Polynomial regression: Difference between revisions

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Find an approximating polynom of known degree for a given data.

Example: For input data:

x = {0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10};
y = {1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321};

The approximating polynom is:

3 x² + 2 x + 1

Here, the polynom's coefficients are (3, 2, 1).

This task is intended as a subtask for Measure relative performance of sorting algorithms implementations.

Ada

<lang ada>with Ada.Numerics.Real_Arrays; use Ada.Numerics.Real_Arrays;

function Fit (X, Y : Real_Vector; N : Positive) return Real_Vector is

  A : Real_Matrix (0..N, X'Range);  -- The plane

begin

  for I in A'Range (2) loop
     for J in A'Range (1) loop
        A (J, I) := X (I)**J;
     end loop;
  end loop;
  return Solve (A * Transpose (A), A * Y);

end Fit;</lang> The function Fit implements least squares approximation of a function defined in the points as specified by the arrays x_i and y_i. The basis φ_j is x^j, j=0,1,..,N. The implementation is straightforward. First the plane matrix A is created. A_ji=φ_j(x_i). Then the linear problem AA^Tc=Ay is solved. The result c_j are the coefficients. Constraint_Error is propagated when dimensions of X and Y differ or else when the problem is ill-defined.

Example

<lang ada>with Fit; with Ada.Float_Text_IO; use Ada.Float_Text_IO;

procedure Fitting is

  C : constant Real_Vector :=
         Fit
         (  (0.0, 1.0,  2.0,  3.0,  4.0,  5.0,   6.0,   7.0,   8.0,   9.0,  10.0),
            (1.0, 6.0, 17.0, 34.0, 57.0, 86.0, 121.0, 162.0, 209.0, 262.0, 321.0),
            2
         );

begin

  Put (C (0), Aft => 3, Exp => 0);
  Put (C (1), Aft => 3, Exp => 0);
  Put (C (2), Aft => 3, Exp => 0);

end Fitting;</lang>

Output:

 1.000 2.000 3.000

ALGOL 68

Translation of: Ada

Works with: ALGOL 68 version Standard - lu decomp and lu solve are from the GSL library

Works with: ALGOL 68G version Any - tested with release mk15-0.8b.fc9.i386

<lang algol68>MODE FIELD = REAL;

MODE

 VEC = [0]FIELD,
 MAT = [0,0]FIELD;

PROC VOID raise index error := VOID: (

 print(("stop", new line));
 stop

);

COMMENT from http://rosettacode.org/wiki/Matrix_Transpose#ALGOL_68 END COMMENT OP ZIP = ([,]FIELD in)[,]FIELD:(

 [2 LWB in:2 UPB in,1 LWB in:1UPB in]FIELD out;
 FOR i FROM LWB in TO UPB in DO
    out[,i]:=in[i,]
 OD;
 out

);

COMMENT from http://rosettacode.org/wiki/Matrix_multiplication#ALGOL_68 END COMMENT OP * = (VEC a,b)FIELD: ( # basically the dot product #

   FIELD result:=0;
   IF LWB a/=LWB b OR UPB a/=UPB b THEN raise index error FI;
   FOR i FROM LWB a TO UPB a DO result+:= a[i]*b[i] OD;
   result
 );

OP * = (VEC a, MAT b)VEC: ( # overload vector times matrix #

   [2 LWB b:2 UPB b]FIELD result;
   IF LWB a/=LWB b OR UPB a/=UPB b THEN raise index error FI;
   FOR j FROM 2 LWB b TO 2 UPB b DO result[j]:=a*b[,j] OD;
   result
 );

OP * = (MAT a, b)MAT: ( # overload matrix times matrix #

    [LWB a:UPB a, 2 LWB b:2 UPB b]FIELD result;
    IF 2 LWB a/=LWB b OR 2 UPB a/=UPB b THEN raise index error FI;
    FOR k FROM LWB result TO UPB result DO result[k,]:=a[k,]*b OD;
    result
  );

COMMENT from http://rosettacode.org/wiki/Pyramid_of_numbers#ALGOL_68 END COMMENT OP / = (VEC a, MAT b)VEC: ( # vector division #

