{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Introduction\n",
    "============\n",
    "\n",
    "\n",
    "Setting up `Python`\n",
    "------------------------------\n",
    "\n",
    "The data we are going to use as illustrations were recorded by Andreas\n",
    "Pippow [Kloppenburg Lab, University of Cologne](http://cecad.uni-koeln.de/Prof-Peter-Kloppenburg.82.0.html) and\n",
    "are freely available in [HDF5](http://www.hdfgroup.org/HDF5/) format at\n",
    "the following URLs:\n",
    "-   <http://xtof.disque.math.cnrs.fr/data/CCD_calibration.hdf5> for the\n",
    "    calibration data.\n",
    "-   <http://xtof.disque.math.cnrs.fr/data/Data_POMC.hdf5> for the POMC\n",
    "    data.\n",
    "\n",
    "`Python 3` is used here so if you want to do the same with `Python 2`\n",
    "you should start with:\n",
    "\n",
    "``` {.python}\n",
    "from __future__ import print_function, division, unicode_literals, absolute_import\n",
    "```\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Loading the data into `Python` requires the installation of\n",
    "[h5py](http://docs.h5py.org/en/latest/index.html), a module available in\n",
    "the [anaconda](http://continuum.io/downloads) distribution.\n",
    "\n",
    "### Loading the POMC data set in `Python`\n",
    "\n",
    "We start by downloading the data set on our local disk. To do that we\n",
    "need the `urllib` module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('Data_POMC.hdf5', <http.client.HTTPMessage at 0x7fbda1e2edd8>)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from urllib.request import urlretrieve\n",
    "urlretrieve(\"http://xtof.disque.math.cnrs.fr/data/Data_POMC.hdf5\",\"Data_POMC.hdf5\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Python 2` users should type instead:\n",
    "\n",
    "``` {.python}\n",
    "import urllib\n",
    "urllib.urlretrieve(\"http://xtof.disque.math.cnrs.fr/data/Data_POMC.hdf5\",\"Data_POMC.hdf5\")\n",
    "```\n",
    "\n",
    "Once the data are on the local disk, they are loaded into `Python` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import h5py"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pomc = h5py.File(\"Data_POMC.hdf5\",\"r\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Information on the content of this `HDF5` file can be found in its\n",
    "`README` attribute:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'POMC data set recorded by Andreas Pippow (Kloppenburg Laboratory Cologne University, http://cecad.uni-koeln.de/Prof-Peter-Kloppenburg.82.0.html). 168 measurements performed with a CCD camera recording Fura-2 fluorescence (excitation wavelength: 340 nm). The size of the CCD chip is 60 x 80 pixels. A stimulation (depolarization induced calcium entry) comes at time 527. Details about this data set can be found in: Joucla et al (2013) Estimating background-subtracted fluorescence transients in calcium imaging experiments: A quantitative approach. Cell Calcium. 54 (2): 71-85.'"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pomc.attrs['README']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We create next variables pointing to the `time` vector and to the image\n",
    "`stack`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "time_pomc = pomc['time'][...]\n",
    "stack_pomc = pomc['stack'][...]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then close the file:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pomc.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Loading the Calibration data set in `Python`\n",
    "\n",
    "We start by downloading the data set on our local disk. To do that we\n",
    "need the `urllib` module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "urlretrieve(\"http://xtof.disque.math.cnrs.fr/data/CCD_calibration.hdf5\",\"CCD_calibration.hdf5\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Python 2` users should type instead:\n",
    "\n",
    "``` {.python}\n",
    "urllib.urlretrieve(\"http://xtof.disque.math.cnrs.fr/data/CCD_calibration.hdf5\",\"CCD_calibration.hdf5\")\n",
    "```\n",
    "\n",
    "Once the data are on the local disk, they are loaded into `Python` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['100ms',\n",
       " '10ms',\n",
       " '20ms',\n",
       " '30ms',\n",
       " '40ms',\n",
       " '50ms',\n",
       " '60ms',\n",
       " '70ms',\n",
       " '80ms',\n",
       " '90ms']"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calibration = h5py.File(\"CCD_calibration.hdf5\",\"r\")\n",
    "list(calibration)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The file is made of 10 `groups` with the above names. Each group\n",
    "contains 2 `datasets`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['stack', 'time']"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(calibration['10ms'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dataset `stack` contains, as its name says, the image stack and its\n",
    "shape is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(60, 80, 100)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calibration['10ms/stack'].shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So the CCD captor is made of 60 x 80 pixels and the 100 exposures take\n",
    "the dimension with index 2 (that is the third dimension).\n",
    "\n",
    "The data set `time` contains a vector of times at which the different\n",
    "exposure were done:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(100,)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calibration['10ms/time'].shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All this information can be found in the `README` attribute of the file:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Imago/SensiCam CCD camera (Till Photonics) calibration data set. Fluorescence measurments were made using a fluorescent plastic slide. 10 exposure times from 10 to 100 ms (each making an HDF5 group) were used. For each exposure time 100 exposures were performed (with 200 ms between each). The fluorescence measured in each of the 60 x 80 pixels of the camera are stored in the stack data set of each group. The time data set (a vector) of each group contains the time at which each illumination was done. These recordings were done by Andreas Pippow (Kloppenburg Laboratory Cologne University, http://cecad.uni-koeln.de/Prof-Peter-Kloppenburg.82.0.html). They were used in: Sébastien Joucla, Andreas Pippow, Peter Kloppenburg and Christophe Pouzat (2010) Quantitative estimation of calcium dynamics from ratiometric measurements: A direct, non-ratioing, method. Journal of Neurophysiology 103: 1130-1144.'"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "calibration.attrs['README']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### loading key modules\n",
    "\n",
    "We are going to use the usual scientific python modules plus `SymPy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import sympy as sy\n",
    "import scipy\n",
    "plt.ion()\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Some function definitions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plotSignal(stack,lw=1):\n",
    "    import numpy as np\n",
    "    import matplotlib.pyplot as plt\n",
    "    n_x, n_y, n_t = stack.shape\n",
    "    amp_min = np.min(stack)\n",
    "    amp_max = np.max(stack)\n",
    "    amp_diff = np.ptp(stack)\n",
    "    x_domain = np.arange(n_t)/n_t\n",
    "    y_domain = (0,n_y)\n",
    "    for r_idx in range(n_x):\n",
    "        for c_idx in range(n_y):\n",
    "            y_min = n_x - r_idx - 1\n",
    "            sig = stack[r_idx,c_idx,:]\n",
    "            Y = (sig-amp_min)/amp_diff + y_min\n",
    "            X = x_domain + c_idx\n",
    "            plt.plot(X,Y,lw=lw,color='black')\n",
    "            plt.ylim([0,n_y-1])\n",
    "    plt.axis('off')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The variability inherent to fluorescence imaging data\n",
    "------------------------------------------------------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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       "AGCQAAUAMEiAAgAY9AOYplEXtj/nrQAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a66eaac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plotSignal(stack_pomc[20:33,33:44,:],lw=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ADU counts (raw data) from Fura-2 excited at 340 nm. Each square\n",
    "corresponds to a pixel. 25.05 s of data are shown. Same scale on each\n",
    "sub-plot. Data recorded by Andreas Pippow (Kloppenburg Lab. Cologne\n",
    "University)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(525, 550)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a6323940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.plot(time_pomc,stack_pomc[27,39,:],lw=2)\n",
    "plt.xlabel(\"Time (s)\",fontsize=25)\n",
    "plt.ylabel(\"ADU count\",fontsize=25)\n",
    "plt.grid()\n",
    "plt.xlim([525,550])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the central pixels of the previous figure."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What do we want? (1)\n",
    "-----------------------------\n",
    "\n",
    "Given the data set illustrated on the last two slides we might want to\n",
    "estimate parameters like:\n",
    "-   the peak amplitude\n",
    "-   the decay time constant(s)\n",
    "-   the baseline level\n",
    "-   the whole time course (strictly speaking, a function).\n",
    "\n",
    "If we have a model linking the calcium dynamics---the time course of the\n",
    "free calcium concentration in the cell---to the fluorescence intensity\n",
    "like:\n",
