The data to be fitted is included in the file tgdata.dat and represents weight loss (in wt. %) as a function of time. The weight loss is due to hydrogen desorption from LiAlH4, a potential material for on-board hydrogen storage in future fuel cell powered vehicles (thank you Ben for mentioning hydrogen power in the Laundrette in issue #114). The data is actually the same as in example 1 of LG#114. For some reason, I suspect that the data may be explained by the following function:
f(t) = A1*(1-exp(-(k1*t)^n1)) + A2*(1-exp(-(k2*t)^n2))
There are different mathematical methods available for finding the parameters that give an optimal fit to real data, but the most widely used is probably the Levenberg-Marquandt algorithm for non-linear least-squares optimization. The algorithm works by minimizing the sum of squares (squared residuals) defined for each data point as
y is the measured dependent variable and
f(t) is the calculated. The Scipy package has the Levenberg-Marquandt algorithm included as the function
The fitting routine is in the file kinfit.py and the python code is listed below. Line numbers have been added for readability.
1 from scipy import *
2 from scipy.optimize import leastsq
3 import scipy.io.array_import
4 from scipy import gplt
6 def residuals(p, y, x):
7 err = y-peval(x,p)
8 return err
10 def peval(x, p):
11 return p*(1-exp(-(p*x)**p)) + p*(1-exp(-(p*(x))**p ))
14 data = scipy.io.array_import.read_array(filename)
16 y = data[:,1]
17 x = data[:,0]
25 pname = (['A1','A2','k1','k2','n1','n2'])
26 p0 = array([A1_0 , A2_0, k1_0, k2_0,n1_0,n2_0])
27 plsq = leastsq(residuals, p0, args=(y, x), maxfev=2000)
28 gplt.plot(x,y,'title "Meas" with points',x,peval(x,plsq),'title "Fit" with lines lt -1')
29 gplt.yaxis((0, 7))
30 gplt.legend('right bottom Left')
31 gplt.xtitle('Time [h]')
32 gplt.ytitle('Hydrogen release [wt. %]')
34 gplt.output('kinfit.png','png medium transparent size 600,400')
36 print "Final parameters"
37 for i in range(len(pname)):
38 print "%s = %.4f " % (pname[i], p0[i])
In order to run the code download the kinfit.py.txt file as kinfit.py (or use another name of your preference), also download the datafile tgdata.dat and run the script with
python kinfit.py. Besides Python, you need to have SciPy and gnuplot installed (vers. 4.0 was used throughout this article). The output of the program is plotted to the screen as shown below. A hard copy is also made. The gnuplot png option
size is a little tricky. The example shown above works with gnuplot compiled against
libgd. If you have
libpng + zlib installed, instead of
picsize and the specified width and height should not be comma separated. As shown in the figure below, the proposed model fit the data very well (sometimes you get lucky :-).
Now, let us go through the code of the example.
- Line 1-4
- all the needed packages are imported. The first is basic SciPy functionality, the second is the Levenberg-Marquandt algorithm, the third is ASCII data file import, and finally the fourth is the gnuplot interface.
- Line 6-11
- First, the function used to calculate the residuals (not the squared ones, squaring will be handled by
leastsq) is defined; second, the fitting function is defined.
- Line 13-17
- The data file name is stored, and the data file is read using
scipy.io.array_import.read_array. For convenience x (time) and y (weight loss) values are stores in separate variables.
- Line 19-26
- All parameters are given some initial guesses. An array with the names of the parameters is created for printing the results and all initial guesses are also stored in an array. I have chosen initial guesses that are quite close to the optimal parameters. However, chosing reasonable starting parameters is not always easy. In the worst case, poor initial parameters might result in the fitting procedure not being able to find a converged solution. In this case, a starting point can be to try and plot the data along with the model predictions and then "tune" the initial parameters to give just a crude description (but better than the initial parameters that did not lead to convergence), so that the model just captures the essential features of the data before starting the fitting procedure.
- Line 27
- Here the Levenberg-Marquandt algorithm (
lestsq) is called. The input parameters are the name of the function defining the residuals, the array of initial guesses, the x- and y-values of the data, and the maximum number of function evaluation are also specified. The values of the optimized parameters are stored in
plsq(actually the initial guesses in
p0are also overwritten with the optimized ones). In order to learn more about the usage of
info(optimize.leastsq)in an interactive python session - remember that the SciPy package should be imported first - or read the tutorial (see references in the end of this article).
- Line 28-34
- This is the plotting of the data and the model calculations (by evaluating the function defining the fitting model with the final parameters as input).
- Line 36-38
- The final parameters are printed to the console as:
A1 = 4.1141
A2 = 2.4435
k1 = 0.6240
k2 = 0.1227
n1 = 1.7987
n2 = 1.5120
Gnuplot also uses the Levenberg-Marquandt algorithm for its built-in curve fitting procedure. Actually, for many cases where the fitting function is somewhat simple and the data does not need heavy pre-processing, I prefer gnuplot over Python - simply due to the fact that Gnuplot also prints standard error estimates of the individual parameters. The advantage of Python over Gnuplot is the availability of many different optimization algorithms in addition to the Levenberg-Marquandt procedure, e.g. the Simplex algorithm, the Powell's method, the Quasi-Newton method, Conjugated-gradient method, etc. One only has to supply a function calculating the sum of squares (with
lestsq squaring and summing of the residuals were performed on-the-fly).