# NOT RUN {
library(gamair); library(mgcv)
## code from Chapter 5 solutions...
## Q.3
pspline.XB <- function(x,q=10,m=2,p.m=2)
# Get model matrix and sqrt Penalty matrix for P-spline
{ # first make knot sequence, k
k <- seq(min(x),max(x),length=q-m)
dk <- k[2]-k[1]
k <- c(k[1]-dk*((m+1):1),k,k[q-m]+dk*(1:(m+1)))
# now get model matrix and root penalty
X <- splineDesign(k,x,ord=m+2)
B <- diff(diag(q),difference=p.m)
list(X=X,B=B)
} ## pspline.XB
## a) and b)
fit.wPs <- function(y,X,B,lambda=0,w=rep(1,length(y)))
# fit to y by weighted penalized least squares, X is
# model matrix, B is sqrt penalty, lambda is smoothing p.
{ w <- as.numeric(w^.5)
n <- nrow(X)
X<-rbind(w*X,sqrt(lambda)*B)
y<-c(w*y,rep(0,nrow(B)))
b <- lm(y~X-1) # actually estimate model
trA <- sum(influence(b)$hat[1:n])
rss <- sum((y-fitted(b))[1:n]^2) ## not really needed here
list(trA=trA,rss=rss,b=coef(b))
}
fitPoiPs <- function(y,X,B,lambda=0)
# Fit Poisson model with log-link by P-IRLS
{ mu <- y;mu[mu==0] <- .1
eta <- log(mu)
converged <- FALSE
dev <- ll.sat <- sum(dpois(y,y,log=TRUE))
while (!converged) {
z <- (y-mu)/mu + eta
w <- mu
fPs <- fit.wPs(z,X,B,lambda,w)
eta <- X%*%fPs$b
mu=exp(eta)
old.dev <- dev
dev <- 2*(ll.sat-sum(dpois(y,mu,log=TRUE)))
if (abs(dev-old.dev)<1e-6*dev) converged <- TRUE
}
list(dev=dev,rss=fPs$rss,trA=fPs$trA,b=fPs$b,fv=mu)
}
## c)
## simulate data as in question...
set.seed(1)
f <- function(x) .04*x^11*(10*(1-x))^6+2*(10*x)^3*(1-x)^10
n <- 100;x <- sort(runif(n))
y <- rpois(rep(1,n),exp(f(x)))
## fitting...
library(splines)
ps <- pspline.XB(x,q=10,m=2,p.m=2)
lambda <- 1e-4;reps <- 60
sp <- trA <- gcv <- rep(0,reps)
for (i in 1:reps) { # loop through trial s.p.s
fps <- fitPoiPs(y,ps$X,ps$B,lambda=lambda)
trA[i] <- fps$trA;sp[i] <- lambda
gcv[i] <- n*fps$dev/(n-trA[i])^2
lambda <- lambda*1.3
}
plot(trA,gcv,type="l")
fps1 <- fitPoiPs(y,ps$X,ps$B,lambda=sp[gcv==min(gcv)])
plot(x,y);lines(x,fps1$fv)
## Q.6 Fellner-Schall for GCV and AIC...
## b)
library(mgcv);library(MASS)
sm <- smoothCon(s(times,k=20),data=mcycle)[[1]]
X <- sm$X; S <- sm$S[[1]]; y <- mcycle$accel
lambda <- 1; n <- length(y)
XX <- crossprod(X);
with(mcycle,plot(times,accel))
for (i in 1:20) {
R <- chol(XX+lambda*S)
b <- backsolve(R,forwardsolve(t(R),t(X) %*% y))
f <- X %*% b
lines(mcycle$times,f,col="grey")
HiS <- backsolve(R,forwardsolve(t(R),S))
HiH <- backsolve(R,forwardsolve(t(R),XX))
tau <- sum(diag(HiH))
if (i>1) { ## convergence test
if (abs(tau-tau0)<1e-5*tau) break
}
tau0 <- tau
dt.dl <- -sum(t(HiH)*HiS)
db.dl <- -HiS %*% b
dD.db <- 2*t(X) %*% (f - y)
lambda <- -sum(2*(y-f)^2)/(n-tau)*dt.dl/sum(db.dl*dD.db) * lambda
}
lines(mcycle$times,f)
## c)
y <- c(12,14,33,50,67,74,123,141,165,204,253,246,240)
t <- 1:13
sm <- smoothCon(s(t),data=data.frame(t=t,y=y))[[1]]
X <- sm$X; S <- sm$S[[1]]; lambda <- .001; n <- length(y)
plot(t,y)
mu <- y; eta <- log(mu)
for (i in 1:50) {
w <- mu; z <- (y-mu)/mu + eta
XWX <- crossprod(sqrt(w)*X)
R <- chol(XWX+lambda*S)
b <- backsolve(R,forwardsolve(t(R),t(X) %*% (w*z)))
eta <- drop(X %*% b);mu <- exp(eta)
lines(t,mu,col="grey")
HiS <- backsolve(R,forwardsolve(t(R),S))
HiH <- backsolve(R,forwardsolve(t(R),XWX))
tau <- sum(diag(HiH))
if (i>1) { ## convergence test
if (abs(tau-tau0)<1e-5*tau) break
}
tau0 <- tau
dt.dl <- -sum(t(HiH)*HiS)
db.dl <- -HiS %*% b
dl.db <- t(X) %*% (y-mu) ## especially simple for this case
lambda <- dt.dl/sum(db.dl*dl.db) * lambda
}
i;tau;lines(t,mu)
## Q.8 log det stabilty (or lack of)
set.seed(1);lam <- 1
A1 <- crossprod(diff(diag(3),diff=1))
A2 <- crossprod(matrix(runif(9),3,3))
A <- matrix(0,5,5);A[1:3,1:3] <- A1
A[3:5,3:5] <- A[3:5,3:5] + lam * A2
ldetA.qr <- ldetA.ev <- ldetA.svd <- ldetA <-
rho <- seq(-40,-25,length=100)
for (i in 1:length(rho)) {
lam <- exp(rho[i])
A <- matrix(0,5,5);A[1:3,1:3] <- A1
A[3:5,3:5] <- A[3:5,3:5] + lam * A2
ea1 <- eigen(A1)
Q <- diag(5);Q[1:3,1:3] <- ea1$vectors
At <- matrix(0,5,5)
At[3:5,3:5] <- At[3:5,3:5] + lam * A2
At <- t(Q)%*%At%*%Q
diag(At)[1:2] <- diag(At)[1:2]+ea1$values[1:2]
ldetA[i] <- sum(log(abs(diag(qr.R(qr(At))))))
ldetA.qr[i] <- sum(log(abs(diag(qr.R(qr(A))))))
ldetA.ev[i] <- sum(log(abs(eigen(A)$values)))
ldetA.svd[i] <- sum(log(abs(svd(A)$d)))
}
plot(rho,ldetA,type="l") ## nice and stable
## not...
lines(rho,ldetA.qr,lty=2)
lines(rho,ldetA.ev,lty=3)
lines(rho,ldetA.svd,lty=4)
# }
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