Polynomial regression
Find an approximating polynomial of known degree for a given data.
You are encouraged to solve this task according to the task description, using any language you may know.
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 polynomial is:
3 x2 + 2 x + 1
Here, the polynomial's coefficients are (3, 2, 1).
This task is intended as a subtask for Measure relative performance of sorting algorithms implementations.
11l
<lang 11l>F average(arr)
R sum(arr) / Float(arr.len)
F poly_regression(x, y)
V xm = average(x) V ym = average(y) V x2m = average(x.map(i -> i * i)) V x3m = average(x.map(i -> i ^ 3)) V x4m = average(x.map(i -> i ^ 4)) V xym = average(zip(x, y).map((i, j) -> i * j)) V x2ym = average(zip(x, y).map((i, j) -> i * i * j)) V sxx = x2m - xm * xm V sxy = xym - xm * ym V sxx2 = x3m - xm * x2m V sx2x2 = x4m - x2m * x2m V sx2y = x2ym - x2m * ym V b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) V c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) V a = ym - b * xm - c * x2m
F abc(xx) R (@a + @b * xx) + (@c * xx * xx)
print("y = #. + #.x + #.x^2\n".format(a, b, c)) print(‘ Input Approximation’) print(‘ x y y1’)
L(i) 0 .< x.len print(‘#2 #3 #3.1’.format(x[i], y[i], abc(i)))
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)</lang>
- Output:
y = 1 + 2x + 3x^2 Input Approximation x y y1 0 1 1.0 1 6 6.0 2 17 17.0 3 34 34.0 4 57 57.0 5 86 86.0 6 121 121.0 7 162 162.0 8 209 209.0 9 262 262.0 10 321 321.0
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 xi and yi. The basis φj is xj, j=0,1,..,N. The implementation is straightforward. First the plane matrix A is created. Aji=φj(xi). Then the linear problem AATc=Ay is solved. The result cj 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
<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
AutoHotkey
<lang AutoHotkey> regression(xa,ya){ n := xa.Count() xm := ym := x2m := x3m := x4m := xym := x2ym := 0
loop % n { i := A_Index 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
result := "Input`tApproximation`nx y`ty1`n" loop % n i := A_Index, result .= xa[i] ", " ya[i] "`t" eval(a, b, c, xa[i]) "`n" return "y = " c "x^2" " + " b "x + " a "`n`n" result } eval(a,b,c,x){ return a + (b + c*x) * x }</lang> Examples:<lang AutoHotkey>xa := [0, 1, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10] ya := [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321] MsgBox % result := regression(xa, ya) return</lang>
- Output:
y = 3.000000x^2 + 2.000000x + 1.000000 Input Approximation x y y1 0, 1 1.000000 1, 6 6.000000 2, 17 17.000000 3, 34 34.000000 4, 57 57.000000 5, 86 86.000000 6, 121 121.000000 7, 162 162.000000 8, 209 209.000000 9, 262 262.000000 10, 321 321.000000
AWK
<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;
} </lang>
- Output:
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
BBC BASIC
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
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++) { 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
C#
<lang csharp> 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<double> 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], Polynomial.Evaluate(x[i], p), y[i] - Polynomial.Evaluate(x[i], p)); Console.ReadKey(true); }</lang>
C++
<lang cpp>#include <algorithm>
- include <iostream>
- include <numeric>
- include <vector>
void polyRegression(const std::vector<int>& x, const std::vector<int>& y) {
int n = x.size(); std::vector<int> r(n); std::iota(r.begin(), r.end(), 0); double xm = std::accumulate(x.begin(), x.end(), 0.0) / x.size(); double ym = std::accumulate(y.begin(), y.end(), 0.0) / y.size(); double x2m = std::transform_reduce(r.begin(), r.end(), 0.0, std::plus<double>{}, [](double a) {return a * a; }) / r.size(); double x3m = std::transform_reduce(r.begin(), r.end(), 0.0, std::plus<double>{}, [](double a) {return a * a * a; }) / r.size(); double x4m = std::transform_reduce(r.begin(), r.end(), 0.0, std::plus<double>{}, [](double a) {return a * a * a * a; }) / r.size();
double xym = std::transform_reduce(x.begin(), x.end(), y.begin(), 0.0, std::plus<double>{}, std::multiplies<double>{}); xym /= fmin(x.size(), y.size());
double x2ym = std::transform_reduce(x.begin(), x.end(), y.begin(), 0.0, std::plus<double>{}, [](double a, double b) { return a * a * b; }); x2ym /= fmin(x.size(), y.size());
double sxx = x2m - xm * xm; double sxy = xym - xm * ym; double sxx2 = x3m - xm * x2m; double sx2x2 = x4m - x2m * x2m; double sx2y = x2ym - x2m * ym;
double b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2); double c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2); double a = ym - b * xm - c * x2m;
auto abc = [a, b, c](int xx) { return a + b * xx + c * xx*xx; };
std::cout << "y = " << a << " + " << b << "x + " << c << "x^2" << std::endl; std::cout << " Input Approximation" << std::endl; std::cout << " x y y1" << std::endl;
