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# Linearizable C-QL
# Common variance for both segments
# Smoothness
llsearch.QL.CDS <- function(x, y, n, jlo, jhi)
{
fj <- matrix(0, n)
fxy <- matrix(0, jhi - jlo + 1)
jgrid <- expand.grid(jlo:jhi)
k.ll <- apply(jgrid, 1, p.estFUN.QL.CDS, x = x, y = y, n = n)
fxy <- matrix(k.ll, nrow = jhi-jlo+1)
rownames(fxy) <- jlo:jhi
z <- findmax(fxy)
jcrit <- z$imax + jlo - 1
list(jhat = jcrit, value = max(fxy))
}
# Function for deriving the ML estimates of the change-points problem.
p.estFUN.QL.CDS <- function(j, x, y, n){
a <- p.est.QL.CDS(x,y,n,j)
s2 <- a$sigma2
t2 <- a$tau2
return(p.ll.CDS(n, j, s2, t2))
}
p.est.QL.CDS <- function(x,y,n,j){
xa <- x[1:j]
ya <- y[1:j]
jp1 <- j+1
xb <- x[jp1:n]
yb <- y[jp1:n]
x1 <- x
x2 <- x^2 + (2*x[j]*(x - x[j]) - (x^2-x[j]^2)) * (x >= x[j])
fun <- lm(y ~ x1 + x2) # points(x, predict(fun), type = "l", col = "red")
a0 <- summary(fun)$coe[1]
a1 <- summary(fun)$coe[2]
a2 <- summary(fun)$coe[3]
b1 <- a1+2*summary(fun)$coe[3]*x[j]
b0 <- a0+(a1-b1)*x[j]+a2*x[j]^2
beta <-c(a0, a1, a2, b0, b1)
s2<- sum((ya-a0 - a1*xa - a2*xa^2)^2)/j
t2 <- sum((yb-b0-b1*xb)^2)/(n-j)
list(a0=beta[1],a1=beta[2],a2=beta[3],b0=beta[4],b1=beta[5],sigma2=s2,tau2 = t2,xj=x[j])
}
# Function to compute the log-likelihood of the change-point problem
p.ll.CDS <- function(n, j, s2, t2){
q1 <- n * log(sqrt(2 * pi))
q2 <- 0.5 * n * (1 + log(s2))
q3 <- 0.5 * (n - j) * (1 + log(t2))
- (q1 + q2 + q3)
}
findmax <-function(a)
{
maxa<-max(a)
imax<- which(a==max(a),arr.ind=TRUE)[1]
jmax<-which(a==max(a),arr.ind=TRUE)[2]
list(imax = imax, jmax = jmax, value = maxa)
}
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