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example-missingdata
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######################################################################
### bayesian nonparametrics for extreme value time series prediction
### using extreme value distribution type-I for maxima
######################################################################
######################################################################
### basic gev functions
######################################################################
source("/home/maha/Dropbox/projeto/R/main-files/basicfunctions-EV-II")
source("/home/maha/Dropbox/projeto/R/main-files/aux-functions")
source("/home/maha/Dropbox/projeto/R/main-files/hmc-rmhmc")
##################################
### loading packages
##################################
require(compiler); enableJIT(3)
require(mvtnorm)
##################################################################
### generating simulated data
##################################################################
n = 100
h = function(t) 2 + t + 1.5*sin(0.5*pi*t) # h = function(t) sin(4*pi*t)*exp(-0.5*t)
t = sort(c(runif(n/2, -7, 3), runif(n/2, 7, 13))) # t = sort(runif(n, 0 , 9))
y = h(t) + rgumax(n, 0, 0.5)
plot(t, y)
#####################
### Euler Constant
#####################
J <- -digamma(1)
###################################
### Matrices that stays unchanged
###################################
T = length(y)
D = T+6
A = list(); A[[1]] = outer(t, t, '*'); A[[2]] = outer(t, t, '+'); A[[3]] = abs(outer(t, t, '-'));
c = 0.0087; I = c*diag(T)
dT = diag(T)
#################################################################
### Covariance function and its derivatives
### using parameterization on the real line
#################################################################
K <- function(theta2, A) {
phi1 = exp(theta2[1])
phi2 = theta2[2]
phi3 = exp(theta2[3])
phi4 = exp(theta2[4])
phi5 = exp(theta2[5])
K = phi1*(A[[1]] - A[[2]]*phi2 + phi2^2) + phi3*exp(-sin(phi4*A[[3]])^2/phi5) + I
return(K)
}
##################################################################
### nabK derivatire of covariance function w.r.t to each parameter
##################################################################
nabK. <- function(theta2, A) {
phi1 = exp(theta2[1])
phi2 = theta2[2]
phi3 = exp(theta2[3])
phi4 = exp(theta2[4])
phi5 = exp(theta2[5])
nab = list()
W1 = sin(phi4*A[[3]])
W2 = W1^2
W3 = W2/phi5
nab[[1]] = phi1*(A[[1]] - A[[2]]*phi2 + phi2^2)
nab[[2]] = - phi1*(A[[2]] - 2*phi2)
nab[[3]] = phi3*exp(-W3)
nab[[4]] = - (2*phi4/phi5)*W1*cos(phi4*A[[3]])*A[[3]]*nab[[3]]
nab[[5]] = W3*nab[[3]]
return(nab)
}
#########################################################
### inverse log-gamma density for
### theta = log sig or phi1 or phi2 using jacobian method
#########################################################
digam <- function(x, a, b) {
b^a / gamma(a) * exp(-x*a - b/exp(x))
}
a = rep(0.1, 6)
b = rep(0.1, 6)
curve(digam(x, a[1], b[1]), -10,20)
################################
### unnormalized log-posterior
################################
ll <- function(theta, x) {
fx = theta[1:T]
sig = exp(theta[T+1])
theta2 = theta[(T+2):D]
ll = sum(log(f(x, fx, sig))) + dmvnorm(fx, rep(0, T), K(theta2, A), log = TRUE) + sum(log(digam(c(theta[T+1], theta2), a, b)))
ifelse(ll != "-Inf", ll, -10000000)
}
ll(c(y, rep(0.1, 6)), y)
##########################################
### gradient of unnormalized log-posterior
##########################################
nll <- function(theta, x) {
fx = theta[1:T]
sig = exp(theta[T+1])
theta2 = theta[(T+2):D]
z = (x - fx)/sig
expz = exp(-z)
invK = solve(K(theta2, A))
nabK = nabK.(theta2, A)
R. = fx %*% t(fx) %*% invK - dT
R = invK %*% (R.)
