19_LSI_Ejemplo_Basico_Topic_Modeling.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 19 - Latent Semantic Index/Analysis (LSI/A)\n",
    "\n",
    "\n",
    "* El LSI es una técnica de factorización matricial aplicable al campo del NLP, que tiene como objetivo analizar las relaciones entre los documentos de un corpus y las palabras. \n",
    "\n",
    "\n",
    "* El LSI asume que las palabras que tienen un significado similar, aparecerán en partes de documentos similares.\n",
    "\n",
    "\n",
    "* El LSI tiene como finalidad extraer una serie de factores latentes que caractericen a los documentos y las palabras del corpus.\n",
    "\n",
    "\n",
    "* Para ello necesita una matriz de frecuencias que represente el número de veces que aparecen las palabras en los documentos:\n",
    "\n",
    "    - ***Filas***: Cada fila representa a una palabra.\n",
    "    - ***Columnas***: Cada columna representa a un documento..\n",
    "    - ***Celda***: Cada celda representa el número de veces que aparece la palabra en el documento.\n",
    "    \n",
    "    \n",
    "* El LSI descompone una matriz “A” (de frecuencias); que esta formada por el numero de apariciones de cada palabra en cada documento, en tres matrices “U”, “S” y “V” cuyo producto matricial es igual a la matriz original “A”: \n",
    "\n",
    "\n",
    "$$ SVD(A) = U \\cdot S \\cdot V^{t}$$\n",
    "\n",
    "\n",
    "* Donde cada una de estas matrices contiene la siguiente información:\n",
    "\n",
    "    - ***Matriz “U”*** contiene los valores de los ***factores latentes de las palabras***. \n",
    "    - ***Matriz “V”*** contiene los valores de los ***factores latentes de los documentos***. \n",
    "    - ***Matriz “S”*** es una matriz en cuya diagonal están los llamados valores singulares que son decrecientes y no negativos. Esta matriz representa la ***importancia que tiene cada uno de los factores latentes de las palabras y de los documentos***. \n",
    "    \n",
    "    \n",
    "<img src=\"./imgs/015_LSI_Matrix.png\" style=\"width: 800px;\"/>\n",
    "\n",
    "\n",
    "* Como la matriz “S” nos informa de la importancia que tiene cada uno de los factores latentes, podemos coger solo los ***K-factores más importantes*** para caracterizar cada una de las palabras y de los items. \n",
    "\n",
    "\n",
    "* De esta forma se trabaja con matrices más reducidas lo que nos permite comprimir la información de la matriz de apariciones.\n",
    "\n",
    "\n",
    "* El SVD tiene una propiedad muy importante que viene dada por el ***teorema de Eckart-Young***, que afirma que la mejor aproximación a la matriz “A” la obtenemos poniendo a ceros los “K” valores singulares de menor a mayor valor; es decir, reduciendo la matriz “S”. Por tanto si multiplicásemos las submatrices “U<sub>k</sub>”, “S<sub>k</sub>” y “V<sub>k</sub>” obtendríamos una matriz A’ de rango K que mejor aproxima (de acuerdo con la norma de Frobenius) a la matriz “A”.\n",
    "\n",
    "\n",
    "<img src=\"./imgs/016_LSI_Matrix_reduc.png\" style=\"width: 800px;\"/>\n",
    "\n",
    "\n",
    "\n",
    "* El valor de K, va a representar el número de temas en que vamos a dividir (o Clusterizar) nuestro corpus, por lo tanto la selección del número de temas es un valor que tenemos que definir a priori. \n",
    "\n",
    "\n",
    "* Para poder trabajar en la extracción de tópicos y ver las relaciones entre los documentos y las palabras, vamos trabajar con las matrices:\n",
    "\n",
    "    - ***U<sub>k</sub>***: Estudiar las relaciones entre palabras\n",
    "    - ***V<sub>k</sub>***: Estudiar las relaciones entre documentos\n",
    "    - ***K***: Número de temas que tendrá a priori el Corpus\n",
    "    \n",
    "\n",
    "<hr>\n",
    "\n",
    "\n",
    "# Ejemplo de LSI con Gensim\n",
    "\n",
    "\n",
    "* Veamos a continuación un ejemplo sencillo sobre el siguiente Corpus del cual podemos ver que habla de 3 temas:\n",
    "    - Fútbol\n",
    "    - Política\n",
    "    - Economía\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import warnings\n",
    "warnings.filterwarnings('ignore')\n",
    "\n",
    "corpus = [\"balon balon balon futbol futbol liga liga liga ronaldo ronaldo ronaldo ronaldo ronaldo messi\",\n",
    "          \"futbol futbol futbol futbol futbol ronaldo ronaldo ronaldo ronaldo messi messi\",\n",
    "          \"balon balon futbol futbol futbol futbol futbol futbol futbol messi messi messi messi messi\",\n",
