ML Project Pairs Trading - K-means.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import / install relevant Python packages\n",
    "import numpy as np\n",
    "import pandas as pd  \n",
    "import datetime as dt\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from sklearn.cluster import (KMeans, MiniBatchKMeans, AgglomerativeClustering, DBSCAN)\n",
    "from sklearn.neighbors import NearestNeighbors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "alpha = 50 #percentile - default is 50 - mediane\n",
    "# Define the key inputs for the K-means model\n",
    "#n_clusters = 5\n",
    "n_max_iter = 1000\n",
    "n_init     = 10\n",
    "init_type  = 'k-means++'\n",
    "#number of clusters list \n",
    "K = [5,10,50, 100, 500, 1000,1500]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "feature_names_list = ['mom1m','absacc','acc','aeavol','age','agr','baspread',\n",
    "'beta','betasq','bm','bm_ia','cash','cashdebt','cashpr','cfp','cfp_ia',\n",
    "'chatoia','chcsho','chempia','chinv','chmom','chpmia','chtx','cinvest',\n",
    "'convind','currat','depr','divi','divo','dolvol','dy','ear','egr','ep',\n",
    "'gma','herf','hire','idiovol','ill','indmom','invest','lev','lgr','maxret',\n",
    "'ms','mve_ia','mvel1','nincr','operprof','pchcapx_ia','pchcurrat','pchdepr',\n",
    "'pchgm_pchsale','pchquick','pchsale_pchrect','pctacc','pricedelay',\n",
    "'ps','quick','rd','retvol','roaq','roeq','roic','rsup','salecash',\n",
    "'salerec','securedind','sgr','sin','sp','std_dolvol','std_turn',\n",
    "'tang','tb','turn','zerotrade']\n",
    "\n",
    "mom_names_list = [\"mom1m\",\"mom2m\",\"mom3m\",\"mom4m\",\"mom5m\",\"mom6m\",\"mom7m\",\"mom8m\",\"mom9m\",\"mom10m\",\"mom11m\",\"mom12m\",\n",
    "                \"mom13m\",\"mom14m\",\"mom15m\",\"mom16m\",\"mom17m\",\"mom18m\",\"mom19m\",\"mom20m\",\"mom21m\",\"mom22m\",\"mom23m\",\"mom24m\",\n",
    "                \"mom25m\",\"mom26m\",\"mom27m\",\"mom28m\",\"mom29m\",\"mom30m\",\"mom31m\",\"mom32m\",\"mom33m\",\"mom34m\",\"mom35m\",\"mom36m\",\n",
    "                \"mom37m\",\"mom38m\",\"mom39m\",\"mom40m\",\"mom41m\",\"mom42m\",\"mom43m\",\"mom44m\",\"mom45m\",\"mom46m\",\"mom47m\",\"mom48m\"]\n",
    "\n",
    "feature_names_list_full =  feature_names_list + mom_names_list[1:]\n",
    "\n",
    "PCA_components_number = len(feature_names_list_full)\n",
    "PCA_components_column_names = ['PCA '+ str(i+1) for i in range(PCA_components_number)]\n",
    "PCA_components_tresholds_column_names=['n_to_reach_90','n_to_reach_95','n_to_reach_99']\n",
    "Original_df_columns_names = ['DATE','permno','sic2']\n",
    "PCA_table_full_columns_names = Original_df_columns_names+PCA_components_column_names+PCA_components_tresholds_column_names\n",
    "KMeans_column_names = ['cluster k=5','distance k=5','outlier k=5','cluster k=10','distance k=10','outlier k=10','cluster k=50','distance k=50','outlier k=50',\n",
    "                        'cluster k=100','distance k=100','outlier k=100','cluster k=500','distance k=500','outlier k=500','cluster k=1000','distance k=1000','outlier k=1000',\n",
    "                        'cluster k=1500','distance k=1500','outlier k=1500']\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_PCA = pd.read_csv('D:/USB Drive/Data/DachengXiu/df_PCA_done.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def distance_of_percentile_to_nearest_neighbors(df_for_determine_percentile_knn,alpha = 50):\n",
    "    k = 2\n",
    "    # initialize model\n",
    "    neigh = NearestNeighbors(n_neighbors=k, n_jobs=-1)\n",
    "    # train for getting nearest neighbour\n",
    "    neigh.fit(df_for_determine_percentile_knn)\n",
    "    dist, ind = neigh.kneighbors(df_for_determine_percentile_knn)\n",
    "    return np.percentile(dist[:,1], alpha)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_KMeans_model(df_for_determine_labels,n_clusters):\n",
    "    # Create an instance of the class KMeans, fit the model, and predict cluster memberships\n",
    "    kmeans_model = KMeans(n_clusters=n_clusters,\n",
    "                        init=init_type,\n",
    "                        n_init=n_init,\n",
    "                        max_iter=n_max_iter,\n",
    "                        random_state=1)\n",
    "    \n",
    "    kmeans = kmeans_model.fit(df_for_determine_labels)\n",
    "    return kmeans"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculate_distances_to_centroide_of_belonging_cluster(kmeans,df_for_determine_labels,n_clusters):\n",
    "    lebels = kmeans.labels_\n",
    "    distances_to_centroides = kmeans.transform(df_for_determine_labels)\n",
    "    distance_to_belonging_cluster_centroide = np.array([distances_to_centroides[i][x] for i,x in enumerate(lebels)])\n",
    "    return distance_to_belonging_cluster_centroide"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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      "DATE  20210129\n",
      "DATE  20210226\n",
      "DATE  20210331\n",
      "DATE  20210430\n",
      "DATE  20210528\n",
