Chapter 8 Dimensionality Reduction.ipynb

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    "# Dimensionality Reduction\n",
    "\n",
    "Sometimes training data has thousands of features. It makes training difficult and slow. To combat this you can reduce the number of features turning an intractable problem into a tractable one. Dimensionality reduction also helps in visualization.\n",
    "<br>Techniques of Dimensionality Reduction:\n",
    "<ul>\n",
    "    <li>PCA</li>\n",
    "    <li>Kernel PCA</li>\n",
    "    <li>LLE</li>\n",
    "</ul>\n",
    "\n",
    "High dimensional datasets run the risk of the training instances being far away from each other. One easy solution would be to get more training data. However, the size of an optimal dataset would grow exponentially as you add more features. "
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    "# Approaches for DR <br>\n",
    "## Projection\n",
    "Many times training instances are not spread out uniformly across all dimensions. Some features are constant while others are correlated. These training instances actually lie within a much lower-dimensional supspace. Projection is where you have a high dimension dataset projected onto something of lower dimension. Think of 3D data being spread out across a plane, so instead of having an x, y, z component you project it onto a 2D plane. \n",
    "<br>\n",
    "When a dataset twists and turns in 3D space, then projection isn't the best approach\n",
    "\n",
    "<br>\n",
    "## Manifold Learning\n",
    "A manifold is a shape that can be bent and twisted in a higher dimensional space. Manifold learning works by modeling the manifold on which the training instances lie. \n"
   ]
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   "cell_type": "markdown",
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   "source": [
    "# PCA Principal Component Analysis\n",
    "First identify the hyperplane that lies closest to the data, and then project data on to it. It's important to choose the right hyperplane. The idea with PCA is to preseve the variance between the training instances. Or the minimize the mean squared distance between the original dataset and its projection onto that axis.\n",
    "\n",
    "<br>\n",
    "PCA finds the axis that accounts for the largest amount of variance in the training set, and a second axis which is orthogonal to the first one, that accounts for the largest amount of remaining variance.\n",
    "\n",
    "To find the principal component axes, use the SVD function that returns the dot product of three matricies. The principal component is orthogonal axes for all dimensions. Once you have identified it you can reduce the dimensions by projecting it onto the hyperplane defined by the first d principal components.\n",
    "\n",
    "## Explained Variance Ratio\n",
    "The EVR indicated the proportion of the dataset's variance that lies along the axis of each principal component.\n",
    "\n",
    "## How to choose the right number of dimensions?\n",
    "For visualization: 2 to 3. Usually the best amount is the dimension that preserves at least 95% of the variance.\n",
    "you can also plot the cumulative sum of all the explained variance ratios based on the dimension, and find the inflection point. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "from sklearn.datasets import load_breast_cancer\n",
    "from sklearn.decomposition import PCA\n",
    "from sklearn.preprocessing import StandardScaler\n",
    "from sklearn.cross_validation import train_test_split\n",
    "from sklearn.neighbors import KNeighborsClassifier\n",
    "import plotly\n",
    "import plotly.graph_objs as go\n",
    " \n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
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       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>mean radius</th>\n",
       "      <th>mean texture</th>\n",
       "      <th>mean perimeter</th>\n",
       "      <th>mean area</th>\n",
       "      <th>mean smoothness</th>\n",
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       "      <th>...</th>\n",
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       "      <th>worst perimeter</th>\n",
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       "      <th>worst compactness</th>\n",
       "      <th>worst concavity</th>\n",
       "      <th>worst concave points</th>\n",
       "      <th>worst symmetry</th>\n",
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       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>...</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
       "      <td>569.000000</td>\n",
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       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>14.127292</td>\n",
       "      <td>19.289649</td>\n",
       "      <td>91.969033</td>\n",
       "      <td>654.889104</td>\n",
       "      <td>0.096360</td>\n",
       "      <td>0.104341</td>\n",
       "      <td>0.088799</td>\n",
       "      <td>0.048919</td>\n",
       "      <td>0.181162</td>\n",
       "      <td>0.062798</td>\n",
       "      <td>...</td>\n",
       "      <td>16.269190</td>\n",
       "      <td>25.677223</td>\n",
       "      <td>107.261213</td>\n",
       "      <td>880.583128</td>\n",
       "      <td>0.132369</td>\n",
       "      <td>0.254265</td>\n",
       "      <td>0.272188</td>\n",
       "      <td>0.114606</td>\n",
       "      <td>0.290076</td>\n",
       "      <td>0.083946</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>3.524049</td>\n",
       "      <td>4.301036</td>\n",
       "      <td>24.298981</td>\n",
       "      <td>351.914129</td>\n",
       "      <td>0.014064</td>\n",
       "      <td>0.052813</td>\n",
       "      <td>0.079720</td>\n",
       "      <td>0.038803</td>\n",
       "      <td>0.027414</td>\n",
       "      <td>0.007060</td>\n",
       "      <td>...</td>\n",
       "      <td>4.833242</td>\n",
       "      <td>6.146258</td>\n",
       "      <td>33.602542</td>\n",
       "      <td>569.356993</td>\n",
       "      <td>0.022832</td>\n",
       "      <td>0.157336</td>\n",
       "      <td>0.208624</td>\n",
       "      <td>0.065732</td>\n",
       "      <td>0.061867</td>\n",
       "      <td>0.018061</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>6.981000</td>\n",
       "      <td>9.710000</td>\n",
       "      <td>43.790000</td>\n",
       "      <td>143.500000</td>\n",
       "      <td>0.052630</td>\n",
       "      <td>0.019380</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.106000</td>\n",
       "      <td>0.049960</td>\n",
       "      <td>...</td>\n",
       "      <td>7.930000</td>\n",
       "      <td>12.020000</td>\n",
       "      <td>50.410000</td>\n",
       "      <td>185.200000</td>\n",
       "      <td>0.071170</td>\n",
       "      <td>0.027290</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.156500</td>\n",
       "      <td>0.055040</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>11.700000</td>\n",
       "      <td>16.170000</td>\n",
       "      <td>75.170000</td>\n",
       "      <td>420.300000</td>\n",
       "      <td>0.086370</td>\n",
       "      <td>0.064920</td>\n",
       "      <td>0.029560</td>\n",
       "      <td>0.020310</td>\n",
       "      <td>0.161900</td>\n",
       "      <td>0.057700</td>\n",
       "      <td>...</td>\n",
       "      <td>13.010000</td>\n",
       "      <td>21.080000</td>\n",
       "      <td>84.110000</td>\n",
       "      <td>515.300000</td>\n",
       "      <td>0.116600</td>\n",
       "      <td>0.147200</td>\n",
       "      <td>0.114500</td>\n",
       "      <td>0.064930</td>\n",
       "      <td>0.250400</td>\n",
       "      <td>0.071460</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>13.370000</td>\n",
       "      <td>18.840000</td>\n",
       "      <td>86.240000</td>\n",
       "      <td>551.100000</td>\n",
       "      <td>0.095870</td>\n",
       "      <td>0.092630</td>\n",
       "      <td>0.061540</td>\n",
       "      <td>0.033500</td>\n",
       "      <td>0.179200</td>\n",
       "      <td>0.061540</td>\n",
       "      <td>...</td>\n",
       "      <td>14.970000</td>\n",
       "      <td>25.410000</td>\n",
       "      <td>97.660000</td>\n",
       "      <td>686.500000</td>\n",
       "      <td>0.131300</td>\n",
       "      <td>0.211900</td>\n",
       "      <td>0.226700</td>\n",
       "      <td>0.099930</td>\n",
       "      <td>0.282200</td>\n",
       "      <td>0.080040</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>15.780000</td>\n",
       "      <td>21.800000</td>\n",
       "      <td>104.100000</td>\n",
       "      <td>782.700000</td>\n",
       "      <td>0.105300</td>\n",
       "      <td>0.130400</td>\n",
       "      <td>0.130700</td>\n",
       "      <td>0.074000</td>\n",
       "      <td>0.195700</td>\n",
       "      <td>0.066120</td>\n",
       "      <td>...</td>\n",
       "      <td>18.790000</td>\n",
       "      <td>29.720000</td>\n",
