Principal Component Analysis (PCA) on Stock Returns in R.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Intro"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Principal Component Analysis is a statistical process that distills measurement variation into vectors with greater ability to predict outcomes utilizing a process of scaling, covariance, and eigendecomposition.\n",
    "\n",
    "An outline of the following page:\n",
    "\n",
    "* Supporting Material\n",
    " - Pluralsight\n",
    " - Explained Visually\n",
    " - Wikipedia\n",
    "* Load Data: Format Data & Sort\n",
    "* Prep Data: Create Returns\n",
    "* Generate Principal Components\n",
    " - Eigen Decomposition and Scree Plot\n",
    " - Create Principal Components\n",
    "* Analysis\n",
    " - FVX using PCA\n",
    " - FVX using Logisitic Regression\n",
    "* Alternative Libraries: Psych for the Social Sciences"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Supporting Material"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following two (2) subsections provide links to bette runderstand the process."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plurasight"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The work is part of a training course from Pluralsight, Understanding and Applying Factor Analysis and PCA:\n",
    "\n",
    "\n",
    "* https://www.pluralsight.com/courses/understanding-applying-factor-analysis-pca"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Explained Visually"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The site Explained Visally has some great visually-oriented descriptions, below:\n",
    "\n",
    "\n",
    "* Principal Component Analysis: http://setosa.io/ev/principal-component-analysis/\n",
    "\n",
    "* Eigenvectors and Eigenvalues: http://setosa.io/ev/eigenvectors-and-eigenvalues/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Wikipedia"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Principal Component Analysis : https://en.wikipedia.org/wiki/Principal_component_analysis\n",
    "\n",
    "* Eigen Vectors: https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors\n",
    "\n",
    "* Eigendecomposition: https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Load Data: Format Data & Sort"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To prep the data, it is loaded, the date formatted, and sorted."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>AsOfDate</th><th scope=col>AAPL</th><th scope=col>ADBE</th><th scope=col>CVX</th><th scope=col>GOOG</th><th scope=col>IBM</th><th scope=col>MDLZ</th><th scope=col>MSFT</th><th scope=col>NFLX</th><th scope=col>ORCL</th><th scope=col>SBUX</th><th scope=col>FVX</th><th scope=col>SP500</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>121</th><td>2007-01-03</td><td>11.10714  </td><td>38.87     </td><td>50.77735  </td><td>251.0010  </td><td>79.24250  </td><td>17.51952  </td><td>24.11848  </td><td>3.258571  </td><td>15.69632  </td><td>15.75219  </td><td>4.819     </td><td>1438.24   </td></tr>\n",
       "\t<tr><th scope=row>120</th><td>2007-02-01</td><td>10.96203  </td><td>39.25     </td><td>48.08294  </td><td>224.9500  </td><td>74.50320  </td><td>16.01943  </td><td>22.09246  </td><td>3.218571  </td><td>15.02859  </td><td>13.93081  </td><td>4.503     </td><td>1406.82   </td></tr>\n",
