air_quality_multivariate-var model.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vector Autoregression (VAR)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import display, Markdown\n",
    "from math import sqrt\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "from sklearn.metrics import mean_squared_error"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Air Quality Dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Load in the dataset\n",
    "data = pd.read_csv(\"AirQualityUCI.csv\", sep=';')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Clean empty columns\n",
    "data = data.drop(['Unnamed: 15', 'Unnamed: 16'], axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Clean empty rows\n",
    "data = data.dropna(axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def report(df):\n",
    "    display(Markdown('<b>head():</b>'))\n",
    "    display(df.head())\n",
    "    display(Markdown('<b>describe():</b>'))\n",
    "    display(df.describe())\n",
    "    display(Markdown('<b>info():</b>'))\n",
    "    display(df.info(verbose=True))\n",
    "    display(Markdown('<b>infer_dtype():</b>'))\n",
    "    display(df.apply(lambda x: pd.api.types.infer_dtype(x.values)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "<b>head():</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Date</th>\n",
       "      <th>Time</th>\n",
       "      <th>CO(GT)</th>\n",
       "      <th>PT08.S1(CO)</th>\n",
       "      <th>NMHC(GT)</th>\n",
       "      <th>C6H6(GT)</th>\n",
       "      <th>PT08.S2(NMHC)</th>\n",
       "      <th>NOx(GT)</th>\n",
       "      <th>PT08.S3(NOx)</th>\n",
       "      <th>NO2(GT)</th>\n",
       "      <th>PT08.S4(NO2)</th>\n",
       "      <th>PT08.S5(O3)</th>\n",
       "      <th>T</th>\n",
       "      <th>RH</th>\n",
       "      <th>AH</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>18.00.00</td>\n",
       "      <td>2,6</td>\n",
       "      <td>1360.0</td>\n",
       "      <td>150.0</td>\n",
       "      <td>11,9</td>\n",
       "      <td>1046.0</td>\n",
       "      <td>166.0</td>\n",
       "      <td>1056.0</td>\n",
       "      <td>113.0</td>\n",
       "      <td>1692.0</td>\n",
       "      <td>1268.0</td>\n",
       "      <td>13,6</td>\n",
       "      <td>48,9</td>\n",
       "      <td>0,7578</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>19.00.00</td>\n",
       "      <td>2</td>\n",
       "      <td>1292.0</td>\n",
       "      <td>112.0</td>\n",
       "      <td>9,4</td>\n",
       "      <td>955.0</td>\n",
       "      <td>103.0</td>\n",
       "      <td>1174.0</td>\n",
       "      <td>92.0</td>\n",
       "      <td>1559.0</td>\n",
       "      <td>972.0</td>\n",
       "      <td>13,3</td>\n",
       "      <td>47,7</td>\n",
       "      <td>0,7255</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>20.00.00</td>\n",
       "      <td>2,2</td>\n",
       "      <td>1402.0</td>\n",
       "      <td>88.0</td>\n",
       "      <td>9,0</td>\n",
       "      <td>939.0</td>\n",
       "      <td>131.0</td>\n",
       "      <td>1140.0</td>\n",
       "      <td>114.0</td>\n",
       "      <td>1555.0</td>\n",
       "      <td>1074.0</td>\n",
       "      <td>11,9</td>\n",
       "      <td>54,0</td>\n",
       "      <td>0,7502</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>21.00.00</td>\n",
       "      <td>2,2</td>\n",
       "      <td>1376.0</td>\n",
       "      <td>80.0</td>\n",
       "      <td>9,2</td>\n",
       "      <td>948.0</td>\n",
       "      <td>172.0</td>\n",
       "      <td>1092.0</td>\n",
       "      <td>122.0</td>\n",
       "      <td>1584.0</td>\n",
       "      <td>1203.0</td>\n",
       "      <td>11,0</td>\n",
       "      <td>60,0</td>\n",
       "      <td>0,7867</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>22.00.00</td>\n",
       "      <td>1,6</td>\n",
       "      <td>1272.0</td>\n",
       "      <td>51.0</td>\n",
       "      <td>6,5</td>\n",
       "      <td>836.0</td>\n",
       "      <td>131.0</td>\n",
       "      <td>1205.0</td>\n",
       "      <td>116.0</td>\n",
       "      <td>1490.0</td>\n",
       "      <td>1110.0</td>\n",
       "      <td>11,2</td>\n",
       "      <td>59,6</td>\n",
       "      <td>0,7888</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         Date      Time CO(GT)  PT08.S1(CO)  NMHC(GT) C6H6(GT)  PT08.S2(NMHC)  \\\n",
       "0  10/03/2004  18.00.00    2,6       1360.0     150.0     11,9         1046.0   \n",
       "1  10/03/2004  19.00.00      2       1292.0     112.0      9,4          955.0   \n",
       "2  10/03/2004  20.00.00    2,2       1402.0      88.0      9,0          939.0   \n",
       "3  10/03/2004  21.00.00    2,2       1376.0      80.0      9,2          948.0   \n",
       "4  10/03/2004  22.00.00    1,6       1272.0      51.0      6,5          836.0   \n",
       "\n",
       "   NOx(GT)  PT08.S3(NOx)  NO2(GT)  PT08.S4(NO2)  PT08.S5(O3)     T    RH  \\\n",
       "0    166.0        1056.0    113.0        1692.0       1268.0  13,6  48,9   \n",
       "1    103.0        1174.0     92.0        1559.0        972.0  13,3  47,7   \n",
       "2    131.0        1140.0    114.0        1555.0       1074.0  11,9  54,0   \n",
