Package Test.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dimension Reduction and PCA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create 3D Dataset with 2 Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x</th>\n",
       "      <th>y</th>\n",
       "      <th>z</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-2.765682</td>\n",
       "      <td>1.598498</td>\n",
       "      <td>-11.373190</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.420065</td>\n",
       "      <td>-1.442333</td>\n",
       "      <td>0.739554</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.064037</td>\n",
       "      <td>2.813549</td>\n",
       "      <td>2.610857</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.260346</td>\n",
       "      <td>-8.748258</td>\n",
       "      <td>-6.009919</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-1.246650</td>\n",
       "      <td>-7.425972</td>\n",
       "      <td>-11.835431</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1.901673</td>\n",
       "      <td>6.792714</td>\n",
       "      <td>14.319324</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-4.201777</td>\n",
       "      <td>13.017509</td>\n",
       "      <td>-8.558442</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-7.012180</td>\n",
       "      <td>10.346667</td>\n",
       "      <td>-23.652749</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-1.807960</td>\n",
       "      <td>3.914268</td>\n",
       "      <td>-5.071832</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>9.429336</td>\n",
       "      <td>-8.041289</td>\n",
       "      <td>36.641774</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>6.604769</td>\n",
       "      <td>-6.619715</td>\n",
       "      <td>24.852702</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>4.338221</td>\n",
       "      <td>-5.682696</td>\n",
       "      <td>15.224909</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>1.045023</td>\n",
       "      <td>-4.843347</td>\n",
       "      <td>0.806171</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>4.642484</td>\n",
       "      <td>-10.218060</td>\n",
       "      <td>12.885938</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>3.678371</td>\n",
       "      <td>3.619661</td>\n",
       "      <td>19.858130</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x          y          z\n",
       "0  -2.765682   1.598498 -11.373190\n",
       "1   0.420065  -1.442333   0.739554\n",
       "2   0.064037   2.813549   2.610857\n",
       "3   0.260346  -8.748258  -6.009919\n",
       "4  -1.246650  -7.425972 -11.835431\n",
       "5   1.901673   6.792714  14.319324\n",
       "6  -4.201777  13.017509  -8.558442\n",
       "7  -7.012180  10.346667 -23.652749\n",
       "8  -1.807960   3.914268  -5.071832\n",
       "9   9.429336  -8.041289  36.641774\n",
       "10  6.604769  -6.619715  24.852702\n",
       "11  4.338221  -5.682696  15.224909\n",
       "12  1.045023  -4.843347   0.806171\n",
       "13  4.642484 -10.218060  12.885938\n",
       "14  3.678371   3.619661  19.858130"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd \n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.preprocessing import StandardScaler\n",
    "%matplotlib inline\n",
    "\n",
    "a = np.random.normal(0, 1, 100) \n",
    "b = np.random.normal(0, 1, 100)\n",
    "\n",
    "x = (3*a) - (2*b)\n",
    "y = (-7*a) - b\n",
    "z = (8*a) - (10*b)\n",
    "\n",
    "A = np.row_stack([x, y, z]) #places data into a matrix stacking each variable as a row\n",
    "A = A.T\n",
    "df = pd.DataFrame(data= A\n",
    "                 , columns =['x','y','z'])\n",
    "df.head(15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Standardize Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x</th>\n",
       "      <th>y</th>\n",
       "      <th>z</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-2.745019</td>\n",
       "      <td>1.556798</td>\n",
       "      <td>-11.312724</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.440728</td>\n",
       "      <td>-1.484032</td>\n",
       "      <td>0.800020</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.084700</td>\n",
       "      <td>2.771850</td>\n",
       "      <td>2.671322</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.281009</td>\n",
       "      <td>-8.789957</td>\n",
       "      <td>-5.949453</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-1.225987</td>\n",
       "      <td>-7.467672</td>\n",
       "      <td>-11.774966</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1.922336</td>\n",
       "      <td>6.751015</td>\n",
       "      <td>14.379789</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-4.181114</td>\n",
       "      <td>12.975809</td>\n",
       "      <td>-8.497976</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-6.991517</td>\n",
       "      <td>10.304967</td>\n",
       "      <td>-23.592283</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-1.787297</td>\n",
       "      <td>3.872569</td>\n",
       "      <td>-5.011366</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>9.449999</td>\n",
       "      <td>-8.082988</td>\n",
       "      <td>36.702239</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>6.625432</td>\n",
       "      <td>-6.661415</td>\n",
       "      <td>24.913168</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>4.358884</td>\n",
       "      <td>-5.724396</td>\n",
       "      <td>15.285375</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>1.065686</td>\n",
       "      <td>-4.885047</td>\n",
