WordEmbeddingMatrices.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## WordEmbeddingMatrices\n",
    "\n",
    "A good example of where an x_min must be specified\n",
    "\n",
    "#### Take the Glove Word Embeddings and form a correlation matrix\n",
    "\n",
    "$$\\mathbf{X}=\\mathbf{E}^{T}\\mathbf{E},\\;\\;\\;shape=(300\\times 300)$$  \n",
    "\n",
    "\n",
    "#### Fit the eigenvalue distribution $\\rho(\\lambda)$ to a power law $\\lambda^{-\\alpha}$\n",
    "\n",
    "$$\\mathbf{X}\\mathbf{v}=\\lambda\\mathbf{v}$$\n",
    "\n",
    "$$\\rho(\\lambda)\\sim\\lambda^{-\\alpha}$$\n",
    "\n",
    "#### If we do a raw fit, we get\n",
    "\n",
    "- $\\alpha=5.2$\n",
    "\n",
    "#### But there is rank collapse (lots of very small, near zero eigenvalues)\n",
    "\n",
    "$$\\lambda\\sim 0$$\n",
    "\n",
    "\n",
    "#### If we ignore the rank collapse, and set xmin=0.1, we get\n",
    "\n",
    "- $\\alpha=3.0$\n",
    "\n",
    "#### But on further inspection, its a pretty bad fit\n",
    "\n",
    "### The Big Question...does the Power Law fit and Rank Collapse affect the quality of the Vectors ?\n",
    "\n",
    "Not sure...\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-27T08:06:08.823252Z",
     "start_time": "2018-08-27T08:06:06.545222Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import powerlaw\n",
    "\n",
    "import os \n",
    "from tqdm import tqdm_notebook as tqdm\n",
    "\n",
    "\n",
    "import matplotlib\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-27T08:06:11.499813Z",
     "start_time": "2018-08-27T08:06:08.826030Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "importing Jupyter notebook from RMT_Util.ipynb\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/charleshmartin/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/charleshmartin/anaconda3/lib/python3.6/site-packages/h5py/__init__.py:36: 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",
      "Using TensorFlow backend.\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.6.5 |Anaconda, Inc.| (default, Apr 26 2018, 08:42:37) \n",
      "[GCC 4.2.1 Compatible Clang 4.0.1 (tags/RELEASE_401/final)]\n",
      "numpy version 1.14.3\n",
      "scipy version 1.1.0\n",
      "tensforflow version 1.10.1\n",
      "keras version 2.2.2\n",
      "sklearn version 0.19.1\n",
      "powerlaw version 1.4.3\n"
     ]
    }
   ],
   "source": [
    "import import_ipynb\n",
    "import RMT_Util"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-27T08:06:11.506733Z",
     "start_time": "2018-08-27T08:06:11.501899Z"
