Applied stochastic Eigen-analysis

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2007

[1]  Roland Speicher,et al.  Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces , 2004, math/0405191.

[2]  L. R. Haff.  The Variational Form of Certain Bayes Estimators , 1991 .

[3]  R. Speicher.  Multiplicative functions on the lattice of non-crossing partitions and free convolution , 1994 .

[4]  Olivier Ledoit,et al.  A well-conditioned estimator for large-dimensional covariance matrices , 2004 .

[5]  R. Speicher,et al.  Lectures on the Combinatorics of Free Probability: The free commutator , 2006 .

[6]  Joel E. Cohen,et al.  Random matrices and their applications , 1986 .

[7]  Peter J. Kootsookos,et al.  Threshold behavior of the maximum likelihood estimator of frequency , 1994, IEEE Trans. Signal Process..

[8]  L. Scharf,et al.  Statistical Signal Processing: Detection, Estimation, and Time Series Analysis , 1991 .

[9]  T. W. Anderson.  ASYMPTOTIC THEORY FOR PRINCIPAL COMPONENT ANALYSIS , 1963 .

[10]  J. W. Silverstein.  On the eigenvectors of large dimensional sample covariance matrices , 1989 .

[11]  S.T. Smith,et al.  Covariance, subspace, and intrinsic Crame/spl acute/r-Rao bounds , 2005, IEEE Transactions on Signal Processing.

[12]  Raj Rao Nadakuditi,et al.  Asymptotic Mean Squared Error Performance of Diagonally Loaded Capon-MVDR Processor , 2005, Conference Record of the Thirty-Ninth Asilomar Conference onSignals, Systems and Computers, 2005..

[13]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[14]  Olivier Ledoit,et al.  Honey, I Shrunk the Sample Covariance Matrix , 2003 .

[15]  Philippe Flajolet,et al.  Analytic combinatorics : functional equations, rational and algebraic functions , 2001 .

[16]  R. Kass,et al.  Shrinkage Estimators for Covariance Matrices , 2001, Biometrics.

[17]  D. Lawley.  TESTS OF SIGNIFICANCE FOR THE LATENT ROOTS OF COVARIANCE AND CORRELATION MATRICES , 1956 .

[18]  D. Paul.  ASYMPTOTICS OF SAMPLE EIGENSTRUCTURE FOR A LARGE DIMENSIONAL SPIKED COVARIANCE MODEL , 2007 .

[19]  Leonard W. Cooper.  Multivariate orthogonal polynomials , 1951 .

[20]  J. W. Silverstein,et al.  Eigenvalues of large sample covariance matrices of spiked population models , 2004, math/0408165.

[21]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[22]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[23]  J. W. Silverstein.  Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices , 1995 .

[24]  H. Cox,et al.  Passive sonar limits upon nulling multiple moving ships with large aperture arrays , 1999, Conference Record of the Thirty-Third Asilomar Conference on Signals, Systems, and Computers (Cat. No.CH37020).

[25]  Henry Cox,et al.  Robust adaptive beamforming , 2005, IEEE Trans. Acoust. Speech Signal Process..

[26]  Roland Speicher,et al.  Free Probability Theory and Random Matrices , 2003 .

[27]  Keith Q. T. Zhang.  Probability of resolution of the MUSIC algorithm , 1995, IEEE Trans. Signal Process..

[28]  Ioana Dumitriu,et al.  Path Counting and Random Matrix Theory , 2003, Electron. J. Comb..

[29]  Alexandru Nica,et al.  Free random variables , 1992 .

[30]  Wen Xu,et al.  Bayesian bounds for matched-field parameter estimation , 2004, IEEE Transactions on Signal Processing.

[31]  J. Capon.  High-resolution frequency-wavenumber spectrum analysis , 1969 .

[32]  Nicolas Le Bihan,et al.  Quaternion-MUSIC for vector-sensor array processing , 2006, IEEE Transactions on Signal Processing.

[33]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[34]  R. Kohn,et al.  Parsimonious Covariance Matrix Estimation for Longitudinal Data , 2002 .

[35]  B. Efron,et al.  Limiting the Risk of Bayes and Empirical Bayes Estimators—Part I: The Bayes Case , 1971 .

[36]  J. Wishart.  THE GENERALISED PRODUCT MOMENT DISTRIBUTION IN SAMPLES FROM A NORMAL MULTIVARIATE POPULATION , 1928 .

[37]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[38]  D. Voiculescu.  Addition of certain non-commuting random variables , 1986 .

[39]  Olivier Ledoit,et al.  Improved estimation of the covariance matrix of stock returns with an application to portfolio selection , 2003 .

[40]  Benoit Collins.  Product of random projections, Jacobi ensembles and universality problems arising from free probability , 2005 .

[41]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[42]  C. D. Richmond.  Capon and Bartlett Beamforming: Threshold Effect in Direction-of-Arrival Estimation Error and On the Probability of Resolution , 2005 .

[43]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[44]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[45]  Piotr Sniady,et al.  Second order freeness and fluctuations of random matrices: II. Unitary random matrices , 2007 .

[46]  S. Péché,et al.  Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices , 2004, math/0403022.

[47]  J. Berger,et al.  Estimation of a Covariance Matrix Using the Reference Prior , 1994 .