      "DATE  20210630\n",
      "DATE  20210730\n",
      "DATE  20210831\n",
      "DATE  20210930\n",
      "DATE  20211029\n",
      "DATE  20211130\n",
      "DATE  20211231\n"
     ]
    }
   ],
   "source": [
    "#inicilize df for KMeans - make empty df\n",
    "#initial dataframe\n",
    "df_KMeans=pd.DataFrame(columns=Original_df_columns_names + KMeans_column_names)\n",
    "dates = sorted(df_PCA['DATE'].unique())\n",
    "for date in dates:\n",
    "    print(\"DATE \",date)\n",
    "    df_KMeans_window = df_PCA[df_PCA['DATE'] == date].sort_values('permno')\n",
    "\n",
    "    #determen list of column names of PCA components needed to explain 99% of variance\n",
    "    number_of_PCA_components_for_99_variance_explainability =  df_KMeans_window['n_to_reach_99'].min() #workaround to get single value\n",
    "    PCA_components_for_99_variance_explainability_column_names = ['PCA '+ str(i+1) for i in range(number_of_PCA_components_for_99_variance_explainability)]\n",
    "    \n",
    "    #calculate treshold for all points in data set - needed for outlier detection\n",
    "    #I have some phyton variable type bug ( TypeError: 'numpy.float64' object is not callable ) and I can not understand it. I will put code of function in here\n",
    "    #distance_of_percentile_to_nearest_neighbors = distance_of_percentile_to_nearest_neighbors(df_KMeans_window[PCA_components_for_99_variance_explainability_column_names],alpha)\n",
    "    #code of function distance_of_percentile_to_nearest_neighbors\n",
    "    r = 2 #I will change k into r since there is k already in use\n",
    "    # initialize model\n",
    "    neigh = NearestNeighbors(n_neighbors=r, n_jobs=-1)\n",
    "    # train for getting nearest neighbour\n",
    "    neigh.fit(df_KMeans_window[PCA_components_for_99_variance_explainability_column_names])\n",
    "    dist, ind = neigh.kneighbors(df_KMeans_window[PCA_components_for_99_variance_explainability_column_names])\n",
    "    distance_of_percentile_to_nearest_neighbors = np.percentile(dist[:,1], alpha)\n",
    "    \n",
    "    #Use number of clusters smaler then number of data points \n",
    "    list_of_cluster_numbers = [x for x in K if x < len(df_KMeans_window)]\n",
    "    for k in list_of_cluster_numbers:\n",
    "        #Create KMeans model \n",
    "        kmeans_fited = create_KMeans_model(df_KMeans_window[PCA_components_for_99_variance_explainability_column_names],k)\n",
    "        #determine cluster lables and fill data to df\n",
    "        df_KMeans_window[f'cluster k={k}'] = kmeans_fited.labels_\n",
    "        #calculate distances to centroide of belonging_cluster for every data point\n",
    "        df_KMeans_window[f'distance k={k}'] = calculate_distances_to_centroide_of_belonging_cluster(kmeans_fited,df_KMeans_window[PCA_components_for_99_variance_explainability_column_names],k)\n",
    "        #setting flag 1 if it is outlier\n",
    "        df_KMeans_window[f'outlier k={k}'] = 0\n",
    "        df_KMeans_window[f'outlier k={k}'].values[df_KMeans_window[f'distance k={k}'] > distance_of_percentile_to_nearest_neighbors] = 1\n",
    "    \n",
    "    df_KMeans=pd.concat([df_KMeans,df_KMeans_window],axis=0)\n",
    "df_KMeans = df_KMeans[Original_df_columns_names+KMeans_column_names]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "#df_KMeans.to_csv('D:/USB Drive/Data/DachengXiu/df_KMeans.csv')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Analysis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_KMeans = pd.read_csv('D:/USB Drive/Data/DachengXiu/df_KMeans.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_1990_2000 = df_KMeans[(df_KMeans['DATE'] > 19891231 ) & (df_KMeans['DATE'] < 20210101)]\n",
    "outliers_column_names_list =  ['outlier k=5','outlier k=10','outlier k=50','outlier k=100','outlier k=500','outlier k=1000','outlier k=1500']\n",
    "clusters_column_names_list =  ['cluster k=5','cluster k=10','cluster k=50','cluster k=100','cluster k=500','cluster k=1000','cluster k=1500']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>cluster k=1000</th>\n",
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       "      <td>0.0</td>\n",
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       "      <td>243930</td>\n",
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       "      <td>193.0</td>\n",
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       "      <td>1.0</td>\n",
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       "      <td>3.371748e-07</td>\n",
       "      <td>0</td>\n",
       "      <td>855.0</td>\n",
       "      <td>3.371748e-07</td>\n",
       "      <td>0.0</td>\n",
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       "    <tr>\n",
       "      <th>1466533</th>\n",
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       "      <td>20201231</td>\n",
       "      <td>93436</td>\n",
       "      <td>37.0</td>\n",
       "      <td>2</td>\n",
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       "      <td>238</td>\n",
