Let us say an event is $d $-dimensional if it becomes well-defined by $d $ numerical parameters, $x_1, \cdots, x_d $ and let us say that $f(x_1, \cdots, x_d) $ is a scoring function:
$$\begin {align}f: \Bbb R^d &\to \Bbb R \\ (x_1, \cdots, x_d) &\to s\end {align}$$
that follows some heuristics determined by observation of some events. How can one find, given said heuristics, an analytical expression for $f $? I do not ask for a recipe but rather for some intuition. Does it depend on the heuristics provided? On the dimension?
For example, if $d = 1$ I can find decent expressions if I have heuristics regarding
- the order of growth
- horizontal/vertical asymptotes
- specific values $f(x) $
What for $d = 2$? More specifically, I started thinking of this because I had a situation where I wanted to find an $f $ that:
- would return bigger numbers the bigger the ratio $\frac y x $ was;
- would return bigger numbers the bigger $y$ was, regardless of the ratio $\frac y x $; [What I mean is that $f(10, 50)$ would be better than $f(30, 50) $ but then $f(300, 180) $ would be around as good as $f(10, 50) $: notice that $\frac{50}{30} = \frac{300}{180}$ but $f(180, 300) > f(30, 50), f(10, 50) > f(30, 50), f(10, 50) \approx f(180, 300)$]
- would return bigger numbers the smaller $x $ was, regardless of the ratio $\frac y x $;
If this is too vague, can anyone at least point some functions/combinations of functions that can be shaped to fit some nice curves? I am aware, for example, that one can readily use the exponential function to model population growth, radioactive decay, etc., etc. but that is with $d = 1$. I believe the main barrier here is the dimension.
If any point in the question needs further clarification, please let me know in the comments.
Thanks for your time.
What about something like this:
$f(x,y) = e^{-g(x,y)}$
$g(x,y) = \omega_1x^2 + \omega_2y^{-2} + \omega_3\bigl(\frac{y}{x}\bigr)^{-2}$
The intuition is that as $g \rightarrow 0$ we have $f \rightarrow 1$, and as $g \rightarrow \infty$ we have $f \rightarrow 0$.
In other words, f scores things between 0 and 1. Personally I prefer scoring functions like this because it is easy to tell whether something has a "good" score or not.
The $\omega$'s are just positive weighting terms that can be adjusted according to your problem. You can tune them to ensure that you get a good balance of scores close to 0 and close to 1. You can also decide if you want things like the ratio to "weigh more" than the other categories.
Does this help?