random_forest_guided_example.ipynb

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   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "We've talked about Random Forests. Now it's time to build one.\n",
    "\n",
    "Here we'll use data from Lending Club (2015) to predict the state of a loan given some information about it. You can download the dataset [here](https://www.dropbox.com/s/0so14yudedjmm5m/LoanStats3d.csv?dl=1)"
   ]
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     "text": [
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/IPython/core/interactiveshell.py:3058: DtypeWarning: Columns (0,19,55) have mixed types. Specify dtype option on import or set low_memory=False.\n",
      "  interactivity=interactivity, compiler=compiler, result=result)\n"
     ]
    }
   ],
   "source": [
    "# Replace the path with the correct path for your data.\n",
    "y2015 = pd.read_csv(\n",
    "    'https://www.dropbox.com/s/0so14yudedjmm5m/LoanStats3d.csv?dl=1',\n",
    "    skipinitialspace=True,\n",
    "    header=1\n",
    ")\n",
    "\n",
    "# Note the warning about dtypes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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       "         id   member_id  loan_amnt  funded_amnt  funded_amnt_inv        term  \\\n",
       "0  68009401  72868139.0    16000.0      16000.0          16000.0   60 months   \n",
       "1  68354783  73244544.0     9600.0       9600.0           9600.0   36 months   \n",
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       "3  68466961  73356799.0    28000.0      28000.0          28000.0   36 months   \n",
       "4  68495092  73384866.0     8650.0       8650.0           8650.0   36 months   \n",
       "\n",
       "  int_rate  installment grade sub_grade  ... num_tl_90g_dpd_24m  \\\n",
       "0   14.85%       379.39     C        C5  ...                0.0   \n",
       "1    7.49%       298.58     A        A4  ...                0.0   \n",
       "2    7.49%       777.55     A        A4  ...                0.0   \n",
       "3    6.49%       858.05     A        A2  ...                0.0   \n",
       "4   19.89%       320.99     E        E3  ...                0.0   \n",
       "\n",
       "  num_tl_op_past_12m pct_tl_nvr_dlq  percent_bc_gt_75 pub_rec_bankruptcies  \\\n",
       "0                2.0           78.9               0.0                  0.0   \n",
       "1                2.0          100.0              66.7                  0.0   \n",
       "2                0.0          100.0              20.0                  0.0   \n",
       "3                0.0           91.7              22.2                  0.0   \n",
       "4               12.0          100.0              50.0                  1.0   \n",
       "\n",
       "  tax_liens tot_hi_cred_lim total_bal_ex_mort total_bc_limit  \\\n",
       "0       2.0        298100.0           31329.0       281300.0   \n",
       "1       0.0         88635.0           55387.0        12500.0   \n",
       "2       0.0        373572.0           68056.0        38400.0   \n",
       "3       0.0        304003.0           74920.0        41500.0   \n",
       "4       0.0         38998.0           18926.0         2750.0   \n",
       "\n",
       "  total_il_high_credit_limit  \n",
       "0                    13400.0  \n",
       "1                    75635.0  \n",
       "2                    82117.0  \n",
       "3                    42503.0  \n",
       "4                    18248.0  \n",
       "\n",
       "[5 rows x 111 columns]"
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     },
     "execution_count": 3,
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   ],
   "source": [
    "y2015.head()"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## Data Cleaning\n",
    "\n",
    "Well, `get_dummies` can be a very memory intensive thing, particularly if data are typed poorly. We got a warning about that earlier. Mixed data types get converted to objects, and that could create huge problems. Our dataset is about 400,000 rows. If there's a bad type there its going to see 400,000 distinct values and try to create dummies for all of them. That's bad. Lets look at all our categorical variables and see how many distinct counts there are..."
