10_Modelos_nolineales.ipynb

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    "# Analítica de Negocios y Ciencia de Datos\n",
    "\n",
    "## CIDE - Otoño 2015\n",
    "\n",
    "### Modelos no lineales"
   ]
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    "# Referencias\n",
    "\n",
    "El modelo no lineal de regresión se puede encontrar en [Greene](http://pages.stern.nyu.edu/~wgreene/Text/econometricanalysis.htm), Cap. 7.\n",
    "\n",
    "Los modelos generalizados aditivos (GAMS) y temas relacionados se encuentran en el capítulo 7 de [ISL](http://www-bcf.usc.edu/~gareth/ISL/) y capítulos 5 y 9 de [ESL](http://statweb.stanford.edu/~tibs/ElemStatLearn.1stEd/)"
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    "# Modelos lineales y no lineales\n",
    "\n",
    "\n",
    "* Hasta ahora nos hemos movido en el mundo lineal.\n",
    "\n",
    "\n",
    "* OLS asume que la función que queremos aprender es lineal en los parámetros $f(x) = x'\\beta$.\n",
    "\n",
    "\n",
    "* Aunque ya vimos que por OLS podemos estimar cualquier modelo que tenga la siguiente forma\n",
    "\n",
    "$$\n",
    "f(x) = g(x)'\\beta\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    "* En el mundo de modelos de clasificación vimos el modelo lineal de probabilidad, y los modelos Logit.\n",
    "\n",
    "\n",
    "* Aunque los segundos no son lineales, una transformación rápida permite ver que el *log-odds* es lineal en los párametros\n",
    "\n",
    "$$\n",
    "\\ln \\left(\\frac{Prob(D=1 | X)}{Prob(D=0 | X)} \\right) = \\ln \\left(\\frac{e^{x'\\beta}/(1+e^{x'\\beta})}{1/(1+e^{x'\\beta})} \\right) = x'\\beta\n",
    "$$\n",
    "\n"
   ]
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    "# Modelos de regresión no lineales\n",
    "\n",
    "* La clase de modelos que vamos a analizar ahora se pueden escribir como\n",
    "\n",
    "\n",
    "$$\n",
    "y_i = h(\\mathbf{x}_i, \\mathbf{\\beta}) + \\epsilon_i\n",
    "$$\n",
    "\n",
    "Siguiendo a Greene, vamos a definir:\n",
    "\n",
    "> ** Un modelo no lineal es aquel para el que las condiciones de primer orden de la minimización por mínimos cuadrados son funciones no lineales de los parámetros.**\n",
    "\n",
    "\n",
    "En sus cursos de econometría verán que hay que imponer ciertos supuestos para que todo funcione bien.  Acá los menciono únicamente.  Pueden encontrar mayor información en Greene.\n",
    "\n",
    "* **Supuesto 1**: forma funcional de la esperanza condicional:\n",
    "$$\n",
    "    E[y_i|x_i] = h(x_i,\\beta) \\hspace{3mm}\\text{para $i=1,\\cdots, N$}\n",
    "$$\n",
    "\n",
    "\n",
    "* **Supuesto 2**: Los parámetros son identificables\n",
    "$$\n",
    "\\beta^0 \\neq \\beta \\implies h(\\mathbf{x}_i,\\beta^0) \\neq h(\\mathbf{x}_i,\\beta) \\hspace{3mm}\\text{para todo $\\mathbf{x}_i$}\n",
    "$$\n",
    "\n",
    "\n",
    "* **Supuesto 3**: Errores con media cero\n",
    "$$\n",
    "E[\\epsilon_i|h(\\mathbf{x}_i,\\beta)] = 0\n",
    "$$\n",
    "\n",
    "\n",
    "* **Supuesto 4**: Homoscedasticidad y no autocorrelación *condicional* de los errores.\n",
    "\n",
    "\n",
    "* **Supuesto 5**: exogeneidad de $\\mathbf{x}_i$ y los errores. \n"
   ]
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    "# El modelo linealizado\n",
    "\n",
    "\n",
    "* Cuando estudiamos el modelo Logit, vimos que es fácil de resolver haciendo una aproximación de primer orden de las condiciones de primer orden.\n",
    "\n",
    "\n",
    "* Acá vamos a hacer lo mismo pero con la función $h(\\mathbf{x}_i,\\beta)$\n",
    "\n",
    "\n",
    "$$\n",
    "h(\\mathbf{x},\\beta) \\approx h(\\mathbf{x},\\beta^0) + \\sum_{p=1}^{P+1} \\frac{\\partial h(\\mathbf{x},\\beta^0)}{\\partial \\beta_p} (\\beta_p - \\beta^0_{p})\n",
    "$$\n",
    "\n",
    "\n",
    "* Igual que antes, vamos a asumir que $\\beta_0$ lo conocemos, así que agrupando, tenemos\n",