       "      <td>125.400000</td>\n",
       "      <td>1084.000000</td>\n",
       "      <td>0.146000</td>\n",
       "      <td>0.339100</td>\n",
       "      <td>0.382900</td>\n",
       "      <td>0.161400</td>\n",
       "      <td>0.317900</td>\n",
       "      <td>0.092080</td>\n",
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       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>28.110000</td>\n",
       "      <td>39.280000</td>\n",
       "      <td>188.500000</td>\n",
       "      <td>2501.000000</td>\n",
       "      <td>0.163400</td>\n",
       "      <td>0.345400</td>\n",
       "      <td>0.426800</td>\n",
       "      <td>0.201200</td>\n",
       "      <td>0.304000</td>\n",
       "      <td>0.097440</td>\n",
       "      <td>...</td>\n",
       "      <td>36.040000</td>\n",
       "      <td>49.540000</td>\n",
       "      <td>251.200000</td>\n",
       "      <td>4254.000000</td>\n",
       "      <td>0.222600</td>\n",
       "      <td>1.058000</td>\n",
       "      <td>1.252000</td>\n",
       "      <td>0.291000</td>\n",
       "      <td>0.663800</td>\n",
       "      <td>0.207500</td>\n",
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       "       mean radius  mean texture  mean perimeter    mean area  \\\n",
       "count   569.000000    569.000000      569.000000   569.000000   \n",
       "mean     14.127292     19.289649       91.969033   654.889104   \n",
       "std       3.524049      4.301036       24.298981   351.914129   \n",
       "min       6.981000      9.710000       43.790000   143.500000   \n",
       "25%      11.700000     16.170000       75.170000   420.300000   \n",
       "50%      13.370000     18.840000       86.240000   551.100000   \n",
       "75%      15.780000     21.800000      104.100000   782.700000   \n",
       "max      28.110000     39.280000      188.500000  2501.000000   \n",
       "\n",
       "       mean smoothness  mean compactness  mean concavity  mean concave points  \\\n",
       "count       569.000000        569.000000      569.000000           569.000000   \n",
       "mean          0.096360          0.104341        0.088799             0.048919   \n",
       "std           0.014064          0.052813        0.079720             0.038803   \n",
       "min           0.052630          0.019380        0.000000             0.000000   \n",
       "25%           0.086370          0.064920        0.029560             0.020310   \n",
       "50%           0.095870          0.092630        0.061540             0.033500   \n",
       "75%           0.105300          0.130400        0.130700             0.074000   \n",
       "max           0.163400          0.345400        0.426800             0.201200   \n",
       "\n",
       "       mean symmetry  mean fractal dimension           ...             \\\n",
       "count     569.000000              569.000000           ...              \n",
       "mean        0.181162                0.062798           ...              \n",
       "std         0.027414                0.007060           ...              \n",
       "min         0.106000                0.049960           ...              \n",
       "25%         0.161900                0.057700           ...              \n",
       "50%         0.179200                0.061540           ...              \n",
       "75%         0.195700                0.066120           ...              \n",
       "max         0.304000                0.097440           ...              \n",
       "\n",
       "       worst radius  worst texture  worst perimeter   worst area  \\\n",
       "count    569.000000     569.000000       569.000000   569.000000   \n",
       "mean      16.269190      25.677223       107.261213   880.583128   \n",
       "std        4.833242       6.146258        33.602542   569.356993   \n",
       "min        7.930000      12.020000        50.410000   185.200000   \n",
       "25%       13.010000      21.080000        84.110000   515.300000   \n",
       "50%       14.970000      25.410000        97.660000   686.500000   \n",
       "75%       18.790000      29.720000       125.400000  1084.000000   \n",
       "max       36.040000      49.540000       251.200000  4254.000000   \n",
       "\n",
       "       worst smoothness  worst compactness  worst concavity  \\\n",
       "count        569.000000         569.000000       569.000000   \n",
       "mean           0.132369           0.254265         0.272188   \n",
       "std            0.022832           0.157336         0.208624   \n",
       "min            0.071170           0.027290         0.000000   \n",
       "25%            0.116600           0.147200         0.114500   \n",
       "50%            0.131300           0.211900         0.226700   \n",
       "75%            0.146000           0.339100         0.382900   \n",
       "max            0.222600           1.058000         1.252000   \n",
       "\n",
       "       worst concave points  worst symmetry  worst fractal dimension  \n",
       "count            569.000000      569.000000               569.000000  \n",
       "mean               0.114606        0.290076                 0.083946  \n",
       "std                0.065732        0.061867                 0.018061  \n",
       "min                0.000000        0.156500                 0.055040  \n",
       "25%                0.064930        0.250400                 0.071460  \n",
       "50%                0.099930        0.282200                 0.080040  \n",
       "75%                0.161400        0.317900                 0.092080  \n",
       "max                0.291000        0.663800                 0.207500  \n",
       "\n",
       "[8 rows x 30 columns]"
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       "      <th>mean radius</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.323782</td>\n",
       "      <td>0.997855</td>\n",
       "      <td>0.987357</td>\n",
       "      <td>0.170581</td>\n",
       "      <td>0.506124</td>\n",
       "      <td>0.676764</td>\n",
       "      <td>0.822529</td>\n",
       "      <td>0.147741</td>\n",
       "      <td>-0.311631</td>\n",
       "      <td>...</td>\n",
       "      <td>0.969539</td>\n",
       "      <td>0.297008</td>\n",
       "      <td>0.965137</td>\n",
       "      <td>0.941082</td>\n",
       "      <td>0.119616</td>\n",
       "      <td>0.413463</td>\n",
       "      <td>0.526911</td>\n",
       "      <td>0.744214</td>\n",
       "      <td>0.163953</td>\n",
       "      <td>0.007066</td>\n",
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       "      <th>mean texture</th>\n",
       "      <td>0.323782</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.329533</td>\n",
       "      <td>0.321086</td>\n",
       "      <td>-0.023389</td>\n",
       "      <td>0.236702</td>\n",
       "      <td>0.302418</td>\n",
       "      <td>0.293464</td>\n",
       "      <td>0.071401</td>\n",
       "      <td>-0.076437</td>\n",
       "      <td>...</td>\n",
       "      <td>0.352573</td>\n",
       "      <td>0.912045</td>\n",
       "      <td>0.358040</td>\n",
       "      <td>0.343546</td>\n",
       "      <td>0.077503</td>\n",
       "      <td>0.277830</td>\n",
       "      <td>0.301025</td>\n",
       "      <td>0.295316</td>\n",
       "      <td>0.105008</td>\n",
       "      <td>0.119205</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean perimeter</th>\n",
       "      <td>0.997855</td>\n",
       "      <td>0.329533</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.986507</td>\n",
       "      <td>0.207278</td>\n",
       "      <td>0.556936</td>\n",
       "      <td>0.716136</td>\n",
       "      <td>0.850977</td>\n",
       "      <td>0.183027</td>\n",
       "      <td>-0.261477</td>\n",
       "      <td>...</td>\n",
       "      <td>0.969476</td>\n",
       "      <td>0.303038</td>\n",
       "      <td>0.970387</td>\n",
       "      <td>0.941550</td>\n",
       "      <td>0.150549</td>\n",
       "      <td>0.455774</td>\n",
       "      <td>0.563879</td>\n",
       "      <td>0.771241</td>\n",
       "      <td>0.189115</td>\n",
       "      <td>0.051019</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean area</th>\n",
       "      <td>0.987357</td>\n",
       "      <td>0.321086</td>\n",
       "      <td>0.986507</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.177028</td>\n",
       "      <td>0.498502</td>\n",
       "      <td>0.685983</td>\n",
       "      <td>0.823269</td>\n",
       "      <td>0.151293</td>\n",
       "      <td>-0.283110</td>\n",
       "      <td>...</td>\n",
       "      <td>0.962746</td>\n",
       "      <td>0.287489</td>\n",
       "      <td>0.959120</td>\n",
       "      <td>0.959213</td>\n",
       "      <td>0.123523</td>\n",
       "      <td>0.390410</td>\n",
       "      <td>0.512606</td>\n",
       "      <td>0.722017</td>\n",
       "      <td>0.143570</td>\n",
       "      <td>0.003738</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean smoothness</th>\n",
