ebay_auction.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd \n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "from matplotlib               import pyplot as plt\n",
    "from sklearn.decomposition    import PCA\n",
    "from mlxtend.plotting         import plot_decision_regions\n",
    "from tabulate                 import tabulate\n",
    "\n",
    "from sklearn.preprocessing    import StandardScaler\n",
    "from sklearn.metrics.pairwise import rbf_kernel\n",
    "from sklearn.model_selection  import train_test_split\n",
    "from sklearn                  import linear_model\n",
    "from sklearn                  import metrics\n",
    "from sklearn.metrics          import classification_report # Produces nice report on classification\n",
    "from sklearn.metrics          import confusion_matrix      # Returns the confusion matrix\n",
    "\n",
    "from sklearn                  import svm\n",
    "from sklearn.linear_model     import LinearRegression      # Linear Regression Class\n",
    "from sklearn.linear_model     import LogisticRegression    # Logistic Regression Class\n",
    "from sklearn.svm              import SVC                   # SVM Regression Class\n",
    "from sklearn.svm              import SVR                   # SVM Regression Class\n",
    "from sklearn.ensemble         import RandomForestRegressor\n",
    "from sklearn.ensemble         import GradientBoostingRegressor\n",
    "from sklearn.model_selection  import GridSearchCV\n",
    "\n",
    "from sklearn.neighbors        import KNeighborsClassifier  # k-Nearest Neighbors Classification Class\n",
    "from sklearn.neural_network   import MLPClassifier\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Περιγραφή και εκκαθάριση δεδομένων\n",
    "<br>Η εργασία βασίζεται σε δεδομένα δημοπρασιών της μορφής που διεξάγονται στο ebay. Οι μεταβλητές που περιλαμβάνονται στο αρχείο παρουσιάζονται παρακάτω. <br> \n",
    "Αρχικά, πραγματοποιείται έλεγχος των δεδομένων κατά τον οποίο δεν εμφανίζονται κενές τιμές στο dataset. Παρ΄ όλα αυτά παρουσιάζονται κενά στους τίτλους των στηλών τα οποία αφαιρούνται."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "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>SellerId</th>\n",
       "      <th>SellerRating</th>\n",
       "      <th>ItemId</th>\n",
       "      <th>StartingBid</th>\n",
       "      <th>Price</th>\n",
       "      <th>NumBidders</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>144.0</td>\n",
       "      <td>84.11</td>\n",
       "      <td>6.0</td>\n",
       "      <td>137.0</td>\n",
       "      <td>145.5</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>40.0</td>\n",
       "      <td>93.76</td>\n",
       "      <td>0.0</td>\n",
       "      <td>30.0</td>\n",
       "      <td>30.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>102.0</td>\n",
       "      <td>94.52</td>\n",
       "      <td>12.0</td>\n",
       "      <td>364.0</td>\n",
       "      <td>1278.0</td>\n",
       "      <td>8.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>77.0</td>\n",
       "      <td>97.41</td>\n",
       "      <td>2.0</td>\n",
       "      <td>82.0</td>\n",
       "      <td>106.5</td>\n",
       "      <td>3.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>54.0</td>\n",
       "      <td>80.32</td>\n",
       "      <td>4.0</td>\n",
       "      <td>173.0</td>\n",
       "      <td>173.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18837</th>\n",
       "      <td>40.0</td>\n",
       "      <td>93.76</td>\n",
       "      <td>2.0</td>\n",
       "      <td>72.0</td>\n",
       "      <td>92.0</td>\n",
       "      <td>3.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18838</th>\n",
       "      <td>112.0</td>\n",
       "      <td>84.30</td>\n",
       "      <td>3.0</td>\n",
       "      <td>100.0</td>\n",
       "      <td>139.5</td>\n",
       "      <td>3.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18839</th>\n",
       "      <td>116.0</td>\n",
       "      <td>99.26</td>\n",
       "      <td>8.0</td>\n",
       "      <td>243.0</td>\n",
       "      <td>498.0</td>\n",
       "      <td>5.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18840</th>\n",
       "      <td>10.0</td>\n",
       "      <td>87.26</td>\n",
       "      <td>4.0</td>\n",
       "      <td>167.0</td>\n",
       "      <td>169.5</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18841</th>\n",
       "      <td>133.0</td>\n",
