2016-fall/seminars/sem03-linregr.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Предобработка данных и функции потерь в линейной регрессии"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Данные\n",
    "На семинаре мы будем работать с набором данных [Automobile Data Set](https://archive.ics.uci.edu/ml/datasets/Automobile). В данных присутствуют категориальные, целочисленные и вещественнозначные признаки."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "X_raw = pd.read_csv(\"cars.csv\", header=None, na_values=[\"?\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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       "      <td>audi</td>\n",
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       "      <td>four</td>\n",
       "      <td>sedan</td>\n",
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       "   0      1            2    3    4     5            6    7      8     9   \\\n",
       "0   3    NaN  alfa-romero  gas  std   two  convertible  rwd  front  88.6   \n",
       "1   3    NaN  alfa-romero  gas  std   two  convertible  rwd  front  88.6   \n",
       "2   1    NaN  alfa-romero  gas  std   two    hatchback  rwd  front  94.5   \n",
       "3   2  164.0         audi  gas  std  four        sedan  fwd  front  99.8   \n",
       "4   2  164.0         audi  gas  std  four        sedan  4wd  front  99.4   \n",
       "\n",
       "    ...      16    17    18    19    20     21      22  23  24       25  \n",
       "0   ...     130  mpfi  3.47  2.68   9.0  111.0  5000.0  21  27  13495.0  \n",
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       "\n",
       "[5 rows x 26 columns]"
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     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_raw.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Предобработка данных\n",
    "Предобработка данных важна при применении любых методов машинного обучения, а в особенности для линейных моделей. В sklearn предобработку удобно делать с помощью модуля [preprocessing](http://scikit-learn.org/stable/modules/classes.html#module-sklearn.preprocessing)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn import preprocessing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Преобразование нечисловых признаков\n",
    "Практически все методы машинного обучения требуют, чтобы на вход функции обучения подавалась вещественная матрица. В процессе обучения используются свойства вещественных чисел, в частности, возможность сравнения и применения арифметических операций. Поэтому, даже если формально в матрице объекты-признаки записаны числовые значения, нужно всегда анализировать, можно ли относиться к ним как к числам. \n",
    "\n",
    "__Пример:__ некоторые признаки могут задаваться целочисленными хешами или id (например, id пользователя соц. сети), однако нельзя сложить двух пользователей и получить третьего, исходя из их id (как это может сделать линейная модель).\n",
    "\n",
    "Это пример категориального признака, принимающего значения из неупорядоченного конечного множества $K$. К таким признакам обычно применяют [one-hot encoding](http://scikit-learn.org/stable/modules/preprocessing.html#encoding-categorical-features) (вместо одного признака создают $K$ бинарных признаков - по одному на каждое возможное значение исходного признака). В sklearn это можно сделать следующим образом:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# для удобства работы с нашим датасетом создаем маску, указывающую на столбцы с категориальными признаками\n",
    "cat_features_mask = (X_raw.dtypes == \"object\").values # категориальные признаки имеют тип \"object\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python2.7/site-packages/numpy/lib/arraysetops.py:200: FutureWarning: numpy not_equal will not check object identity in the future. The comparison did not return the same result as suggested by the identity (`is`)) and will change.\n",
      "  flag = np.concatenate(([True], aux[1:] != aux[:-1]))\n"
     ]
    }
   ],
   "source": [
    "# кодирование категорий-строк натуральными числами\n",
    "label_enc = preprocessing.LabelEncoder()\n",
    "for feature in X_raw.columns[cat_features_mask]: \n",
    "    X_raw[feature] = label_enc.fit_transform(X_raw[feature])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
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   "outputs": [
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       "      <td>99.8</td>\n",
       "      <td>...</td>\n",
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       "      <td>5</td>\n",
       "      <td>3.19</td>\n",
       "      <td>3.40</td>\n",
       "      <td>10.0</td>\n",
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       "   0      1   2   3   4   5   6   7   8     9    ...      16  17    18    19  \\\n",
       "0   3    NaN   0   1   0   2   0   2   0  88.6   ...     130   5  3.47  2.68   \n",
       "1   3    NaN   0   1   0   2   0   2   0  88.6   ...     130   5  3.47  2.68   \n",
       "2   1    NaN   0   1   0   2   2   2   0  94.5   ...     152   5  2.68  3.47   \n",
       "3   2  164.0   1   1   0   1   3   1   0  99.8   ...     109   5  3.19  3.40   \n",
       "4   2  164.0   1   1   0   1   3   0   0  99.4   ...     136   5  3.19  3.40   \n",
       "\n",
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       "0   9.0  111.0  5000.0  21  27  13495.0  \n",
