Introduction to Smoothing Splines

Tongtong WuFeb 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 Choosing the smoothing parameter Available software

Key Words roughness penalty penalized sum of squares natural cubic splines

Motivation51015246810

Index(y18)

Motivation51015246810

Indexy18

Motivation51015246810

Indexy18

Motivation51015246810

Index(y18)

Spline(y18)

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

Roughness penalty approach A method for relaxing the model

assumptions in classical linear regression along lines a little different from polynomial regression.

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

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

{ }∫b

adttg 2)(''

∫ 2''g

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

{ } { }∫∑ +−==

b

a

n

iii dttgtgYgS 2

1

2 )('')()( α

α

)(minargˆ gSg =

Roughness penalty approachCurve for a large value of α51015

246810Indexy18

Roughness penalty approachCurve for a small value of α51015

246810Indexy18

Interpolating and Smoothing Splines Cubic splines

Interpolating splines

Smoothing splines

Choosing the smoothing parameter

Cubic Splines Given a<t1<t2<…<tn<b, a function g is a

cubic spline if

1. On each interval (a,t1), (t1,t2), …, (tn,b), g is a

cubic polynomial

2. 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]

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].

123 for )()()()( +≤≤+−+−+−= iiiiiiiii tttattbttcttdtg

Natural Cubic SplinesValue-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!

)( where,)',,( 1 iin tggggg == L

)('' where,)',,( 12 iin tg== − γγγγ L

Natural Cubic SplinesValue-second derivative representation Theorem 2.1

The vector and specify a natural spline g if and only if

Then the roughness penalty will satisfy

g γ

γRgQ ='

KggRdttgb

a'')('' 2 ==∫ γγ

Natural Cubic SplinesValue-second derivative representation

)2(

11

13

13

12

12

12

12

11

11

00

00

0

0

00

−×−−

−

−−−

−−−

−

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

=

nnnh

hhhh

hhhh

Q

L

MOMM

L

L

L

L

)2()2(12

322

231

)(3

100

0)(3

1

6

1

06

1)(

3

1

−×−−− ⎥

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

+

+

=

nnnn hh

hhh

hhh

R

L

MOMM

L

L

nitth iii ,,1for 1 K=−= +

Natural Cubic SplinesValue-second derivative representation R is strictly diagonal dominant, i.e.

R is positive definite, so we can define

'1QQRK −=

irrij ijii ∀>∑ ≠

,||||

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

2≥n

niztg ii ,,1for )( K==

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

∫ 2''g

niztgbaSg ii ,,1for )(~ with ],[~2 K==∈

∫∫ ≥ 22 ''''~ gg

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

{ } { }∫∑ +−==

b

a

n

iii dttgtgYgS 2

1

2 )('')()( α

α

)(minargˆ gSg =

Smoothing Splines1. The curve estimator is necessarily a

natural cubic spline with knots at ti, for i=1,…,n.

Proof: suppose g is the NCS

)~()( gSgS ≤⇒

g

{ } { }∑∑==

−=−n

iii

n

iii tgYtgY

1

2

1

2 )(~)(

{ } { }∫∫ ≤b

a

b

adttgdttg 22 )(''~)(''

Smoothing Splines2. Existence and uniqueness

Let then

since be precisely the vector of . Express ,Kggg ''' 2 =∫

{ } )()'()(1

2 gYgYtgYn

iii −−=−∑

=

)',,( 1 nYYY K=

g )( itg

€

S(g) = (Y − g)'(Y − g) +αg'Kg

= g'(I +αK)g− 2Y 'g+Y 'Y

YKIg 1)( settingby achieved is Minimum −+= α

Smoothing Splines2. Theorem 2.4

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

)()ˆ( gSgS ≤

YKIg 1)( −+= αg

Smoothing Splines3. The Reinsch algorithm

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

gQQRIgKIY )()( 1−+=+= αα

)'()1 γγαα RgQQYgQQRYg =−=−=⇒ − Q

γα )'(' QQRYQ +=⇒

)'( QQR α+

'' LDLQQR =+α

Smoothing Splines3. 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 .

γ

YQ'

QQR 'α+

YQLDL '' =γ

γαQYg −=

Smoothing Splines4. 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).

g

α g

Choosing the Smoothing Parameter Two different philosophical

approaches Subjective choice Automatic method – chosen by data

Cross-validation Generalized cross-validation

Choosing the Smoothing Parameter Cross-validation

Generalized cross-validation

{ }

αα

ααα

ith smoother w spline theis ˆ if )(1

)(ˆ

);(ˆ)(min

1

2

1

1

2)(1

gA

tgYn

tgYnCV

n

i ii

ii

n

ii

ii

∑

∑

=

−

=

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−=

−=

( )

{ } 221

1

2

1

df)t (equivalen

squares of sum residual

)(1

)(ˆ

)(min×

=−

−=

−

=−∑ n

trAn

tgYnGCV

n

iii

αα

α

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")

Available SoftwareExample 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)

Available Software

Example 1

Available SoftwareExample 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()

Available Software

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

Additive model approach

εβ ++= ')( xtgY

ε+=∑=

d

jjj tgY

1

)(

ε+= )(tgY

Reference P.J. Green and B.W. Silverman (1994)

Nonparametric Regression and Generalized Linear Models. London: Chapman & Hall