6_Interpolacion.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interpolación y Regresión\n",
    "## Introducción\n",
    "El problema de la interpolacion consiste en producir funciones analiticas a partir de información esparcida. Consiste en generar mas información a partir de un conjunto de datos limitados.\n",
    "Un ejemplo clasico es recontruir una fncion previamente muestreada, y recuperar sus valores de curvatura.\n",
    "El caso mas simple de interpolación es la interpolación linea, y esta aproxima los valores intermedios de los valores conocidos utilizando lineas , planos o hyperplanos. Existen tecnicas de interpolacion que utilizan la curvatura de la funcion como interpolacion cuadratica, cubica o polinomica. Existen casos especializados de interpolacion bidimensiona, como la bilineal o bicubica. Ademas casos de interpolacion con Splines con aplicaciones afuera del cambo de ingenieria, como en el caso del diseño insutrial.\n",
    "La fomra general de interpolacion es crear un funcion analitica o mas precisamente un  polinomio que asegure que pase por los puntos dados. Este caso particular se resuelve por medio de polinomios de lagrange. Existen un problem si la data es muy densa el polinomio generado sera muy irregular y producira un fenomeno conocido como el fenomeno de Runge. Para resolver el problema de inestabilidad se utiliza un metodo de regresión.\n",
    "\n",
    "En el caso de la regresión, son metodos que *aproximan* un modelo(es decir un linea o un polinomio) minimizando algun costo. Los modelos de regresión son muy utilis en sistemas experimentales o en la investigación de nuevos fenomentos. Nos ayudan a econtrar modelos. El caso mas simple es la regresión lineal. y el metodos de minimizacón es\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interpolación Usando Diferencias finitas de Newton\n",
    "Antes de introducir al metodo de difrencias finitas de newton(un metodo bastante practico) vamos a visitar el caso mas simple de interpolación. Interpolacion lineal\n",
    "Para esto vamos  defnir una funcion de prueba\n",
    "$$ f(x) = \\mathrm{ln} (x) $$\n",
    "\n",
    "y graficamos para observar la funcion de pruebas\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pylab as plt\n",
    "\n",
    "x = np.linspace(.5,6,1000)\n",
    "y = np.log(x)\n",
    "plt.plot(x,y)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si deseamo solo guardas información de la curva en dos puntos, despues podemos recontruir una aproximacion utilizando una linea recta.\n",
    "En este caso vamos a guardar la informacion en x=1 x=4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "valor guardado de x:[1 4]\n",
      "valor guardado de y:[0.         1.38629436]\n"
     ]
    }
   ],
   "source": [
    "valor_x = np.array([1,4])\n",
    "valor_y = np.log(valor_x)\n",
    "print(\"valor guardado de x:\" +str(valor_x))\n",
    "print(\"valor guardado de y:\" +str(valor_y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "GRaficamos nuevamente la funcion logaritimica y los puntos muestreados(sampling)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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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,\"-\", label=\"ln(x)\")\n",
    "plt.plot(valor_x, valor_y, \"o\")\n",
    "plt.plot(valor_x, valor_y, \"--\", label=\"aproximación\")\n",
    "plt.grid()\n",
    "plt.legend(loc=\"best\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La interpolacion linea se obtiene por relacion de triangulo. o que en una linea la pendiente es constante en todos sus puntos:\n",
    "\n",
    "$$ \\frac{f_1(x) - f(x_0)}{x-x_0} = \\frac{f(x_1) - f(x_0)}{x_1-x_0} $$\n",
    "\n",
    "Se desapeja $f_1(x)$ :\n",
    "$$f_1(x) =   f(x_0)  + \\frac{f(x_1) - f(x_0)}{x_1-x_0} {x-x_0} $$\n",
    "\n",
    "implementamos esta funcion en python( side note en arduino la funcion map es una interpolacion lineal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lineal(x,y,x_):\n",
    "    return (y[1] - y[0]) / (x[1] - x[0])*(x_ - x[0]) + y[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PAra calcular el valor de $ln(x)$ aproximado podemos utilizar esta funcion lineal con dos puntos guardados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "valor real en x=1.5  ; f(x)=0.9162907318741551\n",
      "valor aprox en x=1.5  ; f_lineal(x)=0.6931471805599453\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "161538.46153846153"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Valor Real\n",
    "y_real = np.log(2.5)\n",
    "print(\"valor real en x=1.5  ; f(x)=\"+str(y_real))\n",
    "\n",
    "y_aprox = lineal([1,4], valor_y, 2.5)\n",
    "print(\"valor aprox en x=1.5  ; f_lineal(x)=\"+str(y_aprox))\n",
