Polynomial regression: Difference between revisions
→{{header|jq}}
| (18 intermediate revisions by 11 users not shown) | |||
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{{trans|Swift}}
<
R sum(arr) / Float(arr.len)
Line 47:
V x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
V y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
poly_regression(x, y)</
{{out}}
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=={{header|Ada}}==
<
function Fit (X, Y : Real_Vector; N : Positive) return Real_Vector is
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end loop;
return Solve (A * Transpose (A), A * Y);
end Fit;</
The function Fit implements least squares approximation of a function defined in the points as specified by the arrays ''x''<sub>''i''</sub> and ''y''<sub>''i''</sub>. The basis φ<sub>''j''</sub> is ''x''<sup>''j''</sup>, ''j''=0,1,..,''N''. The implementation is straightforward. First the plane matrix A is created. A<sub>ji</sub>=φ<sub>''j''</sub>(''x''<sub>''i''</sub>). Then the linear problem AA<sup>''T''</sup>''c''=A''y'' is solved. The result ''c''<sub>''j''</sub> are the coefficients. Constraint_Error is propagated when dimensions of X and Y differ or else when the problem is ill-defined.
===Example===
<
with Ada.Float_Text_IO; use Ada.Float_Text_IO;
Line 97:
Put (C (1), Aft => 3, Exp => 0);
Put (C (2), Aft => 3, Exp => 0);
end Fitting;</
{{out}}
<pre>
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<!-- {{does not work with|ELLA ALGOL 68|Any (with appropriate job cards AND formatted transput statements removed) - tested with release 1.8.8d.fc9.i386 - ELLA has no FORMATted transput}} -->
<
MODE
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);
print polynomial(d)
END # fitting #</
{{out}}
<pre>
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=={{header|AutoHotkey}}==
{{trans|Lua}}
<syntaxhighlight lang="autohotkey">
regression(xa,ya){
n := xa.Count()
Line 273:
eval(a,b,c,x){
return a + (b + c*x) * x
}</
Examples:<
ya := [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
MsgBox % result := regression(xa, ya)
return</
{{out}}
<pre>y = 3.000000x^2 + 2.000000x + 1.000000
Line 294:
9, 262 262.000000
10, 321 321.000000</pre>
=={{header|AWK}}==
{{trans|Lua}}
<syntaxhighlight lang="awk">
BEGIN{
i = 0;
xa[i] = 0; i++;
xa[i] = 1; i++;
xa[i] = 2; i++;
xa[i] = 3; i++;
xa[i] = 4; i++;
xa[i] = 5; i++;
xa[i] = 6; i++;
xa[i] = 7; i++;
xa[i] = 8; i++;
xa[i] = 9; i++;
xa[i] = 10; i++;
i = 0;
ya[i] = 1; i++;
ya[i] = 6; i++;
ya[i] = 17; i++;
ya[i] = 34; i++;
ya[i] = 57; i++;
ya[i] = 86; i++;
ya[i] =121; i++;
ya[i] =162; i++;
ya[i] =209; i++;
ya[i] =262; i++;
ya[i] =321; i++;
exit;
}
{
# (nothing to do)
}
END{
a = 0; b = 0; c = 0; # globals - will change by regression()
regression(xa,ya);
printf("y = %6.2f x^2 + %6.2f x + %6.2f\n",c,b,a);
printf("%-13s %-8s\n","Input","Approximation");
printf("%-6s %-6s %-8s\n","x","y","y^")
for (i=0;i<length(xa);i++) {
printf("%6.1f %6.1f %8.3f\n",xa[i],ya[i],eval(a,b,c,xa[i]));
}
}
function eval(a,b,c,x) {
return a+b*x+c*x*x;
}
# locals
function regression(x,y, n,xm,ym,x2m,x3m,x4m,xym,x2ym,sxx,sxy,sxx2,sx2x2,sx2y) {
n = 0
xm = 0.0;
ym = 0.0;
x2m = 0.0;
