phobson/Lognormref.ipynb

## Lognormref.ipynb
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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Sorting out the ambiguities with lognormal distributions in Python"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notes:\n",
      "\n",
      " + $\\log(x)$ denotes the natural logarith of $x$\n",
      " + This notebook refers to the `scipy.stats` modules as just `stats`\n",
      " + See here for more info: http://en.wikipedia.org/wiki/Log-normal_distribution"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "TL; DR"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Using scipy, you can easily use maximum likelihood estimation (MLE)\n",
      "do determine the parameters describing the distribution of a sample\n",
      "of data. However, you can have to get the terminology straight\n",
      "for each type of distribution. This notebook aims to clear that\n",
      "up for lognormal distributions.\n",
      "\n",
      "If you want to estimate the location $(\\mu)$ and scale $(\\sigma)$\n",
      "(as defined by Wikipedia) that describe a lognormally distributed\n",
      "sample $(X)$ with `stats.lognorm.fit`, do the following:\n",
      "\n",
      " + fit the distribution, making sure to feed a guess that scipy's `loc` is 0\n",
      " + set $\\sigma$ to scipy's shape parameter\n",
      " + set $\\mu$ to the natural log of  scipy's scale parameter\n",
      " + check to confirm that scipy's loc parameter is close to 0\n",
      " \n",
      "If you already know $\\mu$ and $\\sigma$ of your distribution, \n",
      "then you can create that distribution in the manner below:\n",
      "\n",
      "`myDist = stats.lognorm(sigma, loc=0, scale=numpy.exp(mu))`"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "More details"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy\n",
      "from matplotlib import pyplot\n",
      "import scipy\n",
      "from scipy import stats\n",
      "import seaborn\n",
      "\n",
      "seaborn.set()\n",
      "%matplotlib inline\n",
      "\n",
      "print('numpy: {}'.format(numpy.version.full_version))\n",
      "print('scipy: {}'.format(scipy.version.full_version))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "numpy: 1.9.1\n",
        "scipy: 0.14.0\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Basic Concepts"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A (perhaps over-)simplification of the lognormal distirbution would state that\n",
      "a sample $(X)$ is lognormally distributed if $\\log(X)$ is normally\n",
      "distributed. This implies that the value of a sample from a populate\n",
      "that is *truly* distributed lognormally can never be zero. You can\n",
      "get really dang close, but not actually there.\n",
      "\n",
      "So let's define the parameters that describe the distirbution of \n",
      "$\\log(X)$ as $\\mu$ (kind of the mean) and $\\sigma$ (kind of the\n",
      "standard deviation). In other words:\n",
      "\n",