       "      <td>93.54</td>\n",
       "      <td>5.0</td>\n",
       "      <td>55.0</td>\n",
       "      <td>307.0</td>\n",
       "      <td>7.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>18842 rows × 6 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "       SellerId  SellerRating  ItemId  StartingBid   Price  NumBidders\n",
       "0         144.0         84.11     6.0        137.0   145.5         2.0\n",
       "1          40.0         93.76     0.0         30.0    30.0         0.0\n",
       "2         102.0         94.52    12.0        364.0  1278.0         8.0\n",
       "3          77.0         97.41     2.0         82.0   106.5         3.0\n",
       "4          54.0         80.32     4.0        173.0   173.0         0.0\n",
       "...         ...           ...     ...          ...     ...         ...\n",
       "18837      40.0         93.76     2.0         72.0    92.0         3.0\n",
       "18838     112.0         84.30     3.0        100.0   139.5         3.0\n",
       "18839     116.0         99.26     8.0        243.0   498.0         5.0\n",
       "18840      10.0         87.26     4.0        167.0   169.5         1.0\n",
       "18841     133.0         93.54     5.0         55.0   307.0         7.0\n",
       "\n",
       "[18842 rows x 6 columns]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv(\"auction_data.csv\")\n",
    "df.columns = df.columns.str.replace(' ', '')\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['SellerId', 'SellerRating', 'ItemId', 'StartingBid', 'Price', 'NumBidders']"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(df.columns)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "SellerId        False\n",
       "SellerRating    False\n",
       "ItemId          False\n",
       "StartingBid     False\n",
       "Price           False\n",
       "NumBidders      False\n",
       "dtype: bool"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.isnull().any()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Από τις μεταβλητές του dataset αυτές του <b>SellerID</b> καθώς και <b>ItemID</b> είναι αδιάφορες ως προς την πρόβλεψη των τιμών κλεισίματος, του κέρδους αλλά και της ανάδειξη της δημοπρασίας ως ολοκληρωμένη ( πραγματοποίηση πώλησης) και ως εκ τούτου αφαιρούνται.<br><br>\n",
    "Τέλος, προστίθεται μια νέα στήλη η οποία αναφέρει εάν η δημοπρασία έχει καταλήξει σε πώληση ή όχι. Η νέα στήλη δημιουργείται από τον έλεγχο της Τιμής Εκκίνησης και της Τιμής Κλεισίματος\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SellerRating</th>\n",
       "      <th>StartingBid</th>\n",
       "      <th>Price</th>\n",
       "      <th>NumBidders</th>\n",
       "      <th>Sold</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>84.11</td>\n",
       "      <td>137.0</td>\n",
       "      <td>145.5</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>93.76</td>\n",
       "      <td>30.0</td>\n",
       "      <td>30.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>94.52</td>\n",
       "      <td>364.0</td>\n",
       "      <td>1278.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>97.41</td>\n",
       "      <td>82.0</td>\n",
       "      <td>106.5</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>80.32</td>\n",
       "      <td>173.0</td>\n",
       "      <td>173.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>99.97</td>\n",
       "      <td>836.0</td>\n",
       "      <td>1146.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>99.01</td>\n",
       "      <td>207.0</td>\n",
       "      <td>599.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>99.97</td>\n",
       "      <td>348.0</td>\n",
       "      <td>529.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>99.26</td>\n",
       "      <td>18.0</td>\n",
       "      <td>100.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>84.11</td>\n",
       "      <td>8.0</td>\n",
       "      <td>24.5</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   SellerRating  StartingBid   Price  NumBidders Sold\n",
       "0         84.11        137.0   145.5         2.0    1\n",
       "1         93.76         30.0    30.0         0.0    0\n",
       "2         94.52        364.0  1278.0         8.0    1\n",
       "3         97.41         82.0   106.5         3.0    1\n",
       "4         80.32        173.0   173.0         0.0    0\n",
       "5         99.97        836.0  1146.0         4.0    1\n",