    "\n",
    "\n",
    "$$\n",
    "h(\\mathbf{x},\\beta) \\approx \\left[ h(\\mathbf{x},\\beta^0) - \\sum_{p=1}^{P+1} \\frac{\\partial h(\\mathbf{x},\\beta^0)}{\\partial \\beta_p} \\beta^0_{p} \\right] + \\sum_{p=1}^{P+1} \\frac{\\partial h(\\mathbf{x},\\beta^0)}{\\partial \\beta_p} \\beta_p\n",
    "$$\n",
    "\n",
    "\n",
    "* Definamos $x^0_p = \\frac{\\partial h(\\mathbf{x},\\beta^0)}{\\partial \\beta_p}$, así que\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray*}\n",
    "h(\\mathbf{x},\\beta) &\\approx& \\left[ h(\\mathbf{x},\\beta^0) - \\sum_{p=1}^{P+1} x^0_p \\beta^0_{p} \\right] + \\sum_{p=1}^{P+1} x^0_p \\beta_p \\\\\n",
    "&=& (h^0  - {\\mathbf{x}^0}' \\beta^0) + {\\mathbf{x}^0}' \\beta\n",
    "\\end{eqnarray*}\n",
    "$$\n"
   ]
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    "# El modelo linealizado (cont.)\n",
    "\n",
    "\n",
    "* Así que\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray*}\n",
    "y - y^0 &\\approx& {\\mathbf{x}^0}'\\beta + \\epsilon \\\\\n",
    "&=& {\\mathbf{x}^0}'\\beta + \\epsilon^0\n",
    "\\end{eqnarray*}\n",
    "$$\n",
    "\n",
    "donde $y^0 := h^0 - {\\mathbf{x}^0}'\\beta^0$, y $\\epsilon^0$ contiene el término de error estocástico, y el error en la aproximación de Taylor.\n",
    "\n",
    "\n",
    "> **La buena noticia es que si tenemos $\\beta^0$, podemos estimar $\\beta$ en una regresión lineal**.\n",
    "\n",
    "* Este procedimiento debre sonar familiar ya: ¿cuál puede ser $\\beta^0$?\n",
    "\n"
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    "# Actualización de Newton:\n",
    "\n",
    "* Ya sabemos que la estimación actual se puede estimar por OLS:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray*}\n",
    "\\beta_{(k)} &=& ({X^0}'X^0)^{-1}{X^0}'(y-y0)\\\\\n",
    "&=& ({X^0}'X^0)^{-1}{X^0}'(y-h^0 + X^0 \\beta_{(k-1)}) \\\\\n",
    "&=& \\beta_{(k)} + \\Delta_k\n",
    "\\end{eqnarray*}\n",
    "$$\n",
    "\n",
    "* donde $X^0$ es la matriz de **seudoregresores** que se obtiene así\n",
    "\n",
    "$$\n",
    "x^0_{ip} = \\frac{\\partial h(\\mathbf{x}_i,\\beta_{(k-1)})}{\\partial \\beta_p}\n",
    "$$\n",
    "\n",
    "y hay que tener cuidado porque $\\Delta_k = \\Delta_k(\\beta_{(k)})$\n",
    "\n",
    "\n",
    "* No vamos a decir más sobre este tema.  \n",
    "\n",
    "\n",
    "* Pero es un método que es fácil de ejecutar (ya han tenido que programar cosas más difíciles) así que ya debe estar en su caja de herramientas."
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    "# No linealidad y bases polinomiales\n",
    "\n",
    "\n",
    "* Ya vimos que la función que deseamos aprender $f(x)$ se puede aproximar por medio de una base de polinomios:\n",
    "\n",
    "$$\n",
    "\\{1, x_1, \\cdots, x_p, x_1^2, \\cdots, x_p^2, x_1 x_2, \\cdots \\}\n",
    "$$\n",
    "\n",
    "\n",
    "* En la práctica es usual utilizar hasta polinomios de grado dos o tres con las respectivas interacciones.\n",
    "\n",
    "\n",
    "* Este método es flexible y generalmente funciona bien, pero a medida que incrementamos el orden de los polinomios tendemos a perder muchos grados de libertad.\n",
    "\n",
    "\n",
    "* Las personas interesadas pueden ver, por ejemplo, el [Cap.76 del Handbook of Econometrics](http://www.sciencedirect.com/science/article/pii/S157344120706076X)"
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    "# Expansiones más generales\n",
    "\n",
    "* En general, podemos aproximar $f(x)$ así\n",
    "\n",
    "$$\n",
    "f(X) = \\sum_{m=1}^M \\beta_m h_m(X)\n",
    "$$\n",
    "\n",
    "\n",
    "* Donde las funciones $\\{h_m()\\}$ es una base de funciones que aproxima la función $f(x)$.\n",
    "\n",
    "\n",
    "* Alternativamente podemos hacer una interpolación polinomial a trozos (piecewise polynomial)."