    "          \"politica politica politica politica pp pp pp pp pp pp rajoy rajoy rajoy rajoy rajoy\",\n",
    "          \"politica politica politica politica pp pp pp psoe psoe psoe psoe zapatero zapatero zapatero rajoy\",\n",
    "          \"politica politica politica politica psoe psoe psoe psoe psoe psoe zapatero zapatero zapatero zapatero zapatero \",\n",
    "          \"dinero fmi fmi fmi fmi fmi ue ue ue ue pib pib pib ibex ibex\",\n",
    "          \"zapatero rajoy dinero dinero dinero dinero fmi fmi fmi fmi ue ue ue ue pib\",\n",
    "          \"pp psoe zapatero rajoy dinero dinero dinero dinero fmi fmi fmi fmi ue ue ue \",\n",
    "          \"futbol politica pib\",\n",
    "          \"futbol zapatero liga rajoy\"]\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Creamos el Diccionario y la Matriz (Bolsa de Palabras)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Diccionario:\n",
      "{'balon': 0,\n",
      " 'dinero': 10,\n",
      " 'fmi': 11,\n",
      " 'futbol': 1,\n",
      " 'ibex': 12,\n",
      " 'liga': 2,\n",
      " 'messi': 3,\n",
      " 'pib': 13,\n",
      " 'politica': 5,\n",
      " 'pp': 6,\n",
      " 'psoe': 8,\n",
      " 'rajoy': 7,\n",
      " 'ronaldo': 4,\n",
      " 'ue': 14,\n",
      " 'zapatero': 9}\n",
      "\n",
      "Bolsa de Palabras:\n",
      "[[(0, 3), (1, 2), (2, 3), (3, 1), (4, 5)],\n",
      " [(1, 5), (3, 2), (4, 4)],\n",
      " [(0, 2), (1, 7), (3, 5)],\n",
      " [(5, 4), (6, 6), (7, 5)],\n",
      " [(5, 4), (6, 3), (7, 1), (8, 4), (9, 3)],\n",
      " [(5, 4), (8, 6), (9, 5)],\n",
      " [(10, 1), (11, 5), (12, 2), (13, 3), (14, 4)],\n",
      " [(7, 1), (9, 1), (10, 4), (11, 4), (13, 1), (14, 4)],\n",
      " [(6, 1), (7, 1), (8, 1), (9, 1), (10, 4), (11, 4), (14, 3)],\n",
      " [(1, 1), (5, 1), (13, 1)],\n",
      " [(1, 1), (2, 1), (7, 1), (9, 1)]]\n"
     ]
    }
   ],
   "source": [
    "from pprint import pprint\n",
    "from gensim import corpora\n",
    "from collections import defaultdict\n",
    "\n",
    "# Tokenizamos\n",
    "documents = [word.split() for word in corpus]\n",
    "\n",
    "# Creamos el diccionario (vocabulario)\n",
    "frequency = defaultdict(int)\n",
    "for doc in documents:\n",
    "    for token in doc:\n",
    "        frequency[token] += 1\n",
    "        \n",
    "documents = [[token for token in doc] for doc in documents]\n",
    "dictionary = corpora.Dictionary(documents)\n",
    "print('Diccionario:')\n",
    "pprint(dictionary.token2id)\n",
    "\n",
    "\n",
    "# Creamos la Bolsa de Palabras\n",
    "corpus = [dictionary.doc2bow(doc) for doc in documents]\n",
    "print('\\nBolsa de Palabras:')\n",
    "pprint(corpus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Creamos el Modelo:\n",
    "\n",
    "* Gensim tiene implementado el LSI en la clase ***LsiModel***: https://radimrehurek.com/gensim/models/lsimodel.html\n",
    "\n",
    "\n",
    "* Como parámetros relevantes necesita:\n",
    "    1. Corpus\n",
    "    2. Número de Topics\n",
    "    3. Diccionario o Vocabulario del Corpus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from gensim.models import LsiModel\n",
    "\n",
    "lsi_model = LsiModel(corpus=corpus, num_topics=3, id2word=dictionary)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matriz U\n",
    "\n",
    "\n",
    "* En esta matriz obtenemos los ***factores latentes de cada una de las palabras del corpus***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.00764122, -0.23838329, -0.01351154],\n",
       "       [ 0.03238081, -0.76194106, -0.03952318],\n",
       "       [ 0.0080157 , -0.12147793, -0.00477548],\n",
       "       [ 0.01436787, -0.44343298, -0.02506618],\n",
       "       [ 0.01216459, -0.38311209, -0.02178312],\n",
       "       [ 0.45080378, -0.00342558,  0.35923484],\n",
       "       [ 0.34610872,  0.00491806,  0.20925565],\n",
       "       [ 0.26841352,  0.00105663,  0.07889487],\n",
       "       [ 0.42300698,  0.00545779,  0.28893431],\n",
       "       [ 0.37700371,  0.00176928,  0.1817653 ],\n",
       "       [ 0.25466521,  0.03058342, -0.36521087],\n",
       "       [ 0.34588865,  0.04406739, -0.55200635],\n",
       "       [ 0.04561172,  0.00674199, -0.09339774],\n",
       "       [ 0.09993979,  0.00781257, -0.18348589],\n",
       "       [ 0.29289059,  0.03742915, -0.47017643]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_matrix = lsi_model.projection.u\n",
    "u_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Para ver los factores de cada una de las palabras lo vamos a mostrar de la siguiente manera:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Topic 1</th>\n",