       "6         99.01        207.0   599.0         4.0    1\n",
       "7         99.97        348.0   529.0         3.0    1\n",
       "8         99.26         18.0   100.0         7.0    1\n",
       "9         84.11          8.0    24.5         4.0    1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#add 'Sold' column where 0=not sold (aka StartingBid has the same value as Price) and 1=sold\n",
    "df['Sold'] = np.where(df['StartingBid']==df['Price'], '0', '1')\n",
    "\n",
    "#remove irrelevant columns \n",
    "cols_to_drop = [\"SellerId\",\"ItemId\"]\n",
    "df = df.drop(columns=cols_to_drop,axis=1)\n",
    "df.head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 18842 entries, 0 to 18841\n",
      "Data columns (total 5 columns):\n",
      " #   Column        Non-Null Count  Dtype  \n",
      "---  ------        --------------  -----  \n",
      " 0   SellerRating  18842 non-null  float64\n",
      " 1   StartingBid   18842 non-null  float64\n",
      " 2   Price         18842 non-null  float64\n",
      " 3   NumBidders    18842 non-null  float64\n",
      " 4   Sold          18842 non-null  object \n",
      "dtypes: float64(4), object(1)\n",
      "memory usage: 736.1+ KB\n"
     ]
    }
   ],
   "source": [
    "df.info()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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>SellerRating</th>\n",
       "      <th>StartingBid</th>\n",
       "      <th>Price</th>\n",
       "      <th>NumBidders</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>18842.000000</td>\n",
       "      <td>18842.000000</td>\n",
       "      <td>18842.000000</td>\n",
       "      <td>18842.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>90.046164</td>\n",
       "      <td>316.835899</td>\n",
       "      <td>393.979753</td>\n",
       "      <td>2.242172</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>6.009262</td>\n",
       "      <td>350.345270</td>\n",
       "      <td>384.044470</td>\n",
       "      <td>2.016307</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>80.050000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.250000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>84.820000</td>\n",
       "      <td>80.000000</td>\n",
       "      <td>135.500000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>90.460000</td>\n",
       "      <td>181.000000</td>\n",
       "      <td>235.500000</td>\n",
       "      <td>2.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>94.990000</td>\n",
       "      <td>426.000000</td>\n",
       "      <td>568.000000</td>\n",
       "      <td>4.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>99.980000</td>\n",
       "      <td>2299.000000</td>\n",
       "      <td>2299.000000</td>\n",
       "      <td>10.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       SellerRating   StartingBid         Price    NumBidders\n",
       "count  18842.000000  18842.000000  18842.000000  18842.000000\n",
       "mean      90.046164    316.835899    393.979753      2.242172\n",
       "std        6.009262    350.345270    384.044470      2.016307\n",
       "min       80.050000      1.000000      1.250000      0.000000\n",
       "25%       84.820000     80.000000    135.500000      0.000000\n",
       "50%       90.460000    181.000000    235.500000      2.000000\n",
       "75%       94.990000    426.000000    568.000000      4.000000\n",
       "max       99.980000   2299.000000   2299.000000     10.000000"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Παρακάτω παρουσιάζεται η συσχέτιση μεταξύ των μεταβλητών. Όπως παρατηρείται οι τιμές StartingBid και Price έχουν αρκετή συσχέτιση όπως ήταν αναμενόμενο."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = plt.figure(figsize = (10,7))\n",
    "plt.matshow(df.corr(), fignum = f.number)\n",
    "plt.xticks(range(df.shape[1]), df.columns, fontsize=10, rotation=0)\n",
    "plt.yticks(range(df.shape[1]), df.columns, fontsize=10)\n",
    "cb = plt.colorbar()\n",
    "cb.ax.tick_params(labelsize=10)\n",
    "plt.title('Correlation Matrix', fontsize=16);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(10,5))\n",
    "ax.scatter(df['Price'], df['NumBidders'])\n",
    "ax.set_xlabel('Closing Price')\n",
    "ax.set_ylabel('Value of Starting Bid')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Στο παραπάνω διάγραμμα παρατηρούνται κάποιοι outliers οι οποίοι απομονώνονται με τη χρήση IQR.\n",