x3m = 0.0;
x4m = 0.0;
xym = 0.0;
x2ym = 0.0;
for (i in x) {
xm += x[i];
ym += y[i];
x2m += x[i] * x[i];
x3m += x[i] * x[i] * x[i];
x4m += x[i] * x[i] * x[i] * x[i];
xym += x[i] * y[i];
x2ym += x[i] * x[i] * y[i];
n++;
}
xm = xm / n;
ym = ym / n;
x2m = x2m / n;
x3m = x3m / n;
x4m = x4m / n;
xym = xym / n;
x2ym = x2ym / n;
sxx = x2m - xm * xm;
sxy = xym - xm * ym;
sxx2 = x3m - xm * x2m;
sx2x2 = x4m - x2m * x2m;
sx2y = x2ym - x2m * ym;
b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
a = ym - b * xm - c * x2m;
}
</syntaxhighlight>
{{out}}
<pre>
y = 3.00 x^2 + 2.00 x + 1.00
Input Approximation
x y y^
0.0 1.0 1.000
1.0 6.0 6.000
2.0 17.0 17.000
3.0 34.0 34.000
4.0 57.0 57.000
5.0 86.0 86.000
6.0 121.0 121.000
7.0 162.0 162.000
8.0 209.0 209.000
9.0 262.0 262.000
10.0 321.0 321.000
</pre>
=={{header|BBC BASIC}}==
{{works with|BBC BASIC for Windows}}
Line 299 ⟶ 406:
and fits an order-5 polynomial, so the test data for this task
is hardly challenging!
<
Max% = 10000
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FOR term% = 5 TO 0 STEP -1
PRINT ;vector(term%) " * x^" STR$(term%)
NEXT</
{{out}}
<pre>
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'''Include''' file (to make the code reusable easily) named <tt>polifitgsl.h</tt>
<
#define _POLIFITGSL_H
#include <gsl/gsl_multifit.h>
Line 370 ⟶ 477:
bool polynomialfit(int obs, int degree,
double *dx, double *dy, double *store); /* n, p */
#endif</
'''Implementation''' (the examples [http://www.gnu.org/software/gsl/manual/html_node/Fitting-Examples.html here] helped alot to code this quickly):
<
bool polynomialfit(int obs, int degree,
Line 412 ⟶ 519:
return true; /* we do not "analyse" the result (cov matrix mainly)
to know if the fit is "good" */
}</
'''Testing''':
<
#include "polifitgsl.h"
Line 434 ⟶ 541:
}
return 0;
}</
{{out}}
<pre>1.000000
Line 442 ⟶ 549:
=={{header|C sharp|C#}}==
{{libheader|Math.Net}}
<
{
// Vandermonde matrix
Line 456 ⟶ 563:
var p = r.Inverse().Multiply(q.TransposeThisAndMultiply(yv));
return p.Column(0).ToArray();
}</
Example:
<
{
const int degree = 2;
Line 469 ⟶ 576:
Console.WriteLine("{0} => {1} diff {2}", x[i], Polynomial.Evaluate(x[i], p), y[i] - Polynomial.Evaluate(x[i], p));
Console.ReadKey(true);
}</
=={{header|C++}}==
{{trans|Java}}
<
#include <iostream>
#include <numeric>
Line 534 ⟶ 641:
return 0;
}</
{{out}}
<pre>y = 1 + 2x + 3x^2
Line 554 ⟶ 661:
Uses the routine (lsqr A b) from [[Multiple regression]] and (mtp A) from [[Matrix transposition]].
<
(defun polyfit (x y n)
(let* ((m (cadr (array-dimensions x)))
Line 562 ⟶ 669:
(setf (aref A i j)
(expt (aref x 0 i) j))))
(lsqr A (mtp y))))</
Example:
<
(y (make-array '(1 11) :initial-contents '((1 6 17 34 57 86 121 162 209 262 321)))))
(polyfit x y 2))
#2A((0.9999999999999759d0) (2.000000000000005d0) (3.0d0))</
=={{header|D}}==
{{trans|Kotlin}}
<
import std.range;
import std.stdio;
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auto y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321];
polyRegression(x, y);
}</
{{out}}
<pre>y = 1 + 2x + 3x^2
Line 636 ⟶ 743:
9 262 262.0
10 321 321.0</pre>
=={{header|EasyLang}}==
{{trans|Lua}}
<syntaxhighlight lang=easylang>
func eval a b c x .
