Statistical - Cubic Regression (Cubic Spline Fit) ?

[Thomas Klemm]

(06-01-2022 06:26 AM)Werner Wrote:  I have the impression cubic regression and cubic spline regression are mixed up.

We can use the same dataset as in the Cubic Regression Calculator: (0, 1), (2, 0), (3, 3), (4, 5), (5, 4)

Code:

0 1
2 0
3 3
4 5
5 4

Cubic spline interpolation

Equation

\(
f(x) = \begin{cases}3.1105 \cdot 10^{-1}\cdot x^3 -4.8828 \cdot 10^{-61}\cdot x^2 -1.7442\cdot x + 1.0000, & \text{if } x \in [0,2], \\-8.5465 \cdot 10^{-1}\cdot x^3 + 6.9942\cdot x^2 -1.5733 \cdot 10^{1}\cdot x + 1.0326 \cdot 10^{1}, & \text{if } x \in (2,3], \\-4.5930 \cdot 10^{-1}\cdot x^3 + 3.4360\cdot x^2 -5.0581\cdot x -3.4884 \cdot 10^{-1}, & \text{if } x \in (3,4], \\6.9186 \cdot 10^{-1}\cdot x^3 -1.0378 \cdot 10^{1}\cdot x^2 + 5.0198 \cdot 10^{1}\cdot x -7.4023 \cdot 10^{1}, & \text{if } x \in (4,5].\end{cases}
\)

We get piecewise defined cubic polynomials that fit all given points.

Evaluation

\(
f(1)=-0.43314
\)

Graph



Usage

Quote:Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points, or knots.
These were used to make technical drawings for shipbuilding and construction by hand, as illustrated in the figure.

Polynomial interpolation

Equation

\(
f(x)=1.6667 \cdot 10^{-2}\cdot x^{4}-5.6667 \cdot 10^{-1}\cdot x^{3}+ 3.6833\cdot x^{2}-5.7333\cdot x+ 1.0000
\)

We get a single polynomial that fits all given points.

Evaluation

\(
f(1)=-1.6000
\)

Graph



Usage

Quote:In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset.

Conclusion

Use a cubic spline if you want a function that goes smoothly through the given data points.
Disadvantage: The function is defined piecewise.

Use polynomial interpolation if you want a single polynomial passing the given data points.
Disadvantage: Runge's phenomenon shows that for high values of n, the interpolation polynomial may oscillate wildly between the data points.

Use cubic regression when you want to get the cubic polynomial that best fits the given data points.
Disadvantage: The polynomial usually does not pass the given data points.