    "Παρ' όλα αυτά οι τιμές αυτές θεωρούνται χρήσιμες οπότε δεν αφαιρούνται από το dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SellerRating</th>\n",
       "      <th>StartingBid</th>\n",
       "      <th>Price</th>\n",
       "      <th>NumBidders</th>\n",
       "      <th>Sold</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>84.11</td>\n",
       "      <td>137.0</td>\n",
       "      <td>145.5</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>93.76</td>\n",
       "      <td>30.0</td>\n",
       "      <td>30.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>97.41</td>\n",
       "      <td>82.0</td>\n",
       "      <td>106.5</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>80.32</td>\n",
       "      <td>173.0</td>\n",
       "      <td>173.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>99.97</td>\n",
       "      <td>836.0</td>\n",
       "      <td>1146.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18837</th>\n",
       "      <td>93.76</td>\n",
       "      <td>72.0</td>\n",
       "      <td>92.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18838</th>\n",
       "      <td>84.30</td>\n",
       "      <td>100.0</td>\n",
       "      <td>139.5</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18839</th>\n",
       "      <td>99.26</td>\n",
       "      <td>243.0</td>\n",
       "      <td>498.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18840</th>\n",
       "      <td>87.26</td>\n",
       "      <td>167.0</td>\n",
       "      <td>169.5</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18841</th>\n",
       "      <td>93.54</td>\n",
       "      <td>55.0</td>\n",
       "      <td>307.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>17808 rows × 5 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "       SellerRating  StartingBid   Price  NumBidders Sold\n",
       "0             84.11        137.0   145.5         2.0    1\n",
       "1             93.76         30.0    30.0         0.0    0\n",
       "3             97.41         82.0   106.5         3.0    1\n",
       "4             80.32        173.0   173.0         0.0    0\n",
       "5             99.97        836.0  1146.0         4.0    1\n",
       "...             ...          ...     ...         ...  ...\n",
       "18837         93.76         72.0    92.0         3.0    1\n",
       "18838         84.30        100.0   139.5         3.0    1\n",
       "18839         99.26        243.0   498.0         5.0    1\n",
       "18840         87.26        167.0   169.5         1.0    1\n",
       "18841         93.54         55.0   307.0         7.0    1\n",
       "\n",
       "[17808 rows x 5 columns]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# outlier detection and removal\n",
    "Q1 = df['Price'].quantile(0.25)\n",
    "Q3 = df['Price'].quantile(0.75)\n",
    "IQR = Q3 - Q1 \n",
    "filter = (df['Price'] >= Q1 - 1.5 * IQR) & (df['Price'] <= Q3 + 1.5 *IQR)\n",
    "df2 = df.loc[filter]\n",
    "df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#visualize the dataset without the outliers\n",
    "fig, ax = plt.subplots(figsize=(10,5))\n",
    "ax.scatter(df2['Price'], df2['NumBidders'])\n",
    "ax.set_xlabel('Closing Price')\n",
    "ax.set_ylabel('Bidders Number ')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Πρόβλεψη Τιμής Κλεισίματος<br>\n",
    "\n",
    "Για την πρόβλεψη της τιμής κλεισίματος γίνεται χρήση του μοντέλου Multiple Regression και Random Forest.<br> \n",
    "Αρχικά το dataset χωρίζεται σε 70% train και 30% test με εξαρτημένη μεταβλητή το Price.<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### 1.1 Multiple Regression<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SellerRating</th>\n",
       "      <th>StartingBid</th>\n",
       "      <th>NumBidders</th>\n",
       "      <th>Sold</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>10071</th>\n",
       "      <td>87.26</td>\n",
       "      <td>159.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13354</th>\n",
       "      <td>86.84</td>\n",
       "      <td>715.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11650</th>\n",
       "      <td>86.75</td>\n",
       "      <td>70.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1383</th>\n",
       "      <td>99.26</td>\n",
       "      <td>53.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4771</th>\n",
       "      <td>80.31</td>\n",
       "      <td>32.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6181</th>\n",
