CalibrateLargeNPL.ipynb

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   "source": [
    "## Tests of the PowerLaw package for large N (1000, 5000)\n",
    "\n",
    "https://github.com/jeffalstott/powerlaw\n",
    "\n",
    "We generate a empirical PowerLaw distribution\n",
    "\n",
    "$$p(x)\\sim\\dfrac{1}{x^{\\alpha}},\\;\\;\\;\\;\\alpha=1+\\mu$$\n",
    "\n",
    " using the numpy Pareto method\n",
    "\n",
    "https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.random.pareto.html\n",
    "\n",
    "\n",
    "and test how well the _PowerLaw.fit(data)_ method works on this data \n",
    "\n",
    "We find, as expected from the literature, \n",
    "\n",
    "http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0170920\n",
    "\n",
    "this, the _Clauset-Shalizi-Newman_ estimator, only works well \n",
    "\n",
    "for $\\mu\\in[1.5,3.5]$ and really for large N, of order 5000 or more\n",
    "\n"
   ]
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      "/Users/charlesmartin14/anaconda3/lib/python3.6/importlib/_bootstrap.py:219: RuntimeWarning: compiletime version 3.5 of module 'tensorflow.python.framework.fast_tensor_util' does not match runtime version 3.6\n",
      "  return f(*args, **kwds)\n",
      "/Users/charlesmartin14/.local/lib/python3.6/site-packages/h5py/__init__.py:34: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.\n",
      "  from ._conv import register_converters as _register_converters\n"
     ]
    },
    {
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     "text": [
      "1.14.2\n",
      "0.19.1\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import tensorflow as tf\n",
    "import pickle, time\n",
    "from copy import deepcopy\n",
    "from shutil import copy\n",
    "from tqdm import tqdm_notebook as tqdm\n",
    "import pandas as pd\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "import powerlaw\n",
    "\n",
    "import sklearn\n",
    "from sklearn.decomposition import TruncatedSVD\n",
    "from sklearn.random_projection import sparse_random_matrix\n",
    "print(np.__version__)\n",
    "print(sklearn.__version__)"
   ]
  },
  {
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   "source": [
    "def get_pareto(mu, M=5000, Q=1):\n",
    "    W = np.random.pareto(mu,size=(M,int(Q*M)))\n",
    "    N, M = np.max(W.shape),np.min(W.shape)\n",
    "    #evals, _ = RMT_Util.eigenspectrum(W)\n",
    "    \n",
    "    X = (1/N)*np.dot(W,W.T)\n",
    "    svd = TruncatedSVD(n_components=M-1, n_iter=7, random_state=42)\n",
    "    svd.fit(X)  \n",
    "    evals = svd.singular_values_\n",
    "    #evals = sv*sv\n",
    "\n",
    "    return W, evals"
   ]
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      "Calculating best minimal value for power law fit\n",
      "/Users/charlesmartin14/anaconda3/lib/python3.6/site-packages/powerlaw.py:697: RuntimeWarning: invalid value encountered in true_divide\n",
      "  (Theoretical_CDF * (1 - Theoretical_CDF))\n"
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      "data sample: mu+1 3.0824156817525346 alpha 2.3054496799277366\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.113095833368644 alpha 2.60079849716682\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.156308489632096 alpha 2.809400267483465\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.158490837708582 alpha 2.562490414114895\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.214055671158138 alpha 2.402440404223756\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.2340903176828197 alpha 2.64126644072655\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.2517622749977924 alpha 2.878390024864441\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.260439951383759 alpha 2.8339196505929376\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.268411589756883 alpha 2.6271515643925545\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.274893506909943 alpha 3.859523862729182\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.3229377168805874 alpha 2.7565606078868514\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.3519211472117485 alpha 2.2715313502753016\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.357493676086145 alpha 2.3013711836828676\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.3737820699137586 alpha 2.48870448138749\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.437022134936851 alpha 2.5443644390140374\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.4386750589232435 alpha 2.4921383457930197\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.484522305462149 alpha 2.857077402445907\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.498940434384458 alpha 3.227832808256315\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.511999037953056 alpha 2.896891816682904\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.5341250283212227 alpha 2.8050396746123827\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.6075604995150847 alpha 3.422194113759475\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.6123069496836258 