       "      <td>0.170581</td>\n",
       "      <td>-0.023389</td>\n",
       "      <td>0.207278</td>\n",
       "      <td>0.177028</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.659123</td>\n",
       "      <td>0.521984</td>\n",
       "      <td>0.553695</td>\n",
       "      <td>0.557775</td>\n",
       "      <td>0.584792</td>\n",
       "      <td>...</td>\n",
       "      <td>0.213120</td>\n",
       "      <td>0.036072</td>\n",
       "      <td>0.238853</td>\n",
       "      <td>0.206718</td>\n",
       "      <td>0.805324</td>\n",
       "      <td>0.472468</td>\n",
       "      <td>0.434926</td>\n",
       "      <td>0.503053</td>\n",
       "      <td>0.394309</td>\n",
       "      <td>0.499316</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean compactness</th>\n",
       "      <td>0.506124</td>\n",
       "      <td>0.236702</td>\n",
       "      <td>0.556936</td>\n",
       "      <td>0.498502</td>\n",
       "      <td>0.659123</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.883121</td>\n",
       "      <td>0.831135</td>\n",
       "      <td>0.602641</td>\n",
       "      <td>0.565369</td>\n",
       "      <td>...</td>\n",
       "      <td>0.535315</td>\n",
       "      <td>0.248133</td>\n",
       "      <td>0.590210</td>\n",
       "      <td>0.509604</td>\n",
       "      <td>0.565541</td>\n",
       "      <td>0.865809</td>\n",
       "      <td>0.816275</td>\n",
       "      <td>0.815573</td>\n",
       "      <td>0.510223</td>\n",
       "      <td>0.687382</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean concavity</th>\n",
       "      <td>0.676764</td>\n",
       "      <td>0.302418</td>\n",
       "      <td>0.716136</td>\n",
       "      <td>0.685983</td>\n",
       "      <td>0.521984</td>\n",
       "      <td>0.883121</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.921391</td>\n",
       "      <td>0.500667</td>\n",
       "      <td>0.336783</td>\n",
       "      <td>...</td>\n",
       "      <td>0.688236</td>\n",
       "      <td>0.299879</td>\n",
       "      <td>0.729565</td>\n",
       "      <td>0.675987</td>\n",
       "      <td>0.448822</td>\n",
       "      <td>0.754968</td>\n",
       "      <td>0.884103</td>\n",
       "      <td>0.861323</td>\n",
       "      <td>0.409464</td>\n",
       "      <td>0.514930</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean concave points</th>\n",
       "      <td>0.822529</td>\n",
       "      <td>0.293464</td>\n",
       "      <td>0.850977</td>\n",
       "      <td>0.823269</td>\n",
       "      <td>0.553695</td>\n",
       "      <td>0.831135</td>\n",
       "      <td>0.921391</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.462497</td>\n",
       "      <td>0.166917</td>\n",
       "      <td>...</td>\n",
       "      <td>0.830318</td>\n",
       "      <td>0.292752</td>\n",
       "      <td>0.855923</td>\n",
       "      <td>0.809630</td>\n",
       "      <td>0.452753</td>\n",
       "      <td>0.667454</td>\n",
       "      <td>0.752399</td>\n",
       "      <td>0.910155</td>\n",
       "      <td>0.375744</td>\n",
       "      <td>0.368661</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean symmetry</th>\n",
       "      <td>0.147741</td>\n",
       "      <td>0.071401</td>\n",
       "      <td>0.183027</td>\n",
       "      <td>0.151293</td>\n",
       "      <td>0.557775</td>\n",
       "      <td>0.602641</td>\n",
       "      <td>0.500667</td>\n",
       "      <td>0.462497</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.479921</td>\n",
       "      <td>...</td>\n",
       "      <td>0.185728</td>\n",
       "      <td>0.090651</td>\n",
       "      <td>0.219169</td>\n",
       "      <td>0.177193</td>\n",
       "      <td>0.426675</td>\n",
       "      <td>0.473200</td>\n",
       "      <td>0.433721</td>\n",
       "      <td>0.430297</td>\n",
       "      <td>0.699826</td>\n",
       "      <td>0.438413</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean fractal dimension</th>\n",
       "      <td>-0.311631</td>\n",
       "      <td>-0.076437</td>\n",
       "      <td>-0.261477</td>\n",
       "      <td>-0.283110</td>\n",
       "      <td>0.584792</td>\n",
       "      <td>0.565369</td>\n",
       "      <td>0.336783</td>\n",
       "      <td>0.166917</td>\n",
       "      <td>0.479921</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.253691</td>\n",
       "      <td>-0.051269</td>\n",
       "      <td>-0.205151</td>\n",
       "      <td>-0.231854</td>\n",
       "      <td>0.504942</td>\n",
       "      <td>0.458798</td>\n",
       "      <td>0.346234</td>\n",
       "      <td>0.175325</td>\n",
       "      <td>0.334019</td>\n",
       "      <td>0.767297</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>radius error</th>\n",
       "      <td>0.679090</td>\n",
       "      <td>0.275869</td>\n",
       "      <td>0.691765</td>\n",
       "      <td>0.732562</td>\n",
       "      <td>0.301467</td>\n",
       "      <td>0.497473</td>\n",
       "      <td>0.631925</td>\n",
       "      <td>0.698050</td>\n",
       "      <td>0.303379</td>\n",
       "      <td>0.000111</td>\n",
       "      <td>...</td>\n",
       "      <td>0.715065</td>\n",
       "      <td>0.194799</td>\n",
       "      <td>0.719684</td>\n",
       "      <td>0.751548</td>\n",
       "      <td>0.141919</td>\n",
       "      <td>0.287103</td>\n",
       "      <td>0.380585</td>\n",
       "      <td>0.531062</td>\n",
       "      <td>0.094543</td>\n",
       "      <td>0.049559</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>texture error</th>\n",
       "      <td>-0.097317</td>\n",
       "      <td>0.386358</td>\n",
       "      <td>-0.086761</td>\n",
       "      <td>-0.066280</td>\n",
       "      <td>0.068406</td>\n",
       "      <td>0.046205</td>\n",
       "      <td>0.076218</td>\n",
       "      <td>0.021480</td>\n",
       "      <td>0.128053</td>\n",
       "      <td>0.164174</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.111690</td>\n",
       "      <td>0.409003</td>\n",
       "      <td>-0.102242</td>\n",
       "      <td>-0.083195</td>\n",
       "      <td>-0.073658</td>\n",
       "      <td>-0.092439</td>\n",
       "      <td>-0.068956</td>\n",
       "      <td>-0.119638</td>\n",
       "      <td>-0.128215</td>\n",
       "      <td>-0.045655</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>perimeter error</th>\n",
       "      <td>0.674172</td>\n",
       "      <td>0.281673</td>\n",
       "      <td>0.693135</td>\n",
       "      <td>0.726628</td>\n",
       "      <td>0.296092</td>\n",
       "      <td>0.548905</td>\n",
       "      <td>0.660391</td>\n",
       "      <td>0.710650</td>\n",
       "      <td>0.313893</td>\n",
       "      <td>0.039830</td>\n",
       "      <td>...</td>\n",
       "      <td>0.697201</td>\n",
       "      <td>0.200371</td>\n",
       "      <td>0.721031</td>\n",
       "      <td>0.730713</td>\n",
       "      <td>0.130054</td>\n",
       "      <td>0.341919</td>\n",
       "      <td>0.418899</td>\n",
       "      <td>0.554897</td>\n",
       "      <td>0.109930</td>\n",
       "      <td>0.085433</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>area error</th>\n",
       "      <td>0.735864</td>\n",
       "      <td>0.259845</td>\n",
       "      <td>0.744983</td>\n",
       "      <td>0.800086</td>\n",
       "      <td>0.246552</td>\n",
       "      <td>0.455653</td>\n",
       "      <td>0.617427</td>\n",
       "      <td>0.690299</td>\n",
       "      <td>0.223970</td>\n",
       "      <td>-0.090170</td>\n",
       "      <td>...</td>\n",
       "      <td>0.757373</td>\n",
       "      <td>0.196497</td>\n",
       "      <td>0.761213</td>\n",
       "      <td>0.811408</td>\n",
       "      <td>0.125389</td>\n",
       "      <td>0.283257</td>\n",
       "      <td>0.385100</td>\n",
       "      <td>0.538166</td>\n",
       "      <td>0.074126</td>\n",
       "      <td>0.017539</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>smoothness error</th>\n",
       "      <td>-0.222600</td>\n",
       "      <td>0.006614</td>\n",
       "      <td>-0.202694</td>\n",
       "      <td>-0.166777</td>\n",
       "      <td>0.332375</td>\n",
       "      <td>0.135299</td>\n",
       "      <td>0.098564</td>\n",
       "      <td>0.027653</td>\n",
       "      <td>0.187321</td>\n",
       "      <td>0.401964</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.230691</td>\n",
       "      <td>-0.074743</td>\n",
       "      <td>-0.217304</td>\n",
       "      <td>-0.182195</td>\n",
       "      <td>0.314457</td>\n",
       "      <td>-0.055558</td>\n",
       "      <td>-0.058298</td>\n",
       "      <td>-0.102007</td>\n",
       "      <td>-0.107342</td>\n",
       "      <td>0.101480</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>compactness error</th>\n",
       "      <td>0.206000</td>\n",
       "      <td>0.191975</td>\n",
