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:
“Of all the nations in the world, the United States was built in nobody’s image. It was the land of the unexpected, of unbounded hope, of ideals, of quest for an unknown perfection. It is all the more unfitting that we should offer ourselves in images. And all the more fitting that the images which we make wittingly or unwittingly to sell America to the world should come back to haunt and curse us.”
—Daniel J. Boorstin (b. 1914)