Splines Piecewise Polynomial Regression

The position of the knots, for many practical purposes, can be determined intuitively. If the knot positions are known, a standard least-squares equation can be used to model them. If the knots are not known, they can be estimated via nonlinear regression techniques. Additionally, most polynomial splines are subject to serious multicollinearity in the rr, predictors, so the fewer splines, the better. 

For most practical situations, Montgomery et al. (2001) recommend using a cubic spline  

This model is useful, but often, a square spline is also useful. That is. 

Recall from Example 7.1 that the y, data were collected on cells per wound closure and x, was the day of measurement—0 through 12. Because there is a nonlinear comptment that we will keep in that basic model, which is y = ho + bix + b2 , and adding the spline, the model we use is 

Input Data Points, Spline Model of Example 7.1 

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Two approaches for interpolation function have been used. In one, polynomials, e.g., in powers of w", are fit to impedance data. Usually, a piecewise regression is required. While piece-wise polynomials are excellent for smoothing, the best example being splines, they are not very reliable for extrapolation and result in a relatively large number of peirameters. A second approach is to use interpolation... 

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