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       "3  10.0  102.0  5500.0  24  30  13950.0  \n",
       "4   8.0  115.0  5500.0  18  22  17450.0  \n",
       "\n",
       "[5 rows x 26 columns]"
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "X_raw.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# применение one-hot ecnoding\n",
    "enc = preprocessing.OneHotEncoder(sparse=False)\n",
    "X_cat_np = enc.fit_transform(X_raw[X_raw.columns[cat_features_mask]])\n",
    "X_cat_pd = pd.DataFrame(data=X_cat_np)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(205, 61)\n"
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       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 61 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    0    1    2    3    4    5    6    7    8    9  ...    51   52   53   54  \\\n",
       "0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 ...   0.0  0.0  0.0  0.0   \n",
       "1  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 ...   0.0  0.0  0.0  0.0   \n",
       "2  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 ...   0.0  0.0  0.0  0.0   \n",
       "3  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 ...   0.0  0.0  0.0  0.0   \n",
       "4  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 ...   0.0  0.0  0.0  0.0   \n",
       "\n",
       "    55   56   57   58   59   60  \n",
       "0  0.0  0.0  0.0  1.0  0.0  0.0  \n",
       "1  0.0  0.0  0.0  1.0  0.0  0.0  \n",
       "2  0.0  0.0  0.0  1.0  0.0  0.0  \n",
       "3  0.0  0.0  0.0  1.0  0.0  0.0  \n",
       "4  0.0  0.0  0.0  1.0  0.0  0.0  \n",
       "\n",
       "[5 rows x 61 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print X_cat_pd.shape\n",
    "X_cat_pd.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Следует заметить, что в новой матрице очень много нулевых значений. Чтобы не хранить их в памяти, можно задать параметр OneHotEncoder(sparse = True), и метод fit_transform вернет [разреженную матрицу](http://docs.scipy.org/doc/scipy/reference/sparse.html), в которой хранятся только ненулевые значения. Выполнение некоторых операций с такой матрицей может быть неэффективным, однако большинство методов sklearn умеют работать с разреженными матрицами."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Вопрос:__ какая проблема возникнет при применении такого способа кодирования для обучения линейной регрессии?\n",
    "    \n",
    "Необходимо удалить один из столбцов, созданных для каждого признака (его легко вычислить из остальных):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X_cat_pd.drop(enc.feature_indices_[:-1], axis=1, inplace=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Помимо категориальных, преобразования требуют, например, строковые признаки. Их можно превращать в матрицу частот слов [CountVectorizer](http://scikit-learn.org/stable/modules/generated/sklearn.feature_extraction.text.CountVectorizer.html#sklearn.feature_extraction.text.CountVectorizer), матрицу частот буквосочетаний фиксированной длины, можно извлекать другие признаки (например, длина строки)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Заполнение пропусков\n",
    "В матрице объекты-признаки могут быть пропущенные значения, и это вызовет исключение при попытке передать такую матрицу в функцию обучения модели. Если пропусков немного, можно удалить объекты с пропусками из обучающей выборки. Заполнить пропуски можно разными способами:\n",
    "* заполнить средними (mean, median);\n",
    "* предсказывать пропущенные значения по непропущенным.\n",
    "\n",
    "Последний вариант сложный и применяется редко. Замена пропусков средними в вещественных признаках:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X_real = X_raw[X_raw.columns[~cat_features_mask]]\n",
    "mis_replacer = preprocessing.Imputer(strategy=\"mean\")\n",
    "X_no_mis = pd.DataFrame(data=mis_replacer.fit_transform(X_real))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "      <th>10</th>\n",
       "      <th>11</th>\n",
       "      <th>12</th>\n",
       "      <th>13</th>\n",
       "      <th>14</th>\n",
       "      <th>15</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>3.0</td>\n",
       "      <td>122.0</td>\n",
       "      <td>88.6</td>\n",
       "      <td>168.8</td>\n",
       "      <td>64.1</td>\n",
       "      <td>48.8</td>\n",
       "      <td>2548.0</td>\n",
       "      <td>130.0</td>\n",
       "      <td>3.47</td>\n",
       "      <td>2.68</td>\n",
       "      <td>9.0</td>\n",
       "      <td>111.0</td>\n",
       "      <td>5000.0</td>\n",
       "      <td>21.0</td>\n",
       "      <td>27.0</td>\n",
       "      <td>13495.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>3.0</td>\n",
       "      <td>122.0</td>\n",
       "      <td>88.6</td>\n",
       "      <td>168.8</td>\n",
       "      <td>64.1</td>\n",
       "      <td>48.8</td>\n",
       "      <td>2548.0</td>\n",
       "      <td>130.0</td>\n",
       "      <td>3.47</td>\n",
       "      <td>2.68</td>\n",
       "      <td>9.0</td>\n",
       "      <td>111.0</td>\n",
       "      <td>5000.0</td>\n",
       "      <td>21.0</td>\n",
       "      <td>27.0</td>\n",
       "      <td>16500.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>122.0</td>\n",
       "      <td>94.5</td>\n",
       "      <td>171.2</td>\n",
       "      <td>65.5</td>\n",
       "      <td>52.4</td>\n",
       "      <td>2823.0</td>\n",
