Han, Sangdae (2008) *Comparing Spectral Densities in Replicated Time Series by Smoothing Spline ANOVA.* Doctoral Dissertation, University of Pittsburgh.

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## Abstract

Comparing several groups of populations based on replicated data is one of the main concerns in statistical analysis. A specific type of data, time series data, such as waves of earthquakes present difficulties because of the correlations amongst the data. Spectral analysis solves this problem somewhat because the discrete Fourier transform transforms the data to near independence under general conditions.The goal of our research is to develop general, user friendly, statistical methods to compare group spectral density functions. To accomplish this, we consider two main problems: How can we construct an estimation function from replicated time series for each group and what method can be used to compare the estimated functions? For the first part, we present smooth estimates of spectral densities from time series data obtained from replication across subjects (units) (Wahba 1990; Guo et al. 2003). We assume that each spectral density is in some reproducing kernel Hilbert space and apply penalized least squares methods to estimate spectral density in smoothing spline ANOVA. For the second part, we consider confidence intervals to determine the frequencies where the spectrum of one spectral density may differ from another. These confidence intervals are the independent simultaneous confidence interval and the bootstrapping confidence interval (Babu et al. 1983; Olshen et al. 1989). Finally, as an application, we consider the replicated time series data that consist of shear (S) waves of 8 earthquakes and 8 explosions (Shumway & Stoffer 2006).

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## Details | ||||||||||||||||

Item Type: | University of Pittsburgh ETD | |||||||||||||||
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ETD Committee: |
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Title: | Comparing Spectral Densities in Replicated Time Series by Smoothing Spline ANOVA | |||||||||||||||

Status: | Unpublished | |||||||||||||||

Abstract: | Comparing several groups of populations based on replicated data is one of the main concerns in statistical analysis. A specific type of data, time series data, such as waves of earthquakes present difficulties because of the correlations amongst the data. Spectral analysis solves this problem somewhat because the discrete Fourier transform transforms the data to near independence under general conditions.The goal of our research is to develop general, user friendly, statistical methods to compare group spectral density functions. To accomplish this, we consider two main problems: How can we construct an estimation function from replicated time series for each group and what method can be used to compare the estimated functions? For the first part, we present smooth estimates of spectral densities from time series data obtained from replication across subjects (units) (Wahba 1990; Guo et al. 2003). We assume that each spectral density is in some reproducing kernel Hilbert space and apply penalized least squares methods to estimate spectral density in smoothing spline ANOVA. For the second part, we consider confidence intervals to determine the frequencies where the spectrum of one spectral density may differ from another. These confidence intervals are the independent simultaneous confidence interval and the bootstrapping confidence interval (Babu et al. 1983; Olshen et al. 1989). Finally, as an application, we consider the replicated time series data that consist of shear (S) waves of 8 earthquakes and 8 explosions (Shumway & Stoffer 2006). | |||||||||||||||

Date: | 30 October 2008 | |||||||||||||||

Date Type: | Completion | |||||||||||||||

Defense Date: | 30 April 2008 | |||||||||||||||

Approval Date: | 30 October 2008 | |||||||||||||||

Submission Date: | 01 May 2008 | |||||||||||||||

Access Restriction: | No restriction; The work is available for access worldwide immediately. | |||||||||||||||

Patent pending: | No | |||||||||||||||

Institution: | University of Pittsburgh | |||||||||||||||

Thesis Type: | Doctoral Dissertation | |||||||||||||||

Refereed: | Yes | |||||||||||||||

Degree: | PhD - Doctor of Philosophy | |||||||||||||||

URN: | etd-05012008-134047 | |||||||||||||||

Uncontrolled Keywords: | reproducing kernel Hilbert space; spectral analysis; bootstrapping confidence interval; smoothing spline ANOVA | |||||||||||||||

Schools and Programs: | Dietrich School of Arts and Sciences > Statistics | |||||||||||||||

Date Deposited: | 10 Nov 2011 14:43 | |||||||||||||||

Last Modified: | 06 Jun 2012 09:48 | |||||||||||||||

Other ID: | http://etd.library.pitt.edu/ETD/available/etd-05012008-134047/, etd-05012008-134047 |

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