# Transformation to bound a parameter into an unbound parameter?

I want to optimize the parameters that minimize a particular function. These parameters are typically lower and upper bounded (i.e. some can only lie between 0 and 1, some only between 4 and 6, etc.). Some algorithms allow you to set some boundaries, but ideally, I'd like to transform the parameters into a continuous infinite space, so that when I transform back, they'll be bounded. I came across some of these transformations such as $y = x/(1-x)$ (for $x\in [0,1)$ maps into $[0,\infty)$. I also seem to remember a rather convoluted one with trigonometric functions but I can't quite remember it. Ideally, I'd like to transform my parameters so that they go from a closed interval to $(-\infty, \infty)$.

• Clearly $(x - a)/(b - a)$ maps $x$ in $[a, b]$ to $[0,1]$. $\text{logit}(x)$ maps $(0,1)$ to $(-\infty, \infty)$. Feb 20, 2014 at 22:45
• @NickCox +1 -- indeed the inverse of any continuous-strictly-monotonic-cdf-on-$(-\infty,\infty)$ maps $(0,1)$ to $(-\infty,\infty)$. There's uncountable infinities of choices. Feb 20, 2014 at 22:51

Clearly $$(x−a)/(b−a)$$ maps $$x \in [a,b]$$ to $$[0,1]$$. logit(x) maps (0,1) to (−∞,∞). – Nick Cox