auto xit = x.cbegin(); auto xend = x.cend(); auto yit = y.cbegin(); auto yend = y.cend(); while (xit != xend && yit != yend) { printf("%2d %3d %5.1f\n", *xit, *yit, abc(*xit)); xit = std::next(xit); yit = std::next(yit); }
}
int main() {
using namespace std;
vector<int> x(11); iota(x.begin(), x.end(), 0);
vector<int> y{ 1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321 };
polyRegression(x, y);
return 0;
}</lang>
- Output:
y = 1 + 2x + 3x^2 Input Approximation x y y1 0 1 1.0 1 6 6.0 2 17 17.0 3 34 34.0 4 57 57.0 5 86 86.0 6 121 121.0 7 162 162.0 8 209 209.0 9 262 262.0 10 321 321.0
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>
D
<lang D>import std.algorithm; import std.range; import std.stdio;
auto average(R)(R r) {
auto t = r.fold!("a+b", "a+1")(0, 0); return cast(double) t[0] / t[1];
}
void polyRegression(int[] x, int[] y) {
auto n = x.length; auto r = iota(0, n).array; auto xm = x.average(); auto ym = y.average(); auto x2m = r.map!"a*a".average(); auto x3m = r.map!"a*a*a".average(); auto x4m = r.map!"a*a*a*a".average(); auto xym = x.zip(y).map!"a[0]*a[1]".average(); auto x2ym = x.zip(y).map!"a[0]*a[0]*a[1]".average();
auto sxx = x2m - xm * xm; auto sxy = xym - xm * ym; auto sxx2 = x3m - xm * x2m; auto sx2x2 = x4m - x2m * x2m; auto sx2y = x2ym - x2m * ym;
auto b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2); auto c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2); auto a = ym - b * xm - c * x2m;
real abc(int xx) { return a + b * xx + c * xx * xx; }
writeln("y = ", a, " + ", b, "x + ", c, "x^2"); writeln(" Input Approximation"); writeln(" x y y1"); foreach (i; 0..n) { writefln("%2d %3d %5.1f", x[i], y[i], abc(x[i])); }
}
void main() {
auto x = iota(0, 11).array; auto y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]; polyRegression(x, y);
}</lang>
- Output:
y = 1 + 2x + 3x^2 Input Approximation x y y1 0 1 1.0 1 6 6.0 2 17 17.0 3 34 34.0 4 57 57.0 5 86 86.0 6 121 121.0 7 162 162.0 8 209 209.0 9 262 262.0 10 321 321.0
Emacs Lisp
<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)))</lang>
- Output:
"3. x^2 + 1.99999999996 x + 1.00000000006"
Fortran
<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
General regressions for different polynomials, here it is for degree 2, (3 terms). <lang FreeBASIC>#Include "crt.bi" 'for rounding only
Type vector
Dim As Double element(Any)
End Type
Type matrix
Dim As Double element(Any,Any) Declare Function inverse() As matrix Declare Function transpose() As matrix private: Declare Function GaussJordan(As vector) As vector
End Type
'mult operators Operator *(m1 As matrix,m2 As matrix) As matrix
Dim rows As Integer=Ubound(m1.element,1) Dim columns As Integer=Ubound(m2.element,2) If Ubound(m1.element,2)<>Ubound(m2.element,1) Then Print "Can't do" Exit Operator End If Dim As matrix ans Redim ans.element(rows,columns) Dim rxc As Double For r As Integer=1 To rows For c As Integer=1 To columns rxc=0 For k As Integer = 1 To Ubound(m1.element,2) rxc=rxc+m1.element(r,k)*m2.element(k,c) Next k ans.element(r,c)=rxc Next c Next r Operator= ans
End Operator
Operator *(m1 As matrix,m2 As vector) As vector
Dim rows As Integer=Ubound(m1.element,1) Dim columns As Integer=Ubound(m2.element,2) If Ubound(m1.element,2)<>Ubound(m2.element) Then Print "Can't do" Exit Operator End If Dim As vector ans Redim ans.element(rows) Dim rxc As Double For r As Integer=1 To rows rxc=0 For k As Integer = 1 To Ubound(m1.element,2) rxc=rxc+m1.element(r,k)*m2.element(k) Next k ans.element(r)=rxc Next r Operator= ans
End Operator
Function matrix.transpose() As matrix
Dim As matrix b Redim b.element(1 To Ubound(this.element,2),1 To Ubound(this.element,1)) For i As Long=1 To Ubound(this.element,1) For j As Long=1 To Ubound(this.element,2) b.element(j,i)=this.element(i,j) Next Next Return b
End Function
Function matrix.GaussJordan(rhs As vector) As vector
Dim As Integer n=Ubound(rhs.element) Dim As vector ans=rhs,r=rhs Dim As matrix b=This #macro pivot(num) For p1 As Integer = num To n - 1 For p2 As Integer = p1 + 1 To n If Abs(b.element(p1,num))<Abs(b.element(p2,num)) Then Swap r.element(p1),r.element(p2) For g As Integer=1 To n Swap b.element(p1,g),b.element(p2,g) Next g End If Next p2 Next p1 #endmacro For k As Integer=1 To n-1 pivot(k) For row As Integer =k To n-1 If b.element(row+1,k)=0 Then Exit For Var f=b.element(k,k)/b.element(row+1,k) r.element(row+1)=r.element(row+1)*f-r.element(k) For g As Integer=1 To n b.element((row+1),g)=b.element((row+1),g)*f-b.element(k,g) Next g Next row Next k 'back substitute For z As Integer=n To 1 Step -1 ans.element(z)=r.element(z)/b.element(z,z) For j As Integer = n To z+1 Step -1 ans.element(z)=ans.element(z)-(b.element(z,j)*ans.element(j)/b.element(z,z)) Next j Next z Function = ans
End Function