       "      <th>Topic 2</th>\n",
       "      <th>Topic 3</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>balon</th>\n",
       "      <td>0.007641</td>\n",
       "      <td>-0.238383</td>\n",
       "      <td>-0.013512</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>futbol</th>\n",
       "      <td>0.032381</td>\n",
       "      <td>-0.761941</td>\n",
       "      <td>-0.039523</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>liga</th>\n",
       "      <td>0.008016</td>\n",
       "      <td>-0.121478</td>\n",
       "      <td>-0.004775</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>messi</th>\n",
       "      <td>0.014368</td>\n",
       "      <td>-0.443433</td>\n",
       "      <td>-0.025066</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ronaldo</th>\n",
       "      <td>0.012165</td>\n",
       "      <td>-0.383112</td>\n",
       "      <td>-0.021783</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>politica</th>\n",
       "      <td>0.450804</td>\n",
       "      <td>-0.003426</td>\n",
       "      <td>0.359235</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>pp</th>\n",
       "      <td>0.346109</td>\n",
       "      <td>0.004918</td>\n",
       "      <td>0.209256</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rajoy</th>\n",
       "      <td>0.268414</td>\n",
       "      <td>0.001057</td>\n",
       "      <td>0.078895</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>psoe</th>\n",
       "      <td>0.423007</td>\n",
       "      <td>0.005458</td>\n",
       "      <td>0.288934</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>zapatero</th>\n",
       "      <td>0.377004</td>\n",
       "      <td>0.001769</td>\n",
       "      <td>0.181765</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>dinero</th>\n",
       "      <td>0.254665</td>\n",
       "      <td>0.030583</td>\n",
       "      <td>-0.365211</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fmi</th>\n",
       "      <td>0.345889</td>\n",
       "      <td>0.044067</td>\n",
       "      <td>-0.552006</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ibex</th>\n",
       "      <td>0.045612</td>\n",
       "      <td>0.006742</td>\n",
       "      <td>-0.093398</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>pib</th>\n",
       "      <td>0.099940</td>\n",
       "      <td>0.007813</td>\n",
       "      <td>-0.183486</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ue</th>\n",
       "      <td>0.292891</td>\n",
       "      <td>0.037429</td>\n",
       "      <td>-0.470176</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           Topic 1   Topic 2   Topic 3\n",
       "balon     0.007641 -0.238383 -0.013512\n",
       "futbol    0.032381 -0.761941 -0.039523\n",
       "liga      0.008016 -0.121478 -0.004775\n",
       "messi     0.014368 -0.443433 -0.025066\n",
       "ronaldo   0.012165 -0.383112 -0.021783\n",
       "politica  0.450804 -0.003426  0.359235\n",
       "pp        0.346109  0.004918  0.209256\n",
       "rajoy     0.268414  0.001057  0.078895\n",
       "psoe      0.423007  0.005458  0.288934\n",
       "zapatero  0.377004  0.001769  0.181765\n",
       "dinero    0.254665  0.030583 -0.365211\n",
       "fmi       0.345889  0.044067 -0.552006\n",
       "ibex      0.045612  0.006742 -0.093398\n",
       "pib       0.099940  0.007813 -0.183486\n",
       "ue        0.292891  0.037429 -0.470176"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "pd.DataFrame(u_matrix, columns=['Topic 1', 'Topic 2', 'Topic 3'], index=dictionary.token2id.keys()).head(15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matriz S\n",
    "\n",
    "\n",
    "* En esta matriz obtenemos la importancia de los ***factores latentes***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([12.47067133, 11.39302903, 11.08980301])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s_matrix = lsi_model.projection.s\n",
    "s_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matriz V\n",
    "\n",
    "\n",
    "* En esta matriz obtenemos los ***factores latentes de cada uno de los documentos del corpus***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.01498903, -0.43556977, -0.02415635],\n",
       "       [ 0.01918888, -0.54673956, -0.03019718],\n",
       "       [ 0.02516204, -0.70459921, -0.03868565],\n",
       "       [ 0.41873726,  0.00185106,  0.27835911],\n",
