PCA.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "1a00c042-14d8-4922-866f-5ff8fe43dc27",
   "metadata": {},
   "source": [
    "# Data Mining II - Task II - Cory Wilson"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "c268486c",
   "metadata": {},
   "outputs": [],
   "source": [
    "### Initial imports and dataframe creation\n",
    "### Importing required packages\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from sklearn.decomposition import PCA\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "from sklearn.preprocessing import StandardScaler\n",
    "from yellowbrick.cluster import KElbowVisualizer\n",
    "from sklearn.pipeline import Pipeline\n",
    "pd.options.display.float_format = '{:.5f}'.format\n",
    "pd.options.display.max_columns = None\n",
    "\n",
    "### Creating initial data set from the Churn CSV\n",
    "data = pd.read_csv('churn_clean.csv')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83bea652-b199-465b-a59c-c608d36fe702",
   "metadata": {},
   "source": [
    "## Part I"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd55ec0f-06f8-4ac2-aa18-1fd54c8786a3",
   "metadata": {},
   "source": [
    "### A. 1."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ef9c437-9071-4d59-829e-16ad683e5772",
   "metadata": {},
   "source": [
    "Can features be reduced in such a way that algorithms run more efficiently and allow for more plotting options?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8875fce3-827c-46c1-a482-cc6e7188b831",
   "metadata": {},
   "source": [
    "### A. 2."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a5a00db4-61c5-4531-b20b-b0349cc1f576",
   "metadata": {},
   "source": [
    "The goal of this analysis is to perform dimensional reduction on the data set. This will allow algorithms to run more quickly and efficiently while also reducing the number of features in the data set. This will allow for better options with plotting as well."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebfc8d0a-de96-4499-b38e-deeffbed0bce",
   "metadata": {},
   "source": [
    "## Part II"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ade323a-d593-4a05-99c3-7ef42cb786b5",
   "metadata": {},
   "source": [
    "### B. 1."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ece97811",
   "metadata": {},
   "source": [
    "PCA starts by converting a data set into a matrix. Then, the process of finding the mean of that matrix occurs. A covariance matrix is created for the whole data set. This matrix is similar to what we see in section D. 2. Once the covariance matrix is formed the process of finding eigenvectors and eigenvalues beings. While the math behind finding eigenvectors and eigenvalues is somewhat complex, the process is there to simply find the variance expressed by each eigenvector. PCA through SkLearn returns a variety of information when run on a data set, the most important information returned are the principal components themselves. (Alto, 2019)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d33cf78e-be6e-47c7-b849-353bac686906",
   "metadata": {},
   "source": [
    "### B. 2."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9477abd7",
   "metadata": {},
   "source": [
    "One assumption that PCA makes about the data is that it must be linear in nature. This means that the features must have linear relationships amongst themselves. With this data set, previous analysis has shown that this is the case. (Vadapalli, 2020)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca3acbfe-2abe-403e-9619-173aa7f7ab9c",
   "metadata": {},
   "source": [
    "## Part III"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16e63f14-d182-4c8a-8005-13c574b5a92e",
   "metadata": {},
   "source": [
    "### C. 1."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c243a6b",
   "metadata": {},
   "source": [
    "Continuous variables that will be included:\n",
    "- Children \n",
    "- Age \n",
    "- Income \n",
    "- Outage_sec_perweek \n",
    "- Email \n",
    "- Contacts \n",
    "- Yearly_equip_failure \n",
    "- Tenure \n",
    "- MonthlyCharge \n",
    "- Bandwidth_GB_Year "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f2cdeeb-a0f0-40b0-a3fb-64f0f7838902",
   "metadata": {},
   "source": [
    "### C. 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1fb4b8c8-5892-4051-a2e5-f18bacdcf6b2",
   "metadata": {},
   "outputs": [],
   "source": [
    "### Dropping columns that aren't required for clustering\n",