Function matrix.inverse() As matrix
Var ub1=Ubound(this.element,1),ub2=Ubound(this.element,2) Dim As matrix ans Dim As vector temp,null_ Redim temp.element(1 To ub1):Redim null_.element(1 To ub1) Redim ans.element(1 To ub1,1 To ub2) For a As Integer=1 To ub1 temp=null_ temp.element(a)=1 temp=GaussJordan(temp) For b As Integer=1 To ub1 ans.element(b,a)=temp.element(b) Next b Next a Return ans
End Function
'vandermode of x Function vandermonde(x_values() As Double,w As Long) As matrix
Dim As matrix mat Var n=Ubound(x_values) Redim mat.element(1 To n,1 To w) For a As Integer=1 To n For b As Integer=1 To w mat.element(a,b)=x_values(a)^(b-1) Next b Next a Return mat
End Function
'main preocedure Sub regress(x_values() As Double,y_values() As Double,ans() As Double,n As Long)
Redim ans(1 To Ubound(x_values)) Dim As matrix m1= vandermonde(x_values(),n) Dim As matrix T=m1.transpose Dim As vector y Redim y.element(1 To Ubound(ans)) For n As Long=1 To Ubound(y_values) y.element(n)=y_values(n) Next n Dim As vector result=(((T*m1).inverse)*T)*y Redim Preserve ans(1 To n) For n As Long=1 To Ubound(ans) ans(n)=result.element(n) Next n
End Sub
'Evaluate a polynomial at x Function polyeval(Coefficients() As Double,Byval x As Double) As Double
Dim As Double acc For i As Long=Ubound(Coefficients) To Lbound(Coefficients) Step -1 acc=acc*x+Coefficients(i) Next i Return acc
End Function
Function CRound(Byval x As Double,Byval precision As Integer=30) As String
If precision>30 Then precision=30 Dim As zstring * 40 z:Var s="%." &str(Abs(precision)) &"f" sprintf(z,s,x) If Val(z) Then Return Rtrim(Rtrim(z,"0"),".")Else Return "0"
End Function
Function show(a() As Double,places as long=10) As String
Dim As String s,g For n As Long=Lbound(a) To Ubound(a) If n<3 Then g="" Else g="^"+Str(n-1) if val(cround(a(n),places))<>0 then s+= Iif(Sgn(a(n))>=0,"+","")+cround(a(n),places)+ Iif(n=Lbound(a),"","*x"+g)+" " end if Next n Return s
End Function
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 ans() regress(x(),y(),ans(),3)
print show(ans()) sleep</lang>
- Output:
+1 +2*x +3*x^2
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
0 1 1 6 2 17 3 34 4 57 5 86 6 121 7 162 8 209 9 262 10 321
e
print sprintf("\n --- \n Polynomial fit: %.4f x^2 + %.4f x + %.4f\n", a, b, c)</lang>
Go
Library gonum/matrix
<lang go>package main
import ( "fmt" "log"
"gonum.org/v1/gonum/mat" )
func main() { var ( x = []float64{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} y = []float64{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}
degree = 2
a = Vandermonde(x, degree+1) b = mat.NewDense(len(y), 1, y) c = mat.NewDense(degree+1, 1, nil) )
var qr mat.QR qr.Factorize(a)
const trans = false err := qr.SolveTo(c, trans, b) if err != nil { log.Fatalf("could not solve QR: %+v", err) } fmt.Printf("%.3f\n", mat.Formatted(c)) }
func Vandermonde(a []float64, d int) *mat.Dense { x := mat.NewDense(len(a), d, nil) for i := range a { for j, p := 0, 1.0; j < d; j, p = j+1, p*a[i] { x.Set(i, j, p) } } return x }</lang>
- Output:
⎡1.000⎤ ⎢2.000⎥ ⎣3.000⎦
Library go.matrix
Least squares solution using QR decomposition and package go.matrix. <lang go>package main
import (
"fmt"
"github.com/skelterjohn/go.matrix"
)
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> Y=:1 6 17 34 57 86 121 162 209 262 321
(%. ^/~@x:@i.@#) Y
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) ^/~ i.#Y 1 2 3</lang> (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.)
Java
<lang Java>import java.util.Arrays; import java.util.function.IntToDoubleFunction; import java.util.stream.IntStream;
public class PolynomialRegression {
private static void polyRegression(int[] x, int[] y) { int n = x.length; int[] r = IntStream.range(0, n).toArray(); double xm = Arrays.stream(x).average().orElse(Double.NaN); double ym = Arrays.stream(y).average().orElse(Double.NaN); double x2m = Arrays.stream(r).map(a -> a * a).average().orElse(Double.NaN); double x3m = Arrays.stream(r).map(a -> a * a * a).average().orElse(Double.NaN); double x4m = Arrays.stream(r).map(a -> a * a * a * a).average().orElse(Double.NaN); double xym = 0.0; for (int i = 0; i < x.length && i < y.length; ++i) { xym += x[i] * y[i]; } xym /= Math.min(x.length, y.length); double x2ym = 0.0; for (int i = 0; i < x.length && i < y.length; ++i) { x2ym += x[i] * x[i] * y[i]; } x2ym /= Math.min(x.length, y.length);
double sxx = x2m - xm * xm; double sxy = xym - xm * ym; double sxx2 = x3m - xm * x2m; double sx2x2 = x4m - x2m * x2m; double sx2y = x2ym - x2m * ym;
double b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2); double c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2); double a = ym - b * xm - c * x2m;
IntToDoubleFunction abc = (int xx) -> a + b * xx + c * xx * xx;
