       Re: LS fit to a plane

• To: mathgroup at smc.vnet.net
• Subject: [mg16854] Re: [mg16828] LS fit to a plane
• From: Maarten.vanderBurgt at icos.be
• Date: Thu, 1 Apr 1999 21:35:25 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```Virgil,

The distance from a point (x0,y0) to a line y = a + b x is D = ( a+ b x0 -
y0)/sqrt(1 + b^2).
For doing a least squares fit (lsq) to a line you can use this distance
instead of the vertical distance (i.e. D = y0 - a - b x0; this last D is
used in ordinary linear regression).
This is worked out in the attached notebook (in ascii text at the bottom of
this message; I deleted all output; the FullSimplify[] which is used to
make the expressions more or less readable can take some time if you do not
have a fast computer).
If you execute everything in this notebook you will see, for a nearly
vertical line, the lsq fit which uses minimal distances is much better than
the ordinary lsq fit (I use Regression[] here, Fit[] would have done as
well).

You can work out something similar for a fit to a plane if you know that
the distance from point (x0,y0,z0) to plane a x + b y + c z + d = 0 is D =
(a x0 + b y0 + c z0 + d)/sqrt(a^2 + b^2 + c^2).

hope this helps a bit

Maarten

Virgil Stokes <virgil.stokes at neuro.ki.se> on 30-03-99 05:46:38 PM

cc:

Subject: [mg16854]  Re: [mg16828] LS fit to a plane

Maarten.vanderBurgt at icos.be wrote:

> Virgil,
>
> Have a look at the following:
>
> In:= z[x_,y_]:= 3x+5y-9;
>
> In:= xyzdata =
> Flatten[Table[{u,v,z[u,v]+Random[]},{u,-1,+1},{v,-1,1}],1]
>
> Out=
> {{-1,-1,-16.9183},{-1,0,-11.6465},{-1,1,-6.13617},{0,-1,-13.8047},{0,
>
> 0,-8.3627},{0,1,-3.00167},{1,-1,-10.0657},{1,0,-5.43228},{1,1,-0.174832}}
>
> In:= Fit[xyzdata,{1,x,y},{x,y}]
>
> Out= -8.39366+3.17135 x+5.24602 y
>
> The equation of the plane is z = 3 x + 5 y - 9.
>
> In xyzdata I add some random noise to nine points in this plane.
>
> Then I fit the leas mean square plane using the Fit[] command.
>
> If you want more regression data have look in the package
> Statistics`LinearRegression`.

Thanks Maarten,

But, I am not asking about a regression fit. I have a set of measured
space coordinates where ALL contained measurement error and
I would like to find the "best" fitting plane to this set of noisy 3D
points
data.

It seems to me that a reasonable function to use for minimization
is the perpendicular distances to the plane (to be determined) from
each point. This is not ordinary regression!

--V. Stokes

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