Week_4/Day_4/gradient_descent_challenge.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optimization exercise"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Goal: Train the 2nd order polynomial predictor using both gradient descent and stochastic gradient descent. Optimize the stepsizes and compare against scikit-learn implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Download data from https://drive.google.com/file/d/0Bz9_0VdXvv9bUUNlUTVrMF9VcVU/view?usp=sharing.\n",
    "2. Create a function psi(x), which transforms features AST (assists), REB (rebounds) and STL (steals) into 2nd order polynomial features (add each feature squared and each pair of features multiplied with every other)\n",
    "3. Create a transformed data matrix X, where each x is mapped to psi(x).\n",
    "4. Create a function p2(x,w), which outputs the value of the polynomial at x for given parameters w.\n",
    "5. Create a function Loss(X,y,w), which computes the squared loss of predicting y from X by p2(x,w) using parameters w. Take variable PTS as y. We will predict scored points based on assists, rebounds and steals.\n",
    "6. Code up the gradient descent. It should input a point w and a stepsize.\n",
    "7. Choose an arbitrary point and stepsize. Run gradient descent for 100 iterations and compute the Loss after each iteration. How does the loss behave? Does it converge to something?\n",
    "8. Can you find the stepsize, for which the loss is smallest after 100 iterations?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 348,
   "metadata": {},
   "outputs": [],
   "source": [
    "# IMPORT PACKAGES\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np \n",
    "import pandas as pd \n",
    "import random\n",
    "from matplotlib.pylab import rcParams\n",
    "from sklearn.preprocessing import PolynomialFeatures\n",
    "from sklearn.linear_model import LinearRegression\n",
    "from sklearn.linear_model import Ridge\n",
    "from sklearn.linear_model import Lasso\n",
    "import copy\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 389,
   "metadata": {},
   "outputs": [],
   "source": [
    "nb = pd.read_csv('nba_games_2013_2015.csv', delimiter=';')\n",
    "x = nb[['AST','REB','STL']]\n",
    "y = nb['PTS']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 421,
   "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>AST</th>\n",
       "      <th>REB</th>\n",
       "      <th>STL</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>41</td>\n",
       "      <td>43</td>\n",
       "      <td>14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>23</td>\n",
       "      <td>43</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>20</td>\n",
       "      <td>39</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>19</td>\n",
       "      <td>47</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>21</td>\n",
       "      <td>43</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      ],
      "text/plain": [
       "   AST  REB  STL\n",
       "0   41   43   14\n",
       "1   23   43    8\n",
       "2   20   39    7\n",
       "3   19   47    6\n",
       "4   21   43    4"
      ]
     },
     "execution_count": 421,
     "metadata": {},
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   ],
   "source": [
    "x.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 350,
   "metadata": {},
   "outputs": [],
   "source": [
    "def psi(x):\n",
    "    X = copy.deepcopy(x)\n",
    "    length = len(X.columns)\n",
    "    for i in range(len(X.columns)):\n",
    "        X[f'{X.columns[i]}_squared'] = X[X.columns[i]]**2\n",
    "        for j in range(i+1,length):\n",
    "            X[f'{X.columns[i]}_{X.columns[j]}'] = X[X.columns[i]]*X[X.columns[j]]\n",
    "    return(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 351,
   "metadata": {},
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>AST</th>\n",
       "      <th>REB</th>\n",
       "      <th>STL</th>\n",
       "      <th>AST_squared</th>\n",
       "      <th>AST_REB</th>\n",
       "      <th>AST_STL</th>\n",
       "      <th>REB_squared</th>\n",
       "      <th>REB_STL</th>\n",
       "      <th>STL_squared</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>41</td>\n",
       "      <td>43</td>\n",
       "      <td>14</td>\n",
       "      <td>1681</td>\n",
       "      <td>1763</td>\n",
       "      <td>574</td>\n",
       "      <td>1849</td>\n",
       "      <td>602</td>\n",
       "      <td>196</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>23</td>\n",
       "      <td>43</td>\n",
       "      <td>8</td>\n",
       "      <td>529</td>\n",