       "      <td>90.57</td>\n",
       "      <td>24.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4044</th>\n",
       "      <td>91.90</td>\n",
       "      <td>277.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10529</th>\n",
       "      <td>80.18</td>\n",
       "      <td>17.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14107</th>\n",
       "      <td>94.26</td>\n",
       "      <td>14.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18129</th>\n",
       "      <td>97.09</td>\n",
       "      <td>25.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5653 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "       SellerRating  StartingBid  NumBidders Sold\n",
       "10071         87.26        159.0         1.0    1\n",
       "13354         86.84        715.0         4.0    1\n",
       "11650         86.75         70.0         0.0    0\n",
       "1383          99.26         53.0         5.0    1\n",
       "4771          80.31         32.0         4.0    1\n",
       "...             ...          ...         ...  ...\n",
       "6181          90.57         24.0         2.0    1\n",
       "4044          91.90        277.0         0.0    0\n",
       "10529         80.18         17.0         3.0    1\n",
       "14107         94.26         14.0         4.0    1\n",
       "18129         97.09         25.0         1.0    1\n",
       "\n",
       "[5653 rows x 4 columns]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#define x and y variables\n",
    "#split the dataset in 70% train and 30% test\n",
    "\n",
    "y = df.Price\n",
    "X = df[['SellerRating', 'StartingBid', 'NumBidders', 'Sold']]\n",
    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state = 42) \n",
    "X_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(13189, 4) (13189,)\n",
      "(5653, 4) (5653,)\n"
     ]
    }
   ],
   "source": [
    "print(X_train.shape, y_train.shape)\n",
    "print(X_test.shape, y_test.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# sc_x = StandardScaler()\n",
    "# X_train = sc_x.fit_transform(X_train)\n",
    "# X_test = sc_x.fit_transform(X_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Παρατηρήθηκε πως μετά την κανονικοποίηση των δεδομένων παρ' όλο που το accuracy δεν παρουσιάζει μεγάλη διαφορά οι τιμές των Coefficient μεταβάλλονται αρκετά. <br> Συγκεκριμένα παράγονται τα παρακάτω αποτελέσματα:\n",
    "\n",
    "**Coefficient**<br>\n",
    "SellerRating:\t-0.309745<br>\n",
    "StartingBid:\t381.599133<br>\n",
    "NumBidders:\t87.062954<br>\n",
    "Sold:\t-4.325168<br>\n",
    "\n",
    "Στον παραπάνω πίνακα η μεταβλητή Price διαθέτει την μεγαλύτερη εξάρτηση με την μεταβλητή StartingBid και στη συνέχεια με την NumBidders, κάτι το οποίο δεν ισχύει. Συνεπώς δεν θα γίνει κανονικοποίηση στα δεδομένα.<br>\n",
    "Σε αυτή την περίπτωση όπως φαίνεται παρακάτω τον μεγαλύτερο συντελεστή coeffiency διαθέτει το NumBidders "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "linear = LinearRegression() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Coefficients: [-0.05142089  1.08143884 43.26921197 -9.43944952]\n",
      "Intercept: -33.46612452230835\n"
     ]
    }
   ],
   "source": [
    "linear.fit(X_train, y_train) \n",
    "print('Coefficients:', linear.coef_)\n",
    "print('Intercept:', linear.intercept_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Coefficient</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>SellerRating</th>\n",
       "      <td>-0.051421</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>StartingBid</th>\n",
       "      <td>1.081439</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>NumBidders</th>\n",
       "      <td>43.269212</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Sold</th>\n",
       "      <td>-9.439450</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              Coefficient\n",
       "SellerRating    -0.051421\n",
       "StartingBid      1.081439\n",
       "NumBidders      43.269212\n",
       "Sold            -9.439450"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coeff_df = pd.DataFrame(linear.coef_, X.columns, columns=['Coefficient'])  \n",
    "coeff_df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x125ec382c48>]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(coeff_df, linestyle = 'none', marker = '*', markersize = 7, color = 'red', zorder = 7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Παρακάτω απεικονίζονται συγκριτικοι πίνακες και διαγράμματα των πραγματικών και την προβλεπόμενων τιμών."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Actual</th>\n",
