Skewed Student-t Hansen (1994).ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import gamma, sqrt, pi\n",
    "from scipy.stats import t\n",
    "import numpy as np\n",
    "\n",
    "import plotly.express as px\n",
    "import plotly.figure_factory as ff\n",
    "\n",
    "import matplotlib.pylab as plt\n",
    "import seaborn as sns\n",
    "\n",
    "import plotly.graph_objects as go\n",
    "from plotly.subplots import make_subplots\n",
    "\n",
    "import pandas as pd\n",
    "\n",
    "from ipywidgets import FloatSlider, IntSlider, VBox, HBox, Checkbox\n",
    "\n",
    "import qgrid"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Skewed Student-t Inverse CDF\n",
    "<img src=\"Images/skt_definitions.png\"/>\n",
    "<img  src=\"Images/skt_inv.png\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sktinv(u,nu,_lambda):\n",
    "    '''\n",
    "    % sktinv(u,nu,_lambda)\n",
    "    % u      = Cdf value (Between 0 and 1)\n",
    "    % nu     = Kurtosis Parameter (>2 to inf)\n",
    "    % lambda = Assymetry Parameter (-1 to 1)\n",
    "    %\n",
    "    % returns z = random Variable (between -inf and inf)\n",
    "    %\n",
    "    % Calculates the inverse CDF for the skewed Student-t. Hansen (1994) version.\n",
    "    % Author: Christian Contino\n",
    "    % Date:   2 May 2021\n",
    "    '''\n",
    "    \n",
    "    u  = np.atleast_1d(u)\n",
    "    \n",
    "    c = gamma((nu+1)/2)/(sqrt(pi*(nu-2))*gamma(nu/2))\n",
    "    a = 4*_lambda*c*((nu-2)/(nu-1))\n",
    "    b = sqrt(1 + 3 * _lambda**2 - a**2)\n",
    "    \n",
    "    inv1 = (1-_lambda)/b * sqrt((nu-2) / nu) * t.ppf(u / (1-_lambda), nu) - a/b\n",
    "    inv2 = (1+_lambda)/b * sqrt((nu-2) / nu) * t.ppf(0.5 + (1 / (1+_lambda)) * (u - (1-_lambda)/2) , nu) - a/b\n",
    "    \n",
    "    limit_variable = (1-_lambda)/2\n",
    "    \n",
    "    z = u.copy()\n",
    "    \n",
    "    lt = u < limit_variable\n",
    "    gt = u >= limit_variable\n",
    "    \n",
    "    z[lt] = inv1[lt]\n",
    "    z[gt] = inv2[gt]\n",
    "\n",
    "    return z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Skewed Student-t CDF\n",
    "<img src=\"Images/skt_cdf.png\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sktcdf(z,nu,_lambda):\n",
    "    '''\n",
    "    % sktcdf(z,nu,lambda)\n",
    "    % z      = Random Variable value (Between -inf and inf)\n",
    "    % nu     = Kurtosis Parameter (>2 to inf)\n",
    "    % lambda = Assymetry Parameter (-1 to 1)\n",
    "    %\n",
    "    % returns u = random Variable (between 0 and 1)\n",
    "    %\n",
    "    % Calculates the CDF for the skewed Student-t. Hansen (1994) version.\n",
    "    % Author: Christian Contino\n",
    "    % Date:   2 May 2021\n",
    "    '''\n",
    "    \n",
    "    z  = np.atleast_1d(z)\n",
    "    \n",
    "    c = gamma((nu+1)/2)/(sqrt(pi*(nu-2))*gamma(nu/2))\n",
    "    a = 4*_lambda*c*((nu-2)/(nu-1))\n",
    "    b = sqrt(1 + 3 * _lambda**2 - a**2)\n",
    "    \n",
    "    limit_variable = -a/b\n",
    "    lt = z < limit_variable\n",
    "    gt = z >= limit_variable\n",
    "    \n",
    "    y_1 = (b*z+a) / (1-_lambda) * sqrt(nu/(nu-2))\n",
    "    y_2 = (b*z+a) / (1+_lambda) * sqrt(nu/(nu-2))\n",
    "    \n",
    "    pdf1 = (1-_lambda) * t.cdf(y_1, nu)\n",
    "    pdf2 = (1-_lambda)/2 + (1+_lambda) * (t.cdf(y_2, nu)-0.5)\n",
    "\n",
    "    u = z.copy()\n",
    "\n",
    "    u[lt] = pdf1[lt]\n",
    "    u[gt] = pdf2[gt]\n",
    "\n",
    "    return u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Skewed Student-t PDF\n",
    "<img src=\"Images/skt_pdf.png\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    " def sktpdf(z, nu, _lambda):\n",
    "    '''\n",
    "    % sktpdf(z,nu,lambda)\n",
    "    % z      = Random Variable value (Between -inf and inf)\n",
    "    % nu     = Kurtosis Parameter (>2 to inf)\n",
    "    % lambda = Assymetry Parameter (-1 to 1)\n",
    "    %\n",
    "    % returns u = random Variable > 0\n",
    "    %\n",
    "    % Calculates the PDF for the skewed Student-t. Hansen (1994) version.\n",
