What is monotonic transformation? (quick answer needed!)

[Drox]

The textbook says it's a way of transforming a set of numbers into another set that preserves the order. But I don't understand what that means.

Here are a few examples. The question was: do these functions represent a monotonic transformation.


u = 2v - 13 (yes)

u = -1/v^2 (no)

u = ln(v) (yes)

u = v^2 (no)



I have no clue what you're supposed to look for.

[harry_hardcore_hoedown]

What it means is that the function is always either increasing or decreasing. You look at the first derivative and the domain for which the function is defined, and if it's always increasing or decreasing in that domain, it's monotonic:

u = 2v - 13

In this case, du/dv = 2. This is constant, so it's always increasing, and therefore it's monotonic.

u = -1/v² = -v^-2

du/dv = 2v^-3

The function is defined for all real numbers, but when v is negative, du/dv is negative, and when v is positive, du/dv is positive, so it's decreasing for v < 0, and increasing for v > 0. Therefore it's not monotonic.

u = ln(v)

du/dv = 1/v

The function ln(v) is only defined for v > 0, and in that domain, du/dv > 0, so this function is monotonic too.

u = v²

du/dv = 2v

This function is defined for all real numbers, but when v < 0, du/dv < 0, and when v > 0, du/dv > 0, so it's not monotonic.

[General Butt.Naked]

STAR SCREAM

[Drox]

Thanks dude.

What if there are two variables? Like (x^2)y.

Do you take partial derivatives of both?

[harry_hardcore_hoedown]

Originally Posted by Drox

Thanks dude.

What if there are two variables? Like (x^2)y.

Do you take partial derivatives of both?

I think so. I'm not entirely sure about that so if you need a definite answer you'll have to wait for somebody who knows more calculus to read this thread, but I would guess that a multivariable function is monotonic or not monotonic with respect to each variable.

[Bilbo]

For multivariate cases, I'm assuming the same conditions must be met:

Without using derivatives, the definition of a strictly monotonically increasing function is "if x<y then f(x)<f(y)". You can extrapolate to decreasing and non-strict conditions.

Restated, we have "If a1<a2,b1<b2,c1<c2... then f(a1,b1...)<f(a2,b2)". However, I think this goes into deeper mathematics such as boolean functions and order theory, which I know nothing of. Wikipedia doesn't help much lol