PyMC3 approach.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PyMC3 implementation of the model behind https://rt.live"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notebook by Maciek Zdanowicz & Grzegorz Kossakowski (@gkossakoski)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  % Total    % Received % Xferd  Average Speed   Time    Time     Time  Current\n",
      "                                 Dload  Upload   Total   Spent    Left  Speed\n",
      "100  691k  100  691k    0     0   553k      0  0:00:01  0:00:01 --:--:--  553k\n"
     ]
    }
   ],
   "source": [
    "!curl -o euData0420.csv https://opendata.ecdc.europa.eu/covid19/casedistribution/csv/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import pymc3 as pm\n",
    "import theano\n",
    "import theano.tensor as tt\n",
    "\n",
    "from matplotlib import pyplot as plt\n",
    "from matplotlib.dates import date2num, num2date\n",
    "from matplotlib import dates as mdates\n",
    "from matplotlib import ticker\n",
    "from matplotlib.colors import ListedColormap\n",
    "from matplotlib.patches import Patch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "filename = 'euData0420.csv'\n",
    "dataTotal = pd.read_csv(filename,\n",
    "                     usecols=['dateRep', 'countriesAndTerritories', 'cases'],\n",
    "                     parse_dates=['dateRep'],infer_datetime_format=True,\n",
    "                     index_col=['countriesAndTerritories', 'dateRep'],\n",
    "                     squeeze=True).sort_index()\n",
    "dataTotal.index.rename(names=['country', 'date'], inplace=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "smoothed = pd.read_pickle('smoothed_france')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "date\n",
       "2020-02-27      12.0\n",
       "2020-02-28      18.0\n",
       "2020-02-29      23.0\n",
       "2020-03-01      30.0\n",
       "2020-03-02      42.0\n",
       "2020-03-03      59.0\n",
       "2020-03-04      73.0\n",
       "2020-03-05     106.0\n",
       "2020-03-06     138.0\n",
       "2020-03-07     179.0\n",
       "2020-03-08     229.0\n",
       "2020-03-09     289.0\n",
       "2020-03-10     362.0\n",
       "2020-03-11     441.0\n",
       "2020-03-12     528.0\n",
       "2020-03-13     640.0\n",
       "2020-03-14     736.0\n",
       "2020-03-15     857.0\n",
       "2020-03-16    1003.0\n",
       "2020-03-17    1138.0\n",
       "2020-03-18    1282.0\n",
       "2020-03-19    1399.0\n",
       "2020-03-20    1641.0\n",
       "2020-03-21    1837.0\n",
       "2020-03-22    2053.0\n",
       "2020-03-23    2341.0\n",
       "2020-03-24    2626.0\n",
       "2020-03-25    2945.0\n",
       "2020-03-26    3163.0\n",
       "2020-03-27    3415.0\n",
       "2020-03-28    3851.0\n",
       "2020-03-29    4089.0\n",
       "2020-03-30    4231.0\n",
       "2020-03-31    4434.0\n",
       "2020-04-01    4500.0\n",
       "2020-04-02    4374.0\n",
       "2020-04-03    4230.0\n",
       "2020-04-04    4133.0\n",
       "2020-04-05    3947.0\n",
       "2020-04-06    3710.0\n",
       "2020-04-07    3708.0\n",
       "2020-04-08    3788.0\n",
       "2020-04-09    3609.0\n",
       "2020-04-10    3493.0\n",
       "2020-04-11    3603.0\n",
       "2020-04-12    3468.0\n",
       "2020-04-13    3346.0\n",
       "2020-04-14    3089.0\n",
       "2020-04-15    2895.0\n",
       "2020-04-16    2594.0\n",
       "2020-04-17    2386.0\n",
       "2020-04-18    2247.0\n",
       "2020-04-19    2052.0\n",
       "2020-04-20    1733.0\n",
       "2020-04-21    1642.0\n",
       "Name: France cases, dtype: float64"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smoothed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "country = 'France'\n",
    "dataEx = dataTotal.xs(country)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "dataEx = smoothed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Gamma is 1/serial interval\n",
    "# https://wwwnc.cdc.gov/eid/article/26/7/20-0282_article\n",
