Applied stochastic eigen-analysis
Nadakuditi, Rajesh Rao
MetadataShow full item record
The first part of the dissertation investigates the application of the theory of large random matrices to high-dimensional inference problems when the samples are drawn from a multivariate normal distribution. A longstanding problem in sensor array processing is addressed by designing an estimator for the number of signals in white noise that dramatically outperforms that proposed by Wax and Kailath. This methodology is extended to develop new parametric techniques for testing and estimation. Unlike techniques found in the literature, these exhibit robustness to high-dimensionality, sample size constraints and eigenvector misspecification. By interpreting the eigenvalues of the sample covariance matrix as an interacting particle system, the existence of a phase transition phenomenon in the largest (“signal”) eigenvalue is derived using heuristic arguments. This exposes a fundamental limit on the identifiability of low-level signals due to sample size constraints when using the sample eigenvalues alone. The analysis is extended to address a problem in sensor array processing, posed by Baggeroer and Cox, on the distribution of the outputs of the Capon-MVDR beamformer when the sample covariance matrix is diagonally loaded. The second part of the dissertation investigates the limiting distribution of the eigenvalues and eigenvectors of a broader class of random matrices. A powerful method is proposed that expands the reach of the theory beyond the special cases of matrices with Gaussian entries; this simultaneously establishes a framework for computational (non-commutative) “free probability” theory. The class of “algebraic” random matrices is defined and the generators of this class are specified. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue distribution and, for a subclass, the limiting conditional “eigenvector distribution.” The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. The method is applied to predict the deterioration in the quality of the sample eigenvectors of large algebraic empirical covariance matrices due to sample size constraints.
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
Suggested CitationThesis: Nadakuditi, Rajesh Rao, "Applied stochastic eigen-analysis", 2007-02, DOI:10.1575/1912/1647, https://hdl.handle.net/1912/1647
Showing items related by title, author, creator and subject.
Effects of climate change on an emperor penguin population : analysis of coupled demographic and climate models Jenouvrier, Stephanie; Holland, Marika M.; Stroeve, Julienne; Barbraud, Christophe; Weimerskirch, Henri; Serreze, Mark; Caswell, Hal (2012-06-21)Sea ice conditions in the Antarctic affect the life cycle of the emperor penguin (Aptenodytes forsteri). We present a population projection for the emperor penguin population of Terre Adelie, Antarctica, by linking ...
Hunter, Christine M.; Caswell, Hal; Runge, Michael C.; Regehr, Eric V.; Amstrup, Steve C.; Stirling, Ian (Ecological Society of America, 2010-10)The polar bear (Ursus maritimus) depends on sea ice for feeding, breeding, and movement. Significant reductions in Arctic sea ice are forecast to continue because of climate warming. We evaluated the impacts of climate ...
Analysis of lethal and sublethal impacts of environmental disasters on sperm whales using stochastic modeling Ackleh, Azmy; Chiquet, Ross A.; Ma, Baoling; Tang, Tingting; Caswell, Hal; Veprauskas, Amy; Sidorovskaia, Natalia (Springer, 2017-05-12)Mathematical models are essential for combining data from multiple sources to quantify population endpoints. This is especially true for species, such as marine mammals, for which data on vital rates are difficult to obtain. ...