Time series: An overview

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  • Date: 26 Sep 2012

A time series is created when the status of an observational unit is recorded through time. This definition is sufficiently broad to include most of our everyday existence, and further qualification is necessary...Our basic definition of a time series is thus that we observe a sequence of random variables or  equally spaced in time at , after a suitable rescaling of the time axis if necessary.The traditional approach to time series modeling recognizes four components of a time series: trend, or long-term movements in the series (T); seasonal fluctuations, which are of known periodicity (S); cyclic variations of a nonseasonal variety, whose periodicity is unknown (C); and R, the random error component.

The models developed within this framework are typically of the forms

When the trend is the component of primary concern, it may be modeled directly in the form  where  denotes the deterministic component of the series and  is the error term (or nondeterministic component). Alternatively, trend might be removed by a combination of transformations and differencing to produce a stationary series. When all four components are relevant, a variety of moving averages and filtering operations may be applied to the data, usually to estimate the seasonal component and thus to produce a deseasonalized series.

Data analysis for time series may be performed either in the time domain or the frequency domain. Time domain methods represent the random variable Y(t) in terms of its past history and uncorrelated error terms, the class of linear autoregressive moving average (ARMA) schemes being the main foundation upon which such analyses are based. The structure of such processes may be described in terms of the autocorrelations.

In the frequency domain, the stationary process is described by a set of cosine waves which vary in angular frequency and amplitude. The wavelength, or distance between successive peaks, is . The spectral density is the plot of the squared amplitude against , which indicates the power (or “variance explained”) at frequency .

A fundamental result in time series is that the autocorrelation function and the spectral density function form a Fourier transform pair so that the information contained in one function is formally equivalent to that contained in the other. Nevertheless, given a finite data record, one approach may do better. In the physical sciences, the study of wave-like phenomena often leads to the use of frequency domain analysis, whereas in the social and economic sciences, the regression type structure of the time domain is usually preferred.

Taken from: N. Balakrishnan, Time Series, Encyclopedia of Statistical Sciences, 2011.

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