Spectral Density - Preliminary Conventions On Notations For Time Series

Preliminary Conventions On Notations For Time Series

The phrase time series has been defined as "... a collection of observations made sequentially in time." But it is also used to refer to a stochastic process that would be the underlying theoretical model for the process that generated the data (and thus include consideration of all the other possible sequences of data that might have been observed, but weren't). Furthermore, time can be either continuous or discrete. There are, therefore, four different but closely related definitions and formulas for the power spectrum of a time series.

If (discrete time) or (continuous time) is a stochastic process, we will refer to a possible time series of data coming from it as a sample or path or signal of the stochastic process. To avoid confusion, we will reserve the word process for a stochastic process, and use one of the words signal, or sample, to refer to a time series of data.

For X any random variable, standard notations of angle brackets or E will be used for ensemble average, also known as statistical expectation, and Var for the theoretical variance.