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    "# Piecewise polynomials\n",
    "\n",
    "* Una interpolación a trozos se obtiene al dividir el dominio de la función en intervalos contiguos.\n",
    "\n",
    "\n",
    "* En cada intervalo se ajusta un polinomio diferente.\n",
    "\n",
    "\n",
    "* Por ejemplo, si dividemos el dominio con dos puntos de corte $\\xi_1, \\xi_2$, una interpolación constante se obtiene así:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray*}\n",
    "h_1(X) &=& I(X<\\xi_1)\\\\\n",
    "h_2(X) &=& I(\\xi_1 \\geq X \\geq \\xi_2)\\\\\n",
    "h_3(X) &=& I(X\\geq \\xi_2)\\\\\n",
    "\\end{eqnarray*}\n",
    "$$\n",
    "\n",
    "\n",
    "* En estos casos, $\\hat{\\beta}_k = \\overline{y}_{I_k}$.\n",
    "\n",
    "donde $I_k$ es la correspondiente variable indicadora definida arriba.\n",
    "\n",
    "\n",
    "* Pero cómo lo estimamos?\n",
    "\n",
    "\n",
    "* **Método 1**: Tres regresiones. $y_i = \\beta_k I_{ik} + \\epsilon_i$\n",
    "\n",
    "\n",
    "* **Método 2**: Una regresión:  $y_i = \\beta_1 I_{i1} + \\beta_2 I_{i2} + \\beta_3 I_{i3} + \\epsilon_i$"
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       "qrTf7Cqk7RvnA+dk9wn4drbZzHTgrKzs8BLgQw3PN9zV/YnA7yPiGkm3AtdL+mVE3NX2b8JsEJeS\n",
       "NstImhARS5T2jP0d8PGBDe/NepVbAmbLfV/S9qRtGH/oBGD9wC0BM7M+5oFhM7M+5iRgZtbHnATM\n",
       "zPqYk4CZWR9zEjAz62NOAmZmfez/A20vve2zKkb/AAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc15e9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Simulemos un modelo\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "np.random.seed(6181015)\n",
    "\n",
    "# vamos a simular una función trigonométrica\n",
    "N = 100\n",
    "x = np.linspace(0,2*np.pi,N).reshape((N,1))\n",
    "eps = 0.5*np.random.randn(N,1)\n",
    "y = np.sin(x) + eps\n",
    "plt.scatter(x,y, color='b',alpha=0.5)\n",
    "plt.plot(x,np.sin(x), color='r', alpha=0.5)\n",
    "plt.xlabel('regresor x')\n",
    "plt.ylabel('y', rotation=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0xc8b0550>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x40cc7b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Dividamos el dominio en tres intervalos iguales\n",
    "pctiles = np.percentile(x, [33.3,66.6])\n",
    "# booleanos para obtener la interpolación\n",
    "ind1    = x<pctiles[0]\n",
    "ind2    = np.logical_and(x>=pctiles[0],x<=pctiles[1])\n",
    "ind3    = x>pctiles[1]\n",
    "# Obtengamos la interpolación\n",
    "h1 = y[ind1].mean()\n",
    "h2 = y[ind2].mean()\n",
    "h3 = y[ind3].mean()\n",
    "plt.scatter(x,y, color='b',alpha=0.5)\n",
    "# primera función\n",
    "plt.plot([x.min(), pctiles[0]],[h1,h1], color='r')\n",
    "# segunda función\n",
    "plt.plot([pctiles[0],pctiles[1]],[h2,h2], color='r')\n",
    "# tercera función\n",
    "plt.plot([pctiles[1],x.max()],[h3,h3], color='r')\n",
    "# las líneas divisorias\n",
    "plt.plot([pctiles[0],pctiles[0]],[-2,2],ls='--', color='0.75')\n",
    "plt.plot([pctiles[1],pctiles[1]],[-2,2],ls='--', color='0.75')\n",
    "plt.axis([-0.5,6.5,-2,2])\n",
    "plt.title(u'Interpolación con funciones constantes')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Interpolación con polinomios de orden uno\n",
    "\n",
    "\n",
    "* La anterior interpolación es útil, pero podemos hacer algo mejor.\n",
    "\n",
    "\n",
    "* Queremos buscar funciones $h_k(X) = \\alpha_k + \\beta_k X$, así que podemos hacer regresiones lineales en cada intervalo.\n",
    "\n",
    "\n",
    "* O podemos hacer una única regresión:\n",
    "\n",
    "$$\n",
    "y_i = \\alpha_0 + \\alpha_1 X_i + \\sum_{k=1}^{K-1} \\beta_k D_{ik} + \\sum_{k=1}^{K-1} \\delta_k D_{ik} X_{i} + \\epsilon_i \n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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       "AElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xd922b70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import sys\n",
    "#sys.path.append('//Users//danielvaughan//Documents//Python Scripts//')\n",
    "sys.path.append('C:\\\\Users\\\\a3940004.EDIFICIOS\\\\Documents\\\\Python Scripts\\\\')\n",
    "import olsdan as ols\n",
    "\n",
    "plt.scatter(x,y, color='b',alpha=0.5)\n",
    "# las líneas divisorias\n",
    "plt.plot([pctiles[0],pctiles[0]],[-2,2],ls='--', color='0.75')\n",
    "plt.plot([pctiles[1],pctiles[1]],[-2,2],ls='--', color='0.75')\n",
    "plt.axis([-0.5,6.5,-2,2])\n",
    "plt.title(u'Interpolación con polinomios de orden 1')\n",
