docs/examples/ex_mexican_hat.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "source": "# Global minimization: Skewed Mexican hat\n\nIn this example we will demonstrate the ease of performing global minimization using ``symfit``. In order to do this we will have a look at a simple skewed mexican hat potential, which has a local minimum and a global minimum. We will then use ``DifferentialEvolution`` to find the global minimum. ",
      "metadata": {
        "pycharm": {
          "metadata": false,
          "name": "#%% md\n"
        }
      }
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "pycharm": {
          "is_executing": false
        }
      },
      "outputs": [],
      "source": "from symfit import Parameter, Variable, Model, Fit, solve, diff, N, re\nfrom symfit.core.minimizers import DifferentialEvolution, BFGS\nimport numpy as np\nimport matplotlib.pyplot as plt\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pycharm": {
          "metadata": false,
          "name": "#%% md\n"
        }
      },
      "source": "First we define a model for the skewed mexican hat.\n"
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "pycharm": {
          "is_executing": false,
          "metadata": false,
          "name": "#%%\n"
        }
      },
      "outputs": [
        {
          "name": "stdout",
          "text": [
            "[y(; x) \u003d x**4 - 10*x**2 + 5*x]\n"
          ],
          "output_type": "stream"
        }
      ],
      "source": "x \u003d Parameter(\u0027x\u0027)\nx.min, x.max \u003d -100, 100\ny \u003d Variable(\u0027y\u0027)\nmodel \u003d Model({y: x**4 - 10 * x**2 + 5 * x})  # Skewed Mexican hat\nprint(model)\n"
    },
    {
      "cell_type": "markdown",
      "source": "Let us visualize what this potential looks like.\n",
      "metadata": {
        "pycharm": {
          "metadata": false
        }
      }
    },
    {
      "cell_type": "code",
      "execution_count": 16,
      "metadata": {
        "pycharm": {
          "is_executing": false,
          "metadata": false,
          "name": "#%%\n"
        }
      },
      "outputs": [
        {
          "data": {
            "text/plain": "\u003cmatplotlib.legend.Legend at 0x225cf47ab00\u003e"
          },
          "metadata": {},
          "output_type": "execute_result",
          "execution_count": 16
        },
        {
          "data": {
            "text/plain": "\u003cFigure size 432x288 with 1 Axes\u003e",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": "xdata \u003d np.linspace(-4, 4, 201)\nydata \u003d model(x\u003dxdata).y\n\nplt.axhline(0, color\u003d\u0027black\u0027)\nplt.axvline(0, color\u003d\u0027black\u0027)\nplt.plot(xdata, ydata, label\u003dr\u0027$f(x)$\u0027)\nplt.xlabel(\u0027x\u0027)\nplt.ylabel(\u0027f(x)\u0027)\nplt.ylim(1.1 * ydata.min(), 1.1 * ydata.max())\nplt.legend()\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pycharm": {
          "metadata": false
        }
      },
      "source": "Using ``sympy``, it is easy to solve the solution analytically, by finding the places where the gradient is zero. \n"
    },
    {
      "cell_type": "code",
      "execution_count": 17,
      "metadata": {
        "pycharm": {
          "is_executing": false,
          "metadata": false,
          "name": "#%%\n"
        }
      },
      "outputs": [
        {
          "data": {
            "text/plain": "[0.253248404857807, 2.09866205647752, -2.35191046133532]"
          },
          "metadata": {},
          "output_type": "execute_result",
          "execution_count": 17
        }
      ],
      "source": "sol \u003d solve(diff(model[y], x), x)\n# Give numerical value\nsol \u003d [re(N(s)) for s in sol]\nsol"
    },
    {
      "cell_type": "markdown",
      "source": "Without providing any initial guesses, ``symfit`` finds the local minimum. This is because the initial guess is set to ``1`` by default.",
      "metadata": {
        "pycharm": {
          "metadata": false,
          "name": "#%% md\n"
        }
      }
    },
    {
      "cell_type": "code",
      "execution_count": 18,
      "metadata": {
        "pycharm": {
          "is_executing": false,
          "metadata": false,
          "name": "#%%\n"
        }
      },
      "outputs": [
        {
          "name": "stdout",
          "text": [
            "exact value 2.09866205647752\nnum  value  2.09866205722533\n"
          ],
          "output_type": "stream"
        }
      ],
      "source": "fit \u003d Fit(model)\nfit_result \u003d fit.execute()\nprint(\u0027exact value\u0027, sol[1])\nprint(\u0027num  value \u0027, fit_result.value(x))\n  "
    },
    {
      "cell_type": "markdown",
      "source": "Let\u0027s use ``DifferentialEvolution`` instead.\n",
      "metadata": {
        "pycharm": {
          "metadata": false,
          "name": "#%% md\n"
        }
      }
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {
        "pycharm": {
          "is_executing": false,
          "metadata": false,
          "name": "#%%\n"
        }
      },
      "outputs": [
        {
          "name": "stdout",
          "text": [
            "exact value -2.35191046133532\nnum  value  -2.3500650028722148\n"
          ],
          "output_type": "stream"
        }
      ],
      "source": "fit \u003d Fit(model, minimizer\u003dDifferentialEvolution)\nfit_result \u003d fit.execute()\nprint(\u0027exact value\u0027, sol[2])\nprint(\u0027num  value \u0027, fit_result.value(x))\n"
    },
    {
      "cell_type": "markdown",
      "source": "Using ``DifferentialEvolution``, we find the correct global minimum. However, it is not exactly the same as the analytical solution. This is because ``DifferentialEvolution`` is expensive to perform, and therefore does not solve to high precision by default. We could demand a higher precission from ``DifferentialEvolution``, but this isn\u0027t worth the high computational cost. Instead, we will just tell ``symfit`` to perform ``DifferentialEvolution``, followed by ``BFGS``.\n",
      "metadata": {
        "pycharm": {
          "metadata": false
        }
      }
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {
        "pycharm": {
          "is_executing": false,
          "metadata": false,
          "name": "#%%\n"
        }
      },
      "outputs": [
        {
          "name": "stdout",
          "text": [
            "exact value -2.35191046133532\nnum  value  -2.351910461335324\n"
          ],
          "output_type": "stream"
        }
      ],
      "source": "fit \u003d Fit(model, minimizer\u003d[DifferentialEvolution, BFGS])\nfit_result \u003d fit.execute()\nprint(\u0027exact value\u0027, sol[2])\nprint(\u0027num  value \u0027, fit_result.value(x))"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pycharm": {
          "metadata": false,
          "name": "#%% md\n"
        }
      },
      "source": "We see that now the proper solution has been found to much higher precision.\n"
    }
  ],
  "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": "3.7.1"
    },
    "stem_cell": {
      "cell_type": "raw",
      "source": "",
      "metadata": {
        "pycharm": {
          "metadata": false
        }
      }
    }
  },
  "nbformat": 4,
  "nbformat_minor": 1
}