Nonlinear curve-fitting example

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Jody Dorsey
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1 Nonlinear curve-fitting example Implementation of curve-fitting in Python. Compare with results of Mathematica for same data sets: see pythontest.nb. In [1]: import scipy as sp from scipy.optimize import curve_fit import matplotlib as mpl # As of July 2017 Bucknell computers use v. 2.x import matplotlib.pyplot as plt # Following is an Ipython magic command that puts figures in the notebook. # For figures in separate windows, comment out following line and uncomment # the next line # Must come before defaults are changed. %matplotlib notebook #%matplotlib # As of Aug reverting to 1.x defaults. # In 2.x text.ustex requires dvipng, texlive-latex-extra, and texlive-fonts-recommended # which don't seem to be universal # See mpl.style.use('classic') # M.L. modifications of matplotlib defaults using syntax of v.2.0 # More info at # Changes can also be put in matplotlibrc file, or effected using mpl.rcparams[] plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures plt.rc('axes', labelsize=16, titlesize=14) plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new si Read in data In [2]: data = sp.loadtxt("sample2.dat") # Each line in file corresponds to # single data point: x,y,u x = data.t[0] # The.T gives transpose of array y = data.t[1] u = data.t[2] In [3]: # More "pythonic" reading of data # The "unpack = True" reads columns. x, y, u = sp.loadtxt("sample2.dat", unpack=true) Plot raw data 1/7

2 In [4]: # "quasi-continuous" set of x's for plotting of function: xfine = sp.linspace(min(x), max(x), 201) plt.figure(1) plt.title('data') # Pad x-range on plot: plt.xlim(min(x) *(max(x) - min(x)), max(x) *(max(x) - min(x))) plt.errorbar(x, y, yerr=u, fmt='o'); Figure 1 Define function to be fit Determine initial parameters for search In [5]: def fun(x, a, b, c, d): return a*sp.exp(-(x-b)**2/2/c**2) + d Initial "guesses" for parameters a,b,c,d In [6]: p0 = 3.5, 105., 8, /7

3 In [7]: # "quasi-continuous" set of x's for plotting of function: xfine = sp.linspace(min(x), max(x), 201) plt.figure(2) plt.title('data with initial "guess"') # Pad x-range on plot: plt.xlim(min(x) *(max(x) - min(x)), max(x) *(max(x) - min(x))) plt.errorbar(x, y, yerr=u, fmt='o') plt.plot(xfine, fun(xfine, *p0)); Figure 2 Fit data Plot fit-function with optimized parameters In [8]: popt, pcov = sp.optimize.curve_fit(fun, x, y, p0, sigma=u) 3/7

4 In [9]: # "quasi-continuous" set of x's for plotting of function: plt.figure(3) xfine = sp.linspace(min(x), max(x), 201) plt.title('data with best fit') # Pad x-range on plot: plt.xlim(min(x) *(max(x) - min(x)), max(x) *(max(x) - min(x))) plt.errorbar(x, y, yerr=u, fmt='o') plt.plot(xfine, fun(xfine, *popt)); Figure 3 In [10]: popt # Best fit parameters Out[10]: array([ , , , ]) In [11]: pcov # Covariance matrix Out[11]: array([[ e-02, e-05, e-03, e-03], [ e-05, e-02, e-05, e-05], [ e-03, e-05, e-01, e-02], [ e-03, e-05, e-02, e-03]]) In [12]: for i in range(len(popt)): print("parameter", i,"=", popt[i], "+/-", sp.sqrt(pcov[i,i])) parameter 0 = / parameter 1 = / parameter 2 = / parameter 3 = / For nicer formatting of output, can use features of sympy. 4/7

5 NOTE: Matrix is from sympy; it's not the same as sp.matrix In [13]: from sympy import * from sympy import init_printing init_printing() In [14]: Matrix(pcov) Out[14]: NOTE: absolute_sigma=true is equivalent to Mathematica VarianceEstimatorFunction-> (1&). False gives covariance matrix based on estimated errors in data (weights are just relative). In [15]: popt, pcov2 = sp.optimize.curve_fit(fun, x, y, p0, sigma=u, absolute_sigma=true) In [16]: Matrix(pcov2) Out[16]: /7

6 In [17]: plt.figure(4) plt.title('normalized residuals') plt.grid() plt.scatter(x, (fun(x, *popt) - y)/u); Figure 4 Calculation of reduced chi-square parameter: χ 2 R 1 =, N c N ( y i f ( x i )) 2 i=1 σi 2 In [18]: sp.sum((y - fun(x, *popt))**2/u**2)/(len(data) - 4) Out[18]: Version details version_information is from J.R. Johansson (jrjohansson at gmail.com) See Introduction to scientific computing with Python: Computing-with-Python.ipynb ( for more information and instructions for package installation. If version_information has been installed system wide (as it has been on Bucknell linux computers with shared file systems), continue with next cell as written. If not, comment out top line in next cell and uncomment the second line 6/7

7 In [19]: %load_ext version_information #%install_ext Loading extensions from ~/.ipython/extensions is deprecated. We recommend managing ex tensions like any other Python packages, in site-packages. In [20]: %version_information scipy, matplotlib, sympy Out[20]: Software Version Python bit [GCC (Red Hat )] IPython OS Linux el7.x86_64 x86_64 with redhat 7.2 Maipo scipy matplotlib sympy 1.1 Tue Aug 01 11:01: EDT In [ ]: 7/7

For examples, documentation, tutorials, etc, see Astropy at ( # For retrieving an image from a URL

Astronomy example 1 Downloading images and writing FITS files For examples, documentation, tutorials, etc, see Astropy at http://www.astropy.org (http://www.astropy.org) In [1]: import scipy as sp import