       "      <td>0.866637</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>4.663147</td>\n",
       "      <td>-10.259759</td>\n",
       "      <td>12.946403</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>3.699034</td>\n",
       "      <td>3.577962</td>\n",
       "      <td>19.918595</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x          y          z\n",
       "0  -2.745019   1.556798 -11.312724\n",
       "1   0.440728  -1.484032   0.800020\n",
       "2   0.084700   2.771850   2.671322\n",
       "3   0.281009  -8.789957  -5.949453\n",
       "4  -1.225987  -7.467672 -11.774966\n",
       "5   1.922336   6.751015  14.379789\n",
       "6  -4.181114  12.975809  -8.497976\n",
       "7  -6.991517  10.304967 -23.592283\n",
       "8  -1.787297   3.872569  -5.011366\n",
       "9   9.449999  -8.082988  36.702239\n",
       "10  6.625432  -6.661415  24.913168\n",
       "11  4.358884  -5.724396  15.285375\n",
       "12  1.065686  -4.885047   0.866637\n",
       "13  4.663147 -10.259759  12.946403\n",
       "14  3.699034   3.577962  19.918595"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "features = ['x', 'y', 'z']\n",
    "\n",
    "# Separating out the features\n",
    "A = df.loc[:, features].values #df.loc accesses a group of rows or columns\n",
    "\n",
    "# Standardizing the features by subtracting the mean\n",
    "mean_vec = np.mean(A, axis=0) #creating vector of means\n",
    "A1 = A - mean_vec #subtracting the means and normalizing\n",
    "\n",
    "pd.DataFrame(data = A1, columns = features).head(15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Find Covariance Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 10.97360854 -17.18127981  36.20020874]\n",
      " [-17.18127981  52.54565017 -35.55886604]\n",
      " [ 36.20020874 -35.55886604 136.81130334]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(3, 3)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = np.cov(A1.T)\n",
    "print(M)\n",
    "M.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Find Eigenvalues and Eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.60467851e+02 3.21964677e-15 3.98627106e+01]\n",
      "[[ 0.25804047 -0.96237642 -0.08512783]\n",
      " [-0.33914995 -0.17273423  0.92473791]\n",
      " [ 0.90465044  0.20974871  0.37096234]]\n"
     ]
    }
   ],
   "source": [
    "e, v = np.linalg.eig(M) \n",
    "# finds the eigenvalues and vectors and stores them separately into e and then v\n",
    "print (e)\n",
    "print (v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sort Eigenvalues/vectors Largest to Smallest"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.25804047 -0.08512783 -0.96237642]\n",
      " [-0.33914995  0.92473791 -0.17273423]\n",
      " [ 0.90465044  0.37096234  0.20974871]]\n"
     ]
    }
   ],
   "source": [
    "idx = np.argsort(e)[::-1]\n",
    "e = e[idx] #orders eigen values based on index of largest to smallest eigenvalues\n",
    "e = np.real_if_close(e) #adjusts eigenvalues to be real numbers\n",
    "v = v[:, idx]#adjusts the eigenvectors to correspond to eigenvalues\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Determine Amount of Variance and Select Principle Components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[8.01015331e-01 1.98984669e-01 1.60716704e-17]\n"
     ]
    }
   ],
   "source": [
    "expvar = e/e.sum() #will give the percent of variance covered by each eigenvalue/vector\n",
    "print(expvar) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Selecting PCs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[160.46785145  39.86271061]\n",
      "[[ 0.25804047 -0.08512783]\n",
      " [-0.33914995  0.92473791]\n",
      " [ 0.90465044  0.37096234]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(3, 2)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nvar = 3 #number of variables\n",
    "ncomp = 2 #number of components\n",
    "remove = nvar-ncomp\n",
    "e = e[0:-remove] #indexes from 0 to everything except for the remove indexes at the end\n",
    "print(e)\n",
    "v = v[:, :-remove] #all rows and all columns except the removed\n",
    "print(v)\n",
    "v.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dimension Reduction and Change of Basis for PCA\n",
    "In this section we will perform matrix multiplication of the standardized data and the vectors of the principal components to get a new data matrix that will have n rows and m columns (where n= # of observations, and m= # of PCs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PC 1</th>\n",
       "      <th>PC 2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-11.470375</td>\n",
       "      <td>-2.523287</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.340774</td>\n",
       "      <td>-1.113082</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.498396</td>\n",
       "      <td>3.546984</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-2.328550</td>\n",
       "      <td>-10.359351</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-8.435922</td>\n",
       "      <td>-11.169342</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>11.215117</td>\n",
       "      <td>11.413635</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-13.167340</td>\n",
       "      <td>9.202723</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-26.641793</td>\n",
       "      <td>1.372718</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-6.308111</td>\n",
       "      <td>1.874232</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>38.382524</td>\n",
       "      <td>5.336045</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>26.506556</td>\n",
