Derivation of The Smoothing Spline
It is useful to think of fitting a smoothing spline in two steps:
- First, derive the values .
- From these values, derive for all x.
Now, treat the second step first.
Given the vector of fitted values, the sum-of-squares part of the spline criterion is fixed. It remains only to minimize, and the minimizer is a natural cubic spline that interpolates the points . This interpolating spline is a linear operator, and can be written in the form
where are a set of spline basis functions. As a result, the roughness penalty has the form
where the elements of A are . The basis functions, and hence the matrix A, depend on the configuration of the predictor variables, but not on the responses or .
Now back to the first step. The penalized sum-of-squares can be written as
where . Minimizing over gives
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Famous quotes containing the word smoothing:
“Generation on generation, your neck rubbed the windowsill
of the stall, smoothing the wood as the sea smooths glass.”
—Donald Hall (b. 1928)