Sequential Detection of Three-Dimensional Signals under Dependent Noise

We study detection methods for multivariable signals under dependent noise. The main focus is on three-dimensional signals, i.e. on signals in the space-time domain. Examples for such signals are multifaceted. They include geographic and climatic data as well as image data, that are observed over a fixed time horizon. We assume that the signal is observed as a finite block of noisy samples whereby we are interested in detecting changes from a given reference signal. Our detector statistic is based on a sequential partial sum process, related to classical signal decomposition and reconstruction approaches applied to the sampled signal. We show that this detector process converges weakly under the no change null hypothesis that the signal coincides with the reference signal, provided that the spatial-temporal partial sum process associated to the random field of the noise terms disturbing the sampled signal con- verges to a Brownian motion. More generally, we also establish the limiting distribution under a wide class of local alternatives that allows for smooth as well as discontinuous changes. Our results also cover extensions to the case that the reference signal is unknown. We conclude with an extensive simulation study of the detection algorithm

Detection limits for close eclipsing and transiting sub-stellar and planetary companions to white dwarfs in the WASP survey

We used photometric data from the WASP (Wide-Angle Search for Planets) survey to explore the possibility of detecting eclipses and transit signals of brown dwarfs, gas giants and terrestrial companions in close orbit around white dwarfs. We performed extensive Monte Carlo simulations and we found that for Gaussian random noise WASP is sensitive to companions as small as the Moon orbiting a $V\sim$12 white dwarf. For fainter stars WASP is sensitive to increasingly larger bodies. Our sensitivity drops in the presence of co-variant noise structure in the data, nevertheless Earth-size bodies remain readily detectable in relatively low S/N data. We searched for eclipses and transit signals in a sample of 194 white dwarfs in the WASP archive however, no evidence for companions was found. We used our results to place tentative upper limits to the frequency of such systems. While we can only place weak limits on the likely frequency of Earth-sized or smaller companions; brown dwarfs and gas giants (radius$\simeq$ R$_{jup}$) with periods $\leq$0.2 days must certainly be rare ($<10\%$). More stringent constraints requires significantly larger white dwarf samples, higher observing cadence and continuous coverage. The short duration of eclipses and transits of white dwarfs compared to the cadence of WASP observations appears to be one of the main factors limiting the detection rate in a survey optimised for planetary transits of main sequence stars.Comment: 8 pages, 3 figure

A two-stage model of orientation integration for Battenberg-modulated micropatterns

The visual system pools information from local samples to calculate textural properties. We used a novel stimulus to investigate how signals are combined to improve estimates of global orientation. Stimuli were 29 × 29 element arrays of 4 c/deg log Gabors, spaced 1° apart. A proportion of these elements had a coherent orientation (horizontal/vertical) with the remainder assigned random orientations. The observer's task was to identify the global orientation. The spatial configuration of the signal was modulated by a checkerboard pattern of square checks containing potential signal elements. The other locations contained either randomly oriented elements (''noise check'') or were blank (''blank check''). The distribution of signal elements was manipulated by varying the size and location of the checks within a fixed-diameter stimulus. An ideal detector would only pool responses from potential signal elements. Humans did this for medium check sizes and for large check sizes when a signal was presented in the fovea. For small check sizes, however, the pooling occurred indiscriminately over relevant and irrelevant locations. For these check sizes, thresholds for the noise check and blank check conditions were similar, suggesting that the limiting noise is not induced by the response to the noise elements. The results are described by a model that filters the stimulus at the potential target orientations and then combines the signals over space in two stages. The first is a mandatory integration of local signals over a fixed area, limited by internal noise at each location. The second is a taskdependent combination of the outputs from the first stage

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 2007The 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.I am grateful to the National Science Foundation for supporting this work via grant DMS-0411962 and the Office of Naval Research Graduate Traineeship awar

Noise limiter Patent

Circuits for amplitude limiting of random noise input

A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise

An encryption of a signal ${\bf s}\in\mathbb{R^N}$ is a random mapping ${\bf s}\mapsto \textbf{y}=(y_1,\ldots,y_M)^T\in \mathbb{R}^M$ which can be corrupted by an additive noise. Given the Encryption Redundancy Parameter (ERP) $\mu=M/N\ge 1$, the signal strength parameter $R=\sqrt{\sum_i s_i^2/N}$, and the ('bare') noise-to-signal ratio (NSR) $\gamma\ge 0$, we consider the problem of reconstructing ${\bf s}$ from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian-distributed random potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap $p_{\infty}\in [0,1]$ between the original signal and its recovered image (known as 'estimator') as $N\to \infty$, which is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full $p_{\infty} (\gamma)$ curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with $p_{\infty}>0$ for any $\mu>1$ and any $\gamma<\infty$, with $p_{\infty}\sim \gamma^{-1/2}$ as $\gamma\to \infty$. In contrast, for the case of purely quadratic nonlinearity, for any ERP $\mu>1$ there exists a threshold NSR value $\gamma_c(\mu)$ such that $p_{\infty}=0$ for $\gamma>\gamma_c(\mu)$ making the reconstruction impossible. The behaviour close to the threshold is given by $p_{\infty}\sim (\gamma_c-\gamma)^{3/4}$ and is controlled by the replica symmetry breaking mechanism.Comment: 33 pages, 5 figure