return a + (b + c * x) * x
.
proc regression xa[] ya[] . .
n = len xa[]
for i = 1 to n
xm = xm + xa[i]
ym = ym + ya[i]
x2m = x2m + xa[i] * xa[i]
x3m = x3m + xa[i] * xa[i] * xa[i]
x4m = x4m + xa[i] * xa[i] * xa[i] * xa[i]
xym = xym + xa[i] * ya[i]
x2ym = x2ym + xa[i] * xa[i] * ya[i]
.
xm = xm / n
ym = ym / n
x2m = x2m / n
x3m = x3m / n
x4m = x4m / n
xym = xym / n
x2ym = x2ym / n
#
sxx = x2m - xm * xm
sxy = xym - xm * ym
sxx2 = x3m - xm * x2m
sx2x2 = x4m - x2m * x2m
sx2y = x2ym - x2m * ym
#
b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
a = ym - b * xm - c * x2m
print "y = " & a & " + " & b & "x + " & c & "x^2"
numfmt 0 3
for i = 1 to n
print xa[i] & " " & ya[i] & " " & eval a b c xa[i]
.
.
xa[] = [ 0 1 2 3 4 5 6 7 8 9 10 ]
ya[] = [ 1 6 17 34 57 86 121 162 209 262 321 ]
regression xa[] ya[]
</syntaxhighlight>
=={{header|Emacs Lisp}}==
{{libheader|Calc}}
<syntaxhighlight lang="lisp">(let ((x '(0 1 2 3 4 5 6 7 8 9 10))
(y '(1 6 17 34 57 86 121 162 209 262 321)))
(calc-eval "fit(a*x^2+b*x+c,[x],[a,b,c],[$1 $2])" nil (cons 'vec x) (cons 'vec y)))</syntaxhighlight>
{{out}}
"3. x^2 + 1.99999999996 x + 1.00000000006"
=={{header|Fortran}}==
{{libheader|LAPACK}}
<
contains
Line 718 ⟶ 865:
end function
end module</
===Example===
<
use fitting
implicit none
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write (*, '(F9.4)') a
end program</
{{out}} (lower powers first, so this seems the opposite of the Python output):
Line 750 ⟶ 897:
=={{header|FreeBASIC}}==
General regressions for different polynomials, here it is for degree 2, (3 terms).
<
Type vector
Line 937 ⟶ 1,084:
print show(ans())
sleep</
{{out}}
<pre>+1 +2*x +3*x^2</pre>
=={{header|GAP}}==
<
local a;
a := List([0 .. n], i -> List(x, s -> s^i));
Line 953 ⟶ 1,100:
# Return coefficients in ascending degree order
PolynomialRegression(x, y, 2);
# [ 1, 2, 3 ]</
=={{header|gnuplot}}==
<
f(x) = a*x**2 + b*x + c
Line 980 ⟶ 1,127:
e
print sprintf("\n --- \n Polynomial fit: %.4f x^2 + %.4f x + %.4f\n", a, b, c)</
=={{header|Go}}==
===Library gonum/matrix===
<
import (
Line 1,024 ⟶ 1,171:
}
return x
}</
{{out}}
<pre>
Line 1,034 ⟶ 1,181:
===Library go.matrix===
Least squares solution using QR decomposition and package [http://github.com/skelterjohn/go.matrix go.matrix].
