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# Fitting data with uncertainties in x and y

This examples shows a general way of fitting a model to y(x) data which has uncertainties in both y and x.

For more in-depth discussion, see

  https://dx.doi.org/10.1021/acs.analchem.0c02178

    ### not including uncertainty in x:
    [[Fit Statistics]]
        # fitting method   = leastsq
        # function evals   = 62
        # data points      = 101
        # variables        = 5
        chi-square         = 344.228513
        reduced chi-square = 3.58571368
        Akaike info crit   = 133.844705
        Bayesian info crit = 146.920308
    [[Variables]]
        amp:     38.7909094 +/- 0.54820333 (1.41%) (init = 50)
        cen:     28.1858429 +/- 0.05334483 (0.19%) (init = 25)
        sig:     4.44210577 +/- 0.05938737 (1.34%) (init = 10)
        slope:   0.01260816 +/- 8.1762e-04 (6.48%) (init = 0.0001)
        offset: -13.0004948 +/- 0.02386032 (0.18%) (init = -5)
    [[Correlations]] (unreported correlations are < 0.100)
        C(slope, offset) = -0.7603
        C(amp, sig)      = +0.7056
        C(amp, offset)   = -0.2735
        C(cen, slope)    = -0.2460
        C(amp, slope)    = -0.2115
        C(sig, offset)   = -0.1924
        C(cen, offset)   = +0.1870
        C(sig, slope)    = -0.1495
    None
    ### including uncertainty in x:
    [[Fit Statistics]]
        # fitting method   = leastsq
        # function evals   = 43
        # data points      = 101
        # variables        = 5
        chi-square         = 223.406586
        reduced chi-square = 2.32715193
        Akaike info crit   = 90.1811577
        Bayesian info crit = 103.256760
    [[Variables]]
        amp:     39.2776438 +/- 0.94011531 (2.39%) (init = 50)
        cen:     28.1731192 +/- 0.12633327 (0.45%) (init = 25)
        sig:     4.44581423 +/- 0.09870523 (2.22%) (init = 10)
        slope:   0.01252275 +/- 6.7955e-04 (5.43%) (init = 0.0001)
        offset: -13.0005237 +/- 0.01980939 (0.15%) (init = -5)
    [[Correlations]] (unreported correlations are < 0.100)
        C(amp, sig)      = +0.8188
        C(slope, offset) = -0.7475
        C(amp, offset)   = -0.2037
        C(cen, slope)    = -0.1715
        C(sig, offset)   = -0.1655
        C(amp, slope)    = -0.1630
        C(cen, offset)   = +0.1381
        C(sig, slope)    = -0.1359
    None
    [0.01438036 0.52896805 1.04666251 1.58348091]

    import matplotlib.pyplot as plt
    import numpy as np

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

    # create data to be fitted
    np.random.seed(17)
    xtrue = np.linspace(0, 50, 101)
    xstep = xtrue[1] - xtrue[0]
    amp, cen, sig, offset, slope = 39, 28.2, 4.4, -13, 0.012

    ytrue = (gaussian(xtrue, amplitude=amp, center=cen, sigma=sig)
             + offset + slope * xtrue)

    ydat = ytrue + np.random.normal(size=xtrue.size, scale=0.1)

    # we add errors to x after y has been created, as if there is
    # an ideal y(x) and we have noise in both x and y.
    # we force the uncertainty away from 'normal', forcing
    # it to be smaller than the step size.
    xerr = np.random.normal(size=xtrue.size, scale=0.1*xstep)
    max_xerr = 0.8*xstep
    xerr[np.where(xerr > max_xerr)] = max_xerr
    xerr[np.where(xerr < -max_xerr)] = -max_xerr
    xdat = xtrue + xerr

    # now we assert that we know the uncertaintits in y and x
    #   we'll pick values that are reesonable but not exactly
    #   what we used to make the noise
    yerr = 0.06
    xerr = xstep

    def peak_model(params, x):
        """Model a peak with a linear background."""
        amp = params['amp'].value
        cen = params['cen'].value
        sig = params['sig'].value
        offset = params['offset'].value
        slope = params['slope'].value
        return offset + slope * x + gaussian(x, amplitude=amp, center=cen, sigma=sig)

    # objective without xerr
    def objective_no_xerr(params, x, y, yerr):
        model = peak_model(params, x)
        return (model - y) / abs(yerr)

    # objective with xerr
    def objective_with_xerr(params, x, y, yerr, xerr):
        model = peak_model(params, x)
        dmodel_dx = np.gradient(model) / np.gradient(x)
        dmodel = np.sqrt(yerr**2 + (xerr*dmodel_dx)**2)
        return (model - y) / dmodel

    # create a set of Parameters
    params = Parameters()
    params.add('amp', value=50, min=0)
    params.add('cen', value=25)
    params.add('sig', value=10)
    params.add('slope', value=1.e-4)
    params.add('offset', value=-5)

    # first fit without xerr
    mini1 = Minimizer(objective_no_xerr, params, fcn_args=(xdat, ydat, yerr))
    result1 = mini1.minimize()
    bestfit1 = peak_model(result1.params, xdat)

    mini2 = Minimizer(objective_with_xerr, params, fcn_args=(xdat, ydat, yerr, xerr))
    result2 = mini2.minimize()

    bestfit2 = peak_model(result2.params, xdat)

    print("### not including uncertainty in x:")
    print(report_fit(result1))
    print("### including uncertainty in x:")
    print(report_fit(result2))

    print(xdat[:4])

    plt.plot(xdat, ydat, 'o', label='data with noise in x and y')
    plt.plot(xtrue, ytrue, '-+', label='true data')
    plt.plot(xdat, bestfit1, label='fit, no x error')
    plt.plot(xdat, bestfit2, label='fit, with x error')
    plt.legend()
    plt.show()

    # # <end examples/doc_uncertainties_in_x_and_y.py>

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