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Change Of Basis With Coordinate Translation
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README.md

Change Of Basis With Coordinate Translation

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change_of_basis_with_translate.ipynb
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"cell_type": "markdown",
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"source": [
"## Imports and Plot-Function"
]
},
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"cell_type": "code",
"execution_count": 1,
"id": "90af060a",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from sklearn import svm\n",
"from sklearn.datasets import make_blobs\n",
"import plotly.express as px\n",
"import plotly.graph_objects as go\n",
"import math"
]
},
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"cell_type": "code",
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"id": "29c46e6f",
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"class ThreeDFigure():\n",
" def __init__(self, trafo_mat=None, intercept=0):\n",
" self.trafo_mat = trafo_mat if trafo_mat is not None else np.eye(3)\n",
" self.inverse_trafo_mat = np.linalg.inv(self.trafo_mat)\n",
" self.intercept = intercept\n",
" self.fig = go.Figure(layout=go.Layout(\n",
" scene=dict(camera=dict(eye=dict(x=1, y=1, z=1)), aspectmode=\"data\")))\n",
" \n",
" def _transform(self, points, inverse=False):\n",
" trafo_mat = self.inverse_trafo_mat if inverse else self.trafo_mat \n",
" points = np.array([trafo_mat.dot(point) for point in points]) \n",
" return points\n",
" \n",
" def add_surface(self, x, y, z_func):\n",
" xy_arr = np.vstack([xx.flatten(), yy.flatten()]).T\n",
" z_arr = np.array([z_func(*i) for i in xy_arr])\n",
" points = np.column_stack([xy_arr, z_arr])\n",
" tmp = points[:,2]\n",
" points = self._transform(points)\n",
" surface_form = lambda x: x.reshape(round(math.sqrt(x.shape[0])),-1)\n",
" self.fig.add_trace(go.Surface(x=surface_form(points[:,0]), y=surface_form(points[:,1]), z=surface_form(points[:,2])))\n",
" \n",
" def add_line(self, point1, point2, width=6):\n",
" point1 = self._transform(np.array([point1])).squeeze()\n",
" point2 = self._transform(np.array([point2])).squeeze()\n",
" self.fig.add_trace(\n",
" go.Scatter3d(x = [point1[0], point2[0]],\n",
" y = [point1[1], point2[1]],\n",
" z = [point1[2], point2[2]],\n",
" marker = dict(size = 1),\n",
" line = dict(width = width)\n",
" )\n",
" )\n",
" \n",
" def add_markers(self, points, color=\"black\", size=2):\n",
" points = np.array(points)\n",
" if points.ndim == 1: points = np.array([points])\n",
" points = self._transform(points)\n",
" self.fig.add_trace(\n",
" go.Scatter3d(\n",
" mode='markers',\n",
" x=points[:,0],\n",
" y=points[:,1],\n",
" z=points[:,2],\n",
" marker={\"color\": color,\n",
" \"size\": size,\n",
" \"line\": {\"width\": 0}\n",
" },\n",
" )\n",
" )\n",
" \n",
" def show(self):\n",
" return self.fig\n",
"\n",
"def make_meshgrid(val=0.1, amount=30):\n",
" lsx = np.linspace(-val, val, amount)\n",
" lsy = np.linspace(-val, val, amount)\n",
" xx, yy = np.meshgrid(lsx,lsy)\n",
" return xx, yy\n",
"\n",
"def generate_trafo_matrices(coefficient, z_func):\n",
" normalize = lambda vec: vec/np.linalg.norm(vec)\n",
" uvec2 = normalize(np.array([1, 0, z_func(1, 0)]))\n",
" uvec3 = normalize(np.cross(coefficient, uvec2))\n",
" back_trafo_matrix = np.array([uvec2, uvec3, coefficient]).T \n",
" #in other order such that its on the xy-plane instead of the yz-plane\n",
