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fitSplicedBayesKDEGPD<-function(cell,prior,burnin=10,niter=100,proposal_scale=evmix::flognormgpd(cell,method="Nelder-Mead")$se,
start=evmix::flognormgpd(cell,method="Nelder-Mead")$optim$par){
# initialization of vectors where values will be assigned later
# burnin length of the burn-in-phase, niter+burnin iterations -> assign values to function
Sample_xi=numeric(niter+burnin)
Sample_tau=numeric(niter+burnin)
Sample_beta=numeric(niter+burnin)
# prior distributions (usually quite flat)
prior_xi<-function(x){
fun=truncnorm::dtruncnorm(x,0,Inf,prior$xi[1],prior$xi[2])
return(fun)
}
prior_tau<-function(x){ # preferably informative, otherwise too large distortion can occur
fun=truncnorm::dtruncnorm(x,0,Inf,prior$tau[1],prior$tau[2])
return(fun)
}
prior_beta<-function(x){
fun=truncnorm::dtruncnorm(x,0,Inf,prior$beta[1],prior$beta[2])
return(fun)
}
## set bandwith based on the density
bandwidth=density(log(cell),kernel="gaussian")$bw
## gaussian kernel density function with boundary correction
gkernellog<-function(x,vdata,h){
fun=(1/length(vdata))*sum(dlnorm(x,log(vdata),h))
return(fun)}
## distribution function
Fgkernellog<-function(x,vdata,h){
fun=(1/length(vdata))*sum(plnorm(x,log(vdata),h))
return(fun)}
## combound density: kernel density + GPD
dkerneldensitygpd=function(x,vtau,vbeta,vxi){
if(x<=vtau){
return(gkernellog(x,cell,bandwidth))
}else{
return((1-Fgkernellog(vtau,cell,bandwidth))*evmix::dgpd(x,vtau,vbeta,vxi))
}
}
## vectorized
vdkerneldensitygpd=Vectorize(dkerneldensitygpd)
## log-likelihood function for the distribution model
loglikelihood_kerneldensitygpd<-function(vdata,vtau,vbeta,vxi){
likeli=sum(log(vdkerneldensitygpd(vdata,vtau,vbeta,vxi)))
return(likeli)}
## logarithmic acceptance rate for the symmetric proposal densities
acceptance_ratesymm<-function(vtheta_prop,vtheta_old,vlogposterior){
a=min(log(1),(vlogposterior(vtheta_prop)-vlogposterior(vtheta_old)))
return(a)
}
#### MH-step 1: sampling xi
mhstep1<-function(vinitialval,cell){
xi=vinitialval[1] # current parameter values
tau=vinitialval[2]
beta=vinitialval[3]
xi_old=xi # current value for xi
## log-likelihood function depending on xi
loglikelihoodfunction_xi<-function(x){
fun=loglikelihood_kerneldensitygpd(cell,tau,beta,x)
return(fun)
}
## logarithmic function proportional to posterior density
logposterior_xi<-function(x){
fun=log(prior_xi(x))+loglikelihoodfunction_xi(x)
return(fun)
}
## parameter for the proposal density
m=xi_old
v=proposal_scale[5] # was assigned to the function
xi_prop=rnorm(1,m,v) # proposed value for the next iteration
acc=acceptance_ratesymm(xi_prop,xi_old,logposterior_xi)
if(is.nan(acc)){acc=-Inf # if NaN occurs, then refuse
}
## comparing the logarithmic acceptance rate with the logarithm of a uniform distributed rv
u=log(runif(1))
if (u<acc){
xi_old=xi_prop # accept if u<acc, else keep old values
}
return(xi_old) # returns the current value
}
#### MH-step 2: sampling tau
mhstep2<-function(vinitialval,cell){
xi=vinitialval[1] # current parameter values
tau=vinitialval[2]
beta=vinitialval[3]
tau_old=tau
loglikelihoodfunction_tau<-function(x){ # log-likelihood function depending on tau
fun=loglikelihood_kerneldensitygpd(cell,x,beta,xi)
return(fun)
}
logposterior_tau<-function(x){
fun=log(prior_tau(x))+loglikelihoodfunction_tau(x)
return(fun)
}
# truncated normal distribution as proposal density
m=tau_old
v=proposal_scale[3]
# logarithmic proposal density for acceptance rate (since now not symmetric anymore)
logproposal<-function(x,m){
fun=log(truncnorm::dtruncnorm(x,min(cell),max(cell),m,v))
return(fun)
}
# proposed value
tau_prop=truncnorm::rtruncnorm(1,min(cell),max(cell),m,v)
# logarithmic acceptance rate
acc=min(log(1),(logposterior_tau(tau_prop)+logproposal(tau_old,tau_prop)-logposterior_tau(tau_old)-logproposal(tau_prop,tau_old)))
if(is.nan(acc)){acc=-Inf
}
u=log(runif(1))
if (u<acc){
tau_old=tau_prop
}
return(tau_old)
}
#### MH-step 3: sampling beta
mhstep3<-function(vinitialval,cell){
xi=vinitialval[1] # current parameter values
tau=vinitialval[2]
beta=vinitialval[3]
beta_old=beta
loglikelihoodfunction_beta<-function(x){
fun=loglikelihood_kerneldensitygpd(cell,tau,x,xi)
return(fun)
}
logposterior_beta<-function(x){
fun=log(prior_beta(x))+loglikelihoodfunction_beta(x)
return(fun)
}
# log-normal proposal density
m=log(beta_old)
v=proposal_scale[4]
logproposal<-function(x,m){
fun=dlnorm(x,m,v,log=TRUE)
return(fun)
}
# proposed value
beta_prop=rlnorm(1,m,v)
# logarithmic acceptance rate
acc=min(log(1),(logposterior_beta(beta_prop)+logproposal(beta_old,log(beta_prop))-logposterior_beta(beta_old)-logproposal(beta_prop,log(beta_old))))
if(is.nan(acc)){acc=-Inf
}
u=log(runif(1))
if(u<acc){
beta_old=beta_prop
}
return(beta_old)
}
##########################################################################################################
# ACTUAL ALGORITHM
##########################################################################################################
#initial values determined by flognormgpd
initialval=c(start[5],start[3],start[4])
## actual MH-algorithm with number of iterations = niter + burnin
# current process of updating of the vector initialval
# cell is data given to the function
for(i in seq(1:(niter+burnin))){
initialval[1] = mhstep1(initialval, cell)
Sample_xi[i] = initialval[1]
initialval[2] = mhstep2(initialval, cell)
Sample_tau[i] = initialval[2]
initialval[3] = mhstep3(initialval,cell)
Sample_beta[i] = initialval[3]
}
# determining and saving sample values without burn-in-phase
xi_estimator=mean(Sample_xi[(burnin+1):(niter+burnin)])
tau_estimator=mean(Sample_tau[(burnin+1):(niter+burnin)])
beta_estimator=mean(Sample_beta[(burnin+1):(niter+burnin)])
# final output
buildSplicedSevdist("lnorm",c(start[1],start[2]),"gpd",c(tau_estimator,beta_estimator,xi_estimator),tau_estimator,0.5)
}
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