    "clean_data = data.drop(columns=['CaseOrder','Customer_id',\n",
    "                                'Interaction','Churn','UID','Population',\n",
    "                                'Zip','Job','City','Lat','Lng','State',\n",
    "                                'County','TimeZone','item1','item2',\n",
    "                                'item3','item4','item5','item6','item7',\n",
    "                                'item8','Area','Marital','Gender',\n",
    "                                'Techie','Contract','Port_modem',\n",
    "                                'Tablet','InternetService','Phone',\n",
    "                                'Multiple','OnlineSecurity',\n",
    "                                'OnlineBackup','DeviceProtection',\n",
    "                                'TechSupport','StreamingTV',\n",
    "                                'StreamingMovies','PaperlessBilling',\n",
    "                                'PaymentMethod'])\n",
    "\n",
    "### Creating a scaler object and scaling the data to make it more uniformed\n",
    "scaler = StandardScaler()\n",
    "scaled_data = scaler.fit_transform(clean_data)\n",
    "\n",
    "### Saving a copy of the cleaned data set.\n",
    "clean_data.to_csv('D212 Task 2 Clean Data.csv')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60ee8daf-bebd-4e52-93c5-0d18e8737e6e",
   "metadata": {},
   "source": [
    "## Part IV"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80d969c8-86af-437a-a7e1-1f96dab01c0f",
   "metadata": {},
   "source": [
    "### D. 1."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aea87334",
   "metadata": {},
   "source": [
    "The code below presents a covariance matrix and heat plot of the matrix with all principle components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "bd364ac8-b8a1-41f3-b643-d5a7da454a83",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.00010001 -0.02973451  0.00994335  0.00191611  0.00447925 -0.02077811\n",
      "   0.00732132 -0.00509183 -0.00978238  0.02558738]\n",
      " [-0.02973451  1.00010001 -0.00409101 -0.00799559  0.00158808  0.01506913\n",
      "   0.00857821  0.01698097  0.01072958 -0.01472512]\n",
      " [ 0.00994335 -0.00409101  1.00010001 -0.00989    -0.00926842  0.00123332\n",
      "   0.00542382  0.00211458 -0.00301427  0.00367392]\n",
      " [ 0.00191611 -0.00799559 -0.00989     1.00010001  0.0040835   0.01516813\n",
      "   0.00298848  0.00288075  0.02044865  0.00412502]\n",
      " [ 0.00447925  0.00158808 -0.00926842  0.0040835   1.00010001  0.00304067\n",
      "  -0.01635598 -0.01446932  0.00199675 -0.01458061]\n",
      " [-0.02077811  0.01506913  0.00123332  0.01516813  0.00304067  1.00010001\n",
      "  -0.00603285  0.00282037  0.00425907  0.00329905]\n",
      " [ 0.00732132  0.00857821  0.00542382  0.00298848 -0.01635598 -0.00603285\n",
      "   1.00010001  0.01243615 -0.00717299  0.0120349 ]\n",
      " [-0.00509183  0.01698097  0.00211458  0.00288075 -0.01446932  0.00282037\n",
      "   0.01243615  1.00010001 -0.00333714  0.99159435]\n",
      " [-0.00978238  0.01072958 -0.00301427  0.02044865  0.00199675  0.00425907\n",
      "  -0.00717299 -0.00333714  1.00010001  0.06041247]\n",
      " [ 0.02558738 -0.01472512  0.00367392  0.00412502 -0.01458061  0.00329905\n",
      "   0.0120349   0.99159435  0.06041247  1.00010001]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x396 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "features = scaled_data.T\n",
    "matrix = np.cov(features)\n",
    "print(matrix)\n",
    "sns.heatmap(matrix)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a53402e-f9ff-4d44-bb25-80f1d447750c",
   "metadata": {},
   "source": [
    "### D. 2."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ad9b5763",
   "metadata": {},
   "source": [
    "The two code blocks below run the PCA on the scaled data and display an elbow plot. To be able to retain at least 80% of the variance, 7 principal components will work. Additionally, to retain at least 90% of the data 8 components should be used."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "30677007",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
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       "    .dataframe thead th {\n",
       "        text-align: right;\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
