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@QuentinAndre
Created January 21, 2019 15:11
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{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import seaborn as sns\n",
"sns.set_style(\"whitegrid\")\n",
"import matplotlib.colors as colors\n",
"import matplotlib as mpl\n",
"from matplotlib import rc\n",
"rc('animation', html='html5')\n",
"from matplotlib import collections\n",
"from matplotlib.animation import FuncAnimation\n",
"from ipywidgets import interact, interactive, fixed, interact_manual, IntSlider"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"pal = sns.color_palette()\n",
"def tand(x):\n",
" return np.tan(x * 3.141592653589793 / 180)\n",
"\n",
"def sind(x):\n",
" return np.sin(x * 3.141592653589793 / 180)\n",
"\n",
"def cosd(x):\n",
" return np.cos(x * 3.141592653589793 / 180)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [],
"source": [
"np.random.seed(22)\n",
"mat_obs = np.random.multivariate_normal([0, 0], [[2, 0.7], [0.7, 1]], size=50)\n",
"mat_obs = mat_obs - mat_obs.mean(axis=0)"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [],
"source": [
"class InteractiveProjection(object):\n",
" def __init__(self, obs, model_type=\"PCA\"):\n",
" self.obs = obs - obs.mean(axis=0)\n",
" self.model_type = model_type\n",
" self.vcv_obs = self.obs.T.dot(self.obs) # Sum of Square Matrix\n",
" eigval, eigvec = np.linalg.eigh(self.vcv_obs) # Eigenvalues and Eigenvectors\n",
" idx = eigval.argsort()[::-1] \n",
" self.eigval = eigval[idx]\n",
" self.eigvec = eigvec[idx,:]\n",
" \n",
" self.angle = 0\n",
" self.minerror = 100000\n",
" self.vec_proj, self.obs_proj, self.error = self.update_projection(self.angle)\n",
" self.fig, self.ax, self.proj_dots, self.linecoll, self.proj_x, self.proj_y, self.annot1, self.annot2 = self.init_figure()\n",
" \n",
" def init_figure(self):\n",
" eigvec, eigval = self.eigvec, self.eigval\n",
" obs = self.obs\n",
" pal = sns.color_palette()\n",
" a1 = -5.3\n",
" a2 = -4.3\n",
" \n",
" fig, ax = plt.subplots(1, 1, figsize=(5, 5), sharey=True)\n",
" plt.autoscale(False)\n",
" sns.despine()\n",
" ax.set_xlim(-4, 4)\n",
" ax.set_ylim(-4, 4)\n",
" ax.scatter(0, 0, s=50, color=\"white\", zorder=100, edgecolors=\"black\", linewidths=2)\n",
" ax.scatter(*obs.T, color=pal[1], s=20)\n",
" e1, = ax.plot(eigvec[0, 0]*np.array([-a2, -a1]), eigvec[0, 1]*np.array([-a2, -a1]), color=pal[4], lw=2)\n",
" e2, = ax.plot(eigvec[1, 0]*np.array([-a2, -a1]), eigvec[1, 1]*np.array([-a2, -a1]), color=pal[3], lw=2)\n",
" ax.plot(eigvec[0, 0]*np.array([a2, a1]), eigvec[0, 1]*np.array([a2, a1]), color=pal[4], lw=2)\n",
" ax.plot(eigvec[1, 0]*np.array([a2, a1]), eigvec[1, 1]*np.array([a2, a1]), color=pal[3], lw=2)\n",
" \n",
"\n",
" linecoll = collections.LineCollection([], color=pal[2], lw=1, linestyle=\":\", alpha=0.7)\n",
" ax.add_collection(linecoll)\n",
"\n",
" annot1 = ax.annotate(\"\\n\", (1, -2.5), va=\"center\")\n",
