# 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