{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The era of BLS planet searches is over!\n",
    "\n",
    "The \"box function\" does not have analytical maximim likelihood estimators for all its parameters, requiring the use of a grid search to sample $t_0$ and $w$. Let's see if a sum of two logistic functions offers better properties."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a simplified version of the [generalised logistic function](https://en.wikipedia.org/wiki/Generalised_logistic_function) as:\n",
    "\n",
    "$$GL(t, a, k, t_0) = a + \\frac{k-a}{1 + e^{-b(t-t_0)}}$$\n",
    "\n",
    "where $t$ is time, $a$ is the lower asymptote, and $k$ is the upper asymptote. The growth rate $b$ can be assumed constant, we'll use $b=5$.\n",
    "\n",
    "We can then define a box-like function as:\n",
    "\n",
    "$$box(t) = GL(t, h/2, h/2+d, t_0+w) + GL(t, h/2, h/2-d, t_0)$$\n",
    "\n",
    "where $h$ is the lightcurve baseline height, $d$ is the transit depth, $w$ is the transit width, and $t_0$ is the transit start. The maths work out as:\n",
    "\n",
    "$$box(t) = h + \\frac{d}{1 + e^{-b(t-t_0-w)}} - \\frac{d}{1 + e^{-b(t-t_0)}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assuming a lightcurve $Y_i$ is subject to uncorreclted Gaussian noise:\n",
    "\n",
    "$$Y_i \\sim \\mathcal{N}(box(t), \\sigma^2)$$\n",
    "\n",
    "the joint probability density function of $\\boldsymbol{Y}$ can be written as:\n",
    "$$\n",
    "\\begin{align} p(\\boldsymbol{y}) = \\prod_{i=1}^{n} p(y_i) = TBD\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function may or may not have an analytical maximum likelihood solution. This is where we need Ze's help."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from numpy import e\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as pl\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8a679d1668>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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wopclmlCWkCkQZNYVF+SxqKZEgTBNh0+eoX8ozoqG8qhLkTlGgSCROK++nNc6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\nL/jJnCqO5wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8a67a41f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def geerts_logistic(t, a, k, t0, b=5.):\n",
    "    return a + (k - a) / (1 + e**(-(b*(t-t0))))\n",
    "\n",
    "def transit_model(t, t0=1, width=4, baseline=2., depth=0.2):\n",
    "    x = geerts_logistic(t, a=baseline/2, k=baseline/2+depth, t0=t0+width)\n",
    "    x += geerts_logistic(t, a=baseline/2, k=baseline/2-depth, t0=t0)\n",
    "    return x\n",
    "\n",
    "t = np.arange(-3, 10, 0.01)\n",
    "x = transit_model(t)\n",
    "pl.plot(t, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}