{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Example: Generating data and fitting a curve" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "source": [ "# Our first step is to import any required modules.\n", "# I will use a shortcut to import commonly-used modules in scientific computing:\n", "%pylab inline \n", "# A 'cleaner' way would be to import only those modules that we really need." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$1$. Generate a quadratic with random coefficients and plot." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0.68711024 0.12545652 0.08283838]\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "coef = np.random.random(3)\n", "print(coef)\n", "x = np.linspace(-5, 5, 100)\n", "y0 = coef[0]*x**2 + coef[1]*x + coef[2]\n", "plt.plot(x, y0);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$2$. Add in some Gaussian noise, representing random error in a measurement." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sigma = 1.0\n", "y = y0 + sigma*np.random.normal(size=len(x))\n", "plt.scatter(x, y);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$3$. Save the data to a text file. Here we will use NumPy instead of the core library." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "fname = \"quadratic_with_noise.dat\"\n", "np.savetxt(fname, [x, y])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$4$. Read the data from a text file (just to show we can), and plot once more." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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4mjPZHDF6fFlmu1G7/XjShFOM4MQ8CHAsUDTcGsyREnpPkKTZR132909qn2d6WMwjc4Cb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\ngg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x,y = np.loadtxt(fname)\n", "plt.plot(x, y, 's');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$5$. Now attempt to fit the data set using \"curve_fit\" from SciPy." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from scipy.optimize import curve_fit\n", "def func(x, a, b, c):\n", " return a*x**2 + b*x + c\n", "co, pcov = curve_fit(func, x, y)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$6$. Check that the fitted coefficients are reasonable estimates." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.6871102410208844\t0.6928882627799449\n", "\n", "0.12545652055739098\t0.12328103840410409\n", "\n", "0.08283837709236319\t0.05912538836928338\n", "\n" ] }, { "data": { "image/png": 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C/ngjTU3XSvLj3Ozas4ap183kvBPPY2OVeyvOdTd5bSJwJMCjxYgRKOASbfId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\nEGFKLmKKqCETdUSkkYk8Qgg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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for i in range(3):\n", " print(\"{}\\t{}\\n\".format(coef[i], co[i]))\n", "\n", "ytrue = coef[0]*x**2 + coef[1]*x + coef[2]\n", "yest = co[0]*x**2 + co[1]*x + co[2]\n", "plt.plot(x, y, 's')\n", "plt.plot(x, ytrue, '-')\n", "plt.plot(x, yest, '+');" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "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": "3.4.3" } }, "nbformat": 4, "nbformat_minor": 0 }