 [LWB a:UPB a,1]FIELD transpose a;
 transpose a[,1]:=a;
 (transpose a/b)[,1]

);

OP / = (MAT a, MAT b)MAT:( # matrix division #

 [LWB b:UPB b]INT p ;
 INT sign;
 [,]FIELD lu = lu decomp(b, p, sign);
 [LWB a:UPB a, 2 LWB a:2 UPB a]FIELD out;
 FOR col FROM 2 LWB a TO 2 UPB a DO
   out[,col] := lu solve(b, lu, p, a[,col]) [@LWB out[,col]]
 OD;
 out

);

FORMAT int repr = $g(0)$,

      real repr = $g(-7,4)$;

PROC fit = (VEC x, y, INT order)VEC: BEGIN

  [0:order, LWB x:UPB x]FIELD a;  # the plane #
  FOR i FROM 2 LWB a TO 2 UPB a  DO
     FOR j FROM LWB a TO UPB a DO
        a [j, i] := x [i]**j
     OD
  OD;
  ( y * ZIP a ) / ( a * ZIP a )

END # fit #;

PROC print polynomial = (VEC x)VOID: (

  BOOL empty := TRUE;
  FOR i FROM UPB x BY -1 TO LWB x DO
    IF x[i] NE 0 THEN
      IF x[i] > 0 AND NOT empty THEN print ("+") FI;
      empty := FALSE;
      IF x[i] NE 1 OR i=0 THEN
        IF ENTIER x[i] = x[i] THEN
          printf((int repr, x[i]))
        ELSE
          printf((real repr, x[i]))
        FI
      FI;
      CASE i+1 IN
        SKIP,print(("x"))
      OUT
        printf(($"x**"g(0)$,i))
      ESAC
    FI
  OD;
  IF empty THEN print("0") FI;
  print(new line)

);

fitting: BEGIN

  VEC c =
         fit
         (  (0.0, 1.0,  2.0,  3.0,  4.0,  5.0,   6.0,   7.0,   8.0,   9.0,  10.0),
            (1.0, 6.0, 17.0, 34.0, 57.0, 86.0, 121.0, 162.0, 209.0, 262.0, 321.0),
            2
         );
  print polynomial(c);
  VEC d =
         fit
         ( (0, 1, 2, 3, 4, 5, 6, 7, 8, 9),
           (2.7, 2.8, 31.4, 38.1, 58.0, 76.2, 100.5, 130.0, 149.3, 180.0),
           2
         );
  print polynomial(d)

END # fitting #</lang>

Output:

3x**2+2x+1
 1.0848x**2+10.3552x-0.6164

BBC BASIC

Works with: BBC BASIC for Windows

The code listed below is good for up to 10000 data points and fits an order-5 polynomial, so the test data for this task is hardly challenging! <lang bbcbasic> INSTALL @lib$+"ARRAYLIB"

     Max% = 10000
     DIM vector(5), matrix(5,5)
     DIM x(Max%), x2(Max%), x3(Max%), x4(Max%), x5(Max%)
     DIM x6(Max%), x7(Max%), x8(Max%), x9(Max%), x10(Max%)
     DIM y(Max%), xy(Max%), x2y(Max%), x3y(Max%), x4y(Max%), x5y(Max%)
     
     npts% = 11
     x() = 0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10
     y() = 1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321
     
     sum_x = SUM(x())
     x2()  = x() * x()   : sum_x2  = SUM(x2())
     x3()  = x() * x2()  : sum_x3  = SUM(x3())
     x4()  = x2() * x2() : sum_x4  = SUM(x4())
     x5()  = x2() * x3() : sum_x5  = SUM(x5())
     x6()  = x3() * x3() : sum_x6  = SUM(x6())
     x7()  = x3() * x4() : sum_x7  = SUM(x7())
     x8()  = x4() * x4() : sum_x8  = SUM(x8())
     x9()  = x4() * x5() : sum_x9  = SUM(x9())
     x10() = x5() * x5() : sum_x10 = SUM(x10())
     
