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# Fit Specifying Different Reduce Function

The `reduce_fcn` specifies how to convert a residual array to a scalar value for the scalar minimizers. The default value is None (i.e., “sum of squares of residual”) - alternatives are: `negentropy`, `neglogcauchy`, or a user-specified `callable`. For more information please refer to: https://lmfit.github.io/lmfit-py/fitting.html#using-the-minimizer-class

Here, we use as an example the Student’s t log-likelihood for robust fitting of data with outliers.

    import matplotlib.pyplot as plt
    import numpy as np

    import lmfit

    def resid(params, x, ydata):
        decay = params['decay'].value
        offset = params['offset'].value
        omega = params['omega'].value
        amp = params['amp'].value

        y_model = offset + amp * np.sin(x*omega) * np.exp(-x/decay)
        return y_model - ydata

Generate synthetic data with noise/outliers and initialize fitting Parameters:

    decay = 5
    offset = 1.0
    amp = 2.0
    omega = 4.0

    np.random.seed(2)
    x = np.linspace(0, 10, 101)
    y = offset + amp * np.sin(omega*x) * np.exp(-x/decay)
    yn = y + np.random.normal(size=y.size, scale=0.250)

    outliers = np.random.randint(int(len(x)/3.0), len(x), int(len(x)/12))
    yn[outliers] += 5*np.random.random(len(outliers))

    params = lmfit.create_params(offset=2.0, omega=3.3, amp=2.5,
                                 decay=dict(value=1, min=0))

Perform fits using the `L-BFGS-B` method with different `reduce_fcn`:

    method = 'L-BFGS-B'
    o1 = lmfit.minimize(resid, params, args=(x, yn), method=method)
    print("# Fit using sum of squares:\n")
    lmfit.report_fit(o1)

    # Fit using sum of squares:

    [[Fit Statistics]]
        # fitting method   = L-BFGS-B
        # function evals   = 130
        # data points      = 101
        # variables        = 4
        chi-square         = 32.1674767
        reduced chi-square = 0.33162347
        Akaike info crit   = -107.560626
        Bayesian info crit = -97.1001440
    ##  Warning: uncertainties could not be estimated:
        this fitting method does not natively calculate uncertainties
        and numdifftools is not installed for lmfit to do this. Use
        `pip install numdifftools` for lmfit to estimate uncertainties
        with this fitting method.
    [[Variables]]
        offset:  1.10392445 (init = 2)
        omega:   3.97313428 (init = 3.3)
        amp:     1.69977002 (init = 2.5)
        decay:   7.65902066 (init = 1)

    o2 = lmfit.minimize(resid, params, args=(x, yn), method=method,
                        reduce_fcn='neglogcauchy')
    print("\n\n# Robust Fit, using log-likelihood with Cauchy PDF:\n")
    lmfit.report_fit(o2)

    # Robust Fit, using log-likelihood with Cauchy PDF:

    [[Fit Statistics]]
        # fitting method   = L-BFGS-B
        # function evals   = 135
        # data points      = 101
        # variables        = 4
        chi-square         = 33.5081335
        reduced chi-square = 0.34544468
        Akaike info crit   = -103.436564
        Bayesian info crit = -92.9760818
    ##  Warning: uncertainties could not be estimated:
        this fitting method does not natively calculate uncertainties
        and numdifftools is not installed for lmfit to do this. Use
        `pip install numdifftools` for lmfit to estimate uncertainties
        with this fitting method.
    [[Variables]]
        offset:  1.02005972 (init = 2)
        omega:   3.98224426 (init = 3.3)
        amp:     1.83231442 (init = 2.5)
        decay:   5.77327309 (init = 1)

    plt.plot(x, y, 'o', label='true function')
    plt.plot(x, yn, '--*', label='with noise+outliers')
    plt.plot(x, yn+o1.residual, '-', label='sum of squares fit')
    plt.plot(x, yn+o2.residual, '-', label='robust fit')
    plt.legend()
    plt.show()

Total running time of the script: (0 minutes 0.452 seconds)

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