# Time series

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There are three steps in the revelation of any truth:
In the first, it is ridiculed;
In the second, resisted;
In the third, it is considered self-evident.

—Schopenhauer (1788-1860)


Discrete-time signals are sequences of numbers, with each number being identified by a fixed time instant. Such a series of data in a time sequence is called a time series. In other words, a time series ${\displaystyle x_{n}}$ is a series of data, with each data value ${\displaystyle x_{n}}$ being associated with a discrete, equally spaced time index n. The time index n is taken to be a whole number, or integer.

Time series occur in all branches of science (Wold, 1938[1]; Kolmogorov, 1941[2]; Wiener, 1942[3]). Economic data always appear in the form of numerical time series. Some meteorologic data, such as daily temperatures, are numerical time series; other meteorologic data, such as continuous barographic records, are continuous-time signals. Continuous-time signals appear in the engineering, biological, and physical sciences. Such continuous-time signals can be read (or measured, observed, or sampled) at equal intervals of time, thereby generating time series (Robinson and Silvia, 1979[4], 1980[5]).

Because a time series represents only the sampled values of a continuous-time signal, it provides only a limited description of the signal. By taking the sampling instants close enough together, the amount of information that is lost by replacing a well-behaved continuous-time function by a time series can be made small. A time spacing that is too gross would mean substantial information loss in the sampling process. At the other extreme, a time spacing that is too fine would mean substantial redundancy in information produced by the sampling process. Thus, in determining the time interval for sampling, we always must balance redundant information against lost information, with consideration given to their relative costs.

It is worth noting that models of both discrete-time systems and continuous-time systems should take into account uncertainties in the amplitudes of the variables. A signal, whether it is a train of pulses or a function of continuous time, always has amplitude imperfections, and these make its precise measurement uncertain to some degree.

The term time series is a generic one. Often, the term is used to represent a signal continuing over all time, from the remote past to the distant future. Thus, we can say that such a time series has infinite time duration or infinite length. However, in any actual situation, we can obtain the values of a time series only over a finite interval of time. Because a finite number of data values represents a sample of an infinite-length time series, we shall call a finite portion of a time series a sample time series.

Consider two processes: (1) the sampling process that extracts a time series (over all discrete time) from a continuous-time function (over all continuous time) and (2) the sampling process that extracts a sample time series (over a finite interval of discrete time) from a time series (over all discrete time).

A sample time series, then, is a finite portion of a time series — that is, the portion from some fixed time to a later time. For example, for a given time series, we might have available only the portion from time ${\displaystyle n=1}$ to time ${\displaystyle n=15}$, consisting of the 15 values

 {\displaystyle {\begin{aligned}&\left\{x_{\rm {l}}{\rm {,\ }}x_{\rm {2}}{\rm {,\dots ,\ }}x_{\rm {l5}}\right\}.\end{aligned}}} (1)

An example of a time series with 15 values (or readings or measurements or observations or samples) is {8, 20, 29, 34, 33, 26, 18, 11, 3, -3, 9, 25, 25, 15, 5}. In this example, one would have to make a notation that the first value, 8, is the value for time ${\displaystyle n=1}$. The following values, 20, 29, …, automatically would be in sequence — they are the values for ${\displaystyle n=2,3,...,14,15}$.

## References

1. Wold, H., 1938, A study in the analysis of stationary time series: Almqvist and Wiksells.
2. Kolmogorov, A., 1941, Interpolation und Extrapolation von stationaren zufalligen Folgen: Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, 5, 3-14. (In Russian, with a German summary.)
3. Wiener, N., 1942, The extrapolation, interpolation, and smoothing of stationary time series: National Defense Research Council. Reprint, 1949, John Wiley.
4. Robinson, E. A., and M. T. Silvia, 1979, Digital foundations of time series analysis, 1: The Box-Jenkins approach: Holden-Day.
5. Robinson, E. A., and M. T. Silvia, 1980, Digital foundations of time series analysis, 2: Wave-equation space-time processing: Holden-Day.