    "# ahora vamos a hacer regresiones lineales en cada uno\n",
    "ind_mat = np.concatenate((ind1.reshape((N,1)),\n",
    "                         ind2.reshape((N,1)),\n",
    "                         ind3.reshape((N,1))),axis=1)\n",
    "for i in range(3):\n",
    "    # la variable dependiente\n",
    "    y_i = y[ind_mat[:,i]]\n",
    "    n_i = y_i.shape[0]\n",
    "    y_i = y_i.reshape((n_i,1))\n",
    "    x_i = x[ind_mat[:,i]].reshape((n_i,1))\n",
    "    # la matrix x\n",
    "    xmat_i = np.concatenate((np.ones((n_i,1)), x_i),axis=1)\n",
    "    # obtengamos la estimación \n",
    "    olsi = ols.ols_dan(y_i, xmat_i)\n",
    "    premat = pd.DataFrame(np.concatenate((olsi.xmat[:,1], olsi.yest()),axis=1), \n",
    "                          columns=['x','yhat']).sort(columns=['x'], ascending=True)\n",
    "    plt.plot(premat.x, premat.yhat, color='r')\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0xe3a6e10>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "AuLK3XEcp4C4cnccxykg/x9w8/2OEj2GZQAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc6678d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# pero no tenemos que hacer tres regresiones\n",
    "plt.scatter(x,y, color='b',alpha=0.5)\n",
    "# las líneas divisorias\n",
    "plt.plot([pctiles[0],pctiles[0]],[-2,2],ls='--', color='0.75')\n",
    "plt.plot([pctiles[1],pctiles[1]],[-2,2],ls='--', color='0.75')\n",
    "plt.axis([-0.5,6.5,-2,2])\n",
    "plt.title(u'Interpolación con polinomios de orden 1')\n",
    "# ahora vamos a hacer regresiones lineales en cada uno\n",
    "ind_mat = np.concatenate((ind1.reshape((N,1)),\n",
    "                         ind2.reshape((N,1)),\n",
    "                         ind3.reshape((N,1))),axis=1)\n",
    "# tenemos que incluir las dummies y las interacciones\n",
    "# Cuidado: broadcasting\n",
    "int_mat = ind_mat*x.reshape((N,1))\n",
    "xmat = np.concatenate((np.ones((N,1)),ind_mat, int_mat), axis=1)\n",
    "olsx = ols.ols_dan(y,xmat)\n",
    "\n",
    "plt.plot(x,olsx.yest(), color='r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x19576e80>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "/H+nz/yS89hWpwAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xe85ba20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Alternativamente podemos dejar el grupo 1 como grupo base o de referencia\n",
    "\n",
    "plt.scatter(x,y, color='b',alpha=0.5)\n",
    "# las líneas divisorias\n",
    "plt.plot([pctiles[0],pctiles[0]],[-2,2],ls='--', color='0.75')\n",
    "plt.plot([pctiles[1],pctiles[1]],[-2,2],ls='--', color='0.75')\n",
    "plt.axis([-0.5,6.5,-2,2])\n",
    "plt.title(u'Interpolación con polinomios de orden 1')\n",
    "# dejemos el grupo 1 como base\n",
    "ind_mat = np.concatenate((\n",
    "                         ind2.reshape((N,1)),\n",
    "                         ind3.reshape((N,1))),axis=1)\n",
    "# tenemos que incluir las dummies y las interacciones\n",
    "# Cuidado: broadcasting\n",
    "int_mat = ind_mat*x.reshape((N,1))\n",
    "xmat = np.concatenate((np.ones((N,1)),x.reshape((N,1)), ind_mat, int_mat), axis=1)\n",
    "olsx = ols.ols_dan(y,xmat)\n",
    "\n",
    "plt.plot(x,olsx.yest(), color='r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# ¿Podemos restringir a que sea continua?\n",
    "\n",
    "* La anterior regresión que incluimos fue\n",
    "$$\n",
    "y = \\alpha_0 + \\alpha_1 X + \\beta_1 I_2 + \\beta_2 I_3 + \\delta_1 (I_2\\times X) + \\delta_2 (I_3\\times X) + \\epsilon\n",
    "$$\n",
    "donde $I_k$ es la variable indicadora correspondiente.\n",
    "\n",
    "\n",
    "* ¿Qué restricciones hay que imponer para que sea continua?\n",
    "\n",
    "\n",
    "* Tomemos el primer caso:  En el primer intervalo $I_2 = I_3 = 0$.  En el segundo $I_3=0$\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray*}\n",
    "\\alpha_0 + \\alpha_1 \\xi_1 &=& (\\alpha_0 + \\beta_1) + (\\alpha_1 + \\delta_1) \\xi_1 \\\\\n",
    "&\\iff& \\beta_1 = -\\delta_1 \\xi_1\n",
    "\\end{eqnarray*}\n",
    "$$\n",
    "\n",
    "* En el segundo caso pueden verificar que:\n",
    "$$\n",
    "\\beta_2 = \\delta_1 \\underbrace{(\\xi_2 - \\xi_1)}_{\\Delta_{21}} - \\delta_2 \\xi_2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "\n",
    "* Incluyamos estas restricciones:\n",
    "$$\n",
    "\\begin{eqnarray*}\n",
    "y &=& \\alpha_0 + \\alpha_1 X + \\beta_1 I_2 + \\beta_2 I_3 + \\delta_1 (I_2\\times X) + \\delta_2 (I_3\\times X) + \\epsilon\\\\\n",
