Integration of two lines

Hello,

I need to integrate the absolute difference between two lines measured
on different points.

# For example :
x <- seq(0, 1, 1/100)
f_x <- runif(101) + x
y <- seq(0, 1, 1/23)
f_y <- runif(24) + (1 - y)

plot(x, f_x, type="l")
lines(y, f_y)

Then I would like to compute Integral( | f_x - f_y | )dx.
(This is not the same as | Integral(f_x)dx -  Integral(f_y)dx |.)

Computing this integral looks non trivial. I guess I should interpolate
the points of f_y over x and integrate both lines on these intervals.
Even then I would miss points where the lines cross.

There are functions to integrate below *one* line (I'm thinking about
the trapz function in caTools).
Do you know if there is a function to do this integration properly with
two lines (and especially their absolute difference)?

Regards,
Xavier

Hello,

I need to integrate the absolute difference between two lines measured
on different points.

# For example :
x <- seq(0, 1, 1/100)
f_x <- runif(101) + x
y <- seq(0, 1, 1/23)
f_y <- runif(24) + (1 - y)

plot(x, f_x, type="l")
lines(y, f_y)

Then I would like to compute Integral( | f_x - f_y | )dx.
(This is not the same as | Integral(f_x)dx -  Integral(f_y)dx |.)

Computing this integral looks non trivial. I guess I should interpolate
the points of f_y over x and integrate both lines on these intervals.
Even then I would miss points where the lines cross.

There are functions to integrate below *one* line (I'm thinking about
the trapz function in caTools).
Do you know if there is a function to do this integration properly with
two lines (and especially their absolute difference)?

Regards,
Xavier

g <- function(x) abs(f1(x)-f2(x))

now you have one function and you can integrate it.

First define a function from those points:

    fx <- approxfun(x, f_x)
    fy <- approxfun(y, f_y)
    f  <- function(x) abs(fx(x)-fy(x))

and now you can apply integrate() or trapz():

    xx <- sort(c(x, y))
    yy <- f(xx)
    trapz(xx, yy)

trapz() should return the more accurate (i.e. exact) result.