{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\nFit Using Inequality Constraint\n===============================\n\nSometimes specifying boundaries using ``min`` and ``max`` are not sufficient,\nand more complicated (inequality) constraints are needed. In the example below\nthe center of the Lorentzian peak is constrained to be between 0-5 away from\nthe center of the Gaussian peak.\n\nSee also: https://lmfit.github.io/lmfit-py/constraints.html#using-inequality-constraints\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import matplotlib.pyplot as plt\nimport numpy as np\n\nfrom lmfit import Minimizer, Parameters, report_fit\nfrom lmfit.lineshapes import gaussian, lorentzian\n\n\ndef residual(pars, x, data):\n    model = (gaussian(x, pars['amp_g'], pars['cen_g'], pars['wid_g']) +\n             lorentzian(x, pars['amp_l'], pars['cen_l'], pars['wid_l']))\n    return model - data"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Generate the simulated data using a Gaussian and Lorentzian line shape:\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "np.random.seed(0)\nx = np.linspace(0, 20.0, 601)\n\ndata = (gaussian(x, 21, 6.1, 1.2) + lorentzian(x, 10, 9.6, 1.3) +\n        np.random.normal(scale=0.1, size=x.size))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Create the fitting parameters and set an inequality constraint for ``cen_l``.\nFirst, we add a new fitting  parameter ``peak_split``, which can take values\nbetween 0 and 5. Afterwards, we constrain the value for ``cen_l`` using the\nexpression to be ``'peak_split+cen_g'``:\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "pfit = Parameters()\npfit.add(name='amp_g', value=10)\npfit.add(name='amp_l', value=10)\npfit.add(name='cen_g', value=5)\npfit.add(name='peak_split', value=2.5, min=0, max=5, vary=True)\npfit.add(name='cen_l', expr='peak_split+cen_g')\npfit.add(name='wid_g', value=1)\npfit.add(name='wid_l', expr='wid_g')\n\nmini = Minimizer(residual, pfit, fcn_args=(x, data))\nout = mini.leastsq()\nbest_fit = data + out.residual"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Performing a fit, here using the ``leastsq`` algorithm, gives the following\nfitting results:\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "report_fit(out.params)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "and figure:\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "plt.plot(x, data, 'bo')\nplt.plot(x, best_fit, 'r--', label='best fit')\nplt.legend(loc='best')\nplt.show()"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.6"
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