examples/Bootstrap.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bootstrap"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Import and settings\n",
    "In this example, we need to import `numpy`, `pandas`, and `graphviz` in addition to `lingam`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:04.843568Z",
     "start_time": "2021-06-25T01:43:00.402248Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['1.24.4', '2.0.3', '0.20.1', '1.8.3']\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import graphviz\n",
    "import lingam\n",
    "from lingam.utils import print_causal_directions, print_dagc, make_dot\n",
    "\n",
    "print([np.__version__, pd.__version__, graphviz.__version__, lingam.__version__])\n",
    "\n",
    "np.set_printoptions(precision=3, suppress=True)\n",
    "np.random.seed(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test data\n",
    "We create test data consisting of 6 variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:04.916646Z",
     "start_time": "2021-06-25T01:43:04.846013Z"
    }
   },
   "outputs": [
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       "      <th>0</th>\n",
       "      <td>2.239321</td>\n",
       "      <td>15.340724</td>\n",
       "      <td>4.104399</td>\n",
       "      <td>0.548814</td>\n",
       "      <td>14.176947</td>\n",
       "      <td>9.249925</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2.155632</td>\n",
       "      <td>16.630954</td>\n",
       "      <td>4.767220</td>\n",
       "      <td>0.715189</td>\n",
       "      <td>12.775458</td>\n",
       "      <td>9.189045</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>2.284116</td>\n",
       "      <td>15.910406</td>\n",
       "      <td>4.139736</td>\n",
       "      <td>0.602763</td>\n",
       "      <td>14.201794</td>\n",
       "      <td>9.273880</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2.343420</td>\n",
       "      <td>14.921457</td>\n",
       "      <td>3.519820</td>\n",
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       "      <td>7.604743</td>\n",
       "      <td>5.312976</td>\n",
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       "         x0         x1        x2        x3         x4        x5\n",
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    "x3 = np.random.uniform(size=1000)\n",
    "x0 = 3.0*x3 + np.random.uniform(size=1000)\n",
    "x2 = 6.0*x3 + np.random.uniform(size=1000)\n",
    "x1 = 3.0*x0 + 2.0*x2 + np.random.uniform(size=1000)\n",
    "x5 = 4.0*x0 + np.random.uniform(size=1000)\n",
    "x4 = 8.0*x0 - 1.0*x2 + np.random.uniform(size=1000)\n",
    "X = pd.DataFrame(np.array([x0, x1, x2, x3, x4, x5]).T ,columns=['x0', 'x1', 'x2', 'x3', 'x4', 'x5'])\n",
    "X.head()"
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   "source": [
    "m = np.array([[0.0, 0.0, 0.0, 3.0, 0.0, 0.0],\n",
    "              [3.0, 0.0, 2.0, 0.0, 0.0, 0.0],\n",
    "              [0.0, 0.0, 0.0, 6.0, 0.0, 0.0],\n",
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    "              [4.0, 0.0, 0.0, 0.0, 0.0, 0.0]])\n",
    "\n",
    "make_dot(m)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bootstrapping\n",
    "We call `bootstrap()` method instead of `fit()`. Here, the second argument specifies the number of bootstrap sampling."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.155729Z",
     "start_time": "2021-06-25T01:43:05.634360Z"
    }
   },
   "outputs": [],
   "source": [
    "model = lingam.DirectLiNGAM()\n",
    "result = model.bootstrap(X, n_sampling=100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Causal Directions\n",
    "Since `BootstrapResult` object is returned, we can get the ranking of the causal directions extracted by `get_causal_direction_counts()` method. In the following sample code, `n_directions` option is limited to the causal directions of the top 8 rankings, and `min_causal_effect` option is limited to causal directions with a coefficient of 0.01 or more."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.187708Z",
     "start_time": "2021-06-25T01:43:15.157773Z"
    }
   },
   "outputs": [],
   "source": [
    "cdc = result.get_causal_direction_counts(n_directions=8, min_causal_effect=0.01, split_by_causal_effect_sign=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check the result by utility function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.218627Z",
     "start_time": "2021-06-25T01:43:15.189703Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x5 <--- x0 (b>0) (100.0%)\n",
      "x1 <--- x0 (b>0) (100.0%)\n",
      "x1 <--- x2 (b>0) (100.0%)\n",
      "x4 <--- x2 (b<0) (100.0%)\n",
      "x0 <--- x3 (b>0) (98.0%)\n",
      "x4 <--- x0 (b>0) (98.0%)\n",
      "x2 <--- x3 (b>0) (96.0%)\n",
      "x3 <--- x2 (b>0) (4.0%)\n"
     ]
    }
   ],
   "source": [
    "print_causal_directions(cdc, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Directed Acyclic Graphs\n",
    "Also, using the `get_directed_acyclic_graph_counts()` method, we can get the ranking of the DAGs extracted. In the following sample code, `n_dags` option is limited to the dags of the top 3 rankings, and `min_causal_effect` option is limited to causal directions with a coefficient of 0.01 or more."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.261954Z",
