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# Fit Using Bounds

A major advantage of using lmfit is that one can specify boundaries on fitting parameters, even if the underlying algorithm in SciPy does not support this. For more information on how this is implemented, please refer to: https://lmfit.github.io/lmfit-py/bounds.html

The example below shows how to set boundaries using the `min` and `max` attributes to fitting parameters.

    import matplotlib.pyplot as plt
    from numpy import exp, linspace, pi, random, sign, sin

    from lmfit import Parameters, minimize
    from lmfit.printfuncs import report_fit

    p_true = Parameters()
    p_true.add('amp', value=14.0)
    p_true.add('period', value=5.4321)
    p_true.add('shift', value=0.12345)
    p_true.add('decay', value=0.01000)

    def residual(pars, x, data=None):
        argu = (x * pars['decay'])**2
        shift = pars['shift']
        if abs(shift) > pi/2:
            shift = shift - sign(shift)*pi
        model = pars['amp'] * sin(shift + x/pars['period']) * exp(-argu)
        if data is None:
            return model
        return model - data

    random.seed(0)
    x = linspace(0, 250, 1500)
    noise = random.normal(scale=2.80, size=x.size)
    data = residual(p_true, x) + noise

    fit_params = Parameters()
    fit_params.add('amp', value=13.0, max=20, min=0.0)
    fit_params.add('period', value=2, max=10)
    fit_params.add('shift', value=0.0, max=pi/2., min=-pi/2.)
    fit_params.add('decay', value=0.02, max=0.10, min=0.00)

    out = minimize(residual, fit_params, args=(x,), kws={'data': data})
    fit = residual(out.params, x)

This gives the following fitting results:

    report_fit(out, show_correl=True, modelpars=p_true)

Out:

    [[Fit Statistics]]
        # fitting method   = leastsq
        # function evals   = 74
        # data points      = 1500
        # variables        = 4
        chi-square         = 11301.3646
        reduced chi-square = 7.55438813
        Akaike info crit   = 3037.18756
        Bayesian info crit = 3058.44044
    [[Variables]]
        amp:     13.8903937 +/- 0.24412384 (1.76%) (init = 13), model_value = 14
        period:  5.44026441 +/- 0.01416175 (0.26%) (init = 2), model_value = 5.4321
        shift:   0.12464470 +/- 0.02414209 (19.37%) (init = 0), model_value = 0.12345
        decay:   0.00996351 +/- 2.0278e-04 (2.04%) (init = 0.02), model_value = 0.01
    [[Correlations]] (unreported correlations are < 0.100)
        C(period, shift) =  0.800
        C(amp, decay)    =  0.576

and shows the plot below:

    plt.plot(x, data, 'ro')
    plt.plot(x, fit, 'b')
    plt.show()

Total running time of the script: ( 0 minutes 0.176 seconds)

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