     sum_y = SUM(y())
     xy()  = x() * y()   : sum_xy  = SUM(xy())
     x2y() = x2() * y()  : sum_x2y = SUM(x2y())
     x3y() = x3() * y()  : sum_x3y = SUM(x3y())
     x4y() = x4() * y()  : sum_x4y = SUM(x4y())
     x5y() = x5() * y()  : sum_x5y = SUM(x5y())
     
     matrix() = \
     \ npts%,  sum_x,   sum_x2,  sum_x3,  sum_x4,  sum_x5, \
     \ sum_x,  sum_x2,  sum_x3,  sum_x4,  sum_x5,  sum_x6, \
     \ sum_x2, sum_x3,  sum_x4,  sum_x5,  sum_x6,  sum_x7, \
     \ sum_x3, sum_x4,  sum_x5,  sum_x6,  sum_x7,  sum_x8, \
     \ sum_x4, sum_x5,  sum_x6,  sum_x7,  sum_x8,  sum_x9, \
     \ sum_x5, sum_x6,  sum_x7,  sum_x8,  sum_x9,  sum_x10
     
     vector() = \
     \ sum_y,  sum_xy,  sum_x2y, sum_x3y, sum_x4y, sum_x5y
     
     PROC_invert(matrix())
     vector() = matrix().vector()
     
     @% = &2040A
     PRINT "Polynomial coefficients = "
     FOR term% = 5 TO 0 STEP -1
       PRINT ;vector(term%) " * x^" STR$(term%)
     NEXT</lang>

Output:

Polynomial coefficients =
0.0000 * x^5
-0.0000 * x^4
0.0002 * x^3
2.9993 * x^2
2.0012 * x^1
0.9998 * x^0

C

Library: GNU Scientific Library

Include file (to make the code reusable easily) named polifitgsl.h <lang c>#ifndef _POLIFITGSL_H

define _POLIFITGSL_H
include <gsl/gsl_multifit.h>
include <stdbool.h>
include <math.h>

bool polynomialfit(int obs, int degree, double *dx, double *dy, double *store); /* n, p */

endif</lang>

Implementation (the examples here helped alot to code this quickly): <lang c>#include "polifitgsl.h"

bool polynomialfit(int obs, int degree, double *dx, double *dy, double *store) /* n, p */ {

 gsl_multifit_linear_workspace *ws;
 gsl_matrix *cov, *X;
 gsl_vector *y, *c;
 double chisq;

 int i, j;

 X = gsl_matrix_alloc(obs, degree);
 y = gsl_vector_alloc(obs);
 c = gsl_vector_alloc(degree);
 cov = gsl_matrix_alloc(degree, degree);

 for(i=0; i < obs; i++) {
   gsl_matrix_set(X, i, 0, 1.0);
   for(j=0; j < degree; j++) {
     gsl_matrix_set(X, i, j, pow(dx[i], j));
   }
   gsl_vector_set(y, i, dy[i]);
 }

 ws = gsl_multifit_linear_alloc(obs, degree);
 gsl_multifit_linear(X, y, c, cov, &chisq, ws);

 /* store result ... */
 for(i=0; i < degree; i++)
 {
   store[i] = gsl_vector_get(c, i);
 }

 gsl_multifit_linear_free(ws);
 gsl_matrix_free(X);
 gsl_matrix_free(cov);
 gsl_vector_free(y);
 gsl_vector_free(c);
 return true; /* we do not "analyse" the result (cov matrix mainly)

to know if the fit is "good" */ }</lang> Testing: <lang c>#include <stdio.h>

include "polifitgsl.h"

define NP 11

double x[] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; double y[] = {1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321};

define DEGREE 3

double coeff[DEGREE];

int main() {

 int i;

 polynomialfit(NP, DEGREE, x, y, coeff);
 for(i=0; i < DEGREE; i++) {
   printf("%lf\n", coeff[i]);
 }
 return 0;

}</lang>

Output:

1.000000
2.000000
3.000000

Common Lisp

Uses the routine (lsqr A b) from Multiple regression and (mtp A) from Matrix transposition.