       "      <td>152.0</td>\n",
       "      <td>2.68</td>\n",
       "      <td>3.47</td>\n",
       "      <td>9.0</td>\n",
       "      <td>154.0</td>\n",
       "      <td>5000.0</td>\n",
       "      <td>19.0</td>\n",
       "      <td>26.0</td>\n",
       "      <td>16500.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2.0</td>\n",
       "      <td>164.0</td>\n",
       "      <td>99.8</td>\n",
       "      <td>176.6</td>\n",
       "      <td>66.2</td>\n",
       "      <td>54.3</td>\n",
       "      <td>2337.0</td>\n",
       "      <td>109.0</td>\n",
       "      <td>3.19</td>\n",
       "      <td>3.40</td>\n",
       "      <td>10.0</td>\n",
       "      <td>102.0</td>\n",
       "      <td>5500.0</td>\n",
       "      <td>24.0</td>\n",
       "      <td>30.0</td>\n",
       "      <td>13950.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2.0</td>\n",
       "      <td>164.0</td>\n",
       "      <td>99.4</td>\n",
       "      <td>176.6</td>\n",
       "      <td>66.4</td>\n",
       "      <td>54.3</td>\n",
       "      <td>2824.0</td>\n",
       "      <td>136.0</td>\n",
       "      <td>3.19</td>\n",
       "      <td>3.40</td>\n",
       "      <td>8.0</td>\n",
       "      <td>115.0</td>\n",
       "      <td>5500.0</td>\n",
       "      <td>18.0</td>\n",
       "      <td>22.0</td>\n",
       "      <td>17450.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    0      1     2      3     4     5       6      7     8     9     10  \\\n",
       "0  3.0  122.0  88.6  168.8  64.1  48.8  2548.0  130.0  3.47  2.68   9.0   \n",
       "1  3.0  122.0  88.6  168.8  64.1  48.8  2548.0  130.0  3.47  2.68   9.0   \n",
       "2  1.0  122.0  94.5  171.2  65.5  52.4  2823.0  152.0  2.68  3.47   9.0   \n",
       "3  2.0  164.0  99.8  176.6  66.2  54.3  2337.0  109.0  3.19  3.40  10.0   \n",
       "4  2.0  164.0  99.4  176.6  66.4  54.3  2824.0  136.0  3.19  3.40   8.0   \n",
       "\n",
       "      11      12    13    14       15  \n",
       "0  111.0  5000.0  21.0  27.0  13495.0  \n",
       "1  111.0  5000.0  21.0  27.0  16500.0  \n",
       "2  154.0  5000.0  19.0  26.0  16500.0  \n",
       "3  102.0  5500.0  24.0  30.0  13950.0  \n",
       "4  115.0  5500.0  18.0  22.0  17450.0  "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_no_mis.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Всегда нужно осознавать, случайны ли пропуски в каком-то признаке. Иногда факт отсутствия информации о значении признака может сам быть важным признаком, который необходимо добавить к другим признакам.\n",
    "\n",
    "__Пример:__ предсказание возраста пользователя по данным с его телефона. Поскольку люди старшего возраста чаще пользуются простыми телефонами, факт отсутствия каких-то данных (например, истории посещенных интернет-страниц), скорее всего, будет хорошим признаком.\n",
    "\n",
    "Для категориальных признаков рекомендуется создавать отдельную категорию, соответствующую пропущенному значению. В наши данных пропусков в категориальных признаках нет."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# проверка пропусков в категориальных признаках\n",
    "np.any(np.isnan(X_cat_pd))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# проверка пропусков после применения Imputer\n",
    "np.any(X_no_mis.isnull())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Масштабирование признаков\n",
    "При начале работы с данными всегда рекомендуется приводить все признаки к одному масштабу.  Это важно по нескольким причинам:\n",
    "* ускорение обучения модели (пояснение на лекции);\n",
    "* улучшение численной устойчивости при работе с матрицей объекты-признаки (рядом с нулем чисел с плавающей точкой больше, чем с области больших чисел)\n",
    "* для линейных моделей: интерпретация весов при признаках как меры их значимости."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Первый популярный способ масштабирования - нормализация: вычитание среднего из каждого признака и деление на стандартное отклонение. Реализация в sklearn (нормировать бинарные признаки не нужно):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "normalizer = preprocessing.StandardScaler()\n",
    "X_real_norm_np = normalizer.fit_transform(X_no_mis)\n",
    "X_real_norm_pd = pd.DataFrame(data=X_real_norm_np)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Второй популярный способ: вычитание минимума из каждого признака, а затем деление на разницу максимального и минимального значения. Реализация в sklearn:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mm_scaler = preprocessing.MinMaxScaler()\n",
    "X_mm_scaled = mm_scaler.fit_transform(X_no_mis)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Объдиняем категориальные и вещественные признаки:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X = pd.concat([X_real_norm_pd, X_cat_pd], axis=1)\n",
    "X.columns = np.array([\"f\"+str(i) for i in range(X.shape[1])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "      <th>0</th>\n",
       "      <td>1.743470</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-1.690772</td>\n",
       "      <td>-0.426521</td>\n",
       "      <td>-0.844782</td>\n",
       "      <td>-2.020417</td>\n",
       "      <td>-0.014566</td>\n",
       "      <td>0.074449</td>\n",
       "      <td>0.519089</td>\n",
       "      <td>-1.839404</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.743470</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-1.690772</td>\n",
