Now we can apply the method of least squares which is a mathematical procedure for finding the best-fitting line to a given set of points by minimizing the sum of the squares of the offsets of the points from the approximating line. In particular, we consider the following quadratic model: Statistical estimation and inference in linear regression focuses on β.The quadratic regression is a form of nonlinear regression analysis, in which observational data are modeled by a quadratic function. In simple linear regression, p=1, and the coefficient is known as regression slope. Its elements are known as effects or regression coefficients (although the latter term is sometimes reserved for the estimated effects). Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.įormulation In linear regression, the observations ( red) are assumed to be the result of random deviations ( green) from an underlying relationship ( blue) between a dependent variable ( y) and an independent variable ( x). Conversely, the least squares approach can be used to fit models that are not linear models. So a cost functions that are robust to outliers should be used if the dataset has many large outliers. Use of the Mean Squared Error(MSE) as the cost on a dataset that has many large outliers, can result in a model that fits the outliers more than the true data due to the higher importance assigned by MSE to large errors. Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the " lack of fit" in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares cost function as in ridge regression ( L 2-norm penalty) and lasso ( L 1-norm penalty).
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