     "start_time": "2021-06-25T01:43:15.223614Z"
    }
   },
   "outputs": [],
   "source": [
    "dagc = result.get_directed_acyclic_graph_counts(n_dags=3, min_causal_effect=0.01, split_by_causal_effect_sign=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check the result by utility function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.307514Z",
     "start_time": "2021-06-25T01:43:15.261954Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "DAG[0]: 84.0%\n",
      "\tx0 <--- x3 (b>0)\n",
      "\tx1 <--- x0 (b>0)\n",
      "\tx1 <--- x2 (b>0)\n",
      "\tx2 <--- x3 (b>0)\n",
      "\tx4 <--- x0 (b>0)\n",
      "\tx4 <--- x2 (b<0)\n",
      "\tx5 <--- x0 (b>0)\n",
      "DAG[1]: 3.0%\n",
      "\tx0 <--- x3 (b>0)\n",
      "\tx1 <--- x0 (b>0)\n",
      "\tx1 <--- x2 (b>0)\n",
      "\tx3 <--- x2 (b>0)\n",
      "\tx4 <--- x0 (b>0)\n",
      "\tx4 <--- x2 (b<0)\n",
      "\tx5 <--- x0 (b>0)\n",
      "DAG[2]: 2.0%\n",
      "\tx0 <--- x3 (b>0)\n",
      "\tx1 <--- x0 (b>0)\n",
      "\tx1 <--- x2 (b>0)\n",
      "\tx1 <--- x3 (b<0)\n",
      "\tx2 <--- x3 (b>0)\n",
      "\tx4 <--- x0 (b>0)\n",
      "\tx4 <--- x2 (b<0)\n",
      "\tx5 <--- x0 (b>0)\n"
     ]
    }
   ],
   "source": [
    "print_dagc(dagc, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probability\n",
    "Using the `get_probabilities()` method, we can get the probability of bootstrapping."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.323471Z",
     "start_time": "2021-06-25T01:43:15.311503Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.   0.   0.03 0.98 0.02 0.  ]\n",
      " [1.   0.   1.   0.02 0.   0.01]\n",
      " [0.01 0.   0.   0.96 0.   0.01]\n",
      " [0.   0.   0.04 0.   0.   0.  ]\n",
      " [0.98 0.01 1.   0.02 0.   0.02]\n",
      " [1.   0.   0.02 0.02 0.   0.  ]]\n"
     ]
    }
   ],
   "source": [
    "prob = result.get_probabilities(min_causal_effect=0.01)\n",
    "print(prob)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Total Causal Effects\n",
    "Using the `get_total_causal_effects()` method, we can get the list of total causal effect. The total causal effects we can get are dictionary type variable.\n",
    "We can display the list nicely by assigning it to pandas.DataFrame. Also, we have replaced the variable index with a label below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.355386Z",
     "start_time": "2021-06-25T01:43:15.326464Z"
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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>from</th>\n",
       "      <th>to</th>\n",
       "      <th>effect</th>\n",
       "      <th>probability</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>x3</td>\n",
       "      <td>x0</td>\n",
       "      <td>3.004106</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>x0</td>\n",
       "      <td>x1</td>\n",
       "      <td>2.963177</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>x2</td>\n",
       "      <td>x1</td>\n",
       "      <td>2.017539</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>x3</td>\n",
       "      <td>x1</td>\n",
       "      <td>20.928254</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>x0</td>\n",
       "      <td>x5</td>\n",
       "      <td>3.997787</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>x3</td>\n",
       "      <td>x4</td>\n",
       "      <td>18.077943</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>x3</td>\n",
       "      <td>x5</td>\n",
       "      <td>12.012988</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>x2</td>\n",
       "      <td>x4</td>\n",
       "      <td>-1.006362</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>x0</td>\n",
       "      <td>x4</td>\n",
       "      <td>8.011818</td>\n",
       "      <td>0.98</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>x3</td>\n",
       "      <td>x2</td>\n",
       "      <td>5.964879</td>\n",
       "      <td>0.96</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>x2</td>\n",
       "      <td>x5</td>\n",
       "      <td>0.396327</td>\n",
       "      <td>0.09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>x2</td>\n",
       "      <td>x0</td>\n",
       "      <td>0.487915</td>\n",
       "      <td>0.07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>x2</td>\n",
       "      <td>x3</td>\n",
       "      <td>0.164565</td>\n",
       "      <td>0.04</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>x5</td>\n",
       "      <td>x4</td>\n",
       "      <td>0.087437</td>\n",
       "      <td>0.03</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>x4</td>\n",
       "      <td>x5</td>\n",
       "      <td>0.496445</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>x5</td>\n",
       "      <td>x1</td>\n",
       "      <td>-0.064703</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>x4</td>\n",
       "      <td>x1</td>\n",
       "      <td>0.367100</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>x4</td>\n",
       "      <td>x0</td>\n",
       "      <td>0.124114</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>x0</td>\n",
       "      <td>x2</td>\n",
       "      <td>0.056261</td>\n",
