Stochastic Processes, Probability and Statistics

Often when doing an experiment or monitoring a system we end up with a sequence of observations, x1, x2, x3,..., x t .... These may be observations made at discrete time intervals e.g. monthly sales figures, velocity per second or we may prefer to think of them as forming a continuous record as time passes, for example like a pen recorder. Figures 6.1–6.4 give examples of records like these. Such sequences or traces are called time series and we shall use x(t) to denote the observation made at time t. (In almost all applications t is time but one could let t be distance, say down a railway line and x(t) could be the “height” of the rail.) Thus in Fig. 6.1 x(t) denotes the seismic noise at time t. Sometimes xt is used to denote the series, usually when we have discrete time intervals.