# How to model a RV bounded by other RVs?

Hello UQWorld!

I would like to ask the following question, in case someone has faced a similar problem and resolved it using UQLab (or other UQ software, for that matter).

I have a circuit board of variable width, taking values in between [bw_low, bw_high] in a uniform fashion. On the board we place cables, the number of which and the diameter of which can also be modeled as uniform RVs, respectively taking values in [nc_low, nc_high] and [cd_low, cd_high]. The condition board_width >= num_cables x cable_diameter should always be met, otherwise the design in physically unfeasible. It is assumed that bw_high > nc_high x cw_high. Still, the condition imposes a new bw_low every time a parameter realization is needed.

Is there a way to model this dependency? Could UQLab be used for that purpose?

Thank you a priori,
Dimitris

Hi @DiLu,

I attempt a response to your question here and hopefully, others might chime in to give you even more proper answer.

So if I get your question correctly, what you’re trying to do is to have a joint distribution of your inputs with:

1. w, nc and cd have all uniform marginals;
2. \frac{nc \times cd}{w} \leq 1;
3. nc_{high} * cw_{high} < bw_{high}; and
4. the lower bound of w \sim \mathcal{U} (bw_{low}, bw_{high}) is defined by the realization of n_c \times c_d.

I am wondering whether you would insist that w must have a uniform marginal in this particular problem. What I mean is, say you have some a priori requirements on the number of cables and its diameter (say, both uniform with certain bound), then because of condition two, I’m not sure you can have an arbitrary distribution (incl. the bound) on the width of the board.

If you do a simple simulation starting with realizations of nc and cd then follow your own construction to get realizations of w, I think you’ll obtain a marginal distribution of the width that, in general, would not be uniform. The question is whether that is the distribution of width of the board that you actually expect (i.e. if it is realistic or reasonable). I don’t know in which context your simulation is, but I think it’s also fair to ask whether in reality the (lower bound) uncertainty of the board width simply depends only on the number of cables and its diameter that you’re going to put there.

So before resorting to a more complicated statistical dependency modeling of the inputs, with which others here would be able to help you better (like @torree ), perhaps you can make sure that the uncertainties associated with these three parameters are more realistically defined?

As a side note, maybe you can also let us know what you’re actually trying to do with your simulation with regard to UQ? If it’s a generic metamodeling (or as noted here some sensitivity analyses), how you actually define the distribution of your inputs may not matter much as long as the input domain is properly defined and you have enough points. But for some other UQ analyses, the probability distributions of the inputs indeed require more rigor in their definitions because the results and conclusion of the analyses would depend on them.

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Hi @damarginal,

thanks for the reply and for your comments. In the meantime, we analyzed the model a bit further, added a spacing parameter sp between the cables (assumed to be uniformly distributed) and now the board width is always given by nc \times cd + (nc-1) \times sp. For simulation purposes, this solves the problem.

Regarding the side note, the goal is to develop a surrogate model for the heat exchange between the circuit and its environment, where we know that the geometry of the board, the geometry of the cables, and the number of cables play a role and the parametric dependency is given by a smooth functional (hence the choice of a PCE). Originally, the analysis is done with a dedicated electro-thermal FEM model, which is quite computationally expensive, hence the need for a surrogate. So, it is not really a UQ analysis and the uniform distribution is merely used to account for the various designs of our industry partner. Therefore, I believe that more sophistication in the probabilistic modeling is not necessary. Probably taking enough points in the design space should resolve this.

Again, thanks a lot!
Dimitris

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