{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Understanding the SVD" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Useful reference\n", "\n", "- [A Singularly Valuable Decomposition](https://datajobs.com/data-science-repo/SVD-[Dan-Kalman].pdf)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Sketch of lecture" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### Singular value decomposition\n", "\n", "Our goal is to understand the following forms of the SVD.\n", "\n", "$$\n", "A = U \\Sigma V^T\n", "$$\n", "\n", "$$\n", "A = \\begin{bmatrix}\n", "U_1 & U_2\n", "\\end{bmatrix}\\begin{bmatrix}\n", "\\Sigma_1 & 0 \\\\\n", "0 & 0 \n", "\\end{bmatrix}\\begin{bmatrix}\n", "V_1^T \\\\\n", "V_2^T\n", "\\end{bmatrix}\n", "$$\n", "\n", "$$\n", "A = \\sum_{i=1}^r \\sigma u_i v_i^T\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### (1) The matrix A" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### What does a matrix do?\n", "\n", "A linear function is one that satisfies the property that\n", "\n", "$$\n", "f(a_1x_1 + a_2x_2 + \\cdots + a_nx_n) = a_1 f(x_1) + a_2 f(x_2) + \\ldots + a_n f(x_n)\n", "$$\n", "\n", "Let $f(x) = Ax$, where $A$ is a matrix and $x$ is a vector. You can check that the matrix $A$ fulfills the property of being a linear function. If $A$ is $m \\times n$, then it is a linear map from $\\mathbb{R}^n \\mapsto \\mathbb{R}^m$.\n", "\n", "Let's consider: what does a matrix *do* to a vector? Matrix multiplication has a *geometric* interpretation. When we multiply a vector, we either rotate, reflect, dilate or some combination of those three. So multiplying by a matrix *transforms* one vector into another vector. This is known as a *linear transformation*.\n", "\n", "Important Facts: \n", "\n", "* Any matrix defines a linear transformation\n", "* The matrix form of a linear transformation is NOT unique\n", "* We need only define a transformation by saying what it does to a *basis*\n", "\n", "Suppose we have a matrix $A$ that defines some transformation. We can take any invertible matrix $B$ and\n", "\n", "$$BAB^{-1}$$\n", "\n", "defines the same transformation. This operation is called a *change of basis*, because we are simply expressing the transformation with respect to a different basis." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "Let $f(x)$ be the linear transformation that takes $e_1=(1,0)$ to $f(e_1)=(2,3)$ and $e_2=(0,1)$ to $f(e_2) = (1,1)$. A matrix representation of $f$ would be given by:\n", "\n", "$$A = \\left(\\begin{matrix}2 & 1\\\\3&1\\end{matrix}\\right)$$\n", "\n", "This is the matrix we use if we consider the vectors of $\\mathbb{R}^2$ to be linear combinations of the form \n", "\n", "$$c_1 e_1 + c_2 e_2$$\n", "\n", "Now, consider a second pair of (linearly independent) vectors in $\\mathbb{R}^2$, say $v_1=(1,3)$ and $v_2=(4,1)$. We first find the transformation that takes $e_1$ to $v_1$ and $e_2$ to $v_2$. A matrix representation for this is:\n", "\n", "$$B = \\left(\\begin{matrix}1 & 4\\\\3&1\\end{matrix}\\right)$$\n", "\n", "Our original transformation $f$ can be expressed with respect to the basis $v_1, v_2$ via\n", "\n", "$$B^{-1}AB$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Fundamental subspaces of $A$\n", "\n", "- Span and basis\n", "- Inner and outer products of vectors\n", "- Rank of outer product is 1\n", "- $C(A)$, $N(A)$, $(C(A^T))$ and $N(A^T)$ mean\n", "- Dimensions of each space and its rank\n", "- How to find a basis for each subspace given a $m \\times n$ matrix $A$\n", "- Sketch the diagram relating the four fundamental subspaces" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### (2) Orthogonal matrices $U$ and $V^T$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Orthogonal (perpendicular) vectors\n", "- Orthonormal vectors\n", "- Orthogonal matrix\n", "- $Q^TQ = QQ^T = I$\n", "- Orthogonal matrices are rotations (and reflections)\n", "- Orthogonal matrices preserve norms (lengths)\n", "- 2D orthogonal matrix is a rotation matrix\n", "$$ V = \n", "\\begin{bmatrix}\n", "\\cos\\theta & -\\sin \\theta \\\\\n", "\\sin \\theta & \\cos \\theta\n", "\\end{bmatrix}\n", "$$\n", "- $V^T$ rotates the perpendicular frame spanned by $V$ into the standard frame spanned by $e_i$\n", "- $V$ rotates the standard frame into the frame spanned by $V$\n", "- \n", "$$\\text{proj}_v x = \\frac{\\langle x, v \\rangle}{\\langle v, v \\rangle} v\n", "$$\n", "- Matrix form\n", "$$\n", "P = \\frac{vv^T}{v^Tv}\n", "$$\n", "- Gram-Schmidt for converting $A$ into an orthogonal matrix $Q$\n", "- QR decomposition" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### (3) Diagonal matrix $S$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Recall that a matrix $A$ is a transform with respect to some basis\n", "- It is desirable to find the simplest similar matrix $B$ in some other basis\n", "- $A$ and $B$ represent the exact same linear transform, just in different coordinate systems\n", "- $Av = \\lambda v$ defines the eigenvectors and eigenvalues of $A$\n", "- When a square matrix $A$ is real, symmetric and has all non-negative eigenvalues, it has an eigen-space decomposition (ESD) \n", "$$ \n", "A = V \\Lambda V^T\n", "$$\n", "where $V$ is orthogonal and $\\Lambda$ is diagonal\n", "- The columns of $V$ are formed from the eigenvectors of $A$\n", "- The diagonals of $\\Lambda$ are the eigenvalues of $A$ (arrange from large to small in absolute value)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## (4) SVD $U\\Sigma V^T$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- The SVD is a generalization of ESD for general $m \\times n$ matrices $A$\n", "- If $A$ is $(m \\times n)$, we cannot perform an ESD\n", "- $A^TA$ is diagonalizable (note this is the dot product of all pairs of column vectors in $A$)\n", "- \n", "$$\n", "A^TA = V \\Lambda V^T\n", "$$\n", "- Let $\\Lambda = \\Sigma^2$\n", "- Let $U = AV\\Sigma^{-1}$\n", "- The $A = U\\Sigma V^T$\n", "- Show $U$ is orthogonal\n", "- Show $U$ is formed from eigenvectors of $AA^T$\n", "- Geometric interpretation of SVD\n", " - rotate orthogonal frame $V$ onto standard frame\n", " - scale by $\\Sigma$\n", " - rotate standard frame into orthogonal frame $U$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Covariance, PCA and SVD" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Remember the formula for covariance\n", "\n", "$$\n", "\\text{Cov}(X, Y) = \\frac{\\sum_{i=1}^n(X_i - \\bar{X})(Y_i - \\bar{Y})}{n-1}\n", "$$\n", "\n", "where $\\text{Cov}(X, X)$ is the sample variance of $X$." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import scipy.linalg as la" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": true }, "outputs": [], "source": [ "np.set_printoptions(precision=3)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def cov(x, y):\n", " \"\"\"Returns covariance of vectors x and y).\"\"\"\n", " xbar = x.mean()\n", " ybar = y.mean()\n", " return np.sum((x - xbar)*(y - ybar))/(len(x) - 1)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [], "source": [ "X = np.random.random(10)\n", "Y = np.random.random(10)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[0.077, 0.027],\n", " [0.027, 0.097]])" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.array([[cov(X, X), cov(X, Y)], [cov(Y, X), cov(Y,Y)]])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using `numpy` function" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[0.077, 0.027],\n", " [0.027, 0.097]])" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.cov(X, Y)" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[0.077, 0.027, 0.01 ],\n", " [0.027, 0.097, 0.014],\n", " [0.01 , 0.014, 0.06 ]])" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Z = np.random.random(10)\n", "np.cov([X, Y, Z])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Eigendecomposition of the covariance matrix" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": true }, "outputs": [], "source": [ "mu = [0,0]\n", "sigma = [[0.6,0.2],[0.2,0.2]]\n", "n = 1000\n", "x = np.random.multivariate_normal(mu, sigma, n).T" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": true }, "outputs": [], "source": [ "A = np.cov(x)" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 1., -1.],\n", " [-1., 1.]])" