Linear trend estimation

Linear trend estimation is a statistical technique to aid interpretation of data. When a series of measurements of a process are treated as, for example, a sequences or time series, trend estimation can be used to make and justify statements about tendencies in the data, by relating the measurements to the times at which they occurred. This model can then be used to describe the behaviour of the observed data, without explaining it. In particular, it may be useful to determine if measurements exhibit an increasing or decreasing trend which is statistically distinguished from random behaviour. Some examples are determining the trend of the daily average temperatures at a given location from winter to summer, and determining the trend in a global temperature series over the last 100 years. In the latter case, issues of homogeneity are important (for example, about whether the series is equally reliable throughout its length). Given a set of data and the desire to produce some kind of model of those data, there are a variety of functions that can be chosen for the fit. If there is no prior understanding of the data, then the simplest function to fit is a straight line with the data values on the y axis, and time (t = 1, 2, 3, ...) on the x axis. Once it has been decided to fit a straight line, there are various ways to do so, but the most usual choice is a least-squares fit. This method minimizes the sum of the squared errors in the data series y. Given a set of points in time , and data values observed for those points in time, values of and are chosen so that is minimized. Here at + b is the trend line, so the sum of squared deviations from the trend line is what is being minimized. This can always be done in closed form since this is a case of simple linear regression. For the rest of this article, “trend” will mean the slope of the least squares line, since this is a common convention. Before considering trends in real data, it is useful to understand trends in random data.