Understanding Polynomial Regression and Step Function

What is Polynomial Regression?:

Polynomial regression is an extension of linear regression where we model the relationship between a dependent variable and one or more independent variables using a polynomial. Unlike linear regression, which models the relationship using a straight line, polynomial regression models the relationship using a curve.

The general form of a polynomial regression equation of degree $n$ is:

where:

• $y$ is the dependent variable,
• $x$ is the independent variable,
• 0,1,…,β0​,β1​,…,βn​ are the coefficients, and
• $ϵ$ is the error term.

Polynomial regression is used when the relationship between the variables is nonlinear and can be captured better with a curved line.

What is Step Function (Piecewise Constant Regression)?:

Step functions allow us to model the relationship between a dependent variable and one or more independent variables using step-like or piecewise constant functions. In this approach, the range of the independent variable(s) is divided into disjoint segments, and a separate constant value is fit to the dependent variable within each segment.

The general form of a step function model is:

where:

• $y$ is the dependent variable,
• $x$ is the independent variable,
• $I$ is an indicator function that equals 1 if the condition inside the parentheses is true and 0 otherwise,
• 1,2,…,S1​,S2​,…,Sk​ are the disjoint segments, and
• $ϵ$ is the error term.

Step functions are used when the relationship between the variables is better captured by different constants in different ranges of the independent variable.

Certainly! Both these plots provide different ways of understanding the relationships between the rates of obesity and inactivity with the rate of diabetes across different counties.

Polynomial Regression Analysis on the Diabetes Data:
– Our analysis suggested that a linear model (polynomial of degree 1) provided the lowest test error, indicating that the relationships between obesity/inactivity rates and diabetes rates are relatively linear within the examined range.
– However, the analysis didn’t provide a clear indication of a non-linear (polynomial) relationship as higher-degree polynomials did not yield a lower test error.

 Step Function (Piecewise Constant Regression) Analysis on Diabetes data:
– The plots represent piecewise constant models (step functions) that divide the range of obesity and inactivity rates into distinct segments. Within each segment, the rate of diabetes is approximated as a constant value.
– The breakpoints between segments are determined to minimize the within-segment variance in diabetes rates. This suggests that there could be specific thresholds in obesity and inactivity rates where the rate of diabetes changes.
– For example, in the first plot, as the obesity rate increases, we see a step-wise increase in the diabetes rate. Similarly, in the second plot, as the inactivity rate increases, the diabetes rate also increases in a step-wise manner.

Interpretation
– These analyses help to visualize and understand how obesity and physical inactivity may relate to diabetes rates across different counties.
– While the polynomial regression suggested a linear relationship, the step function analysis revealed that there might be specific ranges of obesity and inactivity rates that correspond to different levels of diabetes rates.
– The step function analysis provides a more segmented view which could be indicative of thresholds beyond which the rate of diabetes significantly increases.