Python/clases/3_algebra_lineal/3_minimos_cuadrados.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Notas para contenedor de docker:**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comando de docker para ejecución de la nota de forma local:\n",
    "\n",
    "nota: cambiar `dir_montar` por la ruta de directorio que se desea mapear a `/datos` dentro del contenedor de docker.\n",
    "\n",
    "```\n",
    "dir_montar=<ruta completa de mi máquina a mi directorio>#aquí colocar la ruta al directorio a montar, por ejemplo: \n",
    "#dir_montar=/Users/erick/midirectorio.\n",
    "```\n",
    "\n",
    "Ejecutar:\n",
    "\n",
    "```\n",
    "$docker run --rm -v $dir_montar:/datos --name jupyterlab_prope_r_kernel_tidyverse -p 8888:8888 -d palmoreck/jupyterlab_prope_r_kernel_tidyverse:2.1.4   \n",
    "\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ir a `localhost:8888` y escribir el password para jupyterlab: `qwerty`\n",
    "\n",
    "Detener el contenedor de docker:\n",
    "\n",
    "```\n",
    "docker stop jupyterlab_prope_r_kernel_tidyverse\n",
    "```\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Documentación de la imagen de docker `palmoreck/jupyterlab_prope_r_kernel_tidyverse:2.1.4` en [liga](https://github.com/palmoreck/dockerfiles/tree/master/jupyterlab/prope_r_kernel_tidyverse)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para ejecución de la nota usar:\n",
    "\n",
    "[docker](https://www.docker.com/) (instalación de forma **local** con [Get docker](https://docs.docker.com/install/)) y ejecutar comandos que están al inicio de la nota de forma **local**. \n",
    "\n",
    "O bien dar click en alguno de los botones siguientes:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/palmoreck/dockerfiles-for-binder/jupyterlab_prope_r_kernel_tidyerse?urlpath=lab/tree/Propedeutico/Python/clases/3_algebra_lineal/3_minimos_cuadrados.ipynb) esta opción crea una máquina individual en un servidor de Google, clona el repositorio y permite la ejecución de los notebooks de jupyter."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[![Run on Repl.it](https://repl.it/badge/github/palmoreck/dummy)](https://repl.it/languages/python3) esta opción no clona el repositorio, no ejecuta los notebooks de jupyter pero permite ejecución de instrucciones de Python de forma colaborativa con [repl.it](https://repl.it/). Al dar click se crearán nuevos ***repl*** debajo de sus users de ***repl.it***.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mínimos cuadrados lineales"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Supóngase que se han realizado mediciones de un fenómeno de interés en diferentes puntos $x_i$'s resultando en cantidades $y_i$'s $\\forall i=0,1,\\dots, m$ (se tienen $m+1$ puntos) y además las $y_i$'s contienen un ruido aleatorio causado por errores de medición:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://dl.dropboxusercontent.com/s/z0ksltumd4ibyjp/mcuadrados_1.jpg?dl=0\" heigth=\"400\" width=\"400\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El objetivo de los mínimos cuadrados lineales es construir una curva, $f(x|\\beta)$ que \"mejor\" se ajuste a los datos $(x_i,y_i)$, $\\forall i=0,1,\\dots,m$. El término de \"mejor\" se refiere a que la suma: $$\\displaystyle \\sum_{i=0}^m (y_i -f(x_i|\\beta))^2$$ sea lo más pequeña posible, esto es, a que la suma de las distancias verticales entre $y_i$ y $f(x_i|\\beta)$ $\\forall i=0,1,\\dots,m$ al cuadrado sea mínima. Por ejemplo:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://dl.dropboxusercontent.com/s/z31rni7hrp6w91s/mcuadrados_2.jpg?dl=0\" heigth=\"400\" width=\"400\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Obs:**\n",
    "\n",
    "* La notación $f(x|\\beta)$ se utiliza para denotar que $\\beta$ es un vector de parámetros a estimar, en específico $\\beta_0, \\beta_1, \\dots \\beta_n$, esto es: $n+1$ parámetros a estimar."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Modelo en mínimos cuadrados lineales o también nombrados ordinarios"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En los mínimos cuadrados lineales se supone:  $f(x|\\beta) = \\displaystyle \\sum_{j=0}^n\\beta_j\\phi_j(x)$ con $\\phi_j: \\mathbb{R} \\rightarrow \\mathbb{R}$ funciones conocidas por lo que se tiene una gran flexibilidad para el proceso de ajuste."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Obs:** \n",
    "\n",
    "* Si $n=m$ entonces se tiene un problema de interpolación.\n",
    "* x se nombra variable **regresora**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ¿Cómo ajustar el modelo anterior?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En lo siguiente se **asume** $n+1 \\leq m+1$ (tenemos más puntos $(x_i,y_i)$'s que parámetros a estimar)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para realizar el ajuste de mínimos cuadrados se utilizan las **ecuaciones normales**: $$A^TA\\beta=A^Ty$$ donde: $A$ se construye con las $\\phi_j$'s evaluadas en los puntos $x_i$'s, el vector $\\beta$ contiene a los parámetros $\\beta_j$'s a estimar y el vector $y$, la variable **respuesta**, se construye con los puntos $y_i$'s:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$A = \\left[\\begin{array}{cccc}\n",
