Time Series

Time series are time-ordered observations or measurements. A time series can consist of elements equally spaced in time—for example, annual birth counts in a city during four decades—or measurements collected at irregular periods, for example, a person's weight on consecutive visits to a doctor's office. Time plots, that is, graphical representations of time series, are very useful to provide a general view that often allows us to visualize two basic elements of time series: short-term changes and long-term or secular trends.

Time series are often used in epidemiology and public health as descriptive tools. For instance, time plots of life expectancy at birth in Armenia, Georgia, and Ukraine during the years 1970 to 2003 (σee Figure 1) reveal relatively stagnant health conditions before the 1990s in these three countries of the former USSR, as well as a substantial deterioration of health [Page 1036]in the early to mid-1990s, after the breakdown of the USSR. The sharp trough in 1988 in Armenian life expectancy reflects the impact of the Spivak region earthquake that killed tens of thousands of people, including many children.

Figure 1 Life Expectancy at Birth (Years) in Three Countries Formerly Part of the USSR

Source: Created by the author using data from the European Health for All database (HFA-DB).

In describing a time series in mathematical terms, the observation or measurement is considered a variable, indicated for instance by yt, where the subscript t indicates time. It is customary to set t = 0 for the first observation in the series, so that the entire expanded series will be represented by y0, y1, y2, y3, …, ym for a series containing m + 1 elements. The element for time k in this series will be therefore yk, with k > 0and k < m.

In some cases, time series are well described by a mathematical model that can be exponential, logistic, linear (a straight line), and so on. They often reveal recurring patterns, for instance seasonal ones, associated with different seasons of the year. A series of monthly deaths due to respiratory disorders during several consecutive years will show a yearly peak in winter and a yearly trough in summer. Other patterns may be periodic but not seasonal; for instance, among Jews in Israel, deaths are more frequent on Sundays. Still other patterns are recurrent but are acyclical, that is, not periodical; for instance, in market economies, the unemployment rate reveals successive peaks (recessions) and troughs (periods of economic expansion) that make up what has come to be called the ‘business cycle’ (Figure 2), which is not α ‘cycle’ in the ideal sense because it occurs at irregular intervals.

In most time series, there is strong first-order autocorrelation, which means that consecutive values are highly correlated. That is, the value yk is usually not very far from its neighbors, yk − 1 and yk + 1. Thisis the basis for interpolation and extrapolation, the two techniques used to estimate an unknown value of a time series variable. A missing value inside a time series can be estimated by interpolation, which usually implies an averaging of the values in the neighborhood of the missing one. For instance, if the value for Year 8 in a series of annual values was unobserved, it can be estimated as an average of the observed values for years 6, 7, 9, and 10. In general, the error in the estimation through interpolation will [Page 1037]be smaller than the error in estimating through extrapolation, which implies estimating an unobserved value out of the time range of the time series, usually in the future. (Backward extrapolation to the past is also possible, though usually uninteresting.) Forecasting is a term used for predicting the value of a variable at a later time than the last one observed. When a causal model involving the major determinants of the variable to be predicted is not available, forecasting is done through more or less sophisticated techniques of extrapolation. For instance, if the suicide rates during the past 5 years in a nation were, respectively, 12.7, 13.4, 12.9, 12.2, and 13.1 per million, using a somewhat rough extrapolation we might forecast that the suicide rate next year will probably be around 12 or 13 per million. This conclusion, as any other forecast, has a large uncertainty associated with it, because time series may have sudden upturns or downturns that cannot be predicted. Obviously, the uncertainty grows exponentially as the future we try to forecast becomes more distant. The formal techniques of forecasting imply fitting a mathematical model (linear, exponential, etc.) to the observed data, then computing the expected value in the future with that mathematical model. The auto regressive integrated moving average models or ARIMA models, often used in forecasting (frequently referred to as the Box-Jenkins models or methodology), have been sometimes applied in biomedical sciences and epidemiology, but they constitute a quite specialized field of statistics. Like ARIMA models, spectral analysis is another specialized technique in the field of time series analysis.

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