nlp = rep(0, D)
nlp[1:T] = (rep(1, T) - expz)/sig - as.vector(invK %*% fx)
nlp[T+1] = -T + sum(z) - sum(z*expz) - a[1] + b[1]/exp(theta[T+1])
nlp[T+2] = .5*tr(R, nabK[[1]]) - a[2] + b[2]/exp(theta2[1])
nlp[T+3] = .5*tr(R, nabK[[2]]) - a[3] + b[3]/exp(theta2[2])
nlp[T+4] = .5*tr(R, nabK[[3]]) - a[4] + b[4]/exp(theta2[3])
nlp[T+5] = .5*tr(R, nabK[[4]]) - a[5] + b[5]/exp(theta2[4])
nlp[T+6] = .5*tr(R, nabK[[5]]) - a[6] + b[6]/exp(theta2[5])
return(nlp)
}
nll(c(y, rep(0, 6)), y)
#########################################################
### metric tensor G(theta)
#########################################################
G <- function(theta) {
G = matrix(0, D, D)
fx = theta[1:T]
sig = exp(theta[T+1])
theta2 = theta[(T+2):D]
nabK = nabK.(theta2, A)
K = K(theta2, A)
invK = solve(K)
iKdK1 = invK %*% nabK[[1]]
iKdK2 = invK %*% nabK[[2]]
iKdK3 = invK %*% nabK[[3]]
iKdK4 = invK %*% nabK[[4]]
iKdK5 = invK %*% nabK[[5]]
G[1:T, 1:T] = diag(T)/sig^2 + invK
G[1:T, T+1] = G[T+1, 1:T] = (J - 1)/sig
G[T+1, T+1] = T*((pi^2)/6 + (1-J)^2) + b[1]/exp(theta[T+1])
G[T+2, T+2] = .5*tr(iKdK1, iKdK1) + b[2]/exp(theta[T+2])
G[T+2, T+3] = .5*tr(iKdK1, iKdK2) -> G[T+3, T+2]
G[T+2, T+4] = .5*tr(iKdK1, iKdK3) -> G[T+4, T+2]
G[T+2, T+5] = .5*tr(iKdK1, iKdK4) -> G[T+5, T+2]
G[T+2, T+6] = .5*tr(iKdK1, iKdK5) -> G[T+6, T+2]
G[T+3, T+3] = .5*tr(iKdK2, iKdK2) + b[3]/exp(theta[T+3])
G[T+3, T+4] = .5*tr(iKdK2, iKdK3) -> G[T+4, T+3]
G[T+3, T+5] = .5*tr(iKdK2, iKdK4) -> G[T+5, T+3]
G[T+3, T+6] = .5*tr(iKdK2, iKdK5) -> G[T+6, T+3]
G[T+4, T+4] = .5*tr(iKdK3, iKdK3) + b[4]/exp(theta[T+4])
G[T+4, T+5] = .5*tr(iKdK3, iKdK4) -> G[T+5, T+4]
G[T+4, T+6] = .5*tr(iKdK3, iKdK5) -> G[T+6, T+4]
G[T+5, T+5] = .5*tr(iKdK4, iKdK4) + b[5]/exp(theta[T+5])
G[T+5, T+6] = .5*tr(iKdK4, iKdK5) -> G[T+6, T+5]
G[T+6, T+6] = .5*tr(iKdK5, iKdK5) + b[6]/exp(theta[T+6])
return(G)
}
G. = G(c(rep(0, T), -0.8431403, -0.2840989, -2.097073, 1.4178477, -0.2448429, 1.4828678))
invG = solve(G.)