<
import (
Line 1,074 ⟶ 1,221:
}
fmt.Println(c)
}</
{{out}} (lowest order coefficient first)
<pre>
Line 1,082 ⟶ 1,229:
=={{header|Haskell}}==
Uses module Matrix.LU from [http://hackage.haskell.org/package/dsp hackageDB DSP]
<
import Data.Array
import Control.Monad
Line 1,092 ⟶ 1,239:
polyfit d ry = elems $ solve mat vec where
mat = listArray ((1,1), (d,d)) $ liftM2 concatMap ppoly id [0..fromIntegral $ pred d]
vec = listArray (1,d) $ take d ry</
{{out}} in GHCi:
<
[1.0,2.0,3.0]</
=={{header|HicEst}}==
<
x = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
Line 1,112 ⟶ 1,259:
! called by the solver of the SOLVE function. All variables are global
Theory = p(1)*x(nr)^2 + p(2)*x(nr) + p(3)
END</
{{out}}
<pre>SOLVE performs a (nonlinear) least-square fit (Levenberg-Marquardt):
Line 1,118 ⟶ 1,265:
=={{header|Hy}}==
<
(setv x (range 11))
(setv y [1 6 17 34 57 86 121 162 209 262 321])
(print (polyfit x y 2))</
=={{header|J}}==
<
(%. ^/~@x:@i.@#) Y
1 2 3 0 0 0 0 0 0 0 0</
Note that this implementation does not use floating point numbers,
Line 1,136 ⟶ 1,283:
but for small problems like this it is inconsequential.
The above solution fits a polynomial of order 11 (or, more specifically, a polynomial whose order matches the length of its argument sequence).
If the order of the polynomial is known to be 3
(as is implied in the task description)
then the following solution is probably preferable:
<
1 2 3</
(note that this time we used floating point numbers, so that result is approximate rather than exact - it only looks exact because of how J displays floating point numbers (by default, J assumes six digits of accuracy) - changing (i.3) to (x:i.3) would give us an exact result, if that mattered.)
Line 1,147 ⟶ 1,294:
{{trans|D}}
{{works with|Java|8}}
<
import java.util.function.IntToDoubleFunction;
import java.util.stream.IntStream;
Line 1,154 ⟶ 1,301:
private static void polyRegression(int[] x, int[] y) {
int n = x.length;
double xm = Arrays.stream(x).average().orElse(Double.NaN);
double ym = Arrays.stream(y).average().orElse(Double.NaN);
double x2m = Arrays.stream(
double x3m = Arrays.stream(
double x4m = Arrays.stream(
double xym = 0.0;
for (int i = 0; i < x.length && i < y.length; ++i) {
Line 1,196 ⟶ 1,342:
polyRegression(x, y);
}
}</
{{out}}
<pre>y = 1.0 + 2.0x + 3.0x^2
Line 1,212 ⟶ 1,358:
9 262 262.0
10 321 321.0</pre>
=={{header|jq}}==
'''Adapted from [[#Wren|Wren]]'''
'''Works with jq, the C implementation of jq'''
'''Works with gojq, the Go implementation of jq'''
'''Works with jaq, the Rust implementation of jq'''
<syntaxhighlight lang="jq">
def mean: add/length;
def inner_product($y):
. as $x
| reduce range(0; length) as $i (0; . + ($x[$i] * $y[$i]));
# $x and $y should be arrays of the same length
# Emit { a, b, c, z}
# Attempt to avoid overflow
def polynomialRegression($x; $y):
($x | length) as $length
| ($length * $length) as $l2
| ($x | map(./$length)) as $xs
| ($xs | add) as $xm
| ($y | mean) as $ym
| ($xs | map(. * .) | add * $length) as $x2m
| ($x | map( (./$length) * . * .) | add) as $x3m
| ($xs | map(. * . | (.*.) ) | add * $l2 * $length) as $x4m
| ($xs | inner_product($y)) as $xym
| ($xs | map(. * .) | inner_product($y) * $length) as $x2ym
| ($x2m - $xm * $xm) as $sxx
| ($xym - $xm * $ym) as $sxy
