Ejercicio1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SVD y PCA: Ejercicio 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### En esta página https://www.kite.com/python/examples/250/numpy-compute-the-singular-value-decomposition-of-a-matrix se encuentran las instrucciones para calcular la descomposición en valores singulares de una matriz en python."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1 En una libreta de Jupyter, calcula la SVD de la matriz A = [[1, 2], [2, 3]]. Grafica el efecto sobre el conjunto de los vectores en R2 de longitud 1 (el círculo de centro [0, 0] y radio 1) al aplicarle V T luego Σ y luego U. Compara el efecto de aplicarle al círculo directamente A."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matriz A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 2],\n",
       "        [2, 3]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.matrix([[1, 2], \n",
    "               [2, 3]])\n",
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Descomponemos la Matriz A en valores singulares"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "u, s, v = np.linalg.svd(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.52573111 -0.85065081]\n",
      " [-0.85065081  0.52573111]]\n",
      "[4.23606798 0.23606798]\n",
      "[[-0.52573111 -0.85065081]\n",
      " [ 0.85065081 -0.52573111]]\n"
     ]
    }
   ],
   "source": [
    "print (u)\n",
    "print(s)\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1., 2.],\n",
       "        [2., 3.]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(u, np.dot(np.diag(s), v))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conjunto de los vectores en R2 de longitud 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = np.array([[1,0],[-1,0],[0,-1], [0,1],[0.7,0.714],[0.7,-0.714],[-0.7,-0.714],[-0.7,0.714]])\n",
    "origen = np.array([[0,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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": [
    "ax = plt.axes()\n",
    "plt.quiver(*origen, R[:,0], R[:,1], color=['b'], angles='xy', scale_units='xy', scale=1)\n",
    "ax.set_xlim([-2, 2])\n",
    "ax.set_ylim([-2, 2])\n",
    "plt.draw()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculamos el efecto de u, s, v en R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.    -1.    -2.     2.     2.128 -0.728 -2.128  0.728]\n",
      " [ 2.    -2.    -3.     3.     3.542 -0.742 -3.542  0.742]]\n"
     ]
    }
   ],
   "source": [
    "Rsvd = np.transpose(v)@np.diag(s)@u@np.transpose(R)\n",
    "print(Rsvd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.     2.   ]\n",
      " [-1.    -2.   ]\n",
      " [-2.    -3.   ]\n",
      " [ 2.     3.   ]\n",
      " [ 2.128  3.542]\n",
      " [-0.728 -0.742]\n",
      " [-2.128 -3.542]\n",
      " [ 0.728  0.742]]\n"
     ]
    }
   ],
   "source": [
    "RsvdT= np.transpose(Rsvd)\n",
    "R2= np.array(RsvdT)\n",
    "print(R2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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": [
    "ax = plt.axes()\n",
    "plt.quiver(*origen, R[:,0], R[:,1], color=['b'], angles='xy', scale_units='xy', scale=1)\n",
    "plt.quiver(*origen, R2[:,0], R2[:,1], color=['r'], angles='xy', scale_units='xy', scale=1)\n",
    "ax.set_xlim([-5, 5])\n",
    "ax.set_ylim([-5, 5])\n",
    "plt.draw()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculamos el efecto de A en R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.     2.   ]\n",
      " [-1.    -2.   ]\n",
      " [-2.    -3.   ]\n",
      " [ 2.     3.   ]\n",
      " [ 2.128  3.542]\n",
      " [-0.728 -0.742]\n",
      " [-2.128 -3.542]\n",
      " [ 0.728  0.742]]\n"
     ]
    }
   ],
   "source": [
    "AR= A@np.transpose(R)\n",
    "ARt= np.transpose(AR)\n",
    "ARv= np.array(ARt)\n",
    "print(ARv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "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": [
    "ax = plt.axes()\n",
    "plt.quiver(*origen, R[:,0], R[:,1], color=['b'], angles='xy', scale_units='xy', scale=1)\n",
    "plt.quiver(*origen, ARv[:,0], ARv[:,1], color=['g'], angles='xy', scale_units='xy', scale=1)\n",
    "ax.set_xlim([-5, 5])\n",
    "ax.set_ylim([-5, 5])\n",
    "plt.draw()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Obtenemos el mismo efecto"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2 Calcula la matriz de rank 1 que mejor aproxime a A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1.17082039, 1.89442719],\n",
       "        [1.89442719, 3.06524758]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u1= u[:, :1]\n",
    "s1= np.diag(s)[:, :1]\n",
    "v1= v[:1, :]\n",
    "np.dot(u,(np.dot(s1,v1)))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}