{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solution-05\n",
    "\n",
    "This notebook contains the solution to the Bayesian model of 47 Ursae Majoris.\n",
    "\n",
    "Your task to find the orbital parameters of the first planet in this system, using data from table 1 of [Fischer et al 2002](http://iopscience.iop.org/article/10.1086/324336/meta).\n",
    "\n",
    "The following IPython magic command will create the data file ``47UrsaeMajoris.txt`` in the current directory:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%file 47UrsaeMajoris.txt\n",
    "date      rv     rv_err\n",
    "6959.737  -60.48 14.00\n",
    "7194.912  -53.60 7.49\n",
    "7223.798  -38.36 6.14\n",
    "7964.893  0.60 8.19\n",
    "8017.730  -28.29 10.57\n",
    "8374.771  -40.25 9.37\n",
    "8647.897  42.37 11.41\n",
    "8648.910  32.64 11.02\n",
    "8670.878  55.45 11.45\n",
    "8745.691  51.78 8.76\n",
    "8992.061  4.49 11.21\n",
    "9067.771  -14.63 7.00\n",
    "9096.734  -26.06 6.79\n",
    "9122.691  -47.38 7.91\n",
    "9172.686  -38.22 10.55\n",
    "9349.912  -52.21 9.52\n",
    "9374.964  -48.69 8.67\n",
    "9411.839  -36.01 12.81\n",
    "9481.720  -52.46 13.40\n",
    "9767.918  38.58 5.48\n",
    "9768.908  36.68 5.02\n",
    "9802.789  37.93 3.85\n",
    "10058.079  15.82 3.45\n",
    "10068.980  15.46 4.63\n",
    "10072.012  21.20 4.09\n",
    "10088.994  1.30 4.25\n",
    "10089.947  6.12 3.70\n",
    "10091.900  0.00 4.16\n",
    "10120.918  4.07 4.16\n",
    "10124.905  0.29 3.74\n",
    "10125.823  -1.87 3.79\n",
    "10127.898  -0.68 4.10\n",
    "10144.877  -4.13 5.26\n",
    "10150.797  -8.14 4.18\n",
    "10172.829  -10.79 4.43\n",
    "10173.762  -9.33 5.43\n",
    "10181.742  -23.87 3.28\n",
    "10187.740  -16.70 4.67\n",
    "10199.730  -16.29 3.98\n",
    "10203.733  -21.84 4.92\n",
    "10214.731  -24.51 3.67\n",
    "10422.018  -56.63 4.23\n",
    "10438.001  -39.61 3.91\n",
    "10442.027  -44.62 4.05\n",
    "10502.853  -32.05 4.69\n",
    "10504.859  -39.08 4.65\n",
    "10536.845  -22.46 5.18\n",
    "10537.842  -22.83 4.16\n",
    "10563.673  -17.47 4.03\n",
    "10579.697  -11.01 3.84\n",
    "10610.719  -8.67 3.52\n",
    "10793.957  37.00 3.78\n",
    "10795.039  41.85 4.80\n",
    "10978.684  36.42 5.01\n",
    "11131.066  13.56 6.61\n",
    "11175.027  -3.74 8.17\n",
    "11242.842  -21.85 5.43\n",
    "11303.712  -48.75 4.63\n",
    "11508.070  -51.65 8.37\n",
    "11536.064  -72.44 4.73\n",
    "11540.999  -57.58 5.97\n",
    "11607.916  -43.94 4.94\n",
    "11626.771  -39.14 7.03\n",
    "11627.754  -50.88 6.21\n",
    "11628.727  -51.52 5.87\n",
    "11629.832  -51.86 4.60\n",
    "11700.693  -24.58 5.20\n",
    "11861.049  14.64 5.33\n",
    "11874.068  14.15 5.75\n",
    "11881.045  18.02 4.15\n",
    "11895.068  16.96 4.60\n",
    "11906.014  11.73 4.07\n",
    "11907.011  22.83 4.38\n",
    "11909.042  23.42 3.78\n",
    "11910.955  18.34 4.33\n",
    "11914.067  15.45 5.37\n",
    "11915.048  24.05 3.82\n",
    "11916.033  23.16 3.67\n",
    "11939.969  27.53 5.08\n",
    "11946.960  21.44 4.18\n",
    "11969.902  30.99 4.58\n",
    "11971.894  38.36 5.01\n",
    "11998.779  33.82 3.93\n",
    "11999.820  27.52 3.98\n",
    "12000.858  23.40 4.07\n",
    "12028.740  37.08 4.95\n",
    "12033.746  26.28 5.24\n",
    "12040.759  31.12 3.54\n",
    "12041.719  34.04 3.45\n",
    "12042.695  31.38 3.98\n",
    "12073.723  21.81 4.73"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But first, we'll copy some relevant material from the first notebook."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preliminaries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn; seaborn.set() #nice plot formatting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Radial Velocity Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from collections import namedtuple\n",
    "params = namedtuple('params', ['T', 'e', 'K', 'V', 'omega', 'chi', 's'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy import optimize\n",
    "\n",