       "\t<tr><th scope=row>119</th><td>2007-03-01</td><td>12.03738  </td><td>41.70     </td><td>51.90038  </td><td>229.3093  </td><td>75.56135  </td><td>16.00935  </td><td>21.85719  </td><td>3.312857  </td><td>16.58358  </td><td>14.13820  </td><td>4.537     </td><td>1420.86   </td></tr>\n",
       "\t<tr><th scope=row>118</th><td>2007-04-02</td><td>12.93004  </td><td>41.56     </td><td>54.58803  </td><td>235.9259  </td><td>81.93428  </td><td>16.92461  </td><td>23.48060  </td><td>3.167143  </td><td>17.19644  </td><td>13.98491  </td><td>4.516     </td><td>1482.37   </td></tr>\n",
       "\t<tr><th scope=row>117</th><td>2007-05-01</td><td>15.70132  </td><td>44.06     </td><td>57.59827  </td><td>249.2042  </td><td>85.78606  </td><td>17.11170  </td><td>24.14675  </td><td>3.128572  </td><td>17.72696  </td><td>12.98857  </td><td>4.853     </td><td>1530.62   </td></tr>\n",
       "\t<tr><th scope=row>116</th><td>2007-06-01</td><td>15.81145  </td><td>40.15     </td><td>59.54200  </td><td>261.6116  </td><td>84.69965  </td><td>17.95071  </td><td>23.18686  </td><td>2.770000  </td><td>18.02882  </td><td>11.82992  </td><td>4.936     </td><td>1503.35   </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllllllllllll}\n",
       "  & AsOfDate & AAPL & ADBE & CVX & GOOG & IBM & MDLZ & MSFT & NFLX & ORCL & SBUX & FVX & SP500\\\\\n",
       "\\hline\n",
       "\t121 & 2007-01-03 & 11.10714   & 38.87      & 50.77735   & 251.0010   & 79.24250   & 17.51952   & 24.11848   & 3.258571   & 15.69632   & 15.75219   & 4.819      & 1438.24   \\\\\n",
       "\t120 & 2007-02-01 & 10.96203   & 39.25      & 48.08294   & 224.9500   & 74.50320   & 16.01943   & 22.09246   & 3.218571   & 15.02859   & 13.93081   & 4.503      & 1406.82   \\\\\n",
       "\t119 & 2007-03-01 & 12.03738   & 41.70      & 51.90038   & 229.3093   & 75.56135   & 16.00935   & 21.85719   & 3.312857   & 16.58358   & 14.13820   & 4.537      & 1420.86   \\\\\n",
       "\t118 & 2007-04-02 & 12.93004   & 41.56      & 54.58803   & 235.9259   & 81.93428   & 16.92461   & 23.48060   & 3.167143   & 17.19644   & 13.98491   & 4.516      & 1482.37   \\\\\n",
       "\t117 & 2007-05-01 & 15.70132   & 44.06      & 57.59827   & 249.2042   & 85.78606   & 17.11170   & 24.14675   & 3.128572   & 17.72696   & 12.98857   & 4.853      & 1530.62   \\\\\n",
       "\t116 & 2007-06-01 & 15.81145   & 40.15      & 59.54200   & 261.6116   & 84.69965   & 17.95071   & 23.18686   & 2.770000   & 18.02882   & 11.82992   & 4.936      & 1503.35   \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | AsOfDate | AAPL | ADBE | CVX | GOOG | IBM | MDLZ | MSFT | NFLX | ORCL | SBUX | FVX | SP500 | \n",
       "|---|---|---|---|---|---|\n",
       "| 121 | 2007-01-03 | 11.10714   | 38.87      | 50.77735   | 251.0010   | 79.24250   | 17.51952   | 24.11848   | 3.258571   | 15.69632   | 15.75219   | 4.819      | 1438.24    | \n",
       "| 120 | 2007-02-01 | 10.96203   | 39.25      | 48.08294   | 224.9500   | 74.50320   | 16.01943   | 22.09246   | 3.218571   | 15.02859   | 13.93081   | 4.503      | 1406.82    | \n",
       "| 119 | 2007-03-01 | 12.03738   | 41.70      | 51.90038   | 229.3093   | 75.56135   | 16.00935   | 21.85719   | 3.312857   | 16.58358   | 14.13820   | 4.537      | 1420.86    | \n",
       "| 118 | 2007-04-02 | 12.93004   | 41.56      | 54.58803   | 235.9259   | 81.93428   | 16.92461   | 23.48060   | 3.167143   | 17.19644   | 13.98491   | 4.516      | 1482.37    | \n",
       "| 117 | 2007-05-01 | 15.70132   | 44.06      | 57.59827   | 249.2042   | 85.78606   | 17.11170   | 24.14675   | 3.128572   | 17.72696   | 12.98857   | 4.853      | 1530.62    | \n",