    "&=& \\alpha_0 + \\alpha_1 X + (-\\delta_1 \\xi_1) I_2 + (\\delta_1 \\Delta_{21} - \\delta_2 \\xi_2) I_3 + \\delta_1 (I_2\\times X) + \\delta_2 (I_3\\times X) + \\epsilon\\\\\n",
    "&=& \\eta_0 + \\eta_1 X +  \\eta_2 I_2 (X-\\xi_1) + \\eta_3 I_3 (X - \\xi_2) + \\epsilon\\\\\n",
    "&=& \\eta_0 + \\eta_1 X +  \\eta_2 (X-\\xi_1)_+ + \\eta_3 (X - \\xi_2)_+ + \\epsilon\\\\ \n",
    "\\end{eqnarray*}\n",
    "$$\n",
    "\n",
    "\n",
    "* Así que para incluir las restricciones sólo tenemos que cambiar la base original por:\n",
    "$$\n",
    "h_1(X) = 1, \\hspace{2mm} h_2(X) = X,  \\hspace{2mm} h_3(X) = (X-\\xi_1)_+, \\hspace{2mm} h_4(X) = (X-\\xi_2)_+\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1b362a90>]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1b45bbe0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Alternativamente podemos dejar el grupo 1 como grupo base o de referencia\n",
    "\n",
    "plt.scatter(x,y, color='b',alpha=0.5)\n",
    "# las líneas divisorias\n",
    "plt.plot([pctiles[0],pctiles[0]],[-2,2],ls='--', color='0.75')\n",
    "plt.plot([pctiles[1],pctiles[1]],[-2,2],ls='--', color='0.75')\n",
    "plt.axis([-0.5,6.5,-2,2])\n",
    "plt.title(u'Interpolación con polinomios de orden 1 y continuidad')\n",
    "# Construyamos las nuevas bases\n",
    "dum_1 = x>=pctiles[0]\n",
    "dum_2 = x>=pctiles[1]\n",
    "x = x.reshape((N,1))\n",
    "x_pos1 = dum_1*(x-pctiles[0])\n",
    "x_pos2 = dum_2*(x-pctiles[1])\n",
    "\n",
    "xmat = np.concatenate((np.ones((N,1)),x, x_pos1, x_pos2), axis=1)\n",
    "olsx = ols.ols_dan(y,xmat)\n",
    "\n",
    "plt.plot(x,olsx.yest(), color='r')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Qué pasa si queremos una interpolación más suave?\n",
    "\n",
    "\n",
    "* Podemos ajustar polinomios de orden mayor.  Por ejemplo, polinomios de orden 3.\n",
    "\n",
    "\n",
    "* Un **spline cúbico** es una función cúbica a trozos, que es *continua*, *continuamente diferenciable $C^2$* (primera y segundas derivadas son continuas).\n",
    "\n",
    "\n",
    "* En nuestro caso, partimos del modelo:\n",
    "\n",
    "$$\n",
    "y = \\alpha_0 + \\alpha_1 X + \\alpha_2 X^2 + \\alpha_3 X^3 + \\sum_{k>1} D_k + \\sum_{k>1} D_k X + \\sum_{k>1} D_k X^2 + \\sum_{k>1} D_k X^3 + \\epsilon\n",
    "$$\n",
    "\n",
    "e imponemos las restricciones de continuidad (igual que en el caso anterior) y de continuidad de la primera y segundas derivadas.\n",
    "\n",
    "\n",
    "* Pueden intentarlo por su cuenta, que al final llegarán a un modelo con las siguiente base de funciones:\n",
    "\n",
    "$$\n",
    "h_1(X) = 1, \\hspace{2mm} h_2(X) = X, \\hspace{2mm} h_3(X) = X^2, \\hspace{2mm} h_4(X) = X^3\\\\\n",
    "h_5(X) = (X-\\xi_1)^3_+, \\hspace{2mm}  h_6(X) = (X-\\xi_2)^3_+\n",
    "$$\n",
    "\n",
    "\n",
    "* Es una muy buena noticia:  \n",
    "> Para hacer interpolación con splines cúbicos sólo necesitan incluir los monomios hasta orden 3 $(h_1, \\cdots, h_4)$, tantos términos adicionales como el tamaño del grid que generaron $(\\xi_1, \\xi_2, \\xi_3, \\cdots, \\xi_K)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Annotation at 0x200b65f8>"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "RU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ff25198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Splines cúbicos\n",
    "\n",
    "plt.scatter(x,y, color='b',alpha=0.5)\n",
    "# las líneas divisorias\n",
    "plt.plot([pctiles[0],pctiles[0]],[-2,2],ls='--', color='0.75')\n",
    "plt.plot([pctiles[1],pctiles[1]],[-2,2],ls='--', color='0.75')\n",
    "plt.axis([-0.5,6.5,-2,2])\n",
    "plt.title(u'Interpolación con polinomios de orden 1 y continuidad')\n",
    "# Construyamos las nuevas bases\n",
    "dum_1 = x>=pctiles[0]\n",
    "dum_2 = x>=pctiles[1]\n",
    "x = x.reshape((N,1))\n",
    "x_pos1 = dum_1*(x-pctiles[0])**3\n",
    "x_pos2 = dum_2*(x-pctiles[1])**3\n",
    "\n",
    "xmat = np.concatenate((np.ones((N,1)),x, np.power(x,2), np.power(x,3),\n",
    "                       x_pos1, x_pos2), axis=1)\n",
    "olsx = ols.ols_dan(y,xmat)\n",
    "\n",
    "plt.plot(x,olsx.yest(), color='c', lw=2)\n",
    "plt.plot(x,np.sin(x), color='r', lw=0.5)\n",
    "plt.annotate(\"Verdadera\",\n",
    "            xy=(2.1, np.sin(2.1)), xycoords='data',\n",
    "            xytext=(3, 1.5), color='r',\n",
    "            arrowprops=dict(arrowstyle=\"->\",\n",
    "                            connectionstyle=\"arc3\", ec='r'),\n",