| ($x3m - $xm * $x2m) as $sxx2
| ($x4m - $x2m * $x2m) as $sx2x2
| ($x2ym - $x2m * $ym) as $sx2y
| {z: ([$x,$y] | transpose) }
| .b = ($sxy * $sx2x2 - $sx2y * $sxx2) / ($sxx * $sx2x2 - $sxx2 * $sxx2)
| .c = ($sx2y * $sxx - $sxy * $sxx2) / ($sxx * $sx2x2 - $sxx2 * $sxx2)
| .a = $ym - .b * $xm - .c * $x2m ;
# Input: {a,b,c,z}
def report:
def lpad($len): tostring | ($len - length) as $l | (" " * $l) + .;
def abc($x): .a + .b * $x + .c * $x * $x;
def print($p): "\($p[0] | lpad(3)) \($p[1] | lpad(4)) \(abc($p[0]) | lpad(5))";
"y = \(.a) + \(.b)x + \(.c)x^2\n",
" Input Approximation",
" x y y\u0302",
print(.z[]) ;
def x: [range(0;11)];
def y: [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321];
polynomialRegression(x; y)
| report
</syntaxhighlight>
{{output}}
<pre>
y = 1 + 2x + 3x^2
Input Approximation
x y ŷ
0 1 1
1 6 6
2 17 17
3 34 34
4 57 57
5 86 86
6 121 121
7 162 162
8 209 209
9 262 262
10 321 321
</pre>
=={{header|Julia}}==
Line 1,217 ⟶ 1,439:
The least-squares fit problem for a degree <i>n</i>
can be solved with the built-in backslash operator (coefficients in increasing order of degree):
<
x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
@show polyfit(x, y, 2)</
{{out}}
Line 1,228 ⟶ 1,450:
=={{header|Kotlin}}==
{{trans|REXX}}
<
fun polyRegression(x: IntArray, y: IntArray) {
Line 1,263 ⟶ 1,485:
val y = intArrayOf(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321)
polyRegression(x, y)
}</
{{out}}
Line 1,286 ⟶ 1,508:
=={{header|Lua}}==
{{trans|Modula-2}}
<
return a + (b + c * x) * x
end
Line 1,337 ⟶ 1,559:
local xa = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
local ya = {1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}
regression(xa, ya)</
{{out}}
<pre>y = 1 + 2x + 3x^2
Line 1,353 ⟶ 1,575:
=={{header|Maple}}==
<
PolynomialInterpolation([[0, 1], [1, 6], [2, 17], [3, 34], [4, 57], [5, 86], [6, 121], [7, 162], [8, 209], [9, 262], [10, 321]], 'x');
</syntaxhighlight>
Result:
<pre>3*x^2+2*x+1</pre>
Line 1,361 ⟶ 1,583:
=={{header|Mathematica}}/{{header|Wolfram Language}}==
Using the built-in "Fit" function.
<
Fit[data, {1, x, x^2}, x]</
Second version: using built-in "InterpolatingPolynomial" function.
<
WolframAlpha version:
<syntaxhighlight lang="mathematica">curve fit (0,1), (1,6), (2,17), (3,34), (4,57), (5,86), (6,121), (7,162), (8,209), (9,262), (10,321)</syntaxhighlight>
Result:
<pre>1 + 2x + 3x^2</pre>
Line 1,373 ⟶ 1,597:
The output of this function is the coefficients of the polynomial which best fit these x,y value pairs.
<
>> y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321];
>> polyfit(x,y,2)
Line 1,379 ⟶ 1,603:
ans =
2.999999999999998 2.000000000000019 0.999999999999956</
=={{header|МК-61/52}}==
Part 1:
<syntaxhighlight lang="text">ПC С/П ПD ИП9 + П9 ИПC ИП5 + П5
ИПC x^2 П2 ИП6 + П6 ИП2 ИПC * ИП7
+ П7 ИП2 x^2 ИП8 + П8 ИПC ИПD *
ИПA + ПA ИП2 ИПD * ИПB + ПB ИПD
КИП4 С/П БП 00</
''Input'': В/О x<sub>1</sub> С/П y<sub>1</sub> С/П x<sub>2</sub> С/П y<sub>2</sub> С/П ...
Part 2:
<syntaxhighlight lang="text">ИП5 ПC ИП6 ПD П2 ИП7 П3 ИП4 ИПD *
ИПC ИП5 * - ПD ИП4 ИП7 * ИПC ИП6
* - П7 ИП4 ИПA * ИПC ИП9 * -
Line 1,400 ⟶ 1,624:
ИПD ИП8 * ИП7 ИП3 * - / ПB ИПA
ИПB ИП7 * - ИПD / ПA ИП9 ИПB ИП6
* - ИПA ИП5 * - ИП4 / П9 С/П</
''Result'': Р9 = a<sub>0</sub>, РA = a<sub>1</sub>, РB = a<sub>2</sub>.
