Chapter13_14.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "from pandas import Series, DataFrame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "import patsy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.api as sm\n",
    "import statsmodels.formula.api as smf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = DataFrame({ 'x0': [ 1, 2, 3, 4, 5 ], 'x1': [.01, -.01, .25, -4.1, 0 ], 'y': [ -1.5, 0, 3.6, 1.3, -2 ] })"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "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>x0</th>\n",
       "      <th>x1</th>\n",
       "      <th>y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>0.01</td>\n",
       "      <td>-1.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>0.25</td>\n",
       "      <td>3.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>-4.10</td>\n",
       "      <td>1.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>0.00</td>\n",
       "      <td>-2.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   x0    x1    y\n",
       "0   1  0.01 -1.5\n",
       "1   2 -0.01  0.0\n",
       "2   3  0.25  3.6\n",
       "3   4 -4.10  1.3\n",
       "4   5  0.00 -2.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['x0', 'x1', 'y'], dtype='object')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.  ,  0.01, -1.5 ],\n",
       "       [ 2.  , -0.01,  0.  ],\n",
       "       [ 3.  ,  0.25,  3.6 ],\n",
       "       [ 4.  , -4.1 ,  1.3 ],\n",
       "       [ 5.  ,  0.  , -2.  ]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2 = DataFrame(data.values, columns=[ 'one', 'two', 'three' ])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "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>one</th>\n",
       "      <th>two</th>\n",
       "      <th>three</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.0</td>\n",
       "      <td>0.01</td>\n",
       "      <td>-1.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2.0</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3.0</td>\n",
       "      <td>0.25</td>\n",
       "      <td>3.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4.0</td>\n",
       "      <td>-4.10</td>\n",
       "      <td>1.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5.0</td>\n",
       "      <td>0.00</td>\n",
       "      <td>-2.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   one   two  three\n",
       "0  1.0  0.01   -1.5\n",
       "1  2.0 -0.01    0.0\n",
       "2  3.0  0.25    3.6\n",
       "3  4.0 -4.10    1.3\n",
       "4  5.0  0.00   -2.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "df3 = data.copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "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>x0</th>\n",
       "      <th>x1</th>\n",
       "      <th>y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>0.01</td>\n",
       "      <td>-1.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>0.25</td>\n",
       "      <td>3.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>-4.10</td>\n",
       "      <td>1.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>0.00</td>\n",
       "      <td>-2.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   x0    x1    y\n",
       "0   1  0.01 -1.5\n",
       "1   2 -0.01  0.0\n",
       "2   3  0.25  3.6\n",
       "3   4 -4.10  1.3\n",
       "4   5  0.00 -2.0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.  ,  0.01, -1.5 ],\n",
       "       [ 2.  , -0.01,  0.  ],\n",
       "       [ 3.  ,  0.25,  3.6 ],\n",
       "       [ 4.  , -4.1 ,  1.3 ],\n",
       "       [ 5.  ,  0.  , -2.  ]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df3.values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_cols = [ 'x0', 'x1' ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.  ,  0.01],\n",
       "       [ 2.  , -0.01],\n",
       "       [ 3.  ,  0.25],\n",
       "       [ 4.  , -4.1 ],\n",
       "       [ 5.  ,  0.  ]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.loc[:, model_cols].values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "data['category'] = pd.Categorical([ 'a', 'b', 'a', 'a', 'b' ], categories=[ 'a', 'b' ])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "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>x0</th>\n",
       "      <th>x1</th>\n",
       "      <th>y</th>\n",
       "      <th>category</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>0.01</td>\n",
       "      <td>-1.5</td>\n",
       "      <td>a</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>0.0</td>\n",
       "      <td>b</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>0.25</td>\n",
       "      <td>3.6</td>\n",
       "      <td>a</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>-4.10</td>\n",
       "      <td>1.3</td>\n",
       "      <td>a</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>0.00</td>\n",
       "      <td>-2.0</td>\n",
       "      <td>b</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   x0    x1    y category\n",
       "0   1  0.01 -1.5        a\n",
       "1   2 -0.01  0.0        b\n",
       "2   3  0.25  3.6        a\n",
       "3   4 -4.10  1.3        a\n",
       "4   5  0.00 -2.0        b"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "dummies = pd.get_dummies(data.category, prefix='category')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "data_with_dummies = data.drop('category', axis=1).join(dummies)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "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>x0</th>\n",
       "      <th>x1</th>\n",
       "      <th>y</th>\n",
       "      <th>category_a</th>\n",
       "      <th>category_b</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>0.01</td>\n",
       "      <td>-1.5</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>0.25</td>\n",
       "      <td>3.6</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>-4.10</td>\n",
