IPython notebook for Python bootcamp at eScience institute

finished.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = \"retina\"\n",
    "from matplotlib import rcParams\n",
    "rcParams[\"savefig.dpi\"] = 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import emcee                         # http://dfm.io/emcee\n",
    "import corner                        # https://github.com/dfm/corner.py\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as pl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "print(\"emcee:\", emcee.__version__)\n",
    "print(\"corner:\", corner.__version__)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Generate some fake data:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def mean_model(theta, x):\n",
    "    return theta[0] * x + theta[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "np.random.seed(1234)  # live coding\n",
    "\n",
    "theta_true = np.array([1.234, -0.01, 2*np.log(0.9)])  # m, b, ln(s^2)\n",
    "\n",
    "x = np.sort(np.random.uniform(-5, 5, 20))\n",
    "yerr = np.random.uniform(0.3, 0.5, len(x))\n",
    "y = mean_model(theta_true, x)\n",
    "y += np.sqrt(yerr**2 + np.exp(theta_true[2])) * np.random.randn(len(y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Plot the data and the truth:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x0 = np.linspace(-5, 5, 500)\n",
    "\n",
    "pl.errorbar(x, y, yerr=yerr, fmt=\".k\", capsize=0)\n",
    "pl.plot(x0, mean_model(theta_true, x0), \"g\", lw=1.25)\n",
    "pl.xlabel(\"x\")\n",
    "pl.ylabel(\"y\")\n",
    "pl.xlim(-5.1, 5.1)\n",
    "pl.ylim(-5.8, 5.8);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Compute the maximum likelihood line:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = np.vander(x, 2)\n",
    "print(A[:3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "ATA = np.dot(A.T, A / yerr[:, None]**2)\n",
    "theta_ml = np.linalg.solve(ATA, np.dot(A.T, y / yerr**2))\n",
    "theta_ml_cov = np.linalg.inv(ATA)\n",
    "\n",
    "theta_ml_samples = np.random.multivariate_normal(theta_ml, theta_ml_cov, 50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pl.errorbar(x, y, yerr=yerr, fmt=\".k\", capsize=0)\n",
    "pl.plot(x0, mean_model(theta_true, x0), \"g\", lw=1.25)\n",
    "for t in theta_ml_samples:\n",
    "    pl.plot(x0, mean_model(t, x0), \"r\", alpha=0.1)\n",
    "pl.xlabel(\"x\")\n",
    "pl.ylabel(\"y\")\n",
    "pl.xlim(-5.1, 5.1)\n",
    "pl.ylim(-5.8, 5.8);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The probabilistic model:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def lnlike(theta):\n",
    "    sig2 = yerr**2 + np.exp(theta[2])\n",
    "    mean = mean_model(theta, x)\n",
    "    return -0.5*np.sum((y - mean)**2 / sig2 + np.log(sig2))\n",
    "\n",
    "def lnprior(theta):\n",
    "    if np.any(theta < -10) or np.any(theta > 10):\n",
    "        return -np.inf\n",
    "    m = theta[0]\n",
    "    return -1.5 * np.log(1 + m**2)\n",
    "\n",
    "def lnprob(theta):\n",
    "    lp = lnprior(theta)\n",
    "    if not np.isfinite(lp):\n",
    "        return -np.inf\n",
    "    return lnlike(theta) + lp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Run MCMC (burn in):**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ndim, nwalkers = 3, 36\n",
    "initial_coords = 0.1*np.random.randn(nwalkers, ndim)\n",
    "\n",
    "sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob)\n",
    "coords, _, _ = sampler.run_mcmc(initial_coords, 500)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pl.plot(sampler.chain[:, :, 2].T, alpha=0.5);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Run MCMC (production):**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sampler.reset()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "coords, _, _ = sampler.run_mcmc(coords, 500)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pl.plot(sampler.chain[:, :, 2].T, alpha=0.5);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sampler.get_autocorr_time()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Plot results:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "samples = sampler.flatchain\n",
    "corner.corner(samples, truths=theta_true, quantiles=[0.16, 0.84], labels=[\"$m$\", \"$b$\", \"$\\ln s^2$\"]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pl.errorbar(x, y, yerr=yerr, fmt=\".k\", capsize=0)\n",
    "pl.plot(x0, mean_model(theta_true, x0), \"g\", lw=1.25)\n",
    "for t in samples[np.random.randint(len(samples), size=50)]:\n",
    "    pl.plot(x0, mean_model(t, x0), \"b\", alpha=0.1)\n",
    "pl.xlabel(\"x\")\n",
    "pl.ylabel(\"y\")\n",
    "pl.xlim(-5.1, 5.1)\n",
    "pl.ylim(-5.8, 5.8);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
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   "outputs": [],
   "source": []
  }
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