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copied!<p>A (slight) improvement to this solution, not accounting for a priori knowledge of the data might be the following: Take the inverse-mean of the data set and use that as the "scale factor" to be passed to the underlying leastsq() called by curve_fit(). This allows the fitter to work and returns the parameters on the original scale of the data.</p> <p>The relevant line is:</p> <pre><code>popt, pcov = curve_fit(func, xData, yData) </code></pre> <p>which becomes:</p> <pre><code>popt, pcov = curve_fit(func, xData, yData, diag=(1./xData.mean(),1./yData.mean()) ) </code></pre> <p>Here is the full example which produces this image:</p> <p><img src="https://i.stack.imgur.com/cNaP4.png" alt="curve_fit without manually rescaling the data or results"></p> <pre class="lang-python prettyprint-override"><code>from __future__ import division import numpy from scipy.optimize import curve_fit import matplotlib.pyplot as pyplot def func(x,a,b,c): return a*numpy.exp(-b*x)-c xData = numpy.array([1e-06, 2e-06, 3e-06, 4e-06, 5e-06, 6e-06, 7e-06, 8e-06, 9e-06, 1e-05, 2e-05, 3e-05, 4e-05, 5e-05, 6e-05, 7e-05, 8e-05, 9e-05, 0.0001, 0.0002, 0.0003, 0.0004, 0.0005, 0.0006, 0.0007, 0.0008, 0.0009, 0.001, 0.002, 0.003, 0.004, 0.005 , 0.006, 0.007, 0.008, 0.009, 0.01]) yData = numpy.array([6.37420666067e-09, 1.13082012115e-08, 1.52835756975e-08, 2.19214493931e-08, 2.71258852882e-08, 3.38556130078e-08, 3.55765277358e-08, 4.13818145846e-08, 4.72543475372e-08, 4.85834751151e-08, 9.53876562077e-08, 1.45110636413e-07, 1.83066627931e-07, 2.10138415308e-07, 2.43503982686e-07, 2.72107045549e-07, 3.02911771395e-07, 3.26499455951e-07, 3.48319349445e-07, 5.13187669283e-07, 5.98480176303e-07, 6.57028222701e-07, 6.98347073045e-07, 7.28699930335e-07, 7.50686502279e-07, 7.7015576866e-07, 7.87147246927e-07, 7.99607141001e-07, 8.61398763228e-07, 8.84272900407e-07, 8.96463883243e-07, 9.04105135329e-07, 9.08443443149e-07, 9.12391264185e-07, 9.150842683e-07, 9.16878548643e-07, 9.18389990067e-07]) trialX = numpy.linspace(xData[0],xData[-1],1000) # Fit a polynomial fitted = numpy.polyfit(xData, yData, 10)[::-1] y = numpy.zeros(len(trialX)) for i in range(len(fitted)): y += fitted[i]*trialX**i # Fit an exponential popt, pcov = curve_fit(func, xData, yData, diag=(1./xData.mean(),1./yData.mean()) ) yEXP = func(trialX, *popt) pyplot.figure() pyplot.plot(xData, yData, label='Data', marker='o') pyplot.plot(trialX, yEXP, 'r-',ls='--', label="Exp Fit") pyplot.plot(trialX, y, label = '10 Deg Poly') pyplot.legend() pyplot.show() </code></pre>

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