    "            )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x21619400>]"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "/g/yawASGkwmDAAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x216192e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Qué pasa si aproximamos con más nodos?\n",
    "for kk in range(2,10):\n",
    "    K = kk\n",
    "    nod_grid = np.percentile(x, np.linspace(10, 90, K))\n",
    "    # generamos las funciones\n",
    "    xpos_mat = np.zeros((N,K))\n",
    "    for k in range(K):\n",
    "        xpos_mat[:,k] = ((x>nod_grid[k])*(x-nod_grid[k])**3).flatten()\n",
    "\n",
    "    # lo demás es igual\n",
    "    xmat = np.concatenate((np.ones((N,1)),x, np.power(x,2), np.power(x,3),\n",
    "                           xpos_mat), axis=1)\n",
    "    olsx = ols.ols_dan(y,xmat)\n",
    "\n",
    "    plt.plot(x,olsx.yest(), color='b', lw=1, alpha=0.5)\n",
    "\n",
    "plt.plot(x,np.sin(x), color='r', lw=2, alpha=0.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Modelos Aditivos Generalizados (GAMS)\n",
    "\n",
    "*  Un GAM tiene la siguiente forma:\n",
    "\n",
    "$$\n",
    "E(y|X) = \\alpha_0 + f_1(x_1) + f_2(x_2) + \\cdots f_p(x_p) + \\epsilon \n",
    "$$\n",
    "\n",
    "\n",
    "* El caso de regresión lineal es uno donde $f_k(x_k) = \\alpha_k x_k$.\n",
    "\n",
    "\n",
    "* Así que estos modelos son más generales y pueden capturar relaciones no lineales que son difíciles de capturar con expansiones polinomiales.\n",
    "\n",
    "\n",
    "* **No conocemos** las funciones $f_k()$, aunque impondremos que sean *suaves* (continuamente diferenciables).\n",
    "\n",
    "\n",
    "* Podríamos aproximar cada función $f_k(x_k) = \\sum_m b_m h_m(x_k)$, y estimar los coeficientes por OLS.\n",
    "\n",
    "\n",
    "* Vamos a utilizar un método diferente: queremos estimar simultáneamente las funciones, teniendo en cuenta que cada una captura el efecto *neto* del regresor correspondiente sobre $y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Algoritmo de Backfitting para GAMS\n",
    "\n",
    "1. Inicializamos las funciones que queremos estimar:\n",
    "$$\n",
    "\\hat{\\alpha} = \\bar{y}\\\\\n",
    "\\hat{f}_j = 0 \\hspace{2mm} \\text{para todo $j$}\n",
    "$$\n",
    "\n",
    "2. Repetir para $j=1,2, \\cdots, p, 1,2, \\cdots, p, \\cdots$ hasta converger:\n",
    "\n",
    "$$\n",
    "\\hat{f_j}  \\leftarrow \\mathcal{S}_j \\left[y_i - \\hat{\\alpha} - \\sum_{k \\neq j} \\hat{f}_k(x_{ik}) \\right] \\\\\n",
    "\\hat{f_j} \\leftarrow  \\hat{f_j} - \\frac{1}{N}\\sum_{i} \\hat{f}_j(xij)\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Backfitting para GAMS en palabras\n",
    "\n",
    "\n",
    "* Queremos estimar simultáneamente el parámetro $\\alpha$ y las funciones.\n",
    "\n",
    "\n",
    "* Forzamos a que $\\hat{\\alpha}$ sea el promedio de la variable dependiente:\n",
    "\n",
    "> Esto explica la segunda parte del segundo paso: para que $\\hat{\\alpha} = \\bar{y}$ es necesario $\\bar{f_j} = 0$ para todo $j$.\n",
    "\n",
    "\n",
    "* Vamos a utilizar la estimación actual como si fuera óptima:\n",
    "    \n",
    "    * Obtenemos los residuos *excluyendo* la j-ésima función: $r_{-j}$\n",
    "    \n",
    "    * Utilizamos un *scatterplot smoother* de la relación entre $r_{-j}$ y $x_j$.\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# ¿Qué es un *scatterplot smoother*?\n",
    "\n",
    "\n",
    "* Primero: qué es un scatterplot?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x21936898>"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x20a3fac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x,y, color='b',alpha=0.5)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Segundo: dado un scatterplot, que es un *smoother*?\n",
    "\n",
    "\n",
    "* Justo eso: tenemos una relación entre $y$ y $x$.\n",
    "\n",
    "\n",
    "* Un smoother no es más que una función que suaviza esta relación.\n",
    "\n",
    "\n",
    "* En estas notas ya vimos una: **splines cúbicos**.\n",
    "\n",
    "\n",
    "* Se les ocurre otra?  OLS.  Ya entendemos por qué aprendimos a utilizar esa *primera versión de un algoritmo de backfitting* hace varias clases.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Creemos una función:\n",
    "def spline_smoother(y,x,K):\n",
    "    '''\n",
    "    Cubic-spline smoother.\n",
    "    y: variable dendiente\n",
    "    x: variable independiente\n",
    "    K: número de nodos para hacer \n",
    "    '''\n",
