fitting data with numpy

Question

Let me start by telling that what I get may not be what I expect and perhaps you can help me here. I have the following data:

>>> x
array([ 3.08,  3.1 ,  3.12,  3.14,  3.16,  3.18,  3.2 ,  3.22,  3.24,
    3.26,  3.28,  3.3 ,  3.32,  3.34,  3.36,  3.38,  3.4 ,  3.42,
    3.44,  3.46,  3.48,  3.5 ,  3.52,  3.54,  3.56,  3.58,  3.6 ,
    3.62,  3.64,  3.66,  3.68])

>>> y
array([ 0.000857,  0.001182,  0.001619,  0.002113,  0.002702,  0.003351,
    0.004062,  0.004754,  0.00546 ,  0.006183,  0.006816,  0.007362,
    0.007844,  0.008207,  0.008474,  0.008541,  0.008539,  0.008445,
    0.008251,  0.007974,  0.007608,  0.007193,  0.006752,  0.006269,
    0.005799,  0.005302,  0.004822,  0.004339,  0.00391 ,  0.003481,
    0.003095])

Now, I want to fit these data with, say, a 4 degree polynomial. So I do:

>>> coefs = np.polynomial.polynomial.polyfit(x, y, 4)
>>> ffit = np.poly1d(coefs)

Now I create a new grid for x values to evaluate the fitting function ffit:

>>> x_new = np.linspace(x[0], x[-1], num=len(x)*10)

When I do all the plotting (data set and fitting curve) with the command:

>>> fig1 = plt.figure()                                                                                           
>>> ax1 = fig1.add_subplot(111)                                                                                   
>>> ax1.scatter(x, y, facecolors='None')                                                                     
>>> ax1.plot(x_new, ffit(x_new))                                                                     
>>> plt.show()

I get the following:

fitting_data.png

What I expect is the fitting function to fit correctly (at least near the maximum value of the data). What am I doing wrong?

Thanks in advance.

Answer 1

Unfortunately, np.polynomial.polynomial.polyfit returns the coefficients in the opposite order of that for np.polyfit and np.polyval (or, as you used np.poly1d). To illustrate:

In [40]: np.polynomial.polynomial.polyfit(x, y, 4)
Out[40]: 
array([  84.29340848, -100.53595376,   44.83281408,   -8.85931101,
          0.65459882])

In [41]: np.polyfit(x, y, 4)
Out[41]: 
array([   0.65459882,   -8.859311  ,   44.83281407, -100.53595375,
         84.29340846])

In general: np.polynomial.polynomial.polyfit returns coefficients [A, B, C] to A + Bx + Cx^2 + ..., while np.polyfit returns: ... + Ax^2 + Bx + C.

So if you want to use this combination of functions, you must reverse the order of coefficients, as in:

ffit = np.polyval(coefs[::-1], x_new)

However, the documentation states clearly to avoid np.polyfit, np.polyval, and np.poly1d, and instead to use only the new(er) package.

You're safest to use only the polynomial package:

import numpy.polynomial.polynomial as poly

coefs = poly.polyfit(x, y, 4)
ffit = poly.polyval(x_new, coefs)
plt.plot(x_new, ffit)

Or, to create the polynomial function:

ffit = poly.Polynomial(coefs)    # instead of np.poly1d
plt.plot(x_new, ffit(x_new))

Answer 2

Note that you can use the Polynomial class directly to do the fitting and return a Polynomial instance.

from numpy.polynomial import Polynomial

p = Polynomial.fit(x, y, 4)
plt.plot(*p.linspace())

p uses scaled and shifted x values for numerical stability. If you need the usual form of the coefficients, you will need to follow with

pnormal = p.convert(domain=(-1, 1))