System.out.println("y = " + a + " + " + b + "x + " + c + "x^2"); System.out.println(" Input Approximation"); System.out.println(" x y y1"); for (int i = 0; i < n; ++i) { System.out.printf("%2d %3d %5.1f\n", x[i], y[i], abc.applyAsDouble(x[i])); } }
public static void main(String[] args) { int[] x = IntStream.range(0, 11).toArray(); int[] y = new int[]{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}; polyRegression(x, y); }
}</lang>
- Output:
y = 1.0 + 2.0x + 3.0x^2 Input Approximation x y y1 0 1 1.0 1 6 6.0 2 17 17.0 3 34 34.0 4 57 57.0 5 86 86.0 6 121 121.0 7 162 162.0 8 209 209.0 9 262 262.0 10 321 321.0
Julia
The least-squares fit problem for a degree n can be solved with the built-in backslash operator (coefficients in increasing order of degree): <lang julia>polyfit(x::Vector, y::Vector, deg::Int) = collect(v ^ p for v in x, p in 0:deg) \ y
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)</lang>
- Output:
polyfit(x, y, 2) = [1.0, 2.0, 3.0]
Kotlin
<lang scala>// version 1.1.51
fun polyRegression(x: IntArray, y: IntArray) {
val xm = x.average() val ym = y.average() val x2m = x.map { it * it }.average() val x3m = x.map { it * it * it }.average() val x4m = x.map { it * it * it * it }.average() val xym = x.zip(y).map { it.first * it.second }.average() val x2ym = x.zip(y).map { it.first * it.first * it.second }.average()
val sxx = x2m - xm * xm val sxy = xym - xm * ym val sxx2 = x3m - xm * x2m val sx2x2 = x4m - x2m * x2m val sx2y = x2ym - x2m * ym
val b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) val c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) val a = ym - b * xm - c * x2m
fun abc(xx: Int) = a + b * xx + c * xx * xx
println("y = $a + ${b}x + ${c}x^2\n") println(" Input Approximation") println(" x y y1") for ((xi, yi) in x zip y) { System.out.printf("%2d %3d %5.1f\n", xi, yi, abc(xi)) }
}
fun main() {
val x = IntArray(11) { it } val y = intArrayOf(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321) polyRegression(x, y)
}</lang>
- Output:
y = 1.0 + 2.0x + 3.0x^2 Input Approximation x y y1 0 1 1.0 1 6 6.0 2 17 17.0 3 34 34.0 4 57 57.0 5 86 86.0 6 121 121.0 7 162 162.0 8 209 209.0 9 262 262.0 10 321 321.0
Lua
<lang lua>function eval(a,b,c,x)
return a + (b + c * x) * x
end
function regression(xa,ya)
local n = #xa
local xm = 0.0 local ym = 0.0 local x2m = 0.0 local x3m = 0.0 local x4m = 0.0 local xym = 0.0 local x2ym = 0.0
for i=1,n do 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] end xm = xm / n ym = ym / n x2m = x2m / n x3m = x3m / n x4m = x4m / n xym = xym / n x2ym = x2ym / n
local sxx = x2m - xm * xm local sxy = xym - xm * ym local sxx2 = x3m - xm * x2m local sx2x2 = x4m - x2m * x2m local sx2y = x2ym - x2m * ym
local b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) local c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) local a = ym - b * xm - c * x2m
print("y = "..a.." + "..b.."x + "..c.."x^2") for i=1,n do print(string.format("%2d %3d %3d", xa[i], ya[i], eval(a, b, c, xa[i]))) end
end
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)</lang>
- Output:
y = 1 + 2x + 3x^2 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
Maple
<lang Maple>with(CurveFitting); PolynomialInterpolation([[0, 1], [1, 6], [2, 17], [3, 34], [4, 57], [5, 86], [6, 121], [7, 162], [8, 209], [9, 262], [10, 321]], 'x'); </lang> Result:
3*x^2+2*x+1
Mathematica/Wolfram Language
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> Second version: using built-in "InterpolatingPolynomial" function. <lang Mathematica>Simplify@InterpolatingPolynomial[{{0, 1}, {1, 6}, {2, 17}, {3, 34}, {4, 57}, {5, 86}, {6, 121}, {7, 162}, {8, 209}, {9, 262}, {10, 321}}, 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 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.
Modula-2
<lang modula2>MODULE PolynomialRegression; FROM FormatString IMPORT FormatString; FROM RealStr IMPORT RealToStr; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE Eval(a,b,c,x : REAL) : REAL; BEGIN
RETURN a + b*x + c*x*x;
END Eval;
PROCEDURE Regression(x,y : ARRAY OF INTEGER); VAR
n,i : INTEGER; xm,x2m,x3m,x4m : REAL; ym : REAL; xym,x2ym : REAL; sxx,sxy,sxx2,sx2x2,sx2y : REAL; a,b,c : REAL; buf : ARRAY[0..63] OF CHAR;
BEGIN
n := SIZE(x)/SIZE(INTEGER);
xm := 0.0; ym := 0.0; x2m := 0.0; x3m := 0.0; x4m := 0.0; xym := 0.0; x2ym := 0.0; FOR i:=0 TO n-1 DO xm := xm + FLOAT(x[i]); ym := ym + FLOAT(y[i]); x2m := x2m + FLOAT(x[i]) * FLOAT(x[i]); x3m := x3m + FLOAT(x[i]) * FLOAT(x[i]) * FLOAT(x[i]); x4m := x4m + FLOAT(x[i]) * FLOAT(x[i]) * FLOAT(x[i]) * FLOAT(x[i]); xym := xym + FLOAT(x[i]) * FLOAT(y[i]); x2ym := x2ym + FLOAT(x[i]) * FLOAT(x[i]) * FLOAT(y[i]); END; xm := xm / FLOAT(n); ym := ym / FLOAT(n); x2m := x2m / FLOAT(n); x3m := x3m / FLOAT(n); x4m := x4m / FLOAT(n); xym := xym / FLOAT(n); x2ym := x2ym / FLOAT(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;