       "       [ 0.47575577,  0.00256714,  0.34668193],\n",
       "       [ 0.49927348,  0.00244806,  0.36784891],\n",
       "       [ 0.28440438,  0.03840583, -0.51788127],\n",
       "       [ 0.3463434 ,  0.04028414, -0.49346234],\n",
       "       [ 0.37651704,  0.03722386, -0.38959647],\n",
       "       [ 0.04675966, -0.06649277,  0.01228388],\n",
       "       [ 0.05499413, -0.07729227,  0.01950995]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from gensim.matutils import corpus2dense\n",
    "v_matrix = corpus2dense(lsi_model[corpus], len(lsi_model.projection.s)).T / lsi_model.projection.s\n",
    "v_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Para ver los factores de cada uno de los documentos lo vamos a mostrar de la siguiente manera:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    .dataframe tbody tr th {\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Topic 1</th>\n",
       "      <th>Topic 2</th>\n",
       "      <th>Topic 3</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Doc 1</th>\n",
       "      <td>0.014989</td>\n",
       "      <td>-0.435570</td>\n",
       "      <td>-0.024156</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 2</th>\n",
       "      <td>0.019189</td>\n",
       "      <td>-0.546740</td>\n",
       "      <td>-0.030197</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 3</th>\n",
       "      <td>0.025162</td>\n",
       "      <td>-0.704599</td>\n",
       "      <td>-0.038686</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 4</th>\n",
       "      <td>0.418737</td>\n",
       "      <td>0.001851</td>\n",
       "      <td>0.278359</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 5</th>\n",
       "      <td>0.475756</td>\n",
       "      <td>0.002567</td>\n",
       "      <td>0.346682</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 6</th>\n",
       "      <td>0.499273</td>\n",
       "      <td>0.002448</td>\n",
       "      <td>0.367849</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 7</th>\n",
       "      <td>0.284404</td>\n",
       "      <td>0.038406</td>\n",
       "      <td>-0.517881</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 8</th>\n",
       "      <td>0.346343</td>\n",
       "      <td>0.040284</td>\n",
       "      <td>-0.493462</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 9</th>\n",
       "      <td>0.376517</td>\n",
       "      <td>0.037224</td>\n",
       "      <td>-0.389596</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 10</th>\n",
       "      <td>0.046760</td>\n",
       "      <td>-0.066493</td>\n",
       "      <td>0.012284</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Doc 11</th>\n",
       "      <td>0.054994</td>\n",
       "      <td>-0.077292</td>\n",
       "      <td>0.019510</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         Topic 1   Topic 2   Topic 3\n",
       "Doc 1   0.014989 -0.435570 -0.024156\n",
       "Doc 2   0.019189 -0.546740 -0.030197\n",
       "Doc 3   0.025162 -0.704599 -0.038686\n",
       "Doc 4   0.418737  0.001851  0.278359\n",
       "Doc 5   0.475756  0.002567  0.346682\n",
       "Doc 6   0.499273  0.002448  0.367849\n",
       "Doc 7   0.284404  0.038406 -0.517881\n",
       "Doc 8   0.346343  0.040284 -0.493462\n",
       "Doc 9   0.376517  0.037224 -0.389596\n",
       "Doc 10  0.046760 -0.066493  0.012284\n",
       "Doc 11  0.054994 -0.077292  0.019510"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "index = ['Doc {}'.format(i+1) for i,doc in enumerate(documents)]\n",
    "pd.DataFrame(v_matrix, index=index, columns=['Topic 1', 'Topic 2', 'Topic 3']).head(11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusiones:\n",
    "\n",
    "* Tras mostrar los valores de los factores latentes de las palabras y de los documentos, podemos ver como el valor de los factores guarda cierta relación con el tema del que trata el documento.\n",
    "\n",
    "\n",
    "* Se puede observar como los grupos de documentos (1,2,3), (4,5,6) y (7,8,9) que tratan de temas distintos tienen unos valores de factores latentes diferenciables entre sí (sobre todo los factores 2 y 3).\n",
    "\n",
    "\n",
    "* De la misma manera las palabras que son carácterísticas de cada tema tienen también unos valores en los factores latentes similares entre si y entre los valores de los factores latentes de los documentos a los que pertenecen.\n",
    "\n",
    "\n",