" trafo_matrix = np.linalg.inv(back_trafo_matrix)\n",
" return trafo_matrix, back_trafo_matrix"
]
},
{
"cell_type": "markdown",
"id": "8956be33",
"metadata": {},
"source": [
"## Plotting decision hyperplane without any change of basis"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "5c58723d",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Coefficients: [[-0.13242445 0.33866526 -0.63863541]] Intercept: [0.42649519]\n"
]
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"source": [
"X, y = make_blobs(n_samples=80, centers=2, random_state=2, n_features=3)\n",
"model = svm.LinearSVC(C=1, max_iter=10000).fit(X, y)\n",
"z_func = lambda interc, coef, x, y: (interc-coef[0]*x -coef[1]*y) / coef[2]\n",
"\n",
"print(\"Coefficients:\", model.coef_, \"Intercept:\", model.intercept_)\n",
"\n",
"plane_func = lambda xx, yy: z_func(model.intercept_[0],model.coef_[0],xx,yy)\n",
"\n",
"fig = ThreeDFigure()\n",
"fig.add_markers(X, color=y) #samples\n",
"\n",
"xx, yy = make_meshgrid(X.min()) \n",
"fig.add_surface(xx, yy, plane_func) #decision hyperplane\n",
"\n",
"fig.add_line(X.mean(axis=0)-model.coef_[0], X.mean(axis=0)+model.coef_[0]) #orthogonal of decision hyperplane through mean of points\n",
"fig.add_markers([0,0,0], size=3) #coordinate center\n",
"\n",
"fig.show()"
]
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"id": "1ed87b0a",
"metadata": {},
"source": [
"## Trying to apply the transformation matrix"
]
},
{
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"execution_count": 12,
"id": "44d58c74",
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"source": [
"trafo_matrix, back_trafo_matrix = generate_trafo_matrices(model.coef_[0], plane_func)\n",
"\n",
"fig = ThreeDFigure(trafo_matrix)\n",
"fig.add_markers(X, color=y) #samples\n",
"\n",
"xx, yy = make_meshgrid(X.min()) \n",
"fig.add_surface(xx, yy, plane_func) #decision hyperplane\n",
"\n",
"fig.add_line(X.mean(axis=0)-model.coef_[0], X.mean(axis=0)+model.coef_[0]) #orthogonal of decision hyperplane through mean of points\n",
"fig.add_markers([0,0,0], size=3) #coordinate center\n",
"\n",
"fig.show()"
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},
{
"cell_type": "markdown",
"id": "b26e5f21",
"metadata": {},
"source": [
"## Demonstrating with arbitrary planes: showing that it works with intercept=0"
]
},
{
"cell_type": "code",
"execution_count": 35,
"id": "516d82e6",
"metadata": {},
"outputs": [],
"source": [
"vec = np.array([-0.14, -0.02, -0.05])\n",
"interc = 0\n",
"plane_func = lambda x,y: (-interc-vec[0]*x-vec[1]*y) / vec[2]\n",
"trafo_matrix, back_trafo_matrix = generate_trafo_matrices(vec, plane_func)\n",
"\n",
"xx, yy = make_meshgrid(0.05) "
]
},
{
"cell_type": "code",
"execution_count": 36,
"id": "228eb04f",
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" \n",
"var gd = document.getElementById('98cc9be7-9a50-4798-b50e-3c11d4c5e61f');\n",
"var x = new MutationObserver(function (mutations, observer) {{\n",
" var display = window.getComputedStyle(gd).display;\n",
" if (!display || display === 'none') {{\n",
" console.log([gd, 'removed!']);\n",
" Plotly.purge(gd);\n",
" observer.disconnect();\n",
" }}\n",
"}});\n",
"\n",
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" x.observe(notebookContainer, {childList: true});\n",
"}}\n",
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"var outputEl = gd.closest('.output');\n",
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]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = ThreeDFigure()\n",
"fig.add_surface(xx, yy, plane_func)\n",
