Polynomial Fit with a Single Predictor Variable

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August 2007

NI Part Number:
371361D-01

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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 {x _i, y _i}, 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 x _ij = x _i ^j, as shown by the following equations:

x _i ₀ = x _i ⁰ = 1.
x _i ₁ = x _i.
x _i ₂ = x _i ².
…
x _ik _{– 1} = x _i ^k ^{– 1}.

(B)

Because x _ij = x _i ^j, you can build the observation matrix H as shown by the following equation:

(C)

Instead of using x _ij = x _i ^j, you also can choose another function formula to fit the data sets {x _i, y _i}. In general, you can select x _ij = f _j(x _i). Here, f _j(x _i) is the function model that you choose to fit the observation data. In polynomial fit, f _j(x _i) = x _i ^j.

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

(D)

The following equation defines the fit model:

y _i = b ₀ f ₀(x) + b ₁ f ₁(x) + … + b _k _{– 1} f _k _{– 1}(x)

(E)

Polynomial Fit with a Single Predictor Variable

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