    "    % Author: Christian Contino\n",
    "    % Date:   27 May 2021\n",
    "    '''\n",
    "\n",
    "    z  = np.atleast_1d(z)\n",
    "\n",
    "    c = gamma((nu+1)/2)/(sqrt(pi*(nu-2))*gamma(nu/2))\n",
    "    a = 4*_lambda*c*((nu-2)/(nu-1))\n",
    "    b = sqrt(1 + 3 * _lambda**2 - a**2)\n",
    "\n",
    "    limit_variable = -a/b\n",
    "    lt = z < limit_variable\n",
    "    gt = z >= limit_variable\n",
    "\n",
    "    y_1 = b * c * (1 + (1/(nu-2)) * ((b*z+a) / (1-_lambda))**2 )**-((nu+1)/2)\n",
    "    y_2 = b * c * (1 + (1/(nu-2)) * ((b*z+a) / (1+_lambda))**2 )**-((nu+1)/2)\n",
    "\n",
    "    u = z.copy()\n",
    "\n",
    "    u[lt] = y_1[lt]\n",
    "    u[gt] = y_2[gt]\n",
    "    return u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def update_plots(change):\n",
    "    scatter = f.data[0]\n",
    "    scatter.y = sktinv(u,nu = nu_slider.value, _lambda = lambda_slider.value)\n",
    "    f.layout.title = 'Inverse CDF Skewed Student-t, Nu = ' + str(nu_slider.value) + ' Lambda = ' + str(lambda_slider.value)\n",
    "    \n",
    "    scatter = g.data[0]\n",
    "    scatter.y = sktcdf(z,nu = nu_slider.value, _lambda = lambda_slider.value)\n",
    "    g.layout.title = 'CDF Skewed Student-t, Nu = ' + str(nu_slider.value) + ' Lambda = ' + str(lambda_slider.value)\n",
    "    \n",
    "    scatter = h.data[0]\n",
    "    scatter.y = sktpdf(z,nu = nu_slider.value, _lambda = lambda_slider.value)\n",
    "    update_grid()\n",
    "    if show_student_t_checkbox.value:\n",
    "        scale = (nu_slider.value/(nu_slider.value-2))**.5\n",
    "        \n",
    "        scatter_t = f.data[1]\n",
    "        scatter_t.y = t.ppf(u, nu_slider.value, scale=1/scale)\n",
    "        \n",
    "        scatter_t = g.data[1]\n",
    "        scatter_t.y = t.cdf(z, nu_slider.value, scale=1/scale)\n",
    "        \n",
    "        scatter_t = h.data[1]\n",
    "        scatter_t.y = t.pdf(z, nu_slider.value, scale=1/scale)\n",
    "        \n",
    "    h.layout.title = 'PDF Skewed Student-t, Nu = ' + str(nu_slider.value) + ' Lambda = ' + str(lambda_slider.value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_student_t(change):\n",
    "    if ~show_student_t_checkbox.value:\n",
    "        f.data = [f.data[0]]\n",
    "        g.data = [g.data[0]]\n",
    "        h.data = [h.data[0]]\n",
    "        update_grid()\n",
    "        \n",
    "    if change['new']:\n",
    "        scale = (nu_slider.value/(nu_slider.value-2))**.5\n",
    "        f.add_scatter(x=u, y = t.ppf(u, nu_slider.value, scale=1/scale), name = 'Student-t')\n",
    "        g.add_scatter(x=z, y = t.cdf(z, nu_slider.value, scale=1/scale), name = 'Student-t')\n",
    "        h.add_scatter(x=z, y = t.pdf(z, nu_slider.value, scale=1/scale), name = 'Student-t')\n",
    "        update_grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def update_grid():\n",
    "    df = pd.DataFrame(columns = ['Skew-t'], data = np.random.rand(4,1), index=  ['Mean', 'Variance', 'Skew', 'Kurtosis'])\n",
    "    nu = nu_slider.value\n",
    "    _lambda = lambda_slider.value\n",
    "    c = gamma((nu+1)/2)/(sqrt(pi*(nu-2))*gamma(nu/2))\n",
    "    a = 4*_lambda*c*((nu-2)/(nu-1))\n",
    "    b = sqrt(1 + 3 * _lambda**2 - a**2)\n",
    "    \n",
    "    m2 = 1 + 3 * _lambda**2\n",
    "    \n",
    "    df.loc['Mean', 'Skew-t'] = 0\n",
    "    df.loc['Variance', 'Skew-t'] = 1  \n",
    "    if nu > 3:\n",
    "        m3 = 16 * c * _lambda * (1+_lambda**2) * (nu-2)**2 / ((nu-1)*(nu-3))\n",
    "        df.loc['Skew', 'Skew-t'] = (m3 - 3*a*m2 + 2*a**3) / b**3\n",
    "    else:\n",
    "        df.loc['Skew', 'Skew-t'] = np.nan\n",
    "    if nu > 4:\n",
    "        m4 = 3 * (nu-2)/(nu-4) * (1+10*_lambda**2 + 5*_lambda**4)\n",
    "        df.loc['Kurtosis', 'Skew-t'] = (m4 -4*a*m3 + 6*a**2 * m3 - 3*a**4) / b**4\n",
    "    else:\n",
    "        df.loc['Kurtosis', 'Skew-t'] = np.nan\n",
    "    if show_student_t_checkbox.value:\n",
    "        df.loc['Mean', 'Student-t'] = 0\n",
