Fitting A Trend: Least-squares
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 plotted vertically and values of time (t = 1, 2, 3, ...) plotted horizontally.
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 minimises the sum of the squared errors in the data series, denoted the y variable.
Given a set of points in time, and data values observed for those points in time, values of and are chosen so that
is minimised. Here at + b is the trend line, so the sum of squared deviations from the trend line is what is being minimised. 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.
Read more about this topic: Trend Estimation
Famous quotes containing the word fitting:
“Children’s view of the world and their capacity to understand keep expanding as they mature, and they need to ask the same questions over and over, fitting the information into their new level of understanding.”
—Joanna Cole (20th century)