    "* Si representamos en un espacio de dos dimensiones los factores latentes de los documentos y de las palabras (factores 2 y 3) podemos ver no solo que ***los documentos y las palabras de los mismos temas comparte un mismo \"espacio latente\"***, si no también que los documentos y las palabras del mismo tema están muy próximas entre sí.\n",
    "\n",
    "\n",
    "* ***NOTA***: *Al tratarse de un ejemplo didáctico podemos hacer ciertas suposiciones y representar y mostrar ciertos concepto en un espacio reducido; en este caso en 2 dimensiones. En un caso real con muchos documentos, muchas palabras y muchos temas (topics) sera casi imposible visualizar este tipo de cosas con el LSI*\n",
    "\n",
    "\n",
    "### Representación visual de los documentos en 2D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "df = pd.DataFrame({'f2': v_matrix[:,2], 'f3': v_matrix[:,1],\n",
    "                   'group': ['Doc {}'.format(i+1) for i,doc in enumerate(documents)]})\n",
    "\n",
    "sns.lmplot('f2', 'f3', data=df, fit_reg=False, height=6, aspect=2)\n",
    "\n",
    "plt.title('Representación de los documentos (Factores 2 y 3)')\n",
    "plt.xlabel('Factor 2')\n",
    "plt.ylabel('Factor 3')\n",
    "\n",
    "def label_point(x, y, val, ax):\n",
    "    a = pd.concat({'x': x, 'y': y, 'val': val}, axis=1)\n",
    "    for i, point in a.iterrows():\n",
    "        ax.text(point['x']+.02, point['y'], str(point['val']))\n",
    "        \n",
    "label_point(df.f2, df.f3, df.group, plt.gca()) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Representación visual de las Palabras en 2D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "df = pd.DataFrame({'f2': u_matrix[:,2], 'f3': u_matrix[:,1],\n",
    "                   'group': [word for word in dictionary.token2id.keys()]})\n",
    "\n",
    "sns.lmplot('f2', 'f3', data=df, fit_reg=False, height=6, aspect=2)\n",
    "\n",
    "plt.title('Representación de las palabras (Factores 2 y 3)')\n",
    "plt.xlabel('Factor 2')\n",
    "plt.ylabel('Factor 3')\n",
    "\n",
    "def label_point(x, y, val, ax):\n",
    "    a = pd.concat({'x': x, 'y': y, 'val': val}, axis=1)\n",
    "    for i, point in a.iterrows():\n",
    "        ax.text(point['x']+.02, point['y'], str(point['val']))\n",
    "        \n",
    "label_point(df.f2, df.f3, df.group, plt.gca()) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "\n",
    "## Topics & Terms\n",
    "\n",
    "\n",
    "* Cada uno de los temas (***Topics***) esta definido por una importancia (o peso) que tienen las palabras (***Terms***) del corpus en el tema.\n",
    "\n",
    "\n",
    "* Por eso cada tema tiene unas palabras relevantes que en función de si aparecen o no esas palabra en el documentos hacen que este pertenezca más a un tema que a otro.\n",
    "\n",
    "\n",
    "* Gensim nos devuelve un \"formula\" por tema (Topic) que aplicada a las apariciones de las palabras en los documentos nos indica el grado de pertenencia del nuevo documento a ese tema. El que mayor valor tenga tras aplicar la fórmula del tema al documento significará que tiene mayor propensión a pertenecer a ese tema."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0,\n",
       "  '0.451*\"politica\" + 0.423*\"psoe\" + 0.377*\"zapatero\" + 0.346*\"pp\" + 0.346*\"fmi\" + 0.293*\"ue\" + 0.268*\"rajoy\" + 0.255*\"dinero\" + 0.100*\"pib\" + 0.046*\"ibex\"'),\n",
       " (1,\n",
       "  '-0.762*\"futbol\" + -0.443*\"messi\" + -0.383*\"ronaldo\" + -0.238*\"balon\" + -0.121*\"liga\" + 0.044*\"fmi\" + 0.037*\"ue\" + 0.031*\"dinero\" + 0.008*\"pib\" + 0.007*\"ibex\"'),\n",
       " (2,\n",
       "  '-0.552*\"fmi\" + -0.470*\"ue\" + -0.365*\"dinero\" + 0.359*\"politica\" + 0.289*\"psoe\" + 0.209*\"pp\" + -0.183*\"pib\" + 0.182*\"zapatero\" + -0.093*\"ibex\" + 0.079*\"rajoy\"')]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lsi_model.print_topics()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "\n",
    "# Topics de nuevos documentos\n",
    "\n",
    "\n",
    "* Al tratarse el Topic Modeling de un aprendizaje no supervisado (básicamente es un clustering de documentos), tenemos que ser capaces de poder asignar a un nuevo documento el tema al que pertenece.\n",
    "\n",
    "\n",
    "* Para ello tenemos que:\n",
    "    1. ***Obtener los factores latentes del nuevo documento*** en función del vector de apariciones de palabras\n",
    "    2. Calcular la ***similaridad con el resto de documentos*** en función de sus factores latentes\n",
    "    \n",
    "    \n",