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = np.array([[1,2,3],[6,5,4]])\n", "ms = m - m.mean(1).reshape(2,1)\n", "np.dot(ms, ms.T)/2" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": true }, "outputs": [], "source": [ "e, v = la.eigh(A)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "image/png": 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mGV/5wCFEhP9yYZOLzSFKgevqhdWa77A5iJiueggQJim+Y6NUTnPMLDJZ8Xjn\n8UlWu6vMNXxKgc2hutANUyqeyyAaADFiKSbKHkutIavdkDjLR26Rv/fsMqudgHrFY77mM1XR/uuD\nSMeXudrC4c4dprq9Pdb74WgB+OnFFu0gpuLavLrWB9GzDUsYzSB23mO8Pa80wO7sW1sDoufuPSAa\nzJmnhn3gRs/avNoW8p1hApr9mMefXyFMUmxLOL/e5/efW+Llyx16YRH/u+yyMYhwbWvkWrdza3+a\nQaOkfcFBa4MLzQHtIB6FI3j8xRXyHB49PcUgzhhECVXPYmFzQD9MSZWiXnKpeA4N3+VSc8hDR+os\nNAf8wQsrLLVDjk9VmG/4dIKIX/rMa/zipy+w2g04PlXh4ZNTuLZFN0zIFTy8Y7bQHMS849gE/9Xd\nc1RLDsMkxRYLS6DiOwzinPVuzGY/4shEhemKy1o35Hc+d4nFTS3o00wRxCmDOKUzjJmt+ZycrlyT\nTbpRdmkOE/wxP+8ozfSsofDuqXgOM1Wf5U6I61gMopSnXmtxqRVQct8YCmBney42h1xq6QBi/Sjl\nYnPIF1b7fPj5Faar3ra+dak1JEz0ZqsvrHUZRMkbBkSD0dgN+8CNupBdzZSz04bfGsZM132W2xG2\nHePbNpNlj8VmQJJDJ0g4MllitRsSJtlosNm5tb/m28RpTj9KAa0NWsBM1acfpby61uPT5zZ58sIm\nJ6ar1Msul3sRUZoTJzmTFZey64DAlOXx0LEG6/2IZy51GEYZ7UGC5wqvridkObi2Tc3X5gzftXnp\ncoeaP8N9h+ojD5GdZpHxZzOMMuI0J0pSLrZTUOA5Fv0ooVF2aAcxJddCBCbKLq+u9aiXXII4RRD6\nYUKj5LHRi67ZH/xqO0y3ytcouby80qUfptR8B9e2GMQZVT9nuR1sM8fsbM+tAfbVtR5ZzrZ1kfPr\n/VEsnIutIe0g5thUhUbJZakd8NLlLqdmK28YEL/YMYLdsC/ciAvZbqac9V7ERj/iY2fXeHW9zwOH\n6iMvl2cutqi4Fq80B9w728B1hDgVwiRjvu7z+IsrvOPYJI+emuJCEcXv3kMN6iWH55e7zFQ9jk6W\nOTpZ5tlLbeq+PpRicxDjWro8zyy2WO2FlF2L5VYICHONEo+emGShOeDkTJVza32SPMexLI5Nl1nv\nRVxs9glTxd1zNYZxSq6gNYipeDbTVY/2ICHJh5yerVL29C5Mx5Y9B8PxZxNnKWdXuiiEeskGBc2B\n1vS7QVLWFwoNAAAgAElEQVQIdoepqvYTf3V9wHzdZxA59OOEThHga6kVECUZ/TDlS++euaKQn6x4\nvOfMDL/11CWGSZe679Iou7yw3Obhk1OI6LZaagdUPJs8ywnSnDRXHJss49nCWjfk6cXWnrtltwbY\nxeaAu+fqb1gXaQ7i0cDwvnvmWWwOsS2Le+frdIOYVHHFdYIvRowpxnDb2WnKWeuGPL/UZrawCfuO\nxWdfa/LMxRZJljNT9YlTBTmI5AyijBzFfYfrRElGlmvPnEbZ413Hp3jb0UmW2gEnp6u4FvTDhJdX\nugRxxvGpMidnKrSGMZNll1MzNbpBQj9KqfsellhUfJuy5zCMUoZxzpGJCiemK9o3vuxwZk5744Rx\nVmxSckkzxWTZ4+hEmYmKTzdM2ehHIDBRckmSnKcXm3z2tU1eXe/TC3eP9rf1bJ5favPHr26y1gtp\nDvSsYZBkeLYQphlJluPbFnmW0x0mrPZCpipaCIulyzRf81nY7GNb8MCRBkGc8jufu8TC5t5RE9tD\nveD63nvneOhIg26Yst6PuP9wA9/RUQpfvNxBUMxUfWolj9mqx72H6rSHMYMoZa0XbYu2ebkTst57\n/bCQo5NlOmFCkua6PsW6yNHJ8rZIiFsbl+4/XMe1ZRTn5chEyWjrOzAau+G2s9OUs9GPePuxydHO\nw7tma5xd7eFZFocb5VGQqGOTZSyxODlTIUwy7p2v8/mlNtM7XvLmICLLc8qejWNbLDaHxGmOAH/h\n3ce3LdA+vdhicxCRZFrIxFnOPYfqhHFGO4iJs4w/ec88aa748nvn+OUnFrjcCqj5DtWSw1JnyKmZ\nMr7rstQekisQ9OCTZSGWYxEUgmsYZ0SpKjxNErpBsu3w6a0FxoXNAb/33GXSLKdR8ih7DmGcEaY5\nrX5Mxdfb99McKp6FZ9ssbA65d77G0xdbtAcxeeETX3dtHj09Q630+jN64tzmnlr7c5faLLUC0lyx\n2Y+5/3CdimfTC1OOTlYouw4rnYC2axcnNLk0yg5ZruiGSeF3DxebQ45OlqmXXB3CYLNPraTPRrUt\n4fhUGd8RNgYRM1WfUzNV6iWXYZyOFuGvFuflTtkkdydgBLvhpnKtL9u4KedjZ9dwLOHsSncU3W+y\neMlXeyHDKGWm5hMkKRdbQ45NabPK+fU+FzYGhI2cIMnIlVDzbZbaAVMVd+TX/tCRCcIk0xr0G8ow\nxVo3YqkV4NoWp2cqOJZNzXOYrupwBY5tkamMNFc8dnqKj51dJ1GK9jChVnJ4ea3PkbpPybVIcsVa\nL2KYxNiWx5znFGamkIrnUvczLrUD8lwRp3oAODxRQkSvFczXS2z0YyYrLhVXDx5hknG5HTGMMsqu\nxYmZMiXHpuTquOquI5yerfDKWp+Sa3N0okQuwtnlLn/iwWnm6q9v1a+XtIfRbm01XfV4ZrHFbM3H\nsYTzGwPSrMupmSpTVZ+jYcLCZh/Htrh7rs7RyYxzaz3m6iWUUiy0hgRxyjuOTZJkirMrPe4/XGeu\n7hOm2SimfKPs8r575wC2HYa9NYvb2pl6pTgvN7JJ7iAOCKLUrQ+N/thjj6knn3zylt/XcGvZeWr9\nei/S03YRyp7NvfO1XV3hPv6FNV5Y7jJZ9kbHvL243GG27tMo6Tgn3SBlsx+SK3j3qWmW20Pt8eLa\nfPrcBp5t8cCRBlmueHmly9HJCnP10sg9rlkE6zo04fHIyeltL3N7GPOJV9a51ArwLGGhOSTLc6ar\nPvWyU9jMHR46MsFc3eeJCxs8v9Sl6lpUfJeLrQHr3Zg0z3nX8UkUcLmjN9XEac56PyKMMnKl8Fyb\nQ3WfQZwRxClfdtcs3/Kek1zYGNCLUt51fJL/8Pll+sW2+uVOSJ5D2bNRCBVXKBdnoZY9h7vnahyd\nrBBnOfM1j4nK6/HSP/LyKijFVz54ePTd5c6QfpTxtqON0W7RrQBjn19qk+dgW8JKJ6Q11HVSCmbr\n2rNG5QrLktGi5yBKWO/FdMOYM3NVfMfGsy1Krh6QXFs4MV0ZHcKxW5+5kpDd6/qWl834Os2WNn8l\nP/qdfXRrsLhTd7GKyFNKqceuls5o7IYb4kov4s7t/WdXeqy0Axplj+mKy4vLnTeYH7bQ2+m3lA7F\ndNVlpRPgWhWawwQLoeQ6TNc8Xljq8N5755hvlDi70uWBIxNs9iMWmkNOTlc4MlHhqYUWDx6pc3yq\nQpTmnFvrcddcjTDJeWG5wye+sMY9h+rUfAeltFeJFmgB8w3t594PUyqug9+wUAoWmwPKno3v2IRx\nhqXg8ITLiekavbBDyfWYqnpUfAfbEsIkx7NzNgchShRppsjilKVWTqPkYAu8utHnifNNyp7NZLHQ\nWHZtXr7cY7kTYVsWIop+mGgf/qmK9pF3bKIkZ6k1YLbuI8hop+YWbz/a4I/OrtMZRtRLLgvNIc8s\ntnjoaIOXLnepePaoTvWSq4W1Iyw2A2yxmKv5XGwNCZOMdxyb4Nyajslz33wd0F5FcZYT5xnTVZ+p\nssdERfvXA3i2xcYgZqbm7RlN8WqL8Htdv95Ncrf7YJabhRHshjewc/MI6CBOOwX3laa/AE8vNrGw\nqBVHr/WjlMmqT5Ll5EroBin/pd1kEGXb4rgoBe84NsnlTkg3TKj5Dn/izAx/fG6DYZLphcmKzWy9\nQsW1Wdgc0BxEzDdK9KOUqYrHVMVjpRuS5ehIgXWPpXbI2RXtAnjfoToiwmY/ZqriU/FsPn1uk2OT\nJd5xTB+MkeWKr3rwMM1BzNOLTSZKZe6aq/GF1T6NskNrGPPpc5u0hhGWKKI8ZxBn1DyHe+aqlNzX\nF123NN7NfkSaKVCCiNL+9HlOO4hpVFxA+PxSiyzXQbIcsTg0UWKlqz10HEtIcnAsPfQlWc6DRybo\nhQlxluFaFt1hjOdavHi5y5m5GlXPYRClrPdCjk6VWOvHXGwFXNrU4QkWNoZcaA6oujYPFDHSHzk5\nzXTFpTmMman5RElGkOTM131sS3AsC8eGU9PV0QBydBI2+iF3z9Xwbe3P3o9Sjk1V6AYJG4OIybJ3\nU7Th690kd6sPZrlVGMFu2MZ4bJBLrQEvLHfxXZv33jNLyd1+4s1e2s4fn9tguR2yuDmk4tvMqRLn\n1wfYFtR9B0tgoTnAsy1cW2gH8bZ8G2WXONUxu7cYxiknp6u0g4S7ZmujLeZhkjJXK9Ecas8JS+DV\ntR7DJGcQxjxwZIIgSfFdm4pncbjusdQJWO+HZJnizFyNkmvz2kZMsx9jW8Ig2uTL7p7VIXEL10kL\nizDJ+M8vr9ILEgZxqgWJ6zBT9eiEGWkac3JamzKSLKMbxOQKPejFKf0oo18MVKmtUEoL57z4N1Mr\nIbni80sdJsourmXhORavrvVIs4yKp/3mfU9AKS42A0IyumGMINwzWyPNc5Ic3ndmlt9+eonPX2rj\nu9qcNV3xRovFn72wyXLbIogzhmlO2bUI4pRX1nuEWc4983Wmqz6dMKXqW1hlFxEIk2x0QMZd81Wy\nXO8RKLs259f7KIT5WomVbsArawNKnvarf/DIBDO1myPU4frjrN+uo+tuNkawG7bxemyQAeu9iLl6\niSTN+OxrTb7qgUPbgi3tpu2kWc4nvrDO/Yfq3HeozgtLbRY3A5I0I0gy5ht6YS3JQCxFngEiVD2H\nMMk5PFEqogMGnJmpjey9w1j7XD/+4spos0xrGLPcCfBsi4WNPmGccbE1pBfETFY9giTnD19coRsk\nVAuXxUGSsd4OqZaHgNCPdbTEc6t9Zmouc8rn/Eaf5iCm7NrUStrDY6k95MJGH9fWQrITJAyjlIrn\nsNQKcGyhVna53NY7I0uuzSDOmK6WeHW9SzdIKXsWjm2TKmiUPAZRSqJySmIhYtEZ6gEjSlJKjs1k\nwyVX0OzFJJliquzwwNFJojTjwsaADIVkQDGb6oQJvTBlquIyjLQ923aE5kA/L9e2Rjt713sRw6Lu\nJUebkzZVqM8iVYrzG32OTZb5wEOHWNgc8Mxim+mqx32H6tiWjOzQW32mNdQeQ4fq/ig2vWMpVtsB\nLy13yJWe/dwsu/X1bpK7XUfX3WyMYDdsYzw2SJqDZSm6oY698ulzm3zpXdMjb5JX1/v4jrUt8t6F\nTa2JN8oew1h7pvTCmGGcEsXavDBR9qh6FsvNgGrJ4131Eq1hzB+8sMq7T03qA5HrJS5sDghT7c+8\n9ZJuHce22BrSDfT2eKW0Rv/JV9ZJ8hzPseiEKd0woeLaRFlOHurt+VGuyIFmL8KyLJ5dbDFV8Qmz\njG6oZxvzjRKHG2UutobUSw5nV3o8v9SlG0ZYlk3V1fWzLYsoy5go+5QcmxNTZVZ6AUrBoYbPZMXl\n+aUOL1/u4Vh6E1JWy2kPEyyBqmczTCBOcqaqQtm1afYjLBFEUZg+StR9h3MbfS52A5QISZbTD2Mm\nSh4Vz6IzjLncGeIWgbEsEf71x1/RpygdnmC1G/LAoQaDKOHzlzocmSjj2gJKiLIMxxJc26LquyRp\nhC1ClOYj7frUTJV3Hp8cheTthym1kjM6dWlLwIvA4y+sUnYdXA8ubYakueL+Q3U8x963uPjj3KhH\ny0E9eMMI9rcge3Xm/XDb0qfPd5mraZe7i00dSW+m6tGLUj77WhPXEmZrPg8c0n7jz15q885jE6PT\ne+6aqxKlORdbQ/pxyuGJCkHhz9yPtW9zmAiTVY8oUfzBSytEcTYS0t0wZXlV28L7YfqGkK/f+Ohx\nPvz8CihFEGesdPWZnXGuzQ2z9RIvLHfwbBvXFpTSscwvdxPCNGe67JPZQpIqhklOd3MAKC42M3xH\nC8ayaxEkORv9kOV2SKYUNd+lE8RsxgqlFA8cmWC5EzCMEtqDhLJn8fCJKWaqHq9tDlnYGOC7NmXX\nwbG1lhulOcpS2JbgOhaeY5N4OUopcqWYqft616AIvm2z0Y9QSmGJxV0zFYZhwmY/purb3HuozkzV\n5+XLHZIUNvoRYZIxh0UY56xlMe1gkxMTZZJMUSnOG9Xt7BGmXaIkpTWIEdHlefDoBA+fmBwdvDG+\nY3S6qvvYbM0fabc7XQqjNKNRcvQuXkcg1eakNGPfQ+vu1zkAB/HgDSPY32Ls1Znvmqtxfr1/w51c\nxwZp0Q1iUDlZBjE501UX37FoDmNOFJEUAd55fIrz633OrnZ55OQ0D5+cIkkVi80hS80hZdcmzXJs\nW5goe5z0y/SilPYwZaU9pOI7JEFOlGZ0hzGLzSGzNR8bGIQxL/YjVrshD5+cHLlGTlY8giTl1bU+\nYZKx3gvxbG1zD+IMz7bIc0UvSQhSC4VimGREaYZnWYilsHKo+jZZZnFxOKTkWDiO4Fiw3tczluma\ny+JmTK5yKp6ekVQ9jzhLiXPFF1a7xKmiUXYoFYu4Ajx4tIEIlHybi5sDKp6ld572QnKBmqufZc13\nafZ1oLJqyebYRJm1XsjGICFOcnpWTKYgV4rJskPVFcquT9mzEfQC6ly9RHMY4zk2UTYkSXOW20OG\nUUbJU5Rs7SveCWNsYLLssdYNCeOUM7MVXrrcReUK2xUaJYdBmGjzlmPx1EJzmzns8RdXODNT29OD\npBemNMoOrzX1wRqHGyWOz1QI45yab48WJXdTQLbyeTNKyUH1aNkPjGB/i7FXZ37i3CanZ6s33Mm1\nueMQj7+4QjtIOTldJskVYZJzaqZKLcxw7dcjUdRLLu88rg8efuTk6xtFpqse7SAh6AZUPIcvOT1N\nrhSDSMcnT9OYKM0JEu3NMdcokeVwqTng2GSFTOW8strjvkN1Zms+i00d/e+Rk1N0goRPvLoOOVgi\ntAp792TZo5MnhGlGcxiBgomKpw9WzhQKhaCwbaHs6N2arYFOpxBspU0Srm2RKUWSaZfENFWktiJH\n4TpCP9K7Rp2Kx3TNpR/lOLai7AiX2gHnNvq4IoXLpAXoqIVKBE/Ad21QOhRApeRQ82yiTHF+Y8Ch\nhk+cKTaSiF6YUik5VFwbx7FYbIXUfYfZmg4lPIi1GeVic0Av0Id+6FC2Fo4tDOKc+br2TrnUGmKJ\n8L57a2z0Ix47PcNSe0jFc1hoDlnuBKx2I3p+wkTFZb7u44hwdrXLUtshV3Bxc4hjybazSMeF9eVO\nwHTFBywcW8enH0Yprm1xdLJMkGSI8AbF5BOvrCNw1bj4OzmoHi37gRHsbzH26szrvZAHjzTe8P31\ndHJt7jgxOjxhK2hWveTy7KXW6+7lBeNeBFuHST/+4irHpkp0g5S5ul8cluzSCrTG343ikX+3hUMY\n5+To3ZeODRc3tdZ4dLJCybVZ7YUstQNeWO5wfmNAe5ASxdp9crri0RnGrPVC5mo+gzBFgAzoDCJc\nxyZIM6IkJyFHCdR8Xd52kOLagusAyqIdxtQ8FzuISdKMsqe3vHuuxSBIiVRGkmmXzCDJmbNsDtVd\nBnHGWj8mzxX1sovjWqx2I8LiYIzjk2Vag4RBok1S7WGEZQnzjTKe6zBVs7mw1ue1jSGNisNc3acT\nJLiWEEQp7UGCUoo4zbAtwbb1OsJaL0YUtIMEBCxLR3xMspy5qkdrkOC7FjXf4asfPEyj7DCMM+bq\nPhdbAWfmaqS5ouTYtIexjgq5PuBiK+CeuRpRUf575uvFISId7jvU2HYYtQh8+PkV+mFaLN56JEnG\n5zc6rHZDvurBQwRxhmXpzU47FZNOkICCU8X5sNeqlBxUj5b9wAj2txh7dea5emlfO/lkxeNPv/3w\nG7Z4T5RdBEYubrt5ESxsDig5Fqdnayy1htiWkOaKXpTy2KmpsfgfQsmziZOcbBjhiY2UhY1eSBil\nPHp6mqrv0CzCt7YHEefXh4RphmUpZqqlkSeIY1kESUaqFBMlt/hdjBJQucIRIRXBcyy9WSjOaA21\ne6P+ztYhb1Ekac4AmCj72FZOaxCz1ktJiwN6PEeYrnhEWUZzEHHvoTolz2G9H+KJhWAhQKoUaZaz\n1BnQ8PUaRZikJKlNmudkieJSc8D9Rxp0hjligUKRKaEX6aiN/SDVJycVIXrdzKITJIRJjiVqFGLB\nL+o1iDKUgnIRXmCy4vOuExOUPUfvH8ihH2pXzZqvFzSHcYbnCAh0w4Sqp11Sz28MaJQcfMdGRJit\nl+gXh4NvHci91gsRoB3EnJiq0A4SLmxoE9m7TkyQZDmWJVzYHPCBhw7x2oY+fWqcNNWzqXGuRSk5\nqB4t+4ER7G8x9urMX3r3DOfX+2/4/kY6+W4eA++7d45OkPDEuU3WeyFz9RJfevfMtk1LT5zfJM8V\nWa5wHYskzTk6WcZ3bb727Ud49mKbY5MVPnZ2jZrvFFvS9SEK77lrhsmyy1yjTHMYs9yOOLfeY70f\n0R8m5KKIE0U/TgiinJlaCccSDk/4rPVjHMui6tvYYhGmGccmy7SHaSE4tBBvD3Xc8jDVi5ZhkhKl\nCr2soHAdm2NTZSxRrPdjFIo00zMAG/A9IVcKx7IZJjp6Ya5gGCUMFWBBHRfHsojiGNe1qZe0oPIc\nuwjXGyKWxUzV53InpORadIOUOMsoJSllxybNFf0sI8kUZdfCtXQYgTTPCeKUmarLkckyF5tDOmGC\nKxaTNY80zRnGCXGaM1n1WOuFHJ0o8fxSF0sUJ2cqDGOtWXeGCf1Qa/UikGVwqKFdRde6oVYk4oQw\nybBEeO/dsyy1h6P+MFF29c7bJC9ObtInSzVKHiemy6ODTrbi4e+mmDiO9tAZ51qUkoPq0bIfGMH+\nFuNKnXmi7L7pTn6tHjbvOjE5+v78ut5K/uCRBkGhwW25sT13qU0vSCi7DrWSS5Lm9PMExxbeXdhM\nt2J4z9ZLLGz2WelEuLbw0BF96HGcZmz0E5p9Hcd8tRMyiLVHi+dqTxcB2mFKpgJEhKrncrReouzb\nVD2Hhc2AfpTi2AFRohjEGb6j7b5lz6bqe1Q9m0GcUi+5DOKUJNVxT8qezUY/JslyolRr6q4Neo4B\ncaoIohhbwHdsNnphYY/X6cI4oRumiMopexZJntEcpCilt9XrELY+Sik8gbVBzMC2GEY6TaYSKq5D\nzbeJEq3JhllOyYEsz+kNFYNYm7ZOz9W0rtsa6vULhMOTJda6EY6tmK+XEITNfsLxGZcwSnltc8if\neuAQzUHM0akSgzij7FnEqXZZzBWUXYtDEz4q1ysTri2cmtE+7PON0sjF8WNn1yi7+kDpsytdAJI0\nJ8lzwsQbRY3c0sDfdWLyDYrJtcwCr/w+XPls3IMW4OtaMIL9LchenXn8+9c7dHvkqtYcxNs6+MXm\nkN/83CWCwuZ677yOoX3XXI3nLrXpBAlpqnAc4VJryPvunbuqJ8Ira31OzVRY7cUkqRotRp5fH/CX\nHjvJwuaAZy+2WWwGzFY9Kp5N2bM4MlHmoaMT2GLx8uUOlih81+LV9T79OCXPwbEsfNtC2eCnNmmY\nkeWKimdT8S1yYKUTEiRDqp42qyxsDlFYlF3BEi104ixnuuJqn/BcPztBEBQW4FjaPfJSKybLcygC\nYUWZQgALRZ5DqiBTeiG1WtYzjyDRroyeDYNQjdz/PEfPSvphiljC/YfroGCtH4JlkeU5tZIOU5BF\nOspjmDjaXp4LaaroxQlJqsgKq8UgznhtvQ9KL25HaYpnWbSDlGrJ4f7JCq4j2JaNY8NGL2KyrAfc\nhc0B9ZLL6dkauYI0V0yUMwZRSrNw3XzH0SmCJGGuUdp2ytO4wN0eSrfBcjsgynLKrj06khBe18D3\nmgVu9aH91Lz3yx3yrYgR7AeQnR16vRfx8bNrvP3Y5Mh17cPPX+aJ85tMlFyOTJQYRhmfOb/Be+6a\n5SMvrRKmOZNlj0pZ77S81Ap47lIbVWyRH2fcHiooyp7LXF04t9ajG2aUHIujU2UAHn9xlbrvcO9c\nlZdXe3xusUXV037eSy2XIMl58XKHNMt57z1zVD0H1xE2+yl2sTgYpjpkbr1kY9sW8w3tV90LM+Is\npxvEdEPBtSzCWJGrjDgBkRTXErDg/OaQXPsS0olSlJbf+MVJSlGioxjmOWQ51H2bjJQ0Lb5TWnt3\nbL1+MAhTLAHHhjjNiFKwbHCw8FwbEKZqXnFohGJxU29+ci1hpuSwEcQM4wxLIE4gRZFlCRNlfboT\nAnGihbprU5hNci42h0SpHp1c20YsrdXff7ihN0TlkOdK+8+jODJZwbHgmcU277lrhhNTFUqOPqYv\nsy0dkfLkBG87ovclrPe0OWYvgTtuGqz5ur/N1X2GsZ7JnZmt4tjWtgFhL8Vkv10Uv5jdIY1gv8N5\nM1PJrbRPLzbxbJu75nRMlVbh7dAaxsw3SlQ8h/PrA4ZRxunZGoJQK2kXxlfWerT6EW87NjkKcVty\nbSaUPpPykZPTV1ykvWe+zlMLLdaKQ5QtgUFxMtJzl9r0o5QgSlhoBtobxdIHM7eHKRc2BvTjVIeC\nFeFyJ6QXpsxVSwyCjEGSEA0y0kwLtnLNwbdt6sVuy+V2f3R+qS2KQZJRrHciCnyLwtaQMywOjwZ9\njJjl6A1LqQKVKFA6H88RglR7o2SZFv5SCHXXgSxXJKn+DOA72ic9iDNsW3BEyJX20x9EKUmaocQi\nSvTu09OzZZ5f6pNmGY5tUUwQdH5K56+UUPUtBlGGKN0enmMRZQovyehFGYcaPo1KaRQeebkd6EM5\nXAtbLFx9kh6W6N2m01Vv1IbzjRK1kp4dnJmtjvpbxbd2jbw5zrgGfrE15HIn4KEjE5Rca3Qs4cMn\np26LlvzF7A65L4JdRL4W+HH02tLPKqV+ZD/y/WLnzUwlx9NaaGF5dqXLsckKL690EUARjtwW+1Gi\nD4JIlfaIACq+zeVOSNVzeINPY+EFPq6hpVnOhQ29GeXhk5O0hzGnZqr87tNLXFjv4zn6xCLb0rsi\nP/6FdXKluNyJiNOMlU5MkmckYU6YaA+SMMnphgmNkstyJ8D//9t70yC7zvPO7/ee/dy99240urED\nAkiCAAlK1G6LkinHKsuOMxk7ybjGTkXJB4/HqWQ84zgVZ5vMOK5MpcZJKqPYLmcqnnhSJXvssWVr\ns2xKoiSCKwiCBEnsW6PXu9+zv/nw3nvRALE02A02cPH+qlCFBk7f+57Tt5/znOf9P//HNPAck3LO\nplWNibuR2ncEUQIyTTk511AZNteWHXaDby/gSgnFvKs2QpOMVpiQZOrfPVsdlXW/v+QpPxdLoDLd\nDDqZxLdU9h5l6nXjhGs3ju5bJ1lGEKlrVXRtbEPQilIyITEN1SyVZcqmdyjnsFiPaAUJKeBJtUFp\nm5I0A9OA4bxDmGYEUaJuZrbydTEMQRgnLLdChFCbudV2TEMkZJnkSk1tbH92/wSvnK9S7SjP9a1D\nOd64XOWRqfJ1P90kzTjetVHu7akAXRuBxb6NwJaK/57kYnUGPlpw+zeMx7c6fU/0zSh9PMxyyHUH\ndiGECfzvwOeAi8BRIcSfSilPrPe1H3bu5lFy9bEFzyJOVQnih2eWcC2DuDuxpjfFpuDaCGEQdjV8\ntiWotiIMoYYovHB2GWEIirZF3rdwTIMntw31M7RjF6t8+615VtrKwfD0YpO/PH6FLWWfVMJ0Jcdc\nM0RmGWXfod6OOXaxxoe3DzNfDyi6JqZh4BgGS82I4aJBGKe045ROlOKaBouNACnoTjFSAyVG8iZB\nmhEnkjhNMUyDdpCCAbahnjqS7jVZfWuSQLUZYhpqc9AyTchSsu5mYYbENWFi6QrJ9Axh1y8lk5Kc\nq+rjjm2SJpI0yrj5hFIV9NNI4tmSIEqopb3AIih0h4TITJBKqbTnSIRQTwFhom6fplAlnl6WXbAt\noiTDtZTO3zQEZOpG3AokRc8Coerw0gQhJZWcQybhci3gc49MqGvQNQs7NDuEtarJrBHEvH6pSsGz\n+wnEd99ZQAI52+JqPcAAWmHc17vfLLm43zLkh1kOuREZ+4eBd6WUpwGEEH8IfBHQgX2d3M0vyupj\nlUKhwWIjIkmV1PDUfINdlRyuZXB6scnOMfXIXfJtgihlvqEGDz++tcxIweXUVTXsud2JGZMexW6G\n2bXvURsAACAASURBVPMOeeNSjaWWGgC90AioBwlXax1mhgtgSPK2zZayz1IrJEhShnIurik4frGK\naZistGPaYUoiM7YM+TimSb0TEUSpUsW0UhzTRGZwYSUgjFPMbgruWSaWIbESSRirco9hKHOsfup8\nAxkQpWBkqKEWBrRSgZWlPHH1XQ6/+yp/++WvMlVf5Ne+8Cv8m8c/iymUz4nbfbKRmZLm5QyDWpBd\n9/q9tzS7fw9i6MSqUaoTxjgmqr5tAoZqIMpSiTAEhgFZCqap6vcCkFJiCosolYSx2htIDUFiqCcb\n17G6xl3KkKzaVh4wHqZ6CrNNHpkq4jsmn9o7ft1ae093oDL1595eZKUd8uh0WQ3x9myqbdXwFMYZ\nvq0sE4I4YaUdMTOcu2lycb9lyA+zHHIjAvs0cGHV1xeBj9x4kBDiS8CXAGZnZzfgbQefu/lFWX1s\n0bOZrvi8cHaJNMso+Q5P7RghzSSNICYDvvDE1mt69Czg4HS5r565sNzmwHSFRhBTbScUPZOpss9X\nXr7IVNlnOGfz0vllLGGwnEIjSPFtq6/PLvk2b19eJkX5mRQ9C1OotvKzS21KrkmYZIQyo9lJcC0T\n1xLUu+WGsm/RCjOaUYIJENMvm4RxihQ9DYsqk+Q8G4GkniT9uvmtsE04sHKeQ++8wpFTr3Dk7DEK\nYfu6Y37pO/+KPzv4DIYhERhEiSTvmDTDhDRRlgC2kfUVNatJUTV76NW0VZ3eFGrTV0iDKE1JMwhI\nsaRS69gm5F2LlXZvj0AFesdQ1rtjRYfFVkySZDTDlIpvM5J3cEyDi7UOSSoJkgyk6pIdylkstiLG\nDXHTfZqdYwW+9eYcL51dYakVsW04z1xN2fYemh0iTjMEgmaqVD2gpJ31IL5lcnEvM+T3K1scRIOv\ntfCBbZ5KKb8MfBnUzNMP6n03m/XoaO/mF+XG2vepxSajeYftI3lyrkW1HXUHJOT6Nc+eJWuPPz92\nmfNLLVWGEcpGtuJbSCl4d76h2tQLLmGSsdCMyFkGC82IobyDa6vsttZJgIggSXBtiyRNWW5mVHyb\n6SGfvGMSJAnlnI0dCjpBwnwzoBHYeF2deTvKSJF4lkE7UsG794GJM4iya2PzfAviJO2qPlRdPJHX\nNiAlsLV2lU+cfY1PnH+Nj54/xkhz5bprd2F4ih/uOMSxrfv55W/+Hltr8/zsa3/Jv37yC/iOGkYR\nJapkkmaSVm+Dtvv9vWEZqxEG2GoeBqayiyFKUpJMKYdEd3FZBp6jai6uZeJYCZYhKDgWEoFrm+wY\nLYCQTJRzXFrp4NixMkozBGeX2tgCbEv52+Rck9GCi22ZxKkEId6zT9MrsySpKuPUg4TTi03akU/e\nVROvbNPoPx2FSYZnm4RJSsG1bplc3KsM+WGWLb5fNiKwXwJmVn29tftvDz3r/UDezS/K6mPfuFyl\n6Fp8cs84l6odQKgSzEKT6SG/f2O4cQTe6YWm0mvHKVkGUazULFdrIVGaUPYdzi+3yLkWrimodWJS\nmZFlkqVmpPxZMjVEo+DY3QAsGCnYSARLrYiiZ/POfBPRikgyieuYxKlQJYckxRSqA1JmqM7QTAXJ\nXnCPVkVQJSJUni1dxR+mgMmwyscuvM7TZ17jw6dfZbY6d921WioO84Mdj/PmviO8vOcwc0OTjOTV\n4I7Yz/FP/vB/5Fe+9fs898gnaXojWKYkzTKEEN2yj2pSArr9rNdjim6wl2AKgWNaJFlKK1IDNHKu\nRZZBkKiNxSiVVHyHTqKsEWzTYLKi9ipytkm1HRCnMFZyaYUJ7Sil3knwLEEnSmh1B0oXPZskU3LP\nSs5hS0UNNblxn6ZXZjm/rAZZDxdcFutqDGEmTd64XOOTe8b6NfZzyy3CWA3gGB8p3DYLvxcZ8sMs\nW3y/bERgPwrsEULsQAX0nwX+gw143QeejfhA9n5RekH4tQvVW2b+vWN79XYh1IadGo4gybg2POHG\nm86xi1WSTHK1FlDJOdQ6MZ0oYa7eoB7EpFLimCan5ltU2xFTFZ/T820KrhqpZpoGhiGwLcFiI6Tg\n28wMebSDFCklnTDGNm0yoJJTdfk4k9hS6a7jOMOxDTKUI2C6KkvvSQCNVedqArahNhxzYZunLhzn\n0+df4+lzr7Fv/ux116Xh5Tm26xBHdx7ie9se5/z4NizDQBqqlj1sG0RJimUYvPDEj/LCK1/nwydf\n4L/42r/gH/7Ur+I5Bp5lU/DU5CGQZGlGohJiXMskzTKakex6xKhMHQAhaUcxnm2qGai2Rd6x8ByT\nRmDQ6CTKuyZJME1BxbdBCKIko+TZxGnGSiem5KvvK7qmmjdqCOYbyl7YUApOip4yU7Mdi32TJfZP\nljm33HyPN0uvzFLvqKeDvGNhVXyWmmH3iSTjE6uahoIk7atihgvOB969eb9tyj4IrDuwSykTIcQv\nAV9D/b79npTyjXWvbADYqA/k3Wb+N9bb903a18nOqu2IP3r5AmcX2ziWwcxwjnaUMlH0WGpGeLbZ\nHU8XEKUpQzmbVpix0o6Yqnh0kpSl+ZCSZ1HxXVY6yus7kSqiDeUdxkueyjxLOZYaAbUgYudogQvV\nNs1QDZxwTFPJED2bqoxIuh2elgVp3C1jADHd7NwAkYGTRBy58hZPn3mNj55/jYOX38aS11L5wHJ4\ncfoAR3c+zst7nuDU1r1YrkMQqZb9cs6mYJuYpiBJJUvtmExKip6NbZn8L1/8e/zLf/Yf8+xr3+ZP\nDz/Ly3uewHdMMikpeRadOMN3JZ0owzJUp6pvWwRxjClUsA/Ta1m7zCDO1NNIKeeweyxPrbt5XPQs\nJss+M0M+R8+tkJCRdyzyXW2+IQQWgidmhsg5yjJiz0QeQwiq59QNthOlXcdGjyBKyWTGoZkKhqH6\nCm7cp+mVWco5S43dszIMIRgtuuRtg7GS1/9cHZ51+vYBm8X9tin7ILAhNXYp5VeBr27Eaw0SG/WB\nPHaxyqVqhySFgqt8OW43jeZ2tflqO+Ivj1/hu+8ukbNMXMegFcSYpoE9kkcI1aVY9h0KrsViKySM\nVG215NmcXmhxtdEhyWC0UGaq4mFbBgvNEKurwQ4NSa2thi5UWyGeJXAKHhNlj7fm6rTDpNvko96r\n4FmYQhClGUIYqluyG9QzAUaa8sjVU3z83Gt8/OxrHLl0Ai+5dnNMhMFLWz7ECzsO8b2Zgxyd/hCp\n5eBYSiEk4owkUT7yJd+i4NoUXIsgySh6SnqYZhJhqJpydXKGf/Vjf4df+Orv8o/+7Lf5T/7z36EZ\nxtTaEUIYFD2bsYLLQiOiEcY4qKHcowVXZfCmIEgyap2IOAHfsXBtgUSVThaaEUXHYqToIqTEMJU7\n4nBe6d6HC8pIq5OmVDyLMFUS0KLnMDvsk0mlAnIsNcPUsQ0MAzzLIIwzJkp+P7MG3vNZqORsJDAz\nlCdO1RSqME6ZGvIpdf317yceZtni+0V3nt5DNuIDWW1HvHq+ymjBpeSpjayTcw32ThQIet06N7Ba\na3707DLLzYjhgoNpCJphwhtX6hRdG8cUCCFohAmuLXnx3DJxmvXNsvKuxXDO5VLQppKzaIYJ1XbM\naN4llXB5JVA1ZwStIGak4FDxXYpeSqMTc3FFPRGoQKHq8FuHc9SChHaUEMUSy1I1asdStsCOaeCb\nBjvmz/KRM6/xsXOv8fT51ymFrevO8c2x7Ty/7XG+t+1xXph5lNjLIYWqwVsCHFNp1R3LJOuO2yt6\nJlJJwEkyiWmqsollmhhGRs61qfhqE/dPnvk5PvXC19m1eIGf+vr/w+985udJMolAyXOaYcpowWHX\nWJ44y6gHKa4JRc9Rm6HApZUOmcyYHsoTdSc92Y7FfK1DnFM+OWNlH1MIhnIOQ7mY5VasxuUlUPZt\nPr57hHcXmlytB4CSap5ZauMYgu2jeTphQpBk7BpXY/IsU3Bk+9B1WfaN+zS9Msuxi1VW2hGumZJ6\nSn5pG+K6DfX7gYdZtvh+0YH9HrIRH8gziy2G807f+7rdffyP04yndgzf9nuvVAPVXCIErSDhpXMr\nnF9WY+CmKh5ztQApJdV2xOVagG+bWEIiDLWht2vMZrri0QwTLtUCDCG75RwXidJpL7djiq6JlIKV\ndsyFoEM7SgHJVMlnquKRd3v+3k1sw8AUAtcwKOQNip7FQiNicuUqz558iSNnXuXImdeYaC5fdy7n\nK5Mc3XmIH+54nG9NPcZSvoLJtc5PJJjdbtGCqwK6Ywo6UUq1E5FzLI5sG2Khqdwpw0RJNEuejeMo\nTxmZSeZqHWrtmFaU8k++8Mv8zu//A37xu/+aP93/Sc6NzZB2fVfyrqlcHtOMiYJHKju0OwmJK/se\n5lfrIWEiGco5NMKYMeFT76hhHKmUeI5JzjYZLihvFYmqx9e7NrojeYcolYwWPOJEyRld22IoZ2OZ\nhtpwjVImSh5l3yZKlXVBI0h45fxKvxZ+qw3NT+0dZ9tInm+cuEozTPpSyt//7hn2bymxZ6J437gh\nPqyyxfeLDuz3mPV+IOudmLGi2/cuz7lqLuWJKzWefXTylt937GKV45dr+JZB3rOJE0m1FdPoxOBZ\njAiDoZwasbbYDIliVUt3LZOKb3c3zGIKboGyZ3FhJUNmKovvRGm36chWjUZCNdksNQKiVLkbphlc\naXRwbIPpIZ8gVrVj1zKYKHk0L17h8VOv8PTpV3n0rReZWLheSLWQr/D8tsf5/uzjvLz3EIsjW0jS\njDjNSFLwDBBCKLvc7kZlCuQstSHqWAa+bSjteabUKe/Ot0gyZe5V7yi/8rGCy87RArV2RLUTs9yM\nQWRkEv5my36+cvCz/Myxb/Jf/8X/wX/68/8Trm0Rphl5LHzXxDagEcZMlDwupwFDOWXXcGaxTRAn\nqrkpTfEsE5lJ2pHB9LCnvNrjjEaUQFPdLBabERMlj/mGsjGOElX7nq74TJQ8zi212DqUZ7cJs8M5\nPrV3vL+p3tsTOTBV7hu9rUWBtdyK2DGS5/xyi0zClVqHIEl57u0FLtcCXjq3wucOTNx3Wbzm9ujA\nfp9T8m0uLLfZNV6kEcS0oxTbMjkwVWa5Fd3yF+6d+SaGgLxnIxA4lqDgWt2WekkjjGkFCUN5myu1\nDoWczcxQHgmstFW9txnGvDvfxLEMnt45wnIzoh0nzNUCrG6n59WaqrdbpvKn8R01ICLvCCZKPqYh\neGehSTHqsP/tVzn41ovsev0Ftpx/57r1tv08R2cf4wc7D/HXWx7j7ZHZbiMSuJagaAqEUFJBpZqR\nCASpkBhSbVRaQM61yVLViFXrZGSAZ6iB2hdW2viOgWdbVHxVMqkHMWeXWlQ8m6mSRz2Mma/FeJZF\nYCT8zz/yC3zm7R/ysXPH+Iljf82fHXwGKTOCOFW+8MJk/1SBnG3j2xZhnHGlHlD0TT66a4I35+os\nNkK2j+aV+6OhNikLroXbLc+FcUojUMZch2eHuFzt0AoTZkfyXKl1GC96kGR8aLLEvskSUsr+Bvyt\nfFrWqsC6XO3w5uU6zSilE6kxge0oJc2y7uQp+MaJOf69J2fui8xdszZ0YL/P2TGa5ztvLzBacBnK\nOYSJCip7JwpdC9ibN0EJJDnbIk4yHKsrdxMqEIwUXJrdphTHNvAdk+FuzTdMM9IsY6EeEmcpZd/h\ni4emEULw6oUVjl+qs33Y51It4OxSmxRlItYMYpJM6dh9yyBnSva9/RK7Xz/KE+++wr7zb2Jl1/YE\nQsvh7d2Pcf7QR0l+5DO8Mrqd751Z4WI1UEFbKnljTy0epaqrEkN1nBqGAQgMCY4BrmkQZyrCl32L\nq40AyzQQmaQVJ0SZmuTUCiDvZcrXXQqEAQuNgChOqeRtHCEwTTUmzncsqoUyv/mjv8A//Yt/zj/4\nxv/FX+8+wopToNqJumUpm7lqh6GCZHY4R5pJtnQHXbTClCdmh5mrdXAtk5JnM1n21JxXQ+CYRreb\nM2F6KKHWiTi71GJ2OEczVEMnpJTUOxESNegCbr4B/34UWL0B1CudiLGCy2IzpN6OKPoW5a76apuf\nZ6EZas34A4YO7Pc5lZzDodkK55fb1IOEgmuybaRIJ0pZbIb8+bHLXKl12DFSuO4RfLLsU+skVFsR\nBVcJrqutkJGCy7bhHHEmcbotkbUgZr4RcXapyUpbGT1VcjYFy6LWjnnjSo1Ht1Q4NDPE5VrASjME\nKdha8ai2E1IpiYKEg/OnePrsq3zi/DEOnnsDLw7755EaBq/P7Ofl3Yd5fe8TnNr9OE7RZ99EieG8\nQwWYqHRohBmtMCaRGQYqAFqGgWMI5Y+OclWUZOQdkyxVx5TzLr5jqkEaSUqSgGlKLNMkSRM6Ybcz\n1VCdqo0gxrEMhjKHnKfsEKQULLViPNMkziSmaWCZgn976LP8zPFv8dSFN/iVv/p9fuPHfwlTiL7P\n+WTFx7eV8+Fbcw0W6wG+a1HJOdSDGMsQtKIEy1cmYAcmy7x9tYHb7eosuCYF16TsqTr54dlhrlQ7\nvHRumWonIZUtDs8MU3DVHNObbcC/HwXWmcUWO0YKXKkGtMOUvG0yn2TQSZgq+epGn6ihJL0kQvNg\noAP7A8DBrZXupCCr77d9/FKVR6crLLeUzPD8chu/p8aodqi2YyzDYKTgqnpymjFZybF7vND34t45\nWuDkXINcO+LySoflVtxvy59vBOybLLFjJM+F5Q5ZJnlsusKHJoo0Kj6+Y2GdfIt9J15g5+tHOXzq\nVUpB87p1nxzfzsu7DnFs7xFO7DpIzc0rG0MEJoJcmHJppc1CM2TbcI5MCkaKNpappHuNKCVJMixL\nUPYcHNuk7FqcXmoyX4uodVJsSzlTRnFKz249jFMMAzqBxLGuPSX0Wv+VugWiOGNFqjmkrmXQCCIM\nlELHhv4w6Fo74jd/8u/xG//2n/PHT/0EZd/C6Qb/KJUEcUozTPFtg3fm66QZHKj4yqK4GjBZdBkq\nueydLPK9dxdpRylJKruj4Kx+Jv7DM0sEUcZ3onmu1AJyjsnB6RIr7ZhXLqyw0g67IweV1HXHqDqv\n1TX2HSP5/g3+Tgqs3v7NR3aM8MMzixgGSillKL+DomcTxCmzwzmtGX/AELLfIvfBceTIEfniiy9+\n4O/7ILO63DJXCxgtuIyXPF46t0zJswmTjLhrLuVayo53djjPmaUmU2W/76P92oVqvysVlGXr86cW\nubjSphNlmEAzSsm7BttG8uydKDFXVzX19rtnePTkS2x95Xm2vPwDSsvz163x0tAk39/+OC/sOMQL\n2x6jMTRKxbNxbGX1ilTVlJG8g2WoWaGjRYf9U2VKnsXb800sAW/ONUgytVEaRBmplJR9pR1XPuQZ\nK52QdpBiGoJKzibMJDLNsEyTWiciSCTdwUOA2li1gLxnkkqJIdXACdtS81JnR3I0OylFX7DcjBkp\neCRZxkje4e25Bp5rMlcNsC2j63mvdPeVrvbcNgTDRZckkZiG8m7pPXGMFd1uc5nKqldaIY0g5fyy\nKrvMDhewLLi80mFLJcf55SaL9YhGlLBvosiOsQKNIObcUpsfOzDZD9wLjQAJjBevDdi48ed9u7r4\nK+dXiJKMnGPRCGIuVzu8O9/g4nKH3RMFpis+w3kXw0D7stwnCCFeklIeudNxOmN/QFi9SfY3J+f7\n9dSCqywDzi+3ODXfZNtonp2jBSbL3nWTcXq6ZiGUYqYRJrTDlJxrstgMGCs4SASOadIK1aZqcGUe\n43tf48iLz7Pr9RcYuXL+ujUtFyq8uvsJTuw/wmt7n+ANd4QkzdR8UhuKlkWcSUSSYRiqdOFZquOz\nFcTkHDXdXgg1QPrgdIUTV2p8Ys8o337ranfkm5pD2ghikkzZGkyUPCxLIAtq7JvnGIQxIDJMIVhp\nh0r+yDVvdttQNXshwbdMXNukEShHxZxlslAP8CyTnaND7BqFd+cbrLRCmp2E0aJHLVDt9qaUxKlS\n1riW8tUJgS8+voXFVsyllXZ3eHTESivm6R0jTJV95mod4iyjFSRcrQc8Nl1Botr3L660sE2DSte0\n7dxSm7xvdTd3E84vtTGExDLFdVOwep4v20cKwK0mId3aguLGsXYzwzlGCg4/95HCe+bj6qD+YKED\n+wPI6nqqaQhePLuMZxnkXZMwSnnx7DJfPDxNI1CBptfcMpxXHjALzZCVVoRrChaaAZdWAsZLLo8U\nDZzn/4bJl77P/jePMn325HXv2/ELnH3kSc4eeprLhz/G9/0JlpohiYQolSTtEMsQRFKSM1VjkG0K\ngiTt1qNTRvIuaaoGSgzlXYbzLvunimRSkHcMWlHMQiPENExGchbCUMEs7U49wlMDHyxLuRAWHYsr\n1YBHpktcbUTU2yG2ZYJMCdOu/W23xICgW/u2CKKUsm/TjlI1r9SEXTMF5uoBk2WPobyLIWCpHdOO\nYrJM9Bu3DCHI2xa2ZWIhmRr2mR7K49ohl6ptltsRs0M5iq5FLJUVw0pbSSJbYYJvWbwz32C04FL2\nBTtGC5xaaDBRdHn9UpWyb1FfjnEswWIzJIhTVloRB7YUaYbXSks9z5fV+LbJhZU21XZ0RwuK2/VZ\naHnjg40O7A8gqzOtU/NNpioe1XZMnElqYYxrmfzl8SuMFlx8W3nBREnGN05cZcdInpG8Q5xkZGHE\n1teP8rkTR9lz/CjTb7+OmSb994lth8WDRzjxoSOcOPAU3ynNIGybHSM5LMOgMd9gZrSAY5kIIXn5\n7AqmofTXRd/BMgW1doRlGOQcAzMysC2BZ1uUfJucY2KbgvPLHT40WWQo57B3osTZpTlGCw6+bdGO\nU0whqAcxtU6MIQWObVJrxyBRZl5CKnVKPaAVKa/yhhAQZ/iO0ZVGwljRJU4zGp2EvGfTClUjUMG1\nsE1BtRMzXvIYyjucW2pTD1MKrk0lp3T9MpN04pTRohqeHUvJdNnjkelKd6Zpyr6JEo0gIe+Z+B0T\nIQT1TsxoQV3zVpSyY1TtW1Q7ETNDavhJb5DfStde+cxii6srIQVPDbkI04zldszM8LXSac/zZTWd\nWBl2rVX6qBt/BhMd2DeJGyWKw3lnzY+/qzOtK7VOt2YtGMk7nFpocqnRphkkfGz3SFeRkZJmkiyO\naX//B0x/9zk+9uoPmH3zZZww6L9uZhgsHDjE4oc/zou7DpP79CdpGA6n5hu8OVenGcTkyKgFKUuN\nlhrA0K3pyyxlZsin2lHzSucbIZaATpIyWfLZMZan1k6AjO2jyg7hjUt1hvM2YZzQDGLenW+we7zI\nfD2g3lGBPEmVd7nyA88wDGh2YgqOQdE16YQJYwWHalsF5aVmCKg5q2VPNRE5piBMJF88vJXlVsjz\n7y5xudrBc0y2DfmMF33Gyx7nl5qstCM+OzrBqasN4tTENtT3BxGMFJ2u0sRCCMFjE0UObh1irOhy\ncq5OLYjI2SZ5z2S84PLJ3WNIKfnqscuUfLW2qYqHIQSuLai1Yw7PeoRJxuywTy2IyTtW12LAZqER\n4hgGUkr2jZcIs5Qg7rplrvJ8UZuw1ywrCp71HkdH7Yb4cKED+yZwo1vjQiPkuZPzPDpdWXPXYC/T\nenuu0ZfOLTUjTEvgmgZtU3ClGvAFr8HO773A1EvP8+SLz+M2ate9ztnJ7by5/wjzT32C5JOfojw5\nylIrpNZJeHKkzJRj8u58AzL6weJytY1jGF1fFIPhnM1CQ2WsQZzh24LRvM1cIyTLJBMlj88/MsXR\ns0tcqnVYaoW4lsneySLnV9q8dH6FhWaIZylb2+2jed64XKMdJoSpkj36jsX+ySKiO6PUNJTplu8Y\njBQ8au2IHWN5LAHnljuMFNTMz+VWRDtK2T1eJIhSTs03eXymTColYZTSjFJ255Ql7lTZ52Ktw2Ir\nwrMt9k16XK2Fyqc9lUxXVMfo4dkKiZR87sBk38N+70SRuVrAYjPkkekye8aLFD3lqnlwZojto3nS\nTPLOfIO35xr4tolXMkiSlAjYNpynHSdsG8nx7tUmlmnwuQMT1DpK375vssBEyeNStf0ez5cbSyln\nFlvaDfEhRwf2TeBGn/aVdkTZt6/bGOsdd6fH5Kd3jfCDU4uqnm0JKosLfOzECzx56hX2nniRoZXr\nlSsr41s49dhH+O7sQd5+9CmWchXSVHmQTLXh0SDGNg32TRQ5fqmK75qYBkxWPK5U27TihE6UMtTV\naOc9lYFerYcsNUOKvonnqE7PiYLD9vEih2cqTFV8hgtuVxUS0whCGkFMFGfMDOXIOxbNIOG1iys8\nMTtMzjE5frlOvR0hDOWb8tn9E7w93yCIMw5sKTNedNlS8ZFS8vzpJc4utlhoqnb8iZKPAD40WeTS\nSofJkssrF1bYOuQzXcmz1IyoddQG7qVqh/GiUpLsHM+zY7SAEJJLKx22VDwanZCJokucZAhb8M58\nk20jec4ttfpPTkGc8vE9o10JoVKp9DTnT+8a4fRCk5xjcXhmiK2VHGeW1BOPlPR9zp8c7Xrl74r4\n2vE5qp2YnWN5tlRG+zeJ8ZL3HhvdGz8jO0bf6+io3RAfLnRg3wRu7BLsDRBurNoYW/3ofLvxetsI\n+DuXX6TxF19n3xtHmZy7XrnSLA+z+OGPM/fkR3nrwIeZOnyAPz92hTBJ1HAGBK1I+aacW2xxcGuF\nvd1ss+BafPfUArZhUPYsWjmXCcdkvtah1kkwRMiWis878001kq3kIruqkaGcQ842eWSqRCZVITjv\nmHSiFNcyqAcxQZTiWAY5T00U8l2LRj0gTjMe2zpEEKeESY6JosvVesj5lQ4AW4d8PtnNVhtBzAtn\nl1lsBHiWwcxQjvPLbY5drOJZBodnh/nIzhHKvsNi62p/s3Gq7NMIYiWVjDOGcha1TsxP7N9C2bcx\nDcFKK6YdJtiWyXJbGXo9Nl1m23COlVbEv3n5IqD6DHo+9z2PHoFk93ix/9RV9u1+Zj1ccHhy++2f\nxp59dLL/VLf6JrGW4HzjpqgQ6gnndkNaNIPFQxXY1zN/dCO5sUuw4CodccG79qjce3S+sWwTpFoN\n/QAAF35JREFUVmuc+9pf4r1xFO+5v4ZXX+WJVb0IoZ/n9IEnOff4R1h++pOcm9pJ3rM5NFthJ2ra\nz3jJxbVyZBIuLCtL3k6YIoUkXTV6bqzoMjPks224wPOnFpip+Mq0SggMlFRQOSWqTcrxoo9tCkxD\nMF3x8ByLJMvwHNUaHySqcehDk2XOLrY4mWSYAuhaESy1QsIk5d2FJmNFj+lKjiDOaEcxOVc5IYJk\nrh4yXw/wbIPn3l7gnfkGOdvCzhsst2PCVOJaJsN5m6VWSCVnk0mJY8HZhSbNUE0temLbMBeX22RS\n4jsWP7p/grJvc+xiVXntdP1wtg77hHFGsfvzuVjt4FkGEyWf88tt0kyyc6zQz8qf2jbcz5J73O0m\n5XqdQXvvd26pxTdOXCXNMoZzDkmq3Dy1Ln2weWgC+70YiPt+bxQ3+rQP5RwurbSZGc73N8Z62dnZ\nyytMvHaUkR9+l8J3n8N/9UWM+Fp7t3QcqoefYu7Jj/P89oOc3bGfzLAYL3kI4OOzlesyylfOrzBe\ndLlU7VBrJUiZUcn5tOOAiaLXn560b1LdfHaPF0kz1YXYCmMu11S56KntQxw9s8xySyk7HFvV+OM0\nxbUFtil4ZNqj4tuUumWmKMnYMVrgSq1DrRMx5Nk4tsreHdPANdUQC5llvHm5ylDB7db7I3aNFdVN\nIxFMlFzemquRZupmMV3xWWnFnF/uABkFx6QdJFQ7yqvcs022DuWwTZNOorTuI3mbeidi65DPz35k\nG9tG8lTbEV87foXjl2qYhlBPJQZsKfvK2Mu1uFoPSbIUfAffhreu1EHCmcUmj2ypbOhczvUqVqrt\niG+cmMMSguGCR5j0mqLy2vtlwHloAvtGD8Rdz43ixmxsuODwU09sZbkVsdIMmDj9Fgdf/T7+d5/j\nseeew2y3+98rDYP2oSdZfOpjzP77P8lrswcILSVt2x/ElKsdlloRFd/m2Ucnb6pbNrva8PlGSNG1\nWG5FmELpoluhsiLoPfr36rnz9ZA3LkfsmSwxXcmxUA+ZHsrhOza+Y7LYDonjtDuU2uDsUptP7h3j\nY7tG+zeVd+YbWEIwOVVmquxx4nIdhHKGHC04NEI1g3M47xLGKiAXXJOhXB7TENiWwZ6hEgbwzkKD\nzz8yxYXlNqcXmnTijOV2RBCp7tFUgoEk75iEScobl6o8Ml3h03vGOb2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k0Wg0mvWwrsAu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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(x[0,:], x[1,:], alpha=0.2)\n", "for e_, v_ in zip(e, v.T):\n", " plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)\n", "plt.axis([-3,3,-3,3])\n", "plt.title('Eigenvectors of covariance matrix scaled by eigenvalue.');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### PCA\n", "\n", "Principal Components Analysis (PCA) basically means to find and rank all the eigenvalues and eigenvectors of a covariance matrix. This is useful because high-dimensional data (with $p$ features) may have nearly all their variation in a small number of dimensions $k$, i.e. in the subspace spanned by the eigenvectors of the covariance matrix that have the $k$ largest eigenvalues. If we project the original data into this subspace, we can have a dimension reduction (from $p$ to $k$) with hopefully little loss of information.\n", "\n", "Numerically, PCA is typically done using SVD on the data matrix rather than eigendecomposition on the covariance matrix. The next section explains why this works. Numerically, the condition number for working with the covariance matrix directly is the square of the condition number using SVD, so SVD minimizes errors." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For zero-centered vectors,\n", "\n", "\\begin{align}\n", "\\text{Cov}(X, Y) &= \\frac{\\sum_{i=1}^n(X_i - \\bar{X})(Y_i - \\bar{Y})}{n-1} \\\\\n", " &= \\frac{\\sum_{i=1}^nX_iY_i}{n-1} \\\\\n", " &= \\frac{XY^T}{n-1}\n", "\\end{align}\n", "\n", "and so the covariance matrix for a data set X that has zero mean in each feature vector is just $XX^T/(n-1)$. \n", "\n", "In other words, we can also get the eigendecomposition of the covariance matrix from the positive semi-definite matrix $XX^T$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note: Here $x$ is a matrix of **row** vectors" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[0.027, 0.212, 0.602, 0.276],\n", " [0.118, 0.095, 0.058, 0.736],\n", " [0.595, 0.283, 0.537, 0.475],\n", " [0.39 , 0.952, 0.45 , 0.16 ],\n", " [0.493, 0.958, 0.811, 0.184]])" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X = np.random.random((5,4))\n", "X" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": true }, "outputs": [], "source": [ "Y = X - X.mean(1)[:, None]" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([-0., 0., -0., -0., -0.])" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.around(Y.mean(1), 5)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[-0.252, -0.067, 0.323, -0.003],\n", " [-0.134, -0.157, -0.194, 0.484],\n", " [ 0.123, -0.19 , 0.064, 0.003],\n", " [-0.098, 0.464, -0.038, -0.328],\n", " [-0.119, 0.346, 0.2 , -0.427]])" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Y" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 0.057, -0.007, 0.001, -0.006, 0.024],\n", " [-0.007, 0.105, 0.001, -0.07 , -0.095],\n", " [ 0.001, 0.001, 0.018, -0.034, -0.023],\n", " [-0.006, -0.07 , -0.034, 0.111, 0.102],\n", " [ 0.024, -0.095, -0.023, 0.102, 0.119]])" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.cov(X)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 0.057, -0.007, 0.001, -0.006, 0.024],\n", " [-0.007, 0.105, 0.001, -0.07 , -0.095],\n", " [ 0.001, 0.001, 0.018, -0.034, -0.023],\n", " [-0.006, -0.07 , -0.034, 0.111, 0.102],\n", " [ 0.024, -0.095, -0.023, 0.102, 0.119]])" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.cov(Y)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": true }, "outputs": [], "source": [ "e1, v1 = np.linalg.eig(np.dot(x, x.T)/(n-1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Principal components\n", "\n", "Principal components are simply the eigenvectors of the covariance matrix used as basis vectors. Each of the original data points is expressed as a linear combination of the principal components, giving rise to a new set of coordinates. " ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "image/png": 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w0akqQgjW+jEjJZuSpfPTK2vsbjg8s7sYjJHlkm8+sYOmF3NytkndcTk4UeH9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\nMQ9apDysM+j49a5sD3n1zVvkRUG74pDlys1Tzwudbp6YwP4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Covz3h4WWOWqeRPSO/T4c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\nRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(x[0,:], x[1,:], alpha=0.2)\n", "for e_, v_ in zip(e1, v1.T):\n", " plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)\n", "plt.axis([-3,3,-3,3]);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Change of basis\n", "\n", "Suppose we have a vector $u$ in the standard basis $B$ , and a matrix $A$ that maps $u$ to $v$, also in $B$. We can use the eigenvalues of $A$ to form a new basis $B'$. As explained above, to bring a vector $u$ from $B$-space to a vector $u'$ in $B'$-space, we multiply it by $Q^{-1}$, the inverse of the matrix having the eigenvctors as column vectors. Now, in the eigenvector basis, the equivalent operation to $A$ is the diagonal matrix $\\Lambda$ - this takes $u'$ to $v'$. Finally, we convert $v'$ back to a vector $v$ in the standard basis by multiplying with $Q$.\n", "\n", "![Commuative diagram](data/spectral.png)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### We get the principal components by a change of basis" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "image/png": 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kpWSt6eNYgrd2xsyykqW6R2+W4dkWUinuDGJAD6p9BQIL1xasNDx2hwlF6BHn\nBf1ZTs2zEZYgykp6UUbo2oycnIbv8s9fusP3n9/g0kr9Q9caPoiaDx1Q06xgte4yTvW57k9Siqpi\nMm8F8N7+hKfWmjzXChjFOTe7My6vNBhEGYP5vfPvP7tOM3C5fjDl3b0prcClEzrYljgSUHxSG++V\n9QZv7YznbQtssrKilIrfe2YNz7G4M4zZHSdHcvZf9u/v58mpFfbTVhyhFFzdbLM7TslKXQn5t59e\npZTq2Bt9satRVlKWEsexaPgO33/+DLf7ETujlNv9iFFScmEpQErFL271ubhcw7EsbvdjnU5JSuKs\n4pWtMb4r8GyLm70Znm3xzEaLrKj45fsDkqLk17cTzrZrdEKH/UnFZJKx2Ql5d2+G69o0fBtbQM33\ncGxBVkje70X0pgkKQXeag2LRA922IS0kv3y/T1ZKlusuW/0YhSKvKmqevWhBYDsCx4JhWoFS1Jcc\nFDCOSzxHR+qhY1EgGeYlWSlZbeqUR2+WM7vHlpdVkrySuJYgrxSWJRnEOU3f5ebBjKfW6izXfH52\nvcftQUwrdKi7DuvtgFlasjNMGMQ5nbpPzbOoew7XD2a4tuBcJ8S2BFFeoaTimY0WKw0PEDRCByX0\nex7MMrKyIkpLVM3DswV5KfnhW/v8p985/0Bra3ea8fObPXZGKdO0pO5brDdDrh/M2JukVJWau55s\nsqLizZ0JgWsxS0tCTw+oUV6xUvcRArqzjHOdGt+6uMzT6zqFl5U68r+3buKT9ny5er6zWJcYJzmu\nbXF+qcbXzrUZRDn7k4TAsQk9e2HPhI/+/p6mGfqjcmqF/bQVRxy2az3M64JeTKyUjmTuryb84Vu6\nwGSWFlige7TYgj975a5ujKWgE3qkhWR7mLLW9DiYZNzqzQhsCyWhP8vwHJvQtXWzq6Qg9C3uDHSx\nTH+WMU1LsqoimDeLmmUFngtpXjJJS352c8AkzWh4LptLNYSjo9Xr3Sm+baEAramKcVrgWYJJVhFn\nOl3gWgWDqMC2YJZZ2qGBYJKUzNIK17HxHEVaSBQWvqvb0R5MM0LHogJcz+K3NlfZGSVIR1DzFGfb\nITVPu1/ujhKcezz7s0S/p0khkQqWAoe8kGRlhS20wN7qz3h/EOMIwTAq2CszbvQjlmsevmOxUtc9\n6nsTydPrdVqhw94oY6Vesdb0qfsOf3Mzp+CwBa+i5jrMkpKDaUY2t2RWCma57jo5iDJWGj7/5o09\nfvOplSM6yrT/AAAgAElEQVQtBRxL8NMbPX70zgHTtOCJ5RqNwOJ2P2EUlzR8C6V06qPuO3RqLt2J\nREqJJQQvbw3ZHSU8eabBU6sNikpSlJK7ccKV9eYihbc5r0k4TjA/iY23U/P4nStrR3oYzbKSf/Xy\nNssNnzRXBDV4ZWtII3CQSlD3rCPtjO/lNM7QH4VTW6D0aYojHmc+vEtQj1/e6rMzSj60A85r2yMq\nqXPWgWPTCDwCxyErKrrTDCkhKSWea9EKXQ6mKT+70WOWFUwTHc3qfuoK17Z0f3Slo7Tlms/5pZBp\nmvPDN/e4O0hYb/ikpWK14SKE4mCcMk71jj+FlLR8l0rpboVxXlLIimFUoGCRSskriRCK7ixDyopm\nYGMJwfWDKXFeIRVUlSTwHBwbSgmZlAyjnElckBUVSVHgODaeYyGlpFSQFyVloRgkObYtaPoOm52Q\n9baPY9v4rsPmUkhg37OjlBBIqZvU+jaUpaLmOwTzYydzT7trCYQQWPPcc5pVDKOcQkFaKOJ5TUA/\nLjjbCtloB9iW4O4woj9L2Z+mFJXEFrA7yrjVi9ifpqAUnmtRKd1kTCnFwTTjle0RSsHuOOFPf73N\nYJazVPPY6sf8kx/fZHeUsNH08WxtfxWWxZm2r/ved2d0ZznLdR/fsbjZneHacGmlxvZQu3EaocM0\nKbndj6iUduSsNbyHLqb7pMVFh1H+Ny50qKRiMNNVy7aw5tW7BfvTlL1xSitwiLKS3XF67HHNTkxH\nObUR+2nr8/zBLkEf2AmHiWIYF1yU6sh09Y2dMcs1l91RwlJdfwldRzCMSjzbol1z6Ec5/VnGJC2R\nUreerUYpUV7grzXwHIe6byNRRKneaacdelw/iBAW2ALyQjJKc862AkLXwrZsetOUaV4hlEIJLcah\n5xJ4gqyU2JYAJfAdgecIyvm0/mCaYVuCRuDiOxa9acreOCMtJHW/IvQCKjnfjHuSMk0KbHR0Xlbg\n2gIJ1DztuFmu+Qgh6c4gLSoGs4zQt9kf5biWwLIEpQIpFVfWGwt3BkBZVCB0Dt5xtZ88LySllCAg\nySoGcYGSICwoKoWwQCKZZgo7yslcC8fSA2elFOutgHNLerH19btj7gxjAs9ilORUSiGUbioW5xUN\n3yIrBLYlUUpvXu27NpbQAtYOXTY7ek/Z0LP5ybUus7xkEGkn0KXVkNs9XQAmpd7wZCn0cR3BOMlx\nbB8LwWpTV7i2fR2NV1K3LvBsn51hzGrT50wrPDZCvp/jfOnbw5jffYhWvsdZec91Ql7ZGrHRDigq\nPVtSoFs/H5OOeRxm6I9TKujUCvtJ93l+1It2Ehdd7xL0QQHHS7cHeJ7F9YOpbjU7ryZM8oonlus6\nl5uW1H2HKKsoqwqlBHvjlHOdGq9sDSmkZJyWNHyX1aZPJX3SvKKqJFI6PLfRZFL3udGbkRa6+Ma3\nbWZ5SVkpZnnJNNOLs8O4wHccSiUYFTmuEJxpB8zyiiSXtEKbspJsD2JqnsM4qcjKDBQ0fJvuNOdC\nJ2CWl+SV0i17lRbv7jSjqimW84q40N57z7Go+zZpUSGlHkiyokSiF2VnabmwdNrzqD/KCnxXd2SM\nswLPthklpd4mbk6FwhZguxZZXlIArmtTCxxCx2K16TGKC6ZFQd33kEqRpBVFCUIoHAuUlBRCd6Q8\ntxTw8p0hXzvX1u0LbC36aVEs0kqd0KXta+eKUhalLBcL1oXUn8G5hha5w4ZnB7OMaVpwd5wQpzm7\n4xwFtEIdaR9Mc6RUSGCzE7LWDri2N6E3zWiEDst1j7vDhCfXGvSmOVmpI+KXtgYo4LefWuXp9cYx\nd+KHOc6Xvj3U7Xzv3RTluO/BoSg3fIesrAhcnU4JPBvbZj5wW1xaqdPwnQ+J9SjO2RunvLM3ZaXu\nLZq2fZ4z9MctFXRqhR1Ors/zo160457/aXqgH34BdscJr2+PuXYwwUZg2xa/9dTqoppQpzdKrp7v\n8OqdAd2pTns0ApfAsfAcm8AR1AObNFcIdB8WFASOIMuhUXcRCKaJzgLbtkUlJUJYNANdcZp4Fa5t\n4dm61NwRYp5e0akhXfXqcDlwud2L2J2mnF8KCVzB3iilG+XUPIdWoC2VeVkRF9U8FaJdQJZjURWS\noqrYnST0oozVuk/DsymkIp+nZ7KspKwgLRUNKgqpSKu5+6eSRFmOJSCwYZYUyHmqyXcUu5OYs/cU\nxQSuQ8N32JvEOLaFEjo3HaUluW3xzs6UaV6S5JKiyghdB4XCssG1oeG7tEOHYZTTizKiomKzU3Fp\nucbbexOWax41z+GNuzEoRavmEmUFtcDBskEqsG3BMMqwBFg21Fzd9iEpKn5+c8D7/Yhz7ZCLy3UG\ns5w400I5TXJ604y8khRFhUJ79+O8pBO4PLPR4u4gZq3pczBNKaVkZ5hgWYLtQaL74rdCULqd88bc\nMfNx9+ZxvvS2chf++4/6HtmWWLR0fndvAugU3XLdpRO6XD2/tNjgJM7LI2J9eMzVhk+U6Y6h7+wV\nXFquL4rjToKPC8weN7PGqRb2k+JRL9r9z39QD/SPGyhaocv7/Yif3+jR8B2eWKnz0vtDKlkx3WxD\noJ0gz59t4TqCZuDi2Lq8vTvL6IQuT867OO6MElzLYmuS4Tr2onxdKYVtQ15J2oFHI/DIJYuddiZp\nSZTrIpWs1FurHUwyupOczaWQlbrH/kTQm2UMZilV5dEM9AYSG02fq5sdfnK9R3+W49iQlyXdaUm7\n5tEJde/0UiqSokQq7d+XupZ+MWjklSQpdem7Z9tUqkL4DiovURUUUs0XWdXCKW2hFyKjEnxLp27A\nIi1LslxxM7vXFVNRxLqACHRnS2vuES3KCuHaWOjZRJwrLKvCsQS2sAlci9W5/THKSjqhDwqmqd76\n78JSfV4/oPPreVUx6OV4846VKEUJeI6ebSkkTd9lllcM4pyVuseZZsA0LXl1OmYwy6l7NlGmPeRK\nKfI4oztJWW74rNQ8XEewM05Yb+mOms9vtrCEwHcttnoR7+1rMT3TCpAS0qri0lKd9ZZHMW8e9nH3\n+XG+dFDz33/QCXOUFEeiatBdN+O8pOY5PHPmg4rab1zoIKVa3JfHpVPv/W4d9uPvRxm9WfahaupP\ny8MEdo9DKuhejLA/BI960e5//s4ooRW45Pf0QIePHygur9b5s5fv4tkWlq3z0PG85e2/fn2HZ9Yb\nnO3UaPh6G7jDHW8mScEsL7m8Wl98mc514MJSyLWDKZ3ApagqdgYJwlas1D3OtgKubLT47qVl3tuf\n8ubumN40I8krtocJIAgcWJpvplFK7Yt25wK10vBBwDTNudXVYrnW8Ljdjai5Fo6jI1MlwXEt8lIX\nGnWnubZoVuA4UEn9uEMsodMpSSmRFYSehSUEtqX90J6jiHOpi37u4d6/1QOX5Zqv3TOzjEJWID7Y\nIFpKRZJLhAWhYxH4DqO4WLhK8nk6RFhQs6Hu6i3mpNIi3AlcbvRnqEoyzQpagcuV9Sb704T9id73\nNSsqbEswmZUo9EJlOu/l048yKgkvbLYX1auoEt+xmBUVw6Tg8kqdfpzzzt6Ep1YbrJ71GSUF0wxc\nYdFuezx/toNCMYhzll2brJI0Q93sy7UtVn2Xb19a0S15b/QYJjlXNzusNQPqvh4kxkl+ZFOTezmy\nMxeK/WnGRusDX/okLXj+XHshjLuTFFlJdkcJb+9O+N7lFTbaAWlRPbCi9vA1HpROvfe7ddiP/7A3\nz0mJuh6Qclbq/pEB6d7v6+O2KfdXVtg/Sc77US/a/c+fZXoRs+F/YEp6mIGiU/Oo+TZFKXl/7iVv\nBS5SSnrTnNBziHPtfKkmKVIp1psBjiU4GKe8uzvha5ttrqw32Rkl1H2HJ9ca9KOMg35O6Fvak+25\nTNKSc+1w8T4nccEoznV+N3QYzDIqJcjGKZmsWA498rKirCS7k5TVusdyw+fp9ToNz2FvktEMHF7e\nGrHe9DjXqS9cIU3PoTvLUEqhLO2OAahKKO/7DIpSC2hVgQSiTOLaEHg2CshKBfL+T+6osJdlRW+a\nUOi1UHxHp5kOiTOpHTEC4kIirAolJWmuYBHt6wEnLUCqjNVGQF7pa9ubpdwdxLp7o+vg2BaDOKPm\n2uS5YmsYkxYVriO06wVBVlTcGRTz3Yg8QsemFri0AodcKl67M6KSFbV5+99+lNMOHbq2oBHa2uFT\nVniuxTBW1NBR/0o90H74rOD6/pSrm20iITjT9AlcfT8u1wOeOdNkqxez0Q4X6RTdv9469j6/P4pd\nqQe8vr3P/lhvmHKmFXB+qcbV8x1u9SKk1L2JbNDb/qUlv7jV47efWmO54T0wbfpx6dTPUlAP3+Mo\nKVit++SV5N29Cc9utD6U63/czBonIuxCiD8E/hFgA/+LUup/PInjflZ80pz5R120hxkg7n++YwnG\naTEvLdd0p3r6+JfvHnzkQHNxuc7r2yMavkNvmuNYls5VB45emHN03/Qz7WCxU9B7+1PWW3q/0a1B\nzDQtuDNMcBB0QhfPsqi7LtOkIJMV55ZqnO341H17YbH8jbMtdicJVaWwLEHDc5AKxmlOXipCS2LZ\nAsfSXvNcabGdJiWjuKQZ2Di2hWUJpolOvWRlRVpKorwgzqp5ThhKNU+dHHPtFHoxzRL6p5T6gTW0\nV19K8Fwoiw8nBgA8CyqlmKQS29avY1sWzj12RwnYFihx6G239DaCKAJbYFkWtpLIUqdNyhK6kxRp\ngW9bvLU7oZASlKDja+HpTTMkujI2zkukVIRzYXVsvTBeDx02Wrrh2SjR1aFYgovLNTzHIs4kDd9F\noIuotod6cB4kJf1phGsJ2oHDdG6V7M8y2qGjN8pG4TgWrmPx+vaIwTSgWfNRSt+TDdch8GxGSY6X\n6b74gyjnwnINxxIfqpV4bXvE3WFCKXX7hllWcuVMkzjTA79lWXx9fn+/vDVgZ5iSztNJtpVR93SN\nwa1+xHee+PTid3m1/qGNtTu1D/ZzfRQO0zwrdY+8UouBcGeUcGG5dmTweNw25X5kYRdC2MA/Br4P\nbAO/FEL8mVLqrUc99mfFJ82ZP+iiwfHbit0/QNz//IsrNcbzMn+lFN1pxht3R7yw2fnYgeb5cy3+\ndF7EoVAopcjKiotLNcZpwZmWT+B5lJXi/W7E+70I17HY7NR4er3Je3tTbvcTilJyZbOJEBZ/8c4+\nSzVXOyVsj7pv49kWr90ZcfVCh1FS0PAcWoFHJ/SYpAWz1GEUZSgpsIQilxJL6AZfTc9mkBR6Y4uG\nr/PzWcUwilmre2wNY5o1l0vLIfuTnHdGsW5zYAmdqb0njWKhBdqCRSuESn2QnglsyCudw64UNAN7\nXjWqyO+J3AU66pBzb/hywyMpS7Jcb7Ih+KBAybEgcAWB65AUeuH18F+LUmGLikrq2YSY/2QSXO3k\npCqlbosgK6ZZgWULhjP96GfPNtgZVKRKb9jdCT260xSJoigleamdR4VU7E314mklFZ4lkK6tawyq\nijyWOELwjQtLPHe2yV+8vT8/V8mV9RazrKBQkhvdGS3f1VbCmsv1/QjPtnhjZ8q5dsGltTpJXnE3\nybhypsHdQcSr22MavsvfurzExeU6f/H2Pi9sdhal/T++1uXtXb0IPE1LrncjFJJvnl9itenwnUvL\nxPncDy8V3txFNU31onVgC8ZpiWMJWoHzsZtfH343HxQ8qfkFFvo/uuJ4vn/ro7jQDtM8elF3CoBn\nW/SijJWG96Fo/HHalPskIva/BVxXSt0EEEL8M+CPgMdW2D9Nzvy4i3Yo6pVUvLc/ZZZVODZHuuU9\n6Pn35g57s4wXNjuLdqUfNdCUUvGbT61yux/RLxWhZ/HseovAtXhirc6ZVkhRVVw7mOHbgrxUjKKC\nW92Ic+2AwLX4jbMtxmmBbVmLbov9qKAd2LRCl71RQs3T02nf0bv0XFiu0Y8yfnajB0J7rksFgW+j\nKr3xg43FKNXpGiUVd/ox7+5O+I2zLS6u1NkbSlJVstkOyIqSWaLV2XMsykoPUghLDxRqLsZCi7EE\nxD0huEA7UKQ6jLp1mjzNKzxXsNYIiDO92GtZAiEUNhbNmkNW6A0uPNsjcyqyQqccDrn2P/y9j7p9\nHkv++HN+vb//GR23A3zrmN8f97vDx99/Lv/2zT1++NYeX3+IQOmjOLqXrk5f9qKcTug99hWtJyHs\nm8Cde/6+DXzv/gcJIX4A/ADg4sWLJ/Cyn56TyssdttJ9b3+22LwgLSpe2Rot+mM/iHuF/i/fPfjI\ngebeKOZ6d8ZzG01agcPXzunFtayouNOP8ZwmSVFSVpLlukecVVw7GKPmVaN3hnpLtTN5wVpDLwT9\n7EaP1YaHHRd89/IKkyRnHOtddhq+w9u7E6SSvL075mw7ZBAV+J6NrHQ4nOYlRaVIy4TAcUAociUJ\nHL0gawHXuzOSXDfLGowS1ho+S3UfCbx2Z4RUesCqKrAsSSXnETrgWtpqmZWSYh6BH0bxSsGhmaWq\n5s9RkBeKSaoX/Bq+jevatEOPUZQR2oKygqX51nTjOCPOJXl2XOLH8GXk5TtDutOMTug9cLHzYbg3\nhdrwHS4s11hpfHaifpIFTp/b4qlS6k+APwF48cUXj0t/fm6c1EJHK3R5c2e82HsSdLpgue59opvo\nowaa+9cDfMfiVm/Gk2t6C7m8koyighcudFip+zQCh/1JwvNnW/z4vQM6ocso0c21KqnNZ7+6PeIb\nF3T+s+Y7XG13Fn21X9uZEGcFaa6j2CfXGjRshztxQq0see5sk+v7MyZpQVHpvG2lFAhJWlQEvkXN\nc6l5DoMox3csHCGYZQXDJGe96XN5tcFT63V+dqO3sBSq+Y9UOrfuCGgEFhb6+OqeO0ai0yqHWnyY\nJqnmv7csXfEqgYbn4AqLJNN57VIKlmo+/VmO6wgqKenUPJRU/PZ//+dE8zx/Xlb4jk1eVni2rsws\npESgm3QdpnnuN/kJIHQFgWdTloqG5zBJM4Rl8eRqg6qSTIuKOKuwhF7cniTajupo9+jCkXO2E7LW\n9OlOMiZxTpxXBL69sAC2Qo8raw16UYZjCUZxQV4poqwkL0sCz2G57jOMMlqhQ1kpWqHHubbeTKNS\nihfOtXl2o8U/+uE7tGoek0QHBoHnUFQl++OUFy8t0wxdfv9rZwF4dXvItb0pT59p0pvq3ZH6sxQE\nrDZ8nttok843FJdS0ZgL69445WZvxs4w4dxSyJOrDTbaAc9utIjzkjd3Rrx4aWWxiQzAS+8PUCie\n3Wgd2dKv5jv8R1fP8fLWcNFD/nATmWF3ypmmf2Qj8uMKmz6OzzNvftIFTich7HeBC/f8/fz8d48t\nJ3XBLq/W+fF7B6w2fJSyFm1xnznTfKBF7EHHedBAc/96wJOrDV7dHnEwSbl6vsOFZd2P/d4b4OWt\nIW/eHeM4Nk+seOzNbLoT3bBLSUW77tEOXPKyYmcYM/Jcluq6mCTNtd2kHbrsDBP2ximd0KUVusR5\nxR99c5P//WfvkxS6CCgpSlwLWqH2bgeeQ5qXTFPdVbGUEs9xkFIyiUvaoUPgCvqzDMuyqPkW4wSY\ne9ZBR96W0B51xxJM87mnnQ/+/TC+tvVTEfPFVFuAY1ksNVwsLAJXp3mKuZUQoR8behZpXlEoiW/r\nwcJxLL5ztsWbOxOq+UB36F9H6FlFY556K+cVnfdHKDb6sVUpWWsGZKUEIfAsQTfKsBAkeUlR6eZb\nlZQoKal7FlFeIRAIW+frs0oxS0tGiW5P7Lo2jrDwQns+AJTsTRNdKDXPzyshsSywLAspFZ6j911F\nCixLF4GttwLSvKRUillW8NMbXcZpyXJDBw5Fpd1NZSmpzatBO5a38JK3Q5e1pk9VSS6t1Nho+wwj\nn9uDCNfSTd4ureitEt/Zm9CPMp5YqfOLW3pme+VMk8C12BnFPDPfUjF0bRTiQwGO4wjirPrQln7j\ntGQU50cWT2/1Zswz7bRr3iLYOm6x82H5vPLmJ13gdBJNwH4JXBFCXBZCeMB/BvzZCRz3M+VQ3P/O\ns+ufelTs1Dy+eXEJOe/zcbiHqPMAi9jHnctxzZYmSXFkF/lm4HJ1s01eVQ9szHR5tc4gyhEK7gxj\nkLBS99ho+SgheG6jpVvZVrDaCDiYpnSnGUoqWr7uI+M6NlFWMkkLorxirelxdxhzexAR+jbLDY8z\nTR/Xsmj4WnAC16EodeydFCUWhy6QiryCdt