       "| 116 | 2007-06-01 | 15.81145   | 40.15      | 59.54200   | 261.6116   | 84.69965   | 17.95071   | 23.18686   | 2.770000   | 18.02882   | 11.82992   | 4.936      | 1503.35    | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "    AsOfDate   AAPL     ADBE  CVX      GOOG     IBM      MDLZ     MSFT    \n",
       "121 2007-01-03 11.10714 38.87 50.77735 251.0010 79.24250 17.51952 24.11848\n",
       "120 2007-02-01 10.96203 39.25 48.08294 224.9500 74.50320 16.01943 22.09246\n",
       "119 2007-03-01 12.03738 41.70 51.90038 229.3093 75.56135 16.00935 21.85719\n",
       "118 2007-04-02 12.93004 41.56 54.58803 235.9259 81.93428 16.92461 23.48060\n",
       "117 2007-05-01 15.70132 44.06 57.59827 249.2042 85.78606 17.11170 24.14675\n",
       "116 2007-06-01 15.81145 40.15 59.54200 261.6116 84.69965 17.95071 23.18686\n",
       "    NFLX     ORCL     SBUX     FVX   SP500  \n",
       "121 3.258571 15.69632 15.75219 4.819 1438.24\n",
       "120 3.218571 15.02859 13.93081 4.503 1406.82\n",
       "119 3.312857 16.58358 14.13820 4.537 1420.86\n",
       "118 3.167143 17.19644 13.98491 4.516 1482.37\n",
       "117 3.128572 17.72696 12.98857 4.853 1530.62\n",
       "116 2.770000 18.02882 11.82992 4.936 1503.35"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# load data\n",
    "filePath <- \"Portfolios.csv\"\n",
    "portfolios.dirty <- read.table(filePath, header = TRUE, sep = \",\", stringsAsFactors = FALSE)\n",
    "\n",
    "# convert date\n",
    "# http://www.statmethods.net/input/dates.html\n",
    "portfolios.dirty[, c(\"AsOfDate\")] <- as.Date(portfolios.dirty[, c(\"AsOfDate\")], \"%m/%d/%Y\")\n",
    "\n",
    "# sort: \n",
    "# http://www.statmethods.net/management/sorting.html\n",
    "portfolios.sorted <- portfolios.dirty[order(portfolios.dirty$AsOfDate, decreasing = FALSE),]\n",
    "head(portfolios.sorted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Prep Data: Create Returns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data needs to be based on returns, the change in price and not just price, which also mean that the first row must be deleted, since it has no relative change from a prior row."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "vscode": {
     "languageId": "r"
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>AAPL</th><th scope=col>ADBE</th><th scope=col>CVX</th><th scope=col>GOOG</th><th scope=col>IBM</th><th scope=col>MDLZ</th><th scope=col>MSFT</th><th scope=col>NFLX</th><th scope=col>ORCL</th><th scope=col>SBUX</th><th scope=col>FVX</th><th scope=col>SP500</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>121</th><td> 0.013237326 </td><td>-0.009681554 </td><td> 0.05603676  </td><td> 0.11580823  </td><td> 0.06361198  </td><td> 0.0936424314</td><td> 0.09170634  </td><td> 0.01242788  </td><td> 0.04443085  </td><td> 0.13074434  </td><td> 0.070175439 </td><td> 0.022334091 </td></tr>\n",
       "\t<tr><th scope=row>120</th><td>-0.089333748 </td><td>-0.058753020 </td><td>-0.07355329  </td><td>-0.01901083  </td><td>-0.01400377  </td><td> 0.0006291322</td><td> 0.01076419  </td><td>-0.02846063  </td><td>-0.09376719  </td><td>-0.01466842  </td><td>-0.007493939 </td><td>-0.009881367 </td></tr>\n",
       "\t<tr><th scope=row>119</th><td>-0.069038131 </td><td> 0.003368624 </td><td>-0.04923513  </td><td>-0.02804528  </td><td>-0.07778102  </td><td>-0.0540782983</td><td>-0.06913828  </td><td> 0.04600803  </td><td>-0.03563831  </td><td> 0.01096067  </td><td> 0.004650133 </td><td>-0.041494371 </td></tr>\n",
