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Interpolating unknows from polynomial curve

Hi,

I'm am trying to script an existing workflow and I am struggling with the last part.

The users have a standard curve of points as part of the data (x = percentage, y=value) and in an external software (Prism, GraphPad) utilise a 6th order polynomial non-linear regression to utilise that curve and then interpolate unknowns from that.

I have my x and y in 2 arrays ready to go.
In trying to recreate this I have been having a try with Math::Polynomial but the interpolate function of that seems to be using the curve data to predict the polynomial needed so I am a little confused.

Does anyone have a hint as to how best to approach this in perl by reading in curve data then reading out the unknowns?

Thanks
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Epipgx
Asked:
Epipgx
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1 Solution
 
ozoCommented:
If you give Math::Polynomial->interpolate 7 data points, it can generate a 6th order polynomial that passes through those points, you can then evaluate that polynomial at other points.
If that's not what you want, perhaps you want to use Algorithm::CurveFit or PDL::Fit::Polynomial
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EpipgxAuthor Commented:
Thanks Ozo,

So there isn't a way to specify the the nth order? It is (as reading a bit more on the subject I think it should be) driven from the curve itself and produces the best order fit for your data?
If I have understood that correctly can you point me to an example where once the curve is fit using that value to read unknowns from it?
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ozoCommented:
#!/usr/bin/perl
package  Epipgx;
use strict;
use warnings;
use Carp;
use Math::MatrixReal;
use overload '""' => '_stringify';
sub new{
    my($self,$o) = @_;
    my $class = ref $self || $self;
   $o//=6;  
    croak "order must be positive" if $o<0;
    return bless {order=>$o},$class;
}
sub interpolate {
    my($self, $xvalues, $yvalues) = @_;
    my $o=$self->{order}//6;
    $o=$#$xvalues if $o>$#$xvalues;
    my @ix=map{my$x=$_;[map{$x**$_}0..$o]}@$xvalues;
    my @iy=@$yvalues;
    my @XtX;
    for my $x( @ix ){
      for my $i ( 0..$o ){
          for my $j ( 0..$o ){
              $XtX[$i][$j] += $x->[$i]*$x->[$j];
          }
      }
    }
    my $XtX = Math::MatrixReal->new_from_rows( [@XtX] );
    my $inv = $XtX->inverse() or warn $!;
    my $iY = Math::MatrixReal->new_from_cols( [[@iy]] );
    my $iX = Math::MatrixReal->new_from_rows( [@ix] );
    my $P = $inv * ~$iX;
    my $W = $P * $iY;
    $self->{a}=[map{$W->element($_+1,1)}0..$o];
}
sub evaluate{
    my ($self, $x) = @_;
    my $y=0;
    my $X=1;
    for my $w( @{$self->{a}} ){
      $y+=$w*$X;
      $X*=$x;
    }
    return $y;
}  
sub _stringify{
    my($self)=@_;
    return join(" + ",map{($self->{a}[$_]//0).($_>0&&"*x").($_>1&&"^$_")}reverse 0..$self->{order})=~s/\+ -/- /gr;
}
1;
=head1 NAME

Epipgx - Perl class for Interpolating unknows from polynomial curve

=head1 SYNOPSIS

use Epipgx;

my $p=Epipgx->new(6);   # 6th order polynomial

$p->interpolate([0..8], [0, 1, 1, 2, 3, 5, 8, 13, 21]);

print "$p\n";

$y = $p->evaluate(0.5);

=cut
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EpipgxAuthor Commented:
Thanks Ozo that is fantastic, you have gone way beyond for me there and is much appreciated.
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EpipgxAuthor Commented:
Fantastically thorough answer, I was hoping for guidance but received a fully formed solution to my question, brilliant
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