    "$$\\frac{\\mathrm{d}Ca_t}{\\mathrm{dt}} \\left(1 + \\kappa_{F}(Ca_t) + \\kappa_{E}(Ca_t) \\right) + \\frac{j(Ca_t)}{v} = 0 \\, , $$\n",
    "where $Ca_t$ stands for $[Ca^{2+}]_{free}$ at time t, $v$ is the volume\n",
    "of the neurite---within which diffusion effects can be neglected---and\n",
    "$$j(Ca_t) \\equiv \\gamma (Ca_t - Ca_{steady}) \\, ,$$ is the model of\n",
    "calcium extrusion---$Ca_{steady}$ is the steady state\n",
    "$[Ca^{2+}]_{free}$---\n",
    "$$\\kappa_{F}(Ca_t) \\equiv \\frac{F_{total} \\, K_{F}}{(K_{F} + Ca_t)^2} \\quad \\mathrm{and} \\quad \\kappa_{E}(Ca_t) \\equiv \\frac{E_{total} \\, K_{E}}{(K_{E} + Ca_t)^2} \\, ,$$\n",
    "where $F$ stands for the fluorophore en $E$ for the *endogenous* buffer.\n",
    "\n",
    "In the previous slide, assuming that the fluorophore (Fura) parameters:\n",
    "$F_{total}$ and $K_F$ have been calibrated, we might want to estimate:\n",
    "-   the extrusion parameter: $\\gamma$\n",
    "-   the endogenous buffer parameters: $E_{total}$ and $K_E$\n",
    "\n",
    "using an equation relating measured fluorescence to calcium:\n",
    "$$Ca_t = K_{F} \\, \\frac{S_t - S_{min}}{S_{max} - S_t} \\, ,$$ where $S_t$\n",
    "is the fluorescence (signal) measured at time $t$, $S_{min}$ and\n",
    "$S_{max}$ are *calibrated* parameters corresponding respectively to the\n",
    "fluorescence in the absence of calcium and with saturating $[Ca^{2+}]$\n",
    "(for the fluorophore).\n",
    "\n",
    "-   The variability of our signal---meaning that under replication of\n",
    "    our measurements *under the exact same conditions* we wont get the\n",
    "    exact same signal---implies that our estimated parameters will also\n",
    "    fluctuate upon replication.\n",
    "-   Formally our parameters are modeled as *random variables* and **it\n",
    "    is not enough to summarize a random variable by a single number**.\n",
    "-   If we cannot get the full distribution function for our parameters,\n",
    "    we want to give at least ranges within which the true value of the\n",
    "    parameter should be found with a given probability.\n",
    "-   In other words: **an analysis without confidence intervals is not an\n",
    "    analysis**, it is strictly speaking useless since it can't be\n",
    "    reproduced---if I say that my time constant is 25.76 ms the\n",
    "    probability that upon replication I get again 25.76 is essentially\n",
    "    0; if I say that the actual time constant has a 0.95 probability to\n",
    "    be in the interval [24,26.5], I can make a comparison with\n",
    "    replications.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A proper handling of the \"variability\" matters\n",
    "-----------------------------------------------------------\n",
    "\n",
    "Let us consider a simple data generation model:\n",
    "$$Y_i \\sim \\mathcal{P}(f_i)\\, , \\quad i=0,1,\\ldots,K \\; ,$$ where\n",
    "$\\mathcal{P}(f_i)$ stands for the *Poisson distribution* with parameter\n",
    "$f_i$ :\n",
    "$$\\mathrm{Pr}\\{Y_i = n\\} = \\frac{(f_i)^n}{n!} \\exp (-f_i)\\, , \\quad \\mathrm{for} \\quad n=0,1,2,\\ldots $$\n",
    "and\n",
    "$$f_i = f(\\delta i| f_{\\infty}, \\Delta, \\beta) = f_{\\infty} + \\Delta \\, \\exp (- \\beta \\, \\delta i)\\; ,$$\n",
    "δ is a time step and $f_{\\infty}$, Δ and β are model parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7f79a62d13c8>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a62ff0f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "beta_true = 1.0\n",
    "f_infinite = 100\n",
    "Delta = 900\n",
    "X = np.linspace(0,5*beta_true,51)\n",
    "Theo = Delta*np.exp(-X*beta_true)+f_infinite\n",
    "np.random.seed(20061001)\n",
    "Observations = np.random.poisson(Theo)\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.plot(X,Observations,'o')\n",
    "plt.xlabel(\"Time (s)\",fontsize=25)\n",
    "plt.ylabel(\"Observations\",fontsize=25)\n",
    "plt.plot(X,Theo,'r')\n",
    "plt.plot(X[[3,30]],Theo[[3,30]],'sk')\n",
    "plt.plot([X[3],X[3]],[0,Theo[3]],'--k')\n",
    "plt.plot([0,X[3]],[Theo[3],Theo[3]],'--k')\n",
    "plt.plot([X[30],X[30]],[0,Theo[30]],'--k')\n",
    "plt.plot([0,X[30]],[Theo[30],Theo[30]],'--k')\n",
    "plt.text(0.1,730,r'$y_1$',fontsize=25)\n",
    "plt.text(1.5,110,r'$y_2$',fontsize=25)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Data simulated according to the previous model. We are going to assume\n",
    "that $f_{\\infty}$ and $\\Delta$ are known and that $(t_1,y_1)$ and\n",
    "$(t_2,y_2)$ are given. We want to estimate $\\beta$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Two estimators\n",
    "---------------------------\n",
    "\n",
    "We are going to consider two\n",
    "[estimators](https://en.wikipedia.org/wiki/Estimator) for $\\beta$:\n",
    "-   The \"classical\" least square estimator:\n",
    "    $$ \\tilde{\\beta} = \\arg \\min \\tilde{L}(\\beta) \\; ,$$ where\n",
    "    $$ \\tilde{L}(\\beta) = \\sum_j \\big( y_j - f(t_j \\mid \\beta) \\big)^2 \\; .$$\n",
    "-   The least square estimator applied to the *square root* of the data:\n",
    "    $$\\hat{\\beta} = \\arg \\min \\hat{L}(\\beta) \\; ,$$ where\n",
    "    $$ \\hat{L}(\\beta) = \\sum_j \\big( \\sqrt{y_j} - \\sqrt{f(t_j \\mid \\beta)} \\big)^2 \\; .$$\n",
    "\n",
    "\n",
    "We perform an empirical study as follows:\n",
    "-   We simulate 100,000 experiments such that:\n",
    "    $$ (Y_1,Y_2) \\sim \\big(\\mathcal{P}(f(0.3|\\beta_0), \\mathcal{P}(f(3|\\beta_0)\\big) \\; ,$$\n",
    "    with $\\beta_0=1$.\n",
    "-   For each simulated pair, $(y_1,y_2)^{[k]}$ ($k=1,\\ldots,10^5$), we\n",
    "    minimize $\\tilde{L}(\\beta)$ and $\\hat{L}(\\beta)$ to obtain:\n",
    "    $(\\tilde{\\beta}^{[k]},\\hat{\\beta}^{[k]})$.\n",
    "-   We build histograms for $\\tilde{\\beta}^{[k]}$ and\n",
    "    $\\hat{\\beta}^{[k]}$ as density estimators of our estimators.\n",
    "\n",
    "## Simulations with the two estimators\n",
    "\n",
    "\n",
    "### Definitions of utility functions required for applying Newton's method to the first estimator\n",
    "\n",
    "We want to minimize: $$\\tilde{\\beta} = \\arg \\min \\tilde{L}(\\beta) \\; ,$$\n",
    "where\n",
    "$$\\tilde{L}(\\beta) = \\big( y_1 - f(t_1 \\mid \\beta) \\big)^2 + \\big( y_2 - f(t_2 \\mid \\beta) \\big)^2 \\; .$$\n",
    "In other words, we want the root of the derivative. We can get this\n",
    "derivative by hand or with the `SymPy` module. We will next use the\n",
    "latter.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/latex": [
       "$$\\left(- \\Delta e^{- \\beta x_{1}} - f_{infini} + y_{1}\\right)^{2} + \\left(- \\Delta e^{- \\beta x_{2}} - f_{infini} + y_{2}\\right)^{2}$$"
      ],
      "text/plain": [
       "                            2                               2\n",
       "⎛     -β⋅x₁                ⎞    ⎛     -β⋅x₂                ⎞ \n",
       "⎝- Δ⋅ℯ      - f_infini + y₁⎠  + ⎝- Δ⋅ℯ      - f_infini + y₂⎠ "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.init_printing()\n",
    "x_1,x_2,y_1,y_2,Delta,beta,f_infini = sy.symbols('x_1,x_2,y_1,y_2,Delta,beta,f_infini',real=True)\n",
    "L_tilde = (y_1 - Delta*sy.exp(-beta*x_1)-f_infini)**2 + (y_2 - Delta*sy.exp(-beta*x_2)-f_infini)**2\n",
    "L_tilde"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "NeFRIuyZ8Ln1yMJ+2Htun/cS+lu69p6KFh5XVKHTm2DdGTgLh8UYI9j+65m9Zn8Jl6OVOwLedvbC\n",
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       "zOMfVNNwnKiJtwspU/SSqBQ5yv5ancswWsVVApgRJ2oiwdazHZlJhpF2D33q4G23F7d69FK08SIG\n",
       "aWVLxLs79oXshHgqS5lbbqqpGi0qoVCRmn1JN1SUUSs7YNxWI3wfgklyP0+f1yqwka/S97y4SSrw\n",
       "AH12/FTIL0LE2PcX/spOgs3KiRrsBwX23hrr4kHxMeWZ2ddThNE5fvzB7x4/Xqcdg7sx7l95nUBa\n",
       "4kWxH0EvjI6GCi0m6nrpLODh5xR5PXbTlcsOK1GIa199zGMnYPx4DsEEEE/lTFkinflGqLCxxx+U\n",
       "HTBOzBJ+ZQK+M3WC8AHM21o+V39AxlDqNCPEuBjTiPag+KUZoRYiHQq0AJ4VAEu8eg4gUGG8HcSe\n",
       "Ct57xLY+im3cmWjZCvGGQKYQpxkB5Wcjipwqfi47YJzokG9lgt/DKovbzCtBfFe6i026BOuXJ3RR\n",
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       "oaN10K3g9ZsRvyQGEKiwK4FHxPFLAa/jycIiTtL0NDfkAWGVUhcjIb4VPXelL2nI+nBPobR5a5vW\n",
       "2xWbQ4WMgSj+US6bGXnXrEjwnn03XIfnJoXMVQ51rsD7w/X7hUgFClFoYaocsa80tWkt7l3CS2+D\n",
       "ftqWWj2FjCyq0JGq+7bsL78/DCCQYdZP9l3cLP5MVH3pWfet0UWrS1nv48yRshUCv4Wt3s8Ys9Jl\n",
       "5yycnV7pUDdLX/Lv9KCQCo1BiF42M/L7w4oEDy8v/xtzyyxL2xemN3fD/tK+DUUh0H6MAIOFemV9\n",
       "tmlNN0H9pGeRUmgMglWh9q42LVtev6XDwtWfQIblqd5mLqh6+VN7Trb/QZFORcSL40aJQniFTQIw\n",
       "WKhXG57Fht9AnA2TCBUyBqKXzYwvC09aAs4iz1++MUHaVV+dKjDqY8x3qWmj3Z5CRhZVKHW/Jftr\n",
       "/H4LkNq1F6aZWXkxMK2n7BspxUeAPfgaTT4aUcjI0uMXZTPjRulKP8j+ZmfTh8SQT+W79F/JJ+mX\n",
       "/kilBpFIKTSy2OVg3A3ak8H6zfuBqd9VhSVg3IZbE+a1l12sn6A3xqUa/a0ySX96fLunCCOL4uey\n",
       "A8aHAs8qrkVacyTib6Y1yg8/Q4HPOZhA7mAN4vBSCo2a3C/WC54RNnXGaf70ylXPP0E86yk7d3mT\n",
       "DscW/amCY3+gRvcUMuqFcNnMmPAnnA41NOvSN3xhzBw2XGkNr4qnyfbrcRsKtwB3RBQyanKgXQPO\n",
       "m9tDxjOaY9XKkeVljll72aV2mZbsrrJbx3wXbrOnkFGvh8tmxlJd96xC+9IqsP2g8mDsBxG+Hw0C\n",
       "pPQPqexSJRVfrpYKlgTakWRcve3ENaliSo1UsMGg4ZSdP1MbTEWIv6VCJYLoR9IwZMZPleWdqVAp\n",
       "QMMp21lIQUU/6qupYMmgp5PN58Z60dDS/i/LHvGGVvb/E62+A/8FtzaZqII0rmQAAAAASUVORK5C\n",
       "YII=\n"
      ],
      "text/latex": [
       "$$2 \\Delta x_{1} \\left(- \\Delta e^{- \\beta x_{1}} - f_{infini} + y_{1}\\right) e^{- \\beta x_{1}} + 2 \\Delta x_{2} \\left(- \\Delta e^{- \\beta x_{2}} - f_{infini} + y_{2}\\right) e^{- \\beta x_{2}}$$"
      ],
      "text/plain": [
       "       ⎛     -β⋅x₁                ⎞  -β⋅x₁          ⎛     -β⋅x₂               \n",
       "2⋅Δ⋅x₁⋅⎝- Δ⋅ℯ      - f_infini + y₁⎠⋅ℯ      + 2⋅Δ⋅x₂⋅⎝- Δ⋅ℯ      - f_infini + y\n",
       "\n",
       " ⎞  -β⋅x₂\n",