       "\t<tr><th scope=row>118</th><td>-0.176499724 </td><td>-0.056740807 </td><td>-0.05226260  </td><td>-0.05328276  </td><td>-0.04489980  </td><td>-0.0109338030</td><td>-0.02758781  </td><td> 0.01232863  </td><td>-0.02992780  </td><td> 0.07670954  </td><td>-0.069441583 </td><td>-0.031523174 </td></tr>\n",
       "\t<tr><th scope=row>117</th><td>-0.006964953 </td><td> 0.097384777 </td><td>-0.03264480  </td><td>-0.04742678  </td><td> 0.01282663  </td><td>-0.0467396450</td><td> 0.04139801  </td><td> 0.12944838  </td><td>-0.01674270  </td><td> 0.09794208  </td><td>-0.016815235 </td><td> 0.018139501 </td></tr>\n",
       "\t<tr><th scope=row>116</th><td>-0.073770445 </td><td>-0.003474783 </td><td>-0.01196348  </td><td> 0.02490195  </td><td>-0.04880252  </td><td> 0.0763359886</td><td> 0.01655740  </td><td> 0.12536254  </td><td> 0.03085760  </td><td>-0.01649179  </td><td> 0.072577140 </td><td> 0.033038512 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllllllll}\n",
       "  & AAPL & ADBE & CVX & GOOG & IBM & MDLZ & MSFT & NFLX & ORCL & SBUX & FVX & SP500\\\\\n",
       "\\hline\n",
       "\t121 &  0.013237326  & -0.009681554  &  0.05603676   &  0.11580823   &  0.06361198   &  0.0936424314 &  0.09170634   &  0.01242788   &  0.04443085   &  0.13074434   &  0.070175439  &  0.022334091 \\\\\n",
       "\t120 & -0.089333748  & -0.058753020  & -0.07355329   & -0.01901083   & -0.01400377   &  0.0006291322 &  0.01076419   & -0.02846063   & -0.09376719   & -0.01466842   & -0.007493939  & -0.009881367 \\\\\n",
       "\t119 & -0.069038131  &  0.003368624  & -0.04923513   & -0.02804528   & -0.07778102   & -0.0540782983 & -0.06913828   &  0.04600803   & -0.03563831   &  0.01096067   &  0.004650133  & -0.041494371 \\\\\n",
       "\t118 & -0.176499724  & -0.056740807  & -0.05226260   & -0.05328276   & -0.04489980   & -0.0109338030 & -0.02758781   &  0.01232863   & -0.02992780   &  0.07670954   & -0.069441583  & -0.031523174 \\\\\n",
       "\t117 & -0.006964953  &  0.097384777  & -0.03264480   & -0.04742678   &  0.01282663   & -0.0467396450 &  0.04139801   &  0.12944838   & -0.01674270   &  0.09794208   & -0.016815235  &  0.018139501 \\\\\n",
       "\t116 & -0.073770445  & -0.003474783  & -0.01196348   &  0.02490195   & -0.04880252   &  0.0763359886 &  0.01655740   &  0.12536254   &  0.03085760   & -0.01649179   &  0.072577140  &  0.033038512 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | AAPL | ADBE | CVX | GOOG | IBM | MDLZ | MSFT | NFLX | ORCL | SBUX | FVX | SP500 | \n",
       "|---|---|---|---|---|---|\n",
       "| 121 |  0.013237326  | -0.009681554  |  0.05603676   |  0.11580823   |  0.06361198   |  0.0936424314 |  0.09170634   |  0.01242788   |  0.04443085   |  0.13074434   |  0.070175439  |  0.022334091  | \n",
       "| 120 | -0.089333748  | -0.058753020  | -0.07355329   | -0.01901083   | -0.01400377   |  0.0006291322 |  0.01076419   | -0.02846063   | -0.09376719   | -0.01466842   | -0.007493939  | -0.009881367  | \n",
       "| 119 | -0.069038131  |  0.003368624  | -0.04923513   | -0.02804528   | -0.07778102   | -0.0540782983 | -0.06913828   |  0.04600803   | -0.03563831   |  0.01096067   |  0.004650133  | -0.041494371  | \n",
       "| 118 | -0.176499724  | -0.056740807  | -0.05226260   | -0.05328276   | -0.04489980   | -0.0109338030 | -0.02758781   |  0.01232863   | -0.02992780   |  0.07670954   | -0.069441583  | -0.031523174  | \n",