       "3    172.0        1092.0    122.0        1584.0       1203.0  11,0  60,0   \n",
       "4    131.0        1205.0    116.0        1490.0       1110.0  11,2  59,6   \n",
       "\n",
       "       AH  \n",
       "0  0,7578  \n",
       "1  0,7255  \n",
       "2  0,7502  \n",
       "3  0,7867  \n",
       "4  0,7888  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "<b>describe():</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PT08.S1(CO)</th>\n",
       "      <th>NMHC(GT)</th>\n",
       "      <th>PT08.S2(NMHC)</th>\n",
       "      <th>NOx(GT)</th>\n",
       "      <th>PT08.S3(NOx)</th>\n",
       "      <th>NO2(GT)</th>\n",
       "      <th>PT08.S4(NO2)</th>\n",
       "      <th>PT08.S5(O3)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>1048.990061</td>\n",
       "      <td>-159.090093</td>\n",
       "      <td>894.595276</td>\n",
       "      <td>168.616971</td>\n",
       "      <td>794.990168</td>\n",
       "      <td>58.148873</td>\n",
       "      <td>1391.479641</td>\n",
       "      <td>975.072032</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>329.832710</td>\n",
       "      <td>139.789093</td>\n",
       "      <td>342.333252</td>\n",
       "      <td>257.433866</td>\n",
       "      <td>321.993552</td>\n",
       "      <td>126.940455</td>\n",
       "      <td>467.210125</td>\n",
       "      <td>456.938184</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>921.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>711.000000</td>\n",
       "      <td>50.000000</td>\n",
       "      <td>637.000000</td>\n",
       "      <td>53.000000</td>\n",
       "      <td>1185.000000</td>\n",
       "      <td>700.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>1053.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>895.000000</td>\n",
       "      <td>141.000000</td>\n",
       "      <td>794.000000</td>\n",
       "      <td>96.000000</td>\n",
       "      <td>1446.000000</td>\n",
       "      <td>942.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>1221.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>1105.000000</td>\n",
       "      <td>284.000000</td>\n",
       "      <td>960.000000</td>\n",
       "      <td>133.000000</td>\n",
       "      <td>1662.000000</td>\n",
       "      <td>1255.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>2040.000000</td>\n",
       "      <td>1189.000000</td>\n",
       "      <td>2214.000000</td>\n",
       "      <td>1479.000000</td>\n",
       "      <td>2683.000000</td>\n",
       "      <td>340.000000</td>\n",
       "      <td>2775.000000</td>\n",
       "      <td>2523.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       PT08.S1(CO)     NMHC(GT)  PT08.S2(NMHC)      NOx(GT)  PT08.S3(NOx)  \\\n",
       "count  9357.000000  9357.000000    9357.000000  9357.000000   9357.000000   \n",
       "mean   1048.990061  -159.090093     894.595276   168.616971    794.990168   \n",
       "std     329.832710   139.789093     342.333252   257.433866    321.993552   \n",
       "min    -200.000000  -200.000000    -200.000000  -200.000000   -200.000000   \n",
       "25%     921.000000  -200.000000     711.000000    50.000000    637.000000   \n",
       "50%    1053.000000  -200.000000     895.000000   141.000000    794.000000   \n",
       "75%    1221.000000  -200.000000    1105.000000   284.000000    960.000000   \n",
       "max    2040.000000  1189.000000    2214.000000  1479.000000   2683.000000   \n",
       "\n",
       "           NO2(GT)  PT08.S4(NO2)  PT08.S5(O3)  \n",
       "count  9357.000000   9357.000000  9357.000000  \n",
       "mean     58.148873   1391.479641   975.072032  \n",
       "std     126.940455    467.210125   456.938184  \n",
       "min    -200.000000   -200.000000  -200.000000  \n",
       "25%      53.000000   1185.000000   700.000000  \n",
       "50%      96.000000   1446.000000   942.000000  \n",
       "75%     133.000000   1662.000000  1255.000000  \n",
       "max     340.000000   2775.000000  2523.000000  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/markdown": [
       "<b>info():</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
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     "metadata": {},
     "output_type": "display_data"
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "Int64Index: 9357 entries, 0 to 9356\n",
      "Data columns (total 15 columns):\n",
      "Date             9357 non-null object\n",
      "Time             9357 non-null object\n",
      "CO(GT)           9357 non-null object\n",
      "PT08.S1(CO)      9357 non-null float64\n",
      "NMHC(GT)         9357 non-null float64\n",
      "C6H6(GT)         9357 non-null object\n",
      "PT08.S2(NMHC)    9357 non-null float64\n",
      "NOx(GT)          9357 non-null float64\n",
      "PT08.S3(NOx)     9357 non-null float64\n",
      "NO2(GT)          9357 non-null float64\n",
      "PT08.S4(NO2)     9357 non-null float64\n",
      "PT08.S5(O3)      9357 non-null float64\n",
      "T                9357 non-null object\n",
      "RH               9357 non-null object\n",
      "AH               9357 non-null object\n",
      "dtypes: float64(8), object(7)\n",
