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Re: Surface fitting with PDL

by pryrt (Abbot)
 on Aug 09, 2020 at 21:17 UTC Need Help??

in reply to Surface fitting with PDL

Xilman,

Because you invoked "least squares fitting", I assume what you want is a planar least-squares fitting in 3d, analogous to the linear least-squares in 2d, and that z is supposed be be linearly dependent on the others: ax+by+c = z

You can see the algebra in this math.stackexchange answer, but basically you are solving A*X=B, where A is a matrix of the x and y data, X are the column-vector of the coefficients a,b,c from the above equation, and B is the column-vector of the z values for each x,y input.

I haven't looked up the exact PDL syntax, but: An exact solution for 3 sets of (x,y,z) data would have PDL akin to \$coeff_X = \$matrix_A->inverse * \$column_B;. But since you presumably have many points, not just three, since you invoked best-fit, you have to use the "left pseudo inverse", which would have a PDL implementation akin to \$coeff_X = (\$matrix_A->transpose * \$matrix_A)->inverse * (\$matrix_A->transpose) * \$column_B;

update: I was close.

```#!/usr/bin/perl
use strict;
use warnings;
use PDL;
use PDL::Matrix;

# example: 2x + 3y + 4 = z, plug in known exact values; make sure my c
+oefficients end up 2,3,4

my \$matrix_A = mpdl [
[1,1,1],
[3,7,1],
[5,1,1],
[1,5,1],
];

print "A => ", \$matrix_A;
print "B => ", my \$column_B = vpdl [9,31,17,21];
print "X => ", my \$coeff_X = ((\$matrix_A->transpose x \$matrix_A)->inv
+x \$matrix_A->transpose) x \$column_B;

prints:

```A =>
[
[1 1 1]
[3 7 1]
[5 1 1]
[1 5 1]
]
B =>
[
[ 9]
[31]
[17]
[21]
]
X =>
[
[         2]
[         3]
[         4]
]

Replies are listed 'Best First'.
Re^2: Surface fitting with PDL
by Xilman (Hermit) on Aug 10, 2020 at 07:29 UTC

Excellent! Many thanks.

I actually wish to fit a higher order polynomial, perhaps up to a cubic with cross terms, but the generalization to be made to your code should be straightforward.

Re^2: Surface fitting with PDL
by etj (Deacon) on Apr 21, 2022 at 07:23 UTC
Note for posterity: the go-to module for this stuff is PDL::LinearAlgebra. The obvious solver for Ax=B would be "msolve", and the notes for that give which LAPACK function is used "under the hood". For solving many times, you'd want to do a decomposition (quite possibly an LU) first, then reuse it. Take a look!

LAPACK is a huge and powerful library for linear-algebra stuff, and getting to know it will be useful, regardless of which language environment you use - I don't think there is a single language environment that doesn't have a binding for it.

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