"fig.add_line([0,0,0],vec)\n",
"\n",
"fig.add_line([0,0,0], fig._transform([0.1,0,0], inverse=True))\n",
"fig.add_line([0,0,0], fig._transform([0,0.1,0], inverse=True))\n",
"fig.add_line([0,0,0], fig._transform([0,0,0.1], inverse=True))\n",
"\n",
"#print(fig._transform([0,0,0], inverse=True))\n",
"\n",
"fig.show()"
]
},
{
"cell_type": "code",
"execution_count": 37,
"id": "eb90bff7",
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.plotly.v1+json": {
"config": {
"plotlyServerURL": "https://plot.ly"
},
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]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = ThreeDFigure(trafo_matrix)\n",
"fig.add_surface(xx, yy, plane_func)\n",
"fig.add_line([0,0,0],vec)\n",
"\n",
"fig.add_line([0,0,0], fig._transform([0.1,0,0], inverse=True))\n",
"fig.add_line([0,0,0], fig._transform([0,0.1,0], inverse=True))\n",
"fig.add_line([0,0,0], fig._transform([0,0,0.1], inverse=True))\n",
"\n",
"#print(fig._transform([0,0,0], inverse=True))\n",
"\n",
"fig.show()"
]
},
{
"cell_type": "markdown",
"id": "86fa0b95",
"metadata": {},
"source": [
"## Demonstrating with arbitrary planes: showing that it doesn't work with intercept != 0"
]
},
{
"cell_type": "code",
"execution_count": 38,
"id": "89454a63",
"metadata": {},
"outputs": [],
"source": [
"vec = np.array([-0.14, -0.02, -0.05])\n",
"interc = -0.005\n",
"plane_func = lambda x,y: (-interc-vec[0]*x-vec[1]*y) / vec[2]\n",
"trafo_matrix, back_trafo_matrix = generate_trafo_matrices(vec, plane_func)\n",
"\n",
"xx, yy = make_meshgrid(0.05) "
]
},
{
"cell_type": "code",
"execution_count": 39,
"id": "7cd31e89",
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.plotly.v1+json": {
"config": {
"plotlyServerURL": "https://plot.ly"
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\"ticks\": \"\", \"title\": {\"standoff\": 15}, \"zerolinecolor\": \"white\", \"zerolinewidth\": 2}}}}, {\"responsive\": true} ).then(function(){\n",
" \n",
"var gd = document.getElementById('96a149b0-82f7-4907-9e11-f976c34b7959');\n",
"var x = new MutationObserver(function (mutations, observer) {{\n",
" var display = window.getComputedStyle(gd).display;\n",
" if (!display || display === 'none') {{\n",
" console.log([gd, 'removed!']);\n",
" Plotly.purge(gd);\n",
" observer.disconnect();\n",
" }}\n",
"}});\n",
"\n",
"// Listen for the removal of the full notebook cells\n",
"var notebookContainer = gd.closest('#notebook-container');\n",
"if (notebookContainer) {{\n",
" x.observe(notebookContainer, {childList: true});\n",
"}}\n",
"\n",
"// Listen for the clearing of the current output cell\n",
"var outputEl = gd.closest('.output');\n",
"if (outputEl) {{\n",
" x.observe(outputEl, {childList: true});\n",
"}}\n",
"\n",
" }) }; }); </script> </div>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = ThreeDFigure()\n",
"fig.add_surface(xx, yy, plane_func)\n",
"fig.add_line([0,0,0],vec)\n",
"\n",
"fig.add_line([0,0,0], fig._transform([0.1,0,0], inverse=True))\n",
"fig.add_line([0,0,0], fig._transform([0,0.1,0], inverse=True))\n",
"fig.add_line([0,0,0], fig._transform([0,0,0.1], inverse=True))\n",
"\n",
"#print(fig._transform([0,0,0], inverse=True))\n",
"\n",
"fig.show()"
]
},
{
"cell_type": "code",
"execution_count": 40,
"id": "ef796fac",
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.plotly.v1+json": {
"config": {
"plotlyServerURL": "https://plot.ly"
},
"data": [
{
"type": "surface",
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environment.yml
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