       "      <td>-0.426521</td>\n",
       "      <td>-0.844782</td>\n",
       "      <td>-2.020417</td>\n",
       "      <td>-0.014566</td>\n",
       "      <td>0.074449</td>\n",
       "      <td>0.519089</td>\n",
       "      <td>-1.839404</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.133509</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-0.708596</td>\n",
       "      <td>-0.231513</td>\n",
       "      <td>-0.190566</td>\n",
       "      <td>-0.543527</td>\n",
       "      <td>0.514882</td>\n",
       "      <td>0.604046</td>\n",
       "      <td>-2.404862</td>\n",
       "      <td>0.685920</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.938490</td>\n",
       "      <td>1.328961</td>\n",
       "      <td>0.173698</td>\n",
       "      <td>0.207256</td>\n",
       "      <td>0.136542</td>\n",
       "      <td>0.235942</td>\n",
       "      <td>-0.420797</td>\n",
       "      <td>-0.431076</td>\n",
       "      <td>-0.517248</td>\n",
       "      <td>0.462157</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.938490</td>\n",
       "      <td>1.328961</td>\n",
       "      <td>0.107110</td>\n",
       "      <td>0.207256</td>\n",
       "      <td>0.230001</td>\n",
       "      <td>0.235942</td>\n",
       "      <td>0.516807</td>\n",
       "      <td>0.218885</td>\n",
       "      <td>-0.517248</td>\n",
       "      <td>0.462157</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 67 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "         f0        f1        f2        f3        f4        f5        f6  \\\n",
       "0  1.743470  0.000000 -1.690772 -0.426521 -0.844782 -2.020417 -0.014566   \n",
       "1  1.743470  0.000000 -1.690772 -0.426521 -0.844782 -2.020417 -0.014566   \n",
       "2  0.133509  0.000000 -0.708596 -0.231513 -0.190566 -0.543527  0.514882   \n",
       "3  0.938490  1.328961  0.173698  0.207256  0.136542  0.235942 -0.420797   \n",
       "4  0.938490  1.328961  0.107110  0.207256  0.230001  0.235942  0.516807   \n",
       "\n",
       "         f7        f8        f9 ...   f57  f58  f59  f60  f61  f62  f63  f64  \\\n",
       "0  0.074449  0.519089 -1.839404 ...   0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0   \n",
       "1  0.074449  0.519089 -1.839404 ...   0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0   \n",
       "2  0.604046 -2.404862  0.685920 ...   0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0   \n",
       "3 -0.431076 -0.517248  0.462157 ...   0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0   \n",
       "4  0.218885 -0.517248  0.462157 ...   0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0   \n",
       "\n",
       "   f65  f66  \n",
       "0  0.0  0.0  \n",
       "1  0.0  0.0  \n",
       "2  0.0  0.0  \n",
       "3  0.0  0.0  \n",
       "4  0.0  0.0  \n",
       "\n",
       "[5 rows x 67 columns]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Добавление полиномиальных признаков\n",
    "Генерировать все полиномиальные признаки можно следующим образом:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pol_feat = preprocessing.PolynomialFeatures(degree=2, include_bias=False) # не включать константный признак\n",
    "X_pol = pol_feat.fit_transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((205, 67), (205, 2345))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.shape, X_pol.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Обратите внимание, что новая матрциа может занимать очень много памяти. Генерация полиномиальных признаков нужна, например, в случае, если вы хотите с помощью линейной регрессии настраивать полиномиальную модель зависимости целевого признака от данных."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Функции потерь в регрессии"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Функционал качества в задачах обучения с учителем обычно задается в виде суммы по объектам выборки:\n",
    "$$Q(a) = \\frac 1 \\ell \\sum_{i=1}^\\ell L(y_i, a(x_i)),$$\n",
    "где $L(\\cdot, \\cdot)$ - функция потерь, задающая штраф за разницу между предсказанием и истинным значением целевого признака. Свойства функции потерь:\n",
    "* $L(y_i, a(x_i)) \\geqslant 0$;\n",
    "* $L(y_i, y_i) = 0$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Как отмечалось на первой лекции, функционал качества должен в первую очередь отвечать требованиям заказчика, при этом математические свойства функции потерь могут быть неудобны для оптимизации. \n",
    "\n",
    "__Пример:__ если мы не различаем маленькие ошибки (между 0.01 и 0.1 нет особой разницы), но зато не хотим получать большие ошибки, можно использовать следующую функцию потерь:\n",
    "\n",
    "$$L(y_i, a(x_i)) = [| y_i - a(x_i) | < \\varepsilon],$$ $\\varepsilon$ - допустимая разница между предсказанием и фактом.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Среднеквадратичная и средняя абсолютная ошибка\n",
    "Кроме требований заказчика, функционал качества должен учитывать математические особенности модели, например устойчивость к шумовым объектам. В линейной регрессии Mean Squared Error: $L(y_i, a(x_i)) = (a(x_i) - y_i)^2$ не обладает этим свойством, потому что задает очень большие штрафы за большие отклонения от фактического значения. \n",
    "\n",