<lang lisp>;; Least square fit of a polynomial of order n the x-y-curve. (defun polyfit (x y n)

 (let* ((m (cadr (array-dimensions x)))
        (A (make-array `(,m ,(+ n 1)) :initial-element 0)))
   (loop for i from 0 to (- m 1) do
         (loop for j from 0 to n do
               (setf (aref A i j)
                     (expt (aref x 0 i) j))))
   (lsqr A (mtp y))))</lang>

Example:

<lang lisp>(let ((x (make-array '(1 11) :initial-contents '((0 1 2 3 4 5 6 7 8 9 10))))

     (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))</lang>

C#

Library: Math.Net

<lang C sharp> public static double[] Polyfit(double[] x, double[] y, int degree)

       {
           // Vandermonde matrix
           var v = new DenseMatrix(x.Length, degree + 1);
           for (int i = 0; i < v.RowCount; i++)
               for (int j = 0; j <= degree; j++) v[i, j] = Math.Pow(x[i], j);
           var yv = new DenseVector(y).ToColumnMatrix();
           QR qr = v.QR();
           // Math.Net doesn't have an "economy" QR, so:
           // cut R short to square upper triangle, then recompute Q
           var r = qr.R.SubMatrix(0, degree + 1, 0, degree + 1);
           var q = v.Multiply(r.Inverse());
           var p = r.Inverse().Multiply(q.TransposeThisAndMultiply(yv));
           return p.Column(0).ToArray();
       }</lang>

Example: <lang C sharp> static void Main(string[] args)

       {
           const int degree = 2;
           var x = new[] {0.0, 1.0,  2.0,  3.0,  4.0,  5.0,   6.0,   7.0,   8.0,   9.0,  10.0};
           var y = new[] {1.0, 6.0, 17.0, 34.0, 57.0, 86.0, 121.0, 162.0, 209.0, 262.0, 321.0};
           var p = Polyfit(x, y, degree);
           foreach (var d in p) Console.Write("{0} ",d);
           Console.WriteLine();
           for (int i = 0; i < x.Length; i++ )
               Console.WriteLine("{0} => {1} diff {2}", x[i], Polyval(p,x[i]), y[i] - Polyval(p,x[i]));
           Console.ReadKey(true);
       }</lang>

Fortran

Library: LAPACK

<lang fortran>module fitting contains

 function polyfit(vx, vy, d)
   implicit none
   integer, intent(in)                   :: d
   integer, parameter                    :: dp = selected_real_kind(15, 307)
   real(dp), dimension(d+1)              :: polyfit
   real(dp), dimension(:), intent(in)    :: vx, vy
  
   real(dp), dimension(:,:), allocatable :: X
   real(dp), dimension(:,:), allocatable :: XT
   real(dp), dimension(:,:), allocatable :: XTX
  
   integer :: i, j
  
   integer     :: n, lda, lwork
   integer :: info
   integer, dimension(:), allocatable :: ipiv
   real(dp), dimension(:), allocatable :: work
  
   n = d+1
   lda = n
   lwork = n
  
   allocate(ipiv(n))
   allocate(work(lwork))
   allocate(XT(n, size(vx)))
   allocate(X(size(vx), n))
   allocate(XTX(n, n))
  
   ! prepare the matrix
   do i = 0, d
      do j = 1, size(vx)
         X(j, i+1) = vx(j)**i
      end do
   end do
  
   XT  = transpose(X)
   XTX = matmul(XT, X)
  