=={{header|Modula-2}}==
<
FROM FormatString IMPORT FormatString;
FROM RealStr IMPORT RealToStr;
Line 1,491 ⟶ 1,715:
ReadChar;
END PolynomialRegression.</
=={{header|Nim}}==
{{trans|Kotlin}}
<
proc polyRegression(x, y: openArray[int]) =
Line 1,528 ⟶ 1,752:
let x = toSeq(0..10)
let y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
polyRegression(x, y)</
{{out}}
Line 1,550 ⟶ 1,774:
{{trans|Kotlin}}
{{libheader|Base}}
<
open Stdio
Line 1,590 ⟶ 1,814:
let x = Array.init 11 ~f:Float.of_int in
let y = [| 1.; 6.; 17.; 34.; 57.; 86.; 121.; 162.; 209.; 262.; 321. |] in
regression x y</
{{out}}
Line 1,613 ⟶ 1,837:
=={{header|Octave}}==
<
y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321];
coeffs = polyfit(x, y, 2)</
=={{header|PARI/GP}}==
Lagrange interpolating polynomial:
<
In newer versions, this can be abbreviated:
<
{{out}}
<pre>3*x^2 + 2*x + 1</pre>
Least-squares fit:
<
M=matrix(#V,3,i,j,(i-1)^(j-1));Polrev(matsolve(M~*M,M~*V))</
<small>Code thanks to [http://pari.math.u-bordeaux.fr/archives/pari-users-1105/msg00006.html Bill Allombert]</small>
{{out}}
Line 1,633 ⟶ 1,857:
Least-squares polynomial fit in its own function:
<
lsf([0..10], [1,6,17,34,57,86,121,162,209,262,321], 2)</
=={{header|Perl}}==
This code identical to that of [[Multiple regression]] task.
<
use warnings;
use Statistics::Regression;
Line 1,651 ⟶ 1,875:
my @coeff = $reg->theta();
printf "%-6s %8.3f\n", $model[$_], $coeff[$_] for 0..@model-1;</
{{output}}
<pre>const 1.000
Line 1,658 ⟶ 1,882:
PDL Alternative:
<
use strict;
Line 1,683 ⟶ 1,907:
print "\n" unless($_)
}
</syntaxhighlight>
{{output}}
<pre> 3.00000037788248 * $x**2 + 1.99999750988868 * $x + 1.00000180493936</pre>
Line 1,692 ⟶ 1,916:
{{libheader|Phix/pGUI}}
You can run this online [http://phix.x10.mx/p2js/Polynomial_regression.htm here].
<!--<
<span style="color: #000080;font-style:italic;">-- demo\rosetta\Polynomial_regression.exw</span>
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics
<span style="color: #008080;">constant</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">5</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">,</span><span style="color: #000000;">7</span><span style="color: #0000FF;">,</span><span style="color: #000000;">8</span><span style="color: #0000FF;">,</span><span style="color: #000000;">9</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">},</span>
<span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">,</span><span style="color: #000000;">17</span><span style="color: #0000FF;">,</span><span style="color: #000000;">34</span><span style="color: #0000FF;">,</span><span style="color: #000000;">57</span><span style="color: #0000FF;">,</span><span style="color: #000000;">86</span><span style="color: #0000FF;">,</span><span style="color: #000000;">121</span><span style="color: #0000FF;">,</span><span style="color: #000000;">162</span><span style="color: #0000FF;">,</span><span style="color: #000000;">209</span><span style="color: #0000FF;">,</span><span style="color: #000000;">262</span><span style="color: #0000FF;">,</span><span style="color: #000000;">321</span><span style="color: #0000FF;">},</span>
Line 1,765 ⟶ 1,988:
<span style="color: #7060A8;">IupClose</span><span style="color: #0000FF;">()</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<!--</
{{out}}
(plus a simple graphical plot, as per [[Polynomial_regression#Racket|Racket]])
Line 1,786 ⟶ 2,009:
=={{header|PowerShell}}==
<syntaxhighlight lang="powershell">
function qr([double[][]]$A) {
$m,$n = $A.count, $A[0].count
Line 1,867 ⟶ 2,090:
"X^2 X constant"
"$(polyfit $x $y 2)"
</syntaxhighlight>
{{out}}
<pre>
Line 1,878 ⟶ 2,101:
{{libheader|NumPy}}
<
>>> y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
>>> coeffs = numpy.polyfit(x,y,deg=2)
>>> coeffs
array([ 3., 2., 1.])</
Substitute back received coefficients.