    "Рассмотрим это явление на примере. Выберем один признак, от которого целевой признак (имеющий индекс 15 в матрице X) зависит практически линейно. Добавим к выборке два объекта-выброса и посмотрим, как изменится оптимизированная на MSE прямая."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "%matplotlib inline\n",
    "from sklearn.linear_model import LinearRegression as LR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_subset = X[[\"f7\", \"f15\"]].values\n",
    "X_subset_modified = np.vstack((X_subset, [[1, 90], [2, 50]])) # добавление двух шумовых точек"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def scatter_points_and_plot_line_MSE(X_subset):\n",
    "    plt.scatter(X_subset[:, 0], X_subset[:, 1])   # визуализируем точки\n",
    "    lr = LR()\n",
    "    lr.fit(X_subset[:, 0][:, np.newaxis], X_subset[:, 1])  # найдем веса линейной модели\n",
    "    grid = np.linspace(-2, 6, 100)\n",
    "    line = lr.predict(grid[:, np.newaxis])\n",
    "    plt.plot(grid, line)   # визуализируем прямую"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python2.7/site-packages/scipy/linalg/basic.py:884: RuntimeWarning: internal gelsd driver lwork query error, required iwork dimension not returned. This is likely the result of LAPACK bug 0038, fixed in LAPACK 3.2.2 (released July 21, 2010). Falling back to 'gelss' driver.\n",
      "  warnings.warn(mesg, RuntimeWarning)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10fbe0990>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Pf7iSh/bvu/nyuDrnnC/UtddWff3rVc95znsreU4lv13Jqyu5qX+7W2pk5Jx6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\niiQAAAAAOlEkAQAAANCJIgkAAACATv4/qEv4UmfnLC0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fb01510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(20, 5))\n",
    "plt.subplot(1, 2, 1)\n",
    "scatter_points_and_plot_line_MSE(X_subset)\n",
    "plt.ylim(-20, 100)\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.subplot(1, 2, 2)\n",
    "scatter_points_and_plot_line_MSE(X_subset_modified)\n",
    "plt.ylim(-20, 100)\n",
    "plt.xlabel(\"x\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Из-за шумовых объектов прямая достаточно сильно изменила наклон. Поэтому вместо MSE часто используют Mean Absoulte Error: $L(y_i, a(x_i)) = |a(x_i) - y_i|$:\n",
    "\n",
    "Теперь обучим регрессию, оптимизируя MAE. В sklearn такая регрессия не реализована, но можно использовать модуль statsmodels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import statsmodels.api as sm\n",
    "import statsmodels.formula.api as smf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def scatter_points_and_plot_line_MAE(X_subset):\n",
    "    mod = smf.quantreg('f15 ~ f7', pd.DataFrame(data=X_subset, columns=[\"f7\", \"f15\"])) # задаеем зависимость и передаем данные\n",
    "    res = mod.fit(q=0.5)\n",
    "    plt.scatter(X_subset[:, 0], X_subset[:, 1])   # визуализируем точки\n",
    "    grid = np.linspace(-2, 6, 100)\n",
    "    plt.plot(grid, grid * res.params[\"f7\"] + res.params[\"Intercept\"])   # визуализируем прямую\n",
    "    return mod, res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11080ead0>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ZHXts1cMf/uOaNevLNWvWtyr5TiW39ENGVXJ3JT+rPfb4ZT3oQVUPecjKSr5Y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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110757610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(20, 5))\n",
    "plt.subplot(1, 2, 1)\n",
    "model, result = scatter_points_and_plot_line_MAE(X_subset)\n",
    "plt.ylim(-20, 100)\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.subplot(1, 2, 2)\n",
    "model, result = scatter_points_and_plot_line_MAE(X_subset_modified)\n",
    "plt.ylim(-20, 100)\n",
    "plt.xlabel(\"x\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Прямая не изменила направление из-за выбросов."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Попробуем добавить больше шумовых объектов:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_subset_modified_twice = np.vstack((X_subset_modified, np.random.randint(5, size=60).reshape(-1, 2)*[1, 30])) # добавление двух шумовых точек"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x110c02650>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Nfay6y+dbuHDh9u+bm5tpbu7rYSRJUrlbsWIFK1asyHsYFc0MJkmSuhpI/kpZOqGUUgLm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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110764f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(20, 5))\n",
    "plt.subplot(1, 2, 1)\n",
    "model, result = scatter_points_and_plot_line_MAE(X_subset)\n",
    "plt.ylim(-20, 100)\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.subplot(1, 2, 2)\n",
    "model, result = scatter_points_and_plot_line_MAE(X_subset_modified_twice)\n",
    "plt.ylim(-20, 100)\n",
    "plt.xlabel(\"x\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Прямая немного изменила наклон, когда мы добавили 30 (почти 15%) шумовых точек."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Рассмотрим некоторые свойства MSE и MAE.\n",
    "\n",
    "Пусть алгоритм возвращает константный ответ: $a(x) = C$. Такие алгоритмы используются, когда для одного объекта известны несколько различных значений одного целевого признака (например, суммы покупок в одном магазине).\n",
    "\n",
    "__Задача.__ Найдите $C$, минимизирующий среднеквадратичную ошибку.\n",
    "\n",
    "_Решение._ $$MSE(C) = \\frac{1}{\\ell} \\sum_{i=1}^\\ell (C - y_i)^2$$\n",
