# The Additive Model for a Time Series

# Example 1.1.1. (Unemployed1 Data). The monthly numbers of unemployed workers in the
# building trade in Germany from July 1975 to September 1979.
unemployed1 <- read.table("unemployed1.txt")
unemployed1
colnames(unemployed1) <- c("month","t","unemplyd")
unemployed1
plot(unemplyd~t,data=unemployed1,type="b")

# The logistic function  with different values of the parameters
t <- seq(-10,10,length=100)
flog <- function(t,b1,b2,b3) b3/(1+b2*exp(-t*b1))
plot(range(t),c(0,1),type="n",xlab="t",ylab="flog")
lines(t,flog(t,0.5,0.1,1))
lines(t,flog(t,0.5,1,1),col="red")
lines(t,flog(t,1,0.1,1),col="blue")
lines(t,flog(t,1,1,1),col="green")

# Example 1.1.2. (Population1 Data). The population sizes in millions 
# of the area of North-Rhine-Westphalia in 5 years steps from 1935 to 1980
population1 <- read.table("population1.txt")
colnames(population1) <- c("year","t","pop")
population1
plot(pop~t,data=population1)
fit.pop1 <- nls(pop ~ b3/(1+b2*exp(-t*b1)),data = population1,
                 start = list(b1=0.2,b2=1,b3=17))
fit.pop1
predict(fit.pop1)
lines(predict(fit.pop1)~t,data=population1)

# The Gompertz function  with different values of the parameters
t <- seq(0,4,length=100)
fG <- function(t,b1,b2,b3)exp(b1+b2*b3^t)
plot(range(t),c(1,8),type="n",xlab="t",ylab="fG")
lines(t,fG(t,1,-1,0.05))
lines(t,fG(t,1,-1,0.5),col="red")
lines(t,fG(t,1,1,0.05),col="blue")
lines(t,fG(t,1,1,0.5),col="green")

# Example 1.1.3. (Income Data). The (accumulated) annual average 
# increases of gross and net incomes in thousands DM (deutsche mark) 
# in Germany, starting in 1960.
income <- read.table("income.txt")
colnames(income) <- c("year","t","gross","net")
income
plot(net~t,data=income)
fit.net <- lm(log(income$net[-1]) ~ log(income$t[-1]))
summary(fit.net)
coef(fit.net)
b1 <- coef(fit.net)[2]
b2 <- exp(coef(fit.net)[1])
b1
b2
y <- income$net[-1]
y.hat <- b2*(income$t[-1])^b1
y
y.hat
y-y.hat
1-sum((y-y.hat)^2)/sum((y-mean(y))^2)


predict(fit.pop1)
lines(predict(fit.pop1)~t,data=population1)