    "\n",
    "\n",
    "lineal([68,120], [100000, 200000], 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interpolación de Lagrange\n",
    "El polinomio de lagrange\n",
    "$$ f_n(x) = \\sum^n_{i=0} L_i(x) f(f_i)$$\n",
    "donde:\n",
    "$$ L_i(x) = \\prod^n_{j=0 \\bbox {..} j\\neq 1}{\\frac{x-x_j}{x_i-x_j}} $$\n",
    "\n",
    "Ejemplo:\n",
    "Si queremos utilizar el metodo de lagrange para interpolar los valores de \n",
    "\n",
    "$ x_0 = 1 \\quad f(x_0) = 0$\n",
    "\n",
    "$ x_1 = 4 \\quad f(x_1) = 1.386294$\n",
    "\n",
    "$ x_2 = 6 \\quad f(x_2) = 1.791760$\n",
    "\n",
    "El caso para una interpolacion para x = 2, podemos utilizar un polinomio lineal ($n=0$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def L_i(n,i,x,X):\n",
    "    resultado = 1;\n",
    "    for j in range(n+1):\n",
    "        #print(i,j)\n",
    "        if(not j == i ):\n",
    "            resultado = resultado * (x-X[j])/(X[i]-X[j])\n",
    "    return resultado\n",
    "\n",
    "def lagrange(n, x, X,F):\n",
    "    resultado = 0\n",
    "    for i in range(n+1):\n",
    "        #print(i)\n",
    "        resultado += L_i(n,i,x,X) * F[i]\n",
    "    return resultado\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.46209799999999995"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = [1, 4, 6]\n",
    "F = [0, 1.386294, 1.791760]\n",
    "lagrange(1,2,X,F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5658439999999999"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lagrange(2,2, X,F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pylab as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'f_3_array' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-26-94d8bbb21dab>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      7\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx_array\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_1_array\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      8\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx_array\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_2_array\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 9\u001b[0;31m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx_array\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_3_array\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     10\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mX\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\"or\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNameError\u001b[0m: name 'f_3_array' is not defined"
     ]
    },
    {
     "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": [
    "x_array = np.linspace(0,6)\n",
    "f_1_array = np.array(list(map(lambda x:lagrange(1,x,X,F), x_array)))\n",
    "f_2_array = np.array(list(map(lambda x:lagrange(2,x,X,F), x_array)))\n",
    "#f_3_array = np.array(list(map(lambda x:lagrange(3,x,X,F), x_array)))\n",
    "\n",
    "plt.Figure()\n",
    "plt.plot(x_array, f_1_array)\n",
    "plt.plot(x_array, f_2_array)\n",
    "plt.plot(x_array, f_3_array)\n",
    "plt.plot(X,F,\"or\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f0950618e1334e67b6cb72bd8b84dffb",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=5, description='n', max=9, min=1), Output()), _dom_classes=('widget-inte…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from ipywidgets import interact\n",
    "@interact(n=(1, 9))\n",
    "def f(n):\n",
    "    X = np.linspace(0,4,10)\n",
    "    F = np.sin(X)\n",
    "    x_array = np.linspace(0,4)\n",
    "    f_n_array = np.array(list(map(lambda x:lagrange(n,x,X,F), x_array)))\n",
    "    plt.plot(x_array, f_n_array)\n",
    "    plt.plot(X,F,\"or\")\n",
    "    plt.ylim(-1.5,1.5)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Splines\n",
    "### Splines cuadraticas\n",
    "### Splines cubicas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interpolacion multidimensional\n",
    "### Interpolacion bicubica"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "e29177aaf99542bdbb9a41f73f8331d6",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=5, description='n', max=10, min=1), Output()), _dom_classes=('widget-int…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from ipywidgets import interact\n",
    "@interact(n=(1, 10))\n",
    "def f(n):\n",
    "    X = np.linspace(0,5,10)\n",
    "    F = np.sin(X)\n",
    "    x_array = np.linspace(0,6)\n",
    "\n",
    "    f_n_array = np.array(list(map(lambda x:lagrange(n,x,X,F), x_array)))\n",
    "\n",
    "    plt.plot(x_array, f_n_array)\n",
    "\n",
    "    plt.plot(X,F,\"or\")\n",
    "    plt.ylim(-2,2)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Regresión (o Ajuste de Curvas)\n",
