{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# PCA on artificial data" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from mpl_toolkits.mplot3d import Axes3D\n", "\n", "# seed the pseudo random number generator with 0 \n", "# so that the results are consistent in all discussion sections\n", "np.random.seed(0)\n", "\n", "# Matplotlib produces annoying warnings (see https://github.com/matplotlib/matplotlib/issues/5209)\n", "# ignore warnings for now\n", "import warnings\n", "warnings.filterwarnings('ignore')\n", "\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this example we'll generate two clusters of data points on a plane, combine them into a single dataset, randomly embed it in a higher dimensional space, add quite a lot of noise, and then try to use PCA to separate the data into the two original clusters." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Generating 2D data" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For each cluster we'll draw random points drawn from a Gaussian distribution centered around its mean, with a predefined standard deviation.\n", "\n", "[You will learn more about probability, the gaussian distribution, mean and standard deviation in CS70. For now it is enough to know that the points of each clustered are centered around the mean and at least 95% are within twice the standard deviation from the mean.]" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def cluster(mean, stddev, size):\n", " return np.random.multivariate_normal(mean, np.eye(mean.shape[0])*stddev**2, size).T" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'll create two clusters of 5000 points each, centered around (5,5) and (-5, -5) with standard deviation of 1." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "N = 5000\n", "\n", "mean = np.array([5, 5])\n", "stddev = 1\n", "\n", "cluster1 = cluster(mean, stddev, N)\n", "cluster2 = cluster(-mean, stddev, N)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'll combine both clusters into a single dataset and shuffle it for good measure" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "dataset_2d = np.c_[cluster1, cluster2]\n", "np.random.shuffle(dataset_2d)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The dataset looks pretty much like we expected." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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n8/c3uMjBn82fqyIMiIbw2g8RhmlhcBYFWIucfmc+voUwDDMuMoZ6qGoJyxqS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\npoFtL3Dx4g7Xr9+m2x0Sxy3y+YgwTBIEi5jmMpZlkEyO8f2IKBoRRQH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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(*dataset_2d, alpha=0.1)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Embedding the 2D data into a 25D space" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have a 2D dataset, we'll embed it into a 25 dimensional space by multiplying it with a random 25x2 matrix." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To understand how this works, we'll first embed it in a three dimensional space in two ways." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def embed_and_plot(M):\n", " \n", " dataset_3d = M.dot(dataset_2d)\n", "\n", " ax = plt.subplot(projection='3d')\n", " ax.scatter(*dataset_3d, alpha=0.1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, embed the our points in the X-Y plane:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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FhRTPPHOSQiHG8nKR558vk82eYGqqwOxsgVIpg+omGALmgAoqyc4Ay8DlwEFU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\nVKej4HorvZdIqlEi1pchB4PBvl6Ci3ti5oPu1fQ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "embed_and_plot(np.array([[1, 0],[0, 1], [0, 0]]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next, embed it as a different plane. From our point of view, it is now harder to separate the clusters." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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qq6ujU6dOaW8jFAoRDAYpKys7advaxgY5yFKPoigcPXqUzp07p/3+gUAgrXrF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\n4403csstt+Ts/XOEJbr5IhqNsmPHDlasWMHixYv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Iwj1VuusCCScVQOaF+39uL5i9F30m\ndkAraQBmoArOsw946ZrdN0rKD++FKTIspbQAZWSGfc/vBbNr4lz96YFmn2u2ZoHZIyvFTA1Td9Zu\nXwAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "embed_and_plot(np.array([[1, 0],[1, -1], [1, 0]]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now return to our 25-dimensional space. We'll generate a random 25x2 matrix and multiply the matrix by each point.\n", "\n", "[Note that this matrix has to be of rank 2, otherwise the data will become one dimensional - but the probability of a random 25x2 matrix to be of rank less than 2 is negligible]" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The rank of M is 2\n" ] } ], "source": [ "D = 25\n", "M = np.random.rand(D, 2)\n", "print(\"The rank of M is\", np.linalg.matrix_rank(M))\n", "\n", "dataset_Dd = M.dot(dataset_2d)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "How does this data look like? Let's pick two random axes and view the data projected over these axis." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [], "source": [ "axes = np.random.randint(0, D, 2)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(dataset_Dd[axes[0]], dataset_Dd[axes[1]], alpha=0.1)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Not too bad, even though we picked random axes, the two clusters are clearly distinguishable." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Adding some noise" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "But what happens if we add random noise? let's add quite a lot of noise:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [], "source": [ "noise_stddev = 10\n", "noise = np.random.multivariate_normal(np.zeros(D), np.eye(D)*noise_stddev**2, dataset_Dd.shape[1]).T\n", "\n", "noisy_dataset = dataset_Dd + noise" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's draw it again over the same axes:" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Dqt+k5XPv3Mf/PW7LvR2TPO/z7Qw/3wErYjodMxo5ZDJFgqBBuVxndTVHtZqh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TAAAg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\n3D7zfX0RgkGLSKTB5uYq3//+EqXSHP39AXy+IQyjg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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(noisy_dataset[axes[0]], noisy_dataset[axes[1]], alpha=0.1)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The two separate clusters are pretty much gone now, as the clusters are overshadowed by the noise." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## PCA" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Can we use PCA to recover the clusters from the noise?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In PCA, we perform SVD and project over the columns of U that correspond to the singular values with the highest magnitude. These vectors are called the **principal components**." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "U, S, V = np.linalg.svd(noisy_dataset)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this case, we'll pick the first two principal components. Why two? we started with two dimensional data, so it makes sense to pick two principal components. We'll revisit this decision later." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "PC = U[::,:2]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'll project the data into the first two principal components:" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "projected_dataset = PC.T.dot(noisy_dataset)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Then plot the result:" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "plt.scatter(*projected_dataset, alpha=0.1)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Even though we added a lot of noise, we still get two clear, mostly distinguishable, clusters." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Principal components:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us go back to the signular values and plot them:" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "plt.bar(np.arange(S.shape[0]), S)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The singular value corresponding to the first principal component is larger than the rest, which are all very similar.\n", "\n", "Does it make sense? we started with 2D data, shouldn't we expect the first two signular values to significantly larger then the rest?\n", " \n", "It does. Even in the original data, we could project all the points on the line connecting the two cluster means to clearly distinguish the clusters. \n", "\n", "The variation in the other direction on the plane still exists in the data, but it is very small compared to all the noise we added." ] } ], "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": "3.4.4" } }, "nbformat": 4, "nbformat_minor": 0 }