<
>>> yf
array([ 1., 6., 17., 34., 57., 86., 121., 162., 209., 262., 321.])</
Find max absolute error:
<
'1e-013'</
===Example===
For input arrays `x' and `y':
<
>>> y = [2.7, 2.8, 31.4, 38.1, 58.0, 76.2, 100.5, 130.0, 149.3, 180.0]</
<
>>> print p
2
1.085 N + 10.36 N - 0.6164</
Thus we confirm once more that for already sorted sequences
the considered quick sort implementation has
Line 1,911 ⟶ 2,134:
which will find the least squares solution via a QR decomposition:
<syntaxhighlight lang="r">
x <- c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
y <- c(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321)
coef(lm(y ~ x + I(x^2)))</
{{out}}
Line 1,924 ⟶ 2,147:
Alternately, use poly:
<
<pre> (Intercept) poly(x, 2, raw = T)1 poly(x, 2, raw = T)2
1 2 3</pre>
=={{header|Racket}}==
<
#lang racket
(require math plot)
Line 1,948 ⟶ 2,171:
(plot (list (points (map vector xs ys))
(function (poly (fit xs ys 2)))))
</syntaxhighlight>
{{out}}
[[File:polyreg-racket.png]]
Line 1,954 ⟶ 2,177:
=={{header|Raku}}==
(formerly Perl 6)
We'll use a Clifford algebra library. Very slow.
Rationale (in French for some reason):
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)\in\mathbb{R}^3</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 MultiVector;
constant @x1 = <0 1 2 3 4 5 6 7 8 9 10>;
Line 1,966 ⟶ 2,214:
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);
</syntaxhighlight>
{{out}}
<pre>1
Line 1,983 ⟶ 2,227:
=={{header|REXX}}==
<
* Implementation of http://keisan.casio.com/exec/system/14059932254941
*--------------------------------------------------------------------*/
Line 2,033 ⟶ 2,277:
fun:
Parse Arg x
Return a+b*x+c*x**2 </
{{out}}
<pre>y=1+2*x+3*x**2
Line 2,049 ⟶ 2,293:
9 262 262.000
10 321 321.000</pre>
=={{header|RPL}}==
{{trans|Ada}}
≪ 1 + → x y n
≪ { } n + x SIZE + 0 CON
1 x SIZE '''FOR''' j
1 n '''FOR''' k
{ } k + j + x j GET k 1 - ^ PUT
'''NEXT NEXT'''
DUP y * SWAP DUP TRN * /
<span style="color:grey">@ the following lines convert the resulting vector into a polynomial equation</span>
DUP 'x' STO 1 GET
2 x SIZE '''FOR''' j 'X' * x j GET + '''NEXT'''
EXPAN COLCT
≫ ≫ '<span style="color:blue">FIT</span>' STO
[1 2 3 4 5 6 7 8 9 10] [1 6 17 34 57 86 121 162 209 262 321] 2 <span style="color:blue">FIT</span>
{{out}}
<pre>
1: '3+2*X+1*X^2'
</pre>
=={{header|Ruby}}==
<
def regress x, y, degree
Line 2,060 ⟶ 2,325:
((mx.t * mx).inv * mx.t * my).transpose.to_a[0].map(&:to_f)
end</
'''Testing:'''
<
[1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321],
2)</
{{out}}
<pre>[1.0, 2.0, 3.0]</pre>
Line 2,073 ⟶ 2,338:
{{libheader|Scastie qualified}}
{{works with|Scala|2.13}}
<
private def xy = Seq(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321).zipWithIndex.map(_.swap)
Line 2,106 ⟶ 2,371:
polyRegression(xy)
}</
=={{header|Sidef}}==
{{trans|Ruby}}
<
var A = Matrix.build(x.len, degree+1, {|i,j|
x[i]**j
Line 2,129 ⟶ 2,394:
)
say coeff</
{{out}}
<pre>[1, 2, 3]</pre>
Line 2,135 ⟶ 2,400:
=={{header|Stata}}==
See '''[http://www.stata.com/help.cgi?fvvarlist Factor variables]''' in Stata help for explanations on the ''c.x##c.x'' syntax.