    "El problema de ajustes de curvas es es el proseso inverso de interpolacion, donde en este caso tenemos un conjunto de puntos obtenidos de un (experimento, observacion o analisis) y deseamos construir una funcion analitica o algebraica que se aproxime a estos puntos. Este conjunto de puntos en el caso unidimensional, puede aprooximarse a una funcion de una variable por ejemplo, una linea recta una parabola, una funcion exponencial, logaritmica o incluso un polinomio. A continuación presentaremos un ejemplo  con datos de costos de casas segun su  metros cuadrados en un aregion espesifica del pais.\n",
    "\n",
    "|Casa # | $mts^2$ | costo $|  \n",
    "|-------|----|-----|\n",
    "|Casa 1 | 80 | 100,000.00|\n",
    "|Casa 2 | 99 | 220,000.00|\n",
    "|Casa 3 | 120 | 300,000.00|\n",
    "|Casa 4 | 200 | 500,000.00|\n",
    "|Casa 5 | 250 | 602,000.00|"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[    80, 100000],\n",
       "       [    99, 220000],\n",
       "       [   120, 300000],\n",
       "       [   200, 500000],\n",
       "       [   250, 602000]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([[80,99,120,200,250],[100000,220000,300000,500000,602000]]).T\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10bcbaef0>]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "metros =  a[:,0]\n",
    "costos =  a[:,1]\n",
    "plt.plot(metros,costos,\"o\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El concepto de regresión es ajustar una curva que aproxime los puntos, para definir el problema debemos introducir la funcion de hipotesis en este caso una linea recta, sabemos que la linea recta tiene la funcion:\n",
    "$$ y = b + m x$$\n",
    "El cual b es el punto de cruza el eje $y$ cuando $x=0$ y m es la pendiente de la recta.\n",
    "Una  linea recta  como ejemplo seria "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "efc9c183da82483e8d36276201bf0279",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='b', max=5.0, min=-5.0), FloatSlider(value=0.0, descr…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact(b=(-5.0,5.0),m=(-3.0,3.0))\n",
    "def plot(b,m):\n",
    "    X = np.linspace(-10,10)\n",
    "    Y = b + m * X\n",
    "    plt.plot(X,Y)\n",
    "    plt.grid()\n",
    "    plt.ylim(-10,10)\n",
    "    plt.xlim(0,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si deasemos aproximar una linea recta al los puntos del ejemplo mencionado podemos utilizar dos parametros para manipular o intercatuar, estos \"parametro\" son valores arbritarios. Para fromalizar el problem de buscar la mejor curva tenemos que definir un criterio a obptimizar en este caso el error, Como los puntos se definen por una tupla o vector de valores y la curva como un continuo de infinitos puntos, necesitamos evaluar el error en cada punto, es decir, que tan lejos estamos del valor exacto.  Analiticamne este valor se define cmo la diferencia entre el valo exacto y el valor aprooximado para cada punto\n",
    "\n",
    "$$ e_i  = (y_{i} - y'_i)$$\n",
    "Donde $y'$ seria los valores aproximados\n",
    "$$ y' = b + m x$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "449ab2dfdee8490b832f13d228b803d6",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=5000.5, description='b', max=10000.0, min=1.0), FloatSlider(value=5000…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact(b=(1.0,1e4),m=(1.0,1e4))\n",
    "def plot(b,m):\n",
    "    X = np.linspace(metros[0],metros[-1],metros.size)\n",
    "    Y = b + m * X\n",
    "    plt.plot(metros,costos,\"o\",X,Y)\n",
    "    plt.ylim(0,7e5)\n",
    "    \n",
    "    # calculo de error\n",
    "    #error\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "e909cd48838942b2b8e9657a61e3d660",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=5000.5, description='b', max=10000.0, min=1.0), FloatSlider(value=5000…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact(b=(1.0,1e4),m=(1.0,1e4))\n",
    "def plot_error(b,m):\n",
    "    X = np.linspace(metros[0],metros[-1],metros.size)\n",
    "    Y = b + m * X\n",
    "    plt.figure()\n",
    "    plt.plot(metros,costos,\"o\",X,Y)\n",
    "    plt.ylim(0,7e5)\n",
    "    plt.show()\n",
    "    \n",
    "    # calculo de error\n",
    "    error = Y - costos\n",
    "    plt.figure()\n",
    "    plt.plot(metros,error)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El error claramente depende de los valores escogidos de los parametros de la curva en este caso $b$ y $m$ por lo tanto podemos sumar el cuadrado del error ( por razone de cancelar numeros negativos y psitivos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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": [