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       "      <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>0.01413</td>\n",
       "      <td>0.00171</td>\n",
       "      <td>0.00436</td>\n",
       "      <td>0.00581</td>\n",
       "      <td>-0.02078</td>\n",
       "      <td>0.00417</td>\n",
       "      <td>0.01757</td>\n",
       "      <td>0.70542</td>\n",
       "      <td>0.04042</td>\n",
       "      <td>0.70692</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.55945</td>\n",
       "      <td>0.48012</td>\n",
       "      <td>-0.22341</td>\n",
       "      <td>0.21212</td>\n",
       "      <td>0.10722</td>\n",
       "      <td>0.45907</td>\n",
       "      <td>-0.14319</td>\n",
       "      <td>0.00188</td>\n",
       "      <td>0.34465</td>\n",
       "      <td>-0.00792</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.28548</td>\n",
       "      <td>0.42134</td>\n",
       "      <td>0.26603</td>\n",
       "      <td>-0.47857</td>\n",
       "      <td>-0.44067</td>\n",
       "      <td>0.01254</td>\n",
       "      <td>0.39556</td>\n",
       "      <td>0.02099</td>\n",
       "      <td>-0.29922</td>\n",
       "      <td>-0.01971</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.14309</td>\n",
       "      <td>-0.09000</td>\n",
       "      <td>0.16996</td>\n",
       "      <td>0.57983</td>\n",
       "      <td>-0.45064</td>\n",
       "      <td>0.10750</td>\n",
       "      <td>0.53096</td>\n",
       "      <td>-0.04167</td>\n",
       "      <td>0.32848</td>\n",
       "      <td>-0.01273</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.03058</td>\n",
       "      <td>-0.15891</td>\n",
       "      <td>0.78651</td>\n",
       "      <td>-0.02731</td>\n",
       "      <td>-0.00359</td>\n",
       "      <td>0.46564</td>\n",
       "      <td>-0.37010</td>\n",
       "      <td>-0.00458</td>\n",
       "      <td>0.02493</td>\n",
       "      <td>0.00463</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.05898</td>\n",
       "      <td>0.12062</td>\n",
       "      <td>0.21063</td>\n",
       "      <td>-0.24240</td>\n",
       "      <td>-0.16014</td>\n",
       "      <td>-0.54790</td>\n",
       "      <td>-0.23092</td>\n",
       "      <td>-0.03721</td>\n",
       "      <td>0.70590</td>\n",
       "      <td>0.00262</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.28529</td>\n",
       "      <td>0.40773</td>\n",
       "      <td>0.29769</td>\n",
       "      <td>-0.00486</td>\n",
       "      <td>0.68576</td>\n",
       "      <td>-0.04940</td>\n",
       "      <td>0.42071</td>\n",
       "      <td>-0.00445</td>\n",
       "      <td>0.11830</td>\n",
       "      <td>-0.00082</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.64723</td>\n",
       "      <td>0.20822</td>\n",
       "      <td>-0.30150</td>\n",
       "      <td>-0.36740</td>\n",
       "      <td>-0.22794</td>\n",
       "      <td>0.43944</td>\n",
       "      <td>-0.07758</td>\n",
       "      <td>-0.02974</td>\n",
       "      <td>0.24472</td>\n",
       "      <td>-0.00025</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-0.28226</td>\n",
       "      <td>-0.57788</td>\n",
       "      <td>-0.08906</td>\n",
       "      <td>-0.44186</td>\n",
       "      <td>0.20719</td>\n",
       "      <td>0.25487</td>\n",
       "      <td>0.40925</td>\n",
       "      <td>-0.02222</td>\n",
       "      <td>0.32751</td>\n",
       "      <td>0.00908</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-0.02159</td>\n",
       "      <td>0.02237</td>\n",
       "      <td>-0.00094</td>\n",
       "      <td>0.00028</td>\n",
       "      <td>0.00025</td>\n",
       "      <td>-0.00094</td>\n",
       "      <td>-0.00010</td>\n",
       "      <td>-0.70526</td>\n",
       "      <td>-0.04575</td>\n",
       "      <td>0.70678</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         0        1        2        3        4        5        6        7  \\\n",
       "0  0.01413  0.00171  0.00436  0.00581 -0.02078  0.00417  0.01757  0.70542   \n",
       "1 -0.55945  0.48012 -0.22341  0.21212  0.10722  0.45907 -0.14319  0.00188   \n",
       "2 -0.28548  0.42134  0.26603 -0.47857 -0.44067  0.01254  0.39556  0.02099   \n",
       "3  0.14309 -0.09000  0.16996  0.57983 -0.45064  0.10750  0.53096 -0.04167   \n",
       "4  0.03058 -0.15891  0.78651 -0.02731 -0.00359  0.46564 -0.37010 -0.00458   \n",
       "5 -0.05898  0.12062  0.21063 -0.24240 -0.16014 -0.54790 -0.23092 -0.03721   \n",
