In sunsiyu/optimalr: Understand and Solve Optimization Problems

Simplest Case

Absolute error: $|x - \widehat{x}|$

But cannot address difference importance of differences in relation to x, thus Relative error:

$\frac{|x - \widehat{x}|}{|x|}$

For computing convenience, avoid the case x can be non-zero:

$\frac{|x - \widehat{x}|}{1 + |x|}$

has the property that when $|x|>>1$, similar to relative error; and when $|x| <= 1$, similar to absolute error.

Plots

x <- seq(0, 100, 0.1)
x_e <- x + rnorm(length(x))
par(mfrow = c(2,2), mar = c(1,2,1,1))
plot(x)
lines(x_e)
plot(abs(x-x_e), type="l")
plot(abs(x-x_e)/abs(x), type="l")
plot(abs(x-x_e)/(1+abs(x)), type="l")

Log.

The function that outputs between 0 and 1.

x1 <- seq(0, 1, 0.01)
x2 <- seq(0, 100, 0.1)
y1 <- exp(x1) / (1 + exp(x1))
y2 <- exp(x2) / (1 + exp(x2))
par(mfrow = c(2,2), mar = c(1,2,1,1))
plot(x1, y1)
plot(x2, y2)
plot(log(x1), y1)
plot(log(x2), y2)