" annot2 = ax.annotate(\"\\n\", (-3.5, 2.5), va=\"center\")\n",
" proj_dots = ax.scatter([], [], color=pal[2], s=10, animated=True)\n",
"\n",
" proj_x, = ax.plot([], [], 'k', lw=1, alpha=0.7)\n",
" proj_y, = ax.plot([], [], 'k', lw=1, alpha=0.7)\n",
" \n",
" plt.figlegend((e1, e2), (\"First Eigenvector\", \"Second Eigenvector\"), loc=\"lower center\", ncol=2, \n",
" bbox_to_anchor=(0.5, 0), title=\"\")\n",
" \n",
" plt.tight_layout(rect=(0, 0.1, 1, 1))\n",
" plt.close()\n",
" return fig, ax, proj_dots, linecoll, proj_x, proj_y, annot1, annot2\n",
"\n",
" def update_projection(self, angle):\n",
" obs = self.obs\n",
" # Angle of rotation\n",
" self.angle = angle\n",
" \n",
" # Projections corresponding to angles\n",
" vec_proj = np.array([cosd(angle), sind(angle)]).reshape(2, -1)\n",
" \n",
" \n",
" # Projection matrices\n",
" if self.model_type == \"PCA\":\n",
" mat_proj = vec_proj.dot(vec_proj.T)\n",
" elif self.model_type == \"OLSX\":\n",
" mat_proj = np.array([[0, 0], [cosd(angle)/sind(angle), 1]])\n",
" else:\n",
" mat_proj = np.array([[1, sind(angle)/cosd(angle)], [0, 0]])\n",
"\n",
" # Projected observations\n",
" obs_proj = obs.dot(mat_proj)\n",
" error = obs_proj.dot([[0, 1], [-1, 0]]) * (obs-obs_proj)\n",
" \n",
" return vec_proj, obs_proj, error\n",
" \n",
" def plot_projection(self, angle):\n",
" self.vec_proj, self.obs_proj, self.error = self.update_projection(angle)\n",
" obs = self.obs\n",
" w = self.vec_proj\n",
" obs_proj = self.obs_proj\n",
"\n",
" self.proj_dots.set_offsets(obs_proj)\n",
" segs = np.rollaxis(np.dstack([obs, obs_proj]), 2, 1)\n",
" self.linecoll.set_segments(segs)\n",
" self.proj_x.set_data(w[0]*4.5*[-1, 1], w[1]*4.5*[-1, 1])\n",
" self.proj_y.set_data(-w[1]*1.75*[-1, 1], w[0]*1.75*[-1, 1])\n",
" error = np.square(self.error).sum()\n",
" self.annot1.set_text(f\"Current Error:\\n{error:.1f}\")\n",
" if error < self.minerror:\n",
" self.minerror = error\n",
" self.bestangle = self.angle\n",
" self.annot2.set_text(f\"Min Error So Far:\\n{self.minerror:.1f} at {self.bestangle:.0f}°\")\n",
" return self.fig"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "78861b5bf1a04853b1fe9d0539507840",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"interactive(children=(IntSlider(value=0, description='angle', max=75, min=-30), Output()), _dom_classes=('widg…"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"spring = InteractiveProjection(mat_obs, model_type=\"OLSX\");\n",
"interact(spring.plot_projection, angle=IntSlider(min=-30,max=75,step=1,value=0));"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {},
"outputs": [],
"source": [
"class AnimationProjection(object):\n",
" def __init__(self, obs):\n",
" self.obs = obs - obs.mean(axis=0) # Mean-center, since rod is fixed at 0.\n",
" self.vcv_obs = self.obs.T.dot(self.obs) # Sum of Square Matrix\n",
" eigval, eigvec = np.linalg.eigh(self.vcv_obs) # Eigenvalues and Eigenvectors\n",
" idx = eigval.argsort()[::-1] \n",
