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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Copyright 2020 Erik Härkönen. All rights reserved.\n",
    "# This file is licensed to you under the Apache License, Version 2.0 (the \"License\");\n",
    "# you may not use this file except in compliance with the License. You may obtain a copy\n",
    "# of the License at http://www.apache.org/licenses/LICENSE-2.0\n",
    "\n",
    "# Unless required by applicable law or agreed to in writing, software distributed under\n",
    "# the License is distributed on an \"AS IS\" BASIS, WITHOUT WARRANTIES OR REPRESENTATIONS\n",
    "# OF ANY KIND, either express or implied. See the License for the specific language\n",
    "# governing permissions and limitations under the License.\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.decomposition import PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rotMax(degrees):\n",
    "    theta = np.radians(degrees)\n",
    "    c, s = np.cos(theta), np.sin(theta)\n",
    "    return np.array(((c, -s), (s, c)))\n",
    "\n",
    "pointSize = 6\n",
    "colormap = 'spring'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "z = np.random.normal(size=(2,1000))\n",
    "\n",
    "plt.scatter(z[0,:],z[1,:],c='black', s=pointSize)\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "plt.axis('off')\n",
    "\n",
    "plt.savefig('zplot.pdf')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = rotMax(60)\n",
    "lam = np.diag([4,1])\n",
    "A =  R.dot(lam)\n",
    "w = A[:,0]\n",
    "y = A.dot(z)\n",
    "\n",
    "plt.scatter(y[0,:],y[1,:],c=z[0,:],s=pointSize)\n",
    "plt.gca().set_aspect('equal', adjustable='datalim')\n",
    "plt.arrow(0,0,2*w[0],2*w[1], width = 0.1, head_width = 1)\n",
    "plt.axis('off')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# perturb z coordinates\n",
    "zp = rotMax(-15).dot(z)\n",
    "zp = zp + np.random.normal(size=z.shape)\n",
    "\n",
    "# sigmoid offset\n",
    "lam = np.diag([2,1])\n",
    "z2 = np.array(lam.dot(zp))\n",
    "z2[1,:] = z2[1,:]+6*np.tanh(z2[0,:]*0.5)\n",
    "\n",
    "# rotate data\n",
    "A =  rotMax(15).dot(lam)\n",
    "y = A.dot(z2)\n",
    "\n",
    "# PCA \n",
    "yt = y.transpose()\n",
    "pca = PCA(n_components = 1)\n",
    "x = pca.fit_transform(yt)\n",
    "w = pca.components_[0]\n",
    "if w[0] < 0:\n",
    "    w = -w\n",
    "\n",
    "arrow_scale = 10\n",
    "\n",
    "plt.scatter(y[0,:],y[1,:],s=pointSize,c=x[:,0],cmap=colormap) \n",
    "plt.gca().set_aspect('equal', adjustable='datalim')\n",
    "plt.arrow(0,0,arrow_scale*w[0],arrow_scale*w[1], width = 0.1, head_width = 1)\n",
    "plt.axis('off')\n",
    "\n",
    "plt.savefig('yplot.pdf')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.linalg.pinv(x).dot(z.transpose())[0]\n",
    "if u[0] < 0:\n",
    "    u = -u\n",
    "\n",
    "arrow_scale = 20\n",
    "\n",
    "plt.scatter(z[0,:],z[1,:],c=x[:,0]/np.max(np.abs(x)*0.91) , s=pointSize, cmap=colormap)\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "plt.arrow(0,0,arrow_scale*u[0],arrow_scale*u[1], width = 0.1, head_width = 0.3)\n",
    "plt.axis('off')\n",
    "plt.colorbar()\n",
    "\n",
    "plt.savefig('uplot.pdf')\n",
    "plt.show()"
   ]
  }
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