2lKBVxXjGMCxrz9rGBZ9OquXj23H7ogj8X5DOt8IMo\n2BX49gcLoYdI9CJqK3AWYpsXklmuBbGoJJXSs4ulmodr62ZeQgjagUctcAnme8AGrk090B0tLUtQ\nD1xqrt70WUqFQJIUWtSP+/KU6KKlrJTa1aIUAkHNd6hKRT/KFlWVq02fQVTQTzLSedRpWdrbnxTQ\nn6a6EldBXlVkpf4pygoBzLKC7jRjqe4hLIHnCXzHYrnuU/ccLCGYJLq6N0fS8B3qvsMkLWjXvfkO\nXIrlmscLm23e2Z2yP0molCLJC8ZpyXMbbX7r6TXOtsPF/fa7V9b4+9/cpFTQnaU4luDZjRZXzjT5\nradWeXajtUh/lJVkf5Lw81t9zi/VeXajie9aOLbF/9/em8ZIlqZ3vb/37OfEmhG5V1bW1t3V3dPd\n0z3THre5c8eYGYNByAsCCYRAFh9GfEDABwSXOxIIkCWuLF0hwQcYCUsXybqAZFsgsGWPwfLMYNrM\n1tPTS/VW1V1LVlWuscfZXz68J6KysjOrsiqjKrNi3p9UmomuqIj3RJx4znOe9//8nwvzlXHJbJhk\nPD1fZhCnDGJlIDeIU2qFq6lA4lrGpzxj4M7mqdo7EcwEDmGaESYpjinY6qvhHaON2JPI3t85qIvd\ngySIuzlyxi6lTIUQfwv4XdTv7deklG8f9XWfFF5aUc50gWPds6xzv/rZQZnBfmUayzR4ZbVxoDmT\nuuDU+W/v3iZMlOpjoeLx8daAwDYIHHNs/3vpZhuJZLMXs9YOsU2TZsVloxdRdW0sATOBQ8l1ODdX\n5t2bXXpD9cOzTOXxbhmCNFcTgNQtr/JbsQzBbNklTDJcyxhPHPIsg1aYFlmnkjz6poFrWwSecko0\ngM1BTD+KyaVK100pSJBYpupGzSR4NviOxTDO6McpQkDNc0hlTmCZzFc9ar7D9Z0elmHgORaztsGN\nVkgO1AObZtnj2nafzW7ED67tFF2kalNXSoFlGlhWhplLWmGGbaj3lnvS9d1lmTSDtVaI7wiqvoth\nCBKpLJTjPKc9hKpnFD7r4BiSQZYqKaWhZJx5Bu1+pC4SqbrTiMmJhzmCBM8yqHn2WKmUpjmtYYYl\nci4uVbi01qY1SEEqNUeaSZZrLvXA5sVTdRolZ1zC2OhFnJktsdlTU54yKVhteKw0PAzx6Zm49cDh\nL35+5a5z+qmFRS5v9BjEqpzzoxtqXuyXn13krbUOtgFPz1d4er7Ce7c6quEqTMfBfHQ+776T/j+f\nnsMQotgITyi71l2eMVc2+8xXPM42y3iWqZq5UFYTdqFgeRI2Oyetx59IjV1K+dvAb0/itZ40DlPW\nOUr97GH2A0a2pp1hwsqMj2sJwiTHMuDifI1c3qlhbg8SZgKbbpiyUPGwLYMoy3nnRoeFsqs6Ulsh\n2/2YpZqq13q2WYznyyCXDNMM0zRoBA69SKltTtcD0kw1+pyqezTLHs8v1Rgm6ja/M4i5tjNkkKQg\nVUASAsqOCdLEcy1WXRvLgIWKz7XtAY5VlFssQ0kKU9WZmma5akwSEFgGvm1QLZpmhmnGVj9kvau8\nYs7NVvAci84wIXDU7NNG4HC7rTThgyghsG3CJKIZuDR8hyjLma24tPsx13eGKuAD4W495S4MVJ3f\nNNS+xnzFRgrBZjdEIEkziSkEaZ4xSHLSHExD1c0FEsswyE1VBnENgyxPicnHnbquJQpHSYFnm3TD\nmLJrkdlKU55JdVdUDRwGnRCEYLnuslQvYQrBK6cbrLWHnKr7gPJ56YUpJcdiI48ouTa2Kaj5LkjB\nlc0BQmyyXPf3lR9+9vQdoUDNt7my2eettTaGIfAskw/W+/TCZNzyf3GxysXFKpc3euR8emjH3gRn\nue7fNYsV7njG7Fa4jWSJrqXKUI96s3OSTLrB6ce283SS3K8Od5j62SijX2sN6YUpZc8a/5AeZD9g\n90Xk5dN1rm6rDadnFiv8iQtNLm90sUzG9VLTUN2aMyWbtfaQG9tDAtuk5Bl8vDWgFthYRkwuLd68\nHrJQ87BMwUxgs96N2DRAxGq60XzV4+YO1AKLp+YrrHdCPtzoM0gyTjsWL6+qDdu3brS5dDNHInEt\nE1MYtGQECHKhWvZlmOLaBp1UWdaaBuPmKAuBYZrM+CatQVzcqoNjClzHLLTjsNnrkObKJ77kmHTD\nVAVBz2Kh5mED3Sjh9ctb6kIVp0p37WSYhsF2L2GrF1F2bZ5bqhDGGYYBkONaqtQxjJV+flQCsooa\nvmkYBI5gEGdcbQ1peg6urawELANsS81T7ceZaprKIUslGBLPNjCFoFrMKfUxqPk2m0UZxzZNfFdg\nGwb9KGWjF2MYysdmruLi2xamkAS2yVNzZSxD4FgWIFmuB8RphkB9/1kui/q18kAPi27TeuAQJmqv\nZBCnfP+TiJ1+mfdvdSi5FnMVb98kpR44nJuFP/pwg7WdENdWNffOMGOzH3JhrsIzCxU1tGPm4KEd\nu7nfHtS9BrUfZ5PQgzDpjdqpD+wnwSP5frr5UTDOc7jdHmIIQT9O8SwVuF5ZnTmUJzbcuYhkhUd7\nmGR4tomUkppvszKjNpFGJ8/PPr/If/jOVfphwnLNxxYG17b79KIMQ4CQkkbJo+SYtIcJUZrx9EKN\n9iBhkGQ0Sw7VwObPvrDMWmvIR+s9ru0MeP92jzjNaJQctekpJB/c7hCmkpvtoWqN70IzcDAMVcRo\nDRIVKHPJ2dmyuhuIVYZ/Ya7MrU5IlOZESYbvWIRZNt4odGzVmRInGamU49t201BDMyzLYLHmcbYZ\nsJGxQcUAACAASURBVFD1MIG319q0QjW9xwI6Q1W7llI1JKnhEza9KOa7V1t4lsGZRsDNVkiWqx6C\nfpSpjVdRlGdQJRnPNsabukkqCbOMsqP2LQyh1CJ5LpE5OBZEifJ2V2UgA8OSzJYcWsOUqHB1dExl\n1hY4JghBnKbcChMWqx5Vz2J7kLDVTzg9YzNb9lmsetzqDMlzKHsWs2Xl1JjmOZ87M8MgTrlR+BZ9\nvN2jPUzUHoNr0i284YdJxjPzFZJM3VG8daPN6WbAmaYyk9ubpIzO5c5Q9Q+sdyIQsNIIgJxrOwOu\n7QxYrvuHDlz3CnrnZvnUoPZ7TXk6yUxyo3aqA/tJ8Ui+X/1sFIyvbQ/wHQvPVnr4nYFq6X6QnfE7\n2nqlInh2qcrl9S5/cGmdXqSsfPdq7GfLjhqCIQyWZzxqZTWkozeM6cY5lze6LNY8FqseadEheWqm\nxMXFKr0wGcvZLt3qsN1TQ5EXah6eZdONErrDjKrv8NFGnxdPqXLMOzda4xp1L8ooezbdMCGOVTZr\nCoNBllAvOVgCKp5DybW5vNkjRJUbesNikpGprBSGxaZiXtS/u0PlRT9IUnIp6OcpUZJxszVkdbZM\nN0mReU4qBYM0GytroiTHdZR1wCBMKLk2jqG6WiuuQ73kcqs1ZKsX41gqoKS5pBPlauoRkGYZuRTK\n5MxR2nlTCHzbwrFVpg/qAhCnuVLCAImU2FJZB/Qi5cluCHW+WMIgB8I0xUBpI11TjbzrhhkzvgNC\nsjIT8H88pZrY6kHRKRynLFmCXpjTGsZjg7jLm30qntowny07JCkkWUacqulQwzhDCAPfUZ24piG4\n3Q7vOud821Sb9Cj7AMc0cW2Tm50htiWwTJPNbkTdt1io+SzXDze8Yzf39pK5O+gv1spFMtc69oEX\nx8XUjsaDkzMuazTWbvdu/yBOaZQcfnB1h//xwQZXt/ps9CJcS+2Mu5ZBL8oeeGe8WtQ4lbbeQiDI\nETy1UGGhojpJ944tW6j6nCrKPq5l4psWuZSs95TtgW+bZFIQZ2pQ9GY/puQohUl7qNRAl251QKoN\n1FrgsNFWk5QC26LiK7/yeuAghKAXJsrEK89ph0mhHFFTnhCQCdjqRqSZxChGz613QxqBy1zZxTYF\nWZaPLQaSDPqRstkd/XEtVdbpJynrnQjfEiqLFhBmSqnimyZlVxltpTljHbksPGg8U2AWG5SBbWJg\nkqM2k1ebAfNVjxdO1WmWXeq+Q0mpKcklxKnEtCTNssvTsyVmAjWQI3As3ML7HimxTIEQUHJMqsXY\nuoqn5KWBY1BxTZ5dquA7NghByVZ2D2muVDeuJfBdk7JnYpnKuuFme8hy3cd3LBzLpFqs/1Y7ZHMQ\nYwpj/Bv43OoMnz/b4Jn5SvF8gziXzFUcLKE2xaNMlXgAfNsqNigV3VCVsr71wQZvr7UJYzl24LRN\nA9cylXwyz1mqB5yq+w+t9DiI3Qq3c7MlLm/0PjUu8n5j+qaNqc7YT4pH8kFZxeWNHoFjsVD16Ecp\nm70Ix1RytSjNi6lChzMHG6G09RuFtl6y1hqCVPKwfqzuGm53Qr7+zcsEtsFcxaPiW5xplFSjST/G\nRLDdj0nTXPl757DRDTk941NxlWlY4Fo0yg6/+LkVXv9oiyTLWW2WaIUpgzjDKDl4lkHVc/Ac1bZ/\nZaPHUs3jbLPElc0e7VD5yfSyFGFA2XUwRIpnGViWwLEMoiQjQm3KuY6q16rGIanG1xXljwxlKWAI\nKLkmpgAjU7JDTEmcKk23awqWqi5vr7WVU2amVDo7AzVtKClewzYFUa4kdnNVj1aY0B6qvYaGbyMR\nLNVdGiWX2YrL5fU+c4ZHP04xhEmUpCxWfZplh9myx1Y/wi9MtqJUTQBRPvgGZVftDQhgvuLzwqkK\nmVRTj66s99keRJxu+AgBnWFKZ1BsYMcZw0RZDNueQZypKVKjksRc2QEp2ern1H2ffpxyekY1PY0C\n3vk5dR6uNgOubw9pBup7tQ31mXiWYLHqEThm4UhpMOe6Y+XLmzfarLWGPDVXHs8qdW2lgFrvxjSr\nNk1T+RoZQtAouY/UyvakDbw4LqY6sJ8kj+S9t5KjElHgWJyaCXivGOt1ozVQ/ihA4LiHNgfb/T4v\nr9a5uj2gE6Z0QqVj/nCjR8W1+HC9y/+6vIlpCC6sNuiGqrHj3GyJs7NlNrsRv/vOLTa7MYFj0osy\nRC4pF0M8LBN++YvnONMsjfcvNrohC1XVzQjwrX5ML1I+3p5t0Q1ThnHKfNUbO+Q9u1hloxdjGQLb\nUpldlOdKg9+LMQyJaxps9xNM01Aj+oYpYap8XixDZdjSADO/07VqGmCZgjRXdwBhnJPlkkFhaduP\nc3qJ0oOv1H3eu91TActQXucjw7FBnCNz8G2DHImB4HTTV9l9JonilGeXamz0IuI0wzYhSQU132G+\n6rHeiTAFuJbJTj8qsnPG/uNKgZNiCoMkN9QoPkOwWHW40QrphWojN85ShkUXbJLkREmO71oIDNIs\nI8lz8lzSHaY0Sq4apjJfwrHUxdcJU/7EhTm2+xH9SHm/n5oJxr+J7b7awxmNNsxzyVLVpxpYXFws\nKzfNXKoSnyV4eqHMSyt1tvsxb6+1qLhqk79RchFCbaq/cbVFs+Tg2oJWGCOl4IXlCiszAYMkxQ7F\nOFFplBy2+/HE9sBOSjJ33Ex1YJ+0hGiSG7GfHhBQ4cbOgH6kNgbLnpomdFhzsN2MtPV5Dle3esRJ\nhmUK6oHDf7+0TtkxaJZ9DMOgFqhb7H6cEqUZb6216QxiZgJbzeVMc7I8Y77isVzz+crzC+OgPro4\nje443rvV4VRdbU5GWU7ZU7USzzYYxBLPMgs3ypz2IOGFpSq3Oko7f6ruqZZ522Cu4tIdpiAkzyzW\nGCQpjYpHTohvq0aiTpgW3amGUoRkOXkxus+3TMIkRQqDkmfh2AaOoSSeizVHqVsQqhxWdri5MyRD\ndWyauZIq2qbAM00sQw3afmGpqmapRhkrM6p09eFGj8Ay6RUbjamE5+eqxHlG1bMQhZOkbZnkeY5t\nGAzSlGcXKrx3q0uaSDxPNXe5liDP5di3/kwzIEwyrmwNKTkm54tzuRfFBLZFLTCJEouaYxKnOZmQ\neI7JT5ydoeSZ4w330Tn77s32XRff9251lNsi6nfypWfmx97ph3FWPNO8M6v0/dtdolSpg1qDhLJn\n4jsGEou5isv5+TIlx1Kls0Rd7Hxb3Y1+8731u5wjj7oH9rDJ3EkQWUySqQ7sk5QQHbQRe36u/FAZ\nx94TsOLZrDZLPLVQGW8s3c8c7H7H/btv3aIaOHTDhKWaz0zg0AtjhHTGNdPRe99sD6l4KoN6frlG\nO0zYGST4toFtmnSjmDhzx5/FbvXNMFFukr5tYAiYq3jcaA1ZrPvMl9Wt9/u3O+TIIiBLhmlOI7B5\n8VSNqu+w1hqqxiMJv/TKCqD01R+t9/AsJf/7wrkmAD+81qJkW9Q8JROM04yS7TBfc+kOk6Ll3yKT\nKrvNpbIAiJKMilcCEpyqUqjMlWx6YYJjukTFoJA4k+RSMlt2efVsg7fX2kRZxlNzFVzL4L3bXeIk\nYyZw8B2T6+0hZc9iruziOwZpqDpdB2HKTMVhte6pDVJDzR8tOTYz5QTLNHEs1aTl2wZpDhU356UV\n5fvzxvUdFioejZLNjdaQYZJjIOjHGbMVZXlrmIKFqsfpGV+pULoRC4bLD67ujM/FURIQp3khb+zg\n2SauZZIj73seH5REjM7h0azS9V5UZO3q/Lq4WMU0xLg7+gdXd3Atc3zO7wxiaoVCa77qTaRs8rB9\nHydBZDFJpjqww+QkRPvV7nph+tD2oIc5AY9SSqoHDos1j+eWqvSilLXWkE6Y0Ch7CKDk3nnNbpgw\nV/HoDBMc02C+6mKagkbJoTVIuLY1oF5y+JmL8+PN141uxFYv4o1rLSzTYLbkkOWSH15rsTwT8IVz\nDbb6Mf0iK/5Tzy7w3q0Ozy1VuHSzTZzmXNnsc2G+wpxl8tR8hW6YYBhKiXHpVoe11oA0V3rsD9e7\nrLUGrDYDslzSDmM+utUjtwR138J3LWUxbAk+3gqpBVax6Su52VLafMcy1ACLJFN6eFNdtIQQVFxV\nQ86BimeRZZJOlLLeGeJagrmKz/JMwHcubxUj+JT08OJChc+u1PkfH2ziOQZJproe58ou/kzARj/m\n/fUuixWfim9jG4JLt9q0hjGeZSAxyfKMc3MVtvsJSzWlD+/HGWGcszLjEaWSnV7CQsVDIgu5JcyU\nbDZ6MVJKklwyiFIuLlR45fTMXTX07X7MWmvIlY0eG90YhKRU3BW+fHqGYZyNz2PLELy91uZb76/z\n8urMPV1KR+fw7lmlaZbz7FKVp+YrVLw7Y+rg02WSXqQscbvRnTlZhy2bHJRhP0wyN411+akP7JNi\nv9rddjGb8mFOiMOcgPcL/ve7fbzbT1pdDBolhz/6cIP2IFI/qjDhdifkxdN1PtroFgOWc2bLLr0w\nJYxDmhWXP3VxgeW6uo3vhSlvXN1hs6fq97ZlsNVX3Y9nZsvsDCJmSw7PL9XGxmiWYfDy6gxRmvHD\nay2aJZvMt5C55OOtPjXP4kZ7yOdWZ/je1R22+iEmBp1hovTzSc7WIGK9F/KVZxdV041p0R7G9KOU\n5arPYt3jjU92yPKcxVoZ1zAYFllqI7BZqivb4WGU0eqHhQd8zkLFpTNMAdURmuXFoGzDYHuQkGU5\nG50hwzglTFOqnkOSS2zTYKMbEac5Z+dKuJZ6vFzzKbkWa60BL6/OjMsOb67tMOM7XJgr8fGGwe1e\nyFLN5myzhmeb9MI+rqU88c/OurSHEf0oZ5AoQ7WKr7TqcxUXQyjdf9Wzee3CLLfaQ55drPDCqfrY\np2WUeJxrltnshny40eP69oDzcyVKjjX2Pxidx2qugDLqmi27XN0akOXynomKaQjeXlPWAWdnA840\nyuPSIdydiOxNVMqFXn4kl937/IO4X4b9oMncNNblp1ruOElGJ+VutgcxjeDuk/BB5In3MyLbaw4W\npWoA8g+vtfjm++t8+4ONe8q69pNZVjyLv/raWXzH4mZ7SI7kzGyJZsnl4kKVWuDQj1KyTHV8llyD\nnzzX4KXTtTvH3Y/IgFxKbFMUfww6YUJnGFP3XTXHM81xLaX2uLLV56UVFXQ+d6ah