       "| 117 | -0.006964953  |  0.097384777  | -0.03264480   | -0.04742678   |  0.01282663   | -0.0467396450 |  0.04139801   |  0.12944838   | -0.01674270   |  0.09794208   | -0.016815235  |  0.018139501  | \n",
       "| 116 | -0.073770445  | -0.003474783  | -0.01196348   |  0.02490195   | -0.04880252   |  0.0763359886 |  0.01655740   |  0.12536254   |  0.03085760   | -0.01649179   |  0.072577140  |  0.033038512  | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "    AAPL         ADBE         CVX         GOOG        IBM         MDLZ         \n",
       "121  0.013237326 -0.009681554  0.05603676  0.11580823  0.06361198  0.0936424314\n",
       "120 -0.089333748 -0.058753020 -0.07355329 -0.01901083 -0.01400377  0.0006291322\n",
       "119 -0.069038131  0.003368624 -0.04923513 -0.02804528 -0.07778102 -0.0540782983\n",
       "118 -0.176499724 -0.056740807 -0.05226260 -0.05328276 -0.04489980 -0.0109338030\n",
       "117 -0.006964953  0.097384777 -0.03264480 -0.04742678  0.01282663 -0.0467396450\n",
       "116 -0.073770445 -0.003474783 -0.01196348  0.02490195 -0.04880252  0.0763359886\n",
       "    MSFT        NFLX        ORCL        SBUX        FVX          SP500       \n",
       "121  0.09170634  0.01242788  0.04443085  0.13074434  0.070175439  0.022334091\n",
       "120  0.01076419 -0.02846063 -0.09376719 -0.01466842 -0.007493939 -0.009881367\n",
       "119 -0.06913828  0.04600803 -0.03563831  0.01096067  0.004650133 -0.041494371\n",
       "118 -0.02758781  0.01232863 -0.02992780  0.07670954 -0.069441583 -0.031523174\n",
       "117  0.04139801  0.12944838 -0.01674270  0.09794208 -0.016815235  0.018139501\n",
       "116  0.01655740  0.12536254  0.03085760 -0.01649179  0.072577140  0.033038512"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# new data, preserving original\n",
    "returns <- portfolios.sorted\n",
    "\n",
    "# calculate returns\n",
    "returns[-nrow(returns), -1] <- returns[-nrow(returns), -1] / returns[-1, -1] - 1\n",
    "\n",
    "# removes last row, since it is not a valid return value\n",
    "returns <- returns[-nrow(returns),]\n",
    "\n",
    "# create a data frame of stock names, not the date and not the factors\n",
    "stockNames = names(returns)[-which(names(returns) %in% c('AsOfDate'))]\n",
    "\n",
    "# create a data frame of only stocks\n",
    "stockReturns <- returns[stockNames]\n",
    "head(stockReturns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Generate Principal Components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To generate principal components:\n",
    "\n",
    "- The data is scaled\n",
    "- A covariance matrix generated\n",
    "- An Eigenecomposition performed\n",
    "- A scree plot is generated to evalaute the strength of the eigen values\n",
    "- Eigen vectors are matrix multiplied by the scaled vectors\n",
    "\n",
    "Note: Although we generated a scree plot, one can simply by evaluate the number themselves. In this case the scree plot shows that the first principal component explains over 50 percent of the variance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eigen Decomposition and Scree Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "stockReturns.scaled = scale(stockReturns)\n",
    "\n",
    "stockReturns.cov <- cov(stockReturns.scaled)\n",
    "\n",
    "stockReturns.eigen <- eigen(stockReturns.cov)\n",
    "\n",