       "      <th>Predicted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>10071</th>\n",
       "      <td>161.5</td>\n",
       "      <td>167.825427</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13354</th>\n",
       "      <td>998.0</td>\n",
       "      <td>898.934654</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11650</th>\n",
       "      <td>70.0</td>\n",
       "      <td>37.773832</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1383</th>\n",
       "      <td>173.5</td>\n",
       "      <td>225.652707</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4771</th>\n",
       "      <td>52.0</td>\n",
       "      <td>160.647705</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6181</th>\n",
       "      <td>26.0</td>\n",
       "      <td>64.930192</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4044</th>\n",
       "      <td>277.0</td>\n",
       "      <td>261.366854</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10529</th>\n",
       "      <td>24.5</td>\n",
       "      <td>101.163595</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14107</th>\n",
       "      <td>23.5</td>\n",
       "      <td>140.464484</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18129</th>\n",
       "      <td>26.0</td>\n",
       "      <td>22.407155</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5653 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "       Actual   Predicted\n",
       "10071   161.5  167.825427\n",
       "13354   998.0  898.934654\n",
       "11650    70.0   37.773832\n",
       "1383    173.5  225.652707\n",
       "4771     52.0  160.647705\n",
       "...       ...         ...\n",
       "6181     26.0   64.930192\n",
       "4044    277.0  261.366854\n",
       "10529    24.5  101.163595\n",
       "14107    23.5  140.464484\n",
       "18129    26.0   22.407155\n",
       "\n",
       "[5653 rows x 2 columns]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_pred = linear.predict(X_test)\n",
    "df_new = pd.DataFrame({'Actual': y_test, 'Predicted': y_pred})\n",
    "df_new"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#visualize the difference between actual price and predicted price\n",
    "AvsP = df_new.head(30)\n",
    "AvsP.plot(kind ='bar',figsize = (10,6))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Accuracy: 0.895533787589926\n"
     ]
    }
   ],
   "source": [
    "print('Accuracy:',linear.score(X_test,y_test))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean Absolute Error: 68.87593182097979\n",
      "Mean Squared Error: 14804.330600460356\n",
      "Root Mean Squared Error: 121.67304796239945\n"
     ]
    }
   ],
   "source": [
    "print('Mean Absolute Error:', metrics.mean_absolute_error(y_test, y_pred))  \n",
    "print('Mean Squared Error:', metrics.mean_squared_error(y_test, y_pred))  \n",
    "print('Root Mean Squared Error:', np.sqrt(metrics.mean_squared_error(y_test, y_pred)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.2 Random Forest\n",
    "Στη συνέχεια γίνεται εκπαίδευση του dataset χρησιμοποιώντας Random Forest Regressor.<br>\n",
    "Για την εύρεση του optimal n_estimaros γίνεται χρήση της παραμέτρου GridSearchCV για 10, 100, 1000 δέντρα. Βάση των αποτελεσμάτων η εκπαίδευση γίνεται με n_estimators = 1000 και καταλήγει σε accuracy 89,6%."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "y= df.Price\n",
    "X= df[['SellerRating', 'StartingBid', 'NumBidders', 'Sold']]\n",
    "XR_train, XR_test, yR_train, yR_test = train_test_split(X, y, test_size=0.3, random_state = 42) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'bootstrap': True, 'ccp_alpha': 0.0, 'criterion': 'mse', 'max_depth': None, 'max_features': 'auto', 'max_leaf_nodes': None, 'max_samples': None, 'min_impurity_decrease': 0.0, 'min_impurity_split': None, 'min_samples_leaf': 1, 'min_samples_split': 6, 'min_weight_fraction_leaf': 0.0, 'n_estimators': 1000, 'n_jobs': None, 'oob_score': False, 'random_state': 1, 'verbose': 0, 'warm_start': False}\n"
     ]
    }
   ],
   "source": [
    "model_params = {\n",
    "    'n_estimators': [10, 100, 1000],\n",
    "    'max_features': ['auto'],\n",
    "    'min_samples_split': [2, 4, 6]\n",
    "}\n",
    "rf_model = RandomForestRegressor(random_state = 1)\n",
    "clf = GridSearchCV(rf_model, model_params, cv = 5)\n",