   ! calls to LAPACK subs DGETRF and DGETRI
   call DGETRF(n, n, XTX, lda, ipiv, info)
   if ( info /= 0 ) then
      print *, "problem"
      return
   end if
   call DGETRI(n, XTX, lda, ipiv, work, lwork, info)
   if ( info /= 0 ) then
      print *, "problem"
      return
   end if
   
   polyfit = matmul( matmul(XTX, XT), vy)
   
   deallocate(ipiv)
   deallocate(work)
   deallocate(X)
   deallocate(XT)
   deallocate(XTX)
 
 end function

end module</lang>

Example

<lang fortran>program PolynomalFitting

 use fitting
 implicit none

 ! let us test it
 integer, parameter      :: degree = 2
 integer, parameter      :: dp = selected_real_kind(15, 307)
 integer                 :: i
 real(dp), dimension(11) :: x = (/ (i,i=0,10) /)
 real(dp), dimension(11) :: y = (/ 1,   6,  17,  34, &
                                  57,  86, 121, 162, &
                                  209, 262, 321 /)
 real(dp), dimension(degree+1) :: a

 a = polyfit(x, y, degree)

 write (*, '(F9.4)') a

end program</lang>

Output:

(lower powers first, so this seems the opposite of the Python output)

   1.0000
   2.0000
   3.0000

freebasic

<lang freebasic> Sub GaussJordan(matrix() As Double,rhs() As Double,ans() As Double)

   Dim As Integer n=Ubound(matrix,1)
   Redim ans(0):Redim ans(1 To n)
   Dim As Double b(1 To n,1 To n),r(1 To n)
   For c As Integer=1 To n 'take copies
       r(c)=rhs(c)
       For d As Integer=1 To n
           b(c,d)=matrix(c,d)
       Next d
   Next c
   #macro pivot(num)
   For p1 As Integer  = num To n - 1
       For p2 As Integer  = p1 + 1 To n  
           If Abs(b(p1,num))<Abs(b(p2,num)) Then
               Swap r(p1),r(p2)
               For g As Integer=1 To n
                   Swap b(p1,g),b(p2,g)
               Next g
           End If
       Next p2
   Next p1
   #endmacro
   For k As Integer=1 To n-1
       pivot(k)              'full pivoting 
       For row As Integer =k To n-1
           If b(row+1,k)=0 Then Exit For
           Var f=b(k,k)/b(row+1,k)
           r(row+1)=r(row+1)*f-r(k)
           For g As Integer=1 To n
               b((row+1),g)=b((row+1),g)*f-b(k,g)
           Next g
       Next row
   Next k
   'back substitute 
   For z As Integer=n To 1 Step -1
       ans(z)=r(z)/b(z,z)
       For j As Integer = n To z+1 Step -1
           ans(z)=ans(z)-(b(z,j)*ans(j)/b(z,z))
       Next j
       Next    z
   End Sub

   'Interpolate through points.
   Sub Interpolate(x_values() As Double,y_values() As Double,p() As Double)
       Var n=Ubound(x_values)
       Redim p(0):Redim p(1 To n)
       Dim As Double matrix(1 To n,1 To n),rhs(1 To n)
       For a As Integer=1 To n
           rhs(a)=y_values(a)
           For b As Integer=1 To n
               matrix(a,b)=x_values(a)^(b-1)
           Next b
       Next a
       'Solve the linear equations
       GaussJordan(matrix(),rhs(),p())
   End Sub

'======================== SET UP THE POINTS ===============  

   Dim As Double x(1 To ...)={0,1,2,3,4,5,6,7,8,9,10}
   Dim As Double y(1 To ...)={1,6,17,34,57,86,121,162,209,262,321}

   Redim As Double Poly(0)
   'Get the polynomial Poly()
   Interpolate(x(),y(),Poly())

   'print coefficients to console
   print "Polynomial Coefficients:"
   print
   For z As Integer=1 To Ubound(Poly)
       If z=1 Then
           Print "constant term  ";tab(20);Poly(z)
       Else
           Print tab(8); "x^";z-1;" =  ";tab(20);Poly(z)
       End If
   Next z

   sleep

</lang>

Output:

Polynomial Coefficients:

constant term       1
       x^ 1 =       2
       x^ 2 =       3
       x^ 3 =       0
       x^ 4 =       0
       x^ 5 =       0
       x^ 6 =       0
       x^ 7 =       0
       x^ 8 =       0
       x^ 9 =       0
       x^ 10 =      0