       "      <td>1.3</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>0.00</td>\n",
       "      <td>-2.0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   x0    x1    y  category_a  category_b\n",
       "0   1  0.01 -1.5           1           0\n",
       "1   2 -0.01  0.0           0           1\n",
       "2   3  0.25  3.6           1           0\n",
       "3   4 -4.10  1.3           1           0\n",
       "4   5  0.00 -2.0           0           1"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_with_dummies"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('y ~ x0 + x1', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (5, 1)\n",
       "     y\n",
       "  -1.5\n",
       "   0.0\n",
       "   3.6\n",
       "   1.3\n",
       "  -2.0\n",
       "  Terms:\n",
       "    'y' (column 0)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (5, 3)\n",
       "  Intercept  x0     x1\n",
       "          1   1   0.01\n",
       "          1   2  -0.01\n",
       "          1   3   0.25\n",
       "          1   4  -4.10\n",
       "          1   5   0.00\n",
       "  Terms:\n",
       "    'Intercept' (column 0)\n",
       "    'x0' (column 1)\n",
       "    'x1' (column 2)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.  ,  1.  ,  0.01],\n",
       "       [ 1.  ,  2.  , -0.01],\n",
       "       [ 1.  ,  3.  ,  0.25],\n",
       "       [ 1.  ,  4.  , -4.1 ],\n",
       "       [ 1.  ,  5.  ,  0.  ]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.asarray(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-1.5],\n",
       "       [ 0. ],\n",
       "       [ 3.6],\n",
       "       [ 1.3],\n",
       "       [-2. ]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.asarray(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (5, 2)\n",
       "  x0     x1\n",
       "   1   0.01\n",
       "   2  -0.01\n",
       "   3   0.25\n",
       "   4  -4.10\n",
       "   5   0.00\n",
       "  Terms:\n",
       "    'x0' (column 0)\n",
       "    'x1' (column 1)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "patsy.dmatrices('y ~ x0 + x1 + 0', data)[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:1: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.\n",
      "To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.\n",
      "  \"\"\"Entry point for launching an IPython kernel.\n"
     ]
    }
   ],
   "source": [
    "coef, resid, _, _ = np.linalg.lstsq(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.31290976],\n",
       "       [-0.07910564],\n",
       "       [-0.26546384]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coef"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "coef = Series(coef.squeeze(), index=X.design_info.column_names)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Intercept    0.312910\n",
       "x0          -0.079106\n",
       "x1          -0.265464\n",
       "dtype: float64"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coef"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('y ~ x0 + np.log(np.abs(x1) + 1)', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (5, 3)\n",
       "  Intercept  x0  np.log(np.abs(x1) + 1)\n",
       "          1   1                 0.00995\n",
       "          1   2                 0.00995\n",
       "          1   3                 0.22314\n",
       "          1   4                 1.62924\n",
       "          1   5                 0.00000\n",
       "  Terms:\n",
       "    'Intercept' (column 0)\n",
       "    'x0' (column 1)\n",
       "    'np.log(np.abs(x1) + 1)' (column 2)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('y ~ standardize(x0) + center(x1)', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (5, 3)\n",
       "  Intercept  standardize(x0)  center(x1)\n",
       "          1         -1.41421        0.78\n",
       "          1         -0.70711        0.76\n",
       "          1          0.00000        1.02\n",
       "          1          0.70711       -3.33\n",
       "          1          1.41421        0.77\n",
       "  Terms:\n",
       "    'Intercept' (column 0)\n",
       "    'standardize(x0)' (column 1)\n",
       "    'center(x1)' (column 2)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "new_data = DataFrame({ 'x0': [ 6, 7, 8, 9, ], 'x1': [ 3.1, -.5, 0, 2.3, ], 'y': [ 1, 2, 3, 4, ] })"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "new_X = patsy.build_design_matrices([ X.design_info ], new_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[DesignMatrix with shape (4, 3)\n",
       "   Intercept  standardize(x0)  center(x1)\n",
       "           1          2.12132        3.87\n",
       "           1          2.82843        0.27\n",
       "           1          3.53553        0.77\n",
       "           1          4.24264        3.07\n",
       "   Terms:\n",
       "     'Intercept' (column 0)\n",
       "     'standardize(x0)' (column 1)\n",
       "     'center(x1)' (column 2)]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('y ~ I(x0 + x1)', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (5, 2)\n",
       "  Intercept  I(x0 + x1)\n",
       "          1        1.01\n",
       "          1        1.99\n",
       "          1        3.25\n",
       "          1       -0.10\n",
       "          1        5.00\n",
       "  Terms:\n",
       "    'Intercept' (column 0)\n",
       "    'I(x0 + x1)' (column 1)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = DataFrame({ 'key1': [ 'a', 'a', 'b', 'b', 'a', 'b', 'a', 'b' ], 'key2': [ 0, 1, 0, 1, 0, 1, 0, 0, ], 'v1': [ 1, 2, 3, 4, 5, 6, 7, 8, ], 'v2': [ -1, 0, 2.5, -.5, 4, -1.2, .2, -1.7 ] })"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('v2 ~ key1', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (8, 2)\n",
       "  Intercept  key1[T.b]\n",
       "          1          0\n",