    "# https://www.nejm.org/doi/full/10.1056/NEJMoa2001316\n",
    "\n",
    "sigma=0.25 # some estimation for Gaussian increments, we want Laplace (see Systrom)\n",
    "GAMMA = 1/7\n",
    "MAX = 12.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model:\n",
    "\n",
    "$k_t$ - the number of new cases in time $t$\n",
    "\n",
    "$R_t$ - the reproduction rate in time $t$\n",
    "\n",
    "$\\gamma$ - serial interval - fixed in Systrom's model, Wikipedia states it is somewhere in $[4,8]$\n",
    "\n",
    "We assume that $R_t$ is a Markov process with Laplace increments of mean $0$ and standard deviation $\\sigma$.\n",
    "The distribution of $k_t$ is Poisson with parameter $\\lambda_t$ which satisfies the equation\n",
    "$\\lambda_t = k_{t-1} \\cdot e^{\\gamma (R_t - 1)}$.\n",
    "\n",
    "We may summarize it:\n",
    "\n",
    "$$\n",
    "\\lambda_t = k_{t-1} \\cdot e^{\\gamma (R_t - 1)}, \\quad \\quad k_t = {\\rm Poiss}(\\lambda_t), \\quad \\quad P(R_t | R_{t-1}) = {\\rm Laplace}(0,\\sigma)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Auto-assigning NUTS sampler...\n",
      "Initializing NUTS using jitter+adapt_diag...\n",
      "Multiprocess sampling (4 chains in 4 jobs)\n",
      "NUTS: [Diff_54, Diff_53, Diff_52, Diff_51, Diff_50, Diff_49, Diff_48, Diff_47, Diff_46, Diff_45, Diff_44, Diff_43, Diff_42, Diff_41, Diff_40, Diff_39, Diff_38, Diff_37, Diff_36, Diff_35, Diff_34, Diff_33, Diff_32, Diff_31, Diff_30, Diff_29, Diff_28, Diff_27, Diff_26, Diff_25, Diff_24, Diff_23, Diff_22, Diff_21, Diff_20, Diff_19, Diff_18, Diff_17, Diff_16, Diff_15, Diff_14, Diff_13, Diff_12, Diff_11, Diff_10, Diff_9, Diff_8, Diff_7, Diff_6, Diff_5, Diff_4, Diff_3, Diff_2, Diff_1, R_0]\n",
      "Sampling 4 chains, 0 divergences: 100%|██████████| 12000/12000 [18:36<00:00, 10.75draws/s]\n"
     ]
    }
   ],
   "source": [
    "N = dataEx.size\n",
    "k = dataEx.to_numpy()[:N]\n",
    "\n",
    "with pm.Model() as modelTest:\n",
    "    R = np.empty(N,dtype=object)\n",
    "    K = np.empty(N,dtype=object)\n",
    "    R[0] = pm.Gamma('R_0', alpha=4, beta=1)\n",
    "    K[0] = k[0]\n",
    "    for t in range(1,N):\n",
    "        R[t] = pm.Deterministic('R_{}'.format(t),R[t-1] + pm.Normal(\"Diff_{}\".format(t),0,sigma))\n",
    "        # par_t = pm.Deterministic('par_{}'.format(t),GAMMA*(R[t-1] - 1)) \n",
    "        # lambda_t = pm.Deterministic('lambda_{}'.format(t),K[t-1]*tt.exp(par_t))\n",
    "        # Faster that way I guess, PyMC3 does not keep track of the auxiliary variables\n",
    "        lambda_t = K[t-1]*tt.exp(GAMMA*(R[t-1] - 1))\n",
    "        K[t] = pm.Poisson('k_{}'.format(t), lambda_t, observed=k[t])\n",
    "    \n",
    "with modelTest:\n",
    "    trace = pm.sample(draws=1000, tune=2000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "def take_most_likely(a):\n",
    "    hist_counts,hist_vals = np.histogram(a, bins=100)\n",
    "    return hist_vals[hist_counts.argmax()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 0.1\n",
    "\n",
    "resLow, resHigh, ml = [], [], []\n",
    "\n",
    "for t in range(N):\n",
    "    val = np.sort(trace['R_{}'.format(t)])\n",
    "    LEN = len(val)\n",
    "    low, hi = int(p*LEN), int((1-p)*LEN)\n",
    "    # print(f'Low|Hi quantiles {t}: {val[low],val[hi]}')\n",
    "    resLow.append(val[low])\n",
    "    resHigh.append(val[hi])\n",
    "    ml.append(take_most_likely(val))\n",
    "\n",
    "resLow = np.array(resLow)\n",
    "resHigh = np.array(resHigh)\n",
    "res = pd.DataFrame(index=dataEx.index, data = {'Low_90': resLow, 'High_90': resHigh, 'ML': ml})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Low_90</th>\n",
       "      <th>High_90</th>\n",
       "      <th>ML</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2020-02-27</th>\n",
       "      <td>2.398826</td>\n",
       "      <td>3.698603</td>\n",
       "      <td>3.025561</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-02-28</th>\n",
       "      <td>2.431120</td>\n",
       "      <td>3.624032</td>\n",
