In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modeled as a polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y |x). Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y | x) is linear in the unknown parameters that are estimated from the data. Thus, polynomial regression is a special case of multiple linear regression. The explanatory (independent) variables resulting from the polynomial expansion of the "baseline" variables are known as higher-degree terms. Such variables are also used in classification settings.
History Polynomial regression models are usually fit using the method of least squares. The least-squares method minimizes the variance of the unbiased estimators of the coefficients, under the conditions of the Gauss–Markov theorem. The least-squares method was published in 1805 by Legendre and in 1809 by Gauss. The first design of an experiment for polynomial regression appeared in an 1815 paper of Gergonne. In the twentieth century, polynomial regression played an important role in the development of regression analysis, with a greater emphasis on issues of design and inference. More recently, the use of polynomial models has been complemented by other methods, with non-polynomial models having advantages for some classes of problems.
Definition and example The goal of regression analysis is to model the expected value of a dependent variable y in terms of the value of an independent variable (or vector of independent variables) x. In simple linear regression, the model
y = β 0 + β 1 x + ε , {\displaystyle y=\beta _{0}+\beta _{1}x+\varepsilon ,\,}
is used, where ε is an unobserved random error with mean zero conditioned on a scalar variable x. In this model, for each unit increase in the value of x, the conditional expectation of y increases by β1 units. In many settings, such a linear relationship may not hold. For example, if we are modeling the yield of a chemical synthesis in terms of the temperature at which the synthesis takes place, we may find that the yield improves by a different amount for each unit increase in temperature. Or we may find that the yield decreases with increasing temperature (but only for a certain range of temperatures) and increases with increasing temperature in a different range of temperatures. In this case, we might propose a quadratic model of the form
y = β 0 + β 1 x + β 2 x 2 + ε . {\displaystyle y=\beta _{0}+\beta _{1}x+\beta _{2}x^{2}+\varepsilon .\,}
In this model, when the temperature is increased from x to x + 1 units, the expected yield changes by β 1 + β 2 ( 2 x + 1 ) . {\displaystyle \beta _{1}+\beta _{2}(2x+1).} (This can be seen by obtaining the derivative with respect to x of the regression formula.) For infinitesimal changes in x, the effect on y is given by the total derivative with respect to x: β 1 + 2 β 2 x . {\displaystyle \beta _{1}+2\beta _{2}x.} The fact that the change in yield depends on x is what makes the relationship between x and y nonlinear even though the model is linear in the parameters to be estimated. In general, we can model the expected value of y as an nth degree polynomial, yielding the general polynomial regression model
y = β 0 + β 1 x + β 2 x 2 + β 3 x 3 + ⋯ + β n x n + ε . {\displaystyle y=\beta _{0}+\beta _{1}x+\beta _{2}x^{2}+\beta _{3}x^{3}+\cdots +\beta _{n}x^{n}+\varepsilon .\,}
Conveniently, these models are all linear from the point of view of estimation, since the regression function is linear in terms of the unknown parameters β0, β1, .... Therefore, for least squares analysis, the computational and inferential problems of polynomial regression can be completely addressed using the techniques of multiple regression. This is done by treating x, x2, ... as being distinct independent variables in a multiple regression model.
Matrix form and calculation of estimates The polynomial regression model
… excerpt ends here. Continue reading the full article.