       "      <td>0.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>x1</td>\n",
       "      <td>x4</td>\n",
       "      <td>-0.097108</td>\n",
       "      <td>0.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>x5</td>\n",
       "      <td>x2</td>\n",
       "      <td>-0.111894</td>\n",
       "      <td>0.01</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   from  to     effect  probability\n",
       "0    x3  x0   3.004106         1.00\n",
       "1    x0  x1   2.963177         1.00\n",
       "2    x2  x1   2.017539         1.00\n",
       "3    x3  x1  20.928254         1.00\n",
       "4    x0  x5   3.997787         1.00\n",
       "5    x3  x4  18.077943         1.00\n",
       "6    x3  x5  12.012988         1.00\n",
       "7    x2  x4  -1.006362         1.00\n",
       "8    x0  x4   8.011818         0.98\n",
       "9    x3  x2   5.964879         0.96\n",
       "10   x2  x5   0.396327         0.09\n",
       "11   x2  x0   0.487915         0.07\n",
       "12   x2  x3   0.164565         0.04\n",
       "13   x5  x4   0.087437         0.03\n",
       "14   x4  x5   0.496445         0.02\n",
       "15   x5  x1  -0.064703         0.02\n",
       "16   x4  x1   0.367100         0.02\n",
       "17   x4  x0   0.124114         0.02\n",
       "18   x0  x2   0.056261         0.01\n",
       "19   x1  x4  -0.097108         0.01\n",
       "20   x5  x2  -0.111894         0.01"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "causal_effects = result.get_total_causal_effects(min_causal_effect=0.01)\n",
    "\n",
    "# Assign to pandas.DataFrame for pretty display\n",
    "df = pd.DataFrame(causal_effects)\n",
    "labels = [f'x{i}' for i in range(X.shape[1])]\n",
    "df['from'] = df['from'].apply(lambda x : labels[x])\n",
    "df['to'] = df['to'].apply(lambda x : labels[x])\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can easily perform sorting operations with pandas.DataFrame."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.387302Z",
     "start_time": "2021-06-25T01:43:15.358380Z"
    }
   },
   "outputs": [
    {
     "data": {
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>from</th>\n",
       "      <th>to</th>\n",
       "      <th>effect</th>\n",
       "      <th>probability</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>x3</td>\n",
       "      <td>x1</td>\n",
       "      <td>20.928254</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>x3</td>\n",
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       "      <td>18.077943</td>\n",
       "      <td>1.00</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>x3</td>\n",
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       "      <td>12.012988</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
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       "      <td>8.011818</td>\n",
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       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>x3</td>\n",
       "      <td>x2</td>\n",
       "      <td>5.964879</td>\n",
       "      <td>0.96</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  from  to     effect  probability\n",
       "3   x3  x1  20.928254         1.00\n",
       "5   x3  x4  18.077943         1.00\n",
       "6   x3  x5  12.012988         1.00\n",
       "8   x0  x4   8.011818         0.98\n",
       "9   x3  x2   5.964879         0.96"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.sort_values('effect', ascending=False).head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.417222Z",
     "start_time": "2021-06-25T01:43:15.392289Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    .dataframe tbody tr th:only-of-type {\n",
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       "</style>\n",
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>from</th>\n",
       "      <th>to</th>\n",
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       "      <th>probability</th>\n",
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       "  <tbody>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>x5</td>\n",
       "      <td>x2</td>\n",
       "      <td>-0.111894</td>\n",
       "      <td>0.01</td>\n",
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       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>x0</td>\n",
       "      <td>x2</td>\n",
       "      <td>0.056261</td>\n",
       "      <td>0.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>x1</td>\n",
       "      <td>x4</td>\n",
       "      <td>-0.097108</td>\n",
       "      <td>0.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>x4</td>\n",
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       "      <td>0.124114</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>x4</td>\n",
       "      <td>x1</td>\n",
       "      <td>0.367100</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   from  to    effect  probability\n",
       "20   x5  x2 -0.111894         0.01\n",
       "18   x0  x2  0.056261         0.01\n",
       "19   x1  x4 -0.097108         0.01\n",
       "17   x4  x0  0.124114         0.02\n",
