Polynomial regression

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 approximatting polynom is:

   2
3 x + 2 x + 1

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

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

Python

Interpreter: Python

Library: numpy

>>> 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).