alpha 2.750556640037124\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.674277544958671 alpha 2.551079512324047\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.6762309618868385 alpha 2.439664745309165\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.6781482726723445 alpha 3.3654756646573327\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7010814624708783 alpha 2.5013674561478414\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7120279204129405 alpha 2.531453409080566\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7258061046578996 alpha 2.972369374090385\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7878369662963998 alpha 2.923461670066226\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.861586390225364 alpha 2.8847235786704912\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.8950243812234784 alpha 3.4549503591430537\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.901438139631874 alpha 2.9716656202871516\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.9030769827643113 alpha 3.1708750121659106\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.9206838553970904 alpha 2.647414830290489\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.9411648464936633 alpha 2.558052170266871\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.033379036543615 alpha 2.8954374445469266\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.037473868711566 alpha 2.5062224729641107\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.075174269689864 alpha 3.369237559221706\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.107966895580606 alpha 2.528098094372604\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.139754565816537 alpha 2.5139418889737852\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.178398189242042 alpha 2.600911718250117\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.191513164272639 alpha 2.7512935664979428\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.244255040626495 alpha 2.939347783926018\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.311208830085786 alpha 2.672031686542845\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.3434085823237805 alpha 3.464915483845866\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.39118526457948 alpha 2.9176093985052196\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.434271333710475 alpha 2.487865818097975\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.4400189761166216 alpha 2.9581305989537308\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.451055181969007 alpha 2.8188429854185992\n",
      "\n"
     ]
    }
   ],
   "source": [
    "data_alphas = []\n",
    "M=1000\n",
    "#mus = [0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,2.0,  2.1, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 , 3.0, 3.1, \n",
    "#       3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 4, 4.5, 5.0, 5.5]\n",
    "mus=np.sort(3*np.random.random_sample(100)+.5)\n",
    "for mu in tqdm(mus):\n",
    "    \n",
    "    data = np.random.pareto(mu,size=(M))\n",
    "    \n",
    "    fit = powerlaw.Fit(data)\n",
    "    data_alphas.append(fit.alpha)    \n",
    "    print(\"data sample: mu+1 {} alpha {}\".format(mu+1, fit.alpha))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-26T07:14:15.667988Z",
     "start_time": "2018-08-26T07:14:15.664456Z"
    }
   },
   "outputs": [],
   "source": [
    "mup1 = np.array(mus)+1.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### This plot should be a straight line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-26T07:14:15.859105Z",
     "start_time": "2018-08-26T07:14:15.671390Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'PowerLaw Test: fit vs actual')"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(mup1, data_alphas, color='green', label='fit.alpha')\n",
    "plt.xlabel(\"mu+1\")\n",
    "plt.ylabel('alpha')\n",
    "plt.legend()\n",
    "plt.title(\"PowerLaw Test: fit vs actual\")\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-26T09:12:05.148937Z",
     "start_time": "2018-08-26T07:14:15.862285Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n",
      "/Users/charlesmartin14/anaconda3/lib/python3.6/site-packages/powerlaw.py:697: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  (Theoretical_CDF * (1 - Theoretical_CDF))\n",
      "/Users/charlesmartin14/anaconda3/lib/python3.6/site-packages/powerlaw.py:697: RuntimeWarning: invalid value encountered in true_divide\n",
      "  (Theoretical_CDF * (1 - Theoretical_CDF))\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 1.5225460153202397 alpha 1.515931182990634\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n",
      "/Users/charlesmartin14/anaconda3/lib/python3.6/site-packages/powerlaw.py:1175: RuntimeWarning: overflow encountered in double_scalars\n",
      "  return (self.alpha-1) * self.xmin**(self.alpha-1)\n",
      "/Users/charlesmartin14/anaconda3/lib/python3.6/site-packages/powerlaw.py:822: RuntimeWarning: invalid value encountered in multiply\n",
      "  likelihoods = f*C\n",