      "memory usage: 1.1+ MB\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "None"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "<b>infer_dtype():</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Date               string\n",
       "Time               string\n",
       "CO(GT)             string\n",
       "PT08.S1(CO)      floating\n",
       "NMHC(GT)         floating\n",
       "C6H6(GT)           string\n",
       "PT08.S2(NMHC)    floating\n",
       "NOx(GT)          floating\n",
       "PT08.S3(NOx)     floating\n",
       "NO2(GT)          floating\n",
       "PT08.S4(NO2)     floating\n",
       "PT08.S5(O3)      floating\n",
       "T                  string\n",
       "RH                 string\n",
       "AH                 string\n",
       "dtype: object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "report(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "data['CO(GT)'] = data['CO(GT)'].str.replace(',', '.').astype(float)\n",
    "data['C6H6(GT)'] = data['C6H6(GT)'].str.replace(',','.').astype(float)\n",
    "data['T'] = data['T'].str.replace(',', '.').astype(float)\n",
    "data['RH'] = data['RH'].str.replace(',', '.').astype(float)\n",
    "data['AH'] = data['AH'].str.replace(',', '.').astype(float)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "<b>head():</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Date</th>\n",
       "      <th>Time</th>\n",
       "      <th>CO(GT)</th>\n",
       "      <th>PT08.S1(CO)</th>\n",
       "      <th>NMHC(GT)</th>\n",
       "      <th>C6H6(GT)</th>\n",
       "      <th>PT08.S2(NMHC)</th>\n",
       "      <th>NOx(GT)</th>\n",
       "      <th>PT08.S3(NOx)</th>\n",
       "      <th>NO2(GT)</th>\n",
       "      <th>PT08.S4(NO2)</th>\n",
       "      <th>PT08.S5(O3)</th>\n",
       "      <th>T</th>\n",
       "      <th>RH</th>\n",
       "      <th>AH</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>18.00.00</td>\n",
       "      <td>2.6</td>\n",
       "      <td>1360.0</td>\n",
       "      <td>150.0</td>\n",
       "      <td>11.9</td>\n",
       "      <td>1046.0</td>\n",
       "      <td>166.0</td>\n",
       "      <td>1056.0</td>\n",
       "      <td>113.0</td>\n",
       "      <td>1692.0</td>\n",
       "      <td>1268.0</td>\n",
       "      <td>13.6</td>\n",
       "      <td>48.9</td>\n",
       "      <td>0.7578</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>19.00.00</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1292.0</td>\n",
       "      <td>112.0</td>\n",
       "      <td>9.4</td>\n",
       "      <td>955.0</td>\n",
       "      <td>103.0</td>\n",
       "      <td>1174.0</td>\n",
       "      <td>92.0</td>\n",
       "      <td>1559.0</td>\n",
       "      <td>972.0</td>\n",
       "      <td>13.3</td>\n",
       "      <td>47.7</td>\n",
       "      <td>0.7255</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>20.00.00</td>\n",
       "      <td>2.2</td>\n",
       "      <td>1402.0</td>\n",
       "      <td>88.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>939.0</td>\n",
       "      <td>131.0</td>\n",
       "      <td>1140.0</td>\n",
       "      <td>114.0</td>\n",
       "      <td>1555.0</td>\n",
       "      <td>1074.0</td>\n",
       "      <td>11.9</td>\n",
       "      <td>54.0</td>\n",
       "      <td>0.7502</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>21.00.00</td>\n",
       "      <td>2.2</td>\n",
       "      <td>1376.0</td>\n",
       "      <td>80.0</td>\n",
       "      <td>9.2</td>\n",
       "      <td>948.0</td>\n",
       "      <td>172.0</td>\n",
       "      <td>1092.0</td>\n",
       "      <td>122.0</td>\n",
       "      <td>1584.0</td>\n",
       "      <td>1203.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>60.0</td>\n",
       "      <td>0.7867</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>22.00.00</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1272.0</td>\n",
       "      <td>51.0</td>\n",
       "      <td>6.5</td>\n",
       "      <td>836.0</td>\n",
       "      <td>131.0</td>\n",
       "      <td>1205.0</td>\n",
       "      <td>116.0</td>\n",
       "      <td>1490.0</td>\n",
       "      <td>1110.0</td>\n",
       "      <td>11.2</td>\n",
       "      <td>59.6</td>\n",
       "      <td>0.7888</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         Date      Time  CO(GT)  PT08.S1(CO)  NMHC(GT)  C6H6(GT)  \\\n",
       "0  10/03/2004  18.00.00     2.6       1360.0     150.0      11.9   \n",
       "1  10/03/2004  19.00.00     2.0       1292.0     112.0       9.4   \n",
       "2  10/03/2004  20.00.00     2.2       1402.0      88.0       9.0   \n",
       "3  10/03/2004  21.00.00     2.2       1376.0      80.0       9.2   \n",
       "4  10/03/2004  22.00.00     1.6       1272.0      51.0       6.5   \n",
       "\n",
       "   PT08.S2(NMHC)  NOx(GT)  PT08.S3(NOx)  NO2(GT)  PT08.S4(NO2)  PT08.S5(O3)  \\\n",
       "0         1046.0    166.0        1056.0    113.0        1692.0       1268.0   \n",
       "1          955.0    103.0        1174.0     92.0        1559.0        972.0   \n",
       "2          939.0    131.0        1140.0    114.0        1555.0       1074.0   \n",
       "3          948.0    172.0        1092.0    122.0        1584.0       1203.0   \n",