WriteString("y = "); RealToStr(a, buf); WriteString(buf); WriteString(" + "); RealToStr(b, buf); WriteString(buf); WriteString("x + "); RealToStr(c, buf); WriteString(buf); WriteString("x^2"); WriteLn;
FOR i:=0 TO n-1 DO FormatString("%2i %3i ", buf, x[i], y[i]); WriteString(buf); RealToStr(Eval(a,b,c,FLOAT(x[i])), buf); WriteString(buf); WriteLn; END;
END Regression;
TYPE R = ARRAY[0..10] OF INTEGER; VAR
x,y : R;
BEGIN
x := R{0,1,2,3,4,5,6,7,8,9,10}; y := R{1,6,17,34,57,86,121,162,209,262,321}; Regression(x,y);
ReadChar;
END PolynomialRegression.</lang>
Nim
<lang Nim>import lenientops, sequtils, stats, strformat
proc polyRegression(x, y: openArray[int]) =
let xm = mean(x) let ym = mean(y) let x2m = mean(x.mapIt(it * it)) let x3m = mean(x.mapIt(it * it * it)) let x4m = mean(x.mapIt(it * it * it * it)) let xym = mean(zip(x, y).mapIt(it[0] * it[1])) let x2ym = mean(zip(x, y).mapIt(it[0] * it[0] * it[1]))
let sxx = x2m - xm * xm let sxy = xym - xm * ym let sxx2 = x3m - xm * x2m let sx2x2 = x4m - x2m * x2m let sx2y = x2ym - x2m * ym
let b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) let c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) let a = ym - b * xm - c * x2m
func abc(x: int): float = a + b * x + c * x * x
echo &"y = {a} + {b}x + {c}x²\n" echo " Input Approximation" echo " x y y1" for (xi, yi) in zip(x, y): echo &"{xi:2} {yi:3} {abc(xi):5}"
let x = toSeq(0..10)
let y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
polyRegression(x, y)</lang>
- Output:
y = 1.0 + 2.0x + 3.0x² Input Approximation x y y1 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
OCaml
<lang OCaml>open Base open Stdio
let mean fa =
let open Float in (Array.reduce_exn fa ~f:(+)) / (of_int (Array.length fa))
let regression xs ys =
let open Float in let xm = mean xs in let ym = mean ys in let x2m = Array.map xs ~f:(fun x -> x * x) |> mean in let x3m = Array.map xs ~f:(fun x -> x * x * x) |> mean in let x4m = Array.map xs ~f:(fun x -> let x2 = x * x in x2 * x2) |> mean in let xzipy = Array.zip_exn xs ys in let xym = Array.map xzipy ~f:(fun (x, y) -> x * y) |> mean in let x2ym = Array.map xzipy ~f:(fun (x, y) -> x * x * y) |> mean in
let sxx = x2m - xm * xm in let sxy = xym - xm * ym in let sxx2 = x3m - xm * x2m in let sx2x2 = x4m - x2m * x2m in let sx2y = x2ym - x2m * ym in
let b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) in let c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) in let a = ym - b * xm - c * x2m in
let abc xx = a + b * xx + c * xx * xx in
printf "y = %.1f + %.1fx + %.1fx^2\n\n" a b c; printf " Input Approximation\n"; printf " x y y1\n"; Array.iter xzipy ~f:(fun (xi, yi) -> printf "%2g %3g %5.1f\n" xi yi (abc xi) )
let () =
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</lang>
- Output:
y = 1.0 + 2.0x + 3.0x^2 Input Approximation x y y1 0 1 1.0 1 6 6.0 2 17 17.0 3 34 34.0 4 57 57.0 5 86 86.0 6 121 121.0 7 162 162.0 8 209 209.0 9 262 262.0 10 321 321.0
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> In newer versions, this can be abbreviated: <lang parigp>polinterpolate([0..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> Code thanks to Bill Allombert
- Output:
3*x^2 + 2*x + 1
Least-squares polynomial fit in its own function: <lang parigp>lsf(X,Y,n)=my(M=matrix(#X,n+1,i,j,X[i]^(j-1))); Polrev(matsolve(M~*M,M~*Y~)) lsf([0..10], [1,6,17,34,57,86,121,162,209,262,321], 2)</lang>
Perl
This code identical to that of Multiple regression task. <lang perl>use strict; use warnings; use Statistics::Regression;
my @x = <0 1 2 3 4 5 6 7 8 9 10>; my @y = <1 6 17 34 57 86 121 162 209 262 321>;
my @model = ('const', 'X', 'X**2');
my $reg = Statistics::Regression->new( , [@model] ); $reg->include( $y[$_], [ 1.0, $x[$_], $x[$_]**2 ]) for 0..@y-1; my @coeff = $reg->theta();
printf "%-6s %8.3f\n", $model[$_], $coeff[$_] for 0..@model-1;</lang>
- Output:
const 1.000 X 2.000 X**2 3.000
PDL Alternative: <lang perl>#!/usr/bin/perl -w use strict;
use PDL; use PDL::Math; use PDL::Fit::Polynomial;
my $x = float [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]; my $y = float [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321];
- above will output: 3.00000037788248 * $x**2 + 1.99999750988868 * $x + 1.00000180493936
- $x = float [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9];
- $y = float [ 2.7, 2.8, 31.4, 38.1, 58.0, 76.2, 100.5, 130.0, 149.3, 180.0];
- above correctly returns: " 1.08484845125187 * $x**2 + 10.3551513321297 * $x-0.616363852007752 "
my ($yfit, $coeffs) = fitpoly1d $x, $y, 3; # 3rd degree
foreach (reverse(0..$coeffs->dim(0)-1)) {
print " +" unless(($coeffs->at($_) <0) || $_==$coeffs->dim(0)-1); # let the unary minus replace the + operator print " "; print $coeffs->at($_); print " * \$x" if($_); print "**$_" if($_>1); print "\n" unless($_)
} </lang>
- Output:
3.00000037788248 * $x**2 + 1.99999750988868 * $x + 1.00000180493936
Phix
You can run this online here.
