Date of Award




Document Type


Degree Name

Doctor of Philosophy (PhD)


Department of Mathematics and Statistics

Content Description

1 online resource (vii, 82 pages) : illustrations (some color)

Dissertation/Thesis Chair

Igor G Zurbenko

Committee Members

Robert Henry, Karin Reinhold, Malcolm Sherman


Longitudinal method, Spectral theory (Mathematics), Atmospheric tides, Astrometeorology

Subject Categories

Statistics and Probability


Longitudinal data analysis is an observational study that analyzes data that has been collected over long periods of time where time can be arbitrary. It is a popular method that has been widely used over the time domain approach. An equivalent method to the time domain approach is the frequency domain approach. This method allows researchers to observe the spectral properties of the data. In the world everything exists in continuous time t, but we made observations of a given variable at some specific times. Currently spectral analysis is well developed for time series when observations are made at equal time intervals, however in practice it is often the case that time is not equally spaced and can be considered random. We consider time to be uniformly independently identically discretely distributed on an interval of length N, which provides us with a longitudinal dataset Xtk, where tk represents our random time points. We prove that the spectral density estimation fXtk will converge in mean square to the actual spectral density. This method can be used in practice when time is considered arbitrary or when there are missing values that are randomly distributed. This method is available in the KZA package in R-software (Close & Zurbenko, 2010). We apply this result to an application involving an atmospheric pressure dataset, which helped us to detect hidden short seasonality in atmospheric pressure that is due to astronomic influences. With this we were able to provide an explanation of many unusual atmospheric phenomena, including an explanation of increased strength of hurricanes (Zurbenko & Potrzeba, 2008, 2009). Energies of most astronomic influences in the atmosphere are very small compared with synoptic fluctuations within atmospheric variables. However, astronomic influences are strictly periodic when synoptic fluctuations are basically stochastic with no strictly defined periodicities. This allows the detection and reconstruction of astronomic influences in the atmosphere. The demonstration of some very interesting examples will be provided.