These detectors are derived from likelihood reasoning, but it is their geometries and invariances that qualify them as coherence statistics. A chapter on classical hypothesis tests for covariance structure introduces the next three chapters on matched and adaptive subspace detectors. Then least squares theory and the theory of minimum mean-squared error estimation are developed, with special attention paid to statistics that may be interpreted as coherence statistics. The book begins with a review of classical results in the physical and engineering sciences where coherence plays a fundamental role. The appendices contain a comprehensive account of matrix theory, the SVD, the multivariate normal distribution, and many of the important distributions for coherence statistics. Stochastic representations are emphasized, as these are central to Monte Carlo simulations. Throughout, the transformation invariances of statistics are clarified, geometries are illuminated, and null distributions are given where tractable. The reader will find new results for model fitting for dimension reduction in models and ambient spaces for detection, estimation, and space-time series analysis for subspace averaging and for uncertainty quantification. General results are applied to problems in communications, cognitive radio, passive and active radar and sonar, multi-sensor array processing, spectrum analysis, hyperspectral imaging, subspace clustering, and related. The book contains a wealth of classical and modern methods of inference, some reported here for the first time. This book organizes principles and methods of signal processing and machine learning into the framework of coherence.
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