" self.eigval = eigval[idx]\n",
" self.eigvec = eigvec[idx,:]\n",
" \n",
" self.angles = {\"PCA\": -30,\n",
" \"OLS1\": -30,\n",
" \"OLS2\": -30}\n",
" \n",
" self.vec_proj, self.obs_proj, self.error, self.energy = self.init_system()\n",
" self.fig, self.axes, self.proj_dots, self.linecoll, self.proj_x, self.proj_y, self.annot = self.init_figure()\n",
" return None\n",
" \n",
" def init_figure(self):\n",
" eigvec, eigval = self.eigvec, self.eigval\n",
" obs = self.obs\n",
" pal = sns.color_palette()\n",
" a1 = -5.3\n",
" a2 = -4.3\n",
" titles = [\"PCA\\n(Min. Euclidian Distance)\", \n",
" \"X regressed on Y\\n(Min. Horizontal Distance)\", \n",
" \"Y regressed on X\\n(Min. Vertical Distance)\"]\n",
" \n",
" fig, axes = plt.subplots(1, 3, figsize=(12, 4), sharey=True)\n",
" plt.autoscale(False)\n",
" for ax, title in zip(axes, titles):\n",
" ax.set_title(title)\n",
" sns.despine()\n",
" ax.set_aspect(\"equal\")\n",
" ax.set_xlim(-4, 4)\n",
" ax.set_ylim(-4, 4)\n",
" ax.scatter(0, 0, s=50, color=\"white\", zorder=100, edgecolors=\"black\", linewidths=2)\n",
" ax.scatter(*obs.T, color=pal[1], s=20)\n",
" e1, = ax.plot(eigvec[0, 0]*np.array([-a2, -a1]), eigvec[0, 1]*np.array([-a2, -a1]), color=pal[4], lw=2)\n",
" e2, = ax.plot(eigvec[1, 0]*np.array([-a2, -a1]), eigvec[1, 1]*np.array([-a2, -a1]), color=pal[3], lw=2)\n",
" ax.plot(eigvec[0, 0]*np.array([a2, a1]), eigvec[0, 1]*np.array([a2, a1]), color=pal[4], lw=2)\n",
" ax.plot(eigvec[1, 0]*np.array([a2, a1]), eigvec[1, 1]*np.array([a2, a1]), color=pal[3], lw=2)\n",
" \n",
" linecoll = {}\n",
" proj_dots = {}\n",
" proj_x = {}\n",
" proj_y = {}\n",
" annot = {}\n",
" for ax, label in zip(axes, [\"PCA\", \"OLS1\", \"OLS2\"]):\n",
" lines = collections.LineCollection([], color=pal[2], lw=1, linestyle=\":\", alpha=0.7)\n",
" linecoll[label] = lines\n",
" ax.add_collection(lines)\n",
" \n",
" annot[label] = ax.annotate(\"\", (0, -3))\n",
" proj_dots[label] = ax.scatter([], [], color=pal[2], s=10, animated=True)\n",
" \n",
" proj_x[label], = ax.plot([], [], 'k', lw=1, alpha=0.7)\n",
" proj_y[label], = ax.plot([], [], 'k', lw=1, alpha=0.7)\n",
" \n",
" plt.figlegend((e1, e2), (\"First Eigenvector\", \"Second Eigenvector\"), loc=\"lower center\", ncol=2, \n",
" bbox_to_anchor=(0.5, 0), title=\"\")\n",
" plt.tight_layout(rect=(0, 0.1, 1, 1))\n",
" return fig, axes, proj_dots, linecoll, proj_x, proj_y, annot\n",
"\n",
" def init_system(self):\n",
" obs = self.obs\n",
" # Angle of rotation\n",
" angles = self.angles\n",
" \n",
" # Projections corresponding to angles\n",
" vec_proj = {k:np.array([cosd(v), sind(v)]).reshape(2, -1) for k, v in angles.items()}\n",
" \n",
" # Projection matrices\n",
" mat_proj = {\n",
" \"PCA\": vec_proj[\"PCA\"].dot(vec_proj[\"PCA\"].T),\n",
" \"OLS1\": np.array([[0, 0], [cosd(angles[\"OLS1\"])/sind(angles[\"OLS1\"]), 1]]),\n",
" \"OLS2\": np.array([[1, sind(angles[\"OLS2\"])/cosd(angles[\"OLS2\"])], [0, 0]])\n",
" }\n",
" \n",
