Constraints for multiple intervals

[Qazalbash]

I have implemented three different constraints that involve multiple closed intervals.

• Union of intervals

$$ x \in \bigcup_{i=1}^{n} [a_i, b_i] \implies \bigvee_{i=1}^{n} (x \in [a_i, b_i]) $$

• Intersection of intervals

$$ x \in \bigcap_{i=1}^{n} [a_i, b_i] \implies \bigwedge_{i=1}^{n} (x \in [a_i, b_i]) $$

• Unique intervals. The idea is we have $n$ different parameters and each has its unique constraining intervals.

$$ \bigwedge_{i=1}^{n} (x_i \in [a_i, b_i]) $$

[fehiepsi]

Hi @Qazalbash, do you have any concrete example of those constraints?

[fehiepsi]

I don't understand why this composed transform is bijective. Could you clarify?

[Qazalbash]

I wrote this code snippet to test the constraints of the system. This was the best example I could come up with based on a quick review of the test cases. Please note that this code is not based on any specific logic, so feel free to correct me if I'm wrong or suggest improvements!

[fehiepsi]

I see. Probably the tests pass because the domain is quite loose. Exp transform has positive codomain, not union of intervals.

[Qazalbash]

For the union of intervals, an example. I now realize we can avoid the intersection of intervals to a single interval. For the unique interval case, I was writing transformation defined as,

$$ f:(x,y)\to (x',y') $$

where, $0 < y \leq x$ and,

$$ x'=\frac{(xy)^{3/5}}{(x+y)^{1/5}},\qquad y'=\frac{xy}{(x+y)^{2}} $$

and turns out the bounds for each are,

$$ 0 < x'<\infty \qquad 0 < y' \leq \frac{1}{4} $$

I had to deal with a similar situation where each element in the event_dim=1 represents a quantity and has its bounds and I had to write a separate constraint for them.

[fehiepsi]

For the unique interval case, you can use independent(interval(lower_bound, upper_bound)). You are right that the intersection case is just another interval. The trickiest one is the union of intervals, which I think you can assume they do not overlap and maps interval i to (i/n,i/n+1/n) where n is the number of intervals. I guess it is better to avoid dealing with such complicated constraint.

[Qazalbash]

I got your point. I still don't get independent(interval(lower_bound, upper_bound)). Can you explain it using my mentioned use case? After that, I will close the PR!

[fehiepsi]

In that example, I guess you can use the constraint

constraints.independent(constraints.interval(np.zeros(2), np.array([1e6, 0.25])))

[Qazalbash]

Thank you!