    "    N = y.shape[0]\n",
    "    y = y.reshape((N,1))\n",
    "    x = x.reshape((N,1))\n",
    "    # grid\n",
    "    nod_grid = np.percentile(x, np.linspace(10, 90, K))\n",
    "    # matriz con partes positivas\n",
    "    xpos_mat = np.zeros((N,K))\n",
    "    for k in range(K):\n",
    "        xpos_mat[:,k] = ((x>nod_grid[k])*(x-nod_grid[k])**3).flatten()\n",
    "\n",
    "    # Estamos listos para concatenar\n",
    "    xmat = np.concatenate((np.ones((N,1)),x, np.power(x,2), np.power(x,3),\n",
    "                           xpos_mat), axis=1)\n",
    "    # estimamos\n",
    "    olsx = ols.ols_dan(y,xmat)\n",
    "    # lo único que necesitamos es el yest\n",
    "    yest = np.asarray(olsx.yest())\n",
    "    return yest, x\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2225c470>]"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "6M3M2pwTvZlZm/v/3CoHTpc9gOIAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2225c320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# probemos que funciona:\n",
    "ycub, xcub = spline_smoother(y,x,10)\n",
    "plt.scatter(x,y, alpha=0.5)\n",
    "plt.plot(xcub, ycub, color='r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Unnamed: 0</th>\n",
       "      <th>Income</th>\n",
       "      <th>Limit</th>\n",
       "      <th>Rating</th>\n",
       "      <th>Cards</th>\n",
       "      <th>Age</th>\n",
       "      <th>Education</th>\n",
       "      <th>Gender</th>\n",
       "      <th>Student</th>\n",
       "      <th>Married</th>\n",
       "      <th>Ethnicity</th>\n",
       "      <th>Balance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td> 1</td>\n",
       "      <td>  14.891</td>\n",
       "      <td> 3606</td>\n",
       "      <td> 283</td>\n",
       "      <td> 2</td>\n",
       "      <td> 34</td>\n",
       "      <td> 11</td>\n",
       "      <td>   Male</td>\n",
       "      <td>  No</td>\n",
       "      <td> Yes</td>\n",
       "      <td> Caucasian</td>\n",
       "      <td> 333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td> 2</td>\n",
       "      <td> 106.025</td>\n",
       "      <td> 6645</td>\n",
       "      <td> 483</td>\n",
       "      <td> 3</td>\n",
       "      <td> 82</td>\n",
       "      <td> 15</td>\n",
       "      <td> Female</td>\n",
       "      <td> Yes</td>\n",
       "      <td> Yes</td>\n",
       "      <td>     Asian</td>\n",
       "      <td> 903</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td> 3</td>\n",
       "      <td> 104.593</td>\n",
       "      <td> 7075</td>\n",
       "      <td> 514</td>\n",
       "      <td> 4</td>\n",
       "      <td> 71</td>\n",
       "      <td> 11</td>\n",
       "      <td>   Male</td>\n",
       "      <td>  No</td>\n",
       "      <td>  No</td>\n",
       "      <td>     Asian</td>\n",
       "      <td> 580</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td> 4</td>\n",
       "      <td> 148.924</td>\n",
       "      <td> 9504</td>\n",
       "      <td> 681</td>\n",
       "      <td> 3</td>\n",
       "      <td> 36</td>\n",
       "      <td> 11</td>\n",
       "      <td> Female</td>\n",
       "      <td>  No</td>\n",
       "      <td>  No</td>\n",
       "      <td>     Asian</td>\n",
       "      <td> 964</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td> 5</td>\n",
       "      <td>  55.882</td>\n",
       "      <td> 4897</td>\n",
       "      <td> 357</td>\n",
       "      <td> 2</td>\n",
       "      <td> 68</td>\n",
       "      <td> 16</td>\n",
       "      <td>   Male</td>\n",
       "      <td>  No</td>\n",
       "      <td> Yes</td>\n",
       "      <td> Caucasian</td>\n",
       "      <td> 331</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Unnamed: 0   Income  Limit  Rating  Cards  Age  Education  Gender Student  \\\n",
       "0           1   14.891   3606     283      2   34         11    Male      No   \n",
       "1           2  106.025   6645     483      3   82         15  Female     Yes   \n",
       "2           3  104.593   7075     514      4   71         11    Male      No   \n",
       "3           4  148.924   9504     681      3   36         11  Female      No   \n",
       "4           5   55.882   4897     357      2   68         16    Male      No   \n",
       "\n",
       "  Married  Ethnicity  Balance  \n",
       "0     Yes  Caucasian      333  \n",
       "1     Yes      Asian      903  \n",
       "2      No      Asian      580  \n",
       "3      No      Asian      964  \n",
