Polynomial regression: Difference between revisions
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! let us test it |
! let us test it |
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integer, parameter :: degree = 2 |
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integer :: i |
integer :: i |
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real(8), dimension(11) :: x = (/ (i,i=0,10) /) |
real(8), dimension(11) :: x = (/ (i,i=0,10) /) |
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57, 86, 121, 162, & |
57, 86, 121, 162, & |
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209, 262, 321 /) |
209, 262, 321 /) |
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real(8), dimension( |
real(8), dimension(degree+1) :: a |
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a = polyfit(x, y, |
a = polyfit(x, y, degree) |
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write (*, '(F9.4)'), a |
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end program</pre> |
end program</pre> |
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<pre> |
<pre> |
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1.0000 |
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0.999999999999813 2.00000000000002 2.99999999999998 |
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2.0000 |
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3.0000 |
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</pre> |
</pre> |
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Revision as of 00:11, 19 December 2008
Polynomial regression
You are encouraged to solve this task according to the task description, using any language you may know.
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.
Fortran
module fitting contains function polyfit(vx, vy, d) implicit none integer, intent(in) :: d real(8), dimension(d+1) :: polyfit real(8), dimension(:), intent(in) :: vx, vy real(8), dimension(:,:), allocatable :: X real(8), dimension(:,:), allocatable :: XT real(8), dimension(:,:), allocatable :: XTX integer :: i, j integer :: n, lda, lwork integer :: info integer, dimension(:), allocatable :: ipiv real(8), 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
Example
program PolynomalFitting use fitting implicit none ! let us test it integer, parameter :: degree = 2 integer :: i real(8), dimension(11) :: x = (/ (i,i=0,10) /) real(8), dimension(11) :: y = (/ 1, 6, 17, 34, & 57, 86, 121, 162, & 209, 262, 321 /) real(8), dimension(degree+1) :: a a = polyfit(x, y, degree) write (*, '(F9.4)'), a end program
Output (lower powers first, so this seems the opposite of the Python output):
1.0000 2.0000 3.0000
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.])
Substitute back received coefficients.
>>> yf = numpy.polyval(numpy.poly1d(coeffs), x) >>> yf array([ 1., 6., 17., 34., 57., 86., 121., 162., 209., 262., 321.])
Find max absolute error.
>>> '%.1g' % max(y-yf) '1e-013'
Example
For input arrays `x' and `y':
>>> 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]
>>> p = numpy.poly1d(numpy.polyfit(x, y, deg=2), variable='N') >>> print p 2 1.085 N + 10.36 N - 0.6164
Thus we confirm once more that for already sorted sequences the considered quick sort implementation has quadratic dependence on sequence length (see Example section for Python language on Query Performance page).