    "\\phi_0(x_0) &\\phi_1(x_0)&\\dots&\\phi_n(x_0)\\\\\n",
    "\\phi_0(x_1) &\\phi_1(x_1)&\\dots&\\phi_n(x_1)\\\\\n",
    "\\vdots &\\vdots& \\vdots&\\vdots\\\\\n",
    "\\phi_0(x_n) &\\phi_1(x_n)&\\dots&\\phi_n(x_n)\\\\\n",
    "\\vdots &\\vdots& \\vdots&\\vdots\\\\\n",
    "\\phi_0(x_{m-1}) &\\phi_1(x_{m-1})&\\dots&\\phi_n(x_{m-1})\\\\\n",
    "\\phi_0(x_m) &\\phi_1(x_m)&\\dots&\\phi_n(x_m)\n",
    "\\end{array}\n",
    "\\right] \\in \\mathbb{R}^{(m+1)x(n+1)},\n",
    "\\beta=\n",
    "\\left[\\begin{array}{c}\n",
    "\\beta_0\\\\\n",
    "\\beta_1\\\\\n",
    "\\vdots \\\\\n",
    "\\beta_n\n",
    "\\end{array}\n",
    "\\right] \\in \\mathbb{R}^n,\n",
    "y=\n",
    "\\left[\\begin{array}{c}\n",
    "y_0\\\\\n",
    "y_1\\\\\n",
    "\\vdots \\\\\n",
    "y_m\n",
    "\\end{array}\n",
    "\\right] \\in \\mathbb{R}^{m + 1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "y si $A$ es de $rank$ completo (tiene $n+1$ columnas linealmente independientes) se calcula la factorización $QR$ de $A$ : $A = QR$ y entonces: $$A^TA\\beta = A^Ty$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "y como $A=QR$ se tiene: $A^TA = (R^TQ^T)(QR)$ y $A^T = R^TQ^T$ por lo que:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$(R^TQ^T)(QR) \\beta =  R^TQ^T y$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "y usando que $Q$ tiene columnas ortonormales:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$R^TR\\beta = R^TQ^Ty$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como $A$ tiene $n+1$ columnas linealmente independientes, la matriz $R$ es invertible por lo que $R^T$ también lo es y finalmente se tiene el sistema de ecuaciones por resolver:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$R\\beta = Q^Ty$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ejemplo: regresión lineal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el caso de la regresión lineal se tienen dos modelos que se pueden ajustar un modelo con intercepto y otro sin él. La elección depende de los datos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modelo con intercepto"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se ajusta un modelo de la forma: $f(x|\\beta) = \\beta_0 + \\beta_1 x$ a los datos $(x_i,y_i)$'s $\\forall i=0,1,\\dots,m$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Obs:** En este caso se eligen $\\phi_0(x) = 1$, $\\phi_1(x) =x$. Y tenemos que estimar dos parámetros: $\\beta_0, \\beta_1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Ejemplo numérico en numpy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "import pprint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.set_printoptions(precision = 2) #sólo dos decimales que se muestren"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(1989) #para reproducibilidad\n",
    "mpoints = 20\n",
    "x = np.random.randn(mpoints) \n",
    "y = -3*x + np.random.normal(2,1,mpoints)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Los datos ejemplo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "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": [
    "plt.plot(x,y, 'r*')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title('Puntos ejemplo')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### El ajuste"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Con numpy podemos usar la función `polyfit` en el paquete de `numpy` para realizar el ajuste: (ver [numpy.polyfit](https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyfit.html))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El tercer argumento de polyfit especifica el grado del polinomio a ajustar. Usaremos `ndegree = 1` pues queremos ajustar una recta."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "ndegree = 1\n",
    "coefficients = np.polyfit(x,y,ndegree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Una vez realizado el llamado a la función polyfit se regresan los coeficientes de $x$ ordenados del mayor grado al menor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "array([-2.65,  2.03])\n"
     ]
    }
   ],
   "source": [
    "pprint.pprint(coefficients)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Entonces nuestro polinomio es: $$p_{1}(x) = -2.65x + 2.03$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "y así tenemos nuestras beta's ajustadas $\\hat{\\beta_0} = 2.03$, $\\hat{\\beta_1} = -2.65$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Denotamos como $\\hat{y}_i$ al valor ajustado para el dato $x_i$, esto es: $\\hat{y}_i = f(x_i|\\hat{\\beta}) = \\displaystyle \\sum_{j=0}^n\\hat{\\beta}_j\\phi_j(x_i)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### La gráfica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ahora nos gustaría graficar el modelo en el intervalo $[min(x),max(x)]$ con $min(x)$ la entrada con valor mínimo del numpy array $x$ y $max(x)$ su entrada con valor máximo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para lo anterior debemos obtener los valores ajustados al evaluar $p_1(x)$ los valores de $x$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_hat_numpy = coefficients[1] + coefficients[0] * x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.73,  1.8 ,  0.89, -0.35,  0.55,  0.86,  1.03,  2.43, -0.03,\n",
