Polynomial regression
You are encouraged to solve this task according to the task description, using any language you may know.
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 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> Sample 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
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++) {
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 of the test:
1.000000 2.000000 3.000000
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
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>
Haskell
Uses module Matrix.LU from hackageDB DSP <lang haskell>import Data.List import Data.Array import Control.Monad 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>
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>
Octave
<lang octave>x = [0:10]; y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]; coeffs = polyfit(x, y, 2)</lang>
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
R has several tools for fitting. Here we use a generalized nonlinear least squares method with the polynomial as model:
<lang R>library(nlme) 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) fitted <- gnls(y ~ c0*x^2 + c1*x + c2, start=list(c0=1, c1=1, c2=0)) print(paste(fitted$coeff1, "*x^2 + ",
fitted$coeff2, "*x + ", fitted$coeff3))
- get several info about the fitting process
print(summary(fitted))</lang>
The "base" nls could be used with algorithm "port", since the default Gauss-Newton has problems recognizing the convergence:
<lang R>nls(y ~ c0*x^2 + c1*x + c2, start=list(c0=1, c1=1, c2=0), trace=TRUE)</lang>
gives
5.364254e-29 : 3 2 1 Error in nls(y ~ c0 * x^2 + c1 * x + c2, start = list(c0 = 1, c1 = 1, : number of iterations exceeded maximum of 50
And even increasing the maximum possible iterations, it does not finish properly (even if the result, as we can see with the trace option, is reached!). Instead, the
<lang R>nls(y ~ c0*x^2 + c1*x + c2, start=list(c0=1, c1=1, c2=0), algorithm="port")</lang>
works fine.
Tcl
(which includes a package for performing linear algebra operations)
<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
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>