       "₂⎠⋅ℯ     "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_tilde = sy.diff(L_tilde,beta)\n",
    "G_tilde"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following `print` command is usefull to generate the code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2*Delta*x_1*(-Delta*exp(-beta*x_1) - f_infini + y_1)*exp(-beta*x_1) + 2*Delta*x_2*(-Delta*exp(-beta*x_2) - f_infini + y_2)*exp(-beta*x_2)\n"
     ]
    }
   ],
   "source": [
    "print(G_tilde)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "1Qr8F7ri3cc2tpj4AAAAAElFTkSuQmCC\n"
      ],
      "text/latex": [
       "$$2 \\Delta^{2} x_{1}^{2} e^{- 2 \\beta x_{1}} + 2 \\Delta^{2} x_{2}^{2} e^{- 2 \\beta x_{2}} - 2 \\Delta x_{1}^{2} \\left(- \\Delta e^{- \\beta x_{1}} - f_{infini} + y_{1}\\right) e^{- \\beta x_{1}} - 2 \\Delta x_{2}^{2} \\left(- \\Delta e^{- \\beta x_{2}} - f_{infini} + y_{2}\\right) e^{- \\beta x_{2}}$$"
      ],
      "text/plain": [
       "   2   2  -2⋅β⋅x₁      2   2  -2⋅β⋅x₂         2 ⎛     -β⋅x₁                ⎞  \n",
       "2⋅Δ ⋅x₁ ⋅ℯ        + 2⋅Δ ⋅x₂ ⋅ℯ        - 2⋅Δ⋅x₁ ⋅⎝- Δ⋅ℯ      - f_infini + y₁⎠⋅ℯ\n",
       "\n",
       "-β⋅x₁         2 ⎛     -β⋅x₂                ⎞  -β⋅x₂\n",
       "      - 2⋅Δ⋅x₂ ⋅⎝- Δ⋅ℯ      - f_infini + y₂⎠⋅ℯ     "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_prime_tilde = sy.diff(G_tilde,beta)\n",
    "G_prime_tilde"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2*Delta**2*x_1**2*exp(-2*beta*x_1) + 2*Delta**2*x_2**2*exp(-2*beta*x_2) - 2*Delta*x_1**2*(-Delta*exp(-beta*x_1) - f_infini + y_1)*exp(-beta*x_1) - 2*Delta*x_2**2*(-Delta*exp(-beta*x_2) - f_infini + y_2)*exp(-beta*x_2)\n"
     ]
    }
   ],
   "source": [
    "print(G_prime_tilde)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define next a \"constructor\" function returning the functions required\n",
    "to implement Newton's method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def mk_g_dg_tilde(x_1,y_1,x_2,y_2,Delta=900,f_infini=100):\n",
    "    def g(beta):\n",
    "        return 2*Delta*x_1*(-Delta*np.exp(-beta*x_1) - \\\n",
    "                            f_infini + y_1)*np.exp(-beta*x_1) + \\\n",
    "                            2*Delta*x_2*(-Delta*np.exp(-beta*x_2) - \\\n",
    "                                         f_infini + y_2)*np.exp(-beta*x_2)\n",
    "    def dg(beta):\n",
    "        return 2*Delta**2*x_1**2*np.exp(-2*beta*x_1) + \\\n",
    "            2*Delta**2*x_2**2*np.exp(-2*beta*x_2) - \\\n",
    "            2*Delta*x_1**2*(-Delta*np.exp(-beta*x_1) - \\\n",
    "                            f_infini + y_1)*np.exp(-beta*x_1) - \\\n",
    "                            2*Delta*x_2**2*(-Delta*np.exp(-beta*x_2) - \\\n",
    "                                            f_infini + y_2)*np.exp(-beta*x_2)\n",
    "    return (g,dg)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then create the required functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g_tilde, dg_tilde = mk_g_dg_tilde(X[3],Observations[3],X[30],Observations[30])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define next a function implementing Newton's method given an initial\n",
    "guess, a derivative of the target function and a second derivative of\n",
    "the target:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def newton(initial_guess,\n",
    "           target_d,\n",
    "           target_dd,\n",
    "           tolerance=1e-6,\n",
    "           iter_max=100):\n",
    "    pos = initial_guess\n",
    "    value = target_d(pos)\n",
    "    idx = 0\n",
    "    while idx <= iter_max and abs(value) > tolerance :\n",
    "        pos -= value/target_dd(pos)\n",
    "        idx += 1\n",
    "        value = target_d(pos)\n",
    "    return (pos,value,idx)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A short test:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.97769372257738074, -4.8977199185173959e-07, 3)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton(1.0,g_tilde,dg_tilde)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Definitions of utility functions required for applying Newton's method to the second estimator\n",
    "\n",
    "We now want to minimize: $$\\hat{\\beta} = \\arg \\min \\hat{L}(\\beta) \\; ,$$\n",
    "where\n",
    "$$\\hat{L}(\\beta) = \\big( \\sqrt{y_1} - \\sqrt{f(t_1 \\mid \\beta)} \\big)^2 + \\big( \\sqrt{y_2} - \\sqrt{f(t_2 \\mid \\beta)} \\big)^2 \\; .$$\n",
    "We use `Sympy` since doing these calculations by hand is rather \"heavy\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "fuOd1m/Bw6xcqn2YNfj/zf3f6oH/AP+AP0rqpF2+AAAAAElFTkSuQmCC\n"
      ],
      "text/latex": [
       "$$\\frac{\\Delta x_{1} e^{- \\beta x_{1}}}{\\sqrt{\\Delta e^{- \\beta x_{1}} + f_{infini}}} \\left(\\sqrt{y_{1}} - \\sqrt{\\Delta e^{- \\beta x_{1}} + f_{infini}}\\right) + \\frac{\\Delta x_{2} e^{- \\beta x_{2}}}{\\sqrt{\\Delta e^{- \\beta x_{2}} + f_{infini}}} \\left(\\sqrt{y_{2}} - \\sqrt{\\Delta e^{- \\beta x_{2}} + f_{infini}}\\right)$$"
      ],
      "text/plain": [
       "     ⎛            _____________________⎞               ⎛            __________\n",
       "     ⎜  ____     ╱    -β⋅x₁            ⎟  -β⋅x₁        ⎜  ____     ╱    -β⋅x₂ \n",
       "Δ⋅x₁⋅⎝╲╱ y₁  - ╲╱  Δ⋅ℯ      + f_infini ⎠⋅ℯ        Δ⋅x₂⋅⎝╲╱ y₂  - ╲╱  Δ⋅ℯ      \n",
       "─────────────────────────────────────────────── + ────────────────────────────\n",
       "               _____________________                             _____________\n",
       "              ╱    -β⋅x₁                                        ╱    -β⋅x₂    \n",
       "            ╲╱  Δ⋅ℯ      + f_infini                           ╲╱  Δ⋅ℯ      + f\n",
       "\n",
       "___________⎞       \n",
       "           ⎟  -β⋅x₂\n",
       "+ f_infini ⎠⋅ℯ     \n",
       "───────────────────\n",
       "________           \n",
       "                   \n",
       "_infini            "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L_hat = (sy.sqrt(y_1) - sy.sqrt(Delta*sy.exp(-beta*x_1) + f_infini))**2 + (sy.sqrt(y_2) - sy.sqrt(Delta*sy.exp(-beta*x_2) + f_infini))**2\n",
    "G_hat = sy.diff(L_hat,beta)\n",
    "G_hat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Delta*x_1*(sqrt(y_1) - sqrt(Delta*exp(-beta*x_1) + f_infini))*exp(-beta*x_1)/sqrt(Delta*exp(-beta*x_1) + f_infini) + Delta*x_2*(sqrt(y_2) - sqrt(Delta*exp(-beta*x_2) + f_infini))*exp(-beta*x_2)/sqrt(Delta*exp(-beta*x_2) + f_infini)\n"
     ]
    }
   ],
   "source": [
    "print(G_hat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/latex": [
       "$$\\frac{\\Delta^{2} x_{1}^{2} \\left(\\sqrt{y_{1}} - \\sqrt{\\Delta e^{- \\beta x_{1}} + f_{infini}}\\right) e^{- 2 \\beta x_{1}}}{2 \\left(\\Delta e^{- \\beta x_{1}} + f_{infini}\\right)^{\\frac{3}{2}}} + \\frac{\\Delta^{2} x_{1}^{2} e^{- 2 \\beta x_{1}}}{2 \\Delta e^{- \\beta x_{1}} + 2 f_{infini}} + \\frac{\\Delta^{2} x_{2}^{2} \\left(\\sqrt{y_{2}} - \\sqrt{\\Delta e^{- \\beta x_{2}} + f_{infini}}\\right) e^{- 2 \\beta x_{2}}}{2 \\left(\\Delta e^{- \\beta x_{2}} + f_{infini}\\right)^{\\frac{3}{2}}} + \\frac{\\Delta^{2} x_{2}^{2} e^{- 2 \\beta x_{2}}}{2 \\Delta e^{- \\beta x_{2}} + 2 f_{infini}} - \\frac{\\Delta x_{1}^{2} e^{- \\beta x_{1}}}{\\sqrt{\\Delta e^{- \\beta x_{1}} + f_{infini}}} \\left(\\sqrt{y_{1}} - \\sqrt{\\Delta e^{- \\beta x_{1}} + f_{infini}}\\right) - \\frac{\\Delta x_{2}^{2} e^{- \\beta x_{2}}}{\\sqrt{\\Delta e^{- \\beta x_{2}} + f_{infini}}} \\left(\\sqrt{y_{2}} - \\sqrt{\\Delta e^{- \\beta x_{2}} + f_{infini}}\\right)$$"
      ],
      "text/plain": [
       "       ⎛            _____________________⎞                                    \n",
       " 2   2 ⎜  ____     ╱    -β⋅x₁            ⎟  -2⋅β⋅x₁        2   2  -2⋅β⋅x₁     \n",
       "Δ ⋅x₁ ⋅⎝╲╱ y₁  - ╲╱  Δ⋅ℯ      + f_infini ⎠⋅ℯ              Δ ⋅x₁ ⋅ℯ            \n",
       "─────────────────────────────────────────────────── + ─────────────────────── \n",
       "                                    3/2                 ⎛   -β⋅x₁           ⎞ \n",
       "               ⎛   -β⋅x₁           ⎞                  2⋅⎝Δ⋅ℯ      + f_infini⎠ \n",
       "             2⋅⎝Δ⋅ℯ      + f_infini⎠                                          \n",
       "\n",
       "         ⎛            _____________________⎞                                  \n",
       "   2   2 ⎜  ____     ╱    -β⋅x₂            ⎟  -2⋅β⋅x₂        2   2  -2⋅β⋅x₂   \n",
       "  Δ ⋅x₂ ⋅⎝╲╱ y₂  - ╲╱  Δ⋅ℯ      + f_infini ⎠⋅ℯ              Δ ⋅x₂ ⋅ℯ          \n",
       "+ ─────────────────────────────────────────────────── + ──────────────────────\n",
       "                                      3/2                 ⎛   -β⋅x₂           \n",
       "                 ⎛   -β⋅x₂           ⎞                  2⋅⎝Δ⋅ℯ      + f_infini\n",
       "               2⋅⎝Δ⋅ℯ      + f_infini⎠                                        \n",
       "\n",
       "          ⎛            _____________________⎞                ⎛            ____\n",
       "        2 ⎜  ____     ╱    -β⋅x₁            ⎟  -β⋅x₁       2 ⎜  ____     ╱    \n",
       "    Δ⋅x₁ ⋅⎝╲╱ y₁  - ╲╱  Δ⋅ℯ      + f_infini ⎠⋅ℯ        Δ⋅x₂ ⋅⎝╲╱ y₂  - ╲╱  Δ⋅ℯ\n",
       "─ - ──────────────────────────────────────────────── - ───────────────────────\n",
       "⎞                  _____________________                              ________\n",
       "⎠                 ╱    -β⋅x₁                                         ╱    -β⋅x\n",
       "                ╲╱  Δ⋅ℯ      + f_infini                            ╲╱  Δ⋅ℯ    \n",
       "\n",
       "_________________⎞       \n",
       "-β⋅x₂            ⎟  -β⋅x₂\n",
       "      + f_infini ⎠⋅ℯ     \n",
       "─────────────────────────\n",
       "_____________            \n",
       "₂                        \n",