    "        df.loc['Variance', 'Student-t'] = 1\n",
    "        if nu > 3:\n",
    "            df.loc['Skew', 'Student-t'] = 0\n",
    "        else:\n",
    "            df.loc['Skew', 'Student-t'] = np.nan\n",
    "        if nu > 4:\n",
    "            df.loc['Kurtosis', 'Student-t'] = 3 + 6 / (nu-4)\n",
    "        else:\n",
    "            df.loc['Kurtosis', 'Student-t'] = np.nan\n",
    "    grid.df = df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "nu_slider = FloatSlider(min=2.1, max=200, step=0.1, value=4, description = 'Nu', continuous_update=False)\n",
    "lambda_slider = FloatSlider(min=-0.99, max=0.99, step=0.01, value=0, description = 'Lambda', continuous_update=False)\n",
    "show_student_t_checkbox = Checkbox(description = 'Show Student-t')\n",
    "\n",
    "nu_slider.observe(update_plots, names='value')\n",
    "lambda_slider.observe(update_plots, names='value')\n",
    "show_student_t_checkbox.observe(plot_student_t, names='value')\n",
    "\n",
    "df = pd.DataFrame(columns = ['Skew-t'], data = np.random.rand(4,1), index=  ['Mean', 'Variance', 'Skew', 'Kurtosis'])\n",
    "\n",
    "nu = nu_slider.value\n",
    "_lambda = lambda_slider.value\n",
    "\n",
    "c = gamma((nu+1)/2)/(sqrt(pi*(nu-2))*gamma(nu/2))\n",
    "a = 4*_lambda*c*((nu-2)/(nu-1))\n",
    "b = sqrt(1 + 3 * _lambda**2 - a**2)\n",
    "\n",
    "m2 = 1 + 3 * _lambda**2\n",
    "\n",
    "df.loc['Mean', 'Skew-t'] = 0\n",
    "df.loc['Variance', 'Skew-t'] = 1  \n",
    "if nu > 3:\n",
    "    m3 = 16 * c * _lambda * (1+_lambda**2) * (nu-2)**2 / ((nu-1)*(nu-3))\n",
    "    df.loc['Skew', 'Skew-t'] = (m3 - 3*a*m2 + 2*a**3) / b**3\n",
    "else:\n",
    "    df.loc['Skew', 'Skew-t'] = np.nan\n",
    "if nu > 4:\n",
    "    m4 = 3 * (nu-2)/(nu-4) * (1+10*_lambda**2 + 5*_lambda**4)\n",
    "    df.loc['Kurtosis', 'Skew-t'] = (m4 -4*a*m3 + 6*a**2 * m3 - 3*a**4) / b**4\n",
    "else:\n",
    "    df.loc['Kurtosis', 'Skew-t'] = np.nan\n",
    "\n",
    "grid = qgrid.show_grid(df)\n",
    "grid.layout.width = '250px'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "nu = nu_slider.value\n",
    "_lambda = lambda_slider.value\n",
    "u = np.linspace(.01, .99, 100)\n",
    "z = np.linspace(-3, 3, 100)\n",
    "\n",
    "f = go.FigureWidget()\n",
    "f.add_scatter(x = u, y=sktinv(u,nu = nu, _lambda = _lambda), name = 'Skewed-t');\n",
    "f.layout.title = 'Inverse CDF Skewed Student-t, Nu = ' + str(nu_slider.value) + ' Lambda = ' + str(lambda_slider.value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = go.FigureWidget()\n",
    "g.add_scatter(x = z, y=sktcdf(z,nu = nu, _lambda = _lambda), name = 'Skewed-t');\n",
    "g.layout.title = 'CDF Skewed Student-t, Nu = ' + str(nu_slider.value) + ' Lambda = ' + str(lambda_slider.value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = go.FigureWidget()\n",
    "h.add_scatter(x = z, y=sktpdf(z, nu = nu_slider.value, _lambda = lambda_slider.value), name = 'Skewed-t')\n",
    "h.layout.title = 'PDF Skewed Student-t, Nu = ' + str(nu_slider.value) + ' Lambda = ' + str(lambda_slider.value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "bb454ae31f954ebcada2e3e9e113bc6f",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "VBox(children=(HBox(children=(FigureWidget({\n",
       "    'data': [{'name': 'Skewed-t',\n",
       "              'type': 'scatter'…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "VBox([HBox([f, g]),\n",
    "      HBox([h, VBox([nu_slider,\n",
    "          lambda_slider,\n",
    "         show_student_t_checkbox,\n",
    "         grid])])\n",
    "           ])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.random.uniform(size=(1000,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
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       "       [0.31987125]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sktpdf(u,200,0.2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "       [ 2.53807074e-01]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sktinv(u,5,0.3)"
   ]
  }
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