"Peter Spellucci" <spellucci@[EMAIL PROTECTED]
> wrote in
message news:f4oguq$1l2$1@[EMAIL PROTECTED]
>
> In article <f4ntbq$ejl$1@[EMAIL PROTECTED]
>,
> "Vista" <abc@[EMAIL PROTECTED]
> writes:
> >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!
> >
> >
> first you did not define what are the variables, what the parameters
> since you write (x,y,z) but (5,7) I assume we are in R^3 and there are
> three
> parameters. cou can model a jump discontinuity in the reals from 0 to 1
by
> 0.5*(tanh(a*x)+1) for large a. in R^3 a jump diascontinuity along a
> plane
> could be modeled by
> 0.5*(tanh(penalty*(n1*x+n2*y+n3*z-d))+1.0)
> where n1^2+n2^2+n3^3=1 (Hesse normal form of a plane) and you could try
to
> fit such a model to your data. this gives here a smooth problem using
> least squares (with a lot of data on a fine grid evaluated first)
> you can confine the number of necessary data points by using a three
> dimensional
> grid search having found a point in the neighborhood of which such a
jump
> occurs.
> (search left and right until a jump occurs)
> the function will be badly behaved if penalty is large
> hth
> peter
>
>
Haha I know Peter will have interesting comments. Sorry it should be R^5.
I
just made things easier by inspecting 5 and 7 pair.
Let me describe my problem with an simpler example.
Let's say I have a 2D plane.
I draw a straight line to split the plane to two parts: Left and Right.
The
equation for the straightline is y=a*x+b.
In the left part the values of F(x, y) are 1s, while in the right part the
values of F(x, y) are 0s.
You can query the value of F for any x and y. That's to say, you give me a
location (x, y), I will tell you the function value F(x, y).
What's the most efficient way to discover the value of "a" and "b"?
------------
It looks awfully like some type of optimization problem and/or some types
of
datamining problem, such as sup****t vector machines, etc. I am sure there
are simple and elegant algorithms handling such problem. I just don't know
where to find them.
Thank you!
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