    "* Para obtener los factores latentes de un nuevo documento lo podemos hacer con la técnica del \"***Folding-In***\" que básicamente consiste en el siguiente cálculo teniendo las matrices U<sub>k</sub> y S:\n",
    "\n",
    "\n",
    "$$ Factores Latentes_{new doc} = Vector Palabras_{new doc} \\cdot U_{k} \\cdot S^{-1}$$\n",
    "\n",
    "\n",
    "* Gensim ya tiene implementada esta funcionalidad y también la del cálculo de similaridades entre documentos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "new_doc = \"futbol futbol messi zapatero\"\n",
    "new_vec = dictionary.doc2bow(new_doc.split())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Valor</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Topic 1</th>\n",
       "      <td>0</td>\n",
       "      <td>0.456133</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Topic 2</th>\n",
       "      <td>1</td>\n",
       "      <td>-1.965546</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Topic 3</th>\n",
       "      <td>2</td>\n",
       "      <td>0.077653</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               Valor\n",
       "Topic 1  0  0.456133\n",
       "Topic 2  1 -1.965546\n",
       "Topic 3  2  0.077653"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Obtenemos los factores latentes del nuevo documento\n",
    "vec_lsi = lsi_model[new_vec]\n",
    "\n",
    "pd.DataFrame(vec_lsi, index=['Topic 1', 'Topic 2', 'Topic 3'], columns=['', 'Valor']).head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Ahora pasamos a calcular las similaridades entre documentos y podemos observar como este nuevo documento tiene mayor similaridad con los documentos 1, 2 y 3 que hablan sobre fútbol."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Documento (indice)</th>\n",
       "      <th>Similaridad</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>2</td>\n",
       "      <td>0.978027</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>0.977872</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0</td>\n",
       "      <td>0.977710</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>9</td>\n",
       "      <td>0.905462</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>10</td>\n",
       "      <td>0.897511</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>3</td>\n",
       "      <td>0.210638</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>4</td>\n",
       "      <td>0.206452</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>5</td>\n",
       "      <td>0.206373</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>8</td>\n",
       "      <td>0.075325</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>7</td>\n",
       "      <td>0.045581</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>6</td>\n",
       "      <td>0.022825</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    Documento (indice)  Similaridad\n",
       "0                    2     0.978027\n",
       "1                    1     0.977872\n",
       "2                    0     0.977710\n",
       "3                    9     0.905462\n",
       "4                   10     0.897511\n",
       "5                    3     0.210638\n",
       "6                    4     0.206452\n",
       "7                    5     0.206373\n",
       "8                    8     0.075325\n",
       "9                    7     0.045581\n",
       "10                   6     0.022825"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from gensim import similarities\n",
    "\n",
    "# Calculamos las similaridades\n",
    "index = similarities.MatrixSimilarity(lsi_model[corpus])\n",
    "sims = index[vec_lsi]\n",
    "sims = sorted(enumerate(sims), key=lambda item: -item[1])\n",
    "\n",
    "pd.DataFrame(sims, columns=['Documento (indice)', 'Similaridad']).head(11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "\n",
    "# Bonus Track - LSI, Cálculo Analítico \n",
    "\n",
    "\n",
    "* Dada una matriz que representa una Bolsa de Palabras donde:\n",
    "    - Las filas representan a los documentos ('n' documentos)\n",
    "    - Las columnas representan a las palabras ('m' palabras)\n",
    "    \n",
    "    \n",
    "* Podemos descomponer esa matriz en tres matrices ***U***, ***S*** y ***V***, cuyo producto matricial es la matriz original ***A***.\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    " &  &  &  &  &  & \\\\ \n",
    " &  &  &  &  &  & \\\\ \n",
    " &  &  & A &  &  & \\\\ \n",
    " &  &  & _{nxm} &  &  & \\\\ \n",
    " &  &  &  &  &  & \n",