      "\\begin{equation*} X = e^{(\\mu + \\sigma \\: Z)} \\end{equation*}\n",
      "\n",
      "Where $Z$ is a standard normal variable.\n",
      "\n",
      "Wikipedia calls these the location and scale.\n",
      "For our purposes, let's change that to `wikiLoc` and `wikiScale`.\n",
      "That's a pain in the ass, but it gets worse due to things out of\n",
      "my control, so please bare with me."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Sample statistics"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In addition to $\\mu$ and $\\sigma$, we can also talk about the sample mean\n",
      "and the sample standard deviation. Let's call them `wikiM` and `wikiS`. \n",
      "\n",
      "In python you can compute those very easily with e.g., `numpy.mean(X)`"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Generate a random sample from `numpy`"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "You can generate a random sample from a lognormal distribution\n",
      "using numpy and `wikiLoc` and `wikiScale`. The wording of the \n",
      "arguments (`mean` and `sigma`) is unfortunate, but I don't \n",
      "expect that to change anytime soon."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "numpy.random.seed(0)\n",
      "N = 37000\n",
      "wikiLoc = 2.25\n",
      "wikiScale = 0.875\n",
      "\n",
      "# _n denotes numpy\n",
      "X_n = numpy.random.lognormal(mean=wikiLoc, sigma=wikiScale, size=N)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Plot the histogram"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "bins = numpy.logspace(-1, 3, num=250)\n",
      "\n",
      "fig, ax = pyplot.subplots(figsize=(8, 5))\n",
      "_ = ax.hist(X_n, bins=bins, cumulative=False, histtype='step', \n",
      "            normed=True, lw=1.25, label='Hist of random numpy sample')\n",
      "\n",
      "ax.set_xscale('log')\n",
      "ax.set_xlabel(r'$X$')\n",
      "ax.set_ylabel('PDF')\n",
      "ax.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "<matplotlib.legend.Legend at 0x91d6e80>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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LHMwnkhZ21LfxcNUmGlqCeDwet+OISBpz8ph+MZC4lx6OD/nvbUsct2wGShKWvw1c52A2\nkbTR0RVm3bsNzDHlnHLMaLfjiEgac3JPvwlIPOXYa63de3Fx435tRcAeAGPMEGCKtfbpvr5QIKAz\nm9NZJvdfR/wT4S/04fV6eb56J6ter2X77jaOn+3DE28D+NLiGRTk5fDzP75CVihywPfN7/eRne3F\n58vBl5vl+HubyX03GKj/Mo+TRb8KWAzcb4yZB6xJaKsGJhtjSoFWYkP7N8bbFgJP9OeF6uqaDz+t\nuCIQKMro/quvbwWgpTVIJBLh7ZoGdta3ccoxoxlZmk803gawa1cL+b5sOjq66ApHDvi+tbQECYUi\nBINdRCNhR9/bTO+7dKf+S1+H82XNyaL/ILDIGFMVXz7PGHMO4LfW3mmMuRx4jNghhhXW2u3x9aYA\nG9//dCKDW0cwzLt1LYwaVsiZCyf0uq6taeD2B9Zy/MwRzJ40LEkJRSTdOVb0rbVRYNl+D69LaF8J\nrDzAdj91KpNIqirwZVNa5KO1vYuxFf5e151WWUpuThar3tyBGTMkSQlFZDDQHflEUsDsyQFOnD2q\nT+vOmz6cedOHY2saHE4lIoONir6IS6o376Guod3tGCKSQVT0RVxy65/W0NkVRpfei0iyqOiLuOhr\nn5x50BPxdjV0JCmNiAx2KvoiSfaHJ9bz7s4WOkPhPq3/5Ctbe2zb0xxkc20zXaEI9/1zHY2tnfjz\ncwYqqogMMir6IklWs6OZcCTKx44bR0Vpfq/r/vqbJ3X/nOV9/3GAR1fX8OjqGs4+eSKbaps568QJ\nlBT6eHNT/UDHFpFBQEVfxGHtwRANLbEb7PhysgCYNq6UJSf0fi0+QHZWz3fKvvrzR9MVjvD1nz/X\n/dipHxhLdpZXRV9EDkhFX8Rhr79Tzy8feh2AIycMHbDnzfdlkxOOHHxFEZE4JyfcEZG4wrxsTj6q\nb9fhi4g4RUVfJAk8Hg++3Cy3Y4hIhtPwvsggsG1Xa49twc4w9zzyFgBej4cLzpiRrFgikmK0py+S\n5ny5Wbxo63ocSQhHIqx+aye7Gjt4oXpnktOJSCrRnr5IGsvO8vLLy0/s07rHmACbazWVqkgmU9EX\nGUB3PPxG9+VyXzptKkdNDricSETkPRreFxlAbcEQ40cUEwxFCIWjrmbZUd/Ov1/fTm39e5P6RKJR\n1r69m41bG11MJiJuUdEX6afXNuziheqdvFC9k/Zg6H3tI4YWkpPl5bHVNfziwbU0tXYmPWNhXg47\nG9q5+2/V2Jo9AHjwkOX1cuv9a/jDv9YnPZOIuE9FX6SffvcPy4q/vckvH3qd+ubgAdc5YdYIKkoL\neMnW9fke+wPpcx+ewk0XLyAwJK/7semVpdxxxUksOWF80vOISGpQ0Rc5BB8/vvfCefZJkzhTxVVE\nUoyKvoiISIZQ0RdxWO3utu6ft9S1sL2+rZe1RUSco0v2RByUneXh+Tdq8efnMKbcT31TBwCjAn6X\nk4lIJlLRFzmIprbO7jPwC/Ny+rzdsCH53HHFyfs8dtyM4QOaTUSkP1T0RQ7iuTXb+dNTGwH44NGa\nKU9E0pdjRd8Y4wWWAzOBILDUWrsxoX0xcA0QAu621t4Vf/xqYDGQA9xurf1fpzKK9NXYCj+lfl+P\n7c+u2UbV2lq21rUwcmhhEpMdur89v4m3tzUBsGjOGKaOK3U3kIg4zskT+ZYAudba+cBVwE17G4wx\nOcDNwCLgROACY0y5MeYk4Lj4NicBExzMJ9JnWV4v2dk9f1x2N3awu7GDU+aMYcb4siQmO3Rvb2ti\nV2MHb27aw54e7jcgIoOLk8P7C4BHAay1q4wxcxLapgEbrLWNAMaY54CFwNHAWmPMQ0AxcIWD+UQG\n1LCSvINev59qzJghtLR3uR1DRJLEyaJfDDQlLIeNMV5rbSTelnjz72agBBgGjAM+Rmwv/2FgqoMZ\nRQ7LDfe9TH5uNvNmVLgdpUcPPPO22xFEJEU4WfSbgKKE5b0FH2IFP7GtCGgAdgPV1toQsM4Y02GM\nGWat3dXbCwUCRb01S4pL9f4rLPSRk+PF58smPz8Xb5aXwFA/3/js0Wzc2sgjz2+KrZOblXK/y2Wf\nOZqOztj8AFPHD6UwPyeWNTsr9vsU5OL1eigqzjuk7Kn2+0r/qP8yj5NFv4rYCXn3G2PmAWsS2qqB\nycaYUqCV2ND+jUAHcBlwszFmJFBI7ItAr+rqNEd4ugoEilK+/1pbg3R1RQgGQ7RndxIJRwi2dzJn\n0ghCnSGi0Whsnc5wyv0uw0t8QOwExLaWDtpaOmJZQ+HY79PWSSQSpbmpo9/Z06HvpGfqv/R1OF/W\nnCz6DwKLjDFV8eXzjDHnAH5r7Z3GmMuBx4idTLjCWrsd+JsxZqExZnX88Yuste7OTyoiIjJIOFb0\n48V62X4Pr0toXwmsPMB2VzqVScQJ4XCUNzfvIcvjcTuKiEivdHMekcNQVJDLtPj17aPL0/fWuuu3\nNBAlyuxJAQry9GdBZLDSp1ukH17bsJumtvcucZswspjLPz3bxUSHr6Qwl1c27OKpV7dx/fnFKvoi\ng5g+3SJ9NHvSMIaV5AEwclh63HWvJ6FQlK5Q7GKaa8+dS0NLkMtvryLYFaY9GCLfpz8NIoORPtki\nPWjrCLG9vpXd8ZnxFhw5wuVEA2fzjthZ28PLCvZ5/Pv3vkhebhbLLz/RjVgi4jAVfZEebK5t4sY/\nvArA+BHFLqcZOCcdNYq5U8sByI8P5fvzc/jeuXOxNXt46Ll33IwnIg5S0Rc5iF9940QG04n5/vwc\n/Pn7ThGcneVl3PAi6ps7XEolIsmgoi/SCw+Qm5PldgwRkQHh5Cx7IiIikkJU9EVERDKEir6IiEiG\n0DF9EXmfLTtb2La7FYhd1je2QrOxiQwGKvoiso+OzjDX3r0agCyvhw9/YIyKvsggoaIvAnSFIkSi\nsQkds7wesrMy88jXhBHFXHrWTAD8BTk8rGv2RQYVFX0R4N5H3uL5N3YA8IVTDScfNcrlRO4o8fuY\nPdnndgwRcUhm7s6IHMDsScMYWqyCJyKDl4q+SJy/IIecbN2IZ38tbV1s29VKezDkdhQROUwq+iLS\nq2fXbOe7d63ijXfq3Y4iIodJx/RFpEdf/fgMQuEo37njP25HEZEBoKIvsr9olHAkQiTqdhD3FebF\nJuYZTBMOiWQyFX2R/XR0hTn/hqeA2IQ7IiKDhYq+SA/OPW0qw8sK3I6RMt6q2UN7Z4i5U8vJy9Wf\nDpF0pBP5RHowtsLPlDFD3I6REoYW5/Hq+l3c8/dqWtt1Fr9IulLRF9nPo6tq3I6Qcq778ge48nNH\nux1DRA6TY2N0xhgvsByYCQSBpdbajQnti4FrgBBwt7X2rvjjLwON8dXettZ+xamMIvs7Y0ElbfHr\n0cuK8lxOIyIysJw8MLcEyLXWzjfGHAvcFH8MY0wOcDMwB2gDqowxfwGaAay1JzuYS6RH82YMdzuC\niIhjnBzeXwA8CmCtXUWswO81DdhgrW201nYBzwEnArOAAmPMY8aYJ+JfFkRERGQAOFn0i4GmhOVw\nfMh/b1tjQlszUAK0Ajdaa08FLgR+n7CNiIiIHAYnh/ebgMRJuL3W2kj858b92oqAPcA6YAOAtXa9\nMWY3MALY2tsLBQKa6zudpUL/+fJyyMnypkSWVBX2xr5/lw0tJFAau5RR71d6U/9lHieLfhWwGLjf\nGDMPWJPQVg1MNsaUEtu7XwjcCJxH7MS/i40xI4mNCGw/2AvV1TUPcHRJlkCgKCX6L9jRRSjLmxJZ\nUtXuhnYA6ne34gmFU6bv5NCo/9LX4XxZc3Lo/EGgwxhTRewkvv8yxpxjjDk/fhz/cuAx4N/ACmvt\ndmAFUGyMeQb4A3BewuiAiKSAq+94nm8ur3I7hogcAsf29K21UWDZfg+vS2hfCazcb5sQ8AWnMonI\noSspzOW/PjULW9PAc2u2uR1HRA6B7qUpIn3iy8niyAlDaQ/qjnwi6UpnxouIiGQI7elLRqpv6uDp\nV2ND1MWFuS6nERFJjh739I0x+x+PFxk06puD/PXfm/jPm7U8/WqvV4SKiAwavQ3vX7D3B2PM00nI\nIpJ0Jx01yu0IIiJJ09dj+sWOphARERHH6UQ+ERGRDNHbiXx+Y8xCwLPfz1EAa+0zScgnIiIiA6S3\nor8V+O8D/LyXpr+VtBONRmlo6aS5rdPtKCIiSddj0bfWnpTEHCJJEYlG+cYvdAtZEclMvV6nb4yZ\nBiwFpgJtwJvE7pNfk4RsIo65YPF0KkcUs2bjbrejiIgkTW/X6Z8GPAvkAX8D/gWUAy8aY05KSjoR\nh5QV5zG8rACPx+0k6amjK8xv/v4mVWsPOgmmiKSQ3vb0/wc41Vr7UuKDxph7iM2ad4KTwUQkNRUX\n5DJpVAlPvbyFEWUFLDhyhNuRRKSPertkz7d/wQew1q4GCp2LJCKpbOq4Ur75maNYMHOk21FEpJ96\nK/q9TaWlQVERYXNtM8sfXMsL1TvdjiIifdDX6/Qhfn1+fNnvaCqRJGrtCLGzoZ0RQzWA1R/TKsto\nbO7g5XV1bKpt5qlXtvKJEycwcWSJ29FEpAd9vU5/f1scyCKSdHk5WXg8sKc5yORRQ9yOk1bmzxzJ\n5BFFjKsoYldjO39/vobW9i63Y4lIL3or+p8DbgcmA1XAVdbaPUlJJZIkJ8wayQmzdGz6cCyMv3//\nWP2uy0lE5GB6O6Z/D/AWcAXgA25OSiIRERFxRG97+iOttd8GMMb8E3gtOZFEBlZXKEIkGjslRWeg\nikgm663od9+c3FrbZYwJJiGPyID7/ePreOa1bQCcffJEl9OIiLint6KvnSIZNGaML2NXQ7vbMURE\nXNVb0Z9hjHknYXlkwnLUWjvBwVwiA8qfn0Nzq2bWE5HM1lvRn3I4T2yM8QLLgZlAEFhqrd2Y0L4Y\nuIbYTYDuttbeldBWDrwEfMhau+5wcoiIiEhMb1PrbjrM514C5Fpr5xtjjiV2v/4lAMaYHGJXA8wh\nNntflTHmYWvtznjbr4HWw3x9EUmyXY0dbNnZwohhBWR5e7s4SETc4OSncgHwKIC1dhWxAr/XNGCD\ntbbRWtsFPAcsjLfdCPwS0PRdImnmd/9Yx7V3r6a1o7e7eIuIW5ws+sVAU8JyOD7kv7etMaGtGSgx\nxpwL1Flr/xF/XCcTiqSJn168gKs/fzQAV/7yea65a5XLiURkf70d0z9cTUBRwrLXWhuJ/9y4X1sR\n0ABcCkSNMacAs4H/NcZ83Fq7o7cXCgSKemuWFOdU/z350rvc/fAbtHV0Me/IEWRnZ+Ev9AEwZEiB\n/t0MgMT3MABUlBfx3fM+wGsbdrHq9e16j1Oc+ifzOFn0q4DFwP3GmHnAmoS2amCyMaaU2LH7hcCN\n1to/713BGPMk8NWDFXyAurrmAQ0uyRMIFDnWf7vrW4lEo3z5Y9MoK87jd1stLa2x2000NLRRV5fr\nyOtmip76bkKFn207mgmHo/pspjAnP3virMP5suZk0X8QWGSMqYovn2eMOQfwW2vvNMZcDjxG7BDD\nCmutjuHLgMvLzeID0yrcjiEikhIcK/rW2iiwbL+H1yW0rwRW9rL9yQ5FE5Ek+eu/N/HCW7HBukVz\nx3DCTE1uJOImXVMjIo5paAni8XhoC4ZoadO0uyJuc3J4XyRpbv5/r/LOttjFIhefeSRTx5W6nEj2\nqijNZ3eTLsQRSQXa05dBoSMYZsb4MtqCIcKRqNtxRERSkoq+DBqjAn68Hu1Rioj0REVfREQkQ6jo\ni4iIZAgVfRERkQyhoi8iIpIhVPRFxBEtHV3Ymga3Y4hIAl2nLyIDLjAkj1kThwIwYUQxu5s6XE4k\nIqCiLyIOMGNLMWPfu0HSqrcOOm+WiCSBhvclozz07DtuRxARcY329CVjfOqDk2gPhgAYMbTA5TQi\nIsmnoi8ZY3plmdsRRERcpeF9ERGRDKGiLyIikiE0vC+DTn1zB79/fB1b61rcjiIiklK0py+DTkt7\nF0+8tIXcnCxmjNdxfBGRvbSnL4NO1dpaAC5YPJ2CvByX04iIpA4VfRlUPnj0aKLRKNPHlZKVpYEs\nEZFEKvoyqJxzymS3I4iIpCztComIiGQIFX0REZEM4djwvjHGCywHZgJBYKm1dmNC+2LgGiAE3G2t\nvcsYkwUcMBY3AAAXfklEQVTcCUwBosCF1to3nMooIsn37s4WfvHAWgDKS/O5/NOzXU4kkjmc3NNf\nAuRaa+cDVwE37W0wxuQANwOLgBOBC4wx5cBiIGKtPR74LnC9g/lEJMl+/LuXWLHyTXY2tDOmwq8p\nd0WSzMmivwB4FMBauwqYk9A2DdhgrW201nYBzwELrbUPAV+Nr1MJ7HEwn4gk2botjYwp97Pk+PFM\nHj3E7TgiGcfJol8MNCUsh+ND/nvbGhPamoESAGtt2BhzL/Bz4D4H84lIEj1ctQmA+UeO4Izjx5Od\n5XE3kEgGcvKSvSagKGHZa62NxH9u3K+tiIS9emvtucaYK4FVxphp1tr23l4oECjqrVlS3ED0X05O\nFoWFufq3kGR9fb/PW3wELW1dABwxaSilRXn4/XlkZ3vVZy7Se595nCz6VcSO0d9vjJkHrEloqwYm\nG2NKgVZgIXCjMeYLwGhr7Y+AdiAS/1+v6uqaBzq7JEkgUDQg/dfVFaa1tVP/FpKoP303uiwfyvIB\nCHV0UdfRRUtLB6FQRH3mkoH67EnyHc6XNSeL/oPAImNMVXz5PGPMOYDfWnunMeZy4DFihxhWWGu3\nG2P+BNxrjHkayAEus9YGHcwoaeo/b9Ry58o3AThqcsDlNCIi6cGxom+tjQLL9nt4XUL7SmDlftu0\nA592KpMMHlGgwJeNGVtKNBp1O46ISFrQzXkkbeVkeykr9rkdQw5DfXOQ2x9Yy+Mvvut2FJGMoHvv\ni4grRgf8HH/kCN7cVE9ebpbbcUQygvb0Je21B0N0doXdjiH9NGXMED63aAoTRha7HUUkY6joi6s2\nbW/C1uzB1uyhqbXzkJ6juqaBmp0tA5xMRGTw0fC+uOo3f3+TF97cAcBXz5jBsdMr+rX9JxZO4PT5\nlQD4sjVELCLSG+3pi+s+PHcMxQU5h7RtXm42xQW5FBfk4tNx4bRVXbOH2/68hjUbd7kdRWRQ056+\npAaPbsmaqaZXlpGfm83zb9QyvbLM7Tgig5qKvqSsto4Qwa4w//fPdexsaCfL6+HskyYxcVQJ4bCu\nzR8sjpsxnONmDOetzZpfS8RpKvqScppaO9la18IDz77Nxq2xOZv8+Tl0dIZ5p7aJG/7vFQCG+HPd\njCkiknZU9CXlrN/SwC8efB2AEUMLuPSsmRTkZXPVr5/vXudrZx1JaZFuzCMi0h8q+pKSCvOyue3r\nC9/3+CP/qQGgcnixir6ISD+p6Eva+MyHJhOOxI7lF/j0T3cwCoUj3PHXN7qXl54+Ha9O8hQZMPrL\nKWnjhJkj3Y4gDotEo/znjR2MH1HEO9ubWXr6dLcjiQwquk5fRFLOnKnlbkcQGZRU9EVERDKEir6I\niEiGUNEXERHJECr6IiIiGUJFX0RSxubaZrcjiAxqKvqSMkLhCDU7mqlr6HA7iriguDCX9VsaKS3y\nke3VnyYRJ+g6fUkZTW2dXHfPC0DsjnySWa4456jun21NbPKddTUN5PuyGTe8yK1YIoOK/rJKyvne\nuXMJDMl3O4akgBv+7xXGVRTxvfPmuh1FZFDQGJqknHxfFgXa089oU8YM4a4rT+bskye6HUVkUHHs\nL6sxxgssB2YCQWCptXZjQvti4BogBNxtrb3LGJMD3A2MA3zAD6y1f3Uqo4ikJo/HgweI/b+IDBQn\n9/SXALnW2vnAVcBNexvixf1mYBFwInCBMaYc+BxQZ61dCHwEuN3BfCIiIhnFyaK/AHgUwFq7CpiT\n0DYN2GCtbbTWdgHPAQuB+4FrE7KFHMwnLqpau50b/+8VqjftcTuKiEjGcLLoFwNNCcvh+JD/3rbG\nhLZmoMRa22qtbTHGFBH7AvAdB/OJi3Y3drBjTxsfnV/J9Moyt+OIiGQEJ8+WagISr7PxWmsj8Z8b\n92srAvYAGGPGAA8Av7DW/qEvLxQI6HKedFNQ6GPEMD+fP20aAPc+Wo2/0AdAWZmfwLBCN+NJHzn9\n2fP7fWTnePUZd4je18zjZNGvAhYD9xtj5gFrEtqqgcnGmFKgldjQ/o3GmArgH8BF1ton+/pCdXW6\ni1e6aWsN0tUZO3pTV9dMJBKlpTUIQH19C9nRSG+bSwoIBIoc/+y1tAQJdUX0GXdAMvpPnHE4X9ac\nLPoPAouMMVXx5fOMMecAfmvtncaYy4HHiB1iWGGt3W6MuRUoAa41xuw9tn+atVa3aBskOrvCNLV1\n0hbU6RoiIsnmWNG31kaBZfs9vC6hfSWwcr9tLgMucyqTuK+6poGf3f8aAFNGl7icRkQks+gOKJJ0\n2Vkerj9/HtlZujeUHNzupg7ufaQaM2YIkWiU9vgo0bHTKygqyHU5nUh6UdGXpPN4PLrNrvTJiKEF\nzBhfxjOvbeOZ17bt02bGlqroi/STir6IpKxZk4Yxa9Iwjp1WQW19GwDjRxTxk/tecTmZSHpS0ReR\nlDd78rDun1vauwC44b6X8eVm8dOLFrgVSyTt6KCqiKQVX04WXz1jBifOHkVTa6fbcUTSioq+iKSV\nnGwvx06vYOrYIW5HEUk7KvoiIiIZQkVfREQkQ6joS1KEwhHWb2lg265Wt6PIIBIKR7n01me7b/gk\nIr3T2fuSFO3BED/63ctA7JisyOEaU+5n2ZIjeMnupL456HYckbSgoi9J9b1z5zIqoBn05PCV+H3M\nnVpO7e5WFX2RPtIulziusytMZ1ds1rysLI9uvysDbuPWRpbd/DT3P7XB7SgiKU17+uK47961il2N\nmihRnDF3WgVjyov46783EQpF3Y4jktJU9GXAdXSGaA+GAfDlxPbqT59fyexJw3TPfRlww8sKGF5W\n8L5784vI+6noy4B78pWt3P/kRgBOPmoUAIEheUwYWexmLBGRjKeiL44YFSiktMjndgwREUmgoi+H\n5Nk129i4tQmAedMrmDqudJ/2nCwvebn65yXJ9a+Xt/Dc2m2ce9o05k4tdzuOSMrRadRySNbVNPDm\npnpWv7WDbbt1wx1x38fmj+OiJUfgwUN7MER7MEQkqhP7RBKp6MshM2OGaAhfUsbEkSUcNSVATo6X\nex+p5uJbnqFuT7vbsURSioq+iAwql39qNl8/e5bbMURSkg66yoBpautkXU0DW3a2dD+2c08bwa6w\ni6kk04wp9+PPzwFg++42usIRRgf8LqcSSQ3a05cBs62uleUPvc6ajbvJ92WTl5vF5h0tRKOxE/tE\nku3nf17TPeeDiGhPXw4iGo1yxS//3b38w/PnkZuT1es2t319odOxRHpV4s/l55edwJqNu/j94+vd\njiOSMhwv+sYYL7AcmAkEgaXW2o0J7YuBa4AQcLe19q6EtmOBH1trT3Y6p/SsvinI3KnlvFC9k988\nZikrznPkdXTSlQwUr8eDPz9Hl42K7CcZY65LgFxr7XzgKuCmvQ3GmBzgZmARcCJwgTGmPN72LeBO\nQKeHp4Ajxpdx3IwKtu9u5eV1dY68xlOv6jaqMvCi0SitHV20B0NuRxFxXTK+Bi8AHgWw1q4yxsxJ\naJsGbLDWNgIYY54DFgJ/AjYAnwB+m4SMchDlpfmcMGskT7y0hSdf2Trgz/+zrx0/4M8pAtDRGeZr\nP3uW6ZWlfPMzR7kdR8RVydj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       "text": [
