-valued vector time series setting.

Schneider-Luftman D, Walden AT, 2016, Partial coherence estimation via spectral matrix shrinkage under quadratic loss, *IEEE Transactions on Signal Processing*, Vol: 64, Pages: 5767-5777, ISSN: 1941-0476

Partial coherence is an important quantity derivedfrom spectral or precision matrices and is used in seismology,meteorology, oceanography, neuroscience and elsewhere. If thenumber of complex degrees of freedom only slightly exceedsthe dimension of the multivariate stationary time series, spectralmatrices are poorly conditioned and shrinkage techniques suggestthemselves.When true partial coherencies are quite large then forshrinkage estimators of the diagonal weighting kind it is shownempirically that the minimization of risk using quadratic loss(QL) leads to oracle partial coherence estimators far superiorto those derived by minimizing risk using Hilbert-Schmidt (HS)loss. When true partial coherencies are small the methods behavesimilarly. We derive two new QL estimators for spectral matrices,and new QL and HS estimators for precision matrices. In additionfor the full estimation (non-oracle) case where certain traceexpressions must also be estimated, we examine the behaviour ofthree different QL estimators, the precision matrix one seemingparticularly appealing. For the empirical study we carry outexact simulations derived from real EEG data for two individuals,one having large, and the other small, partial coherencies. Thisensures our study covers cases of real-world relevance.

Walden AT, Schneider-Luftman D, 2015, Random matrix derived shrinkage of spectral precision matrices, *IEEE Transactions on Signal Processing*, Vol: 63, Pages: 4689-4699, ISSN: 1053-587X

There has been much research on shrinkage methods for real-valued covariance matrices and their inverses (precision matrices). In spectral analysis of p-vector-valued time series, complex-valued spectral matrices and precision matrices arise, and good shrinkage methods are often required, most notably when the estimated complex-valued spectral matrix is singular. As an improvement on the Ledoit-Wolf (LW) type of spectral matrix estimator we use random matrix theory to derive a Rao-Blackwell estimator for a spectral matrix, its inverse being a Rao-Blackwellized estimator for the spectral precision matrix. A random matrix method has previously been proposed for complex-valued precision matrices. It was implemented by very costly simulations. We formulate a fast, completely analytic approach. Moreover, we derive a way of selecting an important parameter using predictive risk methodology. We show that both the Rao-Blackwell estimator and the random matrix estimator of the precision matrix can substantially outperform the inverse of the LW estimator in a time series setting. Our new methodology is applied to EEG-derived time series data where it is seen to work well and deliver substantial improvements for precision matrix estimation.

Wolstenholme RJ, Walden AT, 2015, An efficient approach to graphical modelling of time series, *IEEE Transactions on Signal Processing*, Vol: 63, Pages: 3266-3276, ISSN: 1053-587X

Chandna S, Walden AT, 2013, Simulation Methodology for Inference on Physical Parameters of Complex Vector-Valued Signals, *IEEE TRANSACTIONS ON SIGNAL PROCESSING*, Vol: 61, Pages: 5260-5269, ISSN: 1053-587X

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- Citations: 14

Ginzberg P, Walden AT, 2013, Matrix-Valued and Quaternion Wavelets, *IEEE TRANSACTIONS ON SIGNAL PROCESSING*, Vol: 61, Pages: 1357-1367, ISSN: 1053-587X

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- Citations: 18

Walden AT, 2013, Rotary components, random ellipses and polarization: a statistical perspective, *PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 371, ISSN: 1364-503X

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- Citations: 16

Ginzberg P, Walden AT, 2013, Quaternion VAR Modelling and Estimation, *IEEE TRANSACTIONS ON SIGNAL PROCESSING*, Vol: 61, Pages: 154-158, ISSN: 1053-587X

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- Citations: 13

Ginzberg P, Walden AT, 2012, Adaptive Orthogonal Matrix-Valued Wavelets and Compression of Vector-Valued Signals, Birmingham, UK, 9th IMA International Conference on Mathematics in Signal Processing

Wavelet transforms using matrix-valued wavelets (MVWs) can process the components of vector-valued signals jointly, and thus offer potential advantages over scalar wavelets. For every matrix-valued scaling filter, there are infinitely many matrix-valued wavelet filters corresponding to rotated bases. We show how the arbitrary orthogonal factor in the choice of wavelet filter can be selected adaptively with a modified SIMPLIMAX algorithm. The 3×3 orthogonal matrix-valued scaling filters of length 6 with 3 vanishing moments have one intrinsic free scalar parameter in addition to three scalar rotation parameters. Tests suggest that even when optimising over these parameters, no significant improvement is obtained when compared to the naive scalar-based filter. We have found however in an image compression test that, for the naive scaling filter, adaptive basis rotation can decrease the RMSE by over 20%.

Walden AT, Cohen EAK, 2012, Statistical Properties for Coherence Estimators From Evolutionary Spectra, *IEEE TRANSACTIONS ON SIGNAL PROCESSING*, Vol: 60, Pages: 4586-4597, ISSN: 1053-587X

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- Citations: 7

Ginzberg P, Walden AT, 2011, Testing for quaternion propriety, *IEEE Transactions on Signal Processing*, Vol: 59, Pages: 3025-3034

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