       "6  0.28529  0.40773  0.29769 -0.00486  0.68576 -0.04940  0.42071 -0.00445   \n",
       "7  0.64723  0.20822 -0.30150 -0.36740 -0.22794  0.43944 -0.07758 -0.02974   \n",
       "8 -0.28226 -0.57788 -0.08906 -0.44186  0.20719  0.25487  0.40925 -0.02222   \n",
       "9 -0.02159  0.02237 -0.00094  0.00028  0.00025 -0.00094 -0.00010 -0.70526   \n",
       "\n",
       "         8        9  \n",
       "0  0.04042  0.70692  \n",
       "1  0.34465 -0.00792  \n",
       "2 -0.29922 -0.01971  \n",
       "3  0.32848 -0.01273  \n",
       "4  0.02493  0.00463  \n",
       "5  0.70590  0.00262  \n",
       "6  0.11830 -0.00082  \n",
       "7  0.24472 -0.00025  \n",
       "8  0.32751  0.00908  \n",
       "9 -0.04575  0.70678  "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca = PCA()\n",
    "ran_pca = pca.fit(scaled_data)\n",
    "\n",
    "pca_df = pd.DataFrame(ran_pca.components_)\n",
    "pca_df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "35546c85",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the explained variance ratio\n",
    "plt.plot(ran_pca.explained_variance_ratio_)\n",
    "\n",
    "plt.xlabel('Principal component index')\n",
    "plt.ylabel('Explained variance ratio')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57a719e0",
   "metadata": {},
   "source": [
    "### D. 3."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7de13918",
   "metadata": {},
   "source": [
    "The array below shows the variance of each individual component. The first component explains 0.1994 or 19.94% of the variance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "a4468db7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.19941328, 0.10532287, 0.10273989, 0.10124756, 0.09996907,\n",
       "       0.09937176, 0.09889587, 0.09646805, 0.09602493, 0.00054672])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ran_pca.explained_variance_ratio_"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bf53880",
   "metadata": {},
   "source": [
    "### D. 4."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "049a9ce5",
   "metadata": {},
   "source": [
    "The array below shows the cumulative sum of all principal components. This array starts with PC1 and adds each additional PC. These 8 components explain .903 or 90.3% of the variance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "c283b4ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.19941328, 0.30473615, 0.40747604, 0.5087236 , 0.60869267,\n",
       "       0.70806443, 0.8069603 , 0.90342836, 0.99945328, 1.        ])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ran_pca.explained_variance_ratio_.cumsum()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d06e55f3",
   "metadata": {},
   "source": [
    "### D. 5."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "169e81d1",
   "metadata": {},
   "source": [
    "The main takeaway from this analysis is that number of features can be slightly reduced while still retaining a large amount of the variance. One item of note is that only continuous variables were used, leading me to question if some of the categorical variables in addition to these would allow for even more of the variance to be retained with a smaller number of PCs."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a4cbf24-8074-41cd-89dd-e5a9ab71545f",
   "metadata": {},
   "source": [
    "## Part V"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9eb24857-016c-4ed3-a5a8-322405566047",
   "metadata": {},
   "source": [
    "### E. 1."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12df9ea4",
   "metadata": {},
   "source": [
    "## References\n",
    "Vadapalli, P. (2020, November 12). PCA in Machine Learning: Assumptions, steps to apply &amp; applications. upGrad blog. Retrieved October 13, 2021, from https://www.upgrad.com/blog/pca-in-machine-learning/#Assumptions_in_PCA.\n",
    "    \n",
    "Alto, V. (2019, July 13). PCA: Eigenvectors and eigenvalues. Medium. Retrieved October 18, 2021, from https://towardsdatascience.com/pca-eigenvectors-and-eigenvalues-1f968bc6777a. "
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
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  "language_info": {
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    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
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   "nbconvert_exporter": "python",
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