        "<matplotlib.figure.Figure at 0x91d6e48>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Enter ~~scipy~~ the confusion"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can fit distributions to samples using scipy. The problem is that \n",
      "the terminology is pretty ambiguous. Scipy needs three parameters to\n",
      "fully define a lognormal distribution. Here they are with `sp` as a \n",
      "prefix to differentiate them from Wikipedia's parameters:\n",
      "\n",
      " + `spShape`\n",
      " + `spLoc`\n",
      " + `spScale`.\n",
      "\n",
      "You might be thinking: `spShape` is weird, but the rest make sense."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "__They don't make sense__."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      " + `spShape` actually corresponds to `wikiScale`.\n",
      " + Remember when we said that lognormal samples can't \n",
      "   take values at or below 0? Scipy disagrees. `spLoc` is \n",
      "   used to translate the PDF of $X$ anywhere along the x-axis.\n",
      "   In other words, `spLoc` (almost) *always* should be set to 0\n",
      " + `spScale` doesn't directly coorespond to `wikiLoc`. It's\n",
      "   actually  $\\log(\\mathtt{wikiLoc})$\n",
      " \n",
      "The take away here is that if you have real-world sample that you\n",
      "think may be best described by a lognormal distribution, don't\n",
      "just throw it into `scipy.stats.lognorm.fit` and you can't just\n",
      "take the returned parameters at face value."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Fitting lognormal distributions with scipy"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "When using the `.fit` method of `scipy.stats` distributions, you\n",
      "can guide the parameters by specifying a start/guess value. For us,\n",
      "this means doing:\n",
      "\n",
      "`stats.lognorm.fit(mydata, loc=0)`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spShape, spLoc, spScale = stats.lognorm.fit(X_n, loc=0)\n",
      "ln_dist = stats.lognorm(spShape, spLoc, spScale)\n",
      "print(\"\"\"\n",
      "spShape = {:.3f}\n",
      "spLoc = {:.3f}\n",
      "spScale = {:.3f}\n",
      "\"\"\".format(spShape, spLoc, spScale))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "spShape = 0.866\n",
        "spLoc = -0.044\n",
        "spScale = 9.505\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So as we expected, this doesn't  really makes \n",
      "sense at facevalue. But information we can use \n",
      "is in there somewhere."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Another graph, this one with the theoretical distribution we just fit"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = pyplot.subplots(figsize=(8, 5))\n",
      "_ = ax.hist(X_n, bins=bins, cumulative=False, histtype='step', \n",
      "            normed=True, lw=1.25, label='Numpy random number')\n",
