Introduction

 

Time varying frequencies are quite common in speech, biological data, geophysical processes and so on.  Traditional Fourier analysis under the assumption of stationarity may not be applicable to these processes since the frequency content is evolving over time.  The idea of time deformation is expanded to transform the time axis in order to change these types of non-stationary processes to stationary  processes.  Gray and Zhang (1988) developed continuous multiplicative-stationary (M-stationary) processes for the purpose of analyzing non-stationary data with cyclical behavior changing approximately linearly in time.  A continuous M-stationary process can be transformed to a continuous weakly stationary process, which is referred to as the dual process, through a logarithmic time transformation.  One type of M-stationary process, the continuous Euler process, whose corresponding dual is the continuous AR processes, was proposed by Gray and Zhang (1988).  Gray and Vijverberg (2003) extended the M-stationary process to the case of discrete data and developed the corresponding discrete Euler process, while Gray, Vijverberg, and Woodward (2003) introduce the resulting instantaneous spectrum.  The instantaneous spectrum was further developed in Jiang (2003).  Choi, Gray, and Woodward (2003) defined the continuous and discrete mixed M-stationary process called the EARMA processes, whose duals are the regular continuous and discrete ARMA processes respectively.  In Vijverberg (2002) and Gray, Vijverberg, and Woodward (2003), it was shown that when the periodic nature of the process is changing approximately linearly with time, M-stationary models give a much better fit than traditional models from both the spectral analysis and forecast performance point of view.  These results were generalized by Jiang (2003) and Jiang, Gray and Woodward (2003) to include processes whose periodic components in the autocorrelation change wth time like   The case  is the M-stationary case, while  is the stationary case.  These results have all been applied by Cohlmia (2003) to develop effective new methods for filtering data whose frequency structure change with time.  The results are further developed in Cohlmia, Gray and Woodward (2003).