-- demo\rosetta\Polynomial_regression.exw with javascript_semantics constant x = {0,1,2,3,4,5,6,7,8,9,10}, y = {1,6,17,34,57,86,121,162,209,262,321}, n = length(x) function regression() atom xm = 0, ym = 0, x2m = 0, x3m = 0, x4m = 0, xym = 0, x2ym = 0 for i=1 to n do atom xi = x[i], yi = y[i] xm += xi ym += yi x2m += power(xi,2) x3m += power(xi,3) x4m += power(xi,4) xym += xi*yi x2ym += power(xi,2)*yi end for xm /= n ym /= n x2m /= n x3m /= n x4m /= n xym /= n x2ym /= n atom Sxx = x2m-power(xm,2), Sxy = xym-xm*ym, Sxx2 = x3m-xm*x2m, Sx2x2 = x4m-power(x2m,2), Sx2y = x2ym-x2m*ym, B = (Sxy*Sx2x2-Sx2y*Sxx2)/(Sxx*Sx2x2-power(Sxx2,2)), C = (Sx2y*Sxx-Sxy*Sxx2)/(Sxx*Sx2x2-power(Sxx2,2)), A = ym-B*xm-C*x2m return {C,B,A} end function atom {a,b,c} = regression() function f(atom x) return a*x*x+b*x+c end function printf(1,"y=%gx^2+%gx+%g\n",{a,b,c}) printf(1,"\n x y f(x)\n") for i=1 to n do printf(1," %2d %3d %3g\n",{x[i],y[i],f(x[i])}) end for -- And a simple plot (re-using x,y from above) include pGUI.e include IupGraph.e function get_data(Ihandle graph) integer {w,h} = IupGetIntInt(graph,"DRAWSIZE") IupSetInt(graph,"YTICK",iff(h<240?iff(h<150?80:40):20)) return {{x,y,CD_RED}} end function IupOpen() Ihandle graph = IupGraph(get_data,"RASTERSIZE=640x440") IupSetAttributes(graph,"XTICK=1,XMIN=0,XMAX=10") IupSetAttributes(graph,"YTICK=20,YMIN=0,YMAX=320") Ihandle dlg = IupDialog(graph,`TITLE="simple plot"`) IupSetAttributes(dlg,"MINSIZE=245x150") IupShow(dlg) if platform()!=JS then IupMainLoop() IupClose() end if
- Output:
(plus a simple graphical plot, as per Racket)
y=3x^2+2x+1 x y f(x) 0 1 1 1 6 6 2 17 17 3 34 34 4 57 57 5 86 86 6 121 121 7 162 162 8 209 209 9 262 262 10 321 321
PowerShell
<lang PowerShell> function qr([double[][]]$A) {
$m,$n = $A.count, $A[0].count $pm,$pn = ($m-1), ($n-1) [double[][]]$Q = 0..($m-1) | foreach{$row = @(0) * $m; $row[$_] = 1; ,$row} [double[][]]$R = $A | foreach{$row = $_; ,@(0..$pn | foreach{$row[$_]})} foreach ($h in 0..$pn) { [double[]]$u = $R[$h..$pm] | foreach{$_[$h]} [double]$nu = $u | foreach {[double]$sq = 0} {$sq += $_*$_} {[Math]::Sqrt($sq)} $u[0] -= if ($u[0] -lt 1) {$nu} else {-$nu} [double]$nu = $u | foreach {$sq = 0} {$sq += $_*$_} {[Math]::Sqrt($sq)} [double[]]$u = $u | foreach { $_/$nu} [double[][]]$v = 0..($u.Count - 1) | foreach{$i = $_; ,($u | foreach{2*$u[$i]*$_})} [double[][]]$CR = $R | foreach{$row = $_; ,@(0..$pn | foreach{$row[$_]})} [double[][]]$CQ = $Q | foreach{$row = $_; ,@(0..$pm | foreach{$row[$_]})} foreach ($i in $h..$pm) { foreach ($j in $h..$pn) { $R[$i][$j] -= $h..$pm | foreach {[double]$sum = 0} {$sum += $v[$i-$h][$_-$h]*$CR[$_][$j]} {$sum} } } if (0 -eq $h) { foreach ($i in $h..$pm) { foreach ($j in $h..$pm) { $Q[$i][$j] -= $h..$pm | foreach {$sum = 0} {$sum += $v[$i][$_]*$CQ[$_][$j]} {$sum} } } } else { $p = $h-1 foreach ($i in $h..$pm) { foreach ($j in 0..$p) { $Q[$i][$j] -= $h..$pm | foreach {$sum = 0} {$sum += $v[$i-$h][$_-$h]*$CQ[$_][$j]} {$sum} } foreach ($j in $h..$pm) { $Q[$i][$j] -= $h..$pm | foreach {$sum = 0} {$sum += $v[$i-$h][$_-$h]*$CQ[$_][$j]} {$sum} } } } } foreach ($i in 0..$pm) { foreach ($j in $i..$pm) {$Q[$i][$j],$Q[$j][$i] = $Q[$j][$i],$Q[$i][$j]} } [PSCustomObject]@{"Q" = $Q; "R" = $R}
}
function leastsquares([Double[][]]$A,[Double[]]$y) {
$QR = qr $A [Double[][]]$Q = $QR.Q [Double[][]]$R = $QR.R $m,$n = $A.count, $A[0].count [Double[]]$z = foreach ($j in 0..($m-1)) { 0..($m-1) | foreach {$sum = 0} {$sum += $Q[$_][$j]*$y[$_]} {$sum} } [Double[]]$x = @(0)*$n for ($i = $n-1; $i -ge 0; $i--) { for ($j = $i+1; $j -lt $n; $j++) { $z[$i] -= $x[$j]*$R[$i][$j] } $x[$i] = $z[$i]/$R[$i][$i] } $x
}
function polyfit([Double[]]$x,[Double[]]$y,$n) {
$m = $x.Count [Double[][]]$A = 0..($m-1) | foreach{$row = @(1) * ($n+1); ,$row} for ($i = 0; $i -lt $m; $i++) { for ($j = $n-1; 0 -le $j; $j--) { $A[$i][$j] = $A[$i][$j+1]*$x[$i] } } leastsquares $A $y
}
function show($m) {$m | foreach {write-host "$_"}}
$A = @(@(12,-51,4), @(6,167,-68), @(-4,24,-41)) $x = @(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) $y = @(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321) "polyfit " "X^2 X constant" "$(polyfit $x $y 2)" </lang>
- Output:
polyfit X^2 X constant 3 1.99999999999998 1.00000000000005
Python
<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
2
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:
<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)))</lang>
- Output:
(Intercept) x I(x^2) 1 2 3
Alternately, use poly:
<lang R>coef(lm(y ~ poly(x, 2, raw=T)))</lang>
- Output:
(Intercept) poly(x, 2, raw = T)1 poly(x, 2, raw = T)2 1 2 3
Racket
<lang 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:
Raku
(formerly Perl 6) We'll use a Clifford algebra library.