       "4     Yes  Caucasian      331  "
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# importemos los datos\n",
    "data = pd.read_csv(u'D:\\\\Míos\\\\Clase CIDE\\\\Datasets\\\\Credit.csv')\n",
    "data.head(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['Caucasian' 'Asian' 'African American']\n",
      "(400L, 6L)\n"
     ]
    }
   ],
   "source": [
    "# Utilicemos sólamente variables continuas\n",
    "# primero organicemos los datos: las variables cuantitativas están bien.  Las cualitativas toca transformarlas\n",
    "N = data.shape[0]\n",
    "dum_hombre     = np.asarray(data.Gender == ' Male').reshape((N,1))\n",
    "dum_estudiante = np.asarray(data.Student == 'Yes').reshape((N,1))\n",
    "dum_casado     = np.asarray(data.Married == 'Yes').reshape((N,1))\n",
    "# Para etnicidad debemos excluir una variable que sirva como referencia:\n",
    "# tenemos tres categorías: vamos a excluir la tercera: \"African American\"\n",
    "print data.Ethnicity.unique()\n",
    "dum_ethnic = np.asarray(pd.get_dummies(data.Ethnicity).iloc[:,:-1])\n",
    "# etiquetas de X\n",
    "labs_x = np.array(['Score','Ingreso',u'Límite de Crédito',u'Número Tarjetas','Edad',u'Educación'])\n",
    "#,'Hombre','Estudiante','Casado','Caucasian','Asian'])\n",
    "# incluyamos todas las variables:\n",
    "prexmat = np.concatenate((\n",
    "                        np.asarray(data.Rating).reshape((N,1)),\n",
    "                        np.asarray(data.Income).reshape((N,1)),\n",
    "                         np.asarray(data.Limit).reshape((N,1)),\n",
    "                         np.asarray(data.Cards).reshape((N,1)),\n",
    "                         np.asarray(data.Age).reshape((N,1)),\n",
    "                         np.asarray(data.Education).reshape((N,1)),\n",
    "                         ),axis=1)\n",
    "saldo = np.asarray(data.Balance).reshape((N,1))\n",
    "print prexmat.shape\n",
    "nvar = prexmat.shape[1]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Listos para programar el backfitting:\n",
    "# inicicialicemos la distancia\n",
    "dist = 1\n",
    "Diff    = [1]\n",
    "crit = 0.01\n",
    "# Inicializamos las funciones a cero\n",
    "Fmat = np.zeros((N,7))\n",
    "# la primera será la constante\n",
    "Fmat[:,0] = np.nanmean(saldo)*np.ones((N))\n",
    "while dist>crit:\n",
    "    # Actualicemos versión anterior de funciones\n",
    "    old_func = Fmat.copy()\n",
    "    # Ahora:\n",
    "    # 1. Residuos exluyendo k\n",
    "    # 2. Smoother\n",
    "    # 3. Media cero y guardar\n",
    "    for i in range(1,7):\n",
    "        # Excluyamos i\n",
    "        ind_i = np.setdiff1d(range(7),np.array([i]))\n",
    "        # Obtengamos residuos\n",
    "        # Fácil: es un modelo aditivo!\n",
    "        res_i = saldo - Fmat[:,ind_i].sum(axis=1).reshape((N,1))\n",
    "        # Smoother: 2 nodos\n",
    "        K = 2\n",
    "        f_i,xl_i   = spline_smoother(res_i,prexmat[:,i-1].reshape((N,1)),K)\n",
    "        Fmat[:,i]  = f_i.flatten()\n",
    "    # Acabamos iteración actual\n",
    "    # 1. medias:\n",
    "    Fmat[:,1:] -= np.nanmean(Fmat[:,1:],axis=0).reshape((1,nvar))\n",
    "    # Compute square distances across columns and use the max\n",
    "    dist = ((Fmat[:,1:]-old_func[:,1:])**2).sum(axis=0).max()\n",
    "    Diff.append(dist)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
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       "TkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x24f25f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "for k in range(6):\n",
    "    plt.subplot(2,3,k+1)\n",
    "    # organicemos primero\n",
    "    predata = pd.DataFrame(np.concatenate((prexmat[:,k].reshape((N,1)), Fmat[:,k+1].reshape((N,1))),axis=1), columns=['x','fx']\n",
    "                          ).sort(columns='x',ascending=True)\n",
    "    plt.plot(predata.x, saldo.mean() + predata.fx)\n",
    "    # para referencia grafiquemos ols:\n",
    "    olsk = ols.ols_dan(saldo, np.concatenate((np.ones((N,1)),prexmat[:,k].reshape((N,1))),axis=1))\n",
    "    predata = pd.DataFrame(np.concatenate((prexmat[:,k].reshape((N,1)), olsk.yest()),axis=1), columns=['x','yhat']\n",
    "                          ).sort(columns='x',ascending=True)\n",
    "    plt.plot(predata.x,predata.yhat)\n",
    "    tit_txt = u'Saldo y ' + labs_x[k]\n",
    "    plt.title(tit_txt)\n",
    "plt.tight_layout()\n"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}