    "$$\\frac{\\partial MSE (C)}{\\partial C} = \\frac{1}{\\ell} \\sum_{i=1}^\\ell 2 (C - y_i) = 2 C - \\frac{1}{\\ell} \\sum_{i=1}^\\ell 2 y_i = 0$$\n",
    "$$C = \\frac 1 \\ell \\sum_{i=1}^\\ell y_i$$.\n",
    "\n",
    "__Задача.__ Найдите $C$, минимизирующий среднюю абсолютную ошибку.\n",
    "\n",
    "_Решение._ $$MAE(C) =  \\frac{1}{\\ell} \\sum_{i=1}^\\ell |C - y_i|$$\n",
    "\n",
    "Покажем, что минимум MAE достигается при $C = median(y_1, \\dots, y_\\ell) = m.$ Рассмотрим $C < m$.\n",
    "\n",
    "$$|y_i - C| - |y_i - m| = \\begin{cases} C - m, y_i < m  \\\\ - (C + m - 2 y_i), C \\leqslant y_i \\leqslant m \\\\ - (C - m), y_i > m \\end{cases}$$\n",
    "$$|y_i - C| - |y_i - m| \\geqslant - (C - m) + 2(C - m) [y_i \\leqslant m]$$\n",
    "Суммируем по i:\n",
    "$$\\ell MAE(C) - \\ell MAE(m) \\geqslant - \\ell (C - m) + 2 (C - m) \\sum_{i=1}^\\ell [y_i \\leqslant m]$$\n",
    "\n",
    "Так как m - медиана, $\\sum_{i=1}^\\ell [y_i \\leqslant m] \\geqslant \\frac \\ell 2$. Тогда\n",
    "\n",
    "$$\\ell MAE(C) - \\ell MAE(m) \\geqslant - \\ell (C - m) + 2 (C - m) \\frac \\ell 2 = 0.$$\n",
    "\n",
    "Итак, для $C < m$ выполняется $MAE(C) \\geqslant MAE(m)$. Аналогично показывается, что при для $C > m$ выполняется $MAE(C) \\geqslant MAE(m)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Поскольку средняя абсолютная ошибка не является дифференцируемой по $w$ функцией, оптимизировать ее напрямую методом градиентного спуска не удастся. Для этого используются субградиентные или другие методы. Один из способов оптимизации MAE будет рассмотрен ниже."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Huber Loss\n",
    "Иногда используют \"гибрид\" MAE и MSE, который, как и MAE, устойчив к шумовым объектам, и как и MSE, мало штрафует малые отклонения от фактического значени целевого признака - Huber Loss:\n",
    "$$L(y_i, a(x_i)) = \\phi_\\varepsilon(a(x_i) - y_i)$$\n",
    "$$\\phi_\\varepsilon(z) = \\begin{cases} \\frac 1 2 z^2, - \\varepsilon < z < \\varepsilon, \\\\\\varepsilon (|z| - \\frac 1 2 \\varepsilon), иначе \\\\ \\end{cases}$$\n",
    "\n",
    "Легко проверить, что у этой функции существует непрерывная первая проиводная во всех точках.\n",
    "\n",
    "Оптимизация Huber Loss реализована в sklearn:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sklearn.linear_model import HuberRegressor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quantile Loss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "В некоторых задачах штраф за ошибку зависит не только от величины абсолютного отклонения от фактического значения, но и от знака этого отклонения. Например, лучше предсказать спрос больше, чем будет по факту, чем меньше, потому что в первом случае будет потеряна прибыль. В этом случае используется квантильная регрессия со следующей функцией потерь:\n",
    "$$L(y_i, a(x_i)) = \\rho_\\tau(y_i - x_i^T w),$$\n",
    "$$\\rho_\\tau(z) = \\begin{cases} \\tau z, \\quad z > 0, \\\\ (\\tau - 1) z, \\quad z \\leqslant 0 \\end{cases}$$\n",
    "Параметр $\\tau$ влияет на то, насколько различаются штрафы за положительную и отрицательную разницу."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10fc74e10>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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pZdALg5j4yUSKRhRx+29ur3VzYSpFm5E0B8L+GwqU+4qqrgH642zAPxpo8Ng8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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110f3b390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "grid = np.linspace(-3, 3, 100)\n",
    "tau_list = [0.1, 0.5, 0.8, 0.9]\n",
    "for tau in tau_list:\n",
    "    plt.plot(grid, tau * grid * (grid > 0) + (tau - 1) * grid * (grid <= 0), label=r\"$\\tau = \" + str(tau) + r\"$\")\n",
    "plt.xlabel(r'$z = y_i - a(x_i)$', size=20)\n",
    "plt.ylabel(r'$\\mathscr{L}(z)$', size=20)\n",
    "plt.legend(loc=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Изобразим график квантильной функции потерь вместе с другими рассмотренными:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x11137ffd0>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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UFGRkZGDChAlyXwOQ1v+NjIyUfR8ZGQlDQ0OIRKJya/wOHDgQ169fR1xcHOzt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\n9t6MhwkPsbL3Ssx0mInahmV8YhKLgZs3pXf5f/z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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11133fe90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "grid = np.linspace(-3, 3, 100)\n",
    "quantile_tau = 0.2\n",
    "mse_loss = grid ** 2\n",
    "mae_loss = np.abs(grid)\n",
    "huber_loss = 0.5 * mse_loss * (grid >= -1) * (grid <= 1) + (mae_loss - 0.5) * (grid < -1) + (mae_loss - 0.5)  * (grid > 1)\n",
    "quantile_loss = quantile_tau * grid * (grid > 0) + (quantile_tau - 1) * grid * (grid <= 0)\n",
    "plt.plot(grid, mae_loss, label=\"Absolute Loss\")\n",
    "plt.plot(grid, mse_loss, label=\"Quadratic Loss\")\n",
    "plt.plot(grid, huber_loss, label=\"Huber Loss\")\n",
    "plt.plot(grid, quantile_loss, label=\"Quantile Loss\")\n",
    "plt.xlabel(\"$y_i - a(x_i)$\")\n",
    "plt.ylabel(\"$L(y_i, a(x_i))$\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Задача.__ Укажите параметр $\\tau$, при котором обучение квантильной регрессии равносильно оптимизации MAE.\n",
    "\n",
    "_Решение._ При $\\tau = \\frac 1 2$ $$\\rho_\\tau(x) = \\frac 1 2 |x|$$ и $L(y_i, a(x_i)) = \\frac 1 2 MAE$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Обучение квантильной регрессии как задача линейного программирования\n",