       "      <td>0.250744</td>\n",
       "      <td>0.212583</td>\n",
       "      <td>0.318943</td>\n",
       "      <td>0.738722</td>\n",
       "      <td>0.670279</td>\n",
       "      <td>0.490424</td>\n",
       "      <td>0.421659</td>\n",
       "      <td>0.559837</td>\n",
       "      <td>...</td>\n",
       "      <td>0.204607</td>\n",
       "      <td>0.143003</td>\n",
       "      <td>0.260516</td>\n",
       "      <td>0.199371</td>\n",
       "      <td>0.227394</td>\n",
       "      <td>0.678780</td>\n",
       "      <td>0.639147</td>\n",
       "      <td>0.483208</td>\n",
       "      <td>0.277878</td>\n",
       "      <td>0.590973</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>concavity error</th>\n",
       "      <td>0.194204</td>\n",
       "      <td>0.143293</td>\n",
       "      <td>0.228082</td>\n",
       "      <td>0.207660</td>\n",
       "      <td>0.248396</td>\n",
       "      <td>0.570517</td>\n",
       "      <td>0.691270</td>\n",
       "      <td>0.439167</td>\n",
       "      <td>0.342627</td>\n",
       "      <td>0.446630</td>\n",
       "      <td>...</td>\n",
       "      <td>0.186904</td>\n",
       "      <td>0.100241</td>\n",
       "      <td>0.226680</td>\n",
       "      <td>0.188353</td>\n",
       "      <td>0.168481</td>\n",
       "      <td>0.484858</td>\n",
       "      <td>0.662564</td>\n",
       "      <td>0.440472</td>\n",
       "      <td>0.197788</td>\n",
       "      <td>0.439329</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>concave points error</th>\n",
       "      <td>0.376169</td>\n",
       "      <td>0.163851</td>\n",
       "      <td>0.407217</td>\n",
       "      <td>0.372320</td>\n",
       "      <td>0.380676</td>\n",
       "      <td>0.642262</td>\n",
       "      <td>0.683260</td>\n",
       "      <td>0.615634</td>\n",
       "      <td>0.393298</td>\n",
       "      <td>0.341198</td>\n",
       "      <td>...</td>\n",
       "      <td>0.358127</td>\n",
       "      <td>0.086741</td>\n",
       "      <td>0.394999</td>\n",
       "      <td>0.342271</td>\n",
       "      <td>0.215351</td>\n",
       "      <td>0.452888</td>\n",
       "      <td>0.549592</td>\n",
       "      <td>0.602450</td>\n",
       "      <td>0.143116</td>\n",
       "      <td>0.310655</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>symmetry error</th>\n",
       "      <td>-0.104321</td>\n",
       "      <td>0.009127</td>\n",
       "      <td>-0.081629</td>\n",
       "      <td>-0.072497</td>\n",
       "      <td>0.200774</td>\n",
       "      <td>0.229977</td>\n",
       "      <td>0.178009</td>\n",
       "      <td>0.095351</td>\n",
       "      <td>0.449137</td>\n",
       "      <td>0.345007</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.128121</td>\n",
       "      <td>-0.077473</td>\n",
       "      <td>-0.103753</td>\n",
       "      <td>-0.110343</td>\n",
       "      <td>-0.012662</td>\n",
       "      <td>0.060255</td>\n",
       "      <td>0.037119</td>\n",
       "      <td>-0.030413</td>\n",
       "      <td>0.389402</td>\n",
       "      <td>0.078079</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fractal dimension error</th>\n",
       "      <td>-0.042641</td>\n",
       "      <td>0.054458</td>\n",
       "      <td>-0.005523</td>\n",
       "      <td>-0.019887</td>\n",
       "      <td>0.283607</td>\n",
       "      <td>0.507318</td>\n",
       "      <td>0.449301</td>\n",
       "      <td>0.257584</td>\n",
       "      <td>0.331786</td>\n",
       "      <td>0.688132</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.037488</td>\n",
       "      <td>-0.003195</td>\n",
       "      <td>-0.001000</td>\n",
       "      <td>-0.022736</td>\n",
       "      <td>0.170568</td>\n",
       "      <td>0.390159</td>\n",
       "      <td>0.379975</td>\n",
       "      <td>0.215204</td>\n",
       "      <td>0.111094</td>\n",
       "      <td>0.591328</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst radius</th>\n",
       "      <td>0.969539</td>\n",
       "      <td>0.352573</td>\n",
       "      <td>0.969476</td>\n",
       "      <td>0.962746</td>\n",
       "      <td>0.213120</td>\n",
       "      <td>0.535315</td>\n",
       "      <td>0.688236</td>\n",
       "      <td>0.830318</td>\n",
       "      <td>0.185728</td>\n",
       "      <td>-0.253691</td>\n",
       "      <td>...</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.359921</td>\n",
       "      <td>0.993708</td>\n",
       "      <td>0.984015</td>\n",
       "      <td>0.216574</td>\n",
       "      <td>0.475820</td>\n",
       "      <td>0.573975</td>\n",
       "      <td>0.787424</td>\n",
       "      <td>0.243529</td>\n",
       "      <td>0.093492</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst texture</th>\n",
       "      <td>0.297008</td>\n",
       "      <td>0.912045</td>\n",
       "      <td>0.303038</td>\n",
       "      <td>0.287489</td>\n",
       "      <td>0.036072</td>\n",
       "      <td>0.248133</td>\n",
       "      <td>0.299879</td>\n",
       "      <td>0.292752</td>\n",
       "      <td>0.090651</td>\n",
       "      <td>-0.051269</td>\n",
       "      <td>...</td>\n",
       "      <td>0.359921</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.365098</td>\n",
       "      <td>0.345842</td>\n",
       "      <td>0.225429</td>\n",
       "      <td>0.360832</td>\n",
       "      <td>0.368366</td>\n",
       "      <td>0.359755</td>\n",
       "      <td>0.233027</td>\n",
       "      <td>0.219122</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst perimeter</th>\n",
       "      <td>0.965137</td>\n",
       "      <td>0.358040</td>\n",
       "      <td>0.970387</td>\n",
       "      <td>0.959120</td>\n",
       "      <td>0.238853</td>\n",
       "      <td>0.590210</td>\n",
       "      <td>0.729565</td>\n",
       "      <td>0.855923</td>\n",
       "      <td>0.219169</td>\n",
       "      <td>-0.205151</td>\n",
       "      <td>...</td>\n",
       "      <td>0.993708</td>\n",
       "      <td>0.365098</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.977578</td>\n",
       "      <td>0.236775</td>\n",
       "      <td>0.529408</td>\n",
       "      <td>0.618344</td>\n",
       "      <td>0.816322</td>\n",
       "      <td>0.269493</td>\n",
       "      <td>0.138957</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst area</th>\n",
       "      <td>0.941082</td>\n",
       "      <td>0.343546</td>\n",
       "      <td>0.941550</td>\n",
       "      <td>0.959213</td>\n",
       "      <td>0.206718</td>\n",
       "      <td>0.509604</td>\n",
       "      <td>0.675987</td>\n",
       "      <td>0.809630</td>\n",
       "      <td>0.177193</td>\n",
       "      <td>-0.231854</td>\n",
       "      <td>...</td>\n",
       "      <td>0.984015</td>\n",
       "      <td>0.345842</td>\n",
       "      <td>0.977578</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.209145</td>\n",
       "      <td>0.438296</td>\n",
       "      <td>0.543331</td>\n",
       "      <td>0.747419</td>\n",
       "      <td>0.209146</td>\n",
       "      <td>0.079647</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst smoothness</th>\n",
       "      <td>0.119616</td>\n",
       "      <td>0.077503</td>\n",
       "      <td>0.150549</td>\n",
       "      <td>0.123523</td>\n",
       "      <td>0.805324</td>\n",
       "      <td>0.565541</td>\n",
       "      <td>0.448822</td>\n",
       "      <td>0.452753</td>\n",
       "      <td>0.426675</td>\n",
       "      <td>0.504942</td>\n",
       "      <td>...</td>\n",
       "      <td>0.216574</td>\n",
       "      <td>0.225429</td>\n",
       "      <td>0.236775</td>\n",
       "      <td>0.209145</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.568187</td>\n",
       "      <td>0.518523</td>\n",
       "      <td>0.547691</td>\n",
       "      <td>0.493838</td>\n",
       "      <td>0.617624</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst compactness</th>\n",
       "      <td>0.413463</td>\n",
       "      <td>0.277830</td>\n",
       "      <td>0.455774</td>\n",
       "      <td>0.390410</td>\n",
       "      <td>0.472468</td>\n",
       "      <td>0.865809</td>\n",
       "      <td>0.754968</td>\n",
       "      <td>0.667454</td>\n",
       "      <td>0.473200</td>\n",
       "      <td>0.458798</td>\n",
       "      <td>...</td>\n",
       "      <td>0.475820</td>\n",
       "      <td>0.360832</td>\n",
       "      <td>0.529408</td>\n",
       "      <td>0.438296</td>\n",
       "      <td>0.568187</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.892261</td>\n",
       "      <td>0.801080</td>\n",
       "      <td>0.614441</td>\n",
       "      <td>0.810455</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst concavity</th>\n",
       "      <td>0.526911</td>\n",
       "      <td>0.301025</td>\n",
       "      <td>0.563879</td>\n",
       "      <td>0.512606</td>\n",
       "      <td>0.434926</td>\n",
       "      <td>0.816275</td>\n",