    "@np.vectorize\n",
    "def compute_E(M, e):\n",
    "    \"\"\"Solve Kepler's eqns for eccentric anomaly given mean anomaly\"\"\"\n",
    "    f = lambda E, M=M, e=e: E - e * np.sin(E) - M\n",
    "    return optimize.brentq(f, 0, 2 * np.pi)\n",
    "\n",
    "\n",
    "def radial_velocity(t, theta):\n",
    "    \"\"\"Compute radial velocity given orbital parameters\"\"\"\n",
    "    T, e, K, V, omega, chi = theta[:6]\n",
    "    \n",
    "    # compute mean anomaly (0 <= M < 2pi)\n",
    "    M = 2 * np.pi * ((t / T + chi) % 1)\n",
    "    \n",
    "    # solve for eccentric anomaly\n",
    "    E = compute_E(M, e)\n",
    "    \n",
    "    # compute true anomaly\n",
    "    f = 2 * np.arctan2(np.sqrt(1 + e) * np.sin(E / 2),\n",
    "                       np.sqrt(1 - e) * np.cos(E / 2))\n",
    "    \n",
    "    # compute radial velocity\n",
    "    return V - K * (np.sin(f + omega) + e * np.sin(omega))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Prior, Likelihood, and Posterior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "theta_lim = params(T=(0.2, 2000),\n",
    "                   e=(0, 1),\n",
    "                   K=(0.01, 2000),\n",
    "                   V=(-2000, 2000),\n",
    "                   omega=(0, 2 * np.pi),\n",
    "                   chi=(0, 1),\n",
    "                   s=(0.001, 100))\n",
    "theta_min, theta_max = map(np.array, zip(*theta_lim))\n",
    "\n",
    "def log_prior(theta):\n",
    "    if np.any(theta < theta_min) or np.any(theta > theta_max):\n",
    "        return -np.inf # log(0)\n",
    "    \n",
    "    # Jeffreys Prior on T, K, and s\n",
    "    return -np.sum(np.log(theta[[0, 2, 6]]))\n",
    "\n",
    "def log_likelihood(theta, t, rv, rv_err):\n",
    "    sq_err = rv_err ** 2 + theta[6] ** 2\n",
    "    rv_model = radial_velocity(t, theta)\n",
    "    return -0.5 * np.sum(np.log(sq_err) + (rv - rv_model) ** 2 / sq_err)\n",
    "\n",
    "def log_posterior(theta, t, rv, rv_err):\n",
    "    ln_prior = log_prior(theta)\n",
    "    if np.isinf(ln_prior):\n",
    "        return ln_prior\n",
    "    else:\n",
    "        return ln_prior + log_likelihood(theta, t, rv, rv_err)\n",
    "    \n",
    "def make_starting_guess(t, rv, rv_err):\n",
    "    model = LombScargleFast()\n",
    "    model.optimizer.set(period_range=theta_lim.T,\n",
    "                        quiet=True)\n",
    "    model.fit(t, rv, rv_err)\n",
    "\n",
    "    rv_range = 0.5 * (np.max(rv) - np.min(rv))\n",
    "    rv_center = np.mean(rv)\n",
    "    return params(T=model.best_period,\n",
    "                  e=0.1,\n",
    "                  K=rv_range,\n",
    "                  V=rv_center,\n",
    "                  omega=np.pi,\n",
    "                  chi=0.5,\n",
    "                  s=rv_err.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After running the above command, your data will be in a file called ``47UrsaeMajoris.txt``.\n",
    "\n",
    "An easy way to load data in this form is the [pandas](http://pandas.pydata.org) package, which implements a DataFrame object (basically, a labeled data table).\n",
    "Reading the CSV file is a one-line operation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "data = pd.read_csv('47UrsaeMajoris.txt', delim_whitespace=True)\n",
    "t, rv, rv_err = data.values.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Visualize the Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Start by Visualizing the Data\n",
    "plt.errorbar(t, rv, rv_err, fmt='.');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Compute the Periodogram"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Compute the periodogram to look for significant periodicity\n",
    "\n",
    "from gatspy.periodic import LombScargleFast\n",
    "model = LombScargleFast()\n",
    "\n",
    "model.fit(t, rv, rv_err)\n",
    "periods, power = model.periodogram_auto()\n",
    "plt.semilogx(periods, power)\n",
