how to calculate curvature of multivariate function?

EDIT1: tried to clarify the question

Context

In the context of an MCMC investigation of non-linear interaction effects in dichotomous models, I am creating data generating processes based on the logistic CDF, $\Lambda()$, that are not monotone.

The issue that I am facing is that the same functional form can have a very different surface based on the coefficients, and therefore different performance of the simulation. In other words, the extent to which the surface is curved, influences the performance of the simulation, under the same basic functional form. So I cannot just compare between different functional forms, but also between each draw of the monte carlo.

For example:

The generic functional form $y = \Lambda(\beta_1x_1 + \beta_2x_2 + \beta_{12}x_1^2x_2 + \epsilon)$ has very different surfaces based of the betas.

On one hand, $$y = \Lambda(-1.72x_1 -3.41x_2 +0.3x_1^2x_2 + \epsilon)$$ has a nice monotone surface:

On the other hand, $$y = \Lambda(0.72x_1 -0.86x_2 +2.91x_1^2x_2 + \epsilon)$$ has a crazy surface.

way forward

I tried looking at $\frac{\partial^2\Lambda}{\partial x_1 \partial x_2}$ in order to distinguish between them, but I could not a deterministic relationship between the shape of the cross partial for all the cases. This made sense only for the extreme cases. It is clear that there are clues there, but not something I know how to quantify in a defensive way.

Ideally, I would be like to find a scalar that represents how curved the surface is. I think I need something like https://en.wikipedia.org/wiki/Mean_curvature but this is well over my head. Can someone suggest something more applied? Hopefully that is also implemented in R or python?

• Because you don't explain the basis of your comparison, it is unclear whether curvature is even relevant to your situation (and if so, what kind of curvature is needed). Could you back up a bit and explain what these functions represent and your purpose in comparing them to each other? – whuber Jun 18 '18 at 1:56
• Not sure what MCMC us , but In dimensions above 1, curvature is much more complicated ( You basically use derivatives for dimension 1) and requires knowledge of Differential geometry. AFAIK, it cannot be described by a scalar; the curve can bend in many ways, many directions, but instead by a tensor , like the curvature . tensor. And the curvature can be Riemannian or Ricci ( ttps://duckduckgo.com/?q=curvature+tensor&t=h_&atb=v88-7&ia=web) . – gary Jun 18 '18 at 2:27
• @whuber hope I made it better – Elad663 Jun 18 '18 at 2:40
• @gary I understand that one scalar does not have enough information. If I can just order them by curvitude, it will should suffice for my needs. – Elad663 Jun 18 '18 at 2:42
• Still, MCMC equals MC^2 if you fudge the algebra :). – gary Jun 18 '18 at 2:44