<
. input x y
0 1
Line 2,167 ⟶ 2,432:
|
_cons | 1 . . . . .
------------------------------------------------------------------------------</
=={{header|Swift}}==
{{trans|Kotlin}}
<syntaxhighlight lang="swift">
let x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
Line 2,214 ⟶ 2,479:
polyRegression(x: x, y: y)
</syntaxhighlight>
{{out}}
Line 2,239 ⟶ 2,504:
<!-- This implementation from Emiliano Gavilan;
posted here with his explicit permission -->
<
proc build.matrix {xvec degree} {
Line 2,287 ⟶ 2,552:
set coeffs [math::linearalgebra::solveGauss $A $b]
# show results
puts $coeffs</
This will print:
1.0000000000000207 1.9999999999999958 3.0
Line 2,293 ⟶ 2,558:
=={{header|TI-83 BASIC}}==
<
seq(X,X,0,10) → L1
{1,6,17,34,57,86,121,162,209,262,321} → L2
QuadReg L1,L2</
{{out}}
Line 2,307 ⟶ 2,572:
=={{header|TI-89 BASIC}}==
<
seq(x,x,0,10) → xs
{1,6,17,34,57,86,121,162,209,262,321} → ys
QuadReg xs,ys
Disp regeq(x)</
<code>seq(''expr'',''var'',''low'',''high'')</code> evaluates ''expr'' with ''var'' bound to integers from ''low'' to ''high'' and returns a list of the results. <code> →</code> is the assignment operator.
Line 2,329 ⟶ 2,594:
whereby the data can be passed as lists rather than arrays,
and all memory management is handled automatically.
<
#import nat
#import flo
(fit "n") ("x","y") = ..dgelsd\"y" (gang \/*pow float*x iota successor "n")* "x"</
test program:
<
y = <1.,6.,17.,34.,57.,86.,121.,162.,209.,262.,321.>
#cast %eL
example = fit2(x,y)</
{{out}}
<pre><3.000000e+00,2.000000e+00,1.000000e+00></pre>
Line 2,346 ⟶ 2,611:
=={{header|VBA}}==
Excel VBA has built in capability for line estimation.
<
Private Function polynomial_regression(y As Variant, x As Variant, degree As Integer) As Variant
Dim a() As Double
Line 2,375 ⟶ 2,640:
Debug.Print "Degrees of freedom:"; result(4, 2)
Debug.Print "Standard error of y estimate:"; result(3, 2)
End Sub</
<pre>coefficients : 1, 2, 3,
standard errors: 0, 0, 0,
Line 2,389 ⟶ 2,654:
{{libheader|Wren-seq}}
{{libheader|Wren-fmt}}
<
import "./seq" for Lst
import "./fmt" for Fmt
var polynomialRegression = Fn.new { |x, y|
Line 2,424 ⟶ 2,689:
for (i in 1..10) x[i] = i
var y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
polynomialRegression.call(x, y)</
{{out}}
Line 2,447 ⟶ 2,712:
=={{header|zkl}}==
Using the GNU Scientific Library
<
xs:=GSL.VectorFromData(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10);
ys:=GSL.VectorFromData(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321);
Line 2,453 ⟶ 2,718:
v.format().println();
GSL.Helpers.polyString(v).println();
GSL.Helpers.polyEval(v,xs).format().println();</
{{out}}
<pre>
Line 2,465 ⟶ 2,730:
Example:
<
T(T(1.0,6.0,17.0,34.0,57.0,86.0,121.0,162.0,209.0,262.0,321.0)), 2)
.flatten().println();</
{{out}}<pre>L(1,2,3)</pre>
| |||