       "      <td>2.517775</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>16.894118</td>\n",
       "      <td>0.005670</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>2.715757</td>\n",
       "      <td>-4.286618</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>16.394847</td>\n",
       "      <td>-5.081924</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>17.760401</td>\n",
       "      <td>10.382835</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         PC 1       PC 2\n",
       "0  -11.470375  -2.523287\n",
       "1    1.340774  -1.113082\n",
       "2    1.498396   3.546984\n",
       "3   -2.328550 -10.359351\n",
       "4   -8.435922 -11.169342\n",
       "5   11.215117  11.413635\n",
       "6  -13.167340   9.202723\n",
       "7  -26.641793   1.372718\n",
       "8   -6.308111   1.874232\n",
       "9   38.382524   5.336045\n",
       "10  26.506556   2.517775\n",
       "11  16.894118   0.005670\n",
       "12   2.715757  -4.286618\n",
       "13  16.394847  -5.081924\n",
       "14  17.760401  10.382835"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PCdata = A1.dot(v)\n",
    "PCdata.shape\n",
    "principalDf = pd.DataFrame(data = PCdata\n",
    "             , columns = ['PC 1', 'PC 2'])\n",
    "principalDf.head(15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot Data in Reduced Dimensions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize = (8,8))\n",
    "ax = fig.add_subplot(1,1,1) \n",
    "ax.set_xlabel('Principal Component 1', fontsize = 15)\n",
    "ax.set_ylabel('Principal Component 2', fontsize = 15)\n",
    "ax.set_title('2 Component PCA', fontsize = 20)\n",
    "\n",
    "ax.scatter(principalDf.loc[:, 'PC 1']\n",
    "               , principalDf.loc[:, 'PC 2']\n",
    "               , s = 50)\n",
    "ax.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test against PCA packages"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probabilistic PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PC 1</th>\n",
       "      <th>PC 2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.206339</td>\n",
       "      <td>-0.479389</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.228973</td>\n",
       "      <td>-0.111861</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.043412</td>\n",
       "      <td>0.446270</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.370284</td>\n",
       "      <td>-1.271515</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.305359</td>\n",
       "      <td>-1.463145</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.620579</td>\n",
       "      <td>1.535526</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>2.140688</td>\n",
       "      <td>0.892444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>3.253811</td>\n",
       "      <td>-0.250921</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.867205</td>\n",
       "      <td>0.125320</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-4.230819</td>\n",
       "      <td>1.233604</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-2.994679</td>\n",
       "      <td>0.712729</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>-2.008575</td>\n",
       "      <td>0.263587</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>-0.590736</td>\n",
       "      <td>-0.468885</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>-2.267042</td>\n",
       "      <td>-0.350841</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>-1.463295</td>\n",
       "      <td>1.514470</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        PC 1      PC 2\n",
       "0   1.206339 -0.479389\n",
       "1  -0.228973 -0.111861\n",
       "2   0.043412  0.446270\n",
       "3  -0.370284 -1.271515\n",
       "4   0.305359 -1.463145\n",
       "5  -0.620579  1.535526\n",
       "6   2.140688  0.892444\n",
       "7   3.253811 -0.250921\n",
       "8   0.867205  0.125320\n",
       "9  -4.230819  1.233604\n",
       "10 -2.994679  0.712729\n",
       "11 -2.008575  0.263587\n",
       "12 -0.590736 -0.468885\n",
       "13 -2.267042 -0.350841\n",
       "14 -1.463295  1.514470"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.decomposition import PCA\n",
    "\n",
    "A2 = StandardScaler().fit_transform(A) #subtracts the mean\n",
    "\n",
    "#choosing 2 principal components and assigning them to the variable \"pca\"\n",
    "pca = PCA(n_components=2) \n",
    "\n",
    "#gives you 2D data, with 2 principle components\n",
    "principalComponents = pca.fit_transform(A2) \n",
    "principalDf1 = pd.DataFrame(data = principalComponents\n",
    "             , columns = ['PC 1', 'PC 2'])\n",
    "\n",
    "principalDf1.head(15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize = (8,8))\n",
    "ax = fig.add_subplot(1,1,1) \n",
    "ax.set_xlabel('Principal Component 1', fontsize = 15)\n",
    "ax.set_ylabel('Principal Component 2', fontsize = 15)\n",
    "ax.set_title('2 Component PPCA', fontsize = 20)\n",
    "\n",
    "ax.scatter(principalDf1.loc[:, 'PC 1']\n",
    "               , principalDf1.loc[:, 'PC 2']\n",
    "               , s = 50)\n",
    "ax.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### PCA vs. Probablistic PCA\n",
    "\n",
    "Significant difference...\n",
    "- PCA only defines a linear projection of the data so the scope of its application is necessarily somewhat limited \n",
    "- Probabilistic PCA is an attempt to account for that and achieve nonlinear PCA in an effort to retain a greater proportion of the variance using fewer components. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}