## ---------------------------------------------------------------------------------------
x <- 1:10
x
diff(x)
cumsum(x)
# Not so nice in the ends!
cumsum(diff(x))
diff(cumsum(x))
# Put a zero in beginning to fix it
cumsum(diff(c(0,x)))
diff(cumsum(c(0,x)))


## ---------------------------------------------------------------------------------------
lagvec <- function(x, lag){
    if (lag > 0) {
        ## Lag x, i.e. delay x lag steps
        return(c(rep(NA, lag), x[1:(length(x) - lag)]))
    }else if(lag < 0) {
        ## Lag x, i.e. delay x lag steps
        return(c(x[(abs(lag) + 1):length(x)], rep(NA, abs(lag))))
    }else{
        ## lag = 0, return x
        return(x)
    }
}


## ---------------------------------------------------------------------------------------
lagdf <- function(x, lag) {
  # Lag x, i.e. delay x lag steps
  if (lag > 0) {
    x[(lag+1):nrow(x), ] <- x[1:(nrow(x)-lag), ]
    x[1:lag, ] <- NA
  }else if(lag < 0) {
    # Lag x "ahead in time"
    x[1:(nrow(x)-abs(lag)), ] <- x[(abs(lag)+1):nrow(x), ]
    x[(nrow(x)-abs(lag)+1):nrow(x), ] <- NA
  }
  return(x)
}


## ----out.width="0.4\\textwidth", fig.width=2*figwidth-----------------------------------
par(mfrow=c(1,2))
# Generate realizations
n <- 100
x1 <- filter(rnorm(n), 0.9, "recursive")
plot(x1, type="b", ylim=c(-6,6))
x2 <- filter(rnorm(n), 0.9, "recursive")
lines(x2, type="b", col=2)
x3 <- filter(rnorm(n), 0.9, "recursive")
lines(x3, type="b", col=3)

# Generate 1000 realizations
X <- matrix(filter(rnorm(1000*n), 0.9, "recursive"), nrow=n)
# One realization (i.e. a time series)
X[ ,1]
# Realization of the stochastic variable at one time point
hist(X[50, ])






































## ----out.width="0.4\\textwidth"---------------------------------------------------------
# Simulate a process
n <- 100
x <- filter(rnorm(n), 0.9, "recursive")
plot(x)

# The acf and "height" of "lag 1 bar"
val <- acf(x)
val[1]
# The correlation with lag 1
cor(x, lagvec(x,1), use="complete.obs")


## ----echo=FALSE, fig.height=figheight, fig.width=figwidth, out.width="0.6\\textwidth"----
# ACF of white noise
set.seed(8372)
x <- rnorm(1000)
acf(x)


## ----eval=FALSE-------------------------------------------------------------------------
## x <- readline("Write a sequence of numbers from 0 to 9\n")
## x <- as.numeric(strsplit(x,"")[[1]])
## acf(x)


## ----size="tiny", fig.width=2*figwidth, out.width="0.7\\textwidth"----------------------
########
# ACF of a sine
w <- 10
x <- sin((1:1000)*2*pi*(w/1000))


## ----size="tiny", fig.width=2*figwidth, out.width="0.7\\textwidth"----------------------
# Plot it
par(mfrow=c(1,2))
plot(x)
# The ACF is actually also a sine!
acf(x, lag.max=500)

# Can you "screw it" and make it look like white noise up by altering a single value in x?

#x[100] <- ??
#acf(x, lag.max=500)