Smoothing spline


Smoothing spline

The smoothing spline is a method of smoothing, or fitting a smooth curve to a set of noisy observations.

Definition

Let (x_i,Y_i); i=1,dots,n be a sequence of observations, modeled by the relation E(Y_i) = mu(x_i). The smoothing spline estimate hatmu of the function mu is defined to be the minimizer (over the class of twice differentiable functions) of [cite book|title=Generalized Additive Models|last=Hastie|first=T. J.|coauthors=Tibshirani, R. J.|year=1990|publisher=Chapman and Hall|isbn=0-412-34390-8] :sum_{i=1}^n (Y_i - hatmu(x_i))^2 + lambda int hatmu"(x)^2 ,dx.

Remarks:
# lambda ge 0 is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate.
# The integral is evaluated over the range of the x_i.
# As lambda o 0 (no smoothing), the smoothing spline converges to the interpolating spline.
# As lambda oinfty (infinite smoothing), the roughness penalty becomes paramount and the estimate converges to a linear least-squares estimate.
# The roughness penalty based on the second derivative is the most common in modern statistics literature, although the method can easily be adapted to penalties based on other derivatives.
# In early literature, with equally-spaced x_i, second or third-order differences were used in the penalty, rather than derivatives.
# When the sum-of-squares term is replaced by a log-likelihood, the resulting estimate is termed "penalized likelihood". The smoothing spline is the special case of penalized likelihood resulting from a Gaussian likelihood.

Derivation of the smoothing spline

It is useful to think of fitting a smoothing spline in two steps:
# First, derive the values hatmu(x_i);i=1,ldots,n.
# From these values, derive hatmu(x) for all "x".

Now, treat the second step first.

Given the vector hat{m} = (hatmu(x_1),ldots,hatmu(x_n))^T of fitted values, the sum-of-squares part of the spline criterion is fixed. It remains only to minimize int hatmu"(x)^2 , dx, and the minimizer is a natural cubic spline that interpolates the points (x_i,hatmu(x_i)). This interpolating spline is a linear operator, and can be written in the form: hatmu(x) = sum_{i=1}^n hatmu(x_i) f_i(x)where f_i(x) are a set of spline basis functions. As a result, the roughness penalty has the form:int hatmu"(x)^2 dx = hat{m}^T A hat{m}.where the elements of "A" are int f_i(x) f_j(x)dx. The basis functions, and hence the matrix "A", depend on the configuration of the predictor variables x_i, but not on the responses Y_i or hat m.

Now back the first step. The penalized sum-of-squares can be written as:|Y - hat m|^2 + lambda hat{m}^T A hat m,where Y=(Y_1,ldots,Y_n)^T.Minimizing over hat m gives:hat m = (I + lambda A)^{-1} Y.

Related methods

Smoothing splines are related to, but distinct from:
* Regression splines. In this method, the data is fitted to a set of spline basis functions with a reduced set of knots, typically by least squares. No roughness penalty is used.
* Penalized Splines. This combines the reduced knots of regression splines, with the roughness penalty of smoothing splines. [cite book|title=Semiparametric Regression|last=Ruppert|first=David|coauthors=Wand, M. P. and Carroll, R. J.|publisher=Cambridge University Press|year=2003|isbn=0-521-78050-0]

Further reading

* Wahba, G. (1990). "Spline Models for Observational Data". SIAM, Philadelphia.
* Green, P. J. and Silverman, B. W. (1994). "Nonparametric Regression and Generalized Linear Models". CRC Press.

References


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