       "      <td>3.095902</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-02-29</th>\n",
       "      <td>2.474328</td>\n",
       "      <td>3.558240</td>\n",
       "      <td>3.021949</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-01</th>\n",
       "      <td>2.530011</td>\n",
       "      <td>3.499814</td>\n",
       "      <td>2.925539</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-02</th>\n",
       "      <td>2.541681</td>\n",
       "      <td>3.443739</td>\n",
       "      <td>2.957557</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-03</th>\n",
       "      <td>2.528527</td>\n",
       "      <td>3.362808</td>\n",
       "      <td>3.019315</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-04</th>\n",
       "      <td>2.566451</td>\n",
       "      <td>3.326757</td>\n",
       "      <td>2.876124</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-05</th>\n",
       "      <td>2.519564</td>\n",
       "      <td>3.216834</td>\n",
       "      <td>2.807199</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-06</th>\n",
       "      <td>2.474652</td>\n",
       "      <td>3.106205</td>\n",
       "      <td>2.744519</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-07</th>\n",
       "      <td>2.402201</td>\n",
       "      <td>3.011192</td>\n",
       "      <td>2.772409</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-08</th>\n",
       "      <td>2.329247</td>\n",
       "      <td>2.900386</td>\n",
       "      <td>2.602952</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-09</th>\n",
       "      <td>2.245434</td>\n",
       "      <td>2.781843</td>\n",
       "      <td>2.527773</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-10</th>\n",
       "      <td>2.156864</td>\n",
       "      <td>2.659136</td>\n",
       "      <td>2.398748</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-11</th>\n",
       "      <td>2.060297</td>\n",
       "      <td>2.561442</td>\n",
       "      <td>2.326657</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-12</th>\n",
       "      <td>2.007948</td>\n",
       "      <td>2.465715</td>\n",
       "      <td>2.251474</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-13</th>\n",
       "      <td>1.882494</td>\n",
       "      <td>2.315894</td>\n",
       "      <td>2.119667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-14</th>\n",
       "      <td>1.851575</td>\n",
       "      <td>2.279750</td>\n",
       "      <td>2.071121</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-15</th>\n",
       "      <td>1.826948</td>\n",
       "      <td>2.223663</td>\n",
       "      <td>1.980137</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-16</th>\n",
       "      <td>1.716163</td>\n",
       "      <td>2.100804</td>\n",
       "      <td>1.879011</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-17</th>\n",
       "      <td>1.643022</td>\n",
       "      <td>2.015459</td>\n",
       "      <td>1.843521</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-18</th>\n",
       "      <td>1.583964</td>\n",
       "      <td>1.934129</td>\n",
       "      <td>1.758100</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-19</th>\n",
       "      <td>1.793399</td>\n",
       "      <td>2.123233</td>\n",
       "      <td>1.991776</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-20</th>\n",
       "      <td>1.671607</td>\n",
       "      <td>1.993979</td>\n",
       "      <td>1.829710</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-21</th>\n",
       "      <td>1.647964</td>\n",
       "      <td>1.965513</td>\n",
       "      <td>1.801338</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-22</th>\n",
       "      <td>1.721477</td>\n",
       "      <td>2.024090</td>\n",
       "      <td>1.831441</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-23</th>\n",
       "      <td>1.665423</td>\n",
       "      <td>1.952666</td>\n",