    "model = clf.fit(XR_train, yR_train)\n",
    "print(model.best_estimator_.get_params())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "rmr = RandomForestRegressor(n_estimators = 1000, random_state = 42)\n",
    "rmr.fit(XR_train, yR_train)\n",
    "rmr_pred = rmr.predict(XR_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Παρακάτω παρουσιάζεται η σύγκριση των πραγματικών με τις προβλεπόμενες τιμές καθώς και τα Accuracy, Mean Absolute Error, Mean Squared Error και Root Mean Squared Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Actual</th>\n",
       "      <th>Predicted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>10071</th>\n",
       "      <td>161.5</td>\n",
       "      <td>161.498000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13354</th>\n",
       "      <td>998.0</td>\n",
       "      <td>1052.095000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11650</th>\n",
       "      <td>70.0</td>\n",
       "      <td>70.002000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1383</th>\n",
       "      <td>173.5</td>\n",
       "      <td>195.218492</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4771</th>\n",
       "      <td>52.0</td>\n",
       "      <td>735.736000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6181</th>\n",
       "      <td>26.0</td>\n",
       "      <td>27.046289</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4044</th>\n",
       "      <td>277.0</td>\n",
       "      <td>276.967000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10529</th>\n",
       "      <td>24.5</td>\n",
       "      <td>36.187227</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14107</th>\n",
       "      <td>23.5</td>\n",
       "      <td>49.416750</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18129</th>\n",
       "      <td>26.0</td>\n",
       "      <td>25.996500</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5653 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "       Actual    Predicted\n",
       "10071   161.5   161.498000\n",
       "13354   998.0  1052.095000\n",
       "11650    70.0    70.002000\n",
       "1383    173.5   195.218492\n",
       "4771     52.0   735.736000\n",
       "...       ...          ...\n",
       "6181     26.0    27.046289\n",
       "4044    277.0   276.967000\n",
       "10529    24.5    36.187227\n",
       "14107    23.5    49.416750\n",
       "18129    26.0    25.996500\n",
       "\n",
       "[5653 rows x 2 columns]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_new = pd.DataFrame({'Actual': yR_test, 'Predicted': rmr_pred})\n",
    "df_new"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Accuracy: 0.8962817556066124\n",
      "Mean Absolute Error: 54.88854891394231\n",
      "Mean Squared Error: 14698.333019595368\n",
      "Root Mean Squared Error: 121.236681823594\n"
     ]
    }
   ],
   "source": [
    "print('Accuracy:', rmr.score(XR_test, yR_test))\n",
    "print('Mean Absolute Error:', metrics.mean_absolute_error(yR_test, rmr_pred))  \n",
    "print('Mean Squared Error:', metrics.mean_squared_error(yR_test, rmr_pred))  \n",
    "print('Root Mean Squared Error:', np.sqrt(metrics.mean_squared_error(yR_test, rmr_pred)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.00000000e-03, -5.40950000e+01, -2.00000000e-03, ...,\n",
       "       -1.16872274e+01, -2.59167500e+01,  3.50000000e-03])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "residual = yR_test.values-rmr_pred\n",
    "residual"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Τα accuracy των 2 μεθόδων παρουσιάζονται παρακάτω. Όπως φαίνεται οι 2 μέθοδοι έχουν σχεδόν την ίδια αποτελεσματικότητα ως προς την πρόβλεψη της τιμής κλεισίματος"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "random = rmr.score(XR_test, yR_test)\n",
    "random = (f\"{random:.2%}\")\n",
    "lin = linear.score(X_test,y_test)\n",
    "lin = (f\"{lin:.2%}\")\n",
    "print(tabulate([['Multiple Regression', lin], ['Random Forest', random]], headers=['Prediction Method', 'Accuracy']))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3 Διάστημα τιμής κλεισίματος \n",
    "Η πρόβλεψη του διαστήματος της τιμής κλεισίματος γίνεται με χρήση του Gradient Boosting Regressor.\n",
    "Στον παρακάτω πίνακα παρουσιάζεται το προβλεπόμενο διάστημα (Lower, Upper) για την τιμή κλεισίματος."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "LOWER_ALPHA = 0.1\n",
    "UPPER_ALPHA = 0.9\n",