2sqXagTF+L23\nb3Y4P1vm2cUqNd8mTSXbgwjDUKUKKVXWmqU5H270ODUT8MWnZik7Fkt1n/PzFQLHRhqC0zM+rX6M\na6va/SsrdXaGKa5lslT1lK9KrgzQHMtCGMq+0RAGgWMpb/hcIgzULFaUx0mUSlrDhLBQLIVJhhDK\nznex6nFutsR81WO+qlRNF+YrNEquavuPUj6/2qAWOJydLXNmNuDFU1WkRBl9CfjCuSaObXDpZput\nXsjpesDOUHXWrsx4DOMMxxDKUbGqLh4/9dQsX3pmji8+Pcdr52fHQX30PfVCNQ7xk60BplB3Kdd3\nQuqBw0xJlcBG5/Faazi2oPZs9TkcJA8eBVfXMnn1TJMXlmvMVbxxB+9uWe+oVr/3fBzJZWcCZ9/n\nH8SkZcz7/baPcxD1JNAZ+yHZL3s2DSVN3M2kT4hR9rE7S/FtkzdvtOiFyViNsN/dwr3uCu7Mo9y5\ny4fjtfNNyp7JzXbI6kzAmabPmUb57oBRODPWi0HYwzgr9NcWvShltREgBHy0ocaJnZ7xWap6hVtl\ni3OzpXED1WqjRJhkJJnkhVM1hBBqxmQqaQ1Sqp4FCPpRSj9K8W2DQVFaWq77dKOEkmOpckSmlDSl\nwCbOJOfnygyTjDzPeG5RmURt9EI1WMRQQTzJMizTJIpSAsdhtuLQGqRs92JMA+q+zUoj4IPbPRCS\nhqda7bNMslj3SFLVKZtJyY3WkEGc8dxiFSEGVIvPLEqV4cBCxUMIdRHMkZRdGyngM0u1cVPa51Zn\nCJOctVbISsPjF19e5t2bbbb7CY3AZbbsUHItcpnzwkqdv/ZTZ8YJwd6y3XbRZZpLyVo7pFLYVHyy\n1eeH11q8dr5Bp5ik1Si5XNsZFp+3GmRddq0DM9f9yhfzFTU4e5SI7L0L3Xs+juSy2/34gfbAJp1h\nn7RB1JNAB/ZDsl+Q/NnnF8Zudo/6hNj7Q0ozqHp2YaqkSio3dgbc7qiuwLtrjvfyldnGwBh701Q8\nm9fOzbIziPnpi/PjC8ruYzQNWKh5mEKQA66p1C7tMGG7H/H9qy2W6z5nmyVMQ9AexlSKgFEtbFhH\ngxM6haLkheXK+OKxXPdxbeULvj3IC8WLoOyqBqGdQcwb13boRSmBreZ59mPVzPXqmQbfv7pFxXdZ\nbfh0Q1XK+dzZhtLm52AIg82eGkmY53B+LuD0TEMpO/J8XDNe64Q0AgvPsQjcO2WfOM2ZqbkMooTW\nIMG2DT7eHGAbqqlrqxexXPdxTMEwyVlrDYnSnHdudVhtBFxcrLJc9/nh9Rb9OMUxDcJElax2D4X4\n6Yvz4+/pWx9scLM9ZL0dEWUZs2WXn395eRwED0o8LEPQGSaUXTXX1TIEZxoBUZbzwXqPzyzXeO1C\nk8sbPSyT4i5E/e+ZZunAROWg4Bom2T27Svc7H880H8xOd9Iy5pM2iHoS6MD+AOx3Uo7c7B71CbH3\nh1R2lcytF6XF5JgOAlioeuOyzL1qo6OAPfIf2T1tRk3nsXcd894L2iI/ut7i+s6AqmtxebPPzfYQ\nxzK5MFsiR5Jlko83eyzVA+SuQdGjAJTnjAdxm0JQD+5W6TyzUME2Dd64vgO5ZLHmY5kOO4OEQZzS\nj1UH5ummrzZ3A4uFikc3TDg3W2a24nKzPWSu4vEzzy1Q822+9cEGgyRhsztke5CS5TlVzyKTkmbZ\n4edfPsUnW31+562bOJbgfLNEmueUHBvbUi31C1WX9U5EL0owDIOlGQsD1Waf5jnzVY9cSn5wdZuy\naxO4Js/MVwDJm9fb9MKEfpRS9WwagY1rCTb7Mc2Sw5mmuriNnAt3I4DAtjg1YxQmXTa1Xc85KPH4\nxju3+Hijx2zZ5frOEIDFqodlCQLbHNvx1nwb0xC8cbVFo+SMjboOSlSOs0fkUWTYJ2kQ9STQgf2I\nPK4TYu8PaZTxVVyLGzuDwgtc3NV8cq9N3NEdwPm5ciF/U0MaLm/2OFUYNN3rGGu+zf/8aIs/fH+d\nimfRLNdoBA43WkNWmwGDKKcdxrQGMa+db45H79UDh/NzZb7xzi2yHBqBw2LV5+PNHmXXGuuZl+oe\ny3WPWmBzY6dPP84ZFjr3OJXkuZLR/fQz82x0I9673SXPVSD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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(ys[0,:], ys[1,:], alpha=0.2)\n", "for e_, v_ in zip(e1, np.eye(2)):\n", " plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)\n", "plt.axis([-3,3,-3,3]);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For example, if we only use the first column of `ys`, we will have the projection of the data onto the first principal component, capturing the majority of the variance in the data with a single feature that is a linear combination of the original features." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Transform back to original coordinates\n", "\n", "We may need to transform the (reduced) data set to the original feature coordinates for interpretation. This is simply another linear transform (matrix multiplication)." ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": true }, "outputs": [], "source": [ "zs = np.dot(v1, ys)" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "image/png": 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w0akqQgjW+jEjJZuSpfPTK2vsbjg8s7sYjJHlkm8+sYOmF3NytkndcTk4UeH9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\nMQ9apDysM+j49a5sD3n1zVvkRUG74pDlys1Tzwudbp6YwP4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Covz3h4WWOWqeRPSO/T4c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\nRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(zs[0,:], zs[1,:], alpha=0.2)\n", "for e_, v_ in zip(e1, v1.T):\n", " plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)\n", "plt.axis([-3,3,-3,3]);" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[1.00e+00, 1.11e-16],\n", " [1.11e-16, 1.00e+00]])" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u, s, v = np.linalg.svd(x)\n", "u.dot(u.T)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Dimension reduction via PCA\n", "\n", "We have the spectral decomposition of the covariance matrix\n", "\n", "$$\n", "A = Q^{-1}\\Lambda Q\n", "$$\n", "\n", "Suppose $\\Lambda$ is a rank $p$ matrix. To reduce the dimensionality to $k \\le p$, we simply set all but the first $k$ values of the diagonal of $\\Lambda$ to zero. This is equivalent to ignoring all except the first $k$ principal components.\n", "\n", "What does this achieve? Recall that $A$ is a covariance matrix, and the trace of the matrix is the overall variability, since it is the sum of the variances." ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[0.589, 0.191],\n", " [0.191, 0.191]])" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.7798589520198774" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.trace()" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[0.114, 0. ],\n", " [0. , 0.665]])" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e, v = np.linalg.eigh(A)\n", "D = np.diag(e)\n", "D" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.7798589520198773" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "D.trace()" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.14677219641978512" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "D[0,0]/D.trace()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since the trace is invariant under change of basis, the total variability is also unchanged by PCA. By keeping only the first $k$ principal components, we can still \"explain\" $\\sum_{i=1}^k e[i]/\\sum{e}$ of the total variability. Sometimes, the degree of dimension reduction is specified as keeping enough principal components so that (say) $90\\%$ of the total variability is explained." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Using SVD for PCA\n", "\n", "SVD is a decomposition of the data matrix $X = U S V^T$ where $U$ and $V$ are orthogonal matrices and $S$ is a diagonal matrix. \n", "\n", "Recall that the transpose of an orthogonal matrix is also its inverse, so if we multiply on the right by $X^T$, we get the following simplification\n", "\n", "\\begin{align}\n", "X &= U S V^T \\\\\n", "X X^T &= U S V^T (U S V^T)^T \\\\\n", " &= U S V^T V S U^T \\\\\n", " &= U S^2 U^T\n", "\\end{align}\n", "\n", "Compare with the eigendecomposition of a matrix $A = W \\Lambda W^{-1}$, we see that SVD gives us the eigendecomposition of the matrix $XX^T$, which as we have just seen, is basically a scaled version of the covariance for a data matrix with zero mean, with the eigenvectors given by $U$ and eigenvealuse by $S^2$ (scaled by $n-1$).." ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": true }, "outputs": [], "source": [ "u, s, v = np.linalg.svd(x)" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "image/png": 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FgW0YDMIUxzY4f33A6bkaVcciSHK2hzGLUx5bo5Rr3YjVtmpPsLwTcqUTULVN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\nkV//nMIvXiX39huI5NOAIR2H+vHnmX/uRV7bfZSLew6RGRb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mLmyFX8x6bhdkVjohacaW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\nAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "e2 = s**2/(n-1)\n", "v2 = u\n", "plt.scatter(x[0,:], x[1,:], alpha=0.2)\n", "for e_, v_ in zip(e2, v2):\n", " plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)\n", "plt.axis([-3,3,-3,3]);" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 0.928, -0.373],\n", " [ 0.373, 0.928]])" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v1 # from eigenvectors of covariance matrix" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[-0.928, -0.373],\n", " [-0.373, 0.928]])" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v2 # from SVD" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([0.665, 0.115])" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e1 # from eigenvalues of covariance matrix" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([0.665, 0.115])" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "e2 # from SVD" ] } ], "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": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 2 }