SVD.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "484daff8",
   "metadata": {},
   "source": [
    "## Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "f952956b",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5da574a9",
   "metadata": {},
   "source": [
    "## Singular Value Decomposition SVD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3e0fd4db",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plotVectors(vecs, cols, alpha=1):\n",
    "    \"\"\"\n",
    "    Plot set of vectors.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    vecs : array-like\n",
    "        Coordinates of the vectors to plot. Each vectors is in an array. For\n",
    "        instance: [[1, 3], [2, 2]] can be used to plot 2 vectors.\n",
    "    cols : array-like\n",
    "        Colors of the vectors. For instance: ['red', 'blue'] will display the\n",
    "        first vector in red and the second in blue.\n",
    "    alpha : float\n",
    "        Opacity of vectors\n",
    "\n",
    "    Returns:\n",
    "\n",
    "    fig : instance of matplotlib.figure.Figure\n",
    "        The figure of the vectors\n",
    "    \"\"\"\n",
    "    plt.figure()\n",
    "    plt.axvline(x=0, color='#A9A9A9', zorder=0)\n",
    "    plt.axhline(y=0, color='#A9A9A9', zorder=0)\n",
    "\n",
    "    for i in range(len(vecs)):\n",
    "        x = np.concatenate([[0,0],vecs[i]])\n",
    "        plt.quiver([x[0]],\n",
    "                   [x[1]],\n",
    "                   [x[2]],\n",
    "                   [x[3]],\n",
    "                   angles='xy', scale_units='xy', scale=1, color=cols[i],\n",
    "                   alpha=alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "8a645c75",
   "metadata": {},
   "outputs": [],
   "source": [
    "def matrixToPlot(matrix, vectorsCol=['#FF9A13', '#1190FF']):\n",
    "    \"\"\"\n",
    "    Modify the unit circle and basis vector by applying a matrix.\n",
    "    Visualize the effect of the matrix in 2D.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    matrix : array-like\n",
    "        2D matrix to apply to the unit circle.\n",
    "    vectorsCol : HEX color code\n",
    "        Color of the basis vectors\n",
    "\n",
    "    Returns:\n",
    "\n",
    "    fig : instance of matplotlib.figure.Figure\n",
    "        The figure containing modified unit circle and basis vectors.\n",
    "    \"\"\"\n",
    "    # Unit circle\n",
    "    x = np.linspace(-1, 1, 100000)\n",
    "    y = np.sqrt(1-(x**2))\n",
    "\n",
    "    # Modified unit circle (separate negative and positive parts)\n",
    "    x1 = matrix[0,0]*x + matrix[0,1]*y\n",
    "    y1 = matrix[1,0]*x + matrix[1,1]*y\n",
    "    x1_neg = matrix[0,0]*x - matrix[0,1]*y\n",
    "    y1_neg = matrix[1,0]*x - matrix[1,1]*y\n",
    "\n",
    "    # Vectors\n",
    "    u1 = [matrix[0,0],matrix[1,0]]\n",
    "    v1 = [matrix[0,1],matrix[1,1]]\n",
    "\n",
    "    plotVectors([u1, v1], cols=[vectorsCol[0], vectorsCol[1]])\n",
    "\n",
    "    plt.plot(x1, y1, 'g', alpha=0.5)\n",
    "    plt.plot(x1_neg, y1_neg, 'g', alpha=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "9249ce9b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Unit circle:\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Unit circle transformed by A:\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.array([[3, 7], \n",
    "              [5, 2]])\n",
    "\n",
    "print ('Unit circle:')\n",
    "matrixToPlot(np.array([[1, 0], [0, 1]]))\n",
    "plt.xlim(-1.5, 1.5)\n",
    "plt.ylim(-1.5, 1.5)\n",
    "plt.show()\n",
    "\n",
    "print ('Unit circle transformed by A:')\n",
    "matrixToPlot(A)\n",
    "plt.xlim(-8, 8)\n",
    "plt.ylim(-8, 8)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "daf2535a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.85065081, -0.52573111],\n",
       "       [-0.52573111,  0.85065081]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, D, V = np.linalg.svd(A)\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "3a50452c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([8.71337969, 3.32821489])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ed23d36b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.59455781, -0.80405286],\n",
       "       [ 0.80405286, -0.59455781]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f45e4eec",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Unit circle:\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First rotation:\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scaling:\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Second rotation:\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Unit circle\n",
    "print ('Unit circle:')\n",
    "matrixToPlot(np.array([[1, 0], [0, 1]]))\n",
    "plt.xlim(-1.5, 1.5)\n",
    "plt.ylim(-1.5, 1.5)\n",
    "plt.show()\n",
    "\n",
    "print ('First rotation:')\n",
    "matrixToPlot(V)\n",
    "plt.xlim(-1.5, 1.5)\n",
    "plt.ylim(-1.5, 1.5)\n",
    "plt.show()\n",
    "\n",
    "print ('Scaling:')\n",
    "matrixToPlot(np.diag(D).dot(V))\n",
    "plt.xlim(-9, 9)\n",
    "plt.ylim(-9, 9)\n",
    "plt.show()\n",
    "\n",
    "print ('Second rotation:')\n",
    "matrixToPlot(U.dot(np.diag(D)).dot(V))\n",
    "plt.xlim(-8, 8)\n",
    "plt.ylim(-8, 8)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "44d08dac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "matrixToPlot(A)\n",
    "plt.xlim(-8, 8)\n",
    "plt.ylim(-8, 8)\n",
    "plt.show()"
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
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   "id": "dd3201c7",
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   "source": [
    "## Singular values interpretation\n",
    "- The singular values are ordered by descending order. They correspond to a new set of features (that are a linear combination of the original features) with the first feature explaining most of the variance. For instance from the last example we can visualize these new features. The major axis of the elipse will be the first left singular vector (u1) and its norm will be the first singular value (σ1).\n",
    "\n",
    "![image.png](attachment:image.png)"
   ]
  }
 ],
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