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.
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(A) |
Comparing Equation A with the following equation
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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.
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(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.
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(D) |
The following equation defines the fit model.
yi = b0f0(x) + b1f1(x) + … + bk – 1fk – 1(x) | (E) |