One of the key features of wavelet analysis is its potential use for effecting time-frequency decompositions of non-stationary signals. The relationship between wavelet analysis and time-dependent spectral analysis has so far rested mainly on heuristic reasoning: in this paper we examine the relationship in a more precise mathematical form. A crucial feature of this analysis is the need to define carefully the notion of "frequency" when applied to non-stationary signals.The recent explosion of interest in wavelet analysis has stimulated renewed studies of time-dependent spectral analysis. The basic aim of wavelet analysis is to represent a function as a linear superposition of 'wavelets' centred on a sequence of time points, and in this sense it forms a natural tool for the investigation of 'local' properties of time varying functions. The literature on wavelets is replete with terminology such as 'discrete wavelet transforms', 'time-frequency localisation', 'windowed Fourier transforms', all of which indicate that, at some level, there should be a close connection between wavelet analysis and time-dependent spectra. Indeed, phrases such as 'local power spectra' are widely used in the wavelet literature, but so far the motivation for the use of this type of terminology has been highly intuitive and there has been little attempt to investigate the relationship between the two concepts in a formal and mathematically sound fashion. In pursuing this latter approach we encounter two basic difficulties: (a) almost all the literature on wavelet analysis deals with the representation of deterministic square-integrable functions, and (b) any form of time-dependent spectral analysis requires a very carefully constructed background theoretical model if the results of numerically calculated spectra are to have any meaningful physical interpretation. To put it quite simply, what meaning do sample spectra calculated from different time intervals possess? What functions are they estimating? In this paper we try to address these questions and establish a more analytical link between wavelet analysis and time-dependent spectral analysis. We first review briefly the basic ideas underlying wavelet analysis.

We develop an approach to the spectral analysis of non-stationary processes which is based on the concept of "evolutionary spectra"; that is, spectral functions which are time dependent, and have a physical interpretation as local energy distributions over frequency. It is shown that the notion of evolutionary spectra generalizes the usual definition of spectra for stationary processes, and that, under certain conditions, the evolutionary spectrum at each instant of time may be estimated from a single realization of a process. By such means it is possible to study processes with continuously changing "spectral patterns".1. INTRODUCTION IN the classical approach to statistical spectral analysis it is always assumed that the process under study, Xt, is stationary, at least up to the second order. That is, we assume that E(Xt) = 1-', a constant (independent of t) which we may take to be zero, and that, for each sand t, the covariance Rs,t = E{(Xs-I-'HXt-I-')*} (1.1) (* denoting the complex conjugate) is a function ofls-tl only. In this case it is well known that Rs,t has a spectral representation of the formwhere F(w) is some function having the properties of a distribution function, and the range of integration is (-00,00) for a continuous parameter process, and (-7T,7T) in the discrete case.Corresponding to (1.2), {XI} has a spectral representation of the formwhere Z(w) is an orthogonal process with E{ldZ(w)12} = dF(w). When {Xt} represents some physical process, the spectral density function few) = F'(w) (when it exists) describes the distribution (over the frequency range) of the energy (per unit time) dissipated by the process, and given a sample record of {XI}, there are several methods of estimatingf(w) (see, e.g., Grenander and Rosenblatt, 1957, Ch. 4).In practice, however, it often happens that the assumption of stationarity is a very doubtful one. For example, records of atmospheric turbulence exhibit marked changes over periods of time, and in such cases classical spectral analysis based on a stationary model can hardly be carried through with conviction. The question arises, therefore, 1965]

Summary In this note we consider the problem of fitting a general functional relationship between two variables. We require only that the function to be fitted is, in some sense, “smooth”, and do not assume that it has a known mathematical form involving only a finite number of unknown parameters.

We consider the problem of testing a given time-series for stationarity. The approach is based on evolutionary spectral analysis, and the proposed method consists essentially in testing the "homogeneity" of a set of evolutionary spectra evaluated at different instants of time.Using a logarithmic transformation, we show that the mechanics of the test are formally equivalent to a two-factor analysis of variance procedure when the residual variance is known, a priori.In addition to testing stationarity, the analysis provides also a method for testing whether the observed series fits a "uniformly modulated" model, and a test for "randomness" (constancy of spectra).

We construct a general class of non-linear models, called 'state-dependent models', which have a very flexible non-linear structure and which contain, as special cases, bilinear, threshold autoregressive, and exponential autoregressive models. We describe a sequential type of recursive algorithm for identifying state-dependent models, and show how such models may be used for forecasting and for indicating specific types of non-linear behaviour.