       "      <td>989</td>\n",
       "      <td>184</td>\n",
       "      <td>1849</td>\n",
       "      <td>344</td>\n",
       "      <td>64</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>20</td>\n",
       "      <td>39</td>\n",
       "      <td>7</td>\n",
       "      <td>400</td>\n",
       "      <td>780</td>\n",
       "      <td>140</td>\n",
       "      <td>1521</td>\n",
       "      <td>273</td>\n",
       "      <td>49</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>19</td>\n",
       "      <td>47</td>\n",
       "      <td>6</td>\n",
       "      <td>361</td>\n",
       "      <td>893</td>\n",
       "      <td>114</td>\n",
       "      <td>2209</td>\n",
       "      <td>282</td>\n",
       "      <td>36</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>21</td>\n",
       "      <td>43</td>\n",
       "      <td>4</td>\n",
       "      <td>441</td>\n",
       "      <td>903</td>\n",
       "      <td>84</td>\n",
       "      <td>1849</td>\n",
       "      <td>172</td>\n",
       "      <td>16</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "   AST  REB  STL  AST_squared  AST_REB  AST_STL  REB_squared  REB_STL  \\\n",
       "0   41   43   14         1681     1763      574         1849      602   \n",
       "1   23   43    8          529      989      184         1849      344   \n",
       "2   20   39    7          400      780      140         1521      273   \n",
       "3   19   47    6          361      893      114         2209      282   \n",
       "4   21   43    4          441      903       84         1849      172   \n",
       "\n",
       "   STL_squared  \n",
       "0          196  \n",
       "1           64  \n",
       "2           49  \n",
       "3           36  \n",
       "4           16  "
      ]
     },
     "execution_count": 351,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = psi(x)\n",
    "X.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 398,
   "metadata": {},
   "outputs": [],
   "source": [
    "def p2(x,params):\n",
    "    y_pred = np.zeros(x.shape[0])\n",
    "    for i in range(len(x.columns)):\n",
    "        y_pred = y_pred + x[x.columns[i]]*params[i]\n",
    "    return(y_pred)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 457,
   "metadata": {},
   "outputs": [
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   "source": [
    "theta"
   ]
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  {
   "cell_type": "code",
   "execution_count": 458,
   "metadata": {},
   "outputs": [
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     "data": {
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       "0       0.0\n",
       "1       0.0\n",
       "2       0.0\n",
       "3       0.0\n",
       "4       0.0\n",
       "       ... \n",
       "7375    0.0\n",
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   "source": [
    "y_result = p2(X,theta)\n",
    "y_result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 431,
   "metadata": {},
   "outputs": [],
   "source": [
    "def p2(x,params):\n",
    "    y_pred = np.zeros(x.shape[0])\n",
    "    for i in range(len(x.columns)):\n",
    "        y_pred = y_pred + x[x.columns[i]]*params[i]\n",
    "    return(y_pred)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 435,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
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       "      <th></th>\n",
       "      <th>AST</th>\n",
       "      <th>REB</th>\n",
       "      <th>STL</th>\n",
       "      <th>AST_squared</th>\n",
       "      <th>AST_REB</th>\n",
       "      <th>AST_STL</th>\n",
       "      <th>REB_squared</th>\n",
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       "      <th>STL_squared</th>\n",
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       "      <th>1</th>\n",
       "      <td>23</td>\n",
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       "      <td>989</td>\n",
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       "      <th>2</th>\n",
       "      <td>20</td>\n",
       "      <td>39</td>\n",
       "      <td>7</td>\n",
       "      <td>400</td>\n",
       "      <td>780</td>\n",
       "      <td>140</td>\n",
       "      <td>1521</td>\n",
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       "      <td>49</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>19</td>\n",
       "      <td>47</td>\n",
       "      <td>6</td>\n",
       "      <td>361</td>\n",
       "      <td>893</td>\n",
       "      <td>114</td>\n",
       "      <td>2209</td>\n",
       "      <td>282</td>\n",