       "      <td>0.884103</td>\n",
       "      <td>0.752399</td>\n",
       "      <td>0.433721</td>\n",
       "      <td>0.346234</td>\n",
       "      <td>...</td>\n",
       "      <td>0.573975</td>\n",
       "      <td>0.368366</td>\n",
       "      <td>0.618344</td>\n",
       "      <td>0.543331</td>\n",
       "      <td>0.518523</td>\n",
       "      <td>0.892261</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.855434</td>\n",
       "      <td>0.532520</td>\n",
       "      <td>0.686511</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst concave points</th>\n",
       "      <td>0.744214</td>\n",
       "      <td>0.295316</td>\n",
       "      <td>0.771241</td>\n",
       "      <td>0.722017</td>\n",
       "      <td>0.503053</td>\n",
       "      <td>0.815573</td>\n",
       "      <td>0.861323</td>\n",
       "      <td>0.910155</td>\n",
       "      <td>0.430297</td>\n",
       "      <td>0.175325</td>\n",
       "      <td>...</td>\n",
       "      <td>0.787424</td>\n",
       "      <td>0.359755</td>\n",
       "      <td>0.816322</td>\n",
       "      <td>0.747419</td>\n",
       "      <td>0.547691</td>\n",
       "      <td>0.801080</td>\n",
       "      <td>0.855434</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.502528</td>\n",
       "      <td>0.511114</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst symmetry</th>\n",
       "      <td>0.163953</td>\n",
       "      <td>0.105008</td>\n",
       "      <td>0.189115</td>\n",
       "      <td>0.143570</td>\n",
       "      <td>0.394309</td>\n",
       "      <td>0.510223</td>\n",
       "      <td>0.409464</td>\n",
       "      <td>0.375744</td>\n",
       "      <td>0.699826</td>\n",
       "      <td>0.334019</td>\n",
       "      <td>...</td>\n",
       "      <td>0.243529</td>\n",
       "      <td>0.233027</td>\n",
       "      <td>0.269493</td>\n",
       "      <td>0.209146</td>\n",
       "      <td>0.493838</td>\n",
       "      <td>0.614441</td>\n",
       "      <td>0.532520</td>\n",
       "      <td>0.502528</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.537848</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>worst fractal dimension</th>\n",
       "      <td>0.007066</td>\n",
       "      <td>0.119205</td>\n",
       "      <td>0.051019</td>\n",
       "      <td>0.003738</td>\n",
       "      <td>0.499316</td>\n",
       "      <td>0.687382</td>\n",
       "      <td>0.514930</td>\n",
       "      <td>0.368661</td>\n",
       "      <td>0.438413</td>\n",
       "      <td>0.767297</td>\n",
       "      <td>...</td>\n",
       "      <td>0.093492</td>\n",
       "      <td>0.219122</td>\n",
       "      <td>0.138957</td>\n",
       "      <td>0.079647</td>\n",
       "      <td>0.617624</td>\n",
       "      <td>0.810455</td>\n",
       "      <td>0.686511</td>\n",
       "      <td>0.511114</td>\n",
       "      <td>0.537848</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>30 rows × 30 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                         mean radius  mean texture  mean perimeter  mean area  \\\n",
       "mean radius                 1.000000      0.323782        0.997855   0.987357   \n",
       "mean texture                0.323782      1.000000        0.329533   0.321086   \n",
       "mean perimeter              0.997855      0.329533        1.000000   0.986507   \n",
       "mean area                   0.987357      0.321086        0.986507   1.000000   \n",
       "mean smoothness             0.170581     -0.023389        0.207278   0.177028   \n",
       "mean compactness            0.506124      0.236702        0.556936   0.498502   \n",
       "mean concavity              0.676764      0.302418        0.716136   0.685983   \n",
       "mean concave points         0.822529      0.293464        0.850977   0.823269   \n",
       "mean symmetry               0.147741      0.071401        0.183027   0.151293   \n",
       "mean fractal dimension     -0.311631     -0.076437       -0.261477  -0.283110   \n",
       "radius error                0.679090      0.275869        0.691765   0.732562   \n",
       "texture error              -0.097317      0.386358       -0.086761  -0.066280   \n",
       "perimeter error             0.674172      0.281673        0.693135   0.726628   \n",
       "area error                  0.735864      0.259845        0.744983   0.800086   \n",
       "smoothness error           -0.222600      0.006614       -0.202694  -0.166777   \n",
       "compactness error           0.206000      0.191975        0.250744   0.212583   \n",
       "concavity error             0.194204      0.143293        0.228082   0.207660   \n",
       "concave points error        0.376169      0.163851        0.407217   0.372320   \n",
       "symmetry error             -0.104321      0.009127       -0.081629  -0.072497   \n",
       "fractal dimension error    -0.042641      0.054458       -0.005523  -0.019887   \n",
       "worst radius                0.969539      0.352573        0.969476   0.962746   \n",
       "worst texture               0.297008      0.912045        0.303038   0.287489   \n",
       "worst perimeter             0.965137      0.358040        0.970387   0.959120   \n",
       "worst area                  0.941082      0.343546        0.941550   0.959213   \n",
       "worst smoothness            0.119616      0.077503        0.150549   0.123523   \n",
       "worst compactness           0.413463      0.277830        0.455774   0.390410   \n",
       "worst concavity             0.526911      0.301025        0.563879   0.512606   \n",
       "worst concave points        0.744214      0.295316        0.771241   0.722017   \n",
       "worst symmetry              0.163953      0.105008        0.189115   0.143570   \n",
       "worst fractal dimension     0.007066      0.119205        0.051019   0.003738   \n",
       "\n",
       "                         mean smoothness  mean compactness  mean concavity  \\\n",
       "mean radius                     0.170581          0.506124        0.676764   \n",
       "mean texture                   -0.023389          0.236702        0.302418   \n",
       "mean perimeter                  0.207278          0.556936        0.716136   \n",
       "mean area                       0.177028          0.498502        0.685983   \n",
       "mean smoothness                 1.000000          0.659123        0.521984   \n",
       "mean compactness                0.659123          1.000000        0.883121   \n",
       "mean concavity                  0.521984          0.883121        1.000000   \n",
       "mean concave points             0.553695          0.831135        0.921391   \n",
       "mean symmetry                   0.557775          0.602641        0.500667   \n",
       "mean fractal dimension          0.584792          0.565369        0.336783   \n",
       "radius error                    0.301467          0.497473        0.631925   \n",
       "texture error                   0.068406          0.046205        0.076218   \n",
       "perimeter error                 0.296092          0.548905        0.660391   \n",
       "area error                      0.246552          0.455653        0.617427   \n",
       "smoothness error                0.332375          0.135299        0.098564   \n",
       "compactness error               0.318943          0.738722        0.670279   \n",
       "concavity error                 0.248396          0.570517        0.691270   \n",
       "concave points error            0.380676          0.642262        0.683260   \n",
       "symmetry error                  0.200774          0.229977        0.178009   \n",
       "fractal dimension error         0.283607          0.507318        0.449301   \n",
       "worst radius                    0.213120          0.535315        0.688236   \n",
       "worst texture                   0.036072          0.248133        0.299879   \n",
       "worst perimeter                 0.238853          0.590210        0.729565   \n",
       "worst area                      0.206718          0.509604        0.675987   \n",
       "worst smoothness                0.805324          0.565541        0.448822   \n",
       "worst compactness               0.472468          0.865809        0.754968   \n",
       "worst concavity                 0.434926          0.816275        0.884103   \n",
       "worst concave points            0.503053          0.815573        0.861323   \n",
       "worst symmetry                  0.394309          0.510223        0.409464   \n",
       "worst fractal dimension         0.499316          0.687382        0.514930   \n",
       "\n",
       "                         mean concave points  mean symmetry  \\\n",
       "mean radius                         0.822529       0.147741   \n",
       "mean texture                        0.293464       0.071401   \n",
       "mean perimeter                      0.850977       0.183027   \n",
       "mean area                           0.823269       0.151293   \n",
