Polynomial Regression (AI Studio Core)

Description

Polynomial regression is a form of linear regression in which the relationship between the independent variable x and the dependent variable y is modeled as an nth order polynomial. In Altair RapidMiner, y is the label attribute and x is the set of regular attributes that are used for the prediction of y. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y | x), and has been used to describe nonlinear phenomena such as the growth rate of tissues and the progression of disease epidemics. 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. For this reason, polynomial regression is considered to be a special case of multiple linear regression.

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 following model is used:

y = w0 + ( w1 * x )

In this model, for each unit increase in the value of x, the conditional expectation of y increases by w1 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 increasing amounts for each unit increase in temperature. In this case, we might propose a quadratic model of the form:

y = w0 + (w1 * x1 ^1) + (w2 * x2 ^2)

In this model, when the temperature is increased from x to x + 1 units, the expected yield changes by w1 + w2 + 2 (w2 * x). The fact that the change in yield depends on x is what makes the relationship nonlinear (this must not be confused with saying that this is nonlinear regression; on the contrary, this is still a case of linear regression). In general, we can model the expected value of y as an nth order polynomial, yielding the general polynomial regression model:

y = w0 + (w1 * x1 ^1) + (w2 * x2 ^2) + . . . + (wm * xm ^m)

Regression is a technique used for numerical prediction. It is a statistical measure that attempts to determine the strength of the relationship between one dependent variable ( i.e. the label attribute) and a series of other changing variables known as independent variables (regular attributes). Just like Classification is used for predicting categorical labels, Regression is used for predicting a continuous value. For example, we may wish to predict the salary of university graduates with 5 years of work experience, or the potential sales of a new product given its price. Regression is often used to determine how much specific factors such as the price of a commodity, interest rates, particular industries or sectors influence the price movement of an asset.