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.rcParams['figure.figsize'] = [5,5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://blog.keras.io/using-pre-trained-word-embeddings-in-a-keras-model.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "GLOVE_DIR = \"/Users/charleshmartin/work-tmp/glove\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(400000, 300)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "embeddings_mat = np.zeros([400000,300])\n",
    "embeddings_mat.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Read the Glove Embedding Matrix into a large matrix E\n",
    "\n",
    "E = embeddings_mat\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = open(os.path.join(GLOVE_DIR, 'glove.6B.300d.txt'))\n",
    "for iw, line in enumerate(f):\n",
    "    values = line.split()\n",
    "    word = values[0]\n",
    "    coefs = np.asarray(values[1:], dtype='float32')\n",
    "    embeddings_mat[iw] = coefs\n",
    "f.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Form the correlation matrix, which is (300 x 300)\n",
    "\n",
    "- and normalize by 1/N "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "X =  np.dot(embeddings_mat.T,embeddings_mat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = (1.0/400000.0)*X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Get all the eigenvales \n",
    "\n",
    "evals = $[\\lambda]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "evals = np.linalg.eigvals(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot the histogram , which is $\\rho(\\lambda)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUIAAAEyCAYAAACPj9ldAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvhp/UCwAADWNJREFUeJzt3W+IZYV5x/Hvr65iYipqHGXrSseCpC1CGxmMrSBF05KoRF9EMLTpEoR9k6amKcRN30jfaSn5UyiFRS0rFRNRixJDWjFKEJpNZtXEP5tUsdZste6ExCS2L1Kbpy/mJAxm7Ozec+/cnXm+H1juveeeO+c5yH495545mqpCkjr7pXkPIEnzZggltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltbdhCJPcnuRIkqfXLDsjyUNJnhseTx+WJ8nfJHk+ybeSXDjL4SVpGrLRnSVJLgVeB+6oqguGZX8FfL+qbk6yFzi9qm5McgXwMeAK4D3A56rqPRsNceaZZ9bi4uK4PZGkNzl48OD3qmpho/V2bLRCVX01yeKbFl8N/N7wfD/wKHDjsPyOWq3r15KclmRnVb3y/21jcXGR5eXljUaRpGOS5N+PZr1JvyM8+2dxGx7PGpafA3x3zXqHh2XrDbgnyXKS5ZWVlQnHkKTxpn2xJOssW/fcu6r2VdVSVS0tLGx45CpJMzNpCF9NshNgeDwyLD8MnLtmvV3Ay5OPJ0mzN2kIHwB2D893A/evWf7Hw9Xji4EfbvT9oCTN24YXS5LcxeqFkTOTHAZuAm4G7k5yPfAScO2w+pdYvWL8PPDfwEdmMLMkTdXRXDX+0Fu8dfk66xbw0bFDSdJm8s4SSe0ZQkntGUJJ7RlCSe0ZQkntbXjVeCta3Pvgz5+/ePOVc5xE0lbgEaGk9gyhpPa21anx2lNiSTpaHhFKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmrPEEpqb1QIk/xZkmeSPJ3kriQnJzkvyYEkzyX5QpKTpjWsJM3CxCFMcg7wp8BSVV0AnABcB9wCfKaqzgd+AFw/jUElaVbGnhrvAN6WZAfwduAV4DLgnuH9/cA1I7chSTM1cQir6j+AvwZeYjWAPwQOAq9V1RvDaoeBc9b7fJI9SZaTLK+srEw6hiSNNubU+HTgauA84FeAU4D3r7Nqrff5qtpXVUtVtbSwsDDpGJI02phT4/cC/1ZVK1X1P8B9wO8Cpw2nygC7gJdHzihJMzUmhC8BFyd5e5IAlwPPAo8AHxzW2Q3cP25ESZqtMd8RHmD1osjjwFPDz9oH3Ah8IsnzwDuB26YwpyTNzI6NV3lrVXUTcNObFr8AXDTm50rSZvLOEkntGUJJ7RlCSe0ZQkntGUJJ7RlCSe0ZQkntGUJJ7RlCSe0ZQkntGUJJ7RlCSe0ZQkntGUJJ7RlCSe0ZQkntGUJJ7RlCSe1t2RAu7n2Qxb0PznsMSdvAlg2hJE2LIZTUniGU1J4hlNSeIZTUniGU1J4hlNSeIZTUniGU1J4hlNSeIZTUniGU1J4hlNSeIZTUniGU1J4hlNSeIZTUniGU1J4hlNSeIZTUniGU1J4hlNSeIZTUniGU1J4hlNSeIZTU3qgQJjktyT1Jvp3kUJLfSXJGkoeSPDc8nj6tYSVpFsYeEX4O+HJV/TrwW8AhYC/wcFWdDzw8vJak49bEIUxyKnApcBtAVf2kql4Drgb2D6vtB64ZO6QkzdKYI8JfA1aAv0/yRJJbk5wCnF1VrwAMj2et9+Eke5IsJ1leWVkZMYYkjTMmhDuAC4G/q6p3A//FMZwGV9W+qlqqqqWFhYURY0jSOGNCeBg4XFUHhtf3sBrGV5PsBBgej4wbUZJma+IQVtV/At9N8q5h0eXAs8ADwO5h2W7g/lETStKM7Rj5+Y8BdyY5CXgB+Aircb07yfXAS8C1I7chSTM1KoRV9SSwtM5bl4/5uZK0mbyzRFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnuGUFJ7hlBSe4ZQUnujQ5jkhCRPJPni8Pq8JAeSPJfkC0lOGj+mJM3ONI4IbwAOrXl9C/CZqjof+AFw/RS2IUkzMyqESXYBVwK3Dq8DXAbcM6yyH7hmzDYkadbGHhF+Fvgk8NPh9TuB16rqjeH1YeCc9T6YZE+S5STLKysrI8eQpMlNHMIkVwFHqurg2sXrrFrrfb6q9lXVUlUtLSwsTDqGJI22Y8RnLwE+kOQK4GTgVFaPEE9LsmM4KtwFvDx+TEmanYmPCKvqU1W1q6oWgeuAr1TVHwKPAB8cVtsN3D96SkmaoVn8HuGNwCeSPM/qd4a3zWAbkjQ1Y06Nf66qHgUeHZ6/AFw0jZ8rSZvBO0sktWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1ZwgltWcIJbVnCCW1N3EIk5yb5JEkh5I8k+SGYfkZSR5K8tzwePr0xpWk6RtzRPgG8OdV9RvAxcBHk/wmsBd4uKrOBx4eXkvScWviEFbVK1X1+PD8x8Ah4BzgamD/sNp+4JqxQ0rSLE3lO8Iki8C7gQPA2VX1CqzGEjjrLT6zJ8lykuWVlZVpjCFJExkdwiTvAO4FPl5VPzraz1XVvqpaqqqlhYWFsWNI0sRGhTDJiaxG8M6qum9Y/GqSncP7O4Ej40aUpNkac9U4wG3Aoar69Jq3HgB2D893A/dPPp4kzd6OEZ+9BPgw8FSSJ4dlfwHcDNyd5HrgJeDacSNK0mxNHMKqegzIW7x9+aQ/V5I2m3eWSGrPEEpqzxBKas8QSmrPEEpqzxBKas8QSmpv24dwce+DLO59cN5jSDqObfsQStJGDKGk9gyhpPYMoaT2DKGk9sb8Z7iOC14RljSWR4SS2jOEktozhJLaM4SS2jOEktozhJLaM4SS2jOEktozhJLaM4SS2jOEktozhJLaM4SS2jOEktozhJLaM4SS2jOEktozhJLaM4SS2jOEktozhJLaM4SS2jOEktrb8v9f46O19v9//OLNV85xEknHG48IJbVnCCW1ZwgltWcIJbXX5mLJRryYIvXlEaGk9mYSwiTvS/KdJM8n2TuLbUjStEz91DjJCcDfAr8PHAa+keSBqnp22tua1NrT4GP9jKfN0ubajL97szgivAh4vqpeqKqfAJ8Hrp7BdiRpKmYRwnOA7655fXhYJknHpVlcNc46y+oXVkr2AHuGl68n+c4xbudM4HvH+Jmjklsme28CM9uHTbYd9mM77ANsj/1Ydx8m/Lv3q0ez0ixCeBg4d83rXcDLb16pqvYB+ybdSJLlqlqa9PPHg+2wD7A99mM77ANsj/2Yxz7M4tT4G8D5Sc5LchJwHfDADLYjSVMx9SPCqnojyZ8A/wScANxeVc9MezuSNC0zubOkqr4EfGkWP3uNiU+rjyPbYR9ge+zHdtgH2B77sen7kKpfuI4hSa14i52k9gyhpPa2XAi3w33MSW5PciTJ0/OeZVJJzk3ySJJDSZ5JcsO8Z5pEkpOTfD3JN4f9+Mt5zzSpJCckeSLJF+c9y6SSvJjkqSRPJlnetO1upe8Ih/uY/5U19zEDHzqe7mM+GkkuBV4H7qiqC+Y9zySS7AR2VtXjSX4ZOAhcswX/WQQ4papeT3Ii8BhwQ1V9bc6jHbMknwCWgFOr6qp5zzOJJC8CS1W1qb8UvtWOCLfFfcxV9VXg+/OeY4yqeqWqHh+e/xg4xBa8lbJWvT68PHH4s3WODgZJdgFXArfOe5ataKuF0PuYj0NJFoF3AwfmO8lkhlPKJ4EjwENVtRX347PAJ4GfznuQkQr45yQHh9twN8VWC+FR3ceszZPkHcC9wMer6kfznmcSVfW/VfXbrN4OelGSLfV1RZKrgCNVdXDes0zBJVV1IfB+4KPD10gzt9VCeFT3MWtzDN+p3QvcWVX3zXuesarqNeBR4H1zHuVYXQJ8YPh+7fPAZUn+Yb4jTaaqXh4ejwD/yOrXYTO31ULofczHieEiw23Aoar69LznmVSShSSnDc/fBrwX+PZ8pzo2VfWpqtpVVYus/p34SlX90ZzHOmZJThkuvJHkFOAPgE35zYotFcKqegP42X3Mh4C7t+J9zEnuAv4FeFeSw0mun/dME7gE+DCrRx9PDn+umPdQE9gJPJLkW6z+i/ahqtqyv36yxZ0NPJbkm8DXgQer6subseEt9eszkjQLW+qIUJJmwRBKas8QSmrPEEpqzxBKas8QSmrPEEpq7/8A3Qv7DcKL0wMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(evals, bins=100);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Do a simple power law fit of $\\rho(\\lambda)\\sim\\lambda^{-\\alpha}$ to find $\\alpha$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Calculating best minimal value for power law fit\n",
