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# Fit with Algebraic Constraint

Out:

    [[Fit Statistics]]
        # fitting method   = leastsq
        # function evals   = 53
        # data points      = 601
        # variables        = 6
        chi-square         = 71878.3055
        reduced chi-square = 120.803875
        Akaike info crit   = 2887.26503
        Bayesian info crit = 2913.65660
    [[Variables]]
        amp_g:       21.1877703 +/- 0.32191821 (1.52%) (init = 10)
        cen_g:       8.11125922 +/- 0.01162984 (0.14%) (init = 9)
        wid_g:       1.20925846 +/- 0.01170853 (0.97%) (init = 1)
        amp_tot:     30.6003740 +/- 0.36481391 (1.19%) (init = 20)
        amp_l:       9.41260369 +/- 0.61672675 (6.55%) == 'amp_tot - amp_g'
        cen_l:       9.61125922 +/- 0.01162984 (0.12%) == '1.5+cen_g'
        wid_l:       2.41851692 +/- 0.02341706 (0.97%) == '2*wid_g'
        line_slope:  0.49615727 +/- 0.00170178 (0.34%) (init = 0)
        line_off:    0.04128588 +/- 0.02448064 (59.30%) (init = 0)
    [[Correlations]] (unreported correlations are < 0.100)
        C(amp_g, wid_g)         =  0.866
        C(amp_g, cen_g)         =  0.750
        C(line_slope, line_off) = -0.714
        C(cen_g, amp_tot)       = -0.695
        C(cen_g, wid_g)         =  0.623
        C(amp_g, amp_tot)       = -0.612
        C(amp_tot, line_off)    = -0.588
        C(wid_g, amp_tot)       = -0.412
        C(cen_g, line_off)      =  0.387
        C(amp_g, line_off)      =  0.183
        C(amp_g, line_slope)    =  0.183
        C(wid_g, line_slope)    =  0.174

    import matplotlib.pyplot as plt
    from numpy import linspace, random

    from lmfit import Minimizer, Parameters
    from lmfit.lineshapes import gaussian, lorentzian
    from lmfit.printfuncs import report_fit

    def residual(pars, x, sigma=None, data=None):
        yg = gaussian(x, pars['amp_g'], pars['cen_g'], pars['wid_g'])
        yl = lorentzian(x, pars['amp_l'], pars['cen_l'], pars['wid_l'])

        slope = pars['line_slope']
        offset = pars['line_off']
        model = yg + yl + offset + x*slope

        if data is None:
            return model
        if sigma is None:
            return model - data
        return (model - data) / sigma

    random.seed(0)
    x = linspace(0.0, 20.0, 601)

    data = (gaussian(x, 21, 8.1, 1.2) +
            lorentzian(x, 10, 9.6, 2.4) +
            random.normal(scale=0.23, size=x.size) +
            x*0.5)

    pfit = Parameters()
    pfit.add(name='amp_g', value=10)
    pfit.add(name='cen_g', value=9)
    pfit.add(name='wid_g', value=1)
    pfit.add(name='amp_tot', value=20)
    pfit.add(name='amp_l', expr='amp_tot - amp_g')
    pfit.add(name='cen_l', expr='1.5+cen_g')
    pfit.add(name='wid_l', expr='2*wid_g')
    pfit.add(name='line_slope', value=0.0)
    pfit.add(name='line_off', value=0.0)

    sigma = 0.021  # estimate of data error (for all data points)

    myfit = Minimizer(residual, pfit,
                      fcn_args=(x,), fcn_kws={'sigma': sigma, 'data': data},
                      scale_covar=True)

    result = myfit.leastsq()
    init = residual(pfit, x)
    fit = residual(result.params, x)

    report_fit(result)

    plt.plot(x, data, 'r+')
    plt.plot(x, init, 'b--', label='initial fit')
    plt.plot(x, fit, 'k-', label='best fit')
    plt.legend(loc='best')
    plt.show()

Total running time of the script: ( 0 minutes 0.211 seconds)

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