########################################################################
#
# Tendencia deterministica ou estocastica?
#
# y1(t): ruido branco em torno de uma tendencia estocastica, 
#        traduzido aqui como um passeio aleatorico com drift 
#    
# y2(t): ruido branco em torno de uma tendencia deterministica
#        traduzido aqui como uma tendencia linear.
#
########################################################################
rm(list=ls())

# Simulando os dados
set.seed(23566)
n    = 200
N    = 50
beta = 0.5
y1 = matrix(0,n,N)
y2 = matrix(0,n,N)
y1[1,] =  rnorm(N,beta,1)
y2[1,] =  rnorm(N,beta,1)
for (t in 2:n){
 y1[t,] =  beta + y1[t-1,] + rnorm(N)
 y2[t,] =  beta*t + rnorm(N)
}

pdf(file="TD-TS-0.pdf",width=8,height=6)
par(mfrow=c(1,1))
ts.plot(y1)
for (i in 1:N)
  lines(y2[,i],col=2)
legend("topleft",legend=c("y(t) = 0.5*t + rb","y(t) = 0.5 + y(t-1) + rb"),col=2:1,lty=1)
dev.off()


set.seed(23566)
n  = 200
beta = 0.5
y1 = rep(0,n)
y2 = rep(0,n)
y1[1] =  rnorm(1,beta,1)
y2[1] =  rnorm(1,beta,1)
for (t in 2:n){
 y1[t] =  beta + y1[t-1] + rnorm(1)
 y2[t] =  beta*t + rnorm(1)
}

pdf(file="TD-TS-1.pdf",width=8,height=6)
time = 1:n
par(mfrow=c(1,1))
ts.plot(y1,main="",ylim=range(y1,y2),lwd=2,ylab="",xlab="Time")
lines(y2,col=2,lwd=2)
legend("topleft",legend=c("Y1","Y2"),col=1:2,lwd=2,lty=1)
dev.off()


# Ajustando tendencias deterministicas para ambas series 
# e fazendo analise residual
pdf(file="TD-TS-2.pdf",width=12,height=6)
par(mfrow=c(2,4))
ts.plot(y1,main="Modeling y1 with DT")
abline(lm(y1~time)$coef,col=2)
ts.plot(lm(y1~time)$res,main="",ylab="Residuals")
title("Noise doesn't look white")
acf(lm(y1~time)$res,main="FAC+FACP => random walk")
pacf(lm(y1~time)$res,main="")

ts.plot(y2,main="Modeling y2 with DT")
abline(lm(y2~time)$coef,col=2)
ts.plot(lm(y2~time)$res,main="",ylab="Residuals")
title("Noise looks white")
acf(lm(y2~time)$res,main="FAC+FACP => white noise")
pacf(lm(y2~time)$res,main="")
dev.off()

# Ajustando passeio aleatorio com drift para ambas series 
# e fazendo analise residual
pdf(file="TD-TS-3.pdf",width=12,height=6)
par(mfrow=c(2,4))
drift=lm(diff(y1)~1)$coef
ts.plot(y1[2:n],main="y1 : random walk + drift",ylab="y1")
lines(drift+y1[1:(n-1)],col=2)
ts.plot(lm(diff(y1)~1)$res,main="",ylab="Residuals")
title("Noise looks white")

acf(lm(diff(y1)~1)$res,main="FAC+FACP => white noise")
pacf(lm(diff(y1)~1)$res,main="")

drift=lm(diff(y2)~1)$coef
ts.plot(y2[2:n],main="y2 : random walk + drift",ylab="y1")
lines(drift+y2[1:(n-1)],col=2)
ts.plot(lm(diff(y2)~1)$res,main="",ylab="Residuals")
title("Noise doesn't look white")
acf(lm(diff(y2)~1)$res,main="FAC+FACP => not white noise")
pacf(lm(diff(y2)~1)$res,main="")
dev.off()