**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).**

** **

** **