       "  + f_infini             "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_prime_hat = sy.diff(G_hat,beta)\n",
    "G_prime_hat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Delta**2*x_1**2*(sqrt(y_1) - sqrt(Delta*exp(-beta*x_1) + f_infini))*exp(-2*beta*x_1)/(2*(Delta*exp(-beta*x_1) + f_infini)**(3/2)) + Delta**2*x_1**2*exp(-2*beta*x_1)/(2*(Delta*exp(-beta*x_1) + f_infini)) + Delta**2*x_2**2*(sqrt(y_2) - sqrt(Delta*exp(-beta*x_2) + f_infini))*exp(-2*beta*x_2)/(2*(Delta*exp(-beta*x_2) + f_infini)**(3/2)) + Delta**2*x_2**2*exp(-2*beta*x_2)/(2*(Delta*exp(-beta*x_2) + f_infini)) - Delta*x_1**2*(sqrt(y_1) - sqrt(Delta*exp(-beta*x_1) + f_infini))*exp(-beta*x_1)/sqrt(Delta*exp(-beta*x_1) + f_infini) - Delta*x_2**2*(sqrt(y_2) - sqrt(Delta*exp(-beta*x_2) + f_infini))*exp(-beta*x_2)/sqrt(Delta*exp(-beta*x_2) + f_infini)\n"
     ]
    }
   ],
   "source": [
    "print(G_prime_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define next the corresponding constructor function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def mk_g_dg_hat(x_1,y_1,x_2,y_2,Delta=900,f_infini=100):\n",
    "    def g(beta):\n",
    "        return Delta*x_1*(np.sqrt(y_1) - np.sqrt(Delta*np.exp(-beta*x_1) + f_infini))*np.exp(-beta*x_1)/np.sqrt(Delta*np.exp(-beta*x_1) + f_infini) + Delta*x_2*(np.sqrt(y_2) - np.sqrt(Delta*np.exp(-beta*x_2)+ f_infini))*np.exp(-beta*x_2)/np.sqrt(Delta*np.exp(-beta*x_2) +f_infini)\n",
    "    def dg(beta):\n",
    "        return Delta**2*x_1**2*(np.sqrt(y_1) - np.sqrt(Delta*np.exp(-beta*x_1) + f_infini))*np.exp(-2*beta*x_1)/(2*(Delta*np.exp(-beta*x_1) + f_infini)**(3/2)) + Delta**2*x_1**2*np.exp(-2*beta*x_1)/(2*(Delta*np.exp(-beta*x_1) + f_infini)) + Delta**2*x_2**2*(np.sqrt(y_2) - np.sqrt(Delta*np.exp(-beta*x_2) + f_infini))*np.exp(-2*beta*x_2)/(2*(Delta*np.exp(-beta*x_2) + f_infini)**(3/2)) + Delta**2*x_2**2*np.exp(-2*beta*x_2)/(2*(Delta*np.exp(-beta*x_2) + f_infini)) - Delta*x_1**2*(np.sqrt(y_1) - np.sqrt(Delta*np.exp(-beta*x_1) + f_infini))*np.exp(-beta*x_1)/np.sqrt(Delta*np.exp(-beta*x_1) + f_infini) - Delta*x_2**2*(np.sqrt(y_2) - np.sqrt(Delta*np.exp(-beta*x_2) + f_infini))*np.exp(-beta*x_2)/np.sqrt(Delta*np.exp(-beta*x_2) + f_infini)\n",
    "    return (g,dg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A little test:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.0226210475375788, -4.1330077138468369e-09, 3)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g_hat, dg_hat = mk_g_dg_hat(X[3],Observations[3],X[30],Observations[30])    \n",
    "newton(1.0,g_hat,dg_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n_rep = int(1e5)\n",
    "beta_tilde = np.zeros((n_rep,3))\n",
    "beta_hat = np.zeros((n_rep,3))\n",
    "np.random.seed(20110928)\n",
    "for rep_idx in range(n_rep):\n",
    "    Y = np.random.poisson(Theo[[3,30]])\n",
    "    g_tilde, dg_tilde = mk_g_dg_tilde(X[3],Y[0],X[30],Y[1])\n",
    "    beta_tilde[rep_idx,:] = newton(1.0,g_tilde,dg_tilde)\n",
    "    g_hat, dg_hat = mk_g_dg_hat(X[3],Y[0],X[30],Y[1])    \n",
    "    beta_hat[rep_idx,:] = newton(1.0,g_hat,dg_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We check that every simulation ended before the maximal allowed number\n",
    "of iteration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(False, False)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(any(beta_tilde[:,2] == 100),any(beta_hat[:,2] == 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f79a61b34a8>]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a60fe630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def Ffct(beta): \n",
    "    return 900 * np.exp(-X[[3,30]]*beta) + 100\n",
    "\n",
    "def dFfct(beta):\n",
    "    return -X[[3,30]]*900 * np.exp(-X[[3,30]]*beta)\n",
    "\n",
    "sd0 = np.sqrt((np.sum(dFfct(1.0)**2*Ffct(1.0))/np.sum(dFfct(1.0)**2)**2))\n",
    "sd1 = np.sqrt(1.0/np.sum(dFfct(1.0)**2/Ffct(1.0)))\n",
    "beta_vector = np.linspace(0.6,1.6,501)\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "useless_stuff = plt.hist([beta_tilde[:,0],beta_hat[:,0]],bins=50,normed=True,histtype='step',lw=2)\n",
    "plt.xlabel(r'$\\beta$',fontsize=25)\n",
    "plt.ylabel('Density',fontsize=25)\n",
    "plt.title(r'Densities of $\\widetilde{\\beta}$ and $\\widehat{\\beta}$',fontsize=25)\n",
    "plt.plot(beta_vector,np.exp(-0.5*(beta_vector-1)**2/sd0**2)/sd0/np.sqrt(2*np.pi),color='black',lw=2)\n",
    "plt.plot(beta_vector,np.exp(-0.5*(beta_vector-1)**2/sd1**2)/sd1/np.sqrt(2*np.pi),color='black',lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both histograms are built with 50 bins. $\\hat{\\beta}$ (green) is __clearly__\n",
    "better than $\\tilde{\\beta}$ (blue) since its variance is smaller. The\n",
    "derivation of the theoretical (large sample) densities is given in\n",
    "[Joucla et al (2010)](http://intl-jn.physiology.org/cgi/content/short/103/2/1130)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "CCD camera noise\n",
    "==========\n",
    "\n",
    "CCD basics\n",
    "--------------\n",
    "\n",
    "![](zerosept_vanVliet1998F3.png)\n",
    "\n",
    "Source: L. van Vliet et col. (1998) [Digital Fluorescence Imaging Using Cooled CCD Array Cameras](http://homepage.tudelft.nl/e3q6n/publications/1998/AP98LVDSTY/AP98LVDSTY.html) (figure 3).\n",
    "\n",
    "\n",
    "\"Noise\" sources in CCD\n",
    "-----------------------------------\n",
    "\n",
    "-   The \"Photon noise\" or \"shot noise\" arises from the fact the\n",
    "    measuring a fluorescence intensity, λ, implies **counting\n",
    "    photons**---unless one changes the laws of Physics there is nothing\n",
    "    one can do to eliminate this source of variability (improperly\n",
    "    called \"noise\")---:\n",
    "    $$\\mathrm{Pr}\\{N=n\\} = \\frac{\\lambda^n}{n!} \\exp -\\lambda\\, , \\quad n \\, = \\, 0,1,\\ldots\\, , \\quad \\lambda > 0\\; .$$\n",
    "-   The \"thermal noise\" arises from thermal agitation which \"dumps\"\n",
    "    electrons in potential wells; this \"noise\" also follows a Poisson\n",
    "    distribution but it can be made negligible by *cooling down* the\n",
    "    camera.\n",
    "-   The \"read out noise\" arises from the conversion of the number of\n",
    "    photo-electrons into an equivalent tension; it follows a normal\n",
    "    distribution whose variance is independent of the mean (as long as\n",
    "    reading is not done at too high a frequency).\n",
    "-   The \"digitization noise\" arises from the mapping of a continuous\n",
    "    value, the tension, onto a grid; it is negligible as soon as more\n",
    "    than 8 bit are used.\n",
    "\n",
    "A simple CCD model\n",
    "-------------------------------\n",
    "\n",
    "-   We can easily obtain a simple CCD model taking into account the two\n",
    "    main \"noise\" sources (photon and read-out).\n",
    "-   To get this model we are going the fact (a theorem) that when a\n",
    "    **large number of photon are detected**, the Poisson distribution is\n",
    "    well approximated by ([converges in\n",
    "    distribution](http://en.wikipedia.org/wiki/Convergence_in_distribution#Convergence_in_distribution)\n",
    "    to) a normal distribution with identical mean and variance:\n",
    "    $$\\mathrm{Pr}\\{N=n\\} = \\frac{\\lambda^n}{n!} \\exp -\\lambda \\approx \\mathcal{N}(\\lambda,\\lambda) \\; .$$\n",
    "-   In other words:\n",
    "    $$ N \\approx \\lambda + \\sqrt{\\lambda} \\, \\epsilon \\; ,$$ where\n",
    "    $\\epsilon \\sim \\mathcal{N}(0,1)$ (follows a standard normal\n",
    "    distribution).\n",
    "-   A read-out noise is added next following a normal distribution with\n",
    "    0 mean and variance $\\sigma_{R}^2$.\n",
    "-   We are therefore adding to the random variable $N$ a new\n",
    "    **independent** random variable $R \\sim \\mathcal{N}(0,\\sigma_{R}^2)$\n",
    "    giving:\n",
    "    $$M \\equiv N+R \\approx \\lambda + \\sqrt{\\lambda+\\sigma_{R}^2} \\, \\epsilon \\; ,$$\n",
    "    where the fact that the sum of two independent normal random\n",
    "    variables is a normal random variable whose mean is the sum of the\n",
    "    mean and whose variance is the sum of the variances has been used.\n",
    "-   Since the capacity of the photo-electron weels is finite (35000 for\n",
    "    the camera used in the first slides) and since the number of\n",
    "    photon-electrons will be digitized on 12 bit (4096 levels), a \"gain\"\n",
    "    $G$ **smaller than one** must be applied if we want to represent\n",
    "    faithfully (without saturation) an almost full well.\n",
    "-   We therefore get:\n",
    "    $$Y \\equiv G \\cdot M \\approx G \\, \\lambda + \\sqrt{G^2 \\, (\\lambda+\\sigma_{R}^2)} \\, \\epsilon \\; .$$\n",
    "\n",
    "For completeness: Convergence in distribution of a Poisson toward a normal rv\n",
    "-------------------------------------------------------------------------------------------------------\n",
    "\n",
    "We use the [moment-generating function](http://en.wikipedia.org/wiki/Moment-generating_function) and\n",
    "the following theorem (_e.g._ John Rice, 2007, _Mathematical Statistics\n",
    "and Data Analysis_, Chap. 5, Theorem A):\n",
    "-   If the moment-generating function of each element of the rv sequence\n",
    "    $X_n$ is $m_n(t)$,\n",
    "-   if the moment-generating function of the rv $X$ is $m(t)$,\n",
    "-   if $m_n(t) \\rightarrow m(t)$ when $n \\rightarrow \\infty$ for all\n",
    "    $|t| \\le b$ where $b > 0$\n",
    "-   then $X_n \\xrightarrow{D} X$.\n",
    "\n",
    "Lets show that: $$Y_n = \\frac{X_n - n}{\\sqrt{n}} \\; , $$ where $X_n$\n",
    "follows a Poisson distribution with parameter $n$, converges in\n",
    "distribution towards $Z$ standard normal rv.\n",
