Smoothing spline

Smoothing spline

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


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.

# 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.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Spline — can refer to:* Flat spline, a device to draw curves * Rotating spline, a mating mechanism on a driveshaft. * Spline (mathematics), a mathematical function used for interpolation or smoothing. * Spline cord, a type of thin rubber cord used to… …   Wikipedia

  • Smoothing — In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine scale structures/rapid phenomena. Many different… …   Wikipedia

  • Spline (mathematics) — A quadratic spline composed of six polynomial segments. Between point 0 and point 1 a straight line. Between point 1 and point 2 a parabola with second derivative = 4. Between point 2 and point 3 a parabola with second derivative = 2. Between… …   Wikipedia

  • Spline interpolation — See also: Spline (mathematics) In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Spline interpolation is preferred… …   Wikipedia

  • Spline — A&V A curve shape produced on a computer or video device by connecting dots or points at various intervals along the curve. In digital picture manipulators, each key frame becomes a point on a curve and the user can control how straight or curved …   Audio and video glossary

  • Thin plate spline — This is a brief derivation for the closed form solutions for smoothing Thin Plate Spline . Details about these splines can be found in (Wahba, 1990).Thin plate splines (TPS) were introduced to geometric design by Duchon (Duchon, 1976). The name… …   Wikipedia

  • maximum forward rate smoothing — An alternative yield curve smoothing technique. The most accurate yield curve smoothing method for forward rates. The yield curve with the smoothest possible forward rate function, consistent with observable data, is closely related to but… …   Financial and business terms

  • Nonuniform rational B-spline — Non uniform rational B spline (NURBS) is a mathematical model commonly used in computer graphics for generating and representing curves and surfaces. History Development of NURBS (Non Uniform Rational Basis Spline) began in the 1950s by engineers …   Wikipedia

  • cubic spline — A mathematical technique used for yield curve smoothing. A cubic spline fits a different third degree polynomial to each interval between data points (0 to 1 years, 1 to 2 years, 2 to 3 years, etc.) Either yields or prices can be smoothed using… …   Financial and business terms

  • Non-uniform rational B-spline — Three dimensional NURBS surfaces can have complex, organic shapes. Control points influence the directions the surface takes. The outermost square below delineates the X/Y extents of the surface …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”