Feature Transfromations
Business Analytics
Motivation
Our Focus
f(\beta, x) = \sum _{j=1}^d \beta_j x_j
Linear in both Parameters and inputs
Potentially Problematic
f(\beta, x) = \sum _{j=1}^d \beta_j x_j
Linear in both Parameters and inputs
Misspecification Error
Types of Error
Feature Transformations
The Optimization Problem
\hat{\beta} = \underset{\beta}{\text{argmin}} \ \underbrace{\frac{1}{n}\sum_{i=1}^n \Big(Y_i - \sum_{p =1}^d \beta_pX_{ip}\Big)^2}_{\text{Objective Function}}
f(\beta, x) = \sum_{p=1}^d\sum_{j=1}^k \beta_{pk} \phi_k(x_p)
Feature Transformations
Model
\phi(x)
Polynomial Transformations in Python
linear_model1 = smf.ols(f'{dep_var} ~ {rhs_var}', data=df)
results1 = linear_model1.fit()linear_model3 = smf.ols(f'{dep_var} ~ {rhs_var} + I({rhs_var} **2) + I({rhs_var} **3) ', data=df)
results3 = linear_model3.fit()\phi_k(x) = x^k