# Polynomial Fit with a Single Predictor Variable

Polynomial fit with a single predictor variable uses one variable to predict another variable. Polynomial fit with a single predictor variable is a special case of multiple regression. If the observation data sets are {xi, yi}, where i = 0, 1, …, n – 1, Equation A defines the model for polynomial fit.

| (A)

Comparing Equation A with the following equation

Source: General LS Linear Fit Theory

shows that xij – xij, as shown by the following equations.

xi0 = xi0 = 1. xi1 = xi. xi2 = xi2. … xik – 1 = xik – 1. | (B)

Because xij – xij, you can build the observation matrix H as shown by the following equation.

| (C)

Instead of using xij – xij, you also can choose another function formula to fit the data sets {xi, yi}. In general, you can select xij = fj(xi). Here, fj(xi) is the function model that you choose to fit the observation data. In polynomial fit, fj(xi) = xij.

In general, you can build H as shown in the following equation.

| (D)

The following equation defines the fit model.

yi = b0f0(x) + b1f1(x) + … + bk – 1fk – 1(x) | (E)