       "4          836.0    131.0        1205.0    116.0        1490.0       1110.0   \n",
       "\n",
       "      T    RH      AH  \n",
       "0  13.6  48.9  0.7578  \n",
       "1  13.3  47.7  0.7255  \n",
       "2  11.9  54.0  0.7502  \n",
       "3  11.0  60.0  0.7867  \n",
       "4  11.2  59.6  0.7888  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "<b>describe():</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CO(GT)</th>\n",
       "      <th>PT08.S1(CO)</th>\n",
       "      <th>NMHC(GT)</th>\n",
       "      <th>C6H6(GT)</th>\n",
       "      <th>PT08.S2(NMHC)</th>\n",
       "      <th>NOx(GT)</th>\n",
       "      <th>PT08.S3(NOx)</th>\n",
       "      <th>NO2(GT)</th>\n",
       "      <th>PT08.S4(NO2)</th>\n",
       "      <th>PT08.S5(O3)</th>\n",
       "      <th>T</th>\n",
       "      <th>RH</th>\n",
       "      <th>AH</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "      <td>9357.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>-34.207524</td>\n",
       "      <td>1048.990061</td>\n",
       "      <td>-159.090093</td>\n",
       "      <td>1.865683</td>\n",
       "      <td>894.595276</td>\n",
       "      <td>168.616971</td>\n",
       "      <td>794.990168</td>\n",
       "      <td>58.148873</td>\n",
       "      <td>1391.479641</td>\n",
       "      <td>975.072032</td>\n",
       "      <td>9.778305</td>\n",
       "      <td>39.485380</td>\n",
       "      <td>-6.837604</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>77.657170</td>\n",
       "      <td>329.832710</td>\n",
       "      <td>139.789093</td>\n",
       "      <td>41.380206</td>\n",
       "      <td>342.333252</td>\n",
       "      <td>257.433866</td>\n",
       "      <td>321.993552</td>\n",
       "      <td>126.940455</td>\n",
       "      <td>467.210125</td>\n",
       "      <td>456.938184</td>\n",
       "      <td>43.203623</td>\n",
       "      <td>51.216145</td>\n",
       "      <td>38.976670</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>0.600000</td>\n",
       "      <td>921.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>4.000000</td>\n",
       "      <td>711.000000</td>\n",
       "      <td>50.000000</td>\n",
       "      <td>637.000000</td>\n",
       "      <td>53.000000</td>\n",
       "      <td>1185.000000</td>\n",
       "      <td>700.000000</td>\n",
       "      <td>10.900000</td>\n",
       "      <td>34.100000</td>\n",
       "      <td>0.692300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>1.500000</td>\n",
       "      <td>1053.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>7.900000</td>\n",
       "      <td>895.000000</td>\n",
       "      <td>141.000000</td>\n",
       "      <td>794.000000</td>\n",
       "      <td>96.000000</td>\n",
       "      <td>1446.000000</td>\n",
       "      <td>942.000000</td>\n",
       "      <td>17.200000</td>\n",
       "      <td>48.600000</td>\n",
       "      <td>0.976800</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>2.600000</td>\n",
       "      <td>1221.000000</td>\n",
       "      <td>-200.000000</td>\n",
       "      <td>13.600000</td>\n",
       "      <td>1105.000000</td>\n",
       "      <td>284.000000</td>\n",
       "      <td>960.000000</td>\n",
       "      <td>133.000000</td>\n",
       "      <td>1662.000000</td>\n",
       "      <td>1255.000000</td>\n",
       "      <td>24.100000</td>\n",
       "      <td>61.900000</td>\n",
       "      <td>1.296200</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>11.900000</td>\n",
       "      <td>2040.000000</td>\n",
       "      <td>1189.000000</td>\n",
       "      <td>63.700000</td>\n",
       "      <td>2214.000000</td>\n",
       "      <td>1479.000000</td>\n",
       "      <td>2683.000000</td>\n",
       "      <td>340.000000</td>\n",
       "      <td>2775.000000</td>\n",
       "      <td>2523.000000</td>\n",
       "      <td>44.600000</td>\n",
       "      <td>88.700000</td>\n",
       "      <td>2.231000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            CO(GT)  PT08.S1(CO)     NMHC(GT)     C6H6(GT)  PT08.S2(NMHC)  \\\n",
       "count  9357.000000  9357.000000  9357.000000  9357.000000    9357.000000   \n",
       "mean    -34.207524  1048.990061  -159.090093     1.865683     894.595276   \n",
       "std      77.657170   329.832710   139.789093    41.380206     342.333252   \n",
       "min    -200.000000  -200.000000  -200.000000  -200.000000    -200.000000   \n",
       "25%       0.600000   921.000000  -200.000000     4.000000     711.000000   \n",
       "50%       1.500000  1053.000000  -200.000000     7.900000     895.000000   \n",
       "75%       2.600000  1221.000000  -200.000000    13.600000    1105.000000   \n",
       "max      11.900000  2040.000000  1189.000000    63.700000    2214.000000   \n",
       "\n",
       "           NOx(GT)  PT08.S3(NOx)      NO2(GT)  PT08.S4(NO2)  PT08.S5(O3)  \\\n",
       "count  9357.000000   9357.000000  9357.000000   9357.000000  9357.000000   \n",
       "mean    168.616971    794.990168    58.148873   1391.479641   975.072032   \n",
       "std     257.433866    321.993552   126.940455    467.210125   456.938184   \n",
       "min    -200.000000   -200.000000  -200.000000   -200.000000  -200.000000   \n",