       "      <td>36</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>21</td>\n",
       "      <td>43</td>\n",
       "      <td>4</td>\n",
       "      <td>441</td>\n",
       "      <td>903</td>\n",
       "      <td>84</td>\n",
       "      <td>1849</td>\n",
       "      <td>172</td>\n",
       "      <td>16</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   AST  REB  STL  AST_squared  AST_REB  AST_STL  REB_squared  REB_STL  \\\n",
       "0   41   43   14         1681     1763      574         1849      602   \n",
       "1   23   43    8          529      989      184         1849      344   \n",
       "2   20   39    7          400      780      140         1521      273   \n",
       "3   19   47    6          361      893      114         2209      282   \n",
       "4   21   43    4          441      903       84         1849      172   \n",
       "\n",
       "   STL_squared  \n",
       "0          196  \n",
       "1           64  \n",
       "2           49  \n",
       "3           36  \n",
       "4           16  "
      ]
     },
     "execution_count": 435,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 434,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0       6763.0\n",
       "1       4033.0\n",
       "2       3229.0\n",
       "3       3967.0\n",
       "4       3533.0\n",
       "         ...  \n",
       "7375    3199.0\n",
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     "execution_count": 434,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "y_result = p2(X,theta)\n",
    "y_result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 432,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Loss(X,y,w):\n",
    "    y_res = p2(X,w)\n",
    "    err = (y - y_res)**2\n",
    "    SSL = sum(err)\n",
    "    return(SSL)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 433,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "120827940125.0"
      ]
     },
     "execution_count": 433,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "Loss(X,y,theta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 467,
   "metadata": {},
   "outputs": [],
   "source": [
    "alpha = 0.00000001 #learning rate\n",
    "iterations = 100 #No. of iterations\n",
    "m = y.size #No. of data points\n",
    "np.random.seed(0) #Set the seed\n",
    "theta = np.zeros(9) #Pick some random values to start with - we a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 469,
   "metadata": {},
   "outputs": [],
   "source": [
    "def grad_descent(X_mat, y, alpha, w, max_iters):\n",
    "    \n",
    "    df = copy.deepcopy(X_mat)\n",
    "    m = len(y)\n",
    "    \n",
    "    loss_func_history = []\n",
    "    loss_func_history.append(Loss(X,y,w))\n",
    "    \n",
    "    for k in range(max_iters):\n",
    "        predictions = p2(x, w)\n",
    "        w_size = len(w)\n",
    "        \n",
    "        for i in range(w_size):\n",
    "            temp = df.iloc[:, i]\n",
    "            errors_x1 = (predictions - y) * temp\n",
    "            w[i] = w[i] - alpha * (1.0 / m) * errors_x1.sum()\n",
    "            \n",
    "        loss_func_history.append(Loss(X,y,w))\n",
    "        \n",
    "    return loss_func_history"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 459,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gradient(df, y, theta, iterations, alpha):\n",
    "    \n",
    "    x = copy.deepcopy(df)\n",
    "    n = len(y)\n",
    "    \n",
    "    my_loss = [Loss(X,y,theta)]\n",
    "    my_theta = [theta]\n",
    "    cur_theta = theta\n",
    "    \n",
    "    for i in range(iterations):\n",
    "        \n",
    "        predictions = p2(x,theta)\n",
    "        error = y - predictions\n",
    "        my_loss.append(Loss(x,y,theta))\n",
    "        \n",
    "        theta = theta - ( (alpha/n) * np.dot(x.T, error))\n",
    "        \n",
    "        my_theta.append(theta)\n",
    "        \n",
    "    return(my_loss, my_theta)\n",
    "        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7) does not converge...."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8) the initial random theta yields lowest loss...."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 455,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly = PolynomialFeatures(2)\n",
    "X_test = pd.DataFrame(poly.fit_transform(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 456,
   "metadata": {},
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.0</td>\n",
       "      <td>41.0</td>\n",
       "      <td>43.0</td>\n",
       "      <td>14.0</td>\n",
       "      <td>1681.0</td>\n",
       "      <td>1763.0</td>\n",
       "      <td>574.0</td>\n",
