# Smoothing spline

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Smoothing spline

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

Definition

Let $\left(x_i,Y_i\right); i=1,dots,n$ be a sequence of observations, modeled by the relation $E\left(Y_i\right) = mu\left(x_i\right)$. 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] :$sum_\left\{i=1\right\}^n \left(Y_i - hatmu\left(x_i\right)\right)^2 + lambda int hatmu"\left(x\right)^2 ,dx.$

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 $hatmu\left(x_i\right);i=1,ldots,n$.
# From these values, derive $hatmu\left(x\right)$ for all "x".

Now, treat the second step first.

Given the vector $hat\left\{m\right\} = \left(hatmu\left(x_1\right),ldots,hatmu\left(x_n\right)\right)^T$ of fitted values, the sum-of-squares part of the spline criterion is fixed. It remains only to minimize $int hatmu"\left(x\right)^2 , dx$, and the minimizer is a natural cubic spline that interpolates the points $\left(x_i,hatmu\left(x_i\right)\right)$. This interpolating spline is a linear operator, and can be written in the form:$hatmu\left(x\right) = sum_\left\{i=1\right\}^n hatmu\left(x_i\right) f_i\left(x\right)$where $f_i\left(x\right)$ are a set of spline basis functions. As a result, the roughness penalty has the form:$int hatmu"\left(x\right)^2 dx = hat\left\{m\right\}^T A hat\left\{m\right\}.$where the elements of "A" are $int f_i\left(x\right) f_j\left(x\right)dx$. 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:$|Y - hat m|^2 + lambda hat\left\{m\right\}^T A hat m,$where $Y=\left(Y_1,ldots,Y_n\right)^T$.Minimizing over $hat m$ gives:$hat m = \left(I + lambda A\right)^\left\{-1\right\} Y.$

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]

* 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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