       "25%      50.000000    637.000000    53.000000   1185.000000   700.000000   \n",
       "50%     141.000000    794.000000    96.000000   1446.000000   942.000000   \n",
       "75%     284.000000    960.000000   133.000000   1662.000000  1255.000000   \n",
       "max    1479.000000   2683.000000   340.000000   2775.000000  2523.000000   \n",
       "\n",
       "                 T           RH           AH  \n",
       "count  9357.000000  9357.000000  9357.000000  \n",
       "mean      9.778305    39.485380    -6.837604  \n",
       "std      43.203623    51.216145    38.976670  \n",
       "min    -200.000000  -200.000000  -200.000000  \n",
       "25%      10.900000    34.100000     0.692300  \n",
       "50%      17.200000    48.600000     0.976800  \n",
       "75%      24.100000    61.900000     1.296200  \n",
       "max      44.600000    88.700000     2.231000  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "<b>info():</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "Int64Index: 9357 entries, 0 to 9356\n",
      "Data columns (total 15 columns):\n",
      "Date             9357 non-null object\n",
      "Time             9357 non-null object\n",
      "CO(GT)           9357 non-null float64\n",
      "PT08.S1(CO)      9357 non-null float64\n",
      "NMHC(GT)         9357 non-null float64\n",
      "C6H6(GT)         9357 non-null float64\n",
      "PT08.S2(NMHC)    9357 non-null float64\n",
      "NOx(GT)          9357 non-null float64\n",
      "PT08.S3(NOx)     9357 non-null float64\n",
      "NO2(GT)          9357 non-null float64\n",
      "PT08.S4(NO2)     9357 non-null float64\n",
      "PT08.S5(O3)      9357 non-null float64\n",
      "T                9357 non-null float64\n",
      "RH               9357 non-null float64\n",
      "AH               9357 non-null float64\n",
      "dtypes: float64(13), object(2)\n",
      "memory usage: 1.1+ MB\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "None"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "<b>infer_dtype():</b>"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Date               string\n",
       "Time               string\n",
       "CO(GT)           floating\n",
       "PT08.S1(CO)      floating\n",
       "NMHC(GT)         floating\n",
       "C6H6(GT)         floating\n",
       "PT08.S2(NMHC)    floating\n",
       "NOx(GT)          floating\n",
       "PT08.S3(NOx)     floating\n",
       "NO2(GT)          floating\n",
       "PT08.S4(NO2)     floating\n",
       "PT08.S5(O3)      floating\n",
       "T                floating\n",
       "RH               floating\n",
       "AH               floating\n",
       "dtype: object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "report(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multivariate Time Series (MTS)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A <b>multivariate time series</b> has more than one time-dependent variable just like our dataset. Each variable depends not only on its past values but also has some dependency on other variables. This dependency is used for forecasting future values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### to_datetime()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    18:00:00\n",
       "1    19:00:00\n",
       "2    20:00:00\n",
       "3    21:00:00\n",
       "4    22:00:00\n",
       "Name: Time, dtype: object"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data['Time'] = pd.to_datetime(data.Time, format='%H.%M.%S').dt.time\n",
    "data['Time'].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0   2004-10-03 18:00:00\n",
       "1   2004-10-03 19:00:00\n",
       "2   2004-10-03 20:00:00\n",
       "3   2004-10-03 21:00:00\n",
       "4   2004-10-03 22:00:00\n",
       "Name: Date_Time, dtype: datetime64[ns]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data['Date_Time'] = pd.to_datetime(data['Date'].apply(str) + ' ' + data['Time'].apply(str))\n",
    "data['Date_Time'].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Date</th>\n",
       "      <th>Time</th>\n",
       "      <th>CO(GT)</th>\n",
       "      <th>PT08.S1(CO)</th>\n",
       "      <th>NMHC(GT)</th>\n",
       "      <th>C6H6(GT)</th>\n",
       "      <th>PT08.S2(NMHC)</th>\n",
       "      <th>NOx(GT)</th>\n",
       "      <th>PT08.S3(NOx)</th>\n",
       "      <th>NO2(GT)</th>\n",
       "      <th>PT08.S4(NO2)</th>\n",
       "      <th>PT08.S5(O3)</th>\n",
       "      <th>T</th>\n",
       "      <th>RH</th>\n",
       "      <th>AH</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Date_Time</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2004-10-03 18:00:00</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>18:00:00</td>\n",
       "      <td>2.6</td>\n",
       "      <td>1360.0</td>\n",
       "      <td>150.0</td>\n",
       "      <td>11.9</td>\n",
       "      <td>1046.0</td>\n",
       "      <td>166.0</td>\n",
       "      <td>1056.0</td>\n",
       "      <td>113.0</td>\n",
       "      <td>1692.0</td>\n",
       "      <td>1268.0</td>\n",
       "      <td>13.6</td>\n",
       "      <td>48.9</td>\n",
       "      <td>0.7578</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004-10-03 19:00:00</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>19:00:00</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1292.0</td>\n",
