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Note

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# Fit Specifying a Function to Compute the Jacobian

Specifying an analytical function to calculate the Jacobian can speed-up the fitting procedure.

    import matplotlib.pyplot as plt
    import numpy as np

    from lmfit import Minimizer, Parameters

    def func(pars, x, data=None):
        a, b, c = pars['a'], pars['b'], pars['c']
        model = a * np.exp(-b*x) + c
        if data is None:
            return model
        return model - data

    def dfunc(pars, x, data=None):
        a, b = pars['a'], pars['b']
        v = np.exp(-b*x)
        return np.array([v, -a*x*v, np.ones(len(x))])

    def f(var, x):
        return var[0] * np.exp(-var[1]*x) + var[2]

    params = Parameters()
    params.add('a', value=10)
    params.add('b', value=10)
    params.add('c', value=10)

    a, b, c = 2.5, 1.3, 0.8
    x = np.linspace(0, 4, 50)
    y = f([a, b, c], x)
    np.random.seed(2021)
    data = y + 0.15*np.random.normal(size=x.size)

Fit without analytic derivative:

    min1 = Minimizer(func, params, fcn_args=(x,), fcn_kws={'data': data})
    out1 = min1.leastsq()
    fit1 = func(out1.params, x)

Fit with analytic derivative:

    min2 = Minimizer(func, params, fcn_args=(x,), fcn_kws={'data': data})
    out2 = min2.leastsq(Dfun=dfunc, col_deriv=1)
    fit2 = func(out2.params, x)

Comparison of fit to exponential decay with/without analytical derivatives to model = a*exp(-b*x) + c:

    print(f'"true" parameters are: a = {a:.3f}, b = {b:.3f}, c = {c:.3f}\n\n'
          '|=========================================\n'
          '| Statistic/Parameter | Without | With   |\n'
          '|-----------------------------------------\n'
          f'|  N Function Calls   | {out1.nfev:d}      | {out2.nfev:d}     |\n'
          f'|     Chi-square      | {out1.chisqr:.4f}  | {out2.chisqr:.4f} |\n'
          f"|         a           | {out1.params['a'].value:.4f}  | {out2.params['a'].value:.4f} |\n"
          f"|         b           | {out1.params['b'].value:.4f}  | {out2.params['b'].value:.4f} |\n"
          f"|         c           | {out1.params['c'].value:.4f}  | {out2.params['c'].value:.4f} |\n"
          '------------------------------------------')

    "true" parameters are: a = 2.500, b = 1.300, c = 0.800

    |=========================================
    | Statistic/Parameter | Without | With   |
    |-----------------------------------------
    |  N Function Calls   | 39      | 12     |
    |     Chi-square      | 1.0920  | 1.0920 |
    |         a           | 2.5635  | 2.5635 |
    |         b           | 1.3585  | 1.3585 |
    |         c           | 0.8241  | 0.8241 |
    ------------------------------------------

and the best-fit to the synthetic data (with added noise) is the same for both methods:

    plt.plot(x, data, 'o', label='data')
    plt.plot(x, fit1, label='with analytical derivative')
    plt.plot(x, fit2, '--', label='without analytical derivative')
    plt.legend()

Total running time of the script: (0 minutes 0.364 seconds)

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