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# Introduction to Smoothing Splines - PowerPoint PPT Presentation

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

Tongtong Wu

Feb 29, 2004

• 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

• roughness penalty

• penalized sum of squares

• natural cubic splines

Spline(y18)

• 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

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

• 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

• 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

• 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

Curve for a large value of

Curve for a small value of

• Cubic splines

• Interpolating splines

• Smoothing splines

• Choosing the smoothing parameter

• 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]

• 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].

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!

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

Value-second derivative representation

Value-second derivative representation

• R is strictly diagonal dominant, i.e.

 R is positive definite, so we can define

• 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

• 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

• 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

1. The curve estimator is necessarily a natural cubic spline with knots at ti, for i=1,…,n.

Proof: suppose g is the NCS

2. Existence and uniqueness

Let then

since be precisely the vector of . Express ,

2. Theorem 2.4

Let be the natural cubic spline with knots at ti for which . Then for any in S2[a,b]

3. The Reinsch algorithm

The matrix has bandwidth 5 and is symmetric and strictly positive-definite, therefore it has a Cholesky decomposition

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 .

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).

• Two different philosophical approaches

• Subjective choice

• Automatic method – chosen by data

• Cross-validation

• Generalized cross-validation

• Cross-validation

• Generalized cross-validation

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")

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)

Example 1

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()

Example 2

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