Linearization Methods in Time Series Analysis

In this dissertation, we propose a set of computationally efficient methods based on approximating/representing nonlinear processes by linear ones, so-called linearization. Firstly, a linearization method is introduced for estimating the multiple frequencies in sinusoidal processes. It utilizes a regularized autoregressive (AR) approximation, which can be regarded as a "large p - small n" approach in a time series context. An appealing property of regularized AR is that it avoids a model selection step and allows for an efficient updating of the frequency estimates whenever new observations are obtained. The theoretical analysis shows that the regularized AR frequency estimates are consistent and asymptotically normally distributed. Secondly, a sieve bootstrap scheme is proposed using the linear representation of generalized autoregressive conditional heteroscedastic (GARCH) models to construct prediction intervals (PIs) for the returns and volatilities. Our method is simple, fast and distribution-free, while providing sharp and well-calibrated PIs. A similar linear bootstrap scheme can also be used for diagnostic testing. Thirdly, we introduce a robust lagrange multiplier (LM) test, which utilizes either the bootstrap or permutation procedure to obtain critical values, for detecting GARCH effects. We justify that both bootstrap and permutation LM tests are consistent. Intensive numerical studies indicate that the proposed resampling algorithms significantly improve the size and power of the LM test in both skewed and heavy-tailed processes. Moreover, fourthly, we introduce a nonparametric trend test in the presence of GARCH effects (NT-GARCH) based on heteroscedastic ANOVA. Our empirical evidence show that NT-GARCH can effectively detect non-monotonic trends under GARCH, especially in the presence of irregular seasonal components. We suggest to apply the bootstrap procedure for both selecting the window length and finding critical values. The newly proposed methods are illustrated by applications to astronomical data, to foreign currency exchange rates as well as to water and air pollution data. Finally, the dissertation is concluded by an outlook on further extensions of linearization methods, e.g., in model order selection and change point detection

Time delay estimation in stationary and non-stationary environments

We develop computationally efficient iterative algorithms for finding the Maximum Likelihood estimates of the delay and spectral parameters of a noise-like Gaussian signal radiated from a common point source and observed by two or more spatially separated receivers. We first consider the stationary case in which the source is stationary (not moving) and the observed signals are modeled as wide sense stationary processes. We then extend the scope by considering a non-stationary (moving) source radiating a possible non-stationary stochastic signal. In that context, we address the practical problem of estimation given discrete-time observations. We also present efficient methods for calculating the Jog-likelihood gradient (score), the Hessian, and the Fisher's information matrix under stationary and non-stationary conditions.Funding was provided by the Naval Air Systems Command through contract Number N00014-85-K-0272

Dynamic Chemical Shift Imaging for Image-Guided Thermal Therapy

Magnetic resonance temperature imaging (MRTI) is recognized as a noninvasive means to provide temperature imaging for guidance in thermal therapies. The most common method of estimating temperature changes in the body using MR is by measuring the water proton resonant frequency (PRF) shift. Calculation of the complex phase difference (CPD) is the method of choice for measuring the PRF indirectly since it facilitates temperature mapping with high spatiotemporal resolution. Chemical shift imaging (CSI) techniques can provide the PRF directly with high sensitivity to temperature changes while minimizing artifacts commonly seen in CPD techniques. However, CSI techniques are currently limited by poor spatiotemporal resolution. This research intends to develop and validate a CSI-based MRTI technique with intentional spectral undersampling which allows relaxed parameters to improve spatiotemporal resolution. An algorithm based on autoregressive moving average (ARMA) modeling is developed and validated to help overcome limitations of Fourier-based analysis allowing highly accurate and precise PRF estimates. From the determined acquisition parameters and ARMA modeling, robust maps of temperature using the k-means algorithm are generated and validated in laser treatments in ex vivo tissue. The use of non-PRF based measurements provided by the technique is also investigated to aid in the validation of thermal damage predicted by an Arrhenius rate dose model

Efficient and Robust Signal Detection Algorithms for the Communication Applications