<lang perl6>use Clifford;
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;
my $J = $x1 ∧ $x2; my $I = $x0 ∧ $J;
my $I2 = ($I·$I.reversion).Real;
.say for (($y ∧ $J)·$I.reversion)/$I2, (($y ∧ ($x2 ∧ $x0))·$I.reversion)/$I2, (($y ∧ ($x0 ∧ $x1))·$I.reversion)/$I2;</lang>
- Output:
1 2 3
REXX
<lang rexx>/* REXX ---------------------------------------------------------------
- Implementation of http://keisan.casio.com/exec/system/14059932254941
- --------------------------------------------------------------------*/
xl='0 1 2 3 4 5 6 7 8 9 10' yl='1 6 17 34 57 86 121 162 209 262 321' n=11 Do i=1 To n
Parse Var xl x.i xl Parse Var yl y.i yl End
xm=0 ym=0 x2m=0 x3m=0 x4m=0 xym=0 x2ym=0 Do i=1 To n
xm=xm+x.i ym=ym+y.i x2m=x2m+x.i**2 x3m=x3m+x.i**3 x4m=x4m+x.i**4 xym=xym+x.i*y.i x2ym=x2ym+(x.i**2)*y.i End
xm =xm /n ym =ym /n x2m=x2m/n x3m=x3m/n x4m=x4m/n xym=xym/n x2ym=x2ym/n Sxx=x2m-xm**2 Sxy=xym-xm*ym Sxx2=x3m-xm*x2m Sx2x2=x4m-x2m**2 Sx2y=x2ym-x2m*ym B=(Sxy*Sx2x2-Sx2y*Sxx2)/(Sxx*Sx2x2-Sxx2**2) C=(Sx2y*Sxx-Sxy*Sxx2)/(Sxx*Sx2x2-Sxx2**2) A=ym-B*xm-C*x2m Say 'y='a'+'||b'*x+'c'*x**2' Say ' Input "Approximation"' Say ' x y y1' Do i=1 To 11
Say right(x.i,2) right(y.i,3) format(fun(x.i),5,3) End
Exit fun:
Parse Arg x Return a+b*x+c*x**2 </lang>
- Output:
y=1+2*x+3*x**2 Input "Approximation" x y y1 0 1 1.000 1 6 6.000 2 17 17.000 3 34 34.000 4 57 57.000 5 86 86.000 6 121 121.000 7 162 162.000 8 209 209.000 9 262 262.000 10 321 321.000
Ruby
<lang ruby>require 'matrix'
def regress x, y, degree
x_data = x.map { |xi| (0..degree).map { |pow| (xi**pow).to_r } }
mx = Matrix[*x_data] my = Matrix.column_vector(y)
((mx.t * mx).inv * mx.t * my).transpose.to_a[0].map(&:to_f)
end</lang> Testing: <lang ruby>p regress([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
[1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321], 2)</lang>
- Output:
[1.0, 2.0, 3.0]
Scala
- Output:
See it yourself by running in your browser Scastie (remote JVM).
<lang Scala>object PolynomialRegression extends App {
private def xy = Seq(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321).zipWithIndex.map(_.swap)
private def polyRegression(xy: Seq[(Int, Int)]): Unit = { val r = xy.indices
def average[U](ts: Iterable[U])(implicit num: Numeric[U]) = num.toDouble(ts.sum) / ts.size
def x3m: Double = average(r.map(a => a * a * a)) def x4m: Double = average(r.map(a => a * a * a * a)) def x2ym = xy.reduce((a, x) => (a._1 + x._1 * x._1 * x._2, 0))._1.toDouble / xy.size def xym = xy.reduce((a, x) => (a._1 + x._1 * x._2, 0))._1.toDouble / xy.size
val x2m: Double = average(r.map(a => a * a)) val (xm, ym) = (average(xy.map(_._1)), average(xy.map(_._2))) val (sxx, sxy) = (x2m - xm * xm, xym - xm * ym) val sxx2: Double = x3m - xm * x2m val sx2x2: Double = x4m - x2m * x2m val sx2y: Double = x2ym - x2m * ym val c: Double = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) val b: Double = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) val a: Double = ym - b * xm - c * x2m
def abc(xx: Int) = a + b * xx + c * xx * xx
println(s"y = $a + ${b}x + ${c}x^2") println(" Input Approximation") println(" x y y1") xy.foreach {el => println(f"${el._1}%2d ${el._2}%3d ${abc(el._1)}%5.1f")} }
polyRegression(xy)
}</lang>
Sidef
<lang ruby>func regress(x, y, degree) {
var A = Matrix.build(x.len, degree+1, {|i,j| x[i]**j })
var B = Matrix.column_vector(y...) ((A.transpose * A)**(-1) * A.transpose * B).transpose[0]
}
func poly(x) {
3*x**2 + 2*x + 1
}
var coeff = regress(
10.of { _ }, 10.of { poly(_) }, 2
)
say coeff</lang>
- Output:
[1, 2, 3]
Stata
See Factor variables in Stata help for explanations on the c.x##c.x syntax. <lang stata>. clear . input x y 0 1 1 6 2 17 3 34 4 57 5 86 6 121 7 162 8 209 9 262 10 321 end
. regress y c.x##c.x
Source | SS df MS Number of obs = 11
+---------------------------------- F(2, 8) = .
