This chapter is an introduction to the theory of time series analysis. In Section 4.1 we discuss the estimators of the sample mean and the correlation function of a time series. In Section 4.2 we introduce non-parametric methods of the spectral analysis of time series, including the multitapering method. A detailed discussion of the time series spectral analysis can be found in Refs. [153, 154, 155, 156].
In Sections 4.3–4.5 we discuss useful tests of the time series. One type of test is for the presence of periodicities in the data, which we discuss in Section 4.3. In Section 4.4 we introduce two goodness-of-fit tests describing whether the data come from a given probability distribution: Pearson's χ2 test and Kolmogorov–Smirnov test. Other types of tests are tests for Gaussianity and linearity of the data, which are discussed in Section 4.5. Both tests use higher-order spectra of time series, which are also introduced in Section 4.5.
Sample mean and correlation function
We assume that we have N contiguous data samples xk (k = 1, …, N) of the stochastic process. We also assume that the underlying process is stationary and ergodic (i.e. satisfying the ergodic theorem, see Section 3.2). We immediately see that the N samples of the stochastic process that constitute our observation cannot be considered as a stationary process. They would be a stationary sequence only asymptotically as we extend the number of samples N to infinity. As we shall see this has profound consequences on the statistical properties of the estimators of the spectrum.