Editing Linear trend estimation (section)

==Fitting a trend: Least-squares==
Given a set of [[data]], there are a variety of [[Function (mathematics)|functions]] that can be chosen to fit the data. The simplest function is a [[Line (geometry)|straight line]] with the dependent variable (typically the measured data) on the vertical axis and the independent variable (often time) on the horizontal axis.

The [[least-squares]] fit is a common method to fit a straight line through the data. This method [[Optimization problem|minimizes]] the sum of the squared errors in the data series <math>y</math>. Given a set of points in time <math>t</math> and data values <math>y_t</math> observed for those points in time, values of <math>\hat a</math> and <math>\hat b</math> are chosen to minimize the sum of squared errors

:<math>\sum_t \left[ y_t - \left( \hat{a}t + \hat{b} \right) \right]^2</math>.

This formula first calculates the difference between the observed data <math>y_t</math> and the estimate <math>(\hat{a}t + \hat{b})</math>, the difference at each data point is squared, and then added together, giving the "sum of squares" measurement of error. The values of <math>\hat{a}</math> and <math>\hat{b}</math> derived from the data parameterize the simple linear estimator <math>\hat{y} = \hat{a} x + \hat{b}</math>. The term "trend" refers to the slope <math>\hat{a}</math> in the least squares estimator.