Model | 120362 2 60181 Prob > F = . Residual | 0 8 0 R-squared = 1.0000
+---------------------------------- Adj R-squared = 1.0000
Total | 120362 10 12036.2 Root MSE = 0
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
+----------------------------------------------------------------
x | 2 . . . . . | c.x#c.x | 3 . . . . . | _cons | 1 . . . . .
</lang>
Swift
<lang Swift>
let x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] let y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
func average(_ input: [Int]) -> Int {
return input.reduce(0, +) / input.count
}
func polyRegression(x: [Int], y: [Int]) {
let xm = average(x) let ym = average(y) let x2m = average(x.map { $0 * $0 }) let x3m = average(x.map { $0 * $0 * $0 }) let x4m = average(x.map { $0 * $0 * $0 * $0 }) let xym = average(zip(x,y).map { $0 * $1 }) let x2ym = average(zip(x,y).map { $0 * $0 * $1 })
let sxx = x2m - xm * xm let sxy = xym - xm * ym let sxx2 = x3m - xm * x2m let sx2x2 = x4m - x2m * x2m let sx2y = x2ym - x2m * ym let b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) let c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) let a = ym - b * xm - c * x2m
func abc(xx: Int) -> Int { return (a + b * xx) + (c * xx * xx) } print("y = \(a) + \(b)x + \(c)x^2\n") print(" Input Approximation") print(" x y y1") for i in 0 ..< x.count { let result = Double(abc(xx: i)) print(String(format: "%2d %3d %5.1f", x[i], y[i], result)) }
}
polyRegression(x: x, y: y) </lang>
- Output:
y = 1 + 2x + 3x^2 Input Approximation x y y1 0 1 1.0 1 6 6.0 2 17 17.0 3 34 34.0 4 57 57.0 5 86 86.0 6 121 121.0 7 162 162.0 8 209 209.0 9 262 262.0 10 321 321.0
Tcl
<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-83 BASIC
<lang ti83b>DelVar X seq(X,X,0,10) → L1 {1,6,17,34,57,86,121,162,209,262,321} → L2 QuadReg L1,L2</lang>
- Output:
y=ax²+bx+c a=3 b=2 c=1
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
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>
VBA
Excel VBA has built in capability for line estimation. <lang vb>Option Base 1 Private Function polynomial_regression(y As Variant, x As Variant, degree As Integer) As Variant
Dim a() As Double ReDim a(UBound(x), 2) For i = 1 To UBound(x) For j = 1 To degree a(i, j) = x(i) ^ j Next j Next i polynomial_regression = WorksheetFunction.LinEst(WorksheetFunction.Transpose(y), a, True, True)
End Function Public Sub main()
x = [{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}] y = [{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}] result = polynomial_regression(y, x, 2) Debug.Print "coefficients : "; For i = UBound(result, 2) To 1 Step -1 Debug.Print Format(result(1, i), "0.#####"), Next i Debug.Print Debug.Print "standard errors: "; For i = UBound(result, 2) To 1 Step -1 Debug.Print Format(result(2, i), "0.#####"), Next i Debug.Print vbCrLf Debug.Print "R^2 ="; result(3, 1) Debug.Print "F ="; result(4, 1) Debug.Print "Degrees of freedom:"; result(4, 2) Debug.Print "Standard error of y estimate:"; result(3, 2)
End Sub</lang>
- Output:
coefficients : 1, 2, 3, standard errors: 0, 0, 0, R^2 = 1 F = 7,70461300500498E+31 Degrees of freedom: 8 Standard error of y estimate: 2,79482284961344E-14
Wren
<lang ecmascript>import "/math" for Nums import "/seq" for Lst import "/fmt" for Fmt
var polynomialRegression = Fn.new { |x, y|
var xm = Nums.mean(x) var ym = Nums.mean(y) var x2m = Nums.mean(x.map { |e| e * e }) var x3m = Nums.mean(x.map { |e| e * e * e }) var x4m = Nums.mean(x.map { |e| e * e * e * e }) var z = Lst.zip(x, y) var xym = Nums.mean(z.map { |p| p[0] * p[1] }) var x2ym = Nums.mean(z.map { |p| p[0] * p[0] * p[1] })
var sxx = x2m - xm * xm var sxy = xym - xm * ym var sxx2 = x3m - xm * x2m var sx2x2 = x4m - x2m * x2m var sx2y = x2ym - x2m * ym
var b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) var c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) var a = ym - b * xm - c * x2m
var abc = Fn.new { |xx| a + b * xx + c * xx * xx }
System.print("y = %(a) + %(b)x + %(c)x^2\n") System.print(" Input Approximation") System.print(" x y y1") for (p in z) Fmt.print("$2d $3d $5.1f", p[0], p[1], abc.call(p[0]))
}
var x = List.filled(11, 0) 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)</lang>
- Output:
y = 1 + 2x + 3x^2 Input Approximation x y y1 0 1 1.0 1 6 6.0 2 17 17.0 3 34 34.0 4 57 57.0 5 86 86.0 6 121 121.0 7 162 162.0 8 209 209.0 9 262 262.0 10 321 321.0
zkl
Using the GNU Scientific Library <lang zkl>var [const] GSL=Import("zklGSL"); // libGSL (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); v :=GSL.polyFit(xs,ys,2); v.format().println(); GSL.Helpers.polyString(v).println(); GSL.Helpers.polyEval(v,xs).format().println();</lang>
- Output:
1.00,2.00,3.00 1 + 2x + 3x^2 1.00,6.00,17.00,34.00,57.00,86.00,121.00,162.00,209.00,262.00,321.00
Or, using lists:
Uses the code from Multiple regression#zkl.
Example: <lang zkl>polyfit(T(T(0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0)),
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();</lang>
- Output:
L(1,2,3)