" # Projected observations\n",
" obs_proj = {k:obs.dot(v) for k, v in mat_proj.items()}\n",
" error = {k:v.dot([[0, 1], [-1, 0]]) * (obs-v) for k, v in obs_proj.items()}\n",
" energy = {k: v.sum() for k, v in error.items()}\n",
" \n",
" return vec_proj, obs_proj, error, energy\n",
" \n",
" def update_system(self):\n",
" energy = {k: (v + self.error[k].sum())*0.93 for k, v in self.energy.items()}\n",
" angles = {k: (v + self.energy[k]/40) for k, v in self.angles.items()}\n",
" obs = self.obs\n",
" \n",
" # Projections corresponding to angles\n",
" vec_proj = {k:np.array([cosd(v), sind(v)]).reshape(2, -1) for k, v in angles.items()}\n",
" \n",
" # Projection matrices\n",
" mat_proj = {\n",
" \"PCA\": vec_proj[\"PCA\"].dot(vec_proj[\"PCA\"].T),\n",
" \"OLS1\": np.array([[0, 0], [cosd(angles[\"OLS1\"])/sind(angles[\"OLS1\"]), 1]]),\n",
" \"OLS2\": np.array([[1, sind(angles[\"OLS2\"])/cosd(angles[\"OLS2\"])], [0, 0]])\n",
" }\n",
"\n",
" # Projected observations\n",
" obs_proj = {k:obs.dot(v) for k, v in mat_proj.items()}\n",
" error = {k:v.dot([[0, 1], [-1, 0]]) * (obs-v) for k, v in obs_proj.items()}\n",
"\n",
" self.angles, self.vec_proj, self.obs_proj, self.error, self.energy = angles, vec_proj, obs_proj, error, energy\n",
" return None\n",
" \n",
" def init_projection(self):\n",
" for ax, label in zip(self.axes, [\"PCA\", \"OLS1\", \"OLS2\"]):\n",
" obs = self.obs\n",
" w = self.vec_proj[label]\n",
" obs_proj = self.obs_proj[label]\n",
" \n",
" self.proj_dots[label].set_offsets(obs_proj)\n",
" segs = np.rollaxis(np.dstack([obs, obs_proj]), 2, 1)\n",
" self.linecoll[label].set_segments(segs)\n",
" self.proj_x[label].set_data(w[0]*4.5*[-1, 1], w[1]*4.5*[-1, 1])\n",
" if label == \"PCA\":\n",
" self.proj_y[label].set_data(-w[1]*1.75*[-1, 1], w[0]*1.75*[-1, 1])\n",
" self.annot[label].set_text(\"Current Error:\\n{:.1f}\".format(np.square(self.error[label]).sum()))\n",
" return self.linecoll[label],\n",
" \n",
" def plot_projection(self, i):\n",
" self.update_system()\n",
" for ax, label in zip(self.axes, [\"PCA\", \"OLS1\", \"OLS2\"]):\n",
" obs = self.obs\n",
" w = self.vec_proj[label]\n",
" obs_proj = self.obs_proj[label]\n",
" \n",
" self.proj_dots[label].set_offsets(obs_proj)\n",
" segs = np.rollaxis(np.dstack([obs, obs_proj]), 2, 1)\n",
" self.linecoll[label].set_segments(segs)\n",
" self.proj_x[label].set_data(w[0]*4.5*[-1, 1], w[1]*4.5*[-1, 1])\n",
" if label == \"PCA\":\n",
" self.proj_y[label].set_data(-w[1]*1.75*[-1, 1], w[0]*1.75*[-1, 1])\n",
" self.annot[label].set_text(\"Current Error:\\n{:.1f}\".format(np.square(self.error[label]).sum()))\n",
" return self.linecoll[label],\n",
" \n",
" def animate(self, frames, interval):\n",
" anim = FuncAnimation(self.fig, init_func=self.init_projection, \n",
" func=self.plot_projection, frames=frames, blit=True, interval=interval)\n",
" return anim"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
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