Introduction to Smoothing Splines

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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

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
Motivation

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
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
Roughness penalty approach

Curve for a large value of

Roughness penalty approach

Curve for a small value of

Interpolating and Smoothing Splines
• Cubic splines
• Interpolating splines
• Smoothing splines
• Choosing the smoothing parameter
Cubic Splines
• Given a
• 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]
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].
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!
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

Natural Cubic Splines

Value-second derivative representation

Natural Cubic Splines

Value-second derivative representation

• R is strictly diagonal dominant, i.e.

 R is positive definite, so we can define

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<…

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

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
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

Smoothing Splines

2. Existence and uniqueness

Let then

since be precisely the vector of . Express ,

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]

Smoothing Splines

3. The Reinsch algorithm

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

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 .

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).
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
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 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)

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()

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