{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Introducing censoring\n",
    "\n",
    "![img](https://lifelines.readthedocs.io/en/latest/_images/survival_analysis_intro_censoring.png)\n",
    "\n",
    "\n",
    "![img2](https://lifelines.readthedocs.io/en/latest/_images/survival_analysis_intro_censoring_revealed.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All we know is that actual lifetime is greater than some threshold. Mathematically, we know $P(T \\ge t) = 1 - F(t) := S(t)$. We can use this in our log likelihood:\n",
    "\n",
    "No censoring cases:\n",
    "\n",
    "$$l(\\theta, t) = \\sum_{\\text{observed}} \\log{\\text{pdf}(\\theta, t)} $$\n",
    "\n",
    "With censoring cases:\n",
    "$$ \n",
    "\\begin{align}\n",
    "l(\\theta, t) & = \\sum_{\\text{observed}} \\log{\\text{pdf}(t, \\theta)} + \\sum_{\\text{censored}} \\log{\\text{S}(t, \\theta)} \\\\\n",
    "& = \\sum_{\\text{observed}} \\log{\\text{pdf}(t, \\theta)} \\frac{S(t)}{S(t)} + \\sum_{\\text{censored}} \\log{\\text{S}(t, \\theta)} \\\\\n",
    "& = \\sum_{\\text{observed}} (\\log{\\frac{\\text{pdf}(t, \\theta)}{S(t)}} + \\log{S(t)}) + \\sum_{\\text{censored}} \\log{\\text{S}(t, \\theta)} \\\\\n",
    "& = \\sum_{\\text{observed}} \\log{\\frac{\\text{pdf}(t, \\theta)}{S(t)}} + \\sum_{\\text{observed}} \\log{S(t)} + \\sum_{\\text{censored}} \\log{\\text{S}(t, \\theta)} \\\\\n",
    "& = \\sum_{\\text{observed}} \\log{\\frac{\\text{pdf}(t, \\theta)}{S(t)}} + \\sum \\log{S(t)} \n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    "The $-\\log{S(t)}$ is known as the _cumulative hazard_, denoted $H(t)$. \n",
    "\n",
    "$$l(\\theta, t) = \\sum_{\\text{observed}} \\log{\\frac{\\text{pdf}(t, \\theta)}{S(t)}} - \\sum H(t, \\theta) $$\n",
    "\n",
    "Also, $\\frac{dH}{dt} = \\frac{\\text{pdf}(t, \\theta)}{S(t)}$. Denote that $h(t)$. \n",
    "\n",
    "$$l(\\theta, t) = \\sum_{\\text{observed}} \\log{h(t, \\theta}) - \\sum H(t, \\theta) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Phew! Now, instead of working in probability space, we will work in hazard space! Here's a link to all the relatioships: https://lifelines.readthedocs.io/en/latest/Survival%20Analysis%20intro.html#hazard-function \n",
    "\n",
    "\n",
    "## Take away: the likelihood function can be used to \"add\" information about the system (think about how penalizers are used...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# the hazard and cumulative hazard for Weibull are much simplier to implement 😗👌\n",
    "\n",
    "def cumulative_hazard(params, t):\n",
    "    lambda_, rho_ = params\n",
    "    return (t / lambda_) ** rho_\n",
    "\n",
    "def hazard(params, t):\n",
    "    # diff of cumulative hazard w.r.t. t\n",
    "    lambda_, rho_ = params\n",
    "    return rho_ / lambda_ * (t / lambda_) ** (rho_ - 1)\n",
    "\n",
    "def log_hazard(params, t):\n",
    "    return np.log(hazard(params, t))\n",
    "\n",
    "def log_likelihood(params, t, e):\n",
    "    return np.sum(e * log_hazard(params, t)) - np.sum(cumulative_hazard(params, t))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = (np.random.exponential(size=1000)/1.5) ** 2.3\n",
    "E = np.random.binomial(1, 0.95, size=1000)\n",
    "\n",
    "from scipy.optimize import minimize\n",
    "\n",
    "results = minimize(log_likelihood, \n",
    "        x0 = np.array([1.0, 1.0]),\n",
    "        method=None, \n",
    "        args=(T, E),\n",
    "        bounds=((0.00001, None), (0.00001, None)))\n",
    "\n",
    "print(results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# why did this fail? Note the -inf."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def negative_log_likelihood(params, t, e):\n",
    "    return -log_likelihood(params, t, e)\n",
    "\n",
    "results = minimize( # what goes here?, \n",
    "        x0 = np.array([1.0, 1.0]),\n",
    "        method=None, \n",
    "        args=(T, E),\n",
    "        bounds=((0.00001, None), (0.00001, None)))\n",
    "\n",
    "print(results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.linspace(0, 10, 100)\n",
    "plt.plot(t, np.exp(-cumulative_hazard(results.x, t)))\n",
    "plt.ylim(0, 1)\n",
    "plt.title(\"Estimated survival function\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's move to part 3 - automatic differentiation! "
   ]
  }
 ],
 "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": 2
}