Signal detection and estimation has been prevalent in signal processing and communications for many years. The relevant studies deal with the processing of information-bearing signals for the purpose of information extraction. Nevertheless, new robust and efficient signal detection and estimation techniques are still in demand since there emerge more and more practical applications which rely on them. In this dissertation work, we proposed several novel signal detection schemes for wireless communications applications, such as source localization algorithm, spectrum sensing method, and normality test. The associated theories and practice in robustness, computational complexity, and overall system performance evaluation are also provided

Robust Techniques for Bearing Estimation in Contaminated Gaussian Noise

The problem of estimating directions-of-arrival (DOA) of radiating sources from measurements provided by a passive array of sensors is frequently encountered in radar, sonar, radio astronomy and seismology. In this study various robust methods for the DOA estimation problem are developed, where the term robustness refers to insensitivity against small deviation in the underlying Gaussian noise assumption. The first method utilizes an eigenvector method and robust reconstruction of the correlation matrix by time series modeling of the array data; Secondly, a decentralized processing scheme is considered for geographically distributed array sites. The method provides reliable estimates even when a few of the subarray sites are malfunctioning. The above two techniques are useful for narrow band and incoherent sources. The third robust method, which utilizes Radon Transform, is capable of handling both the narrow band and wide band sources as well as the incoherent or coherent sources. The technique is also Useful in situations of very low SNR and colored noise with unknown correlation structure. The fourth method is an efficient narrow band robust maximum likelihood DOA estimation algorithm which is capable of handling coherent signals as well as the single snapshot cases. Furthermore, relationships between eigenvector methods and a ML DOA estimation, where the source signals are treated as sample functions of Gaussian random processes, are investigate

Statistical Spectral Parameter Estimation of Acoustic Signals with Applications to Byzantine Music

Digitized acoustical signals of Byzantine music performed by Iakovos Nafpliotis are used to extract the fundamental frequency of each note of the diatonic scale. These empirical results are then contrasted to the theoretical suggestions and previous empirical findings. Several parametric and non-parametric spectral parameter estimation methods are implemented. These include: (1) Phase vocoder method, (2) McAulay-Quatieri method, (3) Levinson-Durbin algorithm,(4) YIN, (5) Quinn & Fernandes Estimator, (6) Pisarenko Frequency Estimator, (7) MUltiple SIgnal Characterization (MUSIC) algorithm, (8) Periodogram method, (9) Quinn & Fernandes Filtered Periodogram, (10) Rife & Vincent Estimator, and (11) the Fourier transform. Algorithm performance was very precise. The psychophysical aspect of human pitch discrimination is explored. The results of eight (8) psychoacoustical experiments were used to determine the aural just noticeable difference (jnd) in pitch and deduce patterns utilized to customize acceptable performable pitch deviation to the application at hand. These customizations [Acceptable Performance Difference (a new measure of frequency differential acceptability), Perceptual Confidence Intervals (a new concept of confidence intervals based on psychophysical experiment rather than statistics of performance data), and one based purely on music-theoretical asymphony] are proposed, discussed, and used in interpretation of results. The results suggest that Nafpliotis\u27 intervals are closer to just intonation than Byzantine theory (with minor exceptions), something not generally found in Thrasivoulos Stanitsas\u27 data. Nafpliotis\u27 perfect fifth is identical to the just intonation, even though he overstretches his octaveby fifteen (15)cents. His perfect fourth is also more just, as opposed to Stanitsas\u27 fourth which is directionally opposite. Stanitsas\u27 tendency to exaggerate the major third interval A4-F4 is still seen in Nafpliotis, but curbed. This is the only noteworthy departure from just intonation, with Nafpliotis being exactly Chrysanthian (the most exaggerated theoretical suggestion of all) and Stanitsas overstretching it even more than Nafpliotis and Chrysanth. Nafpliotis ascends in the second tetrachord more robustly diatonically than Stanitsas. The results are reported and interpreted within the framework of Acceptable Performance Differences