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 1.5325249231462794 alpha 1.5464911706494595\n",
      "data sample: mu+1 1.5539652722068893 alpha 1.5332052687902054\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 1.606978349787021 alpha 1.633460421506082\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 1.6132514699563432 alpha 1.5877227690330993\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 1.637760067154281 alpha 1.6154100042643726\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.544740545904994 alpha 2.951945129460385\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.567298050933201 alpha 3.0826968334204343\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.5798094761873744 alpha 2.9783602429362954\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.592600038138581 alpha 3.422556751539184\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.649360000279465 alpha 3.1624229062511775\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.6876747677158805 alpha 2.748784169272076\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7038984996317508 alpha 3.2050537521823754\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.737320403673528 alpha 2.893390585232854\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7830599578730437 alpha 3.11485141917704\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7938568889390254 alpha 2.803097510777516\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.8039773260902123 alpha 3.039006815090768\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.826972418727313 alpha 2.9876507545479667\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.8355128612775062 alpha 3.1462874214267984\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.841413351771315 alpha 3.0269798560556023\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.8649803511380356 alpha 2.837596306331771\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.890298486678627 alpha 2.987595107923151\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.9267558606642985 alpha 2.9245935626143016\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.98673598823215 alpha 3.0507835131510554\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.023559685054435 alpha 3.2117304208264303\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.1056372402011405 alpha 2.770240442574036\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.118054120908574 alpha 2.8851943031675145\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.1194255340507215 alpha 3.4176108773668297\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.1408838507556265 alpha 3.5289238852641125\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.163223752445127 alpha 3.3075503867794693\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.163874879258564 alpha 3.729703048964667\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.212663544414928 alpha 2.8675172364231827\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.226871736048526 alpha 2.808164288304651\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.322530095077836 alpha 3.3384090918664993\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.452948544541012 alpha 3.088247448331082\n",
      "\n"
     ]
    }
   ],
   "source": [
    "data_alphas5k = []\n",
    "M=5000\n",
    "mus=np.sort(3*np.random.random_sample(100)+.5)\n",
    "for mu in tqdm(mus):\n",
    "    \n",
    "    data = np.random.pareto(mu,size=(M))\n",
    "    \n",
    "    fit = powerlaw.Fit(data, xmax=np.max(data))\n",
    "    data_alphas5k.append(fit.alpha)    \n",
    "    print(\"data sample: mu+1 {} alpha {}\".format(mu+1, fit.alpha))\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-26T09:12:05.254603Z",
     "start_time": "2018-08-26T09:12:05.150652Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'PowerLaw Test (w/xmax): fit vs actual')"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mup1 = np.array(mus)+1.0\n",
    "\n",
    "plt.scatter(mup1[0:len(data_alphas5k)], data_alphas5k, color='green', label='fit.alpha')\n",
    "plt.xlabel(\"mu+1\")\n",
    "plt.ylabel('alpha')\n",
    "plt.legend()\n",
    "plt.title(\"PowerLaw Test (w/xmax): fit vs actual\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Try KS method instead of Maximum Likeliehood"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-26T09:19:56.555789Z",
     "start_time": "2018-08-26T09:12:05.256449Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n",
      "/Users/charlesmartin14/anaconda3/lib/python3.6/site-packages/powerlaw.py:697: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  (Theoretical_CDF * (1 - Theoretical_CDF))\n",
      "/Users/charlesmartin14/anaconda3/lib/python3.6/site-packages/powerlaw.py:697: RuntimeWarning: invalid value encountered in true_divide\n",
      "  (Theoretical_CDF * (1 - Theoretical_CDF))\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 1.5076815184908532 alpha 1.452288597166365\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n",
      "/Users/charlesmartin14/anaconda3/lib/python3.6/site-packages/powerlaw.py:698: RuntimeWarning: overflow encountered in double_scalars\n",