      "/Users/charleshmartin/anaconda3/lib/python3.6/site-packages/powerlaw.py:700: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  (Theoretical_CDF * (1 - Theoretical_CDF))\n",
      "/Users/charleshmartin/anaconda3/lib/python3.6/site-packages/powerlaw.py:700: RuntimeWarning: invalid value encountered in true_divide\n",
      "  (Theoretical_CDF * (1 - Theoretical_CDF))\n"
     ]
    }
   ],
   "source": [
    "fit = powerlaw.Fit(evals, xmax=np.max(evals))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.231594347585907"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit.alpha"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The exponent is very high $\\alpha > 4$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Because there is Rank Collapse..\n",
    "\n",
    "we need to remove the small eigenvalues...the peak near 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 2)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(evals, bins=200);\n",
    "plt.xlim(0,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Remove all $\\lambda<0.1$ and redo the PowerLaw fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 2)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(evals[evals>=0.1], bins=200);\n",
    "plt.xlim(0,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.933789276058489"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit = powerlaw.Fit(evals[evals>=0.1], xmin=0.1)\n",
    "fit.alpha"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Now the $\\alpha$ is more reasonable\n",
    "\n",
    "We can now check more details"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/charleshmartin/anaconda3/lib/python3.6/site-packages/powerlaw.py:700: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  (Theoretical_CDF * (1 - Theoretical_CDF))\n",
      "Assuming nested distributions\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "alpha 2.93379, D 0.321258, best_pl  PL\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "RMT_Util.fit_and_plot_powerlaw(evals, verbose=True, xmin=0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Not a great fit\n",
    "\n",
    "- The D value is very high...should be much closer to 0\n",
    "\n",
    "- The solid and dotted lines should overlap"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Is this just MP?\n",
    "\n",
    "Q is so large that the MP fit is very spiked"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "E = embeddings_mat\n",