       "        2.09, -2.23,  3.  ,  1.09,  1.58, -4.65,  1.66, -1.05,  0.47,\n",
       "        5.22,  3.02])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_hat_numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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": [
    "plt.plot(x, y_hat_numpy, 'k-',\n",
    "         x, y, 'r*')\n",
    "plt.legend(['modelo lineal','datos'], loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Grafiquemos con $x's$ que no hayamos observado o medido:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.2005252660472876, 2.5180237924466593)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(np.min(x),np.max(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "mpoints_new = 30\n",
    "x_new_values = np.linspace(-3,5,mpoints_new)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_hat_numpy_new = coefficients[1] + coefficients[0] * x_new_values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  9.99,   9.26,   8.53,   7.8 ,   7.06,   6.33,   5.6 ,   4.87,\n",
       "         4.13,   3.4 ,   2.67,   1.94,   1.21,   0.47,  -0.26,  -0.99,\n",
       "        -1.72,  -2.46,  -3.19,  -3.92,  -4.65,  -5.38,  -6.12,  -6.85,\n",
       "        -7.58,  -8.31,  -9.05,  -9.78, -10.51, -11.24])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_hat_numpy_new"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(30,)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_hat_numpy_new.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(30,)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_new_values.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "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": [
    "plt.plot(x_new_values, y_hat_numpy_new, 'k-',\n",
    "         x, y, 'r*')\n",
    "plt.legend(['modelo lineal','datos'], loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_new_values[x_new_values.size-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-11.242432565608524"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_hat_numpy_new[y_hat_numpy_new.size-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**También podemos obtener lo anterior con la factorización QR construyendo a la matriz A de forma explícita**\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "A=np.ones((mpoints,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "A[:,1] = x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.  , -0.26],\n",
       "       [ 1.  ,  0.09],\n",
       "       [ 1.  ,  0.43],\n",
       "       [ 1.  ,  0.9 ],\n",
       "       [ 1.  ,  0.56],\n",
       "       [ 1.  ,  0.44],\n",
       "       [ 1.  ,  0.38],\n",
       "       [ 1.  , -0.15],\n",
       "       [ 1.  ,  0.78],\n",
       "       [ 1.  , -0.02],\n",
       "       [ 1.  ,  1.61],\n",
       "       [ 1.  , -0.37],\n",
       "       [ 1.  ,  0.36],\n",
       "       [ 1.  ,  0.17],\n",
       "       [ 1.  ,  2.52],\n",
       "       [ 1.  ,  0.14],\n",
       "       [ 1.  ,  1.16],\n",
       "       [ 1.  ,  0.59],\n",
       "       [ 1.  , -1.2 ],\n",
       "       [ 1.  , -0.37]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "Q,R = np.linalg.qr(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Resolvemos el sistema $R\\beta = Q^Ty$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "array([ 2.03, -2.65])\n"
     ]
    }
   ],
   "source": [
    "beta = np.linalg.solve(R,Q.T@y)\n",
    "pprint.pprint(beta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_hat_QR = A@beta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Obs: que la línea anterior es equivalente a realizar: `y_hat_QR = beta[0] + beta[1]*x`**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "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": [
    "plt.plot(x, y_hat_QR , 'k-',x, y, 'r*')\n",
    "plt.legend(['modelo lineal','datos'], loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modelo sin intercepto"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se ajusta un modelo de la forma: $f(x|\\beta) = \\beta_1 x$ a los datos $(x_i,y_i)$'s $\\forall i=0,1,\\dots,m$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Obs:** En este caso se elige $\\phi_1(x) =x$ y no hay $\\phi_0$ por lo que sólo se tiene que estimar $\\beta_1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Ejemplo numérico en numpy:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(Tarea) Ejercicio: realizar el ajuste correspondiente para este caso con `QR` con un notebook de jupyter. Hacer gráfica**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Una vez hecho el ajuste..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se realiza un **análisis de residuales**, se hace una **gráfica** del modelo si las dimensiones en las que se está trabajando lo permiten y se calcula el **error cuadrático medio**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El residual $i$ es: $r_i = y_i - \\hat{y}_i$ y representa la discrepancia entre los datos y el modelo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El error cuadrático medio se calcula como: $$ECM(\\hat{y}) = \\frac{1}{m} \\displaystyle \\sum_{i=0}^m(y_i-\\hat{y}_i)^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$m$ es el número de puntos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(Tarea)Ejercicio: crear un módulo de nombre `utils.py` en el que se tenga la función:**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```\n",