    "\n",
    "We have: $$m_n(t) \\equiv \\mathrm{E}\\left[\\exp(Y_n t)\\right] \\; ,$$\n",
    "therefore:\n",
    "$$m_n(t) = \\sum_{k=0}^{\\infty} \\exp\\left(\\frac{k-n}{\\sqrt{n}}t\\right) \\frac{n^k}{k!} \\exp(-n) \\; ,$$\n",
    "\n",
    "\n",
    "$$m_n(t) = \\exp(-n) \\exp(-\\sqrt{n}t) \\sum_{k=0}^{\\infty} \\frac{\\left(n \\exp\\left(t/\\sqrt{n}\\right)\\right)^k}{k!}$$\n",
    "$$m_n(t) = \\exp\\left(-n - \\sqrt{n} t+ n \\exp(t/\\sqrt{n})\\right)$$\n",
    "$$m_n(t) = \\exp\\left(-n - \\sqrt{n} t+ n \\sum_{k=0}^{\\infty}  \\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!}\\right)$$\n",
    "$$m_n(t) = \\exp\\left(-n - \\sqrt{n} t+ n + \\sqrt{n} t + \\frac{t^2}{2} + n \\sum_{k=3}^{\\infty}  \\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!}\\right)$$\n",
    "$$m_n(t) = \\exp\\left( \\frac{t^2}{2} + n \\sum_{k=3}^{\\infty} \\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!}\\right)$$\n",
    "\n",
    "\n",
    "We must show:\n",
    "$$n \\sum_{k=3}^{\\infty}\\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!} \\rightarrow_{n \\rightarrow \\infty} 0 \\quad \\forall\\ |t| \\le b, \\quad \\text{where}\n",
    "      \\quad b > 0\\, ,$$ since $\\exp(-t^2/2)$ is the moment-generating\n",
    "function of a standard normal rv. But\n",
    "$$\\left| n \\sum_{k=3}^{\\infty} \\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!} \\right| \\rightarrow_{n \\rightarrow \\infty} 0 \\quad \\forall\\ |t| \\le b, \\quad \\text{where} \\quad b > 0\\,$$\n",
    "implies that since $$- \\left|n \\sum_{k=3}^{\\infty}\n",
    "      \\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!} \\right| \\le n\n",
    "    \\sum_{k=3}^{\\infty} \n",
    "      \\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!} \\le \\left| n\n",
    "        \\sum_{k=3}^{\\infty} \n",
    "      \\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!} \\right| \\, .$$\n",
    "\n",
    "But for all $|t| \\le b$ where $b > 0$\n",
    "\n",
    "\\begin{displaymath}\n",
    "  \\begin{array}{lcl}\n",
    "    0 \\le \\left| n \\sum_{k=3}^{\\infty}\n",
    "      \\left(\\frac{t}{\\sqrt{n}}\\right)^k \\frac{1}{k!} \\right| & \\le & n\n",
    "    \\sum_{k=3}^{\\infty} \n",
    "      \\left(\\frac{|t|}{\\sqrt{n}}\\right)^k \\frac{1}{k!} \\\\\n",
    "      & \\le & \\frac{|t|^3}{\\sqrt{n}} \\sum_{k=0}^{\\infty} \n",
    "      \\left(\\frac{|t|}{\\sqrt{n}}\\right)^k \\frac{1}{(k+3)!} \\\\\n",
    "      & \\le & \\frac{|t|^3}{\\sqrt{n}} \\sum_{k=0}^{\\infty} \n",
    "      \\left(\\frac{|t|}{\\sqrt{n}}\\right)^k \\frac{1}{k!} \\\\\n",
    "      & \\le & \\frac{|t|^3}{\\sqrt{n}}\n",
    "      \\exp\\left(\\frac{|t|}{\\sqrt{n}}\\right) \\rightarrow_{n \\rightarrow\n",
    "      \\infty} 0 \\, ,\n",
    "  \\end{array}\n",
    "\\end{displaymath}\n",
    "\n",
    "which completes the proof.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Illustrations\n",
    "\n",
    "We define first a utility function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.stats import norm, poisson\n",
    "ZZ = np.linspace(-3,3,501)\n",
    "FZ = norm.cdf(ZZ)\n",
    "\n",
    "def poisson_steps(par,start,end):\n",
    "    xpairs = [[(i-par)/np.sqrt(par),(i+1-par)/np.sqrt(par)] for i in range(start,end+1)]\n",
    "    ypairs = [[poisson.cdf(i,par),poisson.cdf(i,par)] for i in range(start,end+1)]\n",
    "    xlist = []\n",
    "    ylist = []\n",
    "    for xends,yends in zip(xpairs,ypairs):\n",
    "        xlist.extend(xends)\n",
    "        xlist.append(None)\n",
    "        ylist.extend(yends)\n",
    "        ylist.append(None)\n",
    "    return [xlist,ylist]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f79a6477668>]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "AABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a610a9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "FY = poisson_steps(5,0,25)\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.plot(ZZ,FZ,color='red',lw=2)\n",
    "plt.xlabel('Z',fontsize=25)\n",
    "plt.ylabel('CDF',fontsize=25)\n",
    "plt.xlim([-3,3])\n",
    "plt.plot(FY[0],FY[1],color='black',lw=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cumulative distribution functions (CDF) of $Y_5$ (black) and $Z$ a\n",
    "standard normal (red)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f79a63fb390>]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a648dc18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "FY = poisson_steps(50,0,500)\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.plot(ZZ,FZ,color='red',lw=2)\n",
    "plt.xlabel('Z',fontsize=25)\n",
    "plt.ylabel('CDF',fontsize=25)\n",
    "plt.xlim([-3,3])\n",
    "plt.plot(FY[0],FY[1],color='black',lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cumulative distribution functions (CDF) of $Y_{50}$ (black) and $Z$ a\n",
    "standard normal (red)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f799dfcaf60>]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "pJb4/wu4T11/COO2AAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a641a0b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "FY = poisson_steps(500,500-67,500+67)\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.plot(ZZ,FZ,color='red',lw=2)\n",
    "plt.xlabel('Z',fontsize=25)\n",
    "plt.ylabel('CDF',fontsize=25)\n",
    "plt.xlim([-3,3])\n",
    "plt.plot(FY[0],FY[1],color='black',lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cumulative distribution functions (CDF) of $Y_{500}$ (black) and $Z$ a standard normal (red)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f799dfc4a20>]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799dfe00f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "FY = poisson_steps(5000,5000-213,5000+213)\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.plot(ZZ,FZ,color='red',lw=2)\n",
    "plt.xlabel('Z',fontsize=25)\n",
    "plt.ylabel('CDF',fontsize=25)\n",
    "plt.xlim([-3,3])\n",
    "plt.plot(FY[0],FY[1],color='black',lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cumulative distribution functions (CDF) of $Y_{5000}$ (black) and $Z$ a standard normal (red)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "CCD calibration \n",
    "=========\n",
    "\n",
    "If what I just exposed is correct, with the two (main) \"noise\" sources,\n",
    "the observations $Y$ (from a CCD pixel) follow:\n",
    "$$Y \\sim G \\, \\lambda + \\sqrt{G^2 \\, (\\lambda+\\sigma_{R}^2)} \\, \\epsilon \\; ,$$\n",
    "where $G$ is the camera gain, $\\sigma_{R}^2$ is the read-out variance\n",
    "and $\\epsilon$ is a standard normal rv. The values of $G$ and\n",
    "$\\sigma_{R}^2$ are specified by the manufacturer for each camera, but\n",
    "experience shows that manufacturers tend to be overoptimistic when it\n",
    "comes to their product performances---they can for instance give an\n",
    "underestimated $\\sigma_{R}^2$. **Its therefore a good idea to measure\n",
    "these parameters with calibration experiments**. **Such calibration\n",
    "experiments are also the occasion to check that our simple model is\n",
    "relevant**.\n",
    "\n",
    "-   Our problem becomes: How to test\n",
    "    $Y \\sim G \\, \\lambda + \\sqrt{G^2 \\, (\\lambda+\\sigma_{R}^2)} \\, \\epsilon$\n",
    "    ? Or how to set different values for $\\lambda$?\n",
    "-   Let's consider a pixel of our CCD \"looking\" at a fixed volume of a\n",
    "    [fluorescein](http://en.wikipedia.org/wiki/Fluorescein) solution\n",
    "    with a given (and stable) concentration. We have two ways of\n",
    "    modifying λ:\n",
    "    -   Change the intensity $i_{e}$ of the light source exciting the\n",
    "        fluorophore.\n",
    "    -   Change the exposure time $\\tau$.\n",
    "\n",
    "\n",
    "We can indeed write our $\\lambda$ as:\n",
    "$$\\lambda = \\phi v c i_{e} \\tau \\, ,$$ where\n",
    "-   $v$ is the solution's volume \"seen\" by a given pixel,\n",
    "-   $c$ is the fluorophore's concentration,\n",
    "-   $\\phi$ is the [quantum yield](http://en.wikipedia.org/wiki/Quantum_yield).\n",
    "\n",
    "In practice it is easier to vary the exposure time *τ*and that's what\n",
    "was done in the experiments described next... **Question: Can you guess\n",
    "what these experiments are?**\n",
    "\n",
    "\n",
    "Sebastien Joucla and myself asked our collaborators from the\n",
    "[Kloppenburg lab](http://cecad.uni-koeln.de/Prof-Peter-Kloppenburg.82.0.html)\n",
    "(Cologne University) to:\n",
    "-   choose 10 exposure times,\n",
    "-   for each of the 10 times, perform 100 exposures,\n",
    "-   for each of the 10 x 100 exposures, record the value $y_{ij}$ of the\n",
    "    rv $Y_{ij}$ of CCD's pixel $i,j$.\n",
    "\n",
    "We introduce a rv $Y_{ij}$ for each pixel because it is very difficult\n",
    "(impossible) to have a uniform intensity ($i_e$) and a uniform volume\n",
    "($v$) and a uniform quantum yield ($\\phi$). We have therefore for each\n",
    "pixel:\n",
    "$$Y_{i,j} \\sim G \\, p_{i,j} \\tau + \\sqrt{G^2 \\, (p_{i,j} \\tau+\\sigma_{R}^2)} \\, \\epsilon_{i,j}\\; ,$$\n",
    "where $p_{i,j} = c \\phi_{i,j} v_{i,j} i_{e,i,j}$.\n",
    "\n",
    "-   If our model is correct we should have for each pixel $i,j$, for a\n",
    "    given exposure time, a mean value:\n",
    "    $$\\bar{y}_{i,j} = \\frac{1}{100} \\sum_{k=1}^{100} y_{i,j,k} \\approx G \\, p_{i,j} \\tau $$\n",
    "-   and a variance:\n",
    "    $$S_{i,j}^2 = \\frac{1}{99} \\sum_{k=1}^{100} (y_{i,j,k}-\\bar{y}_{i,j})^2 \\approx G^2 \\, (p_{i,j} \\tau+\\sigma_{R}^2) \\; .$$\n",
    "-   The graph of $S_{i,j}^2$ *vs* $\\bar{y}_{i,j}$ should be a straight\n",
