Vista wrote:
> automatic parametric space exploration and boundary detection.
>
> Hi all,
>
> I am not sure where to ask about my "strange" question, so as an initial
> attempt, I post it here.
>
> I have a parameterized function f(x, y, z, u, v), which takes values on
{0,
> 1}, i.e. either 0 or 1.
>
> Through experimenting I've found that when plotting out in 2D there is a
> clear linear cut in the parameter space.
>
> If I plot f(1, y, z, 5, 7) as a mesh of grid size 0.1, the cut is a
straight
> line, the region below this line has f=1, and the region above this line
has
> f=0.
>
> If I plot f(2, y, z, 5, 7) as a mesh of grid size 0.1, the cut is a
straight
> line, the region below this line has f=1, and the region above this line
has
> f=0. The only difference is that this line ****fted.
>
> ...
>
> If I plot f(x, 1, z, 5, 7) and f(x, 2, z, 5, 7) etc., still there is a
> linear cut splitting the space.
>
> ...
>
> If I plot f(x, y, 1, 5, 7) and f(x, y, 2, 5, 7) etc., still there is a
> linear cut splitting the space.
>
> ...
>
> It looks like there is a space-cutting plane in my parameter space.
>
> But the grid size is 0.1, I still have a hard time figuring out what
will
> happen when grid size becomes 0.01 and 0.001, etc.
>
> But still, I would like to discover how to define this cutting plane
> mathematically... ba
>
> My function is too complicated, and can only be computed numerically, so
I
> had a hard time finding a closed form for this cutting plane.
>
> Is there a way to determine the cutting plane using some automatifc
> procedure?
>
> I mean, it looks like some kind of optimization procedure, it seeks the
> finest boundry in the space such that on the left of the cut, f=1 and on
the
> other side of the cut, f=0. It also looks like a regression problem. But
so
> far I am still not sure how to model it.
>
> Can anybody give me some suggestions and comments?
>
> Thanks a lot!
>
>
Assuming you know nothing about your function, except that it takes
value of 0 and 1, and that there is a plane the separated the two
values, then assume the plane has the form
\sum a_i x_i = 1
Pick initial points directions to search in (coordinate directions are
nice for a start) and if that search should cause a transition from 0 to
1 (or vice versa) remember that point as a point on the plane. Clearly
if extreme point on in opposite directions give the same value there is
no point if further searching. Else once such points are found a binary
search could be used. After ac***ulating a bunch of such points you can
solve a least squares problem for the a_i. The residual will give some
idea of how good a job the plane does (assuming you have enough points),
and the plane will be the separator. Obviously this could be extended
to using a more general separating function.
Fred
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