       "      <td>1849.0</td>\n",
       "      <td>602.0</td>\n",
       "      <td>196.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>23.0</td>\n",
       "      <td>43.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>529.0</td>\n",
       "      <td>989.0</td>\n",
       "      <td>184.0</td>\n",
       "      <td>1849.0</td>\n",
       "      <td>344.0</td>\n",
       "      <td>64.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>20.0</td>\n",
       "      <td>39.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>400.0</td>\n",
       "      <td>780.0</td>\n",
       "      <td>140.0</td>\n",
       "      <td>1521.0</td>\n",
       "      <td>273.0</td>\n",
       "      <td>49.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.0</td>\n",
       "      <td>19.0</td>\n",
       "      <td>47.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>361.0</td>\n",
       "      <td>893.0</td>\n",
       "      <td>114.0</td>\n",
       "      <td>2209.0</td>\n",
       "      <td>282.0</td>\n",
       "      <td>36.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.0</td>\n",
       "      <td>21.0</td>\n",
       "      <td>43.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>441.0</td>\n",
       "      <td>903.0</td>\n",
       "      <td>84.0</td>\n",
       "      <td>1849.0</td>\n",
       "      <td>172.0</td>\n",
       "      <td>16.0</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",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7375</th>\n",
       "      <td>1.0</td>\n",
       "      <td>17.0</td>\n",
       "      <td>39.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>289.0</td>\n",
       "      <td>663.0</td>\n",
       "      <td>170.0</td>\n",
       "      <td>1521.0</td>\n",
       "      <td>390.0</td>\n",
       "      <td>100.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7376</th>\n",
       "      <td>1.0</td>\n",
       "      <td>26.0</td>\n",
       "      <td>40.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>676.0</td>\n",
       "      <td>1040.0</td>\n",
       "      <td>260.0</td>\n",
       "      <td>1600.0</td>\n",
       "      <td>400.0</td>\n",
       "      <td>100.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7377</th>\n",
       "      <td>1.0</td>\n",
       "      <td>23.0</td>\n",
       "      <td>52.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>529.0</td>\n",
       "      <td>1196.0</td>\n",
       "      <td>184.0</td>\n",
       "      <td>2704.0</td>\n",
       "      <td>416.0</td>\n",
       "      <td>64.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7378</th>\n",
       "      <td>1.0</td>\n",
       "      <td>23.0</td>\n",
       "      <td>41.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>529.0</td>\n",
       "      <td>943.0</td>\n",
       "      <td>253.0</td>\n",
       "      <td>1681.0</td>\n",
       "      <td>451.0</td>\n",
       "      <td>121.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7379</th>\n",
       "      <td>1.0</td>\n",
       "      <td>17.0</td>\n",
       "      <td>44.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>289.0</td>\n",
       "      <td>748.0</td>\n",
       "      <td>68.0</td>\n",
       "      <td>1936.0</td>\n",
       "      <td>176.0</td>\n",
       "      <td>16.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>7380 rows × 10 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "        0     1     2     3       4       5      6       7      8      9\n",
       "0     1.0  41.0  43.0  14.0  1681.0  1763.0  574.0  1849.0  602.0  196.0\n",
       "1     1.0  23.0  43.0   8.0   529.0   989.0  184.0  1849.0  344.0   64.0\n",
       "2     1.0  20.0  39.0   7.0   400.0   780.0  140.0  1521.0  273.0   49.0\n",
       "3     1.0  19.0  47.0   6.0   361.0   893.0  114.0  2209.0  282.0   36.0\n",
       "4     1.0  21.0  43.0   4.0   441.0   903.0   84.0  1849.0  172.0   16.0\n",
       "...   ...   ...   ...   ...     ...     ...    ...     ...    ...    ...\n",
       "7375  1.0  17.0  39.0  10.0   289.0   663.0  170.0  1521.0  390.0  100.0\n",
       "7376  1.0  26.0  40.0  10.0   676.0  1040.0  260.0  1600.0  400.0  100.0\n",
       "7377  1.0  23.0  52.0   8.0   529.0  1196.0  184.0  2704.0  416.0   64.0\n",
       "7378  1.0  23.0  41.0  11.0   529.0   943.0  253.0  1681.0  451.0  121.0\n",
       "7379  1.0  17.0  44.0   4.0   289.0   748.0   68.0  1936.0  176.0   16.0\n",
       "\n",
       "[7380 rows x 10 columns]"
      ]
     },
     "execution_count": 456,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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