#####################################
### Hamiltonian Monte Carlo
######################################
B = hmc(c(y, -0.5, -0.13, -2, 2, -.2, 1.84), 25000, 2000, 20, .085, 12, G., y)
B = hmc(c(y, -1.4, -0.13, -4, 2, -.8, 1.84), 1100, 100, 2, .019, 14, G., y)
par(mfrow=c(3, 2)); for (i in 1:6) plot(B$theta[, T+i], type = 'l')
par(mfrow=c(3, 2)); r = sample(1:T, 6, replace = FALSE); for (i in 1:6) plot(B$theta[,r[i]], type = 'l')
par(mfrow=c(3, 2)); for (i in 1:6) hist(B$theta[,T+i], breaks = 30)
##########################################################################################################
### Predictionsss
##########################################################################################################
###################################################
### horizon desired and/or points to be predicted
###################################################
SS = 10000
t. = seq(t[1] ,t[T] + 10, length.out = 300)
T. = length(t.)
I. = diag(c, T.)
A. = list(), A.[[1]] = outer(t., t, '*'); A.[[2]] = outer(t., t, '+'); A.[[3]] = abs(outer(t., t, '-'));
A.. = list(), A..[[1]] = outer(t., t., '*'); A..[[2]] = outer(t., t., '+'); A..[[3]] = abs(outer(t., t., '-'));
y. = matrix(, SS, T.)
######################################################
### Drawning values for y*|D
######################################################
ft = as.matrix(M[1:SS,1:T])
sig = exp(M[1:SS,(T+1)])
theta2 = M[1:SS,(T+2):D]
ft. = matrix(, SS, T.)
for (i in 1:SS) {
print(i)
K. = K(theta2[i, ], A.)
a = K. %*% solve(K(theta2[i, ], A) + I)
m = a %*% ft[i, ]
Q = K(theta2[i, ], A..) + I. - a %*% t(K.)
test = try(mvrnorm(1, m, Q))
if (inherits(test, "try-error") == TRUE) { ft.[i, ] = ft.[i-1, ] }
else fx.[i, ] = test
y.[i, ] = rgumax(1, ft.[i, ], sig[i])
}
###################################################
### mean variance and
### credible region 95%
### of predictive distribution y*|D
###################################################
meanft = apply(ft, 2, mean)
modeft = apply(ft, 2, mode)
meanft. = apply(ft., 2, mean)
modeft. = apply(ft., 2, mode)
meansy. = apply(y., 2, mean)
modey. = apply(y., 2, mode)
varsy. = apply(y., 2, var)
quanft = matrix(nrow = 100, ncol = 2)
quanft. = matrix(nrow = T., ncol = 2)
quany. = matrix(nrow = T., ncol = 2)
for(i in 1:T) {
quanft[i, ] = quantile(ft[, i], prob = c(0.05, 0.95))
}
for(i in 1:T.) {
quanft.[i, ] = quantile(ft.[, i], prob = c(0.05, 0.95))
quany.[i, ] = quantile(y.[, i], prob = c(0.05, 0.95))
}
lsc = quany.[, 1]
lic = quany.[, 2]
######################################################################################
### display observed series
### predicted series
### credible region
######################################################################################
png('predev.png', width= 2500, height = 1200)
par(mfrow = c(1, 2))
plot(x = NULL, xlim = c(10, 23), ylim = c(10, 25), xlab = "x", ylab = "y", bty = 'n')
plot(x = NULL, xlim = c(10, 23), ylim = c(10, 25), xlab = "x", ylab = "", bty = 'n', yaxt='n')
plot(x = NULL, xlim = c(3, 7), ylim = c(3, 10), xlab = "x", ylab = "y", bty = 'n')
plot(x = NULL, xlim = c(3, 7), ylim = c(3, 10), xlab = "x", ylab = "y", bty = 'n', yaxt='n')
polygon(x = c(t., t.[length(t.):1]), y = c(quany.[, 2], quany.[, 1][length(quany.[, 1]):1]), col = 'lightgrey', border = NA)