    "    line with slope $G$ ordinate at 0, $G^2 \\sigma_{R}^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar at 0x7f799de98978>"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "HZrjOI6TBP8BFYPbxwj/qnMAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a628b240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.imshow(np.transpose(calibration['10ms/stack'][:,:,0]),origin='lower')\n",
    "plt.set_cmap('gray')\n",
    "plt.colorbar()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first exposure of 10 ms (experiment performed by Andreas Pippow, Kloppenburg lag, Cologne University)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "CCD calibration: Checking the assumptions\n",
    "------------------------------------------------------\n",
    "\n",
    "-   The data are going to be analyzed as if the $Y_{i,j,k}$ were IID,\n",
    "    **but they were sequentially recorded**. It is therefore **strongly\n",
    "    recommended** to check that the IID hypothesis is reasonable.\n",
    "-   The small example of the next figure shows that there are no\n",
    "    (obvious) trends.\n",
    "-   We must also check the correlation function.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "zMysh/w/Qypu0W2c5zMAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799c560cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.subplot(311)\n",
    "plt.plot(calibration['10ms/stack'][31,41,:],lw=2)\n",
    "plt.ylabel(\"ADU\",fontsize=25)\n",
    "plt.grid()\n",
    "plt.subplot(312)\n",
    "plt.plot(calibration['10ms/stack'][31,40,:],lw=2)\n",
    "plt.ylabel(\"ADU\",fontsize=25)\n",
    "plt.grid()\n",
    "plt.subplot(313)\n",
    "plt.plot(calibration['10ms/stack'][31,39,:],lw=2)\n",
    "plt.xlabel(\"Time (1 unit = 100 ms)\",fontsize=25)\n",
    "plt.ylabel(\"ADU\",fontsize=25)\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Counts time evolution for three neighboring pixels (10 ms exposure time)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-   If the $Y_{i,j,k}$ are not IID we expect a more or less linear\n",
    "    trend---due to bleaching of the dye.\n",
    "-   Rather then looking at each individual pixel sequence like on the\n",
    "    previous slide, we can fit the following linear model model to each\n",
    "    pixel: $$Y_{i,j,k} = \\beta_0 + \\beta_1 k + \\sigma \\epsilon_{i,j}$$\n",
    "    where the $\\epsilon_{i,j} \\stackrel{IID}{\\sim} \\mathcal{N}(0,1)$,\n",
    "    and check if $\\beta_1$ can be reasonably considered as null; while a\n",
    "    trend due to bleaching would give a negative $\\beta_1$.\n",
    "-   Without a trend, the theoretical distribution of\n",
    "    $\\hat{\\beta}_1/\\hat{\\sigma}_{\\beta_1}$---$\\hat{\\beta}_1$ is the\n",
    "    estimate of $\\beta_1$ and $\\hat{\\sigma}_{\\beta_1}$ its estimated\n",
    "    standard error---is a Student's t distribution with 98 degrees of\n",
    "    freedom.\n",
    "-   Applying this idea to the central pixel of the previous slide we\n",
    "    get...\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.032043204320430452"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D_matrix = np.transpose(np.array([np.ones(100),np.arange(100)]))\n",
    "P_matrix = np.linalg.solve(np.dot(np.transpose(D_matrix),D_matrix),np.transpose(D_matrix))\n",
    "Y = calibration['10ms/stack'][31,40,:]\n",
    "beta = np.dot(P_matrix,Y)\n",
    "beta[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/xtof/anaconda3/lib/python3.4/site-packages/IPython/kernel/__main__.py:3: RuntimeWarning: invalid value encountered in sqrt\n",
      "  app.launch_new_instance()\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0.025163437413154185"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y_hat = np.dot(D_matrix,beta)\n",
    "s2_hat = np.sum((Y-Y_hat)**2)/98\n",
    "beta_se = np.sqrt(s2_hat*np.linalg.inv(np.dot(np.transpose(D_matrix),D_matrix)))\n",
    "beta_se[1,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.017892818267477018, 0.081979226908337921)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.stats import t\n",
    "(beta[1]-beta_se[1,1]*t.ppf(0.975,98),beta[1]+beta_se[1,1]*t.ppf(0.975,98))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f799c3dfef0>]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f79a63dd9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.plot(D_matrix[:,1],Y,lw=2,color='black')\n",
    "plt.grid()\n",
    "plt.xlabel(\"Time (1 unit = 100 ms)\",fontsize=25)\n",
    "plt.ylabel(\"ADU\",fontsize=25)\n",
    "plt.plot(D_matrix[:,1],Y_hat,lw=2,color='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We get $\\hat{\\beta}_1 = 0.032$ and a 95 % conf. int. for it is: $[-0.018,0.082]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the fact that, under the null hypothesis (no trend):\n",
    "$$\\hat{\\beta}_1/\\hat{\\sigma}_{\\beta_1} \\sim t_{98}$$ by constructing the\n",
    "empirical cumulative distribution function (ECDF) of the 60 x 80 pixels\n",
    "at each exposure time to get the maximal difference (in absolute value)\n",
    "with the theoretical CDF to apply a Kolmogorov test. The critical value\n",
    "of the latter for a 99% level and a sample size of 100 is **0.163**. We\n",
    "get the following values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def linear_fit_stack(stack):\n",
    "    I,J,K = stack.shape\n",
    "    D_matrix = np.transpose(np.array([np.ones(K),np.arange(K)]))\n",
    "    P_matrix = np.linalg.solve(np.dot(np.transpose(D_matrix),D_matrix),np.transpose(D_matrix))\n",
    "    the_inv = np.linalg.inv(np.dot(np.transpose(D_matrix),D_matrix))[1,1]\n",
    "    res = np.zeros((I,J))\n",
    "    for i in range(I):\n",
    "        for j in range(J):\n",
    "            beta = np.dot(P_matrix,stack[i,j,:])\n",
    "            Y_hat = np.dot(D_matrix,beta)\n",
    "            s2_hat = np.sum((stack[i,j,:]-Y_hat)**2)/(K-2)\n",
    "            res[i,j] = beta[1]/np.sqrt(s2_hat*the_inv)\n",
    "    return res\n",
    "\n",
    "b1stats = [linear_fit_stack(calibration[n+'/stack']) for n in list(calibration)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[['100ms',\n",
       "  '10ms',\n",
       "  '20ms',\n",
       "  '30ms',\n",
       "  '40ms',\n",
       "  '50ms',\n",
       "  '60ms',\n",
       "  '70ms',\n",
       "  '80ms',\n",
       "  '90ms'],\n",
       " [0.09, 0.089, 0.116, 0.058, 0.135, 0.209, 0.041, 0.178, 0.153, 0.07]]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[[n for n in list(calibration)], [int(1000*np.max(np.abs(np.arange(1,len(b1s)+1)/len(b1s)-t.cdf(b1s,98))))/1000 for b1s in [np.sort(b1.flatten()) for b1 in b1stats]]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The values at 50 and 70 ms are too large."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f799c30f748>]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "RK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799c42e4e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tt = np.linspace(-4,4,501)\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "junk = plt.hist(np.concatenate([b1.flatten() for b1 in b1stats]),bins=100,normed=True,color='black')\n",
    "plt.xlabel(r'$\\hat{\\beta}_1/\\hat{\\sigma}_{\\beta_1}$',fontsize=25)\n",
    "plt.ylabel('Density',fontsize=25)\n",
    "plt.title(r'Density of all $\\hat{\\beta}_1/\\hat{\\sigma}_{\\beta_1}$',fontsize=25)\n",
    "plt.plot(tt,t.pdf(tt,98),color='orange',lw=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Empirical density in black, theoretical one (t with 98 df) in orange."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-   We now look for potential correlations between recording from\n",
    "    different pixels.\n",
    "-   We do that by computing the empirical correlation between pixels\n",
    "    $(i,j)$ and $(u,v)$.\n",
    "-   We get the empirical mean at each pixel (for a given exposure time)\n",
    "    that is: $\\overline{Y}_{ij} = (1/K) \\sum_{k=1}^K Y_{ijk}$.\n",
    "-   We get the empirical variance:\n",
    "    $S^2_{ij} =  1/(K-1) \\sum_{k=1}^K (Y_{ijk}-\\overline{Y}_{ij})^2$.\n",
    "-   We then obtain the normalized signal or *standard score*:\n",
    "    $N_{ijk} = (Y_{ijk}-\\overline{Y}_{ij})/\\sqrt{S^2_{ij}}$.\n",
    "-   The correlation coefficient is then:\n",
    "    $\\rho(ij,uv) = 1/(K-1) \\sum_{k=1}^K N_{ijk} N_{uvk}$.\n",
    "-   Under the null hypothesis, no correlation,\n",
    "    $\\rho(ij,uv) \\sim \\mathcal{N}(0,1/K)$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def correlations(stack):\n",
    "    n_row, n_col, n_time = stack.shape\n",
    "    n_pixel = n_row*n_col\n",
    "    result = np.zeros((n_pixel*(n_pixel-1)/2,))\n",
    "    stack_score = np.copy(stack)\n",
    "    stack_score = (stack_score - stack_score.mean(2).reshape((n_row,n_col,1)))/stack_score.std(2).reshape((n_row,n_col,1))\n",
    "    idx = 0\n",
    "    for i in range(n_pixel-1):\n",
    "        for j in range(i+1,n_pixel):\n",
    "            pos1 = (i//n_col,i-(i//n_col)*n_col)\n",
    "            pos2 = (j//n_col,j-(j//n_col)*n_col)\n",
    "            coef = np.sum(stack_score[pos1[0],pos1[1],:]*stack_score[pos2[0],pos2[1],:])/n_time\n",
    "            result[idx] = coef\n",
    "            idx += 1 \n",
    "    return result\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "corr10 = correlations(calibration['10ms/stack'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f799c3eaeb8>]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799de68278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "hist_corr10 = plt.hist(corr10,bins=100,normed=True,color='black')\n",
    "plt.xlabel(r'$\\rho(ij,uv)$',fontsize=25)\n",
    "plt.ylabel('Density',fontsize=25)\n",
    "plt.title('Density of correlation coefficients at 10 ms',fontsize=25)\n",
