fitting data with numpy

Question

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?

askewchan · Accepted Answer · 2013-10-07 14:38:40Z

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))

fit and data plot

Charles Harris · Accepted Answer · 2014-04-22 04:19:09Z

38

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))

answered Apr 22, 2014 at 4:19

Charles Harris

1,0947 silver badges4 bronze badges

3 Comments

BenC Over a year ago

+1 for the conversion of coefficients, which is useful if you need to do some computation with other polynomials on the default domain. Note that this can be done in the fit() method directly, with the same domain argument.

Mark H Over a year ago

The need for convert() can also be avoided by using the value [] for the domain argument (as of version 1.5).

user115625 Over a year ago

To get the unscaled and unshifted coefficients, one can also do p.convert().coef

Gabriel Staples · Accepted Answer · 2023-10-16 22:33:00Z

Fitting data with a Chebyshev Series and Polynomial Series least squares best fit curve using numpy and matplotlib

Quick summary

The key lines you need to pay attention to (in the full code below) which perform the data fitting are, for example:

import matplotlib.pyplot as plt
from numpy.polynomial import Polynomial

# ...

cheby_series = Chebyshev.fit(x, y, deg=5)
x_cheby, y_cheby = cheby_series.linspace()

poly_series = Polynomial.fit(x, y, deg=5)
x_poly, y_poly = poly_series.linspace()

# ...

plt.plot(x_cheby, y_cheby, linewidth=6, alpha=0.5,
   label="Chebyshev Series 5th degree\nleast squares best fit curve")
plt.plot(x_poly, y_poly, 'k', linewidth=1,
   label="Polynomial Series 5th degree\nleast squares best fit curve")

Here's the output from my full program below (not from just the partial program snippet above):

Details

Even after reading all of the answers here, due to the changes in Matplotlib, and some other questions I had, I was really struggling with this until I finally found this demo!: https://numpy.org/doc/stable/reference/routines.polynomials.classes.html.

It contains a great example at the very bottom, under the "Fitting" section!:

import numpy as np
import matplotlib.pyplot as plt
from numpy.polynomial import Chebyshev as T

np.random.seed(11)
x = np.linspace(0, 2*np.pi, 20)
y = np.sin(x) + np.random.normal(scale=.1, size=x.shape)
p = T.fit(x, y, 5)
plt.plot(x, y, 'o')
xx, yy = p.linspace()
plt.plot(xx, yy, lw=2)
p.domain
p.window
plt.show()

Here is an enhanced version of it, with my modifications and some helpful comments. I modified my code below from my plot_best_fit_polynomial.py demo in my eRCaGuy_hello_world repo:

import matplotlib.pyplot as plt
from numpy.polynomial import Chebyshev
from numpy.polynomial import Polynomial

# data to fit
x =   [0.        , 0.33069396, 0.66138793, 0.99208189, 1.32277585,
       1.65346982, 1.98416378, 2.31485774, 2.64555171, 2.97624567,
       3.30693964, 3.6376336 , 3.96832756, 4.29902153, 4.62971549,
       4.96040945, 5.29110342, 5.62179738, 5.95249134, 6.28318531]
y =   [ 0.17494547,  0.29609217,  0.5657562 ,  0.57183462,  0.9685718 ,
        0.96462136,  0.86211039,  0.76726418,  0.51805246,  0.05803429,
       -0.25321856, -0.52352074, -0.66675568, -0.85965411, -1.12713934,
       -1.08134779, -0.76348274, -0.45674931, -0.32780698, -0.06834466]

# Obtain a 5th degree (order) least-squares fit curve to the x, y data using a
# Chebyshev Series
cheby_series = Chebyshev.fit(x, y, deg=5)
# Lines-space: get evenly-spaced points to plot a line; see:
# https://numpy.org/doc/stable/reference/generated/numpy.polynomial.chebyshev.Chebyshev.linspace.html
x_cheby, y_cheby = cheby_series.linspace()

# Now do the same things with a 5th order Polynomial Series fit as well!
# see: https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.Polynomial.html
poly_series = Polynomial.fit(x, y, deg=5)
x_poly, y_poly = poly_series.linspace()

# plot all the data

f1 = plt.figure()

plt.plot(x, y, 'o')
plt.plot(x_cheby, y_cheby, linewidth=6, alpha=0.5,
   label="Chebyshev Series 5th degree\nleast squares best fit curve")
plt.plot(x_poly, y_poly, 'k', linewidth=1,
   label="Polynomial Series 5th degree\nleast squares best fit curve")

plt.legend()
plt.show()

References:

Your first example graph doesn't represent the code: the code does a 5th degree polynomial fit whereas the graph says it is a chebyshev fit which is incorrect.
@user115625, I can see the confusion. I've updated my answer for clarity. I now say, "Here's the output from my full program below (not from just the partial program snippet above)". Part of the goal of the Quick summary section was to orient you to where to look in the full code in the Details section.
And the full code in the Details section is what produced that plot.

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