    "eigenValues <- stockReturns.eigen$values\n",
    "eigenVectors <- stockReturns.eigen$vectors\n",
    "\n",
    "# scree plot\n",
    "plot(eigenValues, type = \"b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create Principal Components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>pcaOne</th><th scope=col>pcaTwo</th><th scope=col>pcaThree</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>121</th><td>-3.0017131 </td><td>-0.1933158 </td><td>-0.38658150</td></tr>\n",
       "\t<tr><th scope=row>120</th><td> 1.2911600 </td><td>-0.5510177 </td><td> 0.31662153</td></tr>\n",
       "\t<tr><th scope=row>119</th><td> 1.6752924 </td><td>-0.4436764 </td><td> 0.77358849</td></tr>\n",
       "\t<tr><th scope=row>118</th><td> 1.5729454 </td><td>-0.9424242 </td><td> 0.09784403</td></tr>\n",
       "\t<tr><th scope=row>117</th><td>-0.8051984 </td><td>-1.1248464 </td><td> 0.90201542</td></tr>\n",
       "\t<tr><th scope=row>116</th><td>-0.7377155 </td><td>-0.8601747 </td><td>-0.53989377</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & pcaOne & pcaTwo & pcaThree\\\\\n",
       "\\hline\n",
       "\t121 & -3.0017131  & -0.1933158  & -0.38658150\\\\\n",
       "\t120 &  1.2911600  & -0.5510177  &  0.31662153\\\\\n",
       "\t119 &  1.6752924  & -0.4436764  &  0.77358849\\\\\n",
       "\t118 &  1.5729454  & -0.9424242  &  0.09784403\\\\\n",
       "\t117 & -0.8051984  & -1.1248464  &  0.90201542\\\\\n",
       "\t116 & -0.7377155  & -0.8601747  & -0.53989377\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | pcaOne | pcaTwo | pcaThree | \n",
       "|---|---|---|---|---|---|\n",
       "| 121 | -3.0017131  | -0.1933158  | -0.38658150 | \n",
       "| 120 |  1.2911600  | -0.5510177  |  0.31662153 | \n",
       "| 119 |  1.6752924  | -0.4436764  |  0.77358849 | \n",
       "| 118 |  1.5729454  | -0.9424242  |  0.09784403 | \n",
       "| 117 | -0.8051984  | -1.1248464  |  0.90201542 | \n",
       "| 116 | -0.7377155  | -0.8601747  | -0.53989377 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "    pcaOne     pcaTwo     pcaThree   \n",
       "121 -3.0017131 -0.1933158 -0.38658150\n",
       "120  1.2911600 -0.5510177  0.31662153\n",
       "119  1.6752924 -0.4436764  0.77358849\n",
       "118  1.5729454 -0.9424242  0.09784403\n",
       "117 -0.8051984 -1.1248464  0.90201542\n",
       "116 -0.7377155 -0.8601747 -0.53989377"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>pcaOne</th><th scope=col>pcaTwo</th><th scope=col>pcaThree</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>7</th><td> 0.2160844  </td><td> 0.007583937</td><td>-0.4111759  </td></tr>\n",
       "\t<tr><th scope=row>6</th><td>-0.1499740  </td><td>-0.248947616</td><td>-0.1778504  </td></tr>\n",
       "\t<tr><th scope=row>5</th><td>-0.1333889  </td><td> 0.448253174</td><td>-0.6535593  </td></tr>\n",
       "\t<tr><th scope=row>4</th><td> 0.7875921  </td><td>-1.504781568</td><td>-1.4636208  </td></tr>\n",
       "\t<tr><th scope=row>3</th><td> 0.6493136  </td><td> 0.072674219</td><td> 0.6505752  </td></tr>\n",
       "\t<tr><th scope=row>2</th><td> 1.1604751  </td><td> 0.654635664</td><td>-0.7123009  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & pcaOne & pcaTwo & pcaThree\\\\\n",
       "\\hline\n",
       "\t7 &  0.2160844   &  0.007583937 & -0.4111759  \\\\\n",
       "\t6 & -0.1499740   & -0.248947616 & -0.1778504  \\\\\n",
       "\t5 & -0.1333889   &  0.448253174 & -0.6535593  \\\\\n",
       "\t4 &  0.7875921   & -1.504781568 & -1.4636208  \\\\\n",