    "M, N = np.min(E.shape), np.max(E.shape)\n",
    "Q = N/M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "% spikes outside bulk 0.67\n",
      "% sigma 0.6399\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "RMT_Util.ipynb:7: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  \"## RMT Util\\n\",\n",
      "RMT_Util.ipynb:7: RuntimeWarning: invalid value encountered in sqrt\n",
      "  \"## RMT Util\\n\",\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0, 35)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATwAAAE/CAYAAADbkX+oAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvhp/UCwAAHCRJREFUeJzt3XuYnWV97vHvnUw4mGDDYZDIQSgiJysgU8RSacpBQVSwl7rFw44WGrWlYrFSsHYDVRC9BNpdD4AFQ6sFqYAggpUiKbK1SKLhZBAQo0QSEgRKAiqH/PYfzzPkZZiVWTOzVtZkfvfnutaV9R7X867DvZ7f+6x3oojAzCyDKb1ugJnZ+uLAM7M0HHhmloYDz8zScOCZWRoOPDNLw4GXkKTVkn631+0wW98mfOBJOk/S5xvT0yQ93mLe/pJeJulKSSslPSzpPyTtuo79z5P0ZA2BwdutjeXHSLpL0ipJD0r6pqTNhmy7qt7ukPRJSb8ziuObLykk7TVk/tfr/Nl1+lRJT9X2PSrpe5Je3WKfMyVdKGl5bdfdkv5mcHlEzIiI+9pt4/omaePa/sfqMZwwwvq/K+nqeqwPSfp0Y9mXJS2r+7pb0rFDtj1W0r31ef2WpBc3ls2UdJGkFfV26pBtb6jvs8ck3SrpyMay2ZLWDHlfzRmy/dslLa7v3Z9Kes0wx3ZKfR8c0pj3aUn318f9uaS/HbLNG+t7cXV9n+zRWDZH0sK67dK6r77G8i0kXVHb9HNJ72gsO0LSTfX9t1zSFwc/C3X52+rjPSFp/jperzn1mI5tzPtIbfMqST+T9JFW249LREzoG/BO4M7G9KuBxcAdQ+Y9DkwD9gOOAbao0x8H7lrH/ucBn2ix7I+AB4F96vQWwBxgs6HbApsAvw/cANwBTG/z+OYDPwHOaszbElgOrABm13mnAl+u96cBnwaWARpmn18CLgU2p3yp7Qa8pdev5She808C363t370+F4e1WHcj4KfACcD0+jq8orF8T2Djen+3uq99G6/virrORsAXgP8a8jz+O/ACYMf6OO9tLH8F0FfvvwpYBcyq07OBpes4xkOBnwP719doW2DbIevsDNwOPAAc0pi/6+D7q253J/AndXoX4DHgD4E+4GTg3kY7PwC8ph7vtsBC4KTGvi8GvgrMqPv4H2DPuuwdwGH1+dgcuBY4t7HtIcDbgP8DzG9x3JsDd1E+I8c25p8IvLK2edf63Ly94++tXr+523jzbwesAbZqPDGnAD8bMu8/W2y/BRDAli2Wz6N14P018PV1tO152wKbUYLouDaPb359gywFptZ5x9UP31KGCbw6vWc9rq2G2ecdwFHreMwAXlrvbwl8o35IbgE+Adw0ZN0/B+6hfKA/Xj+I36/bXAps1HgzXw2sBB6p97cbw2v+S+C1jemPA5e0WHcu8N0297trfW3eVqc/A3yusfzF9Xh3rtMPAb/fWP7RVo9F+aL9DbBfnZ7NugPve8AxI7T3WuD1wBIagTdknW0poXhi473zzcbyKcCvgYNbbH8C8I16fzrwJPCyxvJ/Bc5sse2fALcPM/9YWgfeufX9NJ9G4A2z3v8F/mm0752RbhO+pI2IpZS0H+zuH0j59v/ekHk3ttjFgcDyiPjVGB7+ZuB1kk6TdICkjdto7yrgukbb2vEA8GPgtXX6fwP/0mrl2o73UD5QDw2zyn8Dp0t6r6RdRnjsz1F6x9tQeq9zhlnnMGBfSm/kROB8Ss97e+DlwNF1vSmUXtFLgB0oH7TPNtr9+VoODXe7ra6zOSV4nj2tUO/v2aL9+wNLJF1by9n5kn6vuUJ93CcoPYtlwDWDi+qNxjT1mIbOG7zfXEYtpX9Dea/MBxY0Fm+tchrkZ5LOkTS9bjMVGAD6azm9VNJnJW3a2O9bgScj4hqGIekkSaspX4rTgX9bxzE9r90NB1J6iAAvA56JiLsby9f13De3HZGk/SjHfe4I64ny+Wl7323rdIJ240bpSZ1D+UCtoHSp39+Y9wjwR8Nstx2lt3D0CPv+DfBo43ZRY/nhlB7Qo8Bq4GzW9sTmMUzvEDgTuK7NY5tP+UZ8F6Wc2BW4uy4b2sN7srZjBfAdamk2zD43pfRGFgJPUUqawxvLA3gpMLUu37WxbLge3gGN6YXA3zSmzwL+oUU79gYeGeVrvX19zE0a8w4FlrRY/9v1GA6nlGkfAe6j9job602llGgfA6bVeQdTenGvqM/ZeZRq4ui6/MvA5ZRe+0spJe1vh2nDtPr4f9WYtw2wR31/7kT5Qj6vLhvsSS4AZgFbAf8POL0un0HpUe9Up5cwTA+PEmT7AKex9jTLbpQvsNn