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Four contributions to the multitaper method of applied spectrum estimation are presented. These are a generalization of the multitaper method of spectrum estimation to time-series possessing irregularly spaced samples, a robust spectrum estimate suitable for cyclostationary, or quasi cyclostationary time-series, an improvement over the standard, multitaper spectrum estimates using quadratic inverse theory, and finally a method of scan-free spectrum estimation using a rotational shear-interferometer. Each of these topics forms a chapter in this thesis. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-02-05 18:01:45.187

Time-series data that reflect a periodic pattern are often used in psychology. In personality psychology, Brown and Moskowitz (1998) used spectral analysis to study whether fluctuations in the expression of four interpersonal behaviors show a cyclical pattern. Spline smoothing had also been used in the past to track the non-periodic trend, but no research has yet been done that combines spectral analysis and spline smoothing. The present thesis describes a new model which combines these two techniques to capture both periodic and non-periodic trends in the data. / The new model is then applied to Brown and Moskowitz's time-series data to investigate the long-term evolution to the four interpersonal behaviors, and to the GDP data to examine the periodic and non-periodic pattern for the GDP values of the 16 countries. Finally, the extent to which the model is accurate is tested using simulated data.

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Surgical scheduling is complicated by naturally occurring, and human-induced variability in the demand for surgical services. We used time series methods to detect, model and forecast these behaviors in surgical demand time series to help improve the scheduling of scarce surgical resources. With institutional approval, we studied 47,752 surgeries undertaken at a large academic medical center over a six-year time frame. Each daily sample in this time series represented the aggregate total hours of surgeries worked on a given day. Linear terms such as periodic cycles, trends, and serial correlations explained approximately 80 percent of the variance in the raw data. We used a moving variance filter to help explain away a large share of the heteroscedastic behavior mainly attributable to surgical activities on specific US holidays, which we defined as holiday variance. In the course of this research, we made a thoughtful attempt to understand the time series structure within our surgical demand data. We also laid a foundation, for further development, of two time series techniques, the multiwindow variance filter and cyclostatogram that can be applied not only to surgical demand time series, but also to other time series problems from other disciplines. We believe that understanding the non-stationarity, in surgical demand time series, may be an important initial step in helping health care managers save critical health care dollars. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-02-09 11:55:42.494

This thesis was undertaken with the intention of applying forecasting with time series analysis, in a manufacturing context. This involved two phases: the updating of existing forecasting techniques, and the application of these techniques to a manufacturing firm. The existing techniques, developed mainly by Brown in the 1960's, had to be adapted for computer application, to allow fast and objective computation of forecasts. This required an investigation into the derivation of each algebraic model, previously computed by hand, and translating those intuitive steps into routine ones. Furthermore, the revision of each forecast in the light of new data had to be dealt with mechanically. As for the application, the data supplied by the client, a large South African manufacturing firm, did not permit a successful application. This concerned both the manner in which the data were recorded (inconsistent time intervals), and the volume of data readily accessible. This then led the thesis in an unanticipated direction to overcome these difficulties. To do this objectively, it became necessary to generate test data with known characteristics, then to study how many data were required to recover those characteristics. Generating data required an investigation into random number generation, real data consisting of both true changes as well as a percentage of random fluctuations. A random data series was, therefore, added to the series with known characteristics. Such characteristics are unknown for genuine data, such as those supplied by the client. Empirical experimentation with the generated data, led to the determination of the number of data required to recover coefficients of various complexity. This number was found to be contrary to the statements made by Brown on this topic, significantly more data being required than was previously thought. Finally, an attempt was made to select an appropriate model for the client's data, based on the knowledge gained from investigating generated data. / Thesis (M.Soc.Sc.)-University of Natal, Durban, 1998.

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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.

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