       "          1          0\n",
       "          1          1\n",
       "          1          1\n",
       "          1          0\n",
       "          1          1\n",
       "          1          0\n",
       "          1          1\n",
       "  Terms:\n",
       "    'Intercept' (column 0)\n",
       "    'key1' (column 1)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('v2 ~ key1 + 0', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (8, 2)\n",
       "  key1[a]  key1[b]\n",
       "        1        0\n",
       "        1        0\n",
       "        0        1\n",
       "        0        1\n",
       "        1        0\n",
       "        0        1\n",
       "        1        0\n",
       "        0        1\n",
       "  Terms:\n",
       "    'key1' (columns 0:2)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('v2 ~ C(key2)', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (8, 2)\n",
       "  Intercept  C(key2)[T.1]\n",
       "          1             0\n",
       "          1             1\n",
       "          1             0\n",
       "          1             1\n",
       "          1             0\n",
       "          1             1\n",
       "          1             0\n",
       "          1             0\n",
       "  Terms:\n",
       "    'Intercept' (column 0)\n",
       "    'C(key2)' (column 1)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "data['key2'] = data['key2'].map({ 0: 'zero', 1: 'one' })"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "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>key1</th>\n",
       "      <th>key2</th>\n",
       "      <th>v1</th>\n",
       "      <th>v2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>a</td>\n",
       "      <td>zero</td>\n",
       "      <td>1</td>\n",
       "      <td>-1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>a</td>\n",
       "      <td>one</td>\n",
       "      <td>2</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>b</td>\n",
       "      <td>zero</td>\n",
       "      <td>3</td>\n",
       "      <td>2.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>b</td>\n",
       "      <td>one</td>\n",
       "      <td>4</td>\n",
       "      <td>-0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>a</td>\n",
       "      <td>zero</td>\n",
       "      <td>5</td>\n",
       "      <td>4.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>b</td>\n",
       "      <td>one</td>\n",
       "      <td>6</td>\n",
       "      <td>-1.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>a</td>\n",
       "      <td>zero</td>\n",
       "      <td>7</td>\n",
       "      <td>0.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>b</td>\n",
       "      <td>zero</td>\n",
       "      <td>8</td>\n",
       "      <td>-1.7</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  key1  key2  v1   v2\n",
       "0    a  zero   1 -1.0\n",
       "1    a   one   2  0.0\n",
       "2    b  zero   3  2.5\n",
       "3    b   one   4 -0.5\n",
       "4    a  zero   5  4.0\n",
       "5    b   one   6 -1.2\n",
       "6    a  zero   7  0.2\n",
       "7    b  zero   8 -1.7"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('v2 ~ key1 + key2', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (8, 3)\n",
       "  Intercept  key1[T.b]  key2[T.zero]\n",
       "          1          0             1\n",
       "          1          0             0\n",
       "          1          1             1\n",
       "          1          1             0\n",
       "          1          0             1\n",
       "          1          1             0\n",
       "          1          0             1\n",
       "          1          1             1\n",
       "  Terms:\n",
       "    'Intercept' (column 0)\n",
       "    'key1' (column 1)\n",
       "    'key2' (column 2)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "y, X = patsy.dmatrices('v2 ~ key1 + key2 + key1:key2', data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DesignMatrix with shape (8, 4)\n",
       "  Intercept  key1[T.b]  key2[T.zero]  key1[T.b]:key2[T.zero]\n",
       "          1          0             1                       0\n",
       "          1          0             0                       0\n",
       "          1          1             1                       1\n",
       "          1          1             0                       0\n",
       "          1          0             1                       0\n",
       "          1          1             0                       0\n",
       "          1          0             1                       0\n",
       "          1          1             1                       1\n",
       "  Terms:\n",
       "    'Intercept' (column 0)\n",
       "    'key1' (column 1)\n",
       "    'key2' (column 2)\n",
       "    'key1:key2' (column 3)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "def dnorm(mean, variance, size=1):\n",
    "    if isinstance(size, int):\n",
    "        size = size,\n",
    "    return mean + variance * np.random.randn(*size)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(12345)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.c_[ dnorm(0, .4, N), dnorm(0, .6, N), dnorm(0, .2, N) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "eps = dnorm(0, .1, N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "beta = [ .1, .3, .5 ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = np.dot(X, beta) + eps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.08188306, -0.93939438,  0.22549622],\n",
       "       [ 0.19157734, -0.33752411, -0.11367269],\n",
       "       [-0.20777549, -0.01959848,  0.06187243],\n",
       "       [-0.22229212, -0.55740372, -0.11547709],\n",
