Re: how to solve linear curve fitting by casting it as a minimization problem


On Aug 24, 7:13 am, none <n...@xxxxxxxx> wrote:
On Sat, 23 Aug 2008 23:35:06 -0700, e...@xxxxxxxxxxxxxxx wrote:
Also to add: for simplicity, let's say that GA>GB>GC>GD.

On Aug 24, 2:31 am, "e...@xxxxxxxxxxxxxxx" <e...@xxxxxxxxxxxxxxx>
wrote:
Hello,

I'm wondering if anybody can help me with this problem. Let's first
limit ourselves to only the first quadrant of a Cartesian coordinate
system. I have a point G that's located on the x-axis, and it's
position (x_G, 0) is known (x_G>0). Now, I have four points A, B, C,
and D in this quadrant. I know these points are on a straight line
(call it AD). This line has some arbitrary slope. I also know the four
distances GA, GB, GC, and GD. Moreover, these four distances are
periodically sampled, meaning that AB=BC=CD. However, I do not know
the coordinates of these four points in the Cartesian system (denoted
as (x_A, y_A), (x_B, y_B), (x_C, y_C), (x_D, y_D), respectively).

I want to find the slope of the line AD in the Cartesian system.
The way I'm thinking about doing is to cast this as a minimization
problem to find the points' coordinates. That way, the traejctory
slope is simply = (y_A - y_B) / (x_A - x_B).

However, I'm not sure how to formulate the minimization (what
should be the objective function, and what should be the constraint
function). Can anyone help me with this? Thanks in advance.

Is this a homeowrk problem? To me it seems that if you know

(x0, y0)
slope m in y = m x + c -> c can be calculated.
Simpler would be to work in a transformed coordinate so that your
working space is centred on (0, 0) -> c = 0

and spacing Delta-l = constant & known -> spacing Delta x = constant &
unknown

then all that remains is simple trigonometry.

sin theta = Delta x/ Delta l would be useful.

You might want to remember that the slope, m, is given by Delta y/Delta x,
and that tan theta = Delta y/Delta x.

Thanks for the reply. No, this is not a homework problem. It's a part
of a bigger problem that I'm trying to solve in my research work.

I was able to do it in a transformed Cartesian system (with GA as the
x-axis, and a y-axis perpendicular to it; this gets rid of the
intercept, c, as you suggested), and trigonometry was indeed used to
transform (or translate) it into the slope in the original Cartesian
system. However, the solution is not accepted and my adviser is
pushing to solve it by linear curve fitting. And I cannot think of
another approach except doing it by means of some minimization
problem, which is why I posted this message in the first place.

Also, by (x0, y0), I take it you mean the coordinates of point A?
Unfortunately, that knowledge is assumed unknown to me. I don't know
the coordinates of any of the points A, B, C, and D. I agree that
having the knowledge of A's coordinates would help a lot. But the lack
of it is what makes this problem seem so difficult.

I'm looking at the problem again and am now wondering: if all the
information I know is the four distance measurements that I can
measure (periodically) from a single source G on the x-axis of a
global Cartesian system, does that seem to suggest that it could have
an infinite number of solutions for the slope? Because we can
basically rotate all four measurements simultaneously around G, and
that induces infinites solutions. Would you agree?

I appreciate your further comments.
.