       "      <td>112.0</td>\n",
       "      <td>9.4</td>\n",
       "      <td>955.0</td>\n",
       "      <td>103.0</td>\n",
       "      <td>1174.0</td>\n",
       "      <td>92.0</td>\n",
       "      <td>1559.0</td>\n",
       "      <td>972.0</td>\n",
       "      <td>13.3</td>\n",
       "      <td>47.7</td>\n",
       "      <td>0.7255</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004-10-03 20:00:00</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>20:00:00</td>\n",
       "      <td>2.2</td>\n",
       "      <td>1402.0</td>\n",
       "      <td>88.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>939.0</td>\n",
       "      <td>131.0</td>\n",
       "      <td>1140.0</td>\n",
       "      <td>114.0</td>\n",
       "      <td>1555.0</td>\n",
       "      <td>1074.0</td>\n",
       "      <td>11.9</td>\n",
       "      <td>54.0</td>\n",
       "      <td>0.7502</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004-10-03 21:00:00</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>21:00:00</td>\n",
       "      <td>2.2</td>\n",
       "      <td>1376.0</td>\n",
       "      <td>80.0</td>\n",
       "      <td>9.2</td>\n",
       "      <td>948.0</td>\n",
       "      <td>172.0</td>\n",
       "      <td>1092.0</td>\n",
       "      <td>122.0</td>\n",
       "      <td>1584.0</td>\n",
       "      <td>1203.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>60.0</td>\n",
       "      <td>0.7867</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004-10-03 22:00:00</th>\n",
       "      <td>10/03/2004</td>\n",
       "      <td>22:00:00</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1272.0</td>\n",
       "      <td>51.0</td>\n",
       "      <td>6.5</td>\n",
       "      <td>836.0</td>\n",
       "      <td>131.0</td>\n",
       "      <td>1205.0</td>\n",
       "      <td>116.0</td>\n",
       "      <td>1490.0</td>\n",
       "      <td>1110.0</td>\n",
       "      <td>11.2</td>\n",
       "      <td>59.6</td>\n",
       "      <td>0.7888</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                           Date      Time  CO(GT)  PT08.S1(CO)  NMHC(GT)  \\\n",
       "Date_Time                                                                  \n",
       "2004-10-03 18:00:00  10/03/2004  18:00:00     2.6       1360.0     150.0   \n",
       "2004-10-03 19:00:00  10/03/2004  19:00:00     2.0       1292.0     112.0   \n",
       "2004-10-03 20:00:00  10/03/2004  20:00:00     2.2       1402.0      88.0   \n",
       "2004-10-03 21:00:00  10/03/2004  21:00:00     2.2       1376.0      80.0   \n",
       "2004-10-03 22:00:00  10/03/2004  22:00:00     1.6       1272.0      51.0   \n",
       "\n",
       "                     C6H6(GT)  PT08.S2(NMHC)  NOx(GT)  PT08.S3(NOx)  NO2(GT)  \\\n",
       "Date_Time                                                                      \n",
       "2004-10-03 18:00:00      11.9         1046.0    166.0        1056.0    113.0   \n",
       "2004-10-03 19:00:00       9.4          955.0    103.0        1174.0     92.0   \n",
       "2004-10-03 20:00:00       9.0          939.0    131.0        1140.0    114.0   \n",
       "2004-10-03 21:00:00       9.2          948.0    172.0        1092.0    122.0   \n",
       "2004-10-03 22:00:00       6.5          836.0    131.0        1205.0    116.0   \n",
       "\n",
       "                     PT08.S4(NO2)  PT08.S5(O3)     T    RH      AH  \n",
       "Date_Time                                                           \n",
       "2004-10-03 18:00:00        1692.0       1268.0  13.6  48.9  0.7578  \n",
       "2004-10-03 19:00:00        1559.0        972.0  13.3  47.7  0.7255  \n",
       "2004-10-03 20:00:00        1555.0       1074.0  11.9  54.0  0.7502  \n",
       "2004-10-03 21:00:00        1584.0       1203.0  11.0  60.0  0.7867  \n",
       "2004-10-03 22:00:00        1490.0       1110.0  11.2  59.6  0.7888  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = data.set_index('Date_Time')\n",
    "data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CO(GT)</th>\n",
       "      <th>PT08.S1(CO)</th>\n",
       "      <th>NMHC(GT)</th>\n",
       "      <th>C6H6(GT)</th>\n",
       "      <th>PT08.S2(NMHC)</th>\n",
       "      <th>NOx(GT)</th>\n",
       "      <th>PT08.S3(NOx)</th>\n",
       "      <th>NO2(GT)</th>\n",
       "      <th>PT08.S4(NO2)</th>\n",
       "      <th>PT08.S5(O3)</th>\n",
       "      <th>T</th>\n",
       "      <th>RH</th>\n",
       "      <th>AH</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Date_Time</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2004-10-03 18:00:00</th>\n",
       "      <td>2.6</td>\n",
       "      <td>1360.0</td>\n",
       "      <td>150.0</td>\n",
       "      <td>11.9</td>\n",
       "      <td>1046.0</td>\n",
       "      <td>166.0</td>\n",
       "      <td>1056.0</td>\n",
       "      <td>113.0</td>\n",
       "      <td>1692.0</td>\n",
       "      <td>1268.0</td>\n",
       "      <td>13.6</td>\n",
       "      <td>48.9</td>\n",
       "      <td>0.7578</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004-10-03 19:00:00</th>\n",