   ]
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   "execution_count": 4,
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    {
     "name": "stdout",
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     "text": [
      "id\n",
      "421097\n",
      "term\n",
      "2\n",
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      "7\n",
      "sub_grade\n",
      "35\n",
      "emp_title\n",
      "120812\n",
      "emp_length\n",
      "11\n",
      "home_ownership\n",
      "4\n",
      "verification_status\n",
      "3\n",
      "issue_d\n",
      "12\n",
      "loan_status\n",
      "7\n",
      "pymnt_plan\n",
      "1\n",
      "url\n",
      "421095\n",
      "desc\n",
      "34\n",
      "purpose\n",
      "14\n",
      "title\n",
      "27\n",
      "zip_code\n",
      "914\n",
      "addr_state\n",
      "49\n",
      "earliest_cr_line\n",
      "668\n",
      "revol_util\n",
      "1211\n",
      "initial_list_status\n",
      "2\n",
      "last_pymnt_d\n",
      "25\n",
      "next_pymnt_d\n",
      "4\n",
      "last_credit_pull_d\n",
      "26\n",
      "application_type\n",
      "2\n",
      "verification_status_joint\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "categorical = y2015.select_dtypes(include=['object'])\n",
    "for i in categorical:\n",
    "    column = categorical[i]\n",
    "    print(i)\n",
    "    print(column.nunique())"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "Well that right there is what's called a problem. Some of these have over a hundred thousand distinct types. Lets drop the ones with over 30 unique values, converting to numeric where it makes sense. In doing this there's a lot of code that gets written to just see if the numeric conversion makes sense. It's a manual process that we'll abstract away and just include the conversion.\n",
    "\n",
    "You could extract numeric features from the dates, but here we'll just drop them. There's a lot of data, it shouldn't be a huge problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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   "outputs": [],
   "source": [
    "# Convert ID and Interest Rate to numeric.\n",
    "y2015['id'] = pd.to_numeric(y2015['id'], errors='coerce')\n",
    "y2015['int_rate'] = pd.to_numeric(y2015['int_rate'].str.strip('%'), errors='coerce')\n",
    "\n",
    "# Drop other columns with many unique variables\n",
    "y2015.drop(['url', 'emp_title', 'zip_code', 'earliest_cr_line', 'revol_util',\n",
    "            'sub_grade', 'addr_state', 'desc'], 1, inplace=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wonder what was causing the dtype error on the id column, which _should_ have all been integers? Let's look at the end of the file."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
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       "<p>5 rows × 103 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                id   member_id  loan_amnt  funded_amnt  funded_amnt_inv  \\\n",
       "421092  36271333.0  38982739.0    13000.0      13000.0          13000.0   \n",
       "421093  36490806.0  39222577.0    12000.0      12000.0          12000.0   \n",
       "421094  36271262.0  38982659.0    20000.0      20000.0          20000.0   \n",
       "421095         NaN         NaN        NaN          NaN              NaN   \n",
       "421096         NaN         NaN        NaN          NaN              NaN   \n",
       "\n",
       "              term  int_rate  installment grade emp_length  ...  \\\n",
       "421092   60 months     15.99       316.07     D    5 years  ...   \n",
       "421093   60 months     19.99       317.86     E     1 year  ...   \n",
       "421094   36 months     11.99       664.20     B  10+ years  ...   \n",
       "421095         NaN       NaN          NaN   NaN        NaN  ...   \n",
       "421096         NaN       NaN          NaN   NaN        NaN  ...   \n",
       "\n",
       "       num_tl_90g_dpd_24m  num_tl_op_past_12m pct_tl_nvr_dlq percent_bc_gt_75  \\\n",
       "421092                0.0                 3.0          100.0             50.0   \n",
       "421093                1.0                 2.0           95.0             66.7   \n",
       "421094                0.0                 1.0          100.0             50.0   \n",
       "421095                NaN                 NaN            NaN              NaN   \n",
       "421096                NaN                 NaN            NaN              NaN   \n",
       "\n",
       "       pub_rec_bankruptcies tax_liens tot_hi_cred_lim total_bal_ex_mort  \\\n",
       "421092                  1.0       0.0         51239.0           34178.0   \n",
       "421093                  0.0       0.0         96919.0           58418.0   \n",
       "421094                  0.0       1.0         43740.0           33307.0   \n",
       "421095                  NaN       NaN             NaN               NaN   \n",
       "421096                  NaN       NaN             NaN               NaN   \n",
       "\n",
       "        total_bc_limit  total_il_high_credit_limit  \n",
       "421092         10600.0                     33239.0  \n",
       "421093          9700.0                     69919.0  \n",
       "421094         41700.0                         0.0  \n",
       "421095             NaN                         NaN  \n",
       "421096             NaN                         NaN  \n",
       "\n",
       "[5 rows x 103 columns]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y2015.tail()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Remove two summary rows at the end that don't actually contain data.\n",
    "y2015 = y2015[:-2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "source": [
    "Now this should be better. Let's try again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "outputs": [
    {
     "data": {
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       "  </tbody>\n",
       "</table>\n",
       "<p>421095 rows × 237 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                id   member_id  loan_amnt  funded_amnt  funded_amnt_inv  \\\n",
       "0       68009401.0  72868139.0    16000.0      16000.0          16000.0   \n",
       "1       68354783.0  73244544.0     9600.0       9600.0           9600.0   \n",
       "2       68466916.0  73356753.0    25000.0      25000.0          25000.0   \n",
       "3       68466961.0  73356799.0    28000.0      28000.0          28000.0   \n",
       "4       68495092.0  73384866.0     8650.0       8650.0           8650.0   \n",
       "...            ...         ...        ...          ...              ...   \n",
       "421090  36371250.0  39102635.0    10000.0      10000.0          10000.0   \n",
       "421091  36441262.0  39152692.0    24000.0      24000.0          24000.0   \n",
       "421092  36271333.0  38982739.0    13000.0      13000.0          13000.0   \n",
       "421093  36490806.0  39222577.0    12000.0      12000.0          12000.0   \n",
       "421094  36271262.0  38982659.0    20000.0      20000.0          20000.0   \n",
       "\n",
       "        int_rate  installment  annual_inc    dti  delinq_2yrs  ...  \\\n",
       "0          14.85       379.39     48000.0  33.18          0.0  ...   \n",
       "1           7.49       298.58     60000.0  22.44          0.0  ...   \n",
       "2           7.49       777.55    109000.0  26.02          0.0  ...   \n",
       "3           6.49       858.05     92000.0  21.60          0.0  ...   \n",
       "4          19.89       320.99     55000.0  25.49          0.0  ...   \n",
       "...          ...          ...         ...    ...          ...  ...   \n",
       "421090     11.99       332.10     31000.0  28.69          0.0  ...   \n",
       "421091     11.99       797.03     79000.0   3.90          0.0  ...   \n",
       "421092     15.99       316.07     35000.0  30.90          0.0  ...   \n",
       "421093     19.99       317.86     64400.0  27.19          1.0  ...   \n",
       "421094     11.99       664.20    100000.0  10.83          0.0  ...   \n",
       "\n",
       "        last_credit_pull_d_Nov-2016  last_credit_pull_d_Oct-2015  \\\n",
       "0                                 0                            0   \n",
       "1                                 0                            0   \n",
       "2                                 0                            0   \n",
       "3                                 0                            0   \n",
       "4                                 0                            0   \n",
       "...                             ...                          ...   \n",
       "421090                            0                            0   \n",
       "421091                            0                            0   \n",
       "421092                            0                            0   \n",
       "421093                            0                            0   \n",
       "421094                            0                            0   \n",
       "\n",
       "        last_credit_pull_d_Oct-2016  last_credit_pull_d_Sep-2015  \\\n",
       "0                                 0                            0   \n",
       "1                                 0                            0   \n",
       "2                                 0                            0   \n",
       "3                                 0                            0   \n",
       "4                                 0                            0   \n",
       "...                             ...                          ...   \n",
       "421090                            0                            0   \n",
       "421091                            0                            0   \n",
       "421092                            0                            0   \n",
       "421093                            1                            0   \n",
       "421094                            0                            0   \n",
       "\n",
       "        last_credit_pull_d_Sep-2016  application_type_INDIVIDUAL  \\\n",
       "0                                 0                            1   \n",
       "1                                 0                            1   \n",
       "2                                 0                            1   \n",
       "3                                 0                            1   \n",
       "4                                 0                            1   \n",
       "...                             ...                          ...   \n",
       "421090                            0                            1   \n",
       "421091                            0                            1   \n",
       "421092                            0                            1   \n",
       "421093                            0                            1   \n",
       "421094                            0                            1   \n",
       "\n",
       "        application_type_JOINT  verification_status_joint_Not Verified  \\\n",
       "0                            0                                       0   \n",
       "1                            0                                       0   \n",
       "2                            0                                       0   \n",
       "3                            0                                       0   \n",
       "4                            0                                       0   \n",
       "...                        ...                                     ...   \n",
       "421090                       0                                       0   \n",
       "421091                       0                                       0   \n",