1+fi7ekwnD7P9e+t7bLBSeg2lc9Bc588YprdWX5NHaPQGG8ue18Orz/8C4NXN932L1/Q0StBu3In8aN4mfA+vupHybfJ7wH0R8QRwU2PeppRv2GdJ6qd8GD4fERePsP/PRMTMxu3ZXk5EXBsRb6SUxkdSelbHttjPoG2Bh9s9uOpy4CDgLyllxHAure3bOiIOioiFw60UEb+OiDMiYl9KyXop8O+Sthiyaj/lnMn9jXn383wPNu7/epjpGQCSXqAyyPRzSY9RXreZtUfTrtX13xc25r2QUk4P59eUgL42Ip6klKlbUs79PSsinomImyhfgh+o866nnB65jFJFLKmPs7Ru9sG6/3uAKylfSEsZIiKeiohrKdXAm+q85RHx44hYExE/o/SM39JoM5SSbVmUXvrZlPIVygf+X+t2LUXxo7q/0+q8uyi99M9SerNbUaqH57Rb0lGUL+bDY22VsJrnPu8wzHMvaX9Kj/It8dze4Lr8OXBbRHx/XStJOo5S4RwREb9tc99t25ACby/gCEo5C6W7u32dd0tE/GZw5VoWfRu4KiJO70QD6hv3ekrPqlV5gKQZlJO33221Tov9P0E5Z/MBWgfeqEXEY8AZlLJnpyGLVwJPU0Jg0PbjeLgPU3qor4qIF1K+kKCWWJLOHTJq2bzdWdv7COWD2hy13ovW5c1tlN5Su/oo5yCpj/e5iNglIramBF8f5RwoEfFwRLwzIraJiD0pn5cftLvvIYL6PNRjXLqOdh8MfLCOhC6nvCaXqjHSPsIxfS0iXh4RW1IC/SWU87MASDoM+CLwxoi4vbGfu4G+IadBnvPcS9oHuAr40/p5aNfBwJsbx/QHwFmSmqc8/hQ4iXK+8XlfLB3R6S5jt26U0vRB4E2Ned+o805vzHsh5U352Tb3O4/WgxZHAm+nnIwX5cT0SuCdQ7cFNqac5/pPyjfq4CjajpQ39o4tHmM+tWtPKXX+sLGs5aDFCMf0d5QR440oo5Z/Syk/ZtTlzUGLr1K+rV9AKYd+wfNL2pc2pm8C3tOY/gTwz/X+pymhvQmlR3xF3b5vlK/1mcB/1ed9N0oAthql3RV4gvIlMxX4K0rpuRGwdX39ZtRlr6OUe0fWbTehfHmJcs5xPnBGY987U3qLUykl60OsHbHcrc7blFLSvotyyuGVdfnsuk9RAusG4EuNff89JYS2rsf5XeDjddmWlJJ48HY/8NZ6HFOA9/Hc9+Qy4IONfe9b29w/+Po2lh0E/Ao4sMXzeQmlJzsdOIDnjtK+nPJ5+18ttp1an9P3Uzopm7D29MHMIcf0PcqAye/U5e+kjKDv3tUc6ebOO9rQ8iKsoTHaSikTAnhdY96cOu9xShd98LZDi/3Oq2/U5roP1WUHAtfXN/oqyjfgicNsu6o+3p3Ap4CZjXVeQymVprV4/Pm0Ppcx1sD7GKWX8hiltJ4P/EFjeTPw+oFvsnaU9lPA9cOtW6fXFXgvro+1uj5X72NsgbcxcGFt04PACY1lOwx9PSmjhffW9eez9gPaTwnOR+uy24E/a2w3k9JDfLx+2D5JPT9bl7+NMqD0BLBoyPtsd8pplFV1/7cAb24sP4HyJf0EJbD+iXqerS6fBny+brucMiq5SYvnYwn1HB4l8L5VX9fB5/mjNH6eVF+jVXWd82j8RIoSvE/z3Pf7tY3lWwBfr8/JL4B3NJZ9ifIZbG7b/MnYe+rr3bzNa+d9T/nVxVND9n3ucNuO56b6YNYlkj4GrIyI83rdlnZI+hSwTTTOY5pNFg685CTtRin/bqeUwddQvnm/3tOGmXVB30grSNqEUo9vXNf/WkScImke5Zfq/1NXfU9ELOpWQ61rNqOcLngx5ecuZ1FGJM0mnRF7ePVHgNMjYrWkaZTzA8dTTkxeHRFf634zzczGb8QeXpREHPxt1LR6cx1sZhuctn6HJ2mqpEWUkue6iBj8ke/pkm6rl82MeNmVmVkvjWrQQtJMym+r/pLyW57llBPe5wM/jYi/H2abuZQLvJk+ffq+u+22WweabWa21sKFCx+KiP6R1hv1KK2kU4DHI+IzjXmzgb+OiDesa9uBgYFYsGDBulYxMxs1SQsjYmCk9UYsaSX1154d9a85HALcJWlWnSfgKOrlOGZmE9WIgxaUv+ZwUb0AfArlAvarJX2nXqAvyq/Q39/FdpqZjVs7o7S3Uf4EzdD5B3WlRWZmXbKh/LUUM7Nxc+CZWRoOPDNLw4FnZmk48MwsDQeemaXhwDOzNBx4ZpaGA8/M0nDgmVkaDjwzS8OBZ2ZpOPDMLA0Hnpml4cAzszQceGaWhgPPzNJw4JlZGg48M0vDgWdmaTjwzCwNB56ZpeHAM7M0HHhmloYDz8zScOCZWRoOPDNLw4FnZmk48MwsDQeemaUxYuBJ2kTSDyTdKulOSafV+TtJulnSPZK+Kmmj7jfXzGzs2unh/RY4KCL2AvYGDpO0P/Ap4JyI2AV4BDime800Mxu/EQMvitV1clq9BXAQ8LU6/yLgqK600MysQ9o6hydpqqRFwArgOuCnwKMR8XRdZSmwbXeaaGbWGW0FXkQ8ExF7A9sB+wG7D7facNtKmitpgaQFK1euHHtLzczGaVSjtBHxKDAf2B+YKamvLtoOeKDFNudHxEBEDPT394+nrWZm49LOKG2/pJn1/qbAIcBi4AbgLXW1OcCV3WqkmVkn9I28CrOAiyRNpQTkpRFxtaQfA5dI+gTwI+CCLrbTzGzcRgy8iLgN2GeY+fdRzueZmW0QfKWFmaXhwDOzNBx4ZpaGA8/M0nDgmVkaDjwzSyNX4C1fDk