      "  )[1:]\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "data sample: mu+1 3.617956961039144 alpha 2.4266403795570968\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.646188202025778 alpha 2.423324388616212\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.6639249572487746 alpha 2.0997736441535766\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.6982690904163107 alpha 2.681693383361056\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7020852739952255 alpha 2.7002973458355184\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7628903392290907 alpha 2.25282702139942\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.7657587513057718 alpha 2.338267825091893\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.865067461513646 alpha 2.267213477668883\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.871420761447253 alpha 2.5817557580960653\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.9002660292145723 alpha 2.092590641313155\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.9289025437628093 alpha 2.6805027808286535\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.9454220387972665 alpha 2.511990188909106\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 3.9838043775116505 alpha 2.5005373979581496\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.033744897834396 alpha 2.8561370504500814\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.072412515793143 alpha 2.7838412216083617\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.159439539275463 alpha 3.3376979858234246\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.1767872754357 alpha 2.556060934364606\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.205465581018626 alpha 2.446653044534247\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.206404126410209 alpha 2.7812286735230973\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.241456620958899 alpha 2.6963724241843483\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.243569906750377 alpha 2.864532491866533\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.262328905350575 alpha 2.293496660156432\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.312254807411768 alpha 2.977357683248605\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.3139106972688275 alpha 2.424539534893626\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.330635030940133 alpha 2.67677801116693\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.364784603667115 alpha 3.068648669089346\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.375656236137813 alpha 2.6380941343366784\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.393923877936347 alpha 2.7401892929080267\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.396192438419212 alpha 2.648439135057984\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.415724412794598 alpha 2.853795344412776\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.438501313115974 alpha 2.6196608352538115\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.451976385966978 alpha 2.449586293179408\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.471153940125976 alpha 2.5515956593431044\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data sample: mu+1 4.471600858166251 alpha 2.6801256352760596\n",
      "\n"
     ]
    }
   ],
   "source": [
    "data_alphas1k = []\n",
    "M=1000\n",
    "mus=np.sort(3*np.random.random_sample(100)+.5)\n",
    "for mu in tqdm(mus):\n",
    "    \n",
    "    data = np.random.pareto(mu,size=(M))\n",
    "    \n",
    "    fit = powerlaw.Fit(data, xmax=np.max(data), fit_method=\"KS\")\n",
    "    data_alphas1k.append(fit.alpha)    \n",
    "    print(\"data sample: mu+1 {} alpha {}\".format(mu+1, fit.alpha))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-26T09:19:56.662798Z",
     "start_time": "2018-08-26T09:19:56.557466Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'PowerLaw Test (w/xmax, KS): fit vs actual')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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gyRJZW0ldBOxHMkDgt4F9yTCWlFmrVWtVctqBp7Ho/kUDvtE3Y17ldmrBVa5W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\nMDOzTBwwzMwsEwcMMzPLxAHDzMwy+f+znSrsHebiQAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mup1 = np.array(mus)+1.0\n",
    "\n",
    "plt.scatter(mup1[0:len(data_alphas1k)], data_alphas1k, color='green', label='fit.alpha')\n",
    "plt.xlabel(\"mu+1\")\n",
    "plt.ylabel('alpha')\n",
    "plt.legend()\n",
    "plt.title(\"PowerLaw Test (w/xmax, KS): fit vs actual\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.6.5"
  },
  "widgets": {
   "state": {
    "4187c75fe4084c6aa72afd669fb1b041": {
     "views": [
      {
       "cell_index": 10
      }
     ]
    },
    "fd36efc4ab934dceb3878bae68d3fdd9": {
     "views": [
      {
       "cell_index": 7
      }
     ]
    },
    "fd522c9c015144cdbe17b4d4e458c991": {
     "views": [
      {
       "cell_index": 3
      }
     ]
    }
   },
   "version": "1.2.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}