       "      <td>1.809986</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-24</th>\n",
       "      <td>1.615425</td>\n",
       "      <td>1.898017</td>\n",
       "      <td>1.796701</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-25</th>\n",
       "      <td>1.422754</td>\n",
       "      <td>1.687069</td>\n",
       "      <td>1.548545</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-26</th>\n",
       "      <td>1.439813</td>\n",
       "      <td>1.697435</td>\n",
       "      <td>1.585652</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-27</th>\n",
       "      <td>1.608532</td>\n",
       "      <td>1.870639</td>\n",
       "      <td>1.746689</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-28</th>\n",
       "      <td>1.321138</td>\n",
       "      <td>1.567393</td>\n",
       "      <td>1.457926</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-29</th>\n",
       "      <td>1.148918</td>\n",
       "      <td>1.389497</td>\n",
       "      <td>1.258454</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-30</th>\n",
       "      <td>1.169871</td>\n",
       "      <td>1.408014</td>\n",
       "      <td>1.256132</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-03-31</th>\n",
       "      <td>0.975224</td>\n",
       "      <td>1.210280</td>\n",
       "      <td>1.123444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-01</th>\n",
       "      <td>0.716842</td>\n",
       "      <td>0.958299</td>\n",
       "      <td>0.827251</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-02</th>\n",
       "      <td>0.659024</td>\n",
       "      <td>0.902505</td>\n",
       "      <td>0.816163</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-03</th>\n",
       "      <td>0.690805</td>\n",
       "      <td>0.931930</td>\n",
       "      <td>0.793910</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-04</th>\n",
       "      <td>0.570609</td>\n",
       "      <td>0.815398</td>\n",
       "      <td>0.692522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-05</th>\n",
       "      <td>0.515781</td>\n",
       "      <td>0.767876</td>\n",
       "      <td>0.643893</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-06</th>\n",
       "      <td>0.823843</td>\n",
       "      <td>1.081851</td>\n",
       "      <td>0.916734</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-07</th>\n",
       "      <td>0.933019</td>\n",
       "      <td>1.188725</td>\n",
       "      <td>1.023367</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-08</th>\n",
       "      <td>0.614974</td>\n",
       "      <td>0.867776</td>\n",
       "      <td>0.699753</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-09</th>\n",
       "      <td>0.686943</td>\n",
       "      <td>0.942757</td>\n",
       "      <td>0.799503</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-10</th>\n",
       "      <td>0.958107</td>\n",
       "      <td>1.221819</td>\n",
       "      <td>1.113353</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-11</th>\n",
       "      <td>0.654371</td>\n",
       "      <td>0.910222</td>\n",
       "      <td>0.765030</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-12</th>\n",
       "      <td>0.580937</td>\n",
       "      <td>0.847604</td>\n",
       "      <td>0.736318</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-13</th>\n",
       "      <td>0.362737</td>\n",
       "      <td>0.633801</td>\n",
       "      <td>0.473673</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-14</th>\n",
       "      <td>0.355153</td>\n",
       "      <td>0.637246</td>\n",
       "      <td>0.494022</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-15</th>\n",
       "      <td>0.170809</td>\n",
       "      <td>0.456652</td>\n",
       "      <td>0.268063</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-16</th>\n",
       "      <td>0.264531</td>\n",
       "      <td>0.555576</td>\n",
       "      <td>0.399014</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-17</th>\n",
