[R] Area Under a Curve

[Rolf Turner]

On 2/07/2008, at 10:38 AM, stephen sefick wrote:

> I would like to integrate the area under a curve without any  
> smoothing or
> the like- just on the raw numbers.  I looked at integrate() but it  
> requires
> a function which I assume means something like x+x^2+x^3
>
> is there a built in function in R for this?
>
> #let's say
> x <- seq(1:50)
> y <- seq(1:50)
> plot(y~x)
> # the are would be 1250
> # I would like to be able to do this but on more complicated  
> numeric sets of
> points say dissolved oxygen mass


Here's a Simpson's Rule function that should do exactly what you want.

	cheers,

		Rolf Turner

simp <- function (y, a = NULL, b = NULL, x = NULL, n = 200)
{
     if (is.null(a) | is.null(b)) {
         if (is.null(x))
             stop("No x values provided to integrate over.\n")
     }
     else {
         x <- c(a, b)
     }
     fff <- 1
     if (length(x) == 2) {
         if (x[1] == x[2])
             return(0)
         if (x[2] < x[1]) {
             fff <- -1
             x <- rev(x)
         }
         x <- seq(x[1], x[2], length = n)
         if (is.function(y))
             y <- y(x)
         else {
             cat("y must be a function when x is\n")
             cat("of length equal to 2.\n")
             stop("Bailing out.\n")
         }
         equisp <- TRUE
     }
     else {
         if (is.function(y))
             y <- y(x)
         else if (length(y) != length(x))
             stop("Mismatch in lengths of x and y.\n")
         s <- order(x)
         x <- x[s]
         ddd <- diff(x)
         if (any(ddd == 0))
             stop("Gridpoints must be distinct.\n")
         equisp <- isTRUE(all.equal(diff(ddd), rep(0, length(ddd) - 1)))
         y <- y[s]
     }
     n <- length(x) - 1
     if (equisp) {
         old.op <- options(warn = -1)
         on.exit(options(old.op))
         M <- matrix(y, nrow = n + 2, ncol = 4)[1:(n - 2), ]
         h <- x[2] - x[1]
         fc <- h * c(-1, 13, 13, -1)/24
         aa <- apply(t(M) * fc, 2, sum)
         a1 <- h * sum(y[1:3] * c(5, 8, -1))/12
         an <- h * sum(y[(n - 1):(n + 1)] * c(-1, 8, 5))/12
         return(fff * sum(c(a1, aa, an)))
     }
     m <- n%/%2
     i <- 1:(m + 1)
     a <- x[2 * i] - x[2 * i - 1]
     i <- 1:m
     b <- x[2 * i + 1] - x[2 * i]
     o <- (a[i] * b + 2 * a[i] * a[i] - b * b)/(6 * a[i])
     p <- (a[i] + b)^3/(6 * a[i] * b)
     q <- (a[i] * b + 2 * b * b - a[i] * a[i])/(6 * b)
     k <- numeric(n + 1)
     k[1] <- o[1]
     i <- 1:(m - 1)
     k[2 * i] <- p[i]
     k[2 * i + 1] <- q[i] + o[-1]
     if (n > 2 * m) {
         aa <- a[m + 1]
         bb <- b[m]
         den <- 6 * bb * (bb + aa)
         k[2 * m] <- p[m] - (aa^3)/den
         k[2 * m + 1] <- q[m] + (aa^3 + 4 * bb * aa^2 + 3 * aa *
             bb^2)/den
         k[2 * m + 2] <- (2 * bb * aa^2 + 3 * aa * bb^2)/den
     }
     else {
         k[2 * m] <- p[m]
         k[2 * m + 1] <- q[m]
     }
     fff * sum(k * y)
}


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