[48]  A. Edelman,et al.  Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models , 2005, math-ph/0510043.

[49]  Dag Jonsson.  Some limit theorems for the eigenvalues of a sample covariance matrix , 1982 .

[50]  Wen Xu,et al.  A bound on mean-square estimation error with background parameter mismatch , 2004, IEEE Transactions on Information Theory.

[51]  A. Akritas.  Sylvester''s forgotten form of the resultant , 1993 .

[52]  B. McKay.  The expected eigenvalue distribution of a large regular graph , 1981 .

[53]  Olivier Ledoit,et al.  Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size , 2002 .

[54]  R. O. Schmidt,et al.  Multiple emitter location and signal Parameter estimation , 1986 .

[55]  Fumio Hiai,et al.  Eigenvalue Density of the Wishart Matrix and Large Deviations , 1998 .

[56]  John K. Thomas,et al.  The probability of a subspace swap in the SVD , 1995, IEEE Trans. Signal Process..

[57]  J. W. Silverstein,et al.  Analysis of the limiting spectral distribution of large dimensional random matrices , 1995 .

[58]  Percy Deift,et al.  New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field , 1998 .

[59]  Petre Stoica,et al.  Performance breakdown of subspace-based methods: prediction and cure , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[60]  Arno B. J. Kuijlaars,et al.  Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields , 2000 .

[61]  Edward M. Hofstetter,et al.  Barankin Bounds on Parameter Estimation , 1971, IEEE Trans. Inf. Theory.

[62]  K. Wachter.  The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements , 1978 .

[63]  J. W. Silverstein,et al.  COVARIANCE MATRICES , 2022 .

[64]  Patrick L. Combettes,et al.  Signal detection via spectral theory of large dimensional random matrices , 1992, IEEE Trans. Signal Process..

[65]  Michael I. Miller,et al.  On the existence of positive-definite maximum-likelihood estimates of structured covariance matrices , 1988, IEEE Trans. Inf. Theory.

[66]  J. W. Silverstein.  Weak Convergence of random functions defined by the eigenvectors of sample covariance matrices , 1990 .

[67]  Stephanos Venakides,et al.  UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY , 1999 .

[68]  V. Girko.  An Introduction to Statistical Analysis of Random Arrays , 1998 .

[69]  Alan Edelman,et al.  MOPS: Multivariate orthogonal polynomials (symbolically) , 2004, J. Symb. Comput..

[70]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices , 2007 .

[71]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[72]  François Bergeron,et al.  Combinatorial Resolution of Systems of Differential Equations III: a Special Class of Differentially Algebraic Series , 1990, Eur. J. Comb..

[73]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[74]  H. Akaike.  A new look at the statistical model identification , 1974 .

[75]  Simon Haykin,et al.  The Cocktail Party Problem , 2005, Neural Computation.

[76]  Lotharingien de Combinatoire.  Free Probability Theory and Non-crossing Partitions , 1997 .

[77]  Philippe Biane,et al.  Processes with free increments , 1998 .

[78]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[79]  D. Voiculescu.  Symmetries of some reduced free product C*-algebras , 1985 .

[80]  O. Zeitouni,et al.  A CLT for a band matrix model , 2004, math/0412040.

[81]  H. Mewes,et al.  An improved Capon estimator for HF DF using limited sample data sets , 1997 .

[82]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[83]  Luc Knockaert,et al.  The Barankin bound and threshold behavior in frequency estimation , 1997, IEEE Trans. Signal Process..

[84]  J. W. Silverstein.  The Limiting Eigenvalue Distribution of a Multivariate F Matrix , 1985 .

[85]  D. Voiculescu.  Limit laws for Random matrices and free products , 1991 .

[86]  B. Carlson.  Covariance matrix estimation errors and diagonal loading in adaptive arrays , 1988 .

[87]  A. Edelman,et al.  Random matrix theory , 2005, Acta Numerica.

[88]  F. Hiai,et al.  The semicircle law, free random variables, and entropy , 2006 .

[89]  R. Tausworthe.  A Method for Calculating Phase-Locked Loop Performance Near Threshold , 1967, IEEE Transactions on Communication Technology.

[90]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of a class of large dimensional random matrices , 1995 .

[91]  Piotr Sniady,et al.  Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants , 2006, Documenta Mathematica.

[92]  I. Reed,et al.  Rapid Convergence Rate in Adaptive Arrays , 1974, IEEE Transactions on Aerospace and Electronic Systems.

[93]  I. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis , 2001 .

[94]  A. E. Hoerl,et al.  Ridge regression: biased estimation for nonorthogonal problems , 2000 .

[95]  N. R. Goodman,et al.  Probability distributions for estimators of the frequency-wavenumber spectrum , 1970 .

[96]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[97]  Z. Bai.  METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES , A REVIEW , 1999 .

[98]  M. Bartlett.  A Note on the Multiplying Factors for Various χ2 Approximations , 1954 .

[99]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series, with Engineering Applications , 1949 .

[100]  Thomas Kailath,et al.  Detection of signals by information theoretic criteria , 1985, IEEE Trans. Acoust. Speech Signal Process..

[101]  E. Wigner.  Characteristic Vectors of Bordered Matrices with Infinite Dimensions I , 1955 .

[102]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[103]  H. Cramér.  Mathematical methods of statistics , 1947 .