       "\t3 &  0.6493136   &  0.072674219 &  0.6505752  \\\\\n",
       "\t2 &  1.1604751   &  0.654635664 & -0.7123009  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | pcaOne | pcaTwo | pcaThree | \n",
       "|---|---|---|---|---|---|\n",
       "| 7 |  0.2160844   |  0.007583937 | -0.4111759   | \n",
       "| 6 | -0.1499740   | -0.248947616 | -0.1778504   | \n",
       "| 5 | -0.1333889   |  0.448253174 | -0.6535593   | \n",
       "| 4 |  0.7875921   | -1.504781568 | -1.4636208   | \n",
       "| 3 |  0.6493136   |  0.072674219 |  0.6505752   | \n",
       "| 2 |  1.1604751   |  0.654635664 | -0.7123009   | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  pcaOne     pcaTwo       pcaThree  \n",
       "7  0.2160844  0.007583937 -0.4111759\n",
       "6 -0.1499740 -0.248947616 -0.1778504\n",
       "5 -0.1333889  0.448253174 -0.6535593\n",
       "4  0.7875921 -1.504781568 -1.4636208\n",
       "3  0.6493136  0.072674219  0.6505752\n",
       "2  1.1604751  0.654635664 -0.7123009"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# check on correctness\n",
    "#eigenVetors %*% t(eigenVetors)\n",
    "\n",
    "# create principal component analysis\n",
    "pcaOne <- as.matrix(stockReturns.scaled) %*% eigenVectors[,1]\n",
    "pcaTwo <- as.matrix(stockReturns.scaled) %*% eigenVectors[,2]\n",
    "pcaThree <- as.matrix(stockReturns.scaled) %*% eigenVectors[,3]\n",
    "\n",
    "pca <- data.frame(pcaOne, pcaTwo, pcaThree)\n",
    "\n",
    "# examine result\n",
    "head(pca)\n",
    "tail(pca)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To highlight the power of PCA, I've created two (2) comparisons, predicting the FVX (http://www.marketwatch.com/investing/index/fvx?countrycode=xx) from PCA versus using logistic regression against all stocks, the latter to show if there are any correlations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## FVX using PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = stockReturns$FVX ~ pcaOne + pcaTwo + pcaThree)\n",
       "\n",
       "Residuals:\n",
       "      Min        1Q    Median        3Q       Max \n",
       "-0.228370 -0.070722  0.002269  0.059415  0.286974 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)  0.018420   0.009275   1.986   0.0494 *  \n",
       "pcaOne      -0.027117   0.004028  -6.732 6.72e-10 ***\n",
       "pcaTwo       0.045286   0.009129   4.961 2.43e-06 ***\n",
       "pcaThree     0.079448   0.009444   8.413 1.19e-13 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 0.1016 on 116 degrees of freedom\n",
       "Multiple R-squared:  0.5481,\tAdjusted R-squared:  0.5364 \n",
       "F-statistic:  46.9 on 3 and 116 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pcaFVX <- lm(stockReturns$FVX ~ pcaOne + pcaTwo + pcaThree)\n",
    "summary(pcaFVX)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## FVX using Logisitic Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = FVX ~ SP500 + GOOG + AAPL + ADBE + CVX + IBM + MDLZ + \n",
       "    MSFT + NFLX + ORCL + SBUX, data = stockReturns)\n",
       "\n",
       "Residuals:\n",
       "     Min       1Q   Median       3Q      Max \n",
       "-0.33511 -0.07983 -0.00106  0.08197  0.32382 \n",
       "\n",
       "Coefficients:\n",
       "             Estimate Std. Error t value Pr(>|t|)  \n",
       "(Intercept)  0.023793   0.013339   1.784   0.0773 .\n",
       "SP500       -0.691827   0.719053  -0.962   0.3381  \n",
