parameter_estimation_robbins_monro.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy.stats import truncnorm, invgamma, norm\n",
    "import pickle\n",
    "import time\n",
    "import math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The autoreload extension is already loaded. To reload it, use:\n",
      "  %reload_ext autoreload\n"
     ]
    }
   ],
   "source": [
    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "from elec_forecast.bootstrap_filter_model import BootstrapFilterModel\n",
    "from elec_forecast.rm_estimation import RmEstimation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Load temperatures data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "df_temp=pd.read_csv('data/temp_2016.csv',sep=',')\n",
    "df_temp=df_temp.drop(df_temp.columns[0],axis=1)\n",
    "df_temp=df_temp.sort_values(by='date')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#temperatures every day at 3am\n",
    "hour = \"03:00\"\n",
    "temp_day=df_temp[df_temp.hour==hour]\n",
    "temp_day.head()\n",
    "T_h=list(temp_day.t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Load electricity data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mypath=\"data/\"\n",
    "df=pd.read_csv(mypath+\"cdc_conso_daytypes.csv\",sep=\",\",encoding='latin-1')\n",
    "df.index=range(0,len(df))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "df=df.sort_values(by=['Date'])\n",
    "df.index=range(0,len(df))\n",
    "#Valeurs intermédiaires à partir de 2017"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#On se concentre d'abord sur une année afin d'initialiser le filtre\n",
    "df=df[(df.Date >'2016-01-01') & (df.Date <'2017-01-01')]\n",
    "df.index=range(0,len(df))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Robbins Monroe parameters estimation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'g_heat_init': array([-1139373.50765916,  -270209.1419129 ,  -449797.54899782, ...,\n",
       "          -97510.92882799,   -73071.4470343 ,   -88779.62566618]),\n",
       " 'kappa': array([ 0.11111111,  0.11111111,  0.11111111,  0.11111111,  0.11111111,\n",
       "         0.11111111,  0.11111111,  0.11111111,  0.11111111]),\n",
       " 's_init': array([  2.53971204e+08,   1.74479333e+07,   5.59420323e+06, ...,\n",
       "          1.44451618e-06,   5.79654646e-07,   8.18654197e-07]),\n",
       " 'sigma2': 27000000,\n",
       " 'sigma_g_init': 210436.23817816001,\n",
       " 'sigma_s_init': 6.7179250479583413e-07,\n",
       " 'u_h': 7.3,\n",
       " 'w_init': array([  0.00000000e+00,   0.00000000e+00,   0.00000000e+00, ...,\n",