      "\n",
      "# theoretical PDF\n",
      "x_hat = numpy.logspace(-1, 2, num=1000)\n",
      "y_hat = ln_dist.pdf(x_hat)\n",
      "\n",
      "ax.plot(x_hat, y_hat, '-', lw=1.25, label='Fit with scipy')\n",
      "ax.set_xscale('log')\n",
      "ax.set_xlabel(r'$X$')\n",
      "ax.set_ylabel('PDF')\n",
      "ax.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "<matplotlib.legend.Legend at 0xa766358>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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4kjxLSh3T0stMSmSOzLgMMmLT2dAoc/ALMdZI0RdiUEKcHbvt07dEu6cVS1wn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\nERFJEyr6IiIiaeL/AXOILqYEpPiHAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa26dcc0>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "See? Buried in all that nonsense is really good info. Let's get it."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Estimating sample statistics"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So let's estimate our sample statistics both directly\n",
      "and from the fit theoretical formulas in the wiki article.\n",
      "\n",
      "\\begin{equation*} m = e^{\\mu + \\frac{1}{2}\\sigma^2} \\end{equation*}\n",
      "\n",
      "\\begin{equation*} s = \\sqrt{\\left(e^{\\sigma^2} - 1 \\right) e^{2\\mu + \\sigma^2}} \\end{equation*}"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "wikiM = numpy.exp(wikiLoc + 0.5*wikiScale**2)\n",
      "wikiM_est = X_n.mean()\n",
      "\n",
      "wikiS = numpy.sqrt((numpy.exp(wikiScale**2) - 1)  * numpy.exp(2*wikiLoc + wikiScale**2))\n",
      "wikiS_est = X_n.std()\n",
      "\n",
      "summary = \"\"\"\n",
      "\\tTheory\\tSample\n",
      "wikiM\\t{0:.3f}\\t{1:.3f}\n",
      "wikiS\\t{2:.3f}\\t{3:.3f}\n",
      "\"\"\"\n",
      "print(summary.format(wikiM, wikiM_est, wikiS, wikiS_est))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "\tTheory\tSample\n",
        "wikiM\t13.913\t13.804\n",
        "wikiS\t14.922\t14.748\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Everything is pretty close. Cool."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Now let's compute `wikiLoc` and `wikiScale` from our sample estimates"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sampleLoc = numpy.log((wikiM_est**2) / numpy.sqrt(wikiS_est**2 + wikiM_est**2))\n",
      "sampleScale = numpy.sqrt(numpy.log(1 + (wikiS_est/wikiM_est)**2))\n",
      "\n",
      "summary = \"\"\"\n",
      "         \\tTheory\\tSample\n",
      "wikiLoc  \\t{0:.3f}\\t{1:.3f}\n",
      "wikiScale\\t{2:.3f}\\t{3:.3f}\n",
      "\"\"\"\n",
      "\n",
      "print(summary.format(wikiLoc, sampleLoc, wikiScale, sampleScale))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "         \tTheory\tSample\n",
        "wikiLoc  \t2.250\t2.244\n",
        "wikiScale\t0.875\t0.873\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Again. Pretty close so we're cool."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The crux - making sense of the results of scipy's `.fit` method"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Recall what scipy returned:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"\"\"\n",