       "       [ 0.78631223, -0.28954359, -0.23372681]])"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.00253515, -0.27996046, -0.03354503, -0.28177094, -0.08708901])"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_model = sm.add_constant(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.        , -0.08188306, -0.93939438,  0.22549622],\n",
       "       [ 1.        ,  0.19157734, -0.33752411, -0.11367269],\n",
       "       [ 1.        , -0.20777549, -0.01959848,  0.06187243],\n",
       "       [ 1.        , -0.22229212, -0.55740372, -0.11547709],\n",
       "       [ 1.        ,  0.78631223, -0.28954359, -0.23372681]])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_model[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = sm.OLS(y, X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "result = model.fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.13913054, 0.26858106, 0.50067241])"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result.params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                      y   R-squared:                       0.756\n",
      "Model:                            OLS   Adj. R-squared:                  0.749\n",
      "Method:                 Least Squares   F-statistic:                     100.3\n",
      "Date:                Mon, 18 Mar 2019   Prob (F-statistic):           1.28e-29\n",
      "Time:                        21:01:09   Log-Likelihood:                 80.825\n",
      "No. Observations:                 100   AIC:                            -155.6\n",
      "Df Residuals:                      97   BIC:                            -147.8\n",
      "Df Model:                           3                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "x1             0.1391      0.026      5.251      0.000       0.087       0.192\n",
      "x2             0.2686      0.019     14.213      0.000       0.231       0.306\n",
      "x3             0.5007      0.057      8.816      0.000       0.388       0.613\n",
      "==============================================================================\n",
      "Omnibus:                        4.662   Durbin-Watson:                   2.201\n",
      "Prob(Omnibus):                  0.097   Jarque-Bera (JB):                4.098\n",
      "Skew:                           0.481   Prob(JB):                        0.129\n",
      "Kurtosis:                       3.243   Cond. No.                         3.01\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "print(result.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = DataFrame(X, columns=[ 'col0', 'col1', 'col2' ])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "data['y'] = y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "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>col0</th>\n",
       "      <th>col1</th>\n",
       "      <th>col2</th>\n",
       "      <th>y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.081883</td>\n",
       "      <td>-0.939394</td>\n",
       "      <td>0.225496</td>\n",
       "      <td>-0.002535</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.191577</td>\n",
       "      <td>-0.337524</td>\n",
       "      <td>-0.113673</td>\n",
       "      <td>-0.279960</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.207775</td>\n",
       "      <td>-0.019598</td>\n",
       "      <td>0.061872</td>\n",
       "      <td>-0.033545</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.222292</td>\n",
       "      <td>-0.557404</td>\n",
       "      <td>-0.115477</td>\n",
       "      <td>-0.281771</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.786312</td>\n",
       "      <td>-0.289544</td>\n",
       "      <td>-0.233727</td>\n",
       "      <td>-0.087089</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       col0      col1      col2         y\n",
       "0 -0.081883 -0.939394  0.225496 -0.002535\n",
       "1  0.191577 -0.337524 -0.113673 -0.279960\n",
       "2 -0.207775 -0.019598  0.061872 -0.033545\n",
       "3 -0.222292 -0.557404 -0.115477 -0.281771\n",
       "4  0.786312 -0.289544 -0.233727 -0.087089"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "results = smf.ols('y ~ col0 + col1 + col2', data=data).fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Intercept    0.010612\n",
       "col0         0.138074\n",
       "col1         0.269310\n",
       "col2         0.510471\n",
       "dtype: float64"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Intercept     0.952188\n",
       "col0          5.204386\n",
       "col1         14.232764\n",
       "col2          8.840053\n",
       "dtype: float64"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.tvalues"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0   -0.138573\n",
       "1   -0.111861\n",
       "2    0.008230\n",
       "3   -0.229143\n",
       "4   -0.078106\n",
       "dtype: float64"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.predict(data[:5])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "init_x = 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "values = [ init_x ] * 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "b0 = .8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [],
   "source": [
    "b1 = -.4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "noise = dnorm(0, .1, N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(N):\n",
    "    new_x = values[-1] * b0 + values[-2] * b1 + noise[i]\n",
    "    values.append(new_x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [],
   "source": [
    "MAXLAGS = 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = sm.tsa.AR(values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "results = model.fit(MAXLAGS)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.00208441,  0.78358666, -0.40862768, -0.01428997,  0.00320974,\n",
       "        0.01504663])"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.params"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}