       "          2.86678274e-38,   2.86678274e-38,   2.86678274e-38]),\n",
       " 'x_init': array([  2.82190226e+07,   1.93865926e+06,   6.21578137e+05, ...,\n",
       "          1.60501798e-07,   6.44060718e-08,   9.09615775e-08])}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Steps to Load pickle file\n",
    "input_file ='data/parameters_init_20180110-024129.pkl'\n",
    "pkl_file = open(input_file, 'rb')\n",
    "parameters_init = pickle.load(pkl_file)\n",
    "parameters_init"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sigma_init = math.sqrt(np.median(parameters_init[\"sigma2\"]))\n",
    "sigma_g_init = math.sqrt(np.median(parameters_init[\"sigma_g_init\"]))\n",
    "sigma_s_init = math.sqrt(np.median(parameters_init[\"sigma_s_init\"]))\n",
    "u_heat_init = math.sqrt(np.median(parameters_init[\"u_h\"]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sigma_init = 10**3\n",
    "sigma_g_init = 10**3\n",
    "sigma_s_init = 10**3\n",
    "u_heat_init = 13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "variance_epsilon = np.diag(np.array(([10**2,10**2,1,10**2])))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def gamma(n):\n",
    "    return 1/(10*n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0210384756659057, 4228.713860454699)"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test = truncnorm.rvs(a=0,b=np.inf,scale=10**3,size=100000)\n",
    "MIN, MAX = np.min(test), np.max(test)\n",
    "MIN, MAX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f_sigma_s(sigma_s): \n",
    "    sigma_s_min, sigma_s_max = MIN, MAX\n",
    "    if sigma_s < sigma_s_min :\n",
    "        f_sigma_s = sigma_s_min\n",
    "    elif sigma_s > sigma_s_max :\n",
    "        f_sigma_s = sigma_s_max\n",
    "    else :\n",
    "        f_sigma_s = sigma_s\n",
    "    \n",
    "    return f_sigma_s\n",
    "\n",
    "def f_sigma_g(sigma_g):\n",
    "    sigma_g_min, sigma_g_max = MIN, MAX\n",
    "    if sigma_g < sigma_g_min :\n",
    "        f_sigma_g = sigma_g_min\n",
    "    elif sigma_g > sigma_g_max :\n",
    "        f_sigma_g = sigma_g_max\n",
    "    else :\n",
    "        f_sigma_g = 10*sigma_g\n",
    "    \n",
    "    return f_sigma_g \n",
    "\n",
    "def f_u_heat(u_heat):\n",
    "    u_heat_min = 0\n",
    "    u_heat_max = 30\n",
    "    if u_heat < u_heat_min :\n",
    "        f_u_heat = u_heat_min\n",
    "    elif u_heat > u_heat_max :\n",
    "        f_u_heat = np.sqrt(u_heat_max)\n",
    "    else:\n",
    "        f_u_heat = u_heat\n",
    "        \n",