    "plt.xlim(0, 10000);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Initialize and run the MCMC Chain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "theta_guess = make_starting_guess(t, rv, rv_err)\n",
    "theta_guess"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import emcee\n",
    "\n",
    "ndim = len(theta_guess)  # number of parameters in the model\n",
    "nwalkers = 50  # number of MCMC walkers\n",
    "\n",
    "# start with a tight distribution of theta around the initial guess\n",
    "rng = np.random.RandomState(42)\n",
    "starting_guesses = theta_guess * (1 + 0.1 * rng.randn(nwalkers, ndim))\n",
    "\n",
    "# create the sampler; fix the random state for replicability\n",
    "sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=(t, rv, rv_err))\n",
    "sampler.random_state = rng\n",
    "\n",
    "# time and run the MCMC\n",
    "%time pos, prob, state = sampler.run_mcmc(starting_guesses, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot the chains: have they stabilized?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def plot_chains(sampler):\n",
    "    fig, ax = plt.subplots(7, figsize=(8, 10), sharex=True)\n",
    "    for i in range(7):\n",
    "        ax[i].plot(sampler.chain[:, :, i].T, '-k', alpha=0.2);\n",
    "        ax[i].set_ylabel(params._fields[i])\n",
    "\n",
    "plot_chains(sampler)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sampler.reset()\n",
    "%time pos, prob, state = sampler.run_mcmc(pos, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot_chains(sampler)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Make a Corner Plot for the Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import corner\n",
    "corner.corner(sampler.flatchain[:, :2], \n",
    "              labels=params._fields[:2]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot the Model Fits over the Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t_fit = np.linspace(t.min(), t.max(), 1000)\n",
    "rv_fit = [radial_velocity(t_fit, sampler.flatchain[i])\n",
    "          for i in rng.choice(sampler.flatchain.shape[0], 200)]\n",
    "\n",
    "plt.figure(figsize=(14, 6))\n",
    "plt.errorbar(t, rv, rv_err, fmt='.k')\n",
    "plt.plot(t_fit, np.transpose(rv_fit), '-k', alpha=0.01)\n",
    "plt.xlabel('time (days)')\n",
    "plt.ylabel('radial velocity (km/s)');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Results: Report your (joint) uncertainties on period and eccentricity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "mean = sampler.flatchain.mean(0)\n",
    "std = sampler.flatchain.std(0)\n",
    "\n",
    "print(\"Period       = {0:.0f} +/- {1:.0f} days\".format(mean[0], std[0]))\n",
    "print(\"eccentricity = {0:.2f} +/- {1:.2f}\".format(mean[1], std[1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, a simple mean and standard deviation is probably not the best summary of the data, because the posterior has multiple modes. We could isolate the strongest mode and then get a better estimate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "chain = sampler.flatchain[sampler.flatchain[:, 0] > 1050]\n",
    "mean = chain.mean(0)\n",
    "std = chain.std(0)\n",
    "\n",
    "print(\"Period       = {0:.0f} +/- {1:.0f} days\".format(mean[0], std[0]))\n",
    "print(\"eccentricity = {0:.2f} +/- {1:.2f}\".format(mean[1], std[1]))\n",
    "corner.corner(chain[:, :2], \n",
    "              labels=params._fields[:2]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a better representation of the parameter estimates for the most prominant peak."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Extra Credit\n",
    "\n",
    "If you finish early, try tackling this...\n",
    "\n",
    "One reason we see multiple modes in the posterior is that there are actually multiple planets in the system! For example, the source of the above data is a paper which actually reports *two* detected planets.\n",
    "\n",
    "Build a Bayesian model which models two planets simultaneously: can you find signals from both planets in the data?"
   ]
  }
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