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"source": [
"import numpy as np\n",
"from bokeh.plotting import figure, show, output_notebook\n",
"\n",
"output_notebook()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## algorithm"
]
},
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"metadata": {
"collapsed": true
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"source": [
"def PCA(X, n_components):\n",
" # normalize to zero mean\n",
" mu = X.mean(axis=0)\n",
" X = X - mu\n",
" \n",
" # eigenvectors of covariance matrix\n",
" sigma = X.T @ X\n",
" eigvals, eigvecs = np.linalg.eig(sigma)\n",
" \n",
" # principal components\n",
" order = np.argsort(eigvals)[::-1]\n",
" components = eigvecs[:, order[:n_components]]\n",
" \n",
" # projection\n",
" Z = X @ components\n",
" \n",
" # result\n",
" return Z, components"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2D data & principal components"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# generate points\n",
"x = np.linspace(0, 13, num=100)\n",
"y = x + np.sin(x) - np.cos(x)\n",
"\n",
"# 2D data\n",
"X = np.c_[x, y]"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
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"outputs": [
{
"data": {
"text/plain": [
"array([[ 0.72299657, -0.69085162],\n",
" [ 0.69085162, 0.72299657]])"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# PCA\n",
"projection, components = PCA(X, n_components=2)\n",
"\n",
"# principal components\n",
"components"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 2705.081, 0. ],\n",
" [ 0. , 47.716]])"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# convariance matrix of projected data\n",
"(projection.T @ projection).round(3)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# prepare plot data\n",
"mean = np.mean(X, axis=0)\n",
"extent = projection.min(), projection.max()\n",
"angle = np.arctan(components[1] / components[0]) + np.pi * (components[0] < 0)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
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" if (document.readyState != \"loading\") fn();\n",
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" console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
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"</script>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# plot original data & principal components\n",
"plot = figure()\n",
"\n",
"plot.scatter(x, y)\n",
"plot.ray(*mean, length=0, angle=angle[0], line_width=2, line_color='red')\n",
"plot.ray(*mean, length=0, angle=angle[1], line_width=2, line_color='green')\n",
"\n",
"show(plot)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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" \n",
" var NB_LOAD_WARNING = {'data': {'text/html':\n",
" \"<div style='background-color: #fdd'>\\n\"+\n",
" \"<p>\\n\"+\n",
" \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n",
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"output_type": "display_data"
}
],
"source": [
"# plot projected data\n",
"plot = figure(x_range=extent, y_range=extent)\n",
"plot.scatter(projection[:, 0], projection[:, 1])\n",
"show(plot)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## binary vectors & dimensionality reduction"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# generate binary vectors\n",
"X = np.random.rand(90, 30)\n",
"X[:30, :] = X[:30, :] < ([.4] * 10 + [.1] * 10 + [.1] * 10)\n",
"X[30:60, :] = X[30:60, :] < ([.1] * 10 + [.4] * 10 + [.1] * 10)\n",
"X[60:, :] = X[60:, :] < ([.1] * 10 + [.1] * 10 + [.4] * 10)\n",
"\n",
"# define 3 classes\n",
"Y = ['red'] * 30 + ['green'] * 30 + ['blue'] * 30"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# PCA\n",
"projection, _ = PCA(X, n_components=2)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
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],
"source": [
"# plot projected data: 30D -> 2D\n",
"plot = figure()\n",
"plot.scatter(projection[:, 0], projection[:, 1], color=Y)\n",
"show(plot)"
]
},
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}
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