#polygon(x = c(t., t.[length(t.):1]), y = c(quanft.[, 2], quanft.[, 1][length(quanft.[, 1]):1]), col = 'darkgrey', border = NA)
#lines(t., meanft. , col = "black", lty = 1)
#lines(t., meanft. , col = "black", lty = 1)
lines(t., meansy. , col = "black", lty = 6, lwd = 3)
lines(t., modey., col = "black", lty = 6, lwd = 3)
#points(t, y , col = "black", type = 'p', pch = 4)
curve(h, -6.5, 23, add = TRUE, lty = 1, col = 1, lwd = 4, n = 200)
# legend(x = 40, y = 0.75, c('serie temporal predita', 'serie observada', 'região de credibilidade 95%'),
# col = c('black', 'black', 'lightgrey'), lty = c(1, NA, 1), lwd = c(1, NA, 6), pch = c(NA,4,NA), cex = 1.2)
dev.off()
####################################################################################################################################
### plots
####################################################################################################################################
data = read.table(file = '/home/maha/Dropbox/projeto/R/sim-gpGEV/gev-sim2-data.txt', header = FALSE)
t = data[, 2]; y = data[, 1]; T = length(t); D = T + 6
B = read.table(file = '/home/maha/Dropbox/projeto/R/sim-gpGEV/gev-sim2-post.txt', header = FALSE)
ft = B[ ,1:100]
ft. = read.table(file = '/home/maha/Dropbox/projeto/R/sim-gpGEV/gev-sim2-predfx.txt', header = FALSE)
y. = read.table(file = '/home/maha/Dropbox/projeto/R/sim-gpGEV/gev-sim2-pred.txt', header = FALSE)
pdf('exEVI.pdf', width = 10, height = 8)
plot(t, y, main = 'valores simulados')
dev.off()
plot(ft[, 1], ft[, 2])
plot(ft[, 1], B[, T+1])
plot(ft.[, 1], ft.[, 2])
plot(y.[, 1], y.[, 2])
scatterplotMatrix(B[, (T+1):D], diagonal = 'histogram', nclass = 30, pch = 4, var.labels = rep('', 6), col = 'gray', reg.lin = F, smoother = F, upper.panel = NULL)
sig = exp(B[, T+1]); phi1 = exp(B[, T+2]); phi2 = B[, T+3]; phi3 = exp(B[, T+4]); phi4 = exp(B[, T+5]); phi5 = exp(B[, T+6])
hist(sort(phi1)[1:24000], breaks = 700, prob = T, xlim = c(0, 40))
hist(sort(phi3)[1:24000], breaks = 5000, prob = T, xlim = c(0, 100))
hist(sort(phi5)[1:24000], breaks = 2000, prob = T, xlim = c(0, 500))
plot(sig, phi1, ylim = c(0, 200))
png('histEVGP.png', width= 2500, height = 1200)
par(mfrow=c(6, 6), mar = c(.9, .9, .9, .9))
hist(sig, prob = T, main = '', axes = F, xlab = '', breaks = 40)
plot(sig, phi1, ylim = c(0, 2000), axes = F)
plot(sig, phi2, axes = F)
plot(sig, phi3, ylim = c(0, 5500), axes = F)
plot(sig, phi4, axes = F)
plot(sig, phi5, ylim = c(0, 3000), axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(sort(phi1)[1:24000], breaks = 700, prob = T, xlim = c(0, 30), main ='', axes = F)
plot(phi1, phi2, xlim = c(0, 700), axes = F)
plot(phi1, phi3, xlim = c(0, 1000), ylim = c(0, 1000), axes = F)
plot(phi1, phi4, xlim = c(0, 1000), axes = F)
plot(phi1, phi5, xlim = c(0, 1000), ylim = c(0, 1000), axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(phi2, breaks = 40, prob = T, main = '', axes = F)
plot(phi2, phi3, ylim = c(0, 5000), axes = F)
plot(phi2, phi4, axes = F)
plot(phi2, phi5, ylim = c(0, 5000), axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(sort(phi3)[1:24000], breaks = 5000, prob = T, xlim = c(0, 40), main = '', axes = F)
plot(phi3, phi4, xlim = c(0, 5000), axes = F)