    "Как и в случае с MAE, функция потерь не дифференцируема в 0. Однако задачу оптимизации MAE можно свести к задаче линейного программирования, для которой известны эффективные алгоритмы решения ([Симплекс-метод](https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81-%D0%BC%D0%B5%D1%82%D0%BE%D0%B4)). Задача линейного программирования в общем виде задается  следующем виде:\n",
    "$$\\langle c, x \\rangle \\rightarrow \\min_{x}$$\n",
    "$$Ax = b$$\n",
    "$$x \\geqslant 0$$\n",
    "\n",
    "__Задача.__ Рассмотрим задачу оптимизации $$\\sum_{i=1}^\\ell \\rho_\\tau(y_i - x_i^T w) \\rightarrow \\min _{w}.$$\n",
    "Сведите эту задачу к задаче линейного программирования.\n",
    "\n",
    "_Решение._ Введем переменные:\n",
    "$$e_i = y_i - x_i^T w$$\n",
    "$$u_i = max\\{e_i, 0\\} \\geqslant 0$$\n",
    "$$v_i = max\\{-e_i, 0\\} \\geqslant 0$$\n",
    "Тогда $u_i - v_i = e_i$,\n",
    "\n",
    "$$ \\rho_\\tau(y_i - x_i^T w) = \\rho_\\tau(e_i) = \\rho_\\tau(u_i - v_i) = \\tau u_i + (1 - \\tau) v_i. $$\n",
    "Последний переход можно проверить непосрдственно. Например, если $e_i \\geqslant 0$, то $v_i = 0$,\n",
    "$$\\rho_\\tau(e_i) = \\tau (u_i - v_i) = \\tau (u_i - v_i) + v_i = \\tau u_i + (1 - \\tau) v_i.$$ Аналогично при $e_i < 0$.\n",
    "\n",
    "Функционал качества:\n",
    "$$\\sum_{i=1}^\\ell \\rho_\\tau(y_i - x_i^T w) = \\sum_{i=1}^\\ell \\bigl (\\tau u_i + (1 - \\tau) v_i \\bigr) = \\tau \\sum_{i=1}^\\ell u_i + (1 - \\tau) \\sum_{i=1}^\\ell v_i = \\tau \\langle 1, u \\rangle + (1 - \\tau) \\langle 1, v \\rangle ,$$\n",
    "$1 = (1, \\dots, 1) \\in R^\\ell$, $u = (u_1, \\dots, u_\\ell$), $v = (v_1, \\dots, v_\\ell)$.\n",
    "\n",
    "Задача линейного программирования:\n",
    "\n",
    "$$\\tau \\langle 1, u \\rangle + (1 - \\tau) \\langle 1, v \\rangle \\rightarrow \\min_{u, v, w}$$\n",
    "$$y - X w = u - v$$\n",
    "$$u \\geqslant 0, \\quad v \\geqslant 0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Проследим наклон прямой в нашей одномерной задаче регрессии при изменении $\\tau$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Library/Python/2.7/site-packages/statsmodels/regression/quantile_regression.py:189: ConvergenceWarning: Convergence cycle detected\n",
      "  warnings.warn(\"Convergence cycle detected\", ConvergenceWarning)\n",
      "/Library/Python/2.7/site-packages/statsmodels/regression/quantile_regression.py:189: ConvergenceWarning: Convergence cycle detected\n",
      "  warnings.warn(\"Convergence cycle detected\", ConvergenceWarning)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x111452590>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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B0dHRaFvXGR3VVe3fxAFgC4BkAPEAMgAIAHhUka7GpgyGeZdkCbJoxqUZZLbb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\nmvbIwbRkyZKPFNgTeHl5aWBXKovx9fXl7NmzyV47ffo0RYoUSbX85s2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m/v7+Mnjw4DTf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\nAQQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11080ef50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X_subset[:, 0], X_subset[:, 1])\n",
    "grid = np.linspace(-2, 6, 100)\n",
    "mod = smf.quantreg('f15 ~ f7', pd.DataFrame(data=X_subset, columns=[\"f7\", \"f15\"]))\n",
    "for q in np.arange(0.1, 1, 0.1):\n",
    "    res = mod.fit(q=q)\n",
    "    plt.plot(grid, grid * res.params[\"f7\"] + res.params[\"Intercept\"], label=\"q = \"+str(q))\n",
    "plt.legend(loc=(1.1, 0.1))\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vowpal wabbit\n",
    "Мы увидели, что в python для оптимизации разных функционалов качества приходится искать отдельные модули. Кроме того, для очень больших матриц объекты-признаки python-методы могут быть неэффективны. В сообществе специалистов по машинному обучению популярна система [vowpal wabbit](https://github.com/JohnLangford/vowpal_wabbit/wiki), позволяющая оптимизировать разные функции потерь и предоставляющая пользователю интерфейс для командной строки.\n",
    "\n",
    "В vowpal wabbit реализован эффективный и масштабируемый онлайн-алгоритм для оптимизации линейной модели.\n",
    "\n",
    "[Входные данные](https://github.com/JohnLangford/vowpal_wabbit/wiki/Input-format) должны быть представлены с виде текстового файла, каждая строка - отдельный объект. Формат строки:\n",
    "\n",
    "      [Label] [Importance] [Base] [Tag]|Namespace Features |Namespace Features ... |Namespace Features\n",
    "где\n",
    "* Label - значение целевого признака\n",
    "* Importance - вес объекта в функционале качества (коэффициент, на который домножается функция потерь)\n",
    "* Tag - идентификатор\n",
    "* Namespace задает группу признаков (пространство имен)\n",
    "* Признаки перечисляются через пробел в виде признак:значение, :1 можно опускать (если какой-то признак в одном пространстве имен повторяется, то его значения суммируются)\n",
    "\n",
    "**Пример**:\n",
    "123 10 |integer 1:0.43 5:2.1 age:20 |text some raw text here age:120\n",
    "\n",
    "Корректность формата входных данных можно проверить при помощи [VW validator](http://hunch.net/%7Evw/validate.html).\n",
    "\n",
