Fourier Methods In the Analysis of Nonstationary and Nonlinear Stochastic Processes
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The investigator develops new Fourier based methods for analyzing nonstationary and nonlinear time series. Fourier analysis is the well established de facto tool for analyzing linear, stationary time series. There are several reasons for this (i) the discrete Fourier transform asymptotically uncorrelates a stationary time series (ii) if the time series is stationary and linear, then estimates of the spectral density function can be used to identify the underlying linear model (iii) the spectral density function can be used as a means of checking goodness of fit of a linear model. However, it has long been observed that several time series models do not fit well within the stationary, linear model framework. Over long periods of time the assumption of stationarity is often quite unrealistic. Even over short periods of time, the assumption of linearity can be too strong. Applying standard Fourier methods to such data can lead to uninformative and misleading conclusions. But in contrast to linear models, there does not exist universal methods for comparing non-nested, nonlinear models, checking adequacy of any given model, etc. As increasingly complex time series models are introduced, it has become increasingly important to develop such methods, and the investigator addresses these issues. The investigator focuses on three areas where, in applications, nonstationarity and nonlinearity can arise (i) nonstationary discrete time stochastic processes (ii) functional time series with random sampling (iii) nonlinear, stationary time series. These are detailed below. In the first project the investigator exploits the fact that the discrete Fourier transform only decorrelates second order stationary time series to characterize and model nonstationary behavior. In the second project the investigator considers continuous time series, which are only observed at discrete, randomly sampled time points. Here the focus is on functional time series, and the investigator defines a modified version of the discrete Fourier transform to test for stationarity and to develop goodness of fit tests. As mentioned above, often the assumption of linearity can be too strong, and in the third project the investigator considers stationary time series'' which are not necessarily linear. The investigator defines a variant of the spectral density which captures the pair-wise dependence structure of a time series. This transformation allows one to understand the dependence structure of the time series on different parts of the domain of the time series. Using this transformation the investigator checks for model adequacy, tests for equality of pair-wise dependence between two time series and measures the dependence between two time series through an appropriate transformations of the data. The analysis of data which is observed over time (usually called a time series) is studied in several disciplines, including the atmospheric sciences, economics etc. As the observations are over time, usually there is dependence (a simple measure of dependence is correlation) between neighboring observations. Understanding and modeling this dependence allows one to forecast (for example, future global temperatures) and compare various different time series (for example, different financial markets). Under the assumption that the time series is stationary (the overall structure does not change over time), and linear (the transition in the times series is smooth), a rich literature on modeling the correlation structure exists. However, there are several real data examples where there are no realistic reasons that these assumptions should hold true, and indeed they could be an oversimplification of the system or simply wrong. In this project, the investigator develops statistical tools which allows one to check whether a time series satisfies the usual assumptions, and if not, how they may violate these assumptions, what impact this may have on standard statistical analysis and how it may effect the conclusions.