Visualization of a multivariate function This is somewhat vague, but suppose you have a black box function $f(x_1,x_2,\ldots,x_k)$, for which you have code, and you are interested in the behaviour of $f$ when the $x_i$ are i.i.d. standard Gaussian random variables. What are some good ways to visualize this function? To make it easier, we may assume that $k$ is smallish, say less than 10. 
One particular relationship of interest is how $f$ varies with one of the input, say $x_i$. An easy way to visualize this relationship would be to sample the function for fixed values of $x_i$ while varying the other input (either in a structured way, or randomly, say), then box-plotting, which could show how the mean trend is affected by $x_i$, but also whether the scatter is affected (i.e. heteroskedasticity). However, interaction between $x_i$ and the levels of the other input might be masked by this approach.
What I am looking for is somewhat open-ended. I do not have a particular hypothesis that I am testing, but rather am looking for new ways of visualizing the response which might reveal peculiarities of the function.
 A: Given that you are at the initial, exploratory stages of the analysis I would start simple. Consider sampling your inputs using a Latin Hypercube strategy. Then, a tornado chart can be used to get a quick assessment of the multiple,one-way sensitivities f() has to the various input variables. Here is an example chart (from here)

This chart is not that interesting, but an interpretation would be "NPV is most sensitive to Shipments, all other things being equal. But, the sensitivity is mostly on the upside, which is good. The Escalation variable induces sensitivity into NPV, but what looks to be skewed negatively a bit...". 
You could do something similar for Mean(f) on the X-axis as well as Var(f)
Given what you find from some first glance visualizations like this, you could then slice and dice more and focus on specific variables or relationships between variables. Maybe you can revisit this thread in coming months and post the visualizations you found useful :)
A: Just a thought, although I've never tried it.


*

*you could obtain a large number of values from the function across different parameter values

*take a tour of the resulting data in ggobi (check out Mat Kelcey's video)

A: You could apply some sort of dimensionality reduction technique like principal components and plot the value of the function as you vary the first, second, third etc. principal components, holding all others fixed.  This would show you how the function varies in the directions of the maximal variance of the inputs.
