# Least squares principle

Least squares principle is a widely used method for obtaining the estimates of the parameters in a statistical model based on observed data. Suppose that we have measurements \(Y_1,\ldots,Y_n\) which are noisy versions of known functions \(f_1(\beta),\ldots,f_n(\beta)\) of an unknown parameter \(\beta\). This means, we can write

\[ Y_i = f_i(\beta) + \varepsilon_i, i=1,\ldots,n \]

where \(\varepsilon_1,\ldots,\varepsilon_n\) are quantities that measure the departure of the observed measurements from the model, and are typically referred to as *noise*. Then the **least squares estimate** of \(\beta\) from this model is defined as

\[ \widehat\beta = \min_{\beta} \sum_{i=1}^n(Y_i - f_i(\beta))^2 \]

The quantity \(f_i(\widehat\beta)\) is then referred to as the *fitted value* of \(Y_i\), and the difference \(Y_i - f_i(\widehat\beta)\) is referred to as the corresponding *residual*. It should be noted that \(\widehat\beta\) may not be unique. Also, even if it is unique it may not be available in a closed mathematical form. Usually, if each \(f_i\) is a smooth function of \(\beta\), one can obtain the estimate \(\widehat\beta\) by using numerical optimization methods that rely on taking derivatives of the objective function. If the functions \(f_i(\beta)\) are linear functions of \(\beta\), as is the case in a linear regression problem, then one can obtain the estimate \(\widehat\beta\) in a closed form.

### Contributors

- Debashis Paul