Time-frequency analysisG(?) G ( ? ) -stationary processes
Methods such as wavelets, short term Fourier transforms and time deformation [Gray and Zhang, 1988. On a class of nonstationary processes. J. Time Ser. Anal. 9(2), 133154 and Gray, Vijverberg and Woodward, 2005. Nonstationary data analysis by time deformation. Commun. Statist. A, to appear] have been developed to analyze the time-frequency properties of a process where frequency changes with time. When the frequencies of a process are either monotonically increasing or monotonically decreasing with time, a rather general time deformation approach is to apply an appropriate BoxCox transformation to the time axis for the given signal in order to obtain a new stationary data set. This data set can then be analyzed by standard methods. Processes which are transformed to a stationary process after BoxCox transformation of the time scale are called G(?) G ( ? ) -stationary processes, where ? ? is the corresponding parameter of the BoxCox transformation. In this paper it is shown that the standard concept of stationarity can be viewed as stationarity under a linear transformation of the index set, while transforming time (time deformation) by a non-linear monotonic transformation introduces the concept of stationarity on a non-linear index set. Thus the notion of stationarity is broadened considerably to allow stationarity on different scales. The method is illustrated with analysis of both simulated and real data. Finally, it is shown that such processes can be transformed to stationarity by sampling properly. The software for performing the analysis discussed in this paper can be downloaded from the website http://faculty.smu.edu/hgray/research.htm.
time deformation boxcox transformation time-varying frequency wignerville