      "spShape = {:.3f}\n",
      "spLoc = {:.3f}\n",
      "spScale = {:.3f}\n",
      "\"\"\".format(spShape, spLoc, spScale))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "spShape = 0.866\n",
        "spLoc = -0.044\n",
        "spScale = 9.505\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "But we want the estimates of `wikiLoc` (known = 2.25) and `wikiScale` (0.875) from the scipy fit."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sp_wikiLoc = numpy.log(spScale)\n",
      "sp_wikiScale = spShape\n",
      "\n",
      "summary = \"\"\"\n",
      "         \\tTheory\\tSample\\tScipy\n",
      "wikiLoc  \\t{0:.3f}\\t{1:.3f}\\t{2:.3f}\n",
      "wikiScale\\t{3:.3f}\\t{4:.3f}\\t{5:.3f}\n",
      "\"\"\"\n",
      "print(summary.format(wikiLoc, sampleLoc, sp_wikiLoc, wikiScale, sampleScale, sp_wikiScale))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "         \tTheory\tSample\tScipy\n",
        "wikiLoc  \t2.250\t2.244\t2.252\n",
        "wikiScale\t0.875\t0.873\t0.866\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Phew. It all makes sense."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Now work backwards"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So now we can use our original `wikiLoc` and `wikiScale` \n",
      "values to freeze a `stats.lognorm` object."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "frozen_dist = stats.lognorm(wikiScale, loc=0, scale=numpy.exp(wikiLoc))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "And then plot it"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "But the theoretical and fit distributions are so close we can't really see the difference."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = pyplot.subplots(figsize=(8, 5))\n",
      "_ = ax.hist(X_n, bins=bins, cumulative=False, histtype='step', \n",
      "            normed=True, lw=1.25, alpha=0.5,\n",
      "            label='Numpy random number')\n",
      "\n",
      "# theoretical PDF\n",
      "x_hat = numpy.logspace(-1, 2, num=1000)\n",
      "y_hat_1 = ln_dist.pdf(x_hat)\n",
      "y_hat_2 = frozen_dist.pdf(x_hat)\n",
      "\n",
      "ax.plot(x_hat, y_hat_1, '-', lw=1.25, label='Fit with scipy')\n",
      "ax.plot(x_hat, y_hat_2, '--', lw=1.25, label='Theoretical')\n",
      "ax.set_xscale('log')\n",
      "ax.set_xlabel(r'$X$')\n",
      "ax.set_ylabel('PDF')\n",
      "ax.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "<matplotlib.legend.Legend at 0xa71b4e0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7VJmEGA7mirUk9PiYmTsdR2cQgsGIvc99QuIU/u18CbffDcgte0KMduF9Q7EQ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\nrBhRXU4PWypa6HZ6ibf1/eHevv5fJPf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       "text": [
        "<matplotlib.figure.Figure at 0xa922438>"
       ]
      }
     ],
     "prompt_number": 11
    }
   ],
   "metadata": {}
  }
 ]
}