plot(phi3, phi5, xlim = c(0, 5000), ylim = c(0,5000), axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(phi4, prob = T, main = '', axes = F, xlab = '', breaks = 40)
plot(phi4, phi5, ylim = c(0, 5000), axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(sort(phi5)[1:24000], breaks = 2000, prob = T, xlim = c(0, 30), main = '', axes = F)
dev.off()
#########################################################################################################################
### parameterization on the real line
#########################################################################################################################
png('histEVGP2.png', width= 2500, height = 1200)
par(mfrow=c(6, 6), mar = c(.9, .9, .9, .9))
hist(B[ ,T+1], prob = T, main = '', axes = F, xlab = '', breaks = 40)
plot(B[ ,T+1], B[ ,T+2], xlim = c(-5, 5), axes = F)
plot(B[ ,T+1], B[ ,T+3], xlim = c(-5, 5), axes = F)
plot(B[ ,T+1], B[ ,T+4], xlim = c(-5, 5), axes = F)
plot(B[ ,T+1], B[ ,T+5], ylim = c(-.35, -.15), axes = F)
plot(B[ ,T+1], B[ ,T+6], xlim = c(-5, 5), axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(B[ ,T+2], prob = T, main = '', axes = F, xlab = '', breaks = 40)
plot(B[ ,T+2], B[ ,T+3], xlim = c(-2.5, 11), ylim = c(-4, 35), axes = F)
plot(B[ ,T+2], B[ ,T+4], axes = F)
plot(B[ ,T+2], B[ ,T+5], ylim = c(-.30, -.20), axes = F)
plot(B[ ,T+2], B[ ,T+6], axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(B[ ,T+3], prob = T, main = '', axes = F, xlab = '', breaks = 50)
plot(B[ ,T+3], B[ ,T+4], xlim = c(-3.5, 30), ylim = c(-4, 25), axes = F)
plot(B[ ,T+3], B[ ,T+5], ylim = c(-.35, -.15), axes = F)
plot(B[ ,T+3], B[ ,T+6], xlim = c(-4, 25), ylim = c(-2, 20), axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(B[ ,T+4], prob = T, main = '', axes = F, xlab = '', breaks = 50)
plot(B[ ,T+4], B[ ,T+5], ylim = c(-.35, -.15), axes = F)
plot(B[ ,T+4], B[ ,T+6], axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(B[ ,T+5], prob = T, main = '', axes = F, xlab = '', breaks = 50)
plot(B[ ,T+5], B[ ,T+6], xlim = c(-.35, -.15), axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(B[ ,T+6], prob = T, main = '', axes = F, xlab = '', breaks = 50)
dev.off()
#########################################################################################################################
### latent Gaussian process
#########################################################################################################################
png('histEVGP2f.png', width= 1500, height = 800)
par(mfrow=c(4, 4), mar = c(.9, .9, .9, .9))
hist(ft[, 1], prob = T, main = '', axes = F, xlab = '', breaks = 40)
plot(ft[, 1], ft[, 2], axes = F)
plot(ft[, 1], ft[, 99], axes = F)
plot(ft[, 1], ft[, 100], axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(ft[, 2], prob = T, main = '', axes = F, xlab = '', breaks = 40)
plot(ft[, 2], ft[, 99], axes = F)
plot(ft[, 2], ft[, 100], axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(ft[, 99], prob = T, main = '', axes = F, xlab = '', breaks = 50)
plot(ft[, 99], ft[, 100], axes = F)
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
plot(1, type = "n", axes = F, xlab = "", ylab = "")
hist(ft[, 100], prob = T, main = '', axes = F, xlab = '', breaks = 50)
dev.off()