    "\\end{bmatrix} =  \\begin{bmatrix}\n",
    " &  &  &  & \\\\ \n",
    " &  &  &  & \\\\ \n",
    " &  & U &  & \\\\ \n",
    " &  & _{nxn} &  & \\\\ \n",
    " &  &  &  & \n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    " &  &  &  &  &  & \\\\ \n",
    " &  &  &  &  &  & \\\\ \n",
    " &  &  & S &  &  & \\\\ \n",
    " &  &  & _{nxm} &  &  & \\\\ \n",
    " &  &  &  &  &  & \n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    " &  &  &  &  &  & \\\\ \n",
    " &  &  &  &  &  & \\\\ \n",
    " &  &  & V^{t} &  &  & \\\\ \n",
    " &  &  & _{mxm} &  &  & \\\\ \n",
    " &  &  &  &  &  & \\\\ \n",
    " &  &  &  &  &  & \\\\ \n",
    " &  &  &  &  &  & \n",
    "\\end{bmatrix} $$\n",
    "\n",
    "\n",
    "* Cada una de estas matrices va a contener la siguiente información:\n",
    "\n",
    "    * ***Matriz U***: Contiene los valores de los ***factores latentes de las palabras***.\n",
    "    * ***Matriz V***: Contiene los valores de los ***factores latentes de los documentos***.\n",
    "    * ***Matriz S***: Matriz en cuya diagonal estan los llamados ***Valores singulares*** que tienen que ser valores decrecientes y no negativos. Los valores de la diagonal representan la importancia que tienen cada uno de los factores latentes de las palabras y de los documentos.\n",
    "\n",
    "<hr>\n",
    "\n",
    "\n",
    "# Cálculo Analítico: de las matrices U, S y V\n",
    "\n",
    "### Cálculo de la matriz U\n",
    "\n",
    "1. Se multiplica la matriz ***A*** por su traspuesta (***A·A <sup>T</sup>***) para obtener una matriz cuadrada de dimensión ***nxn***\n",
    "\n",
    "\n",
    "2. Se calculan los autovalores de la matriz cuadrada (***A<sub>1</sub> = A·A <sup>T</sup>***) a partir de su polinómio característico:\n",
    "\n",
    "$$det(A_{1}-\\lambda I) = 0$$\n",
    "\n",
    "\n",
    "3. Se calculan los autovectores asociados a cada autovalor (en orden decreciente por autovalor)\n",
    "\n",
    "\n",
    "### Cálculo de la matriz S\n",
    "\n",
    "\n",
    "1. Obtenidos los autovalores tras el cálculo de la matriz ***A***, se pone en la diagonal de la matriz ***S*** la raiz cuadrada del los autovalores en orden decreciente. El resto de elementos de la matriz ***S*** tendrán valor ***0***\n",
    "\n",
    "\n",
    "### Cálculo de la matriz V\n",
    "\n",
    "1. Se multiplica ***A <sup>T</sup>·A*** para obtener una matriz cuadrada de dimensión ***mxm***\n",
    "\n",
    "\n",
    "2. Se calculan los autovectores asociados a cada autovalor (ya calculados) en orden descendente.\n",
    "\n",
    "$$det(A_{2}-\\lambda I) = 0$$\n",
    "\n",
    "<hr>\n",
    "\n",
    "## Ejemplo:\n",
    "\n",
    "\n",
    "* Supongamos que tenemos la siguiente matriz:\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "1 & 0 & 1\\\\ \n",
    "2 & 3 & 0\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "### Cálculo de la matriz U:\n",
    "\n",
    "1. Calculamos ***A<sub>1</sub> = A·A <sup>T</sup>*** para obtener una matriz cuadrada de dimensión ***nxn***\n",
    "\n",
    "$$A_{1} = A \\cdot A' = \\begin{bmatrix}\n",
    "1 & 0 &1 \\\\ \n",
    "2 & 3 & 0\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "1&2 \\\\ \n",
    "0 & 3\\\\ \n",
    "1 & 0\n",
    "\\end{bmatrix} = \\begin{bmatrix}\n",
    "2 & 2\\\\ \n",
    "2 & 13\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "2. Cálculo de los autovalores de la matriz cuadrada (***A<sub>1</sub> = A·A <sup>T</sup>***) a partir de su polinómio característico:\n",
    "<br><br><br>\n",
    "$$det(A_{1}-\\lambda I) = \\begin{bmatrix}\n",
    "2 - \\lambda & 2\\\\ \n",
    "2 & 13 - \\lambda\n",
    "\\end{bmatrix} = (2  - \\lambda) \\cdot (13 - \\lambda) - 4 =  \\lambda^{2} - 15 \\lambda + 22$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$ \\lambda = \\frac{15\\pm \\sqrt{15^2 - 4 \\cdot 1 \\cdot 22}}{2\\cdot1} $$\n",
    "<br>\n",
    "$$\\lambda_{1} = 13,35 $$\n",
    "<br>\n",
    "$$\\lambda_{2} = 1,65 $$\n",
    "\n",
    "\n",
    "3. Se calculan los autovectores asociados a cada autovalor (en orden decreciente por autovalor)\n",
    "\n",
    "  * Autovalor 1 = 13,35\n",
    "\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "2 & 2\\\\ \n",
    "2 & 13\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "x\\\\y\n",
    "\\end{bmatrix} = 13,35 \\cdot \\begin{bmatrix}\n",
    "x\\\\y\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$ \\begin{matrix}\n",
    "2x + 2y = 13,35x\n",
    "\\\\ \n",
    "2x + 13y = 13,35y\n",