       "GOOG         0.312631   0.219104   1.427   0.1565  \n",
       "AAPL        -0.134176   0.170626  -0.786   0.4334  \n",
       "ADBE         0.273048   0.238713   1.144   0.2552  \n",
       "CVX          0.330912   0.304777   1.086   0.2800  \n",
       "IBM          0.450707   0.303746   1.484   0.1408  \n",
       "MDLZ        -0.238541   0.282783  -0.844   0.4008  \n",
       "MSFT         0.186383   0.262271   0.711   0.4788  \n",
       "NFLX        -0.001722   0.077568  -0.022   0.9823  \n",
       "ORCL         0.256440   0.282833   0.907   0.3666  \n",
       "SBUX         0.152309   0.203625   0.748   0.4561  \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 0.1422 on 108 degrees of freedom\n",
       "Multiple R-squared:  0.1763,\tAdjusted R-squared:  0.09236 \n",
       "F-statistic: 2.101 on 11 and 108 DF,  p-value: 0.02604\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pcaCompare <- lm(FVX ~ SP500 + GOOG + AAPL + ADBE + CVX + IBM + MDLZ + MSFT + NFLX + ORCL + SBUX, data=stockReturns)\n",
    "summary(pcaCompare)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Alternative Libraries: Psych for the Social Sciences"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Personality Project (http://personality-project.org/) is most popularly known for its work on the Big Five (https://en.wikipedia.org/wiki/Big_Five_personality_traits), aka OCEAN, personality inventory. As part of its work, they have developed and provide an R package, psych. The package's page is at http://personality-project.org/r/psych/, while the R documentation is at https://www.rdocumentation.org/packages/psych."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Installing package into '/home/nbcommon/R'\n",
      "(as 'lib' is unspecified)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Principal Components Analysis\n",
       "Call: principal(r = stockReturns.cov, nfactors = 3, rotate = \"none\")\n",
       "Standardized loadings (pattern matrix) based upon correlation matrix\n",
       "       PC1   PC2   PC3   h2   u2 com\n",
       "AAPL  0.67 -0.15  0.11 0.48 0.52 1.2\n",
       "ADBE  0.80  0.14 -0.03 0.67 0.33 1.1\n",
       "CVX   0.64 -0.29 -0.36 0.62 0.38 2.0\n",
       "GOOG  0.70 -0.15  0.21 0.56 0.44 1.3\n",
       "IBM   0.68  0.08  0.03 0.48 0.52 1.0\n",
       "MDLZ  0.52  0.29 -0.56 0.68 0.32 2.5\n",
       "MSFT  0.73 -0.10  0.14 0.57 0.43 1.1\n",
       "NFLX  0.25  0.79  0.30 0.78 0.22 1.5\n",
       "ORCL  0.76 -0.16 -0.10 0.61 0.39 1.1\n",
       "SBUX  0.64  0.20  0.22 0.50 0.50 1.5\n",
       "FVX   0.42 -0.31  0.53 0.55 0.45 2.6\n",
       "SP500 0.93  0.06 -0.16 0.89 0.11 1.1\n",
       "\n",
       "                       PC1  PC2  PC3\n",
       "SS loadings           5.35 1.04 0.97\n",
       "Proportion Var        0.45 0.09 0.08\n",
       "Cumulative Var        0.45 0.53 0.61\n",
       "Proportion Explained  0.73 0.14 0.13\n",
       "Cumulative Proportion 0.73 0.87 1.00\n",
       "\n",
       "Mean item complexity =  1.5\n",
       "Test of the hypothesis that 3 components are sufficient.\n",
       "\n",
       "The root mean square of the residuals (RMSR) is  0.08 \n",
       "\n",
       "Fit based upon off diagonal values = 0.96"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "install.packages('psych')\n",
    "library(psych)\n",
    "principal(stockReturns.cov, nfactors = 3, rotate = \"none\")"
   ]
  }
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