examples/Use_prior_knowledge_in_BottomUpParceLiNGAM.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# How to use prior knowledge in BottomUpParceLiNGAM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Import and settings\n",
    "In this example, we need to import `numpy`, `pandas`, and `graphviz` in addition to `lingam`."
   ]
  },
  {
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   "execution_count": 1,
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     "end_time": "2019-12-12T06:27:49.386559Z",
     "start_time": "2019-12-12T06:27:47.059613Z"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['1.16.2', '0.24.2', '0.11.1', '1.5.2']\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import graphviz\n",
    "import lingam\n",
    "from lingam.utils import make_prior_knowledge, 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": [
    "## Utility function\n",
    "We define a utility function to draw the directed acyclic graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-12-12T06:27:49.416479Z",
     "start_time": "2019-12-12T06:27:49.404510Z"
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   "outputs": [],
   "source": [
    "def make_prior_knowledge_graph(prior_knowledge_matrix):\n",
    "    d = graphviz.Digraph(engine='dot')\n",
    "    \n",
    "    labels = [f'x{i}' for i in range(prior_knowledge_matrix.shape[0])]\n",
    "    for label in labels:\n",
    "        d.node(label, label)\n",
    "\n",
    "    dirs = np.where(prior_knowledge_matrix > 0)\n",
    "    for to, from_ in zip(dirs[0], dirs[1]):\n",
    "        d.edge(labels[from_], labels[to])\n",
    "\n",
    "    dirs = np.where(prior_knowledge_matrix < 0)\n",
    "    for to, from_ in zip(dirs[0], dirs[1]):\n",
    "        if to != from_:\n",
    "            d.edge(labels[from_], labels[to], style='dashed')\n",
    "    return d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test data\n",
    "We create test data consisting of 6 variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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    "ExecuteTime": {
     "end_time": "2019-12-12T06:27:49.484297Z",
     "start_time": "2019-12-12T06:27:49.420469Z"
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   },
   "outputs": [],
   "source": [
    "np.random.seed(1000)\n",
    "\n",
    "x6 = np.random.uniform(size=1000)\n",
    "x3 = 2.0*x6 + np.random.uniform(size=1000)\n",
    "x0 = 0.5*x3 + np.random.uniform(size=1000)\n",
    "x2 = 2.0*x6 + np.random.uniform(size=1000)\n",
    "x1 = 0.5*x0 + 0.5*x2 + np.random.uniform(size=1000)\n",
    "x5 = 0.5*x0 + np.random.uniform(size=1000)\n",
    "x4 = 0.5*x0 - 0.5*x2 + np.random.uniform(size=1000)\n",
    "\n",
    "# The latent variable x6 is not included.\n",
    "X = pd.DataFrame(np.array([x0, x1, x2, x3, x4, x5]).T, columns=['x0', 'x1', 'x2', 'x3', 'x4', 'x5'])"
   ]
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     "start_time": "2019-12-12T06:27:49.486292Z"
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    "m = np.array([[0.0, 0.0,    0.0,    0.5, 0.0, 0.0],\n",
    "              [0.5, 0.0,    0.5,    0.0, 0.0, 0.0],\n",
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    "\n",
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   "cell_type": "markdown",
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   "source": [
    "## Make Prior Knowledge Matrix\n",
    "We create prior knowledge so that x0, x1 and x4 are sink variables.\n",
    "\n",
    "The elements of prior knowledge matrix are defined as follows:\n",
    "* ``0`` : :math:`x_i` does not have a directed path to :math:`x_j`\n",
    "* ``1`` : :math:`x_i` has a directed path to :math:`x_j`\n",
    "* ``-1`` : No prior knowledge is available to know if either of the two cases above (0 or 1) is true."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
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   },
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "[[-1  0 -1 -1  0 -1]\n",
      " [ 0 -1 -1 -1  0 -1]\n",
      " [ 0  0 -1 -1  0 -1]\n",
      " [ 0  0 -1 -1  0 -1]\n",
      " [ 0  0 -1 -1 -1 -1]\n",
      " [ 0  0 -1 -1  0 -1]]\n"
     ]
    }
   ],
   "source": [
    "prior_knowledge = make_prior_knowledge(\n",
    "    n_variables=6,\n",
    "    sink_variables=[0, 1, 4],\n",
    ")\n",
    "print(prior_knowledge)"
   ]
  },
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   "cell_type": "code",
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     "execution_count": 6,
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   ],
   "source": [
    "# Draw a graph of prior knowledge\n",
    "make_prior_knowledge_graph(prior_knowledge)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Causal Discovery\n",
    "To run causal discovery using prior knowledge, we create a `DirectLiNGAM` object with the prior knowledge matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-12-12T06:27:50.531152Z",
     "start_time": "2019-12-12T06:27:50.383201Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0, 2, 3, 5], 4, 1]\n",
      "[[ 0.     0.       nan    nan  0.       nan]\n",
      " [ 0.     0.     0.479  0.219  0.     0.212]\n",
      " [   nan  0.     0.       nan  0.       nan]\n",
      " [   nan  0.       nan  0.     0.       nan]\n",
      " [ 0.     0.    -0.494  0.212  0.     0.199]\n",
      " [   nan  0.       nan    nan  0.     0.   ]]\n"
     ]
    }
   ],
   "source": [
    "model = lingam.BottomUpParceLiNGAM(prior_knowledge=prior_knowledge)\n",
    "model.fit(X)\n",
    "print(model.causal_order_)\n",
    "print(model.adjacency_matrix_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that x0, x1, and x4 are output as sink variables, as specified in the prior knowledge."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-12-12T06:27:51.009585Z",
     "start_time": "2019-12-12T06:27:50.572914Z"
    }
   },
   "outputs": [
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    "make_dot(model.adjacency_matrix_)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, let's specify the prior knowledge so that x0 is an exogenous variable."
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  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
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     "execution_count": 9,
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   "source": [
    "prior_knowledge = make_prior_knowledge(\n",
    "    n_variables=6,\n",
    "    exogenous_variables=[0],\n",
    ")\n",
    "\n",
    "model = lingam.BottomUpParceLiNGAM(prior_knowledge=prior_knowledge)\n",
    "model.fit(X)\n",
    "\n",
    "make_dot(model.adjacency_matrix_)"
   ]
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