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       "      <td>138</td>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0</td>\n",
       "      <td>4.0</td>\n",
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       "      <td>0.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1222607 rows × 25 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "         Unnamed: 0      DATE  permno  sic2  cluster k=5  distance k=5  \\\n",
       "243927       243927  19900131   10057  36.0            0      6.120755   \n",
       "243928       243928  19900131   10074  34.0            2      7.910803   \n",
       "243929       243929  19900131   10082  48.0            3     12.843116   \n",
       "243930       243930  19900131   10137  49.0            2      4.574791   \n",
       "243931       243931  19900131   10145  99.0            2      5.942988   \n",
       "...             ...       ...     ...   ...          ...           ...   \n",
       "1466529     1466529  20201231   93423  79.0            0      5.298085   \n",
       "1466530     1466530  20201231   93426  36.0            0      4.044636   \n",
       "1466531     1466531  20201231   93427  36.0            1      4.937171   \n",
       "1466532     1466532  20201231   93434   1.0            0      9.800538   \n",
       "1466533     1466533  20201231   93436  37.0            2     27.355990   \n",
       "\n",
       "         outlier k=5  cluster k=10  distance k=10  outlier k=10  ...  \\\n",
       "243927             0             4       6.181993             0  ...   \n",
       "243928             1             1       7.749312             1  ...   \n",
       "243929             1             5      11.848580             1  ...   \n",
       "243930             0             3       3.682299             0  ...   \n",
       "243931             0             3       5.659142             0  ...   \n",
       "...              ...           ...            ...           ...  ...   \n",
       "1466529            0             4       5.722224             1  ...   \n",
       "1466530            0             4       4.111018             0  ...   \n",
       "1466531            0             1       4.635658             0  ...   \n",
       "1466532            1             4       9.856951             1  ...   \n",
       "1466533            1             3      22.915793             1  ...   \n",
       "\n",
       "         outlier k=100  cluster k=500  distance k=500  outlier k=500  \\\n",
       "243927               0             19        5.334956              0   \n",
       "243928               1            373        7.251274              1   \n",
       "243929               1            255        8.631617              1   \n",
       "243930               0             40        2.403934              0   \n",
       "243931               0             53        4.827178              0   \n",
       "...                ...            ...             ...            ...   \n",
       "1466529              0            200        4.177078              0   \n",
       "1466530              0            411        3.405722              0   \n",
       "1466531              0              5        4.303813              0   \n",
       "1466532              1            206        8.013019              1   \n",
       "1466533              0            238        0.000000              0   \n",
       "\n",
       "         cluster k=1000  distance k=1000  outlier k=1000  cluster k=1500  \\\n",
       "243927              103     4.735991e+00               0          1277.0   \n",
       "243928              410     7.150136e+00               1          1097.0   \n",
       "243929              692     0.000000e+00               0           491.0   \n",
       "243930               32     2.559683e+00               0            15.0   \n",
       "243931              456     4.488611e+00               0           880.0   \n",
       "...                 ...              ...             ...             ...   \n",
       "1466529             197     4.400300e+00               0           439.0   \n",
       "1466530             269     3.737890e+00               0           193.0   \n",
       "1466531             251     3.133092e+00               0           272.0   \n",
       "1466532             642     3.371748e-07               0           855.0   \n",
       "1466533             138     0.000000e+00               0             4.0   \n",
       "\n",
       "         distance k=1500  outlier k=1500  \n",
       "243927      3.138217e+00             0.0  \n",
       "243928      0.000000e+00             0.0  \n",
       "243929      0.000000e+00             0.0  \n",
       "243930      2.049353e+00             0.0  \n",
       "243931      4.355209e+00             0.0  \n",
       "...                  ...             ...  \n",
       "1466529     3.777693e+00             0.0  \n",
       "1466530     3.504675e+00             0.0  \n",
       "1466531     3.472513e+00             0.0  \n",