       "mean smoothness                     0.553695       0.557775   \n",
       "mean compactness                    0.831135       0.602641   \n",
       "mean concavity                      0.921391       0.500667   \n",
       "mean concave points                 1.000000       0.462497   \n",
       "mean symmetry                       0.462497       1.000000   \n",
       "mean fractal dimension              0.166917       0.479921   \n",
       "radius error                        0.698050       0.303379   \n",
       "texture error                       0.021480       0.128053   \n",
       "perimeter error                     0.710650       0.313893   \n",
       "area error                          0.690299       0.223970   \n",
       "smoothness error                    0.027653       0.187321   \n",
       "compactness error                   0.490424       0.421659   \n",
       "concavity error                     0.439167       0.342627   \n",
       "concave points error                0.615634       0.393298   \n",
       "symmetry error                      0.095351       0.449137   \n",
       "fractal dimension error             0.257584       0.331786   \n",
       "worst radius                        0.830318       0.185728   \n",
       "worst texture                       0.292752       0.090651   \n",
       "worst perimeter                     0.855923       0.219169   \n",
       "worst area                          0.809630       0.177193   \n",
       "worst smoothness                    0.452753       0.426675   \n",
       "worst compactness                   0.667454       0.473200   \n",
       "worst concavity                     0.752399       0.433721   \n",
       "worst concave points                0.910155       0.430297   \n",
       "worst symmetry                      0.375744       0.699826   \n",
       "worst fractal dimension             0.368661       0.438413   \n",
       "\n",
       "                         mean fractal dimension           ...             \\\n",
       "mean radius                           -0.311631           ...              \n",
       "mean texture                          -0.076437           ...              \n",
       "mean perimeter                        -0.261477           ...              \n",
       "mean area                             -0.283110           ...              \n",
       "mean smoothness                        0.584792           ...              \n",
       "mean compactness                       0.565369           ...              \n",
       "mean concavity                         0.336783           ...              \n",
       "mean concave points                    0.166917           ...              \n",
       "mean symmetry                          0.479921           ...              \n",
       "mean fractal dimension                 1.000000           ...              \n",
       "radius error                           0.000111           ...              \n",
       "texture error                          0.164174           ...              \n",
       "perimeter error                        0.039830           ...              \n",
       "area error                            -0.090170           ...              \n",
       "smoothness error                       0.401964           ...              \n",
       "compactness error                      0.559837           ...              \n",
       "concavity error                        0.446630           ...              \n",
       "concave points error                   0.341198           ...              \n",
       "symmetry error                         0.345007           ...              \n",
       "fractal dimension error                0.688132           ...              \n",
       "worst radius                          -0.253691           ...              \n",
       "worst texture                         -0.051269           ...              \n",
       "worst perimeter                       -0.205151           ...              \n",
       "worst area                            -0.231854           ...              \n",
       "worst smoothness                       0.504942           ...              \n",
       "worst compactness                      0.458798           ...              \n",
       "worst concavity                        0.346234           ...              \n",
       "worst concave points                   0.175325           ...              \n",
       "worst symmetry                         0.334019           ...              \n",
       "worst fractal dimension                0.767297           ...              \n",
       "\n",
       "                         worst radius  worst texture  worst perimeter  \\\n",
       "mean radius                  0.969539       0.297008         0.965137   \n",
       "mean texture                 0.352573       0.912045         0.358040   \n",
       "mean perimeter               0.969476       0.303038         0.970387   \n",
       "mean area                    0.962746       0.287489         0.959120   \n",
       "mean smoothness              0.213120       0.036072         0.238853   \n",
       "mean compactness             0.535315       0.248133         0.590210   \n",
       "mean concavity               0.688236       0.299879         0.729565   \n",
       "mean concave points          0.830318       0.292752         0.855923   \n",
       "mean symmetry                0.185728       0.090651         0.219169   \n",
       "mean fractal dimension      -0.253691      -0.051269        -0.205151   \n",
       "radius error                 0.715065       0.194799         0.719684   \n",
       "texture error               -0.111690       0.409003        -0.102242   \n",
       "perimeter error              0.697201       0.200371         0.721031   \n",
       "area error                   0.757373       0.196497         0.761213   \n",
       "smoothness error            -0.230691      -0.074743        -0.217304   \n",
       "compactness error            0.204607       0.143003         0.260516   \n",
       "concavity error              0.186904       0.100241         0.226680   \n",
       "concave points error         0.358127       0.086741         0.394999   \n",
       "symmetry error              -0.128121      -0.077473        -0.103753   \n",
       "fractal dimension error     -0.037488      -0.003195        -0.001000   \n",
       "worst radius                 1.000000       0.359921         0.993708   \n",
       "worst texture                0.359921       1.000000         0.365098   \n",
       "worst perimeter              0.993708       0.365098         1.000000   \n",
       "worst area                   0.984015       0.345842         0.977578   \n",
       "worst smoothness             0.216574       0.225429         0.236775   \n",
       "worst compactness            0.475820       0.360832         0.529408   \n",
       "worst concavity              0.573975       0.368366         0.618344   \n",
       "worst concave points         0.787424       0.359755         0.816322   \n",
       "worst symmetry               0.243529       0.233027         0.269493   \n",
       "worst fractal dimension      0.093492       0.219122         0.138957   \n",
       "\n",
       "                         worst area  worst smoothness  worst compactness  \\\n",
       "mean radius                0.941082          0.119616           0.413463   \n",
       "mean texture               0.343546          0.077503           0.277830   \n",
       "mean perimeter             0.941550          0.150549           0.455774   \n",
       "mean area                  0.959213          0.123523           0.390410   \n",
       "mean smoothness            0.206718          0.805324           0.472468   \n",
       "mean compactness           0.509604          0.565541           0.865809   \n",
       "mean concavity             0.675987          0.448822           0.754968   \n",
       "mean concave points        0.809630          0.452753           0.667454   \n",
       "mean symmetry              0.177193          0.426675           0.473200   \n",
       "mean fractal dimension    -0.231854          0.504942           0.458798   \n",
       "radius error               0.751548          0.141919           0.287103   \n",
       "texture error             -0.083195         -0.073658          -0.092439   \n",
       "perimeter error            0.730713          0.130054           0.341919   \n",
       "area error                 0.811408          0.125389           0.283257   \n",
       "smoothness error          -0.182195          0.314457          -0.055558   \n",
       "compactness error          0.199371          0.227394           0.678780   \n",
       "concavity error            0.188353          0.168481           0.484858   \n",
       "concave points error       0.342271          0.215351           0.452888   \n",