    "    return f_u_heat\n",
    "\n",
    "def f_sigma(sigma):\n",
    "    sigma_min, sigma_max = MIN, MAX\n",
    "    if sigma < sigma_min :\n",
    "        f_sigma = sigma_min\n",
    "    elif sigma > sigma_max :\n",
    "        f_sigma = sigma_max\n",
    "    else:\n",
    "        f_sigma = 100*sigma\n",
    "        \n",
    "    return f_sigma\n",
    "\n",
    "def f(sigma_s, sigma_g, u_heat, sigma):\n",
    "    #sigma_s\n",
    "    result_sigma_s = f_sigma_s(sigma_s) \n",
    "    #sigma_g\n",
    "    result_sigma_g = f_sigma_g(sigma_g) \n",
    "    #u_heat\n",
    "    result_u_heat = f_u_heat(u_heat)\n",
    "    #sigma\n",
    "    result_sigma = f_sigma(sigma) \n",
    "    \n",
    "    return result_sigma_s, result_sigma_g, result_u_heat, result_sigma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "iterations = 10000\n",
    "RME = RmEstimation(sigma_s_init, sigma_g_init, u_heat_init, sigma_init, variance_epsilon, iterations, f, gamma)\n",
    "RME.estimation()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {
    "collapsed": true,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "#RME.cm_u_heat, RME.cm_sigma, RME.cm_sigma_g, RME.cm_sigma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {
    "collapsed": true,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "#RME.u_heat[0], RME.sigma, RME.sigma_g, RME.sigma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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AkqQJ5nvMwAZgFXBraUNKqQu4A7h6qjdERGNEtJcWoG2eywRATR5cnFooSdJE8x0GVuXr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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1de54300d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(RME.cm_u_heat[0,1:])\n",
    "plt.title('$u_{heat}$ Cesaro Mean')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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H8ueU7x96P3z/wMDwsgPj3p8YGBgqm/Jt/cWydfv7C69TylZTr5VJDG2vvR/I\nNmbHFMrC8O+SFxusX61sGvycWlmAYn3hrv95WqP+aAwyuFRQS+1SkWNcpF1W8Qe6fyDR1z/0g1v7\nYc3eZ9vqf6y399g2MPQDPjAw/Lk/Jfr7B+hP0D8wQP9A3XPJcQO1oJAK56j/7PpgMZCGfXbpMSnR\n31+oX75d1ZVSavhCqQaXCmprscdFmioDA4mt/QP09g3Q299Pb9/A0Ptt/WzdNkDvtoHB55Hb+gv7\nRpbJyvUP+4zBANI/9CPd1z8w+MNfe9bEtES2ZERrS4x4tLUELRG0tebbYvi+4WVbaGkJWgNaIgYn\nRbRE0NKSTZJoLWyr398SkT9qEyry/S2F1+PYH4VyrS0xvGxxfwsEQ+8Hn8k+v/h+3GWjNot15Pmy\ntt5+2SowuFTQ4BgX/2JTE0spCwzjCQbDwkT/AL19w4PAYLmSMDHatr4mGytW/HFuawlaW4d+eEf8\naMfIH/Da9rbW/Ie8JWhpGf482nGtrfnx2zmu9nmtrS2DZUpDRDEwDKtPy4iyox3fUjhPo/+lr2ow\nuFRQLfHaZarpMDCQ2LR1Gxt7t7GpdxvP9NRe99PTlz229PXT0zcw+L64bUv+vtZjMTyUDA8QVRMB\nnW0tdLa10tHWQmdbS/7cWnjdMlimuG175Try953tLXS0ttDZ3kpHawvtrcODRlvr0A9yW2tLIYjU\nlct/sCWVM7hUUK3HZaB6f++rQVJK9PQNsLE3Cxkb87CRPfrY2Nufb+vLn/vz7fnrnr7BcLKxd1tD\nvkNHWwudrcN/5McOBvXbWoeFi+L+oQDRWjjH8GPa/Fe71PQMLhXkOi67poGBxDO92+je3Ef3luyx\nfsvW7HlzHxtq2zbX9mXbagFlqnvg2lqCOTPamN3RxtwZbczqaGVmRysz21vpbM+eZ7S35M/Fx9C2\n7fdeDPU81AJER2uLPQmSpoTBpYIGg4tdLpXT1z/AMz3b2LClL3vu6Rt8XR9EuvPgsX7L0OsdzR4R\nMKejjdmdbcyZ0caczix0zO4Yej+ns+51/fv8dWdbi70PkpqOwaWCWgfHuDS4IruB3m39rOnu5fHu\nLazu3sLq7h5Wr+/hqU29eTDZxjM9WTB5pqePnr4d/48ys72VeTPb2WNWO10z27PXtedZ2XPXzHb2\nmNVB14w25s5oz8JJZxuz2lvtuZC0WzO4VJDruEydpzdtZeWTm3js6c080d3D6u4eHl+fB5TuHp7c\n2Dupz53II779AAAQ0UlEQVTd0UrXzCxQdOXBYo9ZHUOhoy6IzJvZzrz8dWdb6xR/S0nafRhcKqjN\n4DIhPX39PPLUZlau28jKJzexct0mVj25kVVPbuLpzX1jHt/Z1sLieTNYPG8mi/eYweJ5M1gwpzMP\nJlkomdM5FFDmzmijrdW7ZUhSIxhcKmhwOrSDcwcNDCQe796Sh5Ls8dC6LJz8dv0WRmuqfefNYL89\nZ7HvvBks3mPmUEiZN4N995jJ/FntjvWQpCZhcKmgoenQu19w2di7jRVrN7Ji7UZWPblxWFAZbV2Q\nuTPaOHjBHA7eezYH7z2bgxbM5uC953Dg3rOY1eEfc0naVfg3egW17uKXilJKrO7uYeW6TazMw8lD\n67Kwsrq7p/S49tbg2XvN5qC9Z3PwgiygHLxgDgftPZu9ZnfYayJJuwGDSwXtSkv+d2/u4xePPs0v\nfrOeFWuzMSgPP7mJLX39pccsmNvJIQvm8Jx9ZnPQ3nkvyoLZPGuPmY4tkaTdnMGlgmrToZvxJotP\nbezlxw89xY9//SQ/e+R3PLRu03bLtbUEB+w1i4P3nsNzFmTB5JB95nDIgrnMm9W+k2stSWoWBpcK\nam/Lgktfkyzk8vj6LfzbXY/z9btXc89vu0fsP3CvWRx9wHyW7tvFwQuyXpT95s+k3d4TSdIEGVwq\nqPaDXuW72q7u3sK37n2Cb9yzmv98+Olh+5YsmstJh+zNCc/ZixccMJ89Z3c0qJaSpF2NwaWC2lpq\nwaVaPS6PPLWJb937BN+89wl++ej6we0RcNyBe/Lqo/bl1KUL2WfujAbWUpK0KzO4VFBHhS4VPbRu\nI9+4ezXfuPcJ7l+9YXB7BBz77PmcfsRiXvm8RSyeN7OBtZQk7S6mLbhExNHAlcALgX7gq8AFKaWN\nhTIHAJ8BXgZsBG4ELk4pbSuUORK4Nv+cdcD/SSl9tLB/MfBx4FjgEOBTKaV3TaCeewF3Ac8C5qeU\n1o9xyLSrXSraOsq6JdOpr3+Ab9yzms/9aBV3PTY0ZqW1JXjRwXty+hGLefnShezTZc+KJGnnmpbg\nEhH7ArcD/wi8HegCrgZuAF6fl2kFbgaeAE4EFgNfAPqA9+dluoBb8886D3ge8LmIWJ9Sui4/XSdZ\noPkw8O5JVPd64G6y4FIJjRrjsqGnjy//x2+44ccP83i+nkpbS/D7h+zNq563mFOWLnS8iiSpoaar\nx+UMsgByfkppACAizgPujohDUkorgNOApcApKaU1wC8j4oPAlRFxaUppK/AGoAM4J3+/PCKOAi4A\nrgNIKT0MvDM/xzkTqWRE/DWwB3AZ8Iod/M5TZii47Jwel5XrNvLFf/8N//ifv2HT1mx9lb3ndPCm\nEw7kDccfwF5zOndKPSRJGst0BZdOYGsttOS25M8nASuAE4B78tBScwvZpaPDgV/kZX6Qh5ZimfdG\nxPyU0vDpLBMQEUuBS4DjgYMn+znToaN1ese49A8k7vltN995YC2337eG+wpjV35vnzm85cUH8Zqj\nnsWMdu9iLEmqlukKLt8BroqIC4FPArOBK/J9i/PnRcCauuPWFPbVnleNUmZSwSUiOoF/AC5MKf0m\nIsYVXPLjit0Pcydz/rEMjnGZwuDy9KatfP/BdXzngbX84NfrWF+4a3JLwMsO24e/OOHZvOTQBS6d\nL0mqrAkFl4i4AnjvGMWem1JaHhFvBq4CLicbnPspstDR+KkyWZ3uTyl9cYLHXQz8z2mozzDtbfml\nom07PsblwTXP8InbHuS2+9YMu4XA3M42fv+QvTll6UJedtgCLwdJkprCRHtcPk42wHY0KwFSSsuA\nZRGxENgEJLKxKSvzck8Ax9Udu7Cwr/a8cIwyk3Ey8LyIeH3+vtbF8GREfCSlVBZOLicLYzVzgcd2\noB7bNRVjXHr6+vnkt3/N3/9g5WBgWbJoLicv2YeXLdmHF+y/h/f9kSQ1nQkFl5TSOrIZPBM5Zg0M\nDpztAW7Ld90BfCAi9kkprc23nQpsAO4rlPlIRLSnlPoKZX61I+NbgD8BiguPvBD4HPBi4KFRvksv\n0Ft7P12XVHZ0jMvq7i2c+4U7B5ffP3XpQt5z2qEsWdQ1ZXWUJKkRpnMdl7cDPyFbn+VU4GPA+wrr\npNxKFlBuioiLyMasfBi4Ng8IAMvILs1cHxFXAkeQzSAaNu05n2kEMAdYkL/fmlK6L9//OuDylNIS\ngJTSQ3XH752/vL9S67hMIrg8uOYZ3vy5/2B1dw/zZ7Vz5Z8cyWmHLxr7QEmSmsB0rpx7HPAhsjDx\nAPDWlNJNtZ0ppf6IOINsFtEdZJeTbiSb6VMr0x0Rp5EtQHcn8CRwWWENl5pfFF4fA5wFPAIcmG+b\nBxw2Zd9smk12HZf/WPU73nLjf7KhZxvPWTCbG/7yOPbfc9Z0VFGSpIaYtuCSUnrTOMo8ArxyjDJ3\nk13CGa3MqNdsUko3MMrYnJTS9xga59JwQ4Nzx9/j8o17VvOuf/wlW7cNcOyz5/PZNx/LHrNcLE6S\ntGvxXkUVNNExLl/890f44NfuJSV4+eEL+eSfvcA1WCRJuySDSwVNZIzLj3795GBo+YsXPZtLX304\nrS2V6TySJGlKGVwqaLzTobds7eeif76LlODMY/fnstcc7uJxkqRdmgt5VNB4B+de/6OVPN7dw77z\nZnDpqw0tkqRdn8Glgjraxh7j8tjTm7n2u9ms7ve+YgkzOxzTIkna9RlcKmhwjEvJrKKUEpd8bTlb\n+vo57sA9+aMj992Z1ZMkqWEMLhU01hiXb977BN95YC3trcHf/PERtDgYV5K0mzC4VNBoY1ye6enj\n0v+3HIC/fukhHLLPtNygWpKkSjK4VFDHKD0uV932IGuf6eXAvWbxtpc+Z2dXTZKkhjK4VFB7yeDc\n7z+4js//+GEALnvNES4yJ0na7RhcKmh7g3M39W7jfV+9G4A3vugA/uDQBQ2pmyRJjWRwqaCOujEu\nvdv6ed+/3MPq7h7233Mm/+NVSxtZPUmSGsbgUkH1s4qu/e5D/NtdjxMBH3nt87xEJEnabRlcKqg9\nv8nitoHEwEDi+79aC8BFL1/iJSJJ0m7N4FJB7W1D/1me6d3G8sc3AHDGkYsbVSVJkirB4FJBtTEu\nAL/4zdNsG0jsPaeT/ebPbGCtJElqPINLBbUXgst/rPodAEcfsIc3UZQk7fYMLhXU2hLUVvGvBZcX\nHDC/gTWSJKkaDC4VVet1+dkjTwPwggP2aGR1JEmqBINLRRXHubQEHLnfvAbWRpKkajC4VFRxZtHB\nC+Ywq6OtgbWRJKkaDC4VNbtzaJG5JYu8A7QkSWBwqaw9ZnYMvn7u4q4G1kSSpOowuFTUHrPaB18/\nd7E9LpIkgcGlsnoLd4Y+9sA9G1gTSZKqw+BSUS86KAsri7pm0DWjfYzSkiTtHpyqUlHnn3wIHW0t\nnLxkYaOrIklSZRhcKqqzrZW3n/x7ja6GJEmV4qUiSZLUNAwukiSpaRhcJElS0zC4SJKkpmFwkSRJ\nTcPgIkmSmobBRZIkNQ2DiyRJahoGF0mS1DQMLpIkqWm45P8U2LBhQ6OrIElSU5nsb2eklKa4KruP\niHgW8Fij6yFJUhPbL6X02/EWNrjsgIgIYF/gmSn82LlkYWi/Kf7c3ZltOrVsz6lnm04t23PqTVeb\nzgUeTxMII14q2gF5Q487JY5HloUAeCal5DWoKWCbTi3bc+rZplPL9px609imE/4sB+dKkqSmYXCR\nJElNw+BSPb3Ah/JnTQ3bdGrZnlPPNp1atufUq0ybOjhXkiQ1DXtcJElS0zC4SJKkpmFwkSRJTcPg\nIkmSmobBpWIi4vyIeDgieiLipxFxXKPr1GgRcXFE/GdEPBMRayPiXyPisLoyERGXRcTqiNgSEbdH\nxO/VlZkREddGxFMRsTEivhoRC+vK7BkRX4qIDRGxPiKuj4g5O+N7NkpEvC8iUkRcXdhme05QRDwr\nIr6Yt8eWiLgnIo4t7LdNJyAiWiPif0XEqry9HoqID0ZhJTTbtFxE/EFE/FtEPJ7///3auv07re0i\n4oCIuDkiNkf2d/jHImLyC+CmlHxU5AGcSTbV7C+BpcB1wNPAPo2uW4Pb5VvA2cDhwPOBm4FHgNmF\nMu8F1gOvAY4EvgasBGYUynwG+A1wMnAMcAfw47pzfRP4JXA8cBLwa2BZo9tgGtv2hcAq4C7gattz\n0u04H3gY+DxwHHAQcBrwHNt00m36fuBJ4FXAgcDryZaaf4dtOq72ewXwYeB1QAJeW7d/p7Qd0Arc\nA9wGHJXXax3wN5P+bo1uXB/D/gD8FLim8L6F7JYC72t03ar0ABbk/yP+Qf4+gNXAfy+UmQf0AH9W\neL8VeH2hzJL8c16Uv39u/v7YQpnTgQFg30Z/72loxznAg8ApwPfIg4vtOam2vAL44Sj7bdOJt+nX\ngevrtn0V+KJtOuG2HBZcdmbbkQWVfmBhocx5QDfQMZnv46WiioiIDrJEe3ttW0ppIH9/QqPqVVHz\n8uff5c8HAYsY3nbdZEGw1nbHAO11ZR4g+9dErcwJwPqU0s8K57qd7H/C46f2K1TCtcDNKaXb67bb\nnhP3auBnEfFPeVf4LyLivxb226YT9xPgDyPiUICIeD7Zv+i/me+3TSdvZ7bdCcA9KaU1hTK3AF1k\nvegT5k0Wq2Nvsi61NXXb15ClXAER0QJcTdZdeW++eVH+vL22W1QoszWltH6MMmuLO1NK2yLid4Uy\nu4SI+DPgaLJLRfVsz4k7GPhr4Crgb8ja9VMRsTWldCO26WRcQfbj9kBE9JP9/fiBlNKX8v226eTt\nzLZbVHKeYj0mxOCiZnMtcATZv7w0CRGxP/BJ4NSUUk+j67OLaAF+llJ6f/7+FxFxBFmX+I2Nq1ZT\n+1PgDcBZwHKy8RFXR8TjeRjUbspLRdXxJPl1wLrtC4Endn51qicirgHOAF6WUnqssKvWPqO13RNA\nR0TsMUaZferO2Qbsya713+AYsu/584jYFhHbgJcA78hf1/41ZHuO32rgvrpt9wMH5K/9MzpxHwOu\nTCl9OaV0T0rpJuATwMX5ftt08nZm2z1Rcp5iPSbE4FIRKaWtwJ3AH9a25ZdF/pBsJPduK5+2dw3Z\n6PiTU0qr6oqsIvsfoNh2XWTXWGttdyfQV1fmMLIfllqZO4A9IuKYwmefTPb/yU+n7As13reB55H9\nC7b2+Bnwpfz1SmzPifoxcFjdtkPJZr+Bf0YnYxawrW5bP0O/W7bp5O3MtrsDeF5EFAPOqcAGRob9\n8Wn0aGcfw0Z+n0k2qvvNZKO1/45sOvTCRtetwe3yabJpey8huyZae8wslHlv3lavJvtR/le2P7Xv\nEeBlZL0OPwF+UneubwI/J5vS+vtks26aelrkONv4e4ycDm17jr/9Xkj2l/z7gUPILm9sAt5gm066\nTW8AHmNoOvTryKbRXmmbjqv95jD0D5MEvDt/fcDObDuGpkPfQracxcvJxsU4HXpXeQBvz/+g9JIl\n1uMbXadGP/L/6bb3OLtQJoDLyP4V0UM2sv3Qus+ZQTZG5nf5j8q/AIvqyuwJLCNbL6Ib+Bwwp9Ft\nsBPa+HsMDy6258Tb8Iz8L+gesstE/7Vuv206sfacSzYQ/xFgC/AQ2bokHYUytml5+7205O/NG3Z2\n2wHPBr4BbCYLn/8baJvsd4v8QyVJkirPMS6SJKlpGFwkSVLTMLhIkqSmYXCRJElNw+AiSZKahsFF\nkiQ1DYOLJElqGgYXSZLUNAwukiSpaRhcJElS0zC4SJKkpmFwkSRJTeP/Awnd66aVu0bMAAAAAElF\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1de55611128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(RME.cm_sigma[0,1:])\n",
    "plt.title('$\\sigma$ Cesaro Mean')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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BLwCklFJEXJaXuZcsQLwf2Ax8Oy/TGRGfBz4WETuBLuCTwPXVWiEBhR4GV0lIknSQSgPD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\nSRqXgUGSJI3LwCBJksb1/wG508rOHexe5wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1de55cd0908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(RME.cm_sigma_s[0,1:])\n",
    "plt.title('$\\sigma_s$ Cesaro Mean')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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pEsOEJEmqxDDRAe+AKUlSe4YJSZJUiWGiBOdfSpI0lWFCkiRVYpiQJEmVGCY6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\n/w7RVc4lgg4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1de5627f978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(RME.cm_sigma_g[0,1:])\n",
    "plt.title('$\\sigma_g$ Cesaro Mean')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [conda env:py36]",
   "language": "python",
   "name": "conda-env-py36-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}