       "1466532     3.371748e-07             0.0  \n",
       "1466533     0.000000e+00             0.0  \n",
       "\n",
       "[1222607 rows x 25 columns]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_1990_2000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "218.3279569892473"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of clusters 129\n",
    "df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby([\"DATE\",\"cluster k=500\"])['permno'].count().pipe(lambda  dfx: dfx.loc[dfx>1]).count()/372\n",
    "#.count().max()\n",
    "#['permno'].pipe(lambda  dfx: dfx.loc[dfx>1])\n",
    "#218.3279569892473"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3286.5779569892475"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of stocks in total 3157\n",
    "df_1990_2000.groupby(['DATE'])['permno'].count().mean()\n",
    "#3286.5779569892475"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1989.8172043010752"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of stocks in clusters 1257 (39.78) - from paper\n",
    "df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby(['DATE','cluster k=500']).count()['permno'].pipe(lambda  dfx: dfx.loc[dfx>1]).sum()/372\n",
    "#1989.8172043010752"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1296.760752688172"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of outliers 1900 (60.22) - from paper\n",
    "num_outliers_original =  df_1990_2000[df_1990_2000[\"outlier k=500\"] == 1].count()['permno']/372\n",
    "num_outliers_original_after_cleaned_outiers = df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby(['DATE','cluster k=500']).count()['permno'].pipe(lambda  dfx: dfx.loc[dfx==1]).sum()/372\n",
    "num_outliers_original + num_outliers_original_after_cleaned_outiers\n",
    "#1296.760752688172"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "268.6209677419355"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num_outliers_original_after_cleaned_outiers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "number_in_cluster    67.341398\n",
       "dtype: float64"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of stocks in the biggest cluster 63 (5.18)\n",
    "df = df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby(['DATE','cluster k=500']).count()['permno'].pipe(lambda  dfx: dfx.loc[dfx>1])\n",
    "df = df.to_frame('number_in_cluster')\n",
    "mean1st = df.sort_values(['DATE','number_in_cluster'],ascending=False).groupby('DATE').head(1).mean()\n",
    "mean1st\n",
    "#67.341398"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "number_in_cluster    56.303763\n",
       "dtype: float64"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of stocks in the second biggest cluster 51 (4.13)\n",
    "df = df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby(['DATE','cluster k=500']).count()['permno'].pipe(lambda  dfx: dfx.loc[dfx>1])\n",
    "df = df.to_frame('number_in_cluster')\n",
    "mean2nd = df.sort_values(['DATE','number_in_cluster'],ascending=False).groupby('DATE').head(2).mean()*2-mean1st\n",
    "mean2nd\n",
    "#56.303763"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "number_in_cluster    50.680108\n",
       "dtype: float64"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of stocks in the third biggest cluster 45 (3.63)\n",
    "df = df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby(['DATE','cluster k=500']).count()['permno'].pipe(lambda  dfx: dfx.loc[dfx>1])\n",
    "df = df.to_frame('number_in_cluster')\n",
    "mean3rd = df.sort_values(['DATE','number_in_cluster'],ascending=False).groupby('DATE').head(3).mean()*3-mean1st-mean2nd\n",
    "mean3rd\n",
    "#50.680108"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "268.6209677419355"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby(['DATE','cluster k=500']).count()['permno'].pipe(lambda  dfx: dfx.loc[dfx==1]).sum()/372"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "486.9489247311828"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of clusters 129 - The number of clusters, K, is set to 500, but the outlier removing process eliminates some clusters \n",
    "# whose data points are far from each other and the remaining clusters are less than 200.The k-means clustering forms an average of 129 clusters after\n",
    "#removing outliers, with a range of 80 to 165 clusters\n",
    "df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby(['DATE']).nunique()[\"cluster k=500\"].mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "57"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of stocks in the biggest cluster 63 (5.18)\n",
    "df_1990_2000[df_1990_2000[\"outlier k=1500\"] == 0].groupby([\"DATE\",\"cluster k=1500\"])['permno'].count().max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "119"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Number of stocks in the biggest cluster 63 (5.18)\n",
    "df_1990_2000[df_1990_2000[\"outlier k=500\"] == 0].groupby([\"DATE\",\"cluster k=500\"])['permno'].count().max()"
   ]
  }
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