       "      <td>0.337681</td>\n",
       "      <td>0.642430</td>\n",
       "      <td>0.520424</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-18</th>\n",
       "      <td>0.175322</td>\n",
       "      <td>0.489760</td>\n",
       "      <td>0.277117</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-19</th>\n",
       "      <td>-0.082115</td>\n",
       "      <td>0.251544</td>\n",
       "      <td>0.085771</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-20</th>\n",
       "      <td>0.253306</td>\n",
       "      <td>0.638099</td>\n",
       "      <td>0.458854</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-04-21</th>\n",
       "      <td>0.083025</td>\n",
       "      <td>0.820160</td>\n",
       "      <td>0.423753</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              Low_90   High_90        ML\n",
       "date                                    \n",
       "2020-02-27  2.398826  3.698603  3.025561\n",
       "2020-02-28  2.431120  3.624032  3.095902\n",
       "2020-02-29  2.474328  3.558240  3.021949\n",
       "2020-03-01  2.530011  3.499814  2.925539\n",
       "2020-03-02  2.541681  3.443739  2.957557\n",
       "2020-03-03  2.528527  3.362808  3.019315\n",
       "2020-03-04  2.566451  3.326757  2.876124\n",
       "2020-03-05  2.519564  3.216834  2.807199\n",
       "2020-03-06  2.474652  3.106205  2.744519\n",
       "2020-03-07  2.402201  3.011192  2.772409\n",
       "2020-03-08  2.329247  2.900386  2.602952\n",
       "2020-03-09  2.245434  2.781843  2.527773\n",
       "2020-03-10  2.156864  2.659136  2.398748\n",
       "2020-03-11  2.060297  2.561442  2.326657\n",
       "2020-03-12  2.007948  2.465715  2.251474\n",
       "2020-03-13  1.882494  2.315894  2.119667\n",
       "2020-03-14  1.851575  2.279750  2.071121\n",
       "2020-03-15  1.826948  2.223663  1.980137\n",
       "2020-03-16  1.716163  2.100804  1.879011\n",
       "2020-03-17  1.643022  2.015459  1.843521\n",
       "2020-03-18  1.583964  1.934129  1.758100\n",
       "2020-03-19  1.793399  2.123233  1.991776\n",
       "2020-03-20  1.671607  1.993979  1.829710\n",
       "2020-03-21  1.647964  1.965513  1.801338\n",
       "2020-03-22  1.721477  2.024090  1.831441\n",
       "2020-03-23  1.665423  1.952666  1.809986\n",
       "2020-03-24  1.615425  1.898017  1.796701\n",
       "2020-03-25  1.422754  1.687069  1.548545\n",
       "2020-03-26  1.439813  1.697435  1.585652\n",
       "2020-03-27  1.608532  1.870639  1.746689\n",
       "2020-03-28  1.321138  1.567393  1.457926\n",
       "2020-03-29  1.148918  1.389497  1.258454\n",
       "2020-03-30  1.169871  1.408014  1.256132\n",
       "2020-03-31  0.975224  1.210280  1.123444\n",
       "2020-04-01  0.716842  0.958299  0.827251\n",
       "2020-04-02  0.659024  0.902505  0.816163\n",
       "2020-04-03  0.690805  0.931930  0.793910\n",
       "2020-04-04  0.570609  0.815398  0.692522\n",
       "2020-04-05  0.515781  0.767876  0.643893\n",
       "2020-04-06  0.823843  1.081851  0.916734\n",
       "2020-04-07  0.933019  1.188725  1.023367\n",
       "2020-04-08  0.614974  0.867776  0.699753\n",
       "2020-04-09  0.686943  0.942757  0.799503\n",
       "2020-04-10  0.958107  1.221819  1.113353\n",
       "2020-04-11  0.654371  0.910222  0.765030\n",
       "2020-04-12  0.580937  0.847604  0.736318\n",
       "2020-04-13  0.362737  0.633801  0.473673\n",
       "2020-04-14  0.355153  0.637246  0.494022\n",
       "2020-04-15  0.170809  0.456652  0.268063\n",
       "2020-04-16  0.264531  0.555576  0.399014\n",
       "2020-04-17  0.337681  0.642430  0.520424\n",
       "2020-04-18  0.175322  0.489760  0.277117\n",
       "2020-04-19 -0.082115  0.251544  0.085771\n",
       "2020-04-20  0.253306  0.638099  0.458854\n",
       "2020-04-21  0.083025  0.820160  0.423753"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "res.to_pickle('pymc')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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