    "\\end{matrix} $$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$ -11,35x + 2y = 0; \\: si \\:  x = 1; \\:  y = \\frac{11,35}{2} = 5,675$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$Normalizo \\: \\rightarrow \\sqrt{1^2  + 5,675^2} = 5,848$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$Autovector (1) \\: = \\left [ \\frac{1}{5,848} \\:\\: \\frac{5,675}{5,848} \\right ] = \\left [ 0,17 \\:\\:\\: 0,98\\right ]$$\n",
    "\n",
    "   * Para el autovalor 2 realizo los mismos pasos y obtengo el siguiente resultado:\n",
    "   \n",
    "$$\\begin{bmatrix}\n",
    "2 & 2\\\\ \n",
    "2 & 13\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "x\\\\y\n",
    "\\end{bmatrix} = 1,65 \\cdot \\begin{bmatrix}\n",
    "x\\\\y\n",
    "\\end{bmatrix}$$ \n",
    "\n",
    "<br>\n",
    "\n",
    "$$Autovector (2) \\: = \\left [ 0,98 \\:\\:\\: -0,17 \\right ]$$\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "* Por tanto la matriz U quedaría:\n",
    "\n",
    "$$U = \\begin{bmatrix}\n",
    "0,17 & 0,98\\\\ \n",
    "0,98 & -0,17\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "### Cálculo de la matriz S\n",
    "\n",
    "\n",
    "1. Como ya se tienen calculados los autovalores obtenidos tras el cálculo de la matriz ***U***, nos es facil obtener la matriz ***S*** que tendrá en la diagonal la raiz cuadrada de los autovales colocados de manera decreciente:\n",
    "\n",
    "\n",
    "$$ S = \\begin{bmatrix}\n",
    "\\sqrt{13,35} & 0 & 0 \\\\ \n",
    "0 & \\sqrt{1,65} & 0\n",
    "\\end{bmatrix} = \\begin{bmatrix}\n",
    "3,65 & 0 & 0 \\\\ \n",
    "0 & 1,28 & 0\n",
    "\\end{bmatrix} $$\n",
    "\n",
    "\n",
    "### Cálculo de la matriz V\n",
    "\n",
    "1. Se multiplica ***A <sup>T</sup>·A*** para obtener una matriz cuadrada de dimensión ***mxm***\n",
    "\n",
    "$$A_{2} = A' \\cdot A = \\begin{bmatrix}\n",
    "1&2 \\\\ \n",
    "0 & 3\\\\ \n",
    "1 & 0\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "1 & 0 &1 \\\\ \n",
    "2 & 3 & 0\n",
    "\\end{bmatrix} = \\begin{bmatrix}\n",
    "5 & 6 & 1 \\\\ \n",
    "6 & 9 & 0 \\\\ \n",
    "1 & 0 & 1\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "2. Se calculan los autovectores asociados a cada autovalor (ya calculados) en orden descendente.\n",
    "\n",
    "    - Autovalores:\n",
    "<br>\n",
    "$$\\lambda_{1} = 13,35 $$\n",
    "<br>\n",
    "$$\\lambda_{2} = 1,65 $$\n",
    "\n",
    "* Calculamos los autovectores a partir de su polinomio característico (para estos cálculos no mostramos el procedimiento):\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "5 & 6 & 1 \\\\ \n",
    "6 & 9 & 0 \\\\ \n",
    "1 & 0 & 1\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "x\\\\y\\\\z\n",
    "\\end{bmatrix} = 13,35 \\cdot \\begin{bmatrix}\n",
    "x\\\\y\\\\z\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$Autovector (1) \\: = \\left [ 0,58 \\:\\:\\: 0,49 \\:\\: -0,64\\right ]$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "5 & 6 & 1 \\\\ \n",
    "6 & 9 & 0 \\\\ \n",
    "1 & 0 & 1\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "x\\\\y\\\\z\n",
    "\\end{bmatrix} = 1,65 \\cdot \\begin{bmatrix}\n",
    "x\\\\y\\\\z\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$Autovector (2) \\: = \\left [ 0,81 \\:\\: -0,41 \\:\\: 0,43\\right ]$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "5 & 6 & 1 \\\\ \n",
    "6 & 9 & 0 \\\\ \n",
    "1 & 0 & 1\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "x\\\\y\\\\z\n",
    "\\end{bmatrix} = 0 \\cdot \\begin{bmatrix}\n",
    "x\\\\y\\\\z\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "<br>\n",
    "\n",
    "$$Autovector (3) \\: = \\left [ 0,05 \\:\\: 0,77 \\:\\: 0,64\\right ]$$\n",
    "\n",
    "* Por tanto la matriz V quedaría:\n",
    "\n",
    "$$V = \\begin{bmatrix}\n",
    "0,58 & 0,49 & -0,64 \\\\ \n",
    "0,81 & -0,41 & 0,43 \\\\ \n",
    "0,05 & -0,77 & 0,64\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "\n",
    "## La Descomposición Matricial quedaría:\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "1 & 0 &1 \\\\ \n",
    "2 & 3 & 0\n",
    "\\end{bmatrix} = \\begin{bmatrix}\n",
    "0,17 & 0,98\\\\ \n",
    "0,98 & -0,17\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "3,65 & 0 & 0 \\\\ \n",
    "0 & 1,28 & 0\n",
    "\\end{bmatrix} \\cdot \\begin{bmatrix}\n",
    "0,58 & 0,81 & 0,05 \\\\ \n",
    "0,49 & -0,41 & 0,77 \\\\ \n",
    "-0,64 & 0,43 & 0,64\n",
    "\\end{bmatrix}$$"
   ]
  }
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