{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib notebook\n", "\n", "import pandas as pd\n", "import numpy as np\n", "import sklearn.datasets\n", "\n", "import matplotlib.pyplot as plt\n", "\n", "plt.rcParams['figure.figsize'] = [8, 4]" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "iris = sklearn.datasets.load_iris()\n", "\n", "X = pd.DataFrame(iris.data, columns=iris.feature_names)\n", "\n", "X_zscaled = (X - X.mean()) / X.std(ddof=1)\n", "\n", "Y = pd.DataFrame(iris.target, columns=['target'])\n", "Y['species'] = Y.apply(lambda r: iris.target_names[r])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Multicollinearity Check\n", "\n", "Using $corr(X)$, check to see that we some level of multicollinearity in the data, enough to warrant using PCA of visualization in reduced-rank principal component space. As a rule of thumb, the off-diagonal correlation values (either in the upper- or lower-triangle) should have absolute values of around 0.30 or so." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Correlation matrix:\n", "\n", "[[ 1. -0.10936925 0.87175416 0.81795363]\n", " [-0.10936925 1. -0.4205161 -0.35654409]\n", " [ 0.87175416 -0.4205161 1. 0.9627571 ]\n", " [ 0.81795363 -0.35654409 0.9627571 1. ]]\n", "\n", "\n", "Multicollinearity check using off-diagonal values:\n", "\n", "count 6.000000\n", "mean 0.589816\n", "std 0.341915\n", "min 0.109369\n", "25% 0.372537\n", "50% 0.619235\n", "75% 0.858304\n", "max 0.962757\n", "dtype: float64\n" ] } ], "source": [ "corr = X_zscaled.corr()\n", "\n", "tmp = pd.np.triu(corr) - np.eye(corr.shape[0]) \n", "tmp = tmp.flatten()\n", "tmp = tmp[np.nonzero(tmp)]\n", "tmp = pd.Series(np.abs(tmp))\n", "\n", "print('Correlation matrix:\\n\\n{}\\n\\n'.format(corr.values))\n", "\n", "print('Multicollinearity check using off-diagonal values:\\n\\n{}'.format(tmp.describe()))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## PCA via Eigen-Decomposition" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Obtain eigenvalues, eigenvectors of $cov(X)$ via eigen-decomposition\n", "\n", "From the factorization of symmetric matrix $S$ into orthogonal matrix $Q$ of the eigenvectors and diagonal matrix $\\Lambda$ of the eigenvalues, we can likewise decompose $cov(X)$ (or in our case $corr(X)$ since we standardized our data).\n", "\n", "\\begin{align}\n", " S &= Q \\Lambda Q^\\intercal \\\\\n", " \\Rightarrow X^\\intercal X &= V \\Lambda V^\\intercal \\\\\n", "\\end{align}\n", "\n", "where $V$ are the orthonormal eigenvectors of $X X^\\intercal$.\n", "\n", "We can normalize the eigenvalues to see how much variance is captured per each respective principal component. We will also calculate the cumulative variance explained. This will help inform our decision of how many principal components to keep when reducing dimensions in visualizing the data." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "eigenvalues, eigenvectors = np.linalg.eig(X_zscaled.cov())\n", "\n", "eigenvalues_normalized = eigenvalues / eigenvalues.sum()\n", "\n", "cumvar_explained = np.cumsum(eigenvalues_normalized)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Reduce dimensions and visualize\n", "\n", "To project the original data into principal component space, we obtain score matrix $T$ by taking the dot product of $X$ and the eigenvectors $V$.\n", "\n", "\\begin{align}\n", " T &= X V \\\\\n", "\\end{align}" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "T = pd.DataFrame(X_zscaled.dot(eigenvectors))\n", "\n", "# set column names\n", "T.columns = ['pc1', 'pc2', 'pc3', 'pc4']\n", "\n", "# also add the species label as \n", "T = pd.concat([T, Y.species], axis=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can visualize the original, 4D iris data of $X$ by using the first $k$ eigenvectors of $X$, projecting the original data into a reduced-rank $k$-dimensional principal component space.\n", "\n", "\\begin{align}\n", " T_{rank=k} &= X V_{rank=k}\n", "\\end{align}" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "scrolled": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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