On the Use of Running Trends as Summary Statistics for Univariate Time Series and Time Series Association

Given a time series, running trends analysis (RTA) involves evaluating least squares trends over overlapping time windows of L consecutive time points, with overlap by all but one observation. This produces a new series called the running trends series, which is used as summary statistics of the original series for further analysis. In recent years, RTA has been widely used in climate applied research as summary statistics for time series and time series association. There is no doubt that RTA might be a useful descriptive tool, but, despite its general use in applied research, precisely what it reveals about the underlying time series is unclear and, as a result, its interpretation is unclear too. This paper contributes to such interpretation in two ways: 1) an explicit formula is obtained for the set of time series with a given series of running trends, making it possible to show that running trends, alone, perform very poorly as summary statistics for univariate time series and time series association; and 2) an equivalence is established between RTA and the estimation of a (possibly nonlinear) trend component of the underlying time series using a weighted moving average filter. Such equivalence provides a solid ground for RTA implementation and interpretation/validation. In this respect, the authors propose as diagnostic tools for RTA 1) the plot of the original series, with RTA trend estimation superposed, 2) the average R-2 value and the percentage of statistically significant running trends across windows, and 3) the plot of the running trends series with the corresponding confidence intervals.