    "plt.plot(tt/10,norm.pdf(tt/10,0,0.1),color='orange',lw=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Empirical density in black, theoretical one, $\\mathcal{N}(0,0.01)$, in\n",
    "orange."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The empirical variance (x 100 and rounded to the third decimal) of the\n",
    "samples of correlation coefficients (1 sample per exposure duration)\n",
    "are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[['100ms',\n",
       "  '10ms',\n",
       "  '20ms',\n",
       "  '30ms',\n",
       "  '40ms',\n",
       "  '50ms',\n",
       "  '60ms',\n",
       "  '70ms',\n",
       "  '80ms',\n",
       "  '90ms'],\n",
       " [1.009, 1.01, 1.009, 1.01, 1.01, 1.01, 1.009, 1.01, 1.009, 1.009]]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var_of_corr_list = [np.var(correlations(calibration[n+'/stack'])) for n in list(calibration)]\n",
    "[[n for n in list(calibration)], [int(100000*v)/1000 for v in var_of_corr_list]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We wrote previously :\n",
    "-   If our model is correct we should have for each pixel $i,j$, for a\n",
    "    given exposure time, a mean value:\n",
    "    $$\\bar{y}_{i,j} = \\frac{1}{100} \\sum_{k=1}^{100} y_{i,j,k} \\approx G \\, p_{i,j} \\tau $$\n",
    "-   and a variance:\n",
    "    $$S_{i,j}^2 = \\frac{1}{99} \\sum_{k=1}^{100} (y_{i,j,k}-\\bar{y}_{i,j})^2 \\approx G^2 \\, (p_{i,j} \\tau+\\sigma_{R}^2) \\; .$$\n",
    "-   **The graph of $S_{i,j}^2$ *vs* $\\bar{y}_{i,j}$ should be a straight\n",
    "    line with slope $G$ ordinate at 0, $G^2 \\sigma_{R}^2$.**\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 1000)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "AAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799c1fb6a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "for x,y in zip([np.mean(calibration[n+'/stack'],2) for n in list(calibration)],\n",
    "               [np.var(calibration[n+'/stack'],2) for n in list(calibration)]):\n",
    "    plt.scatter(x.flatten(),y.flatten(),0.05)\n",
    "\n",
    "plt.xlabel(r'$\\overline{ADU}$',fontsize=25)\n",
    "plt.ylabel(\"Var(ADU)\",fontsize=25)\n",
    "plt.xlim([0,4500])\n",
    "plt.ylim([0,1000])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We do see the expected linear relation:\n",
    "$\\mathrm{Var}[ADU] = G^2 \\sigma_{R}^2 + G \\mathrm{E}[ADU]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear fit\n",
    "\n",
    "The [heteroscedasticity](http://en.wikipedia.org/wiki/Heteroscedasticity)\n",
    "(inhomogeneous variance) visible on the graph is also expected since the\n",
    "variance of a variance for an IID sample of size $K$ from a normal\n",
    "distribution with mean $\\mu$ and variance $\\sigma^2$ is:\n",
    "$$\\mathrm{Var}[S^2] = \\frac{2\\sigma^4}{(K-1)} \\; .$$\n",
    "\n",
    "-   This means than when we do our linear fit,\n",
    "    $$y_k = a + b x_k + \\sigma_k \\epsilon_k \\, ,$$ we should use\n",
    "    weights.\n",
    "-   Here $$x_k = \\overline{ADU}_k \\quad y_k = \\mathrm{Var}[ADU]_k \\, ,$$\n",
    "    $$b = G \\quad a = G^2 \\sigma_R^2 \\, .$$\n",
    "-   It's easy to show that the least square estimates are:\n",
    "    $$\\hat{a} = \\frac{1}{Z} \\sum_k \\frac{y_k-\\hat{b} x_k}{\\sigma_k^2} \\quad \\text{where} \\quad Z = \\sum_k \\frac{1}{\\sigma_k^2}$$\n",
    "    and\n",
    "    $$\\hat{b} = \\left(\\sum_k \\frac{x_k}{\\sigma_k^2} \\left(y_k - \\frac{1}{Z}\\sum_j \\frac{y_j}{\\sigma_j^2}\\right)\\right) / \\left(\\sum_k \\frac{x_k}{\\sigma_k^2}\\left(x_k - \\frac{1}{Z}\\sum_j \\frac{x_j}{\\sigma_j^2}\\right)\\right) \\, .$$\n",
    "-   We don't know $\\sigma_k$ but we have an estimation:\n",
    "    $\\hat{\\sigma}_k^2 = \\mathrm{Var}[S_k^2]$ we can \"plug-in\" this value\n",
    "    to get our weights.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_k = np.concatenate([x.flatten() for x in [np.mean(calibration[n+'/stack'],2) for n in list(calibration)]])\n",
    "y_k = np.concatenate([x.flatten() for x in [np.var(calibration[n+'/stack'],2) for n in list(calibration)]])\n",
    "sigma2_k = 2*y_k**2/(calibration['10ms/stack'].shape[2]-1)\n",
    "Z = sum(1/sigma2_k)\n",
    "num1 = sum(X_k/sigma2_k*(y_k-sum(y_k/sigma2_k)/Z))\n",
    "denom1 = sum(X_k/sigma2_k*(X_k-sum(X_k/sigma2_k)/Z))\n",
    "b_hat0 = num1/denom1\n",
    "a_hat0 = sum((y_k-b_hat0*X_k)/sigma2_k)/Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 1000)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "QmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799c0d2e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "for x,y in zip([np.mean(calibration[n+'/stack'],2) for n in list(calibration)],\n",
    "               [np.var(calibration[n+'/stack'],2) for n in list(calibration)]):\n",
    "    plt.scatter(x.flatten(),y.flatten(),0.05)\n",
    "\n",
    "plt.xlabel(r'$\\overline{ADU}$',fontsize=25)\n",
    "plt.ylabel(\"Var(ADU)\",fontsize=25)\n",
    "aa = np.linspace(0,4500)\n",
    "plt.plot(np.linspace(0,4500,101),a_hat0+b_hat0*np.linspace(0,4500,101),color='red',lw=2)\n",
    "plt.xlim([0,4500])\n",
    "plt.ylim([0,1000])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.1384399188990568, 289.85755480280824)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_hat0 = b_hat0\n",
    "S2_hat0 = a_hat0/b_hat0**2 \n",
    "(G_hat0, S2_hat0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have here $\\hat{G} = 0.14$ and $\\hat{\\sigma}_R^2 = 290$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Checking the fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x7f79a6401908>"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799c5540b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "for x,y in zip([np.mean(calibration[n+'/stack'],2) for n in list(calibration)],\n",
    "               [np.var(calibration[n+'/stack'],2) for n in list(calibration)]):\n",
    "    plt.scatter(a_hat0+b_hat0*x.flatten(),(y.flatten()-a_hat0-b_hat0*x.flatten())*np.sqrt(99/2)/(a_hat0+b_hat0*x.flatten()),0.05)\n",
    "\n",
    "plt.xlabel('Fitted value',fontsize=25)\n",
    "plt.ylabel(\"Normalized residuals (Normal scores)\",fontsize=25)\n",
    "plt.axhline(0,color='red',lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some remarks\n",
    "----------------\n",
    "\n",
    "-   When we use a linear regression, we are (implicitly) assuming that\n",
    "    the \"independent\" variable, here $\\overline{ADU}_k$, is *exactly*\n",
    "    known.\n",
    "-   This was clearly not the case here since $\\overline{ADU}_k$ was\n",
    "    measured (with an error).\n",
    "-   We could and will therefore refine our fit.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Error propagation and variance stabilization\n",
    "=========================\n",
    "\n",
    "Error propagation \n",
    "--------------------------\n",
    "\n",
    "-   Let us consider two random variables: $Y$ and $Z$ such that:\n",
    "-   $Y \\approx \\mathcal{N}(\\mu_Y,\\sigma^2_Y)$ or\n",
    "    $Y \\approx \\mu_Y + \\sigma_Y \\, \\epsilon$\n",
    "-   $Z = f(Y)$, with $f$ continuous and differentiable.\n",
    "-   Using a first order Taylor expansion we then\n",
    "    have:$$ \\begin{array}{lcl} Z & \\approx & f(\\mu_Y + \\sigma_Y \\, \\epsilon) \\\\ & \\approx & f(\\mu_Y) + \\sigma_Y \\, \\epsilon \\, \\frac{d f}{d Y}(\\mu_Y) \\end{array}$$\n",
    "-   $\\mathrm{E}Z \\approx f(\\mu_Y) = f(\\mathrm{E}Y)$\n",
    "-   $\\mathrm{Var}Z \\equiv \\mathrm{E}[(Z-\\mathrm{E}Z)^2] \\approx \\sigma^2_Y \\, \\frac{d f}{d Y}^2(\\mu_Y)$\n",
    "-   $Z \\approx f(\\mu_Y) + \\sigma_Y\\left| \\frac{d f}{d Y}(\\mu_Y)\\right| \\, \\epsilon$\n",
    "\n",
    "Variance stabilization\n",
    "------------------------------\n",
    "\n",
    "-   For our CCD model we have (for a given pixel):\n",
    "    $$Y \\sim G \\, \\lambda + \\sqrt{G^2 \\, (\\lambda+\\sigma_{R}^2)} \\, \\epsilon = \\mu_Y + \\sqrt{G \\, \\mu_Y + G^2 \\, \\sigma_{R}^2} \\, \\epsilon \\, .$$\n",
    "-   Then if $Z = f(Y)$ we get:\n",
    "    $$Z \\approx f(\\mu_Y) + \\mid f'(\\mu_Y) \\mid G \\sqrt{\\mu_Y / G+\\sigma_{R}^2} \\, \\epsilon$$\n",
    "-   What happens then if we take: \\$f(x) = 2  \n",
    "    \\sqrt{x/G + \\sigma_{R}^2} \\$?\n",
    "-   We have: $$f'(x) = \\frac{1}{G \\sqrt{ x / G + \\sigma_{R}^2}}$$\n",
    "-   Leading to:\n",
    "    $$Z \\approx 2 \\, \\sqrt{\\mu_Y / G + \\sigma_{R}^2} + \\epsilon$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x7f79a648d9e8>"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "IAeBZmZmZiPIQaCZmZnZCPovcoab8p6VKtcAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799c327358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "for x,y in zip([np.mean(2*np.sqrt(calibration[n+'/stack']/G_hat0+S2_hat0),2) for n in list(calibration)],\n",
    "               [np.var(2*np.sqrt(calibration[n+'/stack']/G_hat0+S2_hat0),2) for n in list(calibration)]):\n",
    "    plt.scatter(x.flatten(),y.flatten(),0.05)\n",
    "\n",
    "plt.xlabel(r'$\\mathrm{E}(2 \\sqrt{ADU/\\hat{G}+\\hat{\\sigma}_R^2})$',fontsize=20)\n",
    "plt.ylabel(r'$\\mathrm{Var}(2 \\sqrt{ADU/\\hat{G}+\\hat{\\sigma}_R^2})$',fontsize=20)\n",
    "plt.xlim([100,350])\n",
    "plt.axhline(1,color='red',lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Application \n",
    "======="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Back to where we started\n",
    "---------------------------------\n",
    "\n",
    "![](figs/POMC-raw-data.png)\n",
    "\n",
    "ADU counts (raw data) from Fura-2 excited at 340 nm. Each square\n",
    "corresponds to a pixel. 25.05 s of data are shown. Same scale on each\n",