8/PfJ6ZjYp5Qq8WbPgH/+x160wsx7JFXgAv/pVr1tgZj2SL/CmTu11C8ysR/IF3po1vW6BmfWIA8/M0nDgmVkaDjwzS8OBZ2Zp5Am8qP/lhgPPLK08gffMM+Xfp57qbTvMrGfyBN5g0A0Gn5ml48AzszTyBN7gHw1w4JmllSfwBnt4/mspZmnlCzz38MzSyhN4LmnN0ssTeO7hmaXnwDOzNPIE3mDQOfDM0soTeIOXljnwzNLKE3iD19A68MzSyhd4/h2eWVr5As89PLO0Rgw8SdtLukHSYkl3Sjq+zj9V0i8lLaq313e/uePgc3hm6fW1sc7TwIcj4oeSNgMWSrquLjsnIj7TveZ1kHt4ZumNGHgRsQxYVu+vkrQY2LbbDes4B55ZeqM6hydpR2Af4OY66zhJt0m6UNLmHW5bZ61ZA1OmOPDMEms78CTNAC4DPhQRjwFfAHYG9qb0AM9qsd1cSQskLVi5cmUHmjxGEdDX58AzS6ytwJM0jRJ2X4mIywEi4sGIeCYi1gBfBPYbbtuIOD8iBiJioL+/v1PtHr01a2CjjfyzFLPE2hmlFXABsDgizm7Mn9VY7c3AHZ1vXgetWQPTprmHZ5ZYO6O0BwDvBm6XtKjO+yhwtKS9gQCWAO/rSgs7Zc0amDp17c9TzCyddkZpbwI0zKJrOt+cLopw4Jkll+tKi74+/7+0Zok58MwsjVyBN3WqA88ssTyB53N4ZunlCTyXtGbp5Qo8l7RmqeULPJe0ZmnlCbzBa2ndwzNLK0/guaQ1Sy9f4LmkNUsrV+C5pDVLLU/g+RyeWXp5As/n8MzSyxd4PodnllaewHNJa5ZensBzSWuWXq7A6+tzSWuWWK7Acw/PLLU8gTf456EceGZp5Qk8l7Rm6eULPPfwzNLKFXguac1SyxN4Podnll6ewPM5PLP0cgXe4KVlDj2zlHIF3pR6uA48s5TyBF5ECbwpUxx4ZknlCbzBHt6UKR64MEsqX+BJDjyzpHIFnuSS1iyxEQNP0vaSbpC0WNKdko6v87eQdJ2ke+q/m3e/uePQPIfnHp5ZSu308J4GPhwRuwP7A38haQ/gJOD6iNgFuL5OT1wuac3SGzHwImJZRPyw3l8FLAa2BY4ELqqrXQQc1a1GdkRz0MIlrVlKozqHJ2lHYB/gZuBFEbEMSigCW7fYZq6kBZIWrFy5cnytHY/mOTz38MxSajvwJM0ALgM+FBGPtbtdRJwfEQMRMdDf3z+WNnaGz+GZpddW4EmaRgm7r0TE5XX2g5Jm1eWzgBXdaWKH+ByeWXrtjNIKuABYHBFnNxZdBcyp9+cAV3a+eR3kn6WYpdfXxjoHAO8Gbpe0qM77KHAmcKmkY4BfAG/tThM7xCWtWXojBl5E3ASoxeKDO9ucLhrs1bmkNUsrz5UWES5pzZLLE3jgn6WYJZcn8FzSmqWXJ/CghJ3kktYsqTyB1+zhOfDMUsoVeO7hmaWWJ/DAgWeWXJ7Ac0lrll6uwBvs4ZlZSnkCD9aGnXt4ZinlCTyXtGbp5Qo8D1qYpZYn8MCBZ5ZcnsBrlrRmllKuwPOghVlqeQIPXNKaJZcn8DxKa5ZersBzD88stTyBB77Swiy5PIHX7NW5h2eWUp7AA5e0ZsnlCTwPWpillyvw3MMzSy1P4IEDzyy5PIHnS8vM0ssVeL60zCy1PIEHLmnNkssTeB6lNUsvV+C5h2eWWp7AA19aZpbciIEn6UJJKyTd0Zh3qqRfSlpUb6/vbjM7wJeWmaXXTg9vHnDYMPPPiYi96+2azjarC1zSmqU3YuBFxI3Aw+uhLd3nwDNLbTzn8I6TdFsteTdvtZKkuZIWSFqwcuXKcTzcOHmU1iy9sQbeF4Cdgb2BZcBZrVaMiPMjYiAiBvr7+8f4cB3QLGnNLKUxBV5EPBgRz0TEGuCLwH6dbVaX+EoLs9TGFHiSZjUm3wzc0WrdCcMlrVl6fSOtIOliYDawlaSlwCnAbEl7AwEsAd7XxTZ2jgctzFIbMfAi4uhhZl/QhbZ0l3t4ZunludLCgxZm6eUJPPCghVlyeQLPJa1ZerkCz4MWZqnlCTxw4JkllyfwXNKapZcr8DxKa5ZansADj9KaJZcn8FzSmqWXK/A8aGGWWp7AAweeWXJ5Aq9Z0ppZSrkCz4MWZqnlCTxwSWuWXJ7A8yitWXq5As89PLPU8gQe+EoLs+TyBF6zV+cenllKeQIPXNKaJZcn8DxoYZZersBzD88stTyBBw48s+TyBJ4vLTNLL1fg+dIys9TyBB64pDVLLk/geZTWLL1cgecenllqeQIPfGmZWXJ5As+XlpmlN2LgSbpQ0gpJdzTmbSHpOkn31H83724zO8AlrVl67fTw5gGHDZl3EnB9ROwCXF+nJz4HnllqIwZeRNwIPDxk9pHARfX+RcBRHW5X53mU1iy9sZ7De1FELAOo/27duSZ1SbOkNbOUuj5oIWmupAWSFqxcubLbDzdSY8q/7uGZpTTWwHtQ0iyA+u+KVitGxPkRMRARA/39/WN8uA5wSWuW3lgD7ypgTr0/B7iyM83pMg9amKXWzs9SLga+D+wqaamkY4AzgUMl3QMcWqcnNvfwzNLrG2mFiDi6xaKDO9yW7vKghVl6ea60AA9amCWXJ/Bc0pqllyvwPGhhllqewAMHnllyeQLP/6eFWXq5As9hZ5ZansADj9KaJZcn8FzSmqWXK/Acdmap5Qk8cElrllyewHNJa5ZersBz2JmllifwwCWtWXJ5As8lrVl6uQLPYWeWWp7AA5e0ZsnlCTyXtGbp5Qk8cNiZJZcn8JplrEtas5