    "intentos = 10;\n",
    "pendientes = np.linspace(0,5000,intentos)\n",
    "errores = np.zeros(intentos)\n",
    "i = 0\n",
    "# calculo de error\n",
    "for pendiente in pendientes.tolist():\n",
    "    errores[i] = ((metros*pendiente - costos).sum(axis=0)**2)\n",
    "    i += 1\n",
    "    \n",
    "plt.plot(pendientes, errores,label=\"parametros de pendiente\")\n",
    "plt.legend(loc=\"best\")\n",
    "plt.xlabel(\"$m$ (pendiente)\")\n",
    "plt.ylabel(\"$\\sum_i error$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Errores:'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "array([2.96528400e+12, 1.70534579e+12, 7.91704494e+11, 2.24360111e+11,\n",
       "       3.31264198e+09, 1.28562086e+11, 6.00108444e+11, 1.41795172e+12,\n",
       "       2.58209190e+12, 4.09252900e+12])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "2222.222222222222"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display(\"Errores:\",errores)\n",
    "mejor = pendientes[4]\n",
    "mejor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "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": [
    "    X = np.linspace(metros[0],metros[-1],metros.size)\n",
    "    Y = 0 + (mejor) * X\n",
    "    plt.figure()\n",
    "    plt.plot(metros,costos,\"o\",X,Y)\n",
    "    plt.ylim(0,7e5)\n",
    "    plt.show()   \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Métodos de minimos cuadrados\n",
    "Sin duda alguna un metodo que minimize el error cuadrado es la forma mas optima para ajustar una curva en este primer ejemplo el caso de la linea recta. definimos los \"parametros\" $[a_0,a_1]$ que se fdeniene como y $a_0=b$ y $a_1=m$.\n",
    "\n",
    "$$ y = a_0 + a_1 x$$\n",
    "\n",
    "El calculo del error como la diferencia entre el valor requerido y el valor aproximado:\n",
    "\n",
    "\n",
    "$$ \\sum_{i=1}^ne_i = \\sum_{i=1}^n (y_i - a_0 - a_1 x_i)$$\n",
    "\n",
    "Un opcion para evitar la cancelacion de errore por los signos es utilizar el valor absoluto, pero esta seleccion crea una discontinuidad de primer orden, y para propositos de optimizacion utilizando derivadas preferimos utilizar el cuadrado del error que crea por lomenos un error suave, o continuo de clase 1. El error total se puede calcular como $S$\n",
    "\n",
    "$$S = \\sum_{i=0}^n e^2_i = \\sum_{i=0}^n (y_{i, medido}-y_{i, modelo})^2 = \\sum_{i=1}^n (y_i - a_0 - a_1 x_i)^2 $$ \n",
    "\n",
    "\n",
    "El objetivo es buscar los valores de $a_i$ para $i=0,1$ que minimize el valor de $S$.\n",
    "#### Caso Lineal\n",
    "En el caso de la linea recta podemos utilizar el concepto de la primera derivada para buscar analiticamente el valor minimo de los valores de los parametros $a_0$ $a_1$.\n",
    "$$ \\frac{\\partial S}{\\partial a_0} = -2 \\sum(y_i -a_0 - a_1 x_i) $$\n",
    "$$ \\frac{\\partial S}{\\partial a_1} = -2 \\sum[(y_i -a_0 - a_1 x_i)x_i ]$$\n",
    "Podemos simplificar un poco, y aprovachar la propiedad lineal de la sumatoria:\n",
    "$$0= \\sum y_i - \\sum a_0 - \\sum a_1 x_i $$\n",
    "$$0= \\sum y_i x_i - \\sum a_0 x_i - \\sum a_1 x_i^2 $$\n",
    "\n",
    "\n",
    "Si $\\sum a_0 = na_0$ podemos escribir las dos ecuaciones como un sistema lineal de dos ecuaciones y dos incognitas:\n",
    "\n",
    "$$ n a_0 + (\\sum x_i) a_1 = \\sum y_i$$\n",
    "$$ ( \\sum x_i)a_0 + (\\sum x_i^2) a_1 = \\sum x_i y_i$$\n",
    "\n",
    "En notación matricial:\n",
    "$$ \\begin{bmatrix} n & \\sum x_i \\\\ \\sum x_i & \\sum x_i^2 \\end{bmatrix}\n",
    "\\begin{bmatrix} a_0\\\\ a_1 \\end{bmatrix} = \n",
    "\\begin{bmatrix} \\sum y_i\\\\ \\sum x_i y_i \\end{bmatrix} \n",
    "$$\n",
    "\n",
    "La solucion directa a este sistema lineal es:\n",
    "$$ a_1 = \\frac{n \\sum{x_i y_i} - \\sum{x_i}\\sum{y_i}}{n \\sum x_i^2 - (\\sum x_i)^2} $$\n",
    "\n",
    "$$ a_0 = \\bar y - a_1 \\bar x $$\n",
    "\n",
    "\n",
    "Donde $\\bar x$ y $\\bar y$ son el promedio de $x$ y $y$ respectivamente.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Practicas\n",
    "1)  definir las funciones para calculo directo de la regresion lineal a_0 y a_1\n",
    "\n",
    "2) resolver el problema de forma directa (organizar el trabajo en una tabla)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Regresion  de polinomios"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'ss' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-21-09dbdbc7f6cd>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mss\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'ss' is not defined"
     ]
    }
   ],
   "source": [
    "ss"
   ]
  }
 ],
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    "version": 3
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