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Introduction to Smoothing Splines

Introduction to Smoothing Splines. Tongtong Wu Feb 29, 2004. Outline. Introduction Linear and polynomial regression, and interpolation Roughness penalties Interpolating and Smoothing splines Cubic splines Interpolating splines Smoothing splines Natural cubic splines

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Introduction to Smoothing Splines

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  1. Introduction to Smoothing Splines Tongtong Wu Feb 29, 2004

  2. Outline • Introduction • Linear and polynomial regression, and interpolation • Roughness penalties • Interpolating and Smoothing splines • Cubic splines • Interpolating splines • Smoothing splines • Natural cubic splines • Choosing the smoothing parameter • Available software

  3. Key Words • roughness penalty • penalized sum of squares • natural cubic splines

  4. Motivation

  5. Motivation

  6. Motivation

  7. Motivation Spline(y18)

  8. Introduction • Linear and polynomial regression : • Global influence • Increasing of polynomial degrees happens in discrete steps and can not be controlled continuously • Interpolation • Unsatisfactory as explanations of the given data

  9. Roughness penalty approach • A method for relaxing the model assumptions in classical linear regression along lines a little different from polynomial regression.

  10. Roughness penalty approach • Aims of curving fitting • A good fit to the data • To obtain a curve estimate that does not display too much rapid fluctuation • Basic idea: making a necessary compromise between the two rather different aims in curve estimation

  11. Roughness penalty approach • Quantifying the roughness of a curve • An intuitive way: (g: a twice-differentiable curve) • Motivation from a formalization of a mechanical device: if a thin piece of flexible wood, called a spline, is bent to the shape of the graph g, then the leading term in the strain energy is proportional to

  12. Roughness penalty approach • Penalized sum of squares • g: any twice-differentiable function on [a,b] • : smoothing parameter (‘rate of exchange’ between residual error and local variation) • Penalized least squares estimator

  13. Roughness penalty approach Curve for a large value of

  14. Roughness penalty approach Curve for a small value of

  15. Interpolating and Smoothing Splines • Cubic splines • Interpolating splines • Smoothing splines • Choosing the smoothing parameter

  16. Cubic Splines • Given a<t1<t2<…<tn<b, a function g is a cubic spline if • On each interval (a,t1), (t1,t2), …, (tn,b), g is a cubic polynomial • The polynomial pieces fit together at points ti (called knots) s.t. g itself and its first and second derivatives are continuous at each ti, and hence on the whole [a,b]

  17. Cubic Splines • How to specify a cubic spline • Natural cubic spline (NCS) if its second and third derivatives are zero at a and b, which implies d0=c0=dn=cn=0, so that g is linear on the two extreme intervals [a,t1] and [tn,b].

  18. Natural Cubic Splines Value-second derivative representation • We can specify a NCS by giving its value and second derivative at each knot ti. • Define which specify the curve g completely. • However, not all possible vectors represent a natural spline!

  19. Natural Cubic Splines Value-second derivative representation • Theorem 2.1 The vector and specify a natural spline g if and only if Then the roughness penalty will satisfy

  20. Natural Cubic Splines Value-second derivative representation

  21. Natural Cubic Splines Value-second derivative representation • R is strictly diagonal dominant, i.e.  R is positive definite, so we can define

  22. Interpolating Splines • To find a smooth curve that interpolate (ti,zi), i.e. g(ti)=zi for all i. • Theorem 2.2 Suppose and t1<…<tn. Given any values z1,…,zn, there is a unique natural cubic spline g with knots ti satisfying

  23. Interpolating Splines • The natural cubic spline interpolant is the unique minimizer of over S2[a,b] that interpolate the data. • Theorem 2.3 Suppose g is the interpolant natural cubic spline, then

  24. Smoothing Splines • Penalized sum of squares • g: any twice-differentiable function on [a,b] • : smoothing parameter (‘rate of exchange’ between residual error and local variation) • Penalized least squares estimator

  25. Smoothing Splines 1. The curve estimator is necessarily a natural cubic spline with knots at ti, for i=1,…,n. Proof: suppose g is the NCS

  26. Smoothing Splines 2. Existence and uniqueness Let then since be precisely the vector of . Express ,

  27. Smoothing Splines 2. Theorem 2.4 Let be the natural cubic spline with knots at ti for which . Then for any in S2[a,b]

  28. Smoothing Splines 3. The Reinsch algorithm The matrix has bandwidth 5 and is symmetric and strictly positive-definite, therefore it has a Cholesky decomposition

  29. Smoothing Splines 3. The Reinsch algorithm for spline smoothing Step 1: Evaluate the vector . Step 2: Find the non-zero diagonals of and hence the Cholesky decomposition factors L and D. Step 3: Solve for by forward and back substitution. Step 4: Find g by .

  30. Smoothing Splines 4. Some concluding remarks • Minimizing curve essentially does not depend on a and b, as long as all the data points lie between a and b. • If n=2, for any , setting to be the straight line through the two points (t1,Y1) and (t2,Y2) will reduce S(g) to zero. • If n=1, the minimizer is no longer unique, since any straight line through (t1,Y1) will yield a zero value S(g).

  31. Choosing the Smoothing Parameter • Two different philosophical approaches • Subjective choice • Automatic method – chosen by data • Cross-validation • Generalized cross-validation

  32. Choosing the Smoothing Parameter • Cross-validation • Generalized cross-validation

  33. Available Software smooth.spline in R • Description: Fits a cubic smoothing spline to the supplied data. • Usage: plot(speed, dist) cars.spl <- smooth.spline(speed, dist) cars.spl2 <- smooth.spline(speed, dist, df=10) lines(cars.spl, col = "blue") lines(cars.spl2, lty=2, col = "red")

  34. Available Software Example 1 library(modreg) y18 <- c(1:3,5,4,7:3,2*(2:5),rep(10,4)) xx <- seq(1,length(y18), len=201) (s2 <- smooth.spline(y18)) # GCV (s02 <- smooth.spline(y18, spar = 0.2)) plot(y18, main=deparse(s2$call), col.main=2) lines(s2, col = "blue"); lines(s02, col = "orange"); lines(predict(s2, xx), col = 2) lines(predict(s02, xx), col = 3); mtext(deparse(s02$call), col = 3)

  35. Available Software Example 1

  36. Available Software Example 2 data(cars) ## N=50, n (# of distinct x) =19 attach(cars) plot(speed, dist, main = "data(cars) & smoothing splines") cars.spl <- smooth.spline(speed, dist) cars.spl2 <- smooth.spline(speed, dist, df=10) lines(cars.spl, col = "blue") lines(cars.spl2, lty=2, col = "red") lines(smooth.spline(cars, spar=0.1)) ## spar: smoothing parameter (alpha) in (0,1] legend(5,120,c(paste("default [C.V.] => df =",round(cars.spl$df,1)), "s( * , df = 10)"), col = c("blue","red"), lty = 1:2, bg='bisque') detach()

  37. Available Software Example 2

  38. Extensions of Roughness penalty approach • Semiparametric modeling: a simple application to multiple regression • Generalized linear models (GLM) • To allow all the explanatory variables to be nonlinear • Additive model approach

  39. Reference • P.J. Green and B.W. Silverman (1994) Nonparametric Regression and Generalized Linear Models. London: Chapman & Hall

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