TyBN4RR8A227iXZ5bYiP+Jz6Rx8sm9boGZ9VieHl6TS1qzlPIFnktas7TyBZ6ZpbVBn8ObP3/t/dmzR7GhS1qzlDbIwGsG3ai5pDVLyyWtmaWRM/Bc0pqllC/wXNKapZUv8MwsrXENWkhaAqwCngGejoiBTjSq61zSmqXUiVHaP46Ihzqwn/XDJa1ZWi5pzSyN8QZeAN+WtFDS3E40aL1wSWuW0nhL2gMi4gFJWwPXSborIm5srlCDcC7ADjvsMM6H6wCXtGZpjauHFxEP1H9XAFcA+w2zzvkRMRARA/39/eN5ODOzcRlz4EmaLmmzwfvAa4E7OtWwrnJJa5bSeEraFwFXqJSIfcC/RcS3OtKqhsHrZkf1xwHWxSWtWVpjDryIuA/Yq4NtWX/cwzNLKd/PUtzDM0srX+CZWVr5Am/zzWH69F63wsx6YIP8A6DjcsYZvW6BmfVIvh6emaXlwDOzNBx4ZpaGA8/M0nDgmVkaDjwzS8OBZ2ZpOPDMLA0Hnpml4cAzszQceGaWhgPPzNJw4JlZGg48M0vDgWdmaTjwzCwNB56ZpeHAM7M0HHhmloYDz8zScOCZWRoOPDNLY9IE3vz55WZm1soG8//SOszMbLwmTQ/PzGwkDjwzS2NcgSfpMEk/kXSvpJM61Sgzs24Yc+BJmgp8Djgc2AM4WtIenWqYmVmnjWfQYj/g3oi4D0DSJcCRwI870bCxag5uzJ7dq1aY2UQ0nsDbFri/Mb0UeNX4mtNZ6xrZbRWGg9s4LM3Wr/Xx2RtP4GmYefG8laS5wNw6uVrST0b5OFsBD41ym4lmMhwDTI7jmAzHAJPjODp5DC9pZ6XxBN5SYPvG9HbAA0NXiojzgfPH+iCSFkTEwFi3nwgmwzHA5DiOyXAMMDmOoxfHMJ5R2luAXSTtJGkj4O3AVZ1plplZ5425hxcRT0s6DvgPYCpwYUTc2bGWmZl12LguLYuIa4BrOtSWVsZcDk8gk+EYYHIcx2Q4Bpgcx7Hej0ERzxtnMDOblHxpmZmlMWEDbzJctibpQkkrJN3R67aMlaTtJd0gabGkOyUd3+s2jYWkTST9QNKt9ThO63WbxkrSVEk/knR1r9syVpKWSLpd0iJJC9bb407EkrZetnY3cCjl5y+3AEdHRE+v4hgtSQcCq4F/iYiX97o9YyFpFjArIn4oaTNgIXDUBvhaCJgeEaslTQNuAo6PiP/ucdNGTdIJwADwwoh4Q6/bMxaSlgADEbFef0s4UXt4z162FhFPAoOXrW1QIuJG4OFet2M8ImJZRPyw3l8FLKZcZbNBiWJ1nZxWbxPv234EkrYDjgD+uddt2RBN1MAb7rK1De5DNtlI2hHYB7i5ty0Zm1oKLgJWANdFxIZ4HP8AnAis6XVDximAb0taWK/GWi8mauC1ddmarT+SZgCXAR+KiMd63Z6xiIhnImJvylVB+0naoE4zSHoDsCIiFva6LR1wQES8kvLXlv6inv7puokaeG1dtmbrRz3ndRnwlYi4vNftGa+IeBSYDxzW46aM1gHAm+r5r0uAgyR9ubdNGpuIeKD+uwK4gnIaq+smauD5srUJop7svwBYHBFn97o9YyWpX9LMen9T4BDgrt62anQi4uSI2C4idqR8Jr4TEe/qcbNGTdL0OgCGpOnAa4H18kuGCRl4EfE0MHjZ2mLg0g3xsjVJFwPfB3aVtFTSMb1u0xgcALyb0ptYVG+v73WjxmAWcIOk2yhfqNdFxAb7s44N3IuAmyTdCvwA+GZEfGt9PPCE/FmKmVk3TMgenplZNzjwzCwNB56ZpeHAM7M0HHhmloYDz8zScOCZWRoOPDNL4/8D0xzcspB58TEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "RMT_Util.plot_ESD_and_fit(model=None, eigenvalues=evals,Q=Q, num_spikes=2)\n",
    "plt.ylim([0,35])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Is this MP?  Very hard to tell with the resolution but doubtful"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "RMT_Util.ipynb:7: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  \"## RMT Util\\n\",\n",
      "RMT_Util.ipynb:7: RuntimeWarning: invalid value encountered in sqrt\n",
      "  \"## RMT Util\\n\",\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "% spikes outside bulk 7.14\n",
      "% sigma 0.4524\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "RMT_Util.plot_ESD_and_fit(model=None, eigenvalues=evals,Q=Q, num_spikes=20)\n",
    "plt.ylim([0,35])\n",
    "plt.xlim([0,1])\n",
    "plt.hist(evals,bins=1000);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
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