GAP

<lang gap>PolynomialRegression := function(x, y, n) local a; a := List([0 .. n], i -> List(x, s -> s^i)); return TransposedMat((a * TransposedMat(a))^-1 * a * TransposedMat([y]))[1]; end;

x := [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]; y := [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321];

Return coefficients in ascending degree order

PolynomialRegression(x, y, 2);

[ 1, 2, 3 ]</lang>

gnuplot

<lang gnuplot># The polynomial approximation f(x) = a*x**2 + b*x + c

Initial values for parameters

a = 0.1 b = 0.1 c = 0.1

Fit f to the following data by modifying the variables a, b, c

fit f(x) '-' via a, b, c

e

print sprintf("\n --- \n Polynomial fit: %.4f x^2 + %.4f x + %.4f\n", a, b, c)</lang>

Go

Least squares solution using QR decomposition and package gomatrix. <lang go>package main

import (

   "code.google.com/p/gomatrix/matrix"
   "fmt"

)

var xGiven = []float64{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} var yGiven = []float64{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321} var degree = 2

func main() {

   m := len(yGiven)
   n := degree + 1
   y := matrix.MakeDenseMatrix(yGiven, m, 1)
   x := matrix.Zeros(m, n)
   for i := 0; i < m; i++ {
       ip := float64(1)
       for j := 0; j < n; j++ {
           x.Set(i, j, ip)
           ip *= xGiven[i]
       }
   }

   q, r := x.QR()
   qty, err := q.Transpose().Times(y)
   if err != nil {
       fmt.Println(err)
       return
   }
   c := make([]float64, n)
   for i := n - 1; i >= 0; i-- {
       c[i] = qty.Get(i, 0)
       for j := i + 1; j < n; j++ {
           c[i] -= c[j] * r.Get(i, j)
       }
       c[i] /= r.Get(i, i)
   }
   fmt.Println(c)

}</lang>

Output:

(lowest order coefficient first)

[0.9999999999999758 2.000000000000015 2.999999999999999]

Haskell

Uses module Matrix.LU from hackageDB DSP <lang haskell>import Data.List import Data.Array import Control.Monad import Control.Arrow import Matrix.LU

ppoly p x = map (x**) p

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</lang>

Output:

in GHCi

<lang haskell>*Main> polyfit 3 [1,6,17,34,57,86,121,162,209,262,321] [1.0,2.0,3.0]</lang>

HicEst

<lang hicest>REAL :: n=10, x(n), y(n), m=3, p(m)

  x = (0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10)
  y = (1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321)

  p = 2 ! initial guess for the polynom's coefficients

  SOLVE(NUL=Theory()-y(nr), Unknown=p, DataIdx=nr, Iters=iterations)

  WRITE(ClipBoard, Name) p, iterations

FUNCTION Theory()

  ! 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</lang>

Output:

SOLVE performs a (nonlinear) least-square fit (Levenberg-Marquardt): 
p(1)=2.997135145; p(2)=2.011348347; p(3)=0.9906627242; iterations=19;

Hy

<lang lisp>(import [numpy [polyfit]])

(setv x (range 11)) (setv y [1 6 17 34 57 86 121 162 209 262 321])

(print (polyfit x y 2))</lang>

J

<lang j> X=:i.# Y=:1 6 17 34 57 86 121 162 209 262 321

  Y (%. (^/ x:@i.@#)) X

1 2 3 0 0 0 0 0 0 0 0</lang>

Note that this implementation does not use floating point numbers, so we do not introduce floating point errors. Using exact arithmetic has a speed penalty, but for small problems like this it is inconsequential.

The above solution fits a polynomial of order 11. 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: <lang j> Y (%. (i.3) ^/~ ]) X 1 2 3</lang>

Julia

The least-squares fit problem for a degree n can be solved with the built-in backslash operator: <lang julia>function polyfit(x, y, n)

 A = [ float(x[i])^p for i = 1:length(x), p = 0:n ]
 A \ y

end</lang>

Output:

julia> x = [0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10]
julia> y = [1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321]
julia> polyfit(x, y, 2)
3-element Array{Float64,1}:
 1.0
 2.0
 3.0

(giving the coefficients in increasing order of degree).