    "def MSE(y, y_hat):\n",
    "    \"\"\"\n",
    "    Compute mean squared error.\n",
    "    See: https://en.wikipedia.org/wiki/Mean_squared_error\n",
    "    Args:\n",
    "        y (numpy 1d array of floats): actual values of data.\n",
    "        y_hat (numpy 1d array of floats): estimated values via model.\n",
    "    Returns:\n",
    "        ecm (float): mean squared error result.\n",
    "    \"\"\"\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**En esta función se implementa la fórmula de error cuadrático medio.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ejemplo: ajuste de un modelo por mínimos cuadrados lineales con funciones $\\phi_j$'s no lineales"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Obsérvese que el modelo que se ha utilizado: $f(x|\\beta) = \\displaystyle \\sum_{j=0}^n\\beta_j\\phi_j(x)$ permite elegir las $\\phi_j$'s como funciones de $\\mathbb{R}$ a $\\mathbb{R}$. Por lo que tenemos una amplia gama de posibilidades de ajuste de curvas a datos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como ejemplo utilizaremos el conjunto de datos **data_for_nbook_3_minimos_cuadrados.txt** el cual lo pueden descargar en el mismo directorio de este *ipynb* o bien dando click [aquí](https://drive.google.com/file/d/1Ht7d2E1LWw7EIrrkULFQ_7-5nGVxHT4P/view?usp=sharing) y ajustaremos tres modelos de la forma:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$f_1(x|\\beta) = \\beta_0 + \\beta_1 \\frac{x}{x+1}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$f_2(x|\\beta) = \\beta_0 + \\beta_1x + \\beta_2x^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$f_3(x|\\beta) = \\beta_0 + \\beta_1\\text{log}(x+1)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "tomando como variable respuesta la segunda columna de los datos etiquetada como $y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "datos = np.loadtxt('data_for_nbook_3_minimos_cuadrados.txt', skiprows=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ver: [numpy.loadtxt](https://docs.scipy.org/doc/numpy/reference/generated/numpy.loadtxt.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = datos[:,0]\n",
    "y = datos[:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "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": [
    "plt.plot(x, y, 'r^')\n",
    "plt.legend(['datos'], loc='best')\n",
    "plt.title('datos')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modelo 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "phi_1 = lambda var: var/(var+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construimos a la matriz A como sigue:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "mpoints, = x.shape\n",
    "A=np.ones((mpoints,2))\n",
    "A[:,1] = phi_1(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculamos la factorización QR y graficamos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "Q,R = np.linalg.qr(A)\n",
    "beta = np.linalg.solve(R,Q.T@y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "beta\n",
      "array([-1.3,  6.2])\n"
     ]
    }
   ],
   "source": [
    "print('beta')\n",
    "pprint.pprint(beta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_hat_QR = A@beta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Obs: que la línea anterior es equivalente a realizar: `y_hat_QR = beta[0] + beta[1]*phi_1(x)`**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "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": [
    "plt.plot(x, y_hat_QR , 'k-',x, y, 'r^')\n",
    "plt.legend(['modelo1','datos'], loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modelo 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No tenemos que construir A pues es un polinomio de grado 2 por lo que usamos `polyfit` de `numpy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "ndegree = 2\n",
    "coefficients = np.polyfit(x,y,ndegree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "array([-0.69,  3.39, -0.92])\n"
     ]
    }
   ],
   "source": [
    "pprint.pprint(coefficients)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_hat_numpy = coefficients[2] + coefficients[1] * x + coefficients[0] * x**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "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": [
    "plt.plot(x, y_hat_numpy, 'k-',x, y, 'r^')\n",
    "plt.legend(['modelo polinomio cuadrático','datos'], loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modelo 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(Tarea) Ejercicio: ajustar el modelo 3. Calcular ECM de cada modelo y realizar en una sola gráfica los tres modelos. ¿Cuál es el modelo con menor ECM? Realizar este ejercicio en un notebook de jupyter.**"
   ]
  }
 ],
 "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}