Editing Least squares (section)

==Example==

Consider a simple example drawn from physics. A spring should obey [[Hooke's law]] which states that the extension of a spring {{mvar|y}} is proportional to the force, ''F'', applied to it.
<math display="block">y = f(F,k) = k F</math>
constitutes the model, where ''F'' is the independent variable. In order to estimate the [[force constant]], ''k'', we conduct a series of ''n'' measurements with different forces to produce a set of data, <math>(F_i, y_i),\ i=1,\dots,n\!</math>, where ''y<sub>i</sub>'' is a measured spring extension.<ref>{{Cite book | title = Mechanics of Materials | last1 = Gere | first1 = James M. | date = 2013 | publisher = Cengage Learning | last2 = Goodno | first2 = Barry J. | isbn = 978-1-111-57773-5 |edition = 8th | location = Stamford, Conn. | oclc = 741541348}}</ref> Each experimental observation will contain some error, <math>\varepsilon</math>, and so we may specify an empirical model for our observations,
<math display="block"> y_i = kF_i + \varepsilon_i. </math>

There are many methods we might use to estimate the unknown parameter ''k''. Since the ''n'' equations in the ''m'' variables in our data comprise an [[overdetermined system]] with one unknown and ''n'' equations, we estimate ''k'' using least squares. The sum of squares to be minimized is<ref name=":1" />
<math display="block"> S = \sum_{i=1}^n \left(y_i - k F_i\right)^2. </math>

The least squares estimate of the force constant, ''k'', is given by
<math display="block">\hat k = \frac{\sum_i F_i y_i}{\sum_i F_i^2}.</math>

We assume that applying force '''''causes''''' the spring to expand. After having derived the force constant by least squares fitting, we predict the extension from Hooke's law.