Polynomial Basis Expansion for Feature Engineering

In machine learning and statistical modeling, polynomial basis expansion is a powerful feature transformation technique that maps input data into a higher-dimensional space by generating polynomial features. This transformation enables linear models to capture nonlinear relationships in the data, effectively fitting curves to data that would otherwise require complex nonlinear algorithms.

Given a set of scalar data points and a specified polynomial degree d, the polynomial basis expansion transforms each data point x into a feature vector containing its powers from 0 to d:

$$\phi(x) = [x^0, x^1, x^2, \ldots, x^d] = [1, x, x^2, \ldots, x^d]$$

This expanded representation allows models like linear regression to learn polynomial relationships. For example, with degree 2, a simple input x becomes the feature vector [1, x, x²], enabling the model to fit quadratic curves.

Why This Matters:

• The constant term x⁰ = 1 serves as the bias/intercept term in regression models
• Higher degrees allow fitting more complex curves but risk overfitting
• This is the foundation of polynomial regression and kernel methods

Your Task: Write a Python function that performs polynomial basis expansion on a list of data points. The function should:

• Transform each data point into a vector of polynomial features from degree 0 to the specified degree
• Return a nested list where each inner list represents the transformed features for one data point
• Return an empty list if the degree is negative (invalid input)