       "      <td>2.0</td>\n",
       "      <td>1292.0</td>\n",
       "      <td>112.0</td>\n",
       "      <td>9.4</td>\n",
       "      <td>955.0</td>\n",
       "      <td>103.0</td>\n",
       "      <td>1174.0</td>\n",
       "      <td>92.0</td>\n",
       "      <td>1559.0</td>\n",
       "      <td>972.0</td>\n",
       "      <td>13.3</td>\n",
       "      <td>47.7</td>\n",
       "      <td>0.7255</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004-10-03 20:00:00</th>\n",
       "      <td>2.2</td>\n",
       "      <td>1402.0</td>\n",
       "      <td>88.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>939.0</td>\n",
       "      <td>131.0</td>\n",
       "      <td>1140.0</td>\n",
       "      <td>114.0</td>\n",
       "      <td>1555.0</td>\n",
       "      <td>1074.0</td>\n",
       "      <td>11.9</td>\n",
       "      <td>54.0</td>\n",
       "      <td>0.7502</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004-10-03 21:00:00</th>\n",
       "      <td>2.2</td>\n",
       "      <td>1376.0</td>\n",
       "      <td>80.0</td>\n",
       "      <td>9.2</td>\n",
       "      <td>948.0</td>\n",
       "      <td>172.0</td>\n",
       "      <td>1092.0</td>\n",
       "      <td>122.0</td>\n",
       "      <td>1584.0</td>\n",
       "      <td>1203.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>60.0</td>\n",
       "      <td>0.7867</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2004-10-03 22:00:00</th>\n",
       "      <td>1.6</td>\n",
       "      <td>1272.0</td>\n",
       "      <td>51.0</td>\n",
       "      <td>6.5</td>\n",
       "      <td>836.0</td>\n",
       "      <td>131.0</td>\n",
       "      <td>1205.0</td>\n",
       "      <td>116.0</td>\n",
       "      <td>1490.0</td>\n",
       "      <td>1110.0</td>\n",
       "      <td>11.2</td>\n",
       "      <td>59.6</td>\n",
       "      <td>0.7888</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                     CO(GT)  PT08.S1(CO)  NMHC(GT)  C6H6(GT)  PT08.S2(NMHC)  \\\n",
       "Date_Time                                                                     \n",
       "2004-10-03 18:00:00     2.6       1360.0     150.0      11.9         1046.0   \n",
       "2004-10-03 19:00:00     2.0       1292.0     112.0       9.4          955.0   \n",
       "2004-10-03 20:00:00     2.2       1402.0      88.0       9.0          939.0   \n",
       "2004-10-03 21:00:00     2.2       1376.0      80.0       9.2          948.0   \n",
       "2004-10-03 22:00:00     1.6       1272.0      51.0       6.5          836.0   \n",
       "\n",
       "                     NOx(GT)  PT08.S3(NOx)  NO2(GT)  PT08.S4(NO2)  \\\n",
       "Date_Time                                                           \n",
       "2004-10-03 18:00:00    166.0        1056.0    113.0        1692.0   \n",
       "2004-10-03 19:00:00    103.0        1174.0     92.0        1559.0   \n",
       "2004-10-03 20:00:00    131.0        1140.0    114.0        1555.0   \n",
       "2004-10-03 21:00:00    172.0        1092.0    122.0        1584.0   \n",
       "2004-10-03 22:00:00    131.0        1205.0    116.0        1490.0   \n",
       "\n",
       "                     PT08.S5(O3)     T    RH      AH  \n",
       "Date_Time                                             \n",
       "2004-10-03 18:00:00       1268.0  13.6  48.9  0.7578  \n",
       "2004-10-03 19:00:00        972.0  13.3  47.7  0.7255  \n",
       "2004-10-03 20:00:00       1074.0  11.9  54.0  0.7502  \n",
       "2004-10-03 21:00:00       1203.0  11.0  60.0  0.7867  \n",
       "2004-10-03 22:00:00       1110.0  11.2  59.6  0.7888  "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = data.drop(['Date', 'Time'], axis=1)\n",
    "data.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Imputation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the missing values in the data are replaced with a value -200, we will have to impute the missing value with a better number. Consider this - if the present dew point value is missing, we can safely assume that it will be close to the value of the previous hour. Makes sense, right? Here, I will impute -200 with the previous value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "cols = data.columns\n",
    "\n",
    "# iterate through all columns\n",
    "for j in cols:\n",
    "    # iterate through all rows\n",
    "    for i in range(0, len(data)):\n",
    "        if data[j][i] == -200:\n",
    "            data[j][i] = data[j][i-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stationarity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A stationary time series will more often than not give us a better set of predictions. Similar to the <b>Augmented Dickey-Fuller test</b> for univariate series, we have <b>Johansen’s test</b> for checking the stationarity of any multivariate time series data. Since the test works for only 12 variables, I have randomly dropped in the next iteration, I would drop another and check the eigenvalues."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### coint_johansen()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`statsmodels.tsa.vector_ar.vecm.coint_johansen(endog, det_order, k_ar_diff)` - perform the Johansen cointegration test for determining the cointegration rank of a VECM. \n",
