raku-rosettacode/programming_tasks/P/Polynomial_regression.md

Polynomial regression

We'll use a Clifford algebra library. Very slow.

Rationale (in French for some reason):

Le système d'équations peut s'écrire :

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<annotation encoding="application/x-tex">{\displaystyle \left(a+bx\_{i}+cx\_{i}^{2}=y\_{i}\right)\_{i=1\ldots N}}</annotation>

, où on cherche

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<annotation encoding="application/x-tex">{\displaystyle (a,b,c)\in \mathbb {R} ^{3}}</annotation>

. On considère

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{N}}</annotation>

et on répartit chaque équation sur chaque dimension:

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<annotation encoding="application/x-tex">{\displaystyle (a+bx\_{i}+cx\_{i}^{2})\mathbf {e} \_{i}=y\_{i}\mathbf {e} \_{i}}</annotation>

Posons alors :

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<annotation encoding="application/x-tex">{\displaystyle \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}}</annotation>

Le système d'équations devient :

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<annotation encoding="application/x-tex">{\displaystyle a\mathbf {x} \_{0}+b\mathbf {x} \_{1}+c\mathbf {x} \_{2}=\mathbf {y} }</annotation>

.

D'où :

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a=\mathbf {y} \land \mathbf {x} \_{1}\land \mathbf {x} \_{2}/(\mathbf {x} \_{0}\land \mathbf {x\_{1}} \land \mathbf {x\_{2}} )\\b=\mathbf {y} \land \mathbf {x} \_{2}\land \mathbf {x} \_{0}/(\mathbf {x} \_{1}\land \mathbf {x\_{2}} \land \mathbf {x\_{0}} )\\c=\mathbf {y} \land \mathbf {x} \_{0}\land \mathbf {x} \_{1}/(\mathbf {x} \_{2}\land \mathbf {x\_{0}} \land \mathbf {x\_{1}} )\\\end{aligned}}}</annotation>

use MultiVector;

constant @x1 = <0 1 2 3 4 5 6 7 8 9 10>;
constant @y = <1 6 17 34 57 86 121 162 209 262 321>;

constant $x0 = [+] @e[^@x1];
constant $x1 = [+] @x1 Z* @e;
constant $x2 = [+] @x1 »**» 2  Z* @e;

constant $y  = [+] @y Z* @e;

.say for
  $y∧$x1∧$x2/($x0∧$x1∧$x2),
  $y∧$x2∧$x0/($x1∧$x2∧$x0),
  $y∧$x0∧$x1/($x2∧$x0∧$x1);

Output:

1
2
3