Mathematica

Using the built-in "Fit" function.

<lang Mathematica>data = Transpose@{Range[0, 10], {1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}}; Fit[data, {1, x, x^2}, x]</lang> Result:

1 + 2x + 3x^2

MATLAB

Matlab has a built-in function "polyfit(x,y,n)" which performs this task. The arguments x and y are vectors which are parametrized by the index suck that $point_{i}=(x_{i},y_{i})$ and the argument n is the order of the polynomial you want to fit. The output of this function is the coefficients of the polynomial which best fit these x,y value pairs.

<lang MATLAB>>> x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]; >> y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]; >> polyfit(x,y,2)

ans =

  2.999999999999998   2.000000000000019   0.999999999999956</lang>

МК-61/52

Part 1: <lang>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</lang>

Input: В/О x₁ С/П y₁ С/П x₂ С/П y₂ С/П ...

Part 2: <lang>ИП5 ПC ИП6 ПD П2 ИП7 П3 ИП4 ИПD * ИПC ИП5 * - ПD ИП4 ИП7 * ИПC ИП6

- П7 ИП4 ИПA * ИПC ИП9 * -

ПA ИП4 ИП3 * ИП2 ИП5 * - П3 ИП4 ИП8 * ИП2 ИП6 * - П8 ИП4 ИПB * ИП2 ИП9 * - ИПD * ИП3 ИПA * - ИПD ИП8 * ИП7 ИП3 * - / ПB ИПA ИПB ИП7 * - ИПD / ПA ИП9 ИПB ИП6

- ИПA ИП5 * - ИП4 / П9 С/П</lang>

Result: Р9 = a₀, РA = a₁, РB = a₂.

Octave

<lang octave>x = [0:10]; y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]; coeffs = polyfit(x, y, 2)</lang>

PARI/GP

Lagrange interpolating polynomial: <lang parigp>polinterpolate([0,1,2,3,4,5,6,7,8,9,10],[1,6,17,34,57,86,121,162,209,262,321])</lang>

Output:

3*x^2 + 2*x + 1

Least-squares fit: <lang parigp>V=[1,6,17,34,57,86,121,162,209,262,321]~; M=matrix(#V,3,i,j,(i-1)^(j-1));Polrev(matsolve(M~*M,M~*V))</lang>

Output:

3*x^2 + 2*x + 1

Code thanks to Bill Allombert

Python

Library: numpy

<lang python>>>> x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] >>> y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321] >>> coeffs = numpy.polyfit(x,y,deg=2) >>> coeffs array([ 3., 2., 1.])</lang> Substitute back received coefficients. <lang python>>>> yf = numpy.polyval(numpy.poly1d(coeffs), x) >>> yf array([ 1., 6., 17., 34., 57., 86., 121., 162., 209., 262., 321.])</lang> Find max absolute error: <lang python>>>> '%.1g' % max(y-yf) '1e-013'</lang>

Example

For input arrays `x' and `y': <lang python>>>> x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] >>> y = [2.7, 2.8, 31.4, 38.1, 58.0, 76.2, 100.5, 130.0, 149.3, 180.0]</lang>

<lang python>>>> p = numpy.poly1d(numpy.polyfit(x, y, deg=2), variable='N') >>> print p

1.085 N + 10.36 N - 0.6164</lang> Thus we confirm once more that for already sorted sequences the considered quick sort implementation has quadratic dependence on sequence length (see Example section for Python language on Query Performance page).

