# Here is a command to simulate an arima process
x <- arima.sim(10000,model=list(ndiff=1,ma=.6))

print(x)
ts.plot(x)

## Exponential smoother function

expsmooth<-function(series=x,alpha=.5,eps=1e-8,forecast=T){
nobs <- length(x)
# Calculate the time lag at which series is said to decay
#Estimate alpha from the ARIMA(0,1,1) perspective
estarima <- arima.mle(x,model=list(order=c(0,1,1)))$model$ma

# Estimate alpha over a grid
for(i in 1:9){

alphatemp <- .36 + i/100
decaylength <- ceiling(log(eps/alphatemp)/log(1 - alphatemp))
runningsum <- 0
for(j in 1:(nobs-decaylength-1)){
pred <- sum(x[j:(j+decaylength)] * alphatemp * (1 - alphatemp)^c((decaylength):0))
runningsum <- runningsum + (pred - x[j + decaylength + 1])^2
}
print(runningsum)
}
return(c(nobs,decaylength,estarima))

}

expsmooth(x)