    "Интерфейс командной строки подробно описан на [Github-Wiki проекта](https://github.com/JohnLangford/vowpal_wabbit/wiki/Command-line-arguments).\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Данные в указанном формате нужно сохранить в файл."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# функция для записи датафрейма в vowpal wabbit\n",
    "def convert_df_to_vw(df, target_col, vw_path):\n",
    "    target = df[target_col]\n",
    "    feature_cols = df.columns.drop(target_col)\n",
    "    with open(vw_path, \"w\") as fout:\n",
    "        for idx in df.index:\n",
    "            fout.write(str(target[idx])+\" |features\")\n",
    "            features = df[feature_cols].loc[idx]\n",
    "            for feature in features.index:\n",
    "                if features[feature] != 0:\n",
    "                    fout.write(\" \"+str(feature)+\":\"+str(features[feature]))\n",
    "            fout.write(\"\\n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sklearn.cross_validation import train_test_split\n",
    "\n",
    "X_train, X_test = train_test_split(X, test_size = 0.2)\n",
    "convert_df_to_vw(X_train, \"f15\", \"train.txt\")\n",
    "convert_df_to_vw(X_test, \"f15\", \"test.txt\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Пример запуска обучения регрессии на MAE vowpal wabbit с сохранением обученной модели в model.vw (! в начале строки сообщает,  что это команда для командной строки):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "final_regressor = model.vw\r\n",
      "Num weight bits = 18\r\n",
      "learning rate = 0.5\r\n",
      "initial_t = 0\r\n",
      "power_t = 0.5\r\n",
      "using no cache\r\n",
      "Reading datafile = train.txt\r\n",
      "num sources = 1\r\n",
      "average  since         example        example  current  current  current\r\n",
      "loss     last          counter         weight    label  predict features\r\n",
      "0.338371 0.338371            1            1.0  -0.6767   0.0000       23\r\n",
      "0.265846 0.193321            2            2.0  -0.8012  -0.4146       24\r\n",
      "0.184785 0.103723            4            4.0  -0.7318  -0.5246       24\r\n",
      "0.232496 0.280207            8            8.0   1.0062   0.3331       24\r\n",
      "0.183575 0.134654           16           16.0   1.1225   0.2475       24\r\n",
      "0.164616 0.145658           32           32.0  -0.0334  -0.6443       24\r\n",
      "0.166911 0.169205           64           64.0   0.3163   0.7487       23\r\n",
      "0.156413 0.145914          128          128.0  -0.9820  -0.7032       24\r\n",
      "\r\n",
      "finished run\r\n",
      "number of examples per pass = 164\r\n",
      "passes used = 1\r\n",
      "weighted example sum = 164.000000\r\n",
      "weighted label sum = -2.828436\r\n",
      "average loss = 0.158514\r\n",
      "total feature number = 3868\r\n"
     ]
    }
   ],
   "source": [
    "! vw --data train.txt --loss_function quantile --quantile_tau 0.5 -f model.vw"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Другие полезные опции:\n",
    "* --passes 10 : число проходов по данным во время обучения\n",
    "* --quadratic ab: сгенерировать все парные признаки, где первый признак берется из пространств с именем \"a\\*\", а второй — из \"b*\"\n",
    "* --cubic abc : кубические признаки\n",
    "* --ignore a : игнорировать признаки из namespace, начинающихся с a\n",
    "* --keep ab : использовать только признаки из namespace, начинающихся с a и b\n",
    "* --l1 1.0 : коэффициент при L1-регуляризаторе\n",
    "* --l2 1.0 : коэффициент при L2-регуляризаторе\n",
    "* --minibatch 5 : число объектов в мини-батче при обучении\n",
    "* параметры градиентного шага: \n",
    "$$w^{(t+1)} = w^t - \\alpha_t \\frac{\\partial}{\\partial w} Q(x_{i_t}),$$\n",
    "$$\\alpha_t = s \\left( \\frac{i}{i + t} \\right)^p,$$\n",
    "где \n",
    "    * -l s\n",
    "    * -initial_t i\n",
    "    * -power_t p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Сделать предсказания для новых объектов:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "only testing\r\n",
      "raw predictions = raw_predictions.txt\r\n",
      "Num weight bits = 18\r\n",
      "learning rate = 0.5\r\n",
      "initial_t = 0\r\n",
      "power_t = 0.5\r\n",
      "using no cache\r\n",
      "Reading datafile = test.txt\r\n",
      "num sources = 1\r\n",
      "average  since         example        example  current  current  current\r\n",
      "loss     last          counter         weight    label  predict features\r\n",
      "0.350830 0.350830            1            1.0  -0.1665   0.5351       24\r\n",
      "0.239463 0.128096            2            2.0  -0.7743  -1.0305       23\r\n",
      "0.143031 0.046598            4            4.0   0.6463   0.6646       24\r\n",
      "0.140188 0.137345            8            8.0   0.7278   0.9080       23\r\n",
      "0.188482 0.236776           16           16.0   0.4214   0.6957       24\r\n",
      "0.219974 0.251466           32           32.0   3.0158   2.2876       23\r\n",
      "\r\n",
      "finished run\r\n",
      "number of examples per pass = 41\r\n",
      "passes used = 1\r\n",
      "weighted example sum = 41.000000\r\n",
      "weighted label sum = 2.828436\r\n",
      "average loss = 0.208477\r\n",
      "total feature number = 973\r\n"
     ]
    }
   ],
   "source": [
    "!`vw --data test.txt -t -i model.vw --loss_function quantile --quantile_tau 0.5 -r raw_predictions.txt`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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