Closure to “Testing Stationarity with Wavelet-Based Surrogates” by Megan McCullough and Ahsan Kareem

Abstract

The writers would like to thank the discusser for the comments regarding stationarity testing in the time-frequency domain and bringing additional research in this area to their attention. The discusser provides insight into different approaches used to solve the same general problem, i.e., determining whether data are stationary or nonstationary and using related wavelet-based approaches. The comments related to the discusser’s discussion points can be summarized as follows: 1. In Keylock (2006, 2007), the discusser suggests using stationary wavelet transform (SWT) (Percival and Walden 2000) as opposed to continuous wavelet transform (CWT), which was the time-frequency tool used in the original paper. The writers have also used SWT for simulation purposes and thus are familiar with this approach (Wang 2007; Wang et al. 2013, 2014). The discusser’s approach proposes the use of an iterative phase-randomization procedure (iterative amplitude adjusted Fourier transform approach) to simulate additional realizations at each decomposition level that results using SWT. The levels are then circularly rotated individually until an error function is minimized. This surrogate generation procedure, while performed directly in the time-frequency domain, involves the simulation of many levels instead of simulating the entire surrogate at once using the classic phase-randomization procedure (Schreiber and Schmitz 1996) used in the writers’ approach. While this approach is performed directly in the time-frequency domain and there is no need to reconstruct the actual surrogate time series, there is still an important trade-off in computational efficiency and ease of implementation between the two approaches. The discusser’s approach requires simulation of many levels and circular rotation steps and the writers’ approach requires simulation followed by CWT. The writers would consider the simulation step in the time domain to be more straightforward than the multiple simulations and rotational matching steps necessary using SWT. The filtering step in the writers’ approach is used to directly assess stationarity; however, it is not clear exactly how the surrogates using the SWT approach would be quantitatively compared with the original time series to assess stationarity, as is defined by the writers for structural engineering applications. However, the additional control the discusser’s approach allows does have important applications for nonlinearity testing. 2. The discusser also suggests using derivative-of-Gaussian (DOG) wavelets with CWT to evaluate nonstationarity of the fractal dimension. DOG wavelets simply offer an alternative to theMorlet wavelet used in thewriters’ approach.Morlet wavelets offer high-frequency resolution and theDOGwavelet offers better time resolution with poor frequency resolution. Therefore, it is not apparent how this approach would outperform the writers’ approach. In addition, the discusser mentions randomization of the modulus maxima to destroy multifractality to test for nonstationarity of the fractal dimension (Keylock 2007). The use of fractal dimension is, again, more important for nonlinearity detection than the nonstationarity detection that was the focus in the original paper. However, these are important extensions of the writers’ approach. 3. Thewriters agreewith the discusser that theflexibility inherent in the discusser’s approaches, including graduate wavelet reconstruction (Keylock 2010), offers new possibilities for quantifying how nonstationary a time series is rather than using a binary testing approach. Deviating from the binary approach was motivation for the filtering technique presented in the writers’ approach. The filtering technique allows for localization of nonstationarity within the original data. However, the different types of nonstationarity that may be present can complicate the use of a quantitative metric or affect the understanding of what the metric actually represents. This approach would offer interesting insight into which frequency levels are characterized by the presence of more or less stationarity, determined by the fraction of coefficients that are pinned. The writers again thank the discusser for the important comments, insights, and suggestions. The writers look forward to continued discussions on similar topics in the future.