Customer service UNILAB

When we have a table of data points (like a list of x and y values), we often want to find a polynomial that describes the relationship between them. There are two main goals we might have, leading to different methods and ways to check how well our polynomial fits.

1. Polynomial Interpolation: Passing Exactly Through the Points

Goal: To find a polynomial that perfectly goes through every data point in your table. Imagine connecting dots with a smooth curve – that's interpolation.

• How it works (Newton's Methods):

  • Newton Forward Interpolation: This method is best when you want to estimate values near the beginning of your data set. It uses "forward differences" calculated from the starting points.

  • Newton Backward Interpolation: This one is useful for estimating values near the end of your data set. It uses "backward differences" calculated from the ending points.

• Both Newton's methods are efficient, especially if your x values are evenly spaced.

• How good is the fit? (R-squared is not used here): Since the polynomial must pass through every original data point, it will perfectly match them. If you were to calculate something like R-squared ($R^2$) for these methods using your original data, it would always be 1 (or 100%). This simply means the polynomial perfectly describes the points it was built from, but it doesn't tell you how well it might predict new, unseen data, or how well it represents an underlying trend if your data has noise.

2. Polynomial Regression: Finding the Best Fit for Noisy Data

Goal: To find a polynomial that best approximates the trend in your data, even if the data points don't lie perfectly on a smooth curve (e.g., if there's measurement error or "noise"). This is like drawing a general trend line, not necessarily hitting every single dot.

• How it works (Stepwise Selection Methods): When doing regression, especially with polynomials (e.g., trying to see if x, x², or x³ terms are important), We often use "stepwise" methods to decide which terms to include in our final polynomial equation:

  • Forward Selection: Start with a very simple polynomial (maybe just a constant). Then, one by one, add the most "helpful" new polynomial terms (like x, then x², then x³) if they significantly improve the fit. You stop when adding more terms doesn't help much.

  • Backward Elimination: Start with a complex polynomial that includes all possible terms you might need (e.g., up to x⁵). Then, one by one, remove the "least helpful" terms until all the remaining terms are important for the fit.

  • Stepwise Regression: This is a combination of Forward and Backward. At each step, it might add a new term or remove an existing one, always looking for the best possible polynomial structure.

• How good is the fit? (Using R-squared): For regression, we use R-squared ($R^2$) to measure how well our polynomial model explains the variation in your y values.

  • An $R^2$ value close to 1 means your polynomial explains a large proportion of the variability in the data (a good fit).

  • An $R^2$ value close to 0 means your polynomial explains very little of the variability.

• We often use Adjusted R-squared ($R^{adj}$), which is better for comparing different regression models. $R^{adj}$ penalizes models that add unnecessary terms, helping us avoid "overfitting" (where the polynomial becomes too complex and fits the noise in the data rather than the true underlying pattern). Other measures like AIC/BIC are also used for this purpose.

In short, interpolation gives you a polynomial that hits every point exactly, useful for precise curve-fitting (where $R^2$ isn't meaningful). Regression gives you a polynomial that best approximates the overall trend in noisy data, and here, $R^2$ (especially $R^{adj}$) is a crucial tool to judge how well your polynomial model captures that trend.