Wavelet Packet Methods for the Analysis of Variance of Time Series With Application to Crack Widths on the Brunelleschi Dome

Abstract

In this article we extend the definition of wavelet variance to wavelet packets. We also adapt to wavelet packets an iterated cumulative sum of squares algorithm for the location of variance change points. Wavelet packets have greater decorrelation properties than standard wavelets in that they induce a finer partitioning of the frequency domain of the process generating the data. This allows our procedure to be applied to a wide class of processes. We show this on simulated data and on a benchmark time series. Our initial interest in wavelet variance change points location was motivated by an application to time series of crack widths on the Brunelleschi dome of the Santa Maria del Fiore cathedral in Florence. The structure of the dome includes an internal thick dome and an external thin one. In an effort to understand the dynamics of the crack widths we apply wavelet packet variance analysis to measurements from instruments located in the different parts of the outer and inner domes, highlighting different features and seasonal behavior. Our findings agree well with the structural functions of the different elements of the dome and also reveal some interesting aspects regarding the dynamics of crack evolution.

Key Words:

• Change points detection
• Nondecimated transforms
• Wavelet variance