R

The easiest (and most robust) approach to solve this in R is to use the base package's lm function which will find the least squares solution via a QR decomposition:

Output:

(Intercept)           x      I(x^2) 
          1           2           3

Racket

lang racket

(require math plot)

(define xs '(0 1 2 3 4 5 6 7 8 9 10)) (define ys '(1 6 17 34 57 86 121 162 209 262 321))

(define (fit x y n)

 (define Y (->col-matrix y))
 (define V (vandermonde-matrix x (+ n 1)))
 (define VT (matrix-transpose V))
 (matrix->vector (matrix-solve (matrix* VT V) (matrix* VT Y))))

(define ((poly v) x)

 (for/sum ([c v] [i (in-naturals)])
   (* c (expt x i))))

(plot (list (points (map vector xs ys))

           (function (poly (fit xs ys 2)))))

</lang>

Output:

Ruby

<lang ruby>require 'matrix'

def regress x, y, degree

 x_data = x.map { |xi| (0..degree).map { |pow| (xi**pow).to_f } }

 mx = Matrix[*x_data]
 my = Matrix.column_vector(y)

 ((mx.t * mx).inv * mx.t * my).transpose.to_a[0]

end</lang> Testing: <lang ruby>betas = regress [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10],

               [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321],
               2

p betas</lang>

Output:

[1.00000000000018, 2.00000000000011, 3.00000000000001]

Tcl

Library: Tcllib (Package: math::linearalgebra)

<lang tcl>package require math::linearalgebra

proc build.matrix {xvec degree} {

   set sums [llength $xvec]
   for {set i 1} {$i <= 2*$degree} {incr i} {
       set sum 0
       foreach x $xvec {
           set sum [expr {$sum + pow($x,$i)}] 
       }
       lappend sums $sum
   }

   set order [expr {$degree + 1}]
   set A [math::linearalgebra::mkMatrix $order $order 0]
   for {set i 0} {$i <= $degree} {incr i} {
       set A [math::linearalgebra::setrow A $i [lrange $sums $i $i+$degree]]
   }
   return $A

}

proc build.vector {xvec yvec degree} {

   set sums [list]
   for {set i 0} {$i <= $degree} {incr i} {
       set sum 0
       foreach x $xvec y $yvec {
           set sum [expr {$sum + $y * pow($x,$i)}] 
       }
       lappend sums $sum
   }

   set x [math::linearalgebra::mkVector [expr {$degree + 1}] 0]
   for {set i 0} {$i <= $degree} {incr i} {
       set x [math::linearalgebra::setelem x $i [lindex $sums $i]]
   }
   return $x

}

Now, to solve the example from the top of this page

set x {0 1 2 3 4 5 6 7 8 9 10} set y {1 6 17 34 57 86 121 162 209 262 321}

build the system A.x=b

set degree 2 set A [build.matrix $x $degree] set b [build.vector $x $y $degree]

solve it

set coeffs [math::linearalgebra::solveGauss $A $b]

show results

puts $coeffs</lang> This will print:

1.0000000000000207 1.9999999999999958 3.0

which is a close approximation to the correct solution.

TI-89 BASIC

<lang ti89b>DelVar x seq(x,x,0,10) → xs {1,6,17,34,57,86,121,162,209,262,321} → ys QuadReg xs,ys Disp regeq(x)</lang>

seq(expr,var,low,high) evaluates expr with var bound to integers from low to high and returns a list of the results. → is the assignment operator. QuadReg, "quadratic regression", does the fit and stores the details in a number of standard variables, including regeq, which receives the fitted quadratic (polynomial) function itself. We then apply that function to the (undefined as ensured by DelVar) variable x to obtain the expression in terms of x, and display it.

Output:

3.·x² + 2.·x + 1.

Ursala

Library: LAPACK

The fit function defined below returns the coefficients of an nth-degree polynomial in order of descending degree fitting the lists of inputs x and outputs y. The real work is done by the dgelsd function from the lapack library. Ursala provides a simplified interface to this library whereby the data can be passed as lists rather than arrays, and all memory management is handled automatically. <lang Ursala>#import std

import nat
import flo

(fit "n") ("x","y") = ..dgelsd\"y" (gang \/*pow float*x iota successor "n")* "x"</lang> test program: <lang Ursala>x = <0.,1.,2.,3.,4.,5.,6.,7.,8.,9.,10.> y = <1.,6.,17.,34.,57.,86.,121.,162.,209.,262.,321.>

cast %eL

example = fit2(x,y)</lang>

Output:

<3.000000e+00,2.000000e+00,1.000000e+00>