Polynomial Regression
Synopsis
This operator generates a polynomial regression model from the given ExampleSet. Polynomial regression is considered to be a special case of multiple linear regression.
Description
Polynomial regression is a form of linear regression in which the relationship between the independent variablexand the dependent variableyis modeled as annthorder polynomial. In RapidMiner,yis the label attribute andx是用于常规属性的设置吗the prediction ofy. Polynomial regression fits a nonlinear relationship between the value ofxand the corresponding conditional mean ofy, denotedE(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 functionE(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 variableyin 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 ofx, the conditional expectation ofyincreases byw1units.
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 + (w1x1 ^1) + (w2x2 ^2)
In this model, when the temperature is increased fromxtox + 1units, the expected yield changes byw1 + w2 + 2 (w2 * x). The fact that the change in yield depends onxis 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 ofyas annthorder polynomial, yielding the general polynomial regression model:
y = w0 + (w1x1 ^1) + (w2x2 ^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.
Differentiation
Linear Regression
Polynomial regression is a form of linear regression in which the relationship between the independent variablexand the dependent variableyis modeled as annthorder polynomial.
Input
training set
This input port expects an ExampleSet. This operator cannot handle nominal attributes; it can be applied on data sets with numeric attributes. Thus often you may have to use the Nominal to Numerical operator before application of this operator.
Output
model
The regression model is delivered from this output port. This model can now be applied on unseen data sets.
example set
The ExampleSet that was given as input is passed without any modifications to the output through this port. This is usually used to reuse the same ExampleSet in further operators or to view the ExampleSet in the Results Workspace.
Parameters
Max iterations
This parameter specifies the maximum number of iterations to be used for the model fitting.
Replication factor
This parameter specifies the amount of times each input variable is replicated, i.e. how many different degrees and coefficients can be applied to each variable.
Max degree
This parameter specifies the maximal degree to be used for the final polynomial.
Min coefficient
This parameter specifies the minimum number to be used for the coefficients and the offset.
Max coefficient
This parameter specifies the maximum number to be used for the coefficients and the offset.
Use local random seed
This parameter indicates if alocal random seedshould be used for randomization. Using the same value of thelocal random seedwill produce the same randomization.
Local random seed
This parameter specifies thelocal random seed. This parameter is only available if theuse local random seedparameter is set to true.