       "symmetry error            -0.110343         -0.012662           0.060255   \n",
       "fractal dimension error   -0.022736          0.170568           0.390159   \n",
       "worst radius               0.984015          0.216574           0.475820   \n",
       "worst texture              0.345842          0.225429           0.360832   \n",
       "worst perimeter            0.977578          0.236775           0.529408   \n",
       "worst area                 1.000000          0.209145           0.438296   \n",
       "worst smoothness           0.209145          1.000000           0.568187   \n",
       "worst compactness          0.438296          0.568187           1.000000   \n",
       "worst concavity            0.543331          0.518523           0.892261   \n",
       "worst concave points       0.747419          0.547691           0.801080   \n",
       "worst symmetry             0.209146          0.493838           0.614441   \n",
       "worst fractal dimension    0.079647          0.617624           0.810455   \n",
       "\n",
       "                         worst concavity  worst concave points  \\\n",
       "mean radius                     0.526911              0.744214   \n",
       "mean texture                    0.301025              0.295316   \n",
       "mean perimeter                  0.563879              0.771241   \n",
       "mean area                       0.512606              0.722017   \n",
       "mean smoothness                 0.434926              0.503053   \n",
       "mean compactness                0.816275              0.815573   \n",
       "mean concavity                  0.884103              0.861323   \n",
       "mean concave points             0.752399              0.910155   \n",
       "mean symmetry                   0.433721              0.430297   \n",
       "mean fractal dimension          0.346234              0.175325   \n",
       "radius error                    0.380585              0.531062   \n",
       "texture error                  -0.068956             -0.119638   \n",
       "perimeter error                 0.418899              0.554897   \n",
       "area error                      0.385100              0.538166   \n",
       "smoothness error               -0.058298             -0.102007   \n",
       "compactness error               0.639147              0.483208   \n",
       "concavity error                 0.662564              0.440472   \n",
       "concave points error            0.549592              0.602450   \n",
       "symmetry error                  0.037119             -0.030413   \n",
       "fractal dimension error         0.379975              0.215204   \n",
       "worst radius                    0.573975              0.787424   \n",
       "worst texture                   0.368366              0.359755   \n",
       "worst perimeter                 0.618344              0.816322   \n",
       "worst area                      0.543331              0.747419   \n",
       "worst smoothness                0.518523              0.547691   \n",
       "worst compactness               0.892261              0.801080   \n",
       "worst concavity                 1.000000              0.855434   \n",
       "worst concave points            0.855434              1.000000   \n",
       "worst symmetry                  0.532520              0.502528   \n",
       "worst fractal dimension         0.686511              0.511114   \n",
       "\n",
       "                         worst symmetry  worst fractal dimension  \n",
       "mean radius                    0.163953                 0.007066  \n",
       "mean texture                   0.105008                 0.119205  \n",
       "mean perimeter                 0.189115                 0.051019  \n",
       "mean area                      0.143570                 0.003738  \n",
       "mean smoothness                0.394309                 0.499316  \n",
       "mean compactness               0.510223                 0.687382  \n",
       "mean concavity                 0.409464                 0.514930  \n",
       "mean concave points            0.375744                 0.368661  \n",
       "mean symmetry                  0.699826                 0.438413  \n",
       "mean fractal dimension         0.334019                 0.767297  \n",
       "radius error                   0.094543                 0.049559  \n",
       "texture error                 -0.128215                -0.045655  \n",
       "perimeter error                0.109930                 0.085433  \n",
       "area error                     0.074126                 0.017539  \n",
       "smoothness error              -0.107342                 0.101480  \n",
       "compactness error              0.277878                 0.590973  \n",
       "concavity error                0.197788                 0.439329  \n",
       "concave points error           0.143116                 0.310655  \n",
       "symmetry error                 0.389402                 0.078079  \n",
       "fractal dimension error        0.111094                 0.591328  \n",
       "worst radius                   0.243529                 0.093492  \n",
       "worst texture                  0.233027                 0.219122  \n",
       "worst perimeter                0.269493                 0.138957  \n",
       "worst area                     0.209146                 0.079647  \n",
       "worst smoothness               0.493838                 0.617624  \n",
       "worst compactness              0.614441                 0.810455  \n",
       "worst concavity                0.532520                 0.686511  \n",
       "worst concave points           0.502528                 0.511114  \n",
       "worst symmetry                 1.000000                 0.537848  \n",
       "worst fractal dimension        0.537848                 1.000000  \n",
       "\n",
       "[30 rows x 30 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "corr_matrix = df.corr()\n",
    "corr_matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.44272026 0.18971182 0.09393163] 0.7263637090894923\n"
     ]
    }
   ],
   "source": [
    "pca = PCA(n_components=3)\n",
    "pca.fit(preprocessed_data)\n",
    "decomposed_data = pca.transform(preprocessed_data)\n",
    "print(pca.explained_variance_ratio_, sum(pca.explained_variance_ratio_))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
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   ],
   "source": [
    "data = go.Heatmap(z=pca.components_, \n",
    "                  x=bunch.feature_names, \n",
    "                  y=['PC 1', 'PC 2', 'PC 3'], \n",
    "                  colorscale='Viridis')\n",
    " \n",
    "# Plot heatmap.\n",
    "plotly.offline.iplot([data], filename='heatmap')\n",
    "#This plot shows how each feature correlates with each principal component"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Add malignant column.\n",
    "decomposed_df = pd.DataFrame(decomposed_data, columns=['x', 'y', 'z'])\n",
    "decomposed_df['malignant'] = 1 - bunch.target\n",
    " \n",
    "# Create individual data sets.\n",
    "malignant = decomposed_df[decomposed_df.malignant == 1]\n",
    "benign = decomposed_df[decomposed_df.malignant == 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
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{\"showLink\": true, \"linkText\": \"Export to plot.ly\"})});</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create line style.\n",
    "line_style = dict(color='rgba(0, 0, 0, 0.14)',width=0.5)\n",
    " \n",
    "# Create scatters.\n",
    "malignant_scatter = go.Scatter3d(\n",
    "    x=malignant['x'],\n",
    "    y=malignant['y'],\n",
    "    z=malignant['z'],\n",
    "    mode='markers',\n",
    "    marker=dict(\n",
    "        color='rgb(181, 20, 37)',\n",
    "        size=12,\n",
    "        opacity=0.8,\n",
    "        line=line_style\n",
    "    ),\n",
    "    name='Malignant'\n",
    ")\n",
    "benign_scatter = go.Scatter3d(\n",
    "    x=benign['x'],\n",
    "    y=benign['y'],\n",
    "    z=benign['z'],\n",
    "    mode='markers',\n",
    "    marker=dict(\n",
    "        color='rgb(5, 99, 226)',\n",
    "        size=12,\n",
    "        opacity=0.8,\n",
    "        line=line_style\n",
    "    ),\n",
    "    name='Benign'\n",
    ")\n",
    " \n",
    "# Create data array. Ensure malignant scatter is rendered above (can we merge layers somehow?).\n",
    "data = [benign_scatter, malignant_scatter]\n",
    " \n",
    "# Create layout.\n",
    "layout = go.Layout(showlegend=True, margin=dict(l=0,r=0,b=0,t=0))\n",
    " \n",
    "# Render (offline).\n",
    "fig = go.Figure(data=data, layout=layout)\n",
    "plotly.offline.iplot(fig, filename='3d-scatter')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>mean radius</th>\n",
       "      <th>mean texture</th>\n",
       "      <th>mean perimeter</th>\n",
       "      <th>mean area</th>\n",
       "      <th>mean smoothness</th>\n",
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       "      <th>...</th>\n",
