Master Polynomial Regression: Fit Non‑Linear Data with Simple Polynomials

1 Polynomial regression

Although linear models can solve many practical problems, many real‑world systems exhibit nonlinear growth trends, requiring nonlinear fitting.

For nonlinear regression, the simplest and most common method is polynomial regression. A polynomial is a basic algebraic expression formed from variables (unknowns) and constant coefficients using a finite number of additions, subtractions, multiplications, and natural‑number exponents.

A polynomial is a type of expression. A polynomial with a single variable is a univariate polynomial (e.g., a simple quadratic). A polynomial with multiple variables is a multivariate polynomial (e.g., a three‑variable cubic).

We can fit scattered data points with a polynomial. A standard univariate high‑order polynomial function is expressed as a sum of terms where the degree indicates the highest power and each term has a coefficient. When fitting data with the polynomial, two key elements must be determined:

Polynomial coefficients

Polynomial degree

These are the two basic elements of a polynomial.

If the polynomial degree is manually specified, only the coefficient values need to be determined. For example, fixing the degree to a certain value turns the polynomial into a specific form with unknown coefficients.

When the coefficient values are set, the problem reduces to minimizing the residual sum of squares, i.e., applying the ordinary least‑squares method learned from linear regression.

2 Other nonlinear function fitting

We can also fit univariate functions using other forms such as exponential, logarithmic, or sinusoidal expressions.

The final choice of model—linear or which nonlinear form to use—depends on understanding the problem and the fitting performance.