    "low = GradientBoostingRegressor(loss=\"quantile\",alpha=LOWER_ALPHA)\n",
    "mid = GradientBoostingRegressor(loss=\"ls\")\n",
    "upper = GradientBoostingRegressor(loss=\"quantile\",alpha=UPPER_ALPHA)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low.fit(X_train, y_train)\n",
    "mid.fit(X_train, y_train)\n",
    "upper.fit(X_train, y_train)\n",
    "predictions = pd.DataFrame(y_test)\n",
    "predictions['Lower'] = low.predict(X_test)\n",
    "predictions['Mid'] = mid.predict(X_test)\n",
    "predictions['Upper'] = upper.predict(X_test)\n",
    "predictions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.Πρόβλεψη του αν η δημοπρασία θα οδηγήσει σε πώληση ή οχι\n",
    "\n",
    "Για την πρόβλεψη της πώλησης χρησιμοποιούνται oι αλγόριθμοι Logistic Regression και kNN.\n",
    "\n",
    "### 2.1 Logistic Regression\n",
    "Γίνεται και πάλι διαχωρισμός των δεδομένων σε train και test data με την μεταβλητή Sold ως εξαρτημένη.\n",
    "Αρχικά γίνεται κανονικοποίηση των ανεξάρτητων μεταβλητών του train και test dataset και στη συνέχεια γίνεται η εκπαίδευση και η πρόβλεψη στις test τιμές."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "yS = df.Sold\n",
    "XS = df[['SellerRating', 'StartingBid', 'NumBidders', 'Price']]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Xtrain, Xtest, ytrain, ytest = train_test_split(XS, yS, test_size=0.3, random_state = 42) # 70% training and 30% test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ssc = StandardScaler()                                                  # Initialize a StandardScaler object\n",
    "Xtrain = ssc.fit_transform(Xtrain)                                      # Fit and transform the training set\n",
    "Xtrain = ssc.transform(Xtrain)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "logistic = LogisticRegression()\n",
    "logistic.fit(Xtrain, ytrain)\n",
    "Lpred = logistic.predict(Xtest)\n",
    "logistic.score(Xtest, ytest)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print('coef:',logistic.coef_)\n",
    "print('intercept:',logistic.intercept_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Όπως φαίνεται παραπάνω το accuracy του μοντέλου είναι 77%. Οι διαφορές των πραγματικών με των προβλεπόμενων τιμών παρουσιάζονται παρακάτω."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_log = pd.DataFrame({'Actual': ytest, 'Predicted': Lpred})\n",
    "df_log"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(metrics.confusion_matrix(ytest, Lpred))\n",
    "print(metrics.classification_report(ytest, Lpred))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2 kNN \n",
    "Η επιλογή των n_neighbours γίνεται με την γραφική απεικόνιση του σφάλματος της τιμής k. Όπως παρατηρείται παρακάτω το μέσο σφάλμα είναι 0 όταν η τιμή k είναι ίση με 5, το οποίο αποφέρει 96,5% accuracy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "knn = KNeighborsClassifier(n_neighbors = 5, algorithm = 'kd_tree', p = 2, metric = 'minkowski', weights = 'distance')\n",
    "knn.fit(Xtrain, ytrain)\n",
    "knn_pred = knn.predict(Xtest)\n",
    "print(knn.score(Xtest, ytest))\n",
    "print(classification_report(ytest, knn_pred))\n",
    "print(confusion_matrix(ytest, knn_pred))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "error = []\n",
    "# Calculating error for K values between 1 and 40\n",
    "for i in range(1, 40):\n",
    "    knn = KNeighborsClassifier(n_neighbors = i)\n",
    "    knn.fit(Xtrain, ytrain)\n",
    "    pred_i = knn.predict(Xtest)\n",
    "    error.append(np.mean(pred_i != ytest))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "plt.figure(figsize = (12, 6))\n",
    "plt.plot(range(1, 40), error, color = 'red', linestyle = 'dashed', marker = 'o', markerfacecolor = 'blue', markersize=10)\n",
    "plt.title('Error Rate K Value')\n",
    "plt.xlabel('K Value')\n",
    "plt.ylabel('Mean Error')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Τέλος παρουσιάζονται τα υπολογιζόμενα accuracy που προέκυψαν απο τις 2 μεθόδους πρόβλεψης πώλησης. Όπως ήταν αναμενόμενο η διαφορά και πάλι δεν είναι μεγάλη."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "log = logistic.score(Xtest, ytest)\n",
    "log = (f\"{log:.2%}\")\n",
    "kNN = knn.score(Xtest,ytest)\n",
    "kNN = (f\"{kNN:.2%}\")\n",
    "print(tabulate([['Logistic Regression', log], ['kNN', kNN]], headers=['Prediction Method', 'Accuracy']))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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