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       "      <td>0.10960</td>\n",
       "      <td>0.15990</td>\n",
       "      <td>0.1974</td>\n",
       "      <td>0.12790</td>\n",
       "      <td>0.2069</td>\n",
       "      <td>0.05999</td>\n",
       "      <td>...</td>\n",
       "      <td>23.57</td>\n",
       "      <td>25.53</td>\n",
       "      <td>152.50</td>\n",
       "      <td>1709.0</td>\n",
       "      <td>0.1444</td>\n",
       "      <td>0.4245</td>\n",
       "      <td>0.4504</td>\n",
       "      <td>0.2430</td>\n",
       "      <td>0.3613</td>\n",
       "      <td>0.08758</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>11.42</td>\n",
       "      <td>20.38</td>\n",
       "      <td>77.58</td>\n",
       "      <td>386.1</td>\n",
       "      <td>0.14250</td>\n",
       "      <td>0.28390</td>\n",
       "      <td>0.2414</td>\n",
       "      <td>0.10520</td>\n",
       "      <td>0.2597</td>\n",
       "      <td>0.09744</td>\n",
       "      <td>...</td>\n",
       "      <td>14.91</td>\n",
       "      <td>26.50</td>\n",
       "      <td>98.87</td>\n",
       "      <td>567.7</td>\n",
       "      <td>0.2098</td>\n",
       "      <td>0.8663</td>\n",
       "      <td>0.6869</td>\n",
       "      <td>0.2575</td>\n",
       "      <td>0.6638</td>\n",
       "      <td>0.17300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>20.29</td>\n",
       "      <td>14.34</td>\n",
       "      <td>135.10</td>\n",
       "      <td>1297.0</td>\n",
       "      <td>0.10030</td>\n",
       "      <td>0.13280</td>\n",
       "      <td>0.1980</td>\n",
       "      <td>0.10430</td>\n",
       "      <td>0.1809</td>\n",
       "      <td>0.05883</td>\n",
       "      <td>...</td>\n",
       "      <td>22.54</td>\n",
       "      <td>16.67</td>\n",
       "      <td>152.20</td>\n",
       "      <td>1575.0</td>\n",
       "      <td>0.1374</td>\n",
       "      <td>0.2050</td>\n",
       "      <td>0.4000</td>\n",
       "      <td>0.1625</td>\n",
       "      <td>0.2364</td>\n",
       "      <td>0.07678</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 30 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "   mean radius  mean texture  mean perimeter  mean area  mean smoothness  \\\n",
       "0        17.99         10.38          122.80     1001.0          0.11840   \n",
       "1        20.57         17.77          132.90     1326.0          0.08474   \n",
       "2        19.69         21.25          130.00     1203.0          0.10960   \n",
       "3        11.42         20.38           77.58      386.1          0.14250   \n",
       "4        20.29         14.34          135.10     1297.0          0.10030   \n",
       "\n",
       "   mean compactness  mean concavity  mean concave points  mean symmetry  \\\n",
       "0           0.27760          0.3001              0.14710         0.2419   \n",
       "1           0.07864          0.0869              0.07017         0.1812   \n",
       "2           0.15990          0.1974              0.12790         0.2069   \n",
       "3           0.28390          0.2414              0.10520         0.2597   \n",
       "4           0.13280          0.1980              0.10430         0.1809   \n",
       "\n",
       "   mean fractal dimension           ...             worst radius  \\\n",
       "0                 0.07871           ...                    25.38   \n",
       "1                 0.05667           ...                    24.99   \n",
       "2                 0.05999           ...                    23.57   \n",
       "3                 0.09744           ...                    14.91   \n",
       "4                 0.05883           ...                    22.54   \n",
       "\n",
       "   worst texture  worst perimeter  worst area  worst smoothness  \\\n",
       "0          17.33           184.60      2019.0            0.1622   \n",
       "1          23.41           158.80      1956.0            0.1238   \n",
       "2          25.53           152.50      1709.0            0.1444   \n",
       "3          26.50            98.87       567.7            0.2098   \n",
       "4          16.67           152.20      1575.0            0.1374   \n",
       "\n",
       "   worst compactness  worst concavity  worst concave points  worst symmetry  \\\n",
       "0             0.6656           0.7119                0.2654          0.4601   \n",
       "1             0.1866           0.2416                0.1860          0.2750   \n",
       "2             0.4245           0.4504                0.2430          0.3613   \n",
       "3             0.8663           0.6869                0.2575          0.6638   \n",
       "4             0.2050           0.4000                0.1625          0.2364   \n",
       "\n",
       "   worst fractal dimension  \n",
       "0                  0.11890  \n",
       "1                  0.08902  \n",
       "2                  0.08758  \n",
       "3                  0.17300  \n",
       "4                  0.07678  \n",
       "\n",
       "[5 rows x 30 columns]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get data.\n",
    "bunch = load_breast_cancer()\n",
    "df = pd.DataFrame(bunch.data, columns=bunch.feature_names)\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "dataset = load_breast_cancer()\n",
    "features = dataset['data']\n",
    "labels = dataset['target']\n",
    "train, test, train_labels, test_labels = train_test_split(features,\n",
    "                                                          labels,\n",
    "                                                          test_size=0.15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9069767441860465\n"
     ]
    }
   ],
   "source": [
    "newClassifier = KNeighborsClassifier()\n",
    "newClassifier.fit(train, train_labels)\n",
    "print(newClassifier.score(test, test_labels))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "modifiedDataset = load_breast_cancer()\n",
    "X_raw = modifiedDataset['data']\n",
    "y = modifiedDataset['target']\n",
    "scaler = StandardScaler()\n",
    "X = scaler.fit_transform(X_raw)\n",
    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.20, random_state=42)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9473684210526315\n"
     ]
    }
   ],
   "source": [
    "pca = PCA(n_components=25)\n",
    "X_train_pca = pca.fit_transform(X_train)\n",
    "nearClassifier = KNeighborsClassifier()\n",
    "X_test_pca = pca.transform(X_test)\n",
    "nearClassifier.fit(X_train_pca, y_train)\n",
    "print(nearClassifier.score(X_test_pca, y_test))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Incremental PCA\n",
    "Regular PCA required the training set to be in memory at once. Incremental PCA allows you to split the training data in mini-batches.\n",
    "\n",
    "# Randomized PCA \n",
    "Is a stochastic algorithm that approximates the first d principal components. It's complexisty is much less than regular PCA.\n",
    "\n",
    "# Kernel PCA\n",
    "Kernel PCA uses the Kernel trick (a mathematical technique that implicitly maps training instaces into a high dimensional space, allowing for nonlinear classification and regression. It allows for linear decision boundaries in high dimensional space which translate to nonlinear boundries in original space. <br>\n",
    "Kernel trick can be applied to PCA to allow for nonlinear projections for dimensionality reduction.\n",
    "It is good at preserving clusters of instances after projection. \n",
    "# LLE\n",
    "Locally Linear Embedding is a manifold technique. LLE works by measuring how each training instance linearly relates to its closest neighbors and then looks for a low-dimensional representation of the training set where these local relationships are best preserved. This algorithm is good at unrolling twisted manifolds, when there isn't noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9473684210526315\n"
     ]
    }
   ],
   "source": [
    "from sklearn.decomposition import KernelPCA\n",
    "rbf_pca = KernelPCA(n_components = 25, kernel=\"rbf\", gamma=0.001)\n",
    "X_reduced = rbf_pca.fit_transform(X_train)\n",
    "secondaryClassifier = KNeighborsClassifier()\n",
    "X_reduced_test = rbf_pca.transform(X_test)\n",
    "secondaryClassifier.fit(X_reduced, y_train)\n",
    "print(secondaryClassifier.score(X_reduced_test, y_test))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9210526315789473\n"
     ]
    }
   ],
   "source": [
    "from sklearn.manifold import LocallyLinearEmbedding\n",
    "lle = LocallyLinearEmbedding(n_components = 25, n_neighbors=20)\n",
    "X_reduced_lle = lle.fit_transform(X_train)\n",
    "thirdClassifier = KNeighborsClassifier()\n",
    "X_reduced_lle_test = lle.transform(X_test)\n",
    "thirdClassifier.fit(X_reduced_lle, y_train)\n",
    "print(thirdClassifier.score(X_reduced_lle_test, y_test))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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