Polynomial regression: Difference between revisions
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(formerly Perl 6)
We'll use a Clifford algebra library. Very slow.
Rationale:
Le système d'équations peut s'écrire :
<math>\left(a + b x_i + cx_i^2 = y_i\right)_{i=1\ldots N}</math>, où on cherche <math>(a,b,c)</math>. On considère <math>\mathbb{R}^N</math> et on répartit chaque équation sur chaque dimension:
<math> (a + b x_i + cx_i^2)\mathbf{e}_i = y_i\mathbf{e}_i</math>
Posons alors :
<math>
\mathbf{x}_0 = \sum_{i=1}^N \mathbf{e}_i,\,
\mathbf{x}_1 = \sum_{i=1}^N x_i\mathbf{e}_i,\,
\mathbf{x}_2 = \sum_{i=1}^N x_i^2\mathbf{e}_i,\,
\mathbf{y} = \sum_{i=1}^N y_i\mathbf{e}_i
</math>
Le système d'équations devient : <math>a\mathbf{x}_0+b\mathbf{x}_1+c\mathbf{x}_2 = \mathbf{y}</math>.
D'où :
<math>\begin{align}
a = \mathbf{y}\and\mathbf{x}_1\and\mathbf{x}_2/(\mathbf{x}_0\and\mathbf{x_1}\and\mathbf{x_2})\\
b = \mathbf{y}\and\mathbf{x}_2\and\mathbf{x}_0/(\mathbf{x}_1\and\mathbf{x_2}\and\mathbf{x_0})\\
c = \mathbf{y}\and\mathbf{x}_0\and\mathbf{x}_1/(\mathbf{x}_2\and\mathbf{x_0}\and\mathbf{x_1})\\
\end{align}</math>
<syntaxhighlight lang="raku" line>use Clifford;
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