       "421092                       0                                       0   \n",
       "421093                       0                                       0   \n",
       "421094                       0                                       0   \n",
       "\n",
       "        verification_status_joint_Source Verified  \\\n",
       "0                                               0   \n",
       "1                                               0   \n",
       "2                                               0   \n",
       "3                                               0   \n",
       "4                                               0   \n",
       "...                                           ...   \n",
       "421090                                          0   \n",
       "421091                                          0   \n",
       "421092                                          0   \n",
       "421093                                          0   \n",
       "421094                                          0   \n",
       "\n",
       "        verification_status_joint_Verified  \n",
       "0                                        0  \n",
       "1                                        0  \n",
       "2                                        0  \n",
       "3                                        0  \n",
       "4                                        0  \n",
       "...                                    ...  \n",
       "421090                                   0  \n",
       "421091                                   0  \n",
       "421092                                   0  \n",
       "421093                                   0  \n",
       "421094                                   0  \n",
       "\n",
       "[421095 rows x 237 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.get_dummies(y2015)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "source": [
    "It finally works! We had to sacrifice sub grade, state address and description, but that's fine. If you want to include them you could run the dummies independently and then append them back to the dataframe.\n",
    "\n",
    "## Second Attempt\n",
    "\n",
    "Now let's try this model again.\n",
    "\n",
    "We're also going to drop NA columns, rather than impute, because our data is rich enough that we can probably get away with it.\n",
    "\n",
    "This model may take a few minutes to run."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([0.97573052, 0.95807409, 0.76725243, 0.93730556, 0.97721336])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn import ensemble\n",
    "from sklearn.model_selection import cross_val_score\n",
    "\n",
    "rfc = ensemble.RandomForestClassifier()\n",
    "X = y2015.drop('loan_status', 1)\n",
    "Y = y2015['loan_status']\n",
    "X = pd.get_dummies(X)\n",
    "X = X.dropna(axis=1)\n",
    "\n",
    "cross_val_score(rfc, X, Y, cv=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "run_control": {
     "frozen": false,
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   },
   "source": [
    "The score cross validation reports is the accuracy of the tree. Here we're about 98% accurate.\n",
    "\n",
    "That works pretty well, but there are a few potential problems. Firstly, we didn't really do much in the way of feature selection or model refinement. As such there are a lot of features in there that we don't really need. Some of them are actually quite impressively useless.\n",
    "\n",
    "There's also some variance in the scores. The fact that one gave us only 93% accuracy while others gave higher than 98 is concerning. This variance could be corrected by increasing the number of estimators. That will make it take even longer to run, however, and it is already quite slow."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DRILL: Third Attempt\n",
    "\n",
    "So here's your task. Get rid of as much data as possible without dropping below an average of 90% accuracy in a 10-fold cross validation.\n",
    "\n",
    "You'll want to do a few things in this process. First, dive into the data that we have and see which features are most important. This can be the raw features or the generated dummies. You may want to use PCA or correlation matrices.\n",
    "\n",
    "Can you do it without using anything related to payment amount or outstanding principal? How do you know?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "from sklearn.decomposition import PCA\n",
    "pca = PCA(n_components=20)\n",
    "pca.fit(X)\n",
    "X_pca = pca.transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([0.77376189, 0.90391831, 0.92904298, 0.91879408, 0.89819037])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rfc = ensemble.RandomForestClassifier()\n",
    "cross_val_score(rfc, X_pca, Y, cv=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using PCA dimensionality reduction, 90% + accuracy can be maintained by reducing the data to 20 principle components. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Can you do it without any information related to payment amount or outstanding principal?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "columns = ['out_prncp', \"out_prncp_inv\", \"total_pymnt\", \"total_pymnt_inv\", \"total_rec_prncp\", \"total_rec_int\", \"last_pymnt_amnt\"]\n",
    "\n",
    "X_trim = X.drop(columns=columns)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n",
      "/Users/yutakatsuyama/anaconda3/lib/python3.7/site-packages/sklearn/ensemble/forest.py:245: FutureWarning: The default value of n_estimators will change from 10 in version 0.20 to 100 in 0.22.\n",
      "  \"10 in version 0.20 to 100 in 0.22.\", FutureWarning)\n"
     ]
    }
   ],
   "source": [
    "rfc = ensemble.RandomForestClassifier()\n",
    "cross_val_score(rfc, X_pca, Y, cv=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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