       "16   x4  x1  0.367100         0.02"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.sort_values('probability', ascending=True).head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And with pandas.DataFrame, we can easily filter by keywords. The following code extracts the causal direction towards x1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:15.449138Z",
     "start_time": "2021-06-25T01:43:15.421214Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\n",
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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>from</th>\n",
       "      <th>to</th>\n",
       "      <th>effect</th>\n",
       "      <th>probability</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>x0</td>\n",
       "      <td>x1</td>\n",
       "      <td>2.963177</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>x2</td>\n",
       "      <td>x1</td>\n",
       "      <td>2.017539</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>x3</td>\n",
       "      <td>x1</td>\n",
       "      <td>20.928254</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>x5</td>\n",
       "      <td>x1</td>\n",
       "      <td>-0.064703</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>x4</td>\n",
       "      <td>x1</td>\n",
       "      <td>0.367100</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   from  to     effect  probability\n",
       "1    x0  x1   2.963177         1.00\n",
       "2    x2  x1   2.017539         1.00\n",
       "3    x3  x1  20.928254         1.00\n",
       "15   x5  x1  -0.064703         0.02\n",
       "16   x4  x1   0.367100         0.02"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df[df['to']=='x1'].head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because it holds the raw data of the total causal effect (the original data for calculating the median), it is possible to draw a histogram of the values of the causal effect, as shown below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:17.468026Z",
     "start_time": "2021-06-25T01:43:15.452132Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 1.,  8., 10., 17., 19., 20., 13.,  9.,  1.,  2.]),\n",
       " array([2.936, 2.951, 2.965, 2.98 , 2.994, 3.008, 3.023, 3.037, 3.051,\n",
       "        3.066, 3.08 ]),\n",
       " <BarContainer object of 10 artists>)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "sns.set()\n",
    "%matplotlib inline\n",
    "\n",
    "from_index = 3 # index of x3\n",
    "to_index = 0 # index of x0\n",
    "plt.hist(result.total_effects_[:, to_index, from_index])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bootstrap Probability of Path\n",
    "Using the `get_paths()` method, we can explore all paths from any variable to any variable and calculate the bootstrap probability for each path. The path will be output as an array of variable indices. For example, the array `[3, 0, 1]` shows the path from variable X3 through variable X0 to variable X1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-06-25T01:43:17.514898Z",
     "start_time": "2021-06-25T01:43:17.470046Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        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>path</th>\n",
       "      <th>effect</th>\n",
       "      <th>probability</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>[3, 0, 1]</td>\n",
       "      <td>8.893562</td>\n",
       "      <td>0.98</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>[3, 2, 1]</td>\n",
       "      <td>12.030408</td>\n",
       "      <td>0.96</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>[3, 2, 0, 1]</td>\n",
       "      <td>2.239175</td>\n",
       "      <td>0.03</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>[3, 1]</td>\n",
       "      <td>-0.639462</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>[3, 2, 4, 0, 1]</td>\n",
       "      <td>-3.194541</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>[3, 4, 0, 1]</td>\n",
       "      <td>9.820705</td>\n",
       "      <td>0.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>[3, 0, 2, 1]</td>\n",
       "      <td>3.061033</td>\n",
       "      <td>0.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>[3, 0, 5, 1]</td>\n",
       "      <td>1.176834</td>\n",
       "      <td>0.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>[3, 0, 5, 2, 1]</td>\n",
       "      <td>-2.719517</td>\n",
       "      <td>0.01</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              path     effect  probability\n",
       "0        [3, 0, 1]   8.893562         0.98\n",
       "1        [3, 2, 1]  12.030408         0.96\n",
       "2     [3, 2, 0, 1]   2.239175         0.03\n",
       "3           [3, 1]  -0.639462         0.02\n",
       "4  [3, 2, 4, 0, 1]  -3.194541         0.02\n",
       "5     [3, 4, 0, 1]   9.820705         0.02\n",
       "6     [3, 0, 2, 1]   3.061033         0.01\n",
       "7     [3, 0, 5, 1]   1.176834         0.01\n",
       "8  [3, 0, 5, 2, 1]  -2.719517         0.01"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from_index = 3 # index of x3\n",
    "to_index = 1 # index of x0\n",
    "\n",
    "pd.DataFrame(result.get_paths(from_index, to_index))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.8.10"
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