TD_PolynomialFeaturesTranform function extracts values of arguments [TargetColumns, Degree, IncludeBias, and InteractionOnlygenerates] from the output of the TD_PolynomialFeaturesFit function and generates a feature matrix of all polynomial combinations of the features.
TD_PolynomialFeaturesTransform is a method used to transform the original input feature matrix X into a new matrix X', where each column represents a polynomial combination of the original features. This can be expressed mathematically as follows:
X' = [1, X, X2, ..., Xd]
where X is the original input feature matrix, d is the degree of the polynomial function, and Xk represents the matrix of all combinations of the original features up to degree k. The first column of the transformed feature matrix represents a constant term, and the remaining columns represent the original input features and their pairwise products up to the specified degree.
For example, if X has two features, x1 and x2, and the degree of the polynomial function is 2, then the transformed input feature matrix X' would be:
This transformation enables you to capture nonlinear relationships between the input features and the target variable. By including polynomial combinations of the original features, you can model more complex relationships than what is possible with linear models.
In addition to polynomial combinations, the TD_PolynomialFeaturesTransform method can also include interaction terms and cross-terms, which are useful in certain applications. Interaction terms involve multiplying two or more features together, while cross-terms involve multiplying features from different inputs.
After obtaining the input feature matrix X', you can fit a polynomial function to the dataset using regression techniques such as ordinary least squares (OLS) regression or regularized regression.
Overall, the TD_PolynomialFeaturesTransform method is a powerful tool for capturing nonlinear relationships between the input features and the target variable and is widely used in fields such as data science, engineering, and economics.