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functions_for_CI.R
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f_left <- function(t, scale, delta) {
return((1 - (1-exp(scale*t))/(1-exp(-scale*delta)))/2)
}
f <- function(t, t_centre=0, scale, delta) {
t <- t-t_centre
rv <-
ifelse(t < -delta,
0,
ifelse(t > delta,
1,
ifelse(t<=0,
f_left(t,scale,delta),
1-f_left(-t,scale,delta)))
)
return(rv)
}
g <-function(t, t_centre, scale, delta) {
f(t_centre-t, scale = scale, delta = delta)
}
sigma_tree <- function(scale, sigma, d) {
delta <- min(3*sigma,d)
var <- sigma^2
(2+(-2-scale*delta*(2+scale*delta))*exp(-scale*delta)) / ( scale^2 * (1 - exp(-scale*delta))) - var
}
likelihood <- function(t, t1 = 0, t2, scale1, delta1, scale2, delta2) {
f(t = t, t_centre = t1, scale = scale1, delta = delta1) * g(t = t, t_centre = t2, scale = scale2, delta = delta2)
}
find_scale_delta <- function(d, sigma) {
if ((delta <- 3*sigma) < d) {
#delta <- 3*sigma
scale <- 1.195554/sigma
} else {
delta <- d
scale <- uniroot(f = sigma_tree, lower = 1/(10*sigma), upper = 10*(1/sigma), sigma = sigma, d = d)$root
}
return(c(delta = delta, scale = scale))
}
t1 <- 0
d1 <- 15
sigma1 <- 0.2*d1
delta1 <- find_scale_delta(d = d1, sigma = sigma1)[[1]]
scale1 <- find_scale_delta(d = d1, sigma = sigma1)[[2]]
t2 <- 25
d2 <- 13
sigma2 <- 0.2*d2
delta2 <- find_scale_delta(d = d2, sigma = sigma2)[[1]]
scale2 <- find_scale_delta(d = d2, sigma = sigma2)[[2]]
plot(x = seq(-20,50,0.1),
y = likelihood(t = seq(-20,50,0.1), t1 = t1, t2 = t2,
scale1 = scale1, delta1 = delta1,
scale2 = scale2, delta2 = delta2), type = "l")
const <- cubature::adaptIntegrate(f = likelihood,
lowerLimit = t1-2*delta1,
upperLimit = t2+2*delta2,
t1 = t1,
t2 = t2,
scale1 = scale1,
delta1 = delta1,
scale2 = scale2,
delta2 = delta2)$integral
posterior_density_prop <- function(t, t1, t2, scale1, delta1, scale2, delta2, const = NULL) {
prop <- cubature::adaptIntegrate(f = likelihood,
lowerLimit = t1-delta1,
upperLimit = t,
t1 = t1,
t2 = t2,
scale1 = scale1,
delta1 = delta1,
scale2 = scale2,
delta2 = delta2)$integral/const
return(prop)
}
posterior_tree <- function(t, t1, t2, scale1, delta1, scale2, delta2, const = NULL, p = 0.025) {
prop <- posterior_density_prop(t = t,
t1 = t1,
t2 = t2,
scale1 = scale1,
delta1 = delta1,
scale2 = scale2,
delta2 = delta2,
const = const)
return(prop - p)
}
find_ci_limits <- function(t1, t2, scale1, delta1, scale2, delta2, const = const, alpha = 0.05) {
ci.lb <- uniroot(f = posterior_tree, lower = t1- 2 * delta1, upper = t2 + 2 * delta2, t1 = t1, t2 = t2, scale1 = scale1, delta1 = delta1, scale2 = scale2, delta2 = delta2, const = const, p = alpha/2)$root
ci.ub <- uniroot(f = posterior_tree, lower = t1- 2 * delta1, upper = t2 + 2 * delta2, t1 = t1, t2 = t2, scale1 = scale1, delta1 = delta1, scale2 = scale2, delta2 = delta2, const = const, p = 1 - alpha/2)$root
return(list(CI.LB = ci.lb, CI.UB = ci.ub))
}
find_ci_limits(t1, t2, scale1, delta1, scale2, delta2, const = const, alpha = 0.05)
# Testing stuff - ignore
posterior_density_prop(t = 47, t1 = 0, t2 = 30,
scale1 = scale1, delta1 = delta1,
scale2 = scale2, delta2 = delta2, const = const)
posterior_tree(t = 47, t1 = 0, t2 = 30,
scale1 = scale1, delta1 = delta1,
scale2 = scale2, delta2 = delta2,
const = const, p = 0.025)
t1 <- 0
d1 <- 15
sigma1 <- 2
delta1 <- min(3*sigma1,d1)
scale1 <- uniroot(f = sigma_tree, lower = 1/(10*sigma1), upper = 5*(1/sigma1), sigma = sigma1, d = d1)$root
print(scale1)
test <- cubature::adaptIntegrate(f = likelihood,
lowerLimit = t1-delta1,
upperLimit = t1+delta1,
t1 = t1,
t2 = t2,
scale1 = scale1,
delta1 = delta1,
scale2 = scale2,
delta2 = delta2)$integral
print(test)
sigma1 <- 5
delta1 <- min(3*sigma1,d1)
scale1 <- uniroot(f = sigma_tree, lower = 1/(10*sigma1), upper = 5*(1/sigma1), sigma = sigma1, d = d1)$root
print(scale1)
test <- cubature::adaptIntegrate(f = likelihood,
lowerLimit = t1-delta1,
upperLimit = t1+delta1,
t1 = t1,
t2 = t2,
scale1 = scale1,
delta1 = delta1,
scale2 = scale2,
delta2 = delta2)$integral
print(test)