    "* <b>endog</b> - the data with presample.\n",
    "* <b>det_order: int</b> -\n",
    "  * <b>-1</b> - no deterministic terms.\n",
    "  * <b>0</b> - constant term.\n",
    "  * <b>1</b> - linear trend.\n",
    "* <b>k_ar_diff: int, nonnegative</b> - number of lagged differences in the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "#from statsmodels.tsa.vector_ar.vecm import coint_johansen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "#johan_test_temp = data.drop(['CO(GT)'], axis=1)\n",
    "#coint_johansen(johan_test_temp, -1, 1).eig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Train-Validation Split"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Creating a validation set for time series problems is tricky because we have to take into account the time component. One cannot directly use the `train_test_split` or k-fold validation since this will disrupt the pattern in the series. The validation set should be created considering the date and time values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "train = data[:int(0.8*(len(data)))]\n",
    "valid = data[int(0.8*(len(data))):]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector Autoregression (VAR)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a <b>VAR</b> model, each variable is a linear function of the past values of itself and the past values of all the other variables. Unlike AR, VAR is able to understand and use the relationship between several variables. This is useful for describing the dynamic behavior of the data and also provides better forecasting results. Additionally, implementing VAR is as simple as using any other univariate technique."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/mosthusnejahan/anaconda3/lib/python3.6/site-packages/statsmodels/compat/pandas.py:56: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.\n",
      "  from pandas.core import datetools\n"
     ]
    }
   ],
   "source": [
    "from statsmodels.tsa.vector_ar.var_model import VAR"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### VAR()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`statsmodels.tsa.vector_ar.var_model.VAR(endog, exog=None, dates=None, freq=None, missing='none')` - fit VAR(p) process and do lag order selection.\n",
    "\\begin{equation*} \n",
    "y_t = A_1y_{t-1} + ... + A_py_{t-p} + u_t\n",
    "\\end{equation*}\n",
    "* <b>endog</b> - 2-d endogenous response variable. The independent variable.\n",
    "* <b>exog</b> - 2-d exogenous variable.\n",
    "* <b>dates</b> - must match number of rows of endog"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/mosthusnejahan/anaconda3/lib/python3.6/site-packages/statsmodels/tsa/vector_ar/var_model.py:461: 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",
      "  params = np.linalg.lstsq(z, y_sample)[0]\n"
     ]
    }
   ],
   "source": [
    "# Fit the model\n",
    "model = VAR(endog=train)\n",
    "model_fit = model.fit()\n",
    "\n",
    "# Make prediction on validation\n",
    "prediction = model_fit.forecast(model_fit.y, steps=len(valid))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Convert array to DataFrame\n",
    "cols = data.columns\n",
    "pred = pd.DataFrame(index=range(0, len(prediction)), columns=[cols])\n",
    "\n",
    "for j in range(0, len(cols)):\n",
    "    for i in range(0, len(prediction)):\n",
    "        pred.iloc[i][j] = prediction[i][j]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RMSE value for CO(GT) is: 1.408688883687256\n",
      "RMSE value for PT08.S1(CO) is: 205.89558284021305\n",
      "RMSE value for NMHC(GT) is: 6.67354871134285\n",
      "RMSE value for C6H6(GT) is: 7.130087248705714\n",
      "RMSE value for PT08.S2(NMHC) is: 277.84844376800464\n",
      "RMSE value for NOx(GT) is: 214.7832234091224\n",
      "RMSE value for PT08.S3(NOx) is: 244.9576966193527\n",
      "RMSE value for NO2(GT) is: 66.69695211710588\n",
      "RMSE value for PT08.S4(NO2) is: 490.08388934110286\n",
      "RMSE value for PT08.S5(O3) is: 446.51541648881795\n",
      "RMSE value for T is: 10.72132579560568\n",
      "RMSE value for RH is: 17.111676248172948\n",
      "RMSE value for AH is: 0.5216247245187544\n"
     ]
    }
   ],
   "source": [
    "# RMSE\n",
    "for i in cols:\n",
    "    print('RMSE value for', i, 'is:', sqrt(mean_squared_error(pred[i], valid[i])))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2.34596328e+00 1.08633212e+03 2.80762173e+02 1.24130779e+01\n",
      "  1.05535947e+03 2.80882233e+02 6.59534851e+02 1.68444418e+02\n",
      "  1.15918056e+03 8.50845529e+02 2.73639014e+01 1.55311062e+01\n",
      "  5.15317053e-01]]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/mosthusnejahan/anaconda3/lib/python3.6/site-packages/statsmodels/tsa/vector_ar/var_model.py:461: 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",
      "  params = np.linalg.lstsq(z, y_sample)[0]\n"
     ]
    }
   ],
   "source": [
    "# Make final predictions\n",
    "model = VAR(endog=data)\n",
    "model_fit = model.fit()\n",
    "\n",
    "yhat = model_fit.forecast(model_fit.y, steps=1)\n",
    "print(yhat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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