    "sub-plot. Data recorded by Andreas Pippow (Kloppenburg Lab. Cologne University)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quick ROI detection: Motivation\n",
    "------------------------------------------\n",
    "\n",
    "-   After variance stabilization:\n",
    "    $Z_{i,j,k} = 2 \\, \\sqrt{ADU_{i,j,k} / G + \\sigma_{R}^2}$, the\n",
    "    variance at each pixel $(i,j)$ at each time, $k$, should be 1.\n",
    "-   If a pixel contains no dynamical signal---that is nothing more than\n",
    "    a constant background signal---the following statistics:\n",
    "    $$RSS_{i,j} \\equiv \\sum_{k=1}^{K} (Z_{i,j,k} - \\overline{Z}_{i,j})^2 \\quad \\mathrm{with} \\quad \\overline{Z}_{i,j} \\equiv \\frac{1}{K} \\sum_{k=1}^{K} Z_{i,j,k}$$\n",
    "    should follow a $\\chi^2$ distribution with $K-1$ degrees of freedom.\n",
    "-   We could therefore compute the values of the complementary\n",
    "    cumulative distribution function of the theoretical $\\chi_{K-1}^2$\n",
    "    distribution:$$1 - F_{\\chi_{K-1}^2}(RSS_{i,j})$$ and look for very\n",
    "    small values---that is very small probabilities---(using a log scale\n",
    "    helps here).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "stack_pomc_stab = 2*np.sqrt(np.copy(stack_pomc)/G_hat0+S2_hat0)\n",
    "stack_pomc_stab_rss = np.sum((stack_pomc_stab-stack_pomc_stab.mean(2).reshape((60,80,1)))**2,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "RU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7997e2b080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import chi2\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.subplot(121)\n",
    "plt.contour(range(60),range(80),\n",
    "            np.transpose(np.log(chi2.sf(stack_pomc_stab_rss,df=stack_pomc_stab.shape[2]-1))),\n",
    "            linewidths=2,linestyles='solid',colors='black')\n",
    "plt.grid()\n",
    "plt.title('Full field', fontsize=20)\n",
    "plt.subplot(122)\n",
    "plt.contour(range(60),range(80),\n",
    "            np.transpose(np.log(chi2.sf(stack_pomc_stab_rss,df=stack_pomc_stab.shape[2]-1))),\n",
    "            linewidths=2,linestyles='solid',colors='black')\n",
    "plt.xlim([23,33])\n",
    "plt.ylim([33,43])\n",
    "plt.title('Cell body', fontsize=20)\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Contour plots of $\\log\\left(1 - F_{\\chi_{K-1}^2}(RSS_{i,j})\\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pointwise time course estimation\n",
    "---------------------------------------------\n",
    "\n",
    "-   We are going to be (very) conservative and keep as our ROI the\n",
    "    pixels having an\n",
    "    $\\log\\left(1 - F_{\\chi_{K-1}^2}(RSS)\\right) \\le -300$.\n",
    "-   We are then left with 12 pixels.\n",
    "-   We are going to model the fluorescence intensity of each of these\n",
    "    pixels by: $$S_{i,j}(t) = \\phi_{i,j} \\, f(t) + b \\; ,$$ where $f(t)$\n",
    "    is a signal time course to all pixels of the ROI, $\\phi_{i,j}$ is a\n",
    "    pixel specific parameter and $b$ is a background fluorescence\n",
    "    assumed identical for each pixel.\n",
    "-   The time $t$ is in fact a discrete variable, $t = \\delta \\, k$\n",
    "    ($\\delta$ = 150 ms) and we are seeking a pointwise estimation:\n",
    "    $\\{f_1,f_2,\\ldots,f_K\\}$ ($K$ = 168) where $f_k = f(\\delta \\, k)$.\n",
    "-   We end up with 12 ($\\phi_{i,j}$) + 168 ($f_k$) + 1 ($b$) = 181\n",
    "    parameters for 12 x 168 = 2016 measurements.\n",
    "-   We need to add a constraint since with our model specification:\n",
    "    $$S_{i,j,k} = \\phi_{i,j} \\, f_k + b \\; ,$$ we can multiply all the\n",
    "    $\\phi_{i,j}$ by 2 and divide all the $f_k$ by 2 and get the same\n",
    "    prediction.\n",
    "-   We are going to set $f_k=1$ for the first 5 time points (the\n",
    "    stimulation comes at the 11th) and our pointwise estimation relates\n",
    "    to what is usually done with this type of data, $\\Delta S(t) / S_0$\n",
    "    (where $S_0$ is a baseline average) through:\n",
    "    $$\\Delta S(t) / S_0 = \\frac{S(t) - S_0}{S_0} = f(t) - 1 + \\mathrm{noise}\\, .$$\n",
    "-   **Notice that no independent background measurement is used**.\n",
    "-   With variance stabilization we end up minimizing:\n",
    "    $$RSS\\left(b,(\\phi_{i,j}),(f_k)_{k=5,\\ldots,168}\\right) = \\sum_{(i,j) \\in \\mathrm{ROI}} \\sum_{k=1}^{168} \\left(Z_{ijk}-F_{ijk}\\right)^2 \\, ,$$\n",
    "    where $$Z_{ijk} = 2*\\sqrt{ADU_{ijk}/\\hat{G}+\\hat{\\sigma}_R^2}$$ and\n",
    "    $$F_{ijk} = 2*\\sqrt{\\phi_{i,j} \\, f_k + b + \\hat{\\sigma}_R^2}\\, .$$\n",
    "-   If our model is correct we should have:\n",
    "    $$RSS\\left(\\hat{b},(\\hat{\\phi}_{i,j}),(\\hat{f}_k)_{k=5,\\ldots,168}\\right) \\sim \\chi^2_{12 \\times 168 - 176}\\, .$$\n",
    "-   The method also generates confidence intervals for the estimated\n",
    "    parameters.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pointwise time course estimation: Technical details\n",
    "----------------------------------------------------------------\n",
    "\n",
    "-   To solve this 176 dimensional optimization problem in a reasonable\n",
    "    time (\\< 10 s) in `Python` we use Newton's method with conjugate\n",
    "    gradients for the inversion of the Hessian matrix.\n",
    "-   That means we have to define a function returning the\n",
    "    gradient---vector of first derivatives---and the Hessian---matrix of\n",
    "    second derivatives---of the `RSS` we just defined (**that's a\n",
    "    painful work**).\n",
    "-   To improve numerical behavior, since all parameters are positive, we\n",
    "    work with the log of the parameters.\n",
    "\n",
    "Doing the fit\n",
    "------------------------\n",
    "\n",
    "### Symbolic derivation setting\n",
    "\n",
    "We want now to define three functions :\n",
    "\n",
    "1.  One that returns the residuals sum of square given the model's\n",
    "    parameters (and the data):\n",
    "    \\$RSS\\left(b,(*ϕ*~i,j~),(f~k~)~k=5,…,168~\\right) =\n",
    "    ∑~(i,j)\\\\  ∈ *R**O**I*~ ∑~k=1~^168^ \\left(Z~ijk~-F~ijk~\\right)^2^  \n",
    "    ,\\$ where $Z_{ijk} = 2*\\sqrt{ADU_{ijk}/\\hat{G}+\\hat{\\sigma}_R^2}$\n",
    "    and $F_{ijk} = 2*\\sqrt{\\phi_{i,j} \\, f_k + b + \\hat{\\sigma}_R^2}$.\n",
    "2.  One that returns the gradient vector of this RSS.\n",
    "3.  One that returns the Hessian matrix.\n",
    "\n",
    "As explained above, since all the model parameters are positive we are\n",
    "going to define these three functions as functions of the log of the\n",
    "parameters. We are going to use the `SymPy` module to get the analytical\n",
    "expressions we need to define our functions.\n",
    "\n",
    "We start by defining a symbolic \"unitary\" rss corresponding to one term\n",
    "$(i,j,k)$ in the sum:\n",
    "$$\\left(2*\\sqrt{ADU_{ijk}/\\hat{G}+\\hat{\\sigma}_R^2} - 2*\\sqrt{\\exp \\log \\phi_{i,j} \\, \\exp \\log f_k + \\exp \\log b + \\hat{\\sigma}_R^2}\\right)^2 \\, .$$\n",
    "\n",
    "In the code that follows:\n",
    "\n",
    "-   `y_k` stands for $2*\\sqrt{ADU_{ijk}/\\hat{G}+\\hat{\\sigma}_R^2}$.\n",
    "-   `logphi_i` stands for $\\log \\phi_{i,j}$.\n",
    "-   `logf_k` stands for $\\log f_k$.\n",
    "-   `logb_sy` stands for $\\log b$.\n",
    "-   `S2_L` stands for $\\hat{\\sigma}_R^2$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/latex": [
       "$$\\left(y_{k} - 2 \\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}\\right)^{2}$$"
      ],
      "text/plain": [
       "                                                  2\n",
       "⎛           _____________________________________⎞ \n",
       "⎜          ╱          logb_sy    logf_k  logphiᵢ ⎟ \n",
       "⎝y_k - 2⋅╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ        ⎠ "
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_k,logphi_i,logf_k,logb_sy,S2_L = sy.symbols('y_k,logphi_i,logf_k,logb_sy,S2_L',real=True)\n",
    "rss_u = (y_k - 2*sy.sqrt(sy.exp(logphi_i)*sy.exp(logf_k)+sy.exp(logb_sy)+S2_L))**2\n",
    "rss_u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symbolic gradient\n",
    "\n",
    "We get the first derivative with respect to `logb_sy` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/latex": [
       "$$- \\frac{2 \\left(y_{k} - 2 \\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}\\right) e^{logb_{sy}}}{\\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}}$$"
      ],
      "text/plain": [
       "   ⎛           _____________________________________⎞          \n",
       "   ⎜          ╱          logb_sy    logf_k  logphiᵢ ⎟  logb_sy \n",
       "-2⋅⎝y_k - 2⋅╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ        ⎠⋅ℯ        \n",
       "───────────────────────────────────────────────────────────────\n",
       "               _____________________________________           \n",
       "              ╱          logb_sy    logf_k  logphiᵢ            \n",
       "            ╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ                   "
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.diff(rss_u,logb_sy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-2*(y_k - 2*sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i)))*exp(logb_sy)/sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i))\n"
     ]
    }
   ],
   "source": [
    "print(sy.diff(rss_u,logb_sy))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over all the pixels $(i,j)$ at all\n",
    "times $k$. We get the first derivative with respect to `logphi_i` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "5p2YJk+G3bQR9A5nuIyfzMisy0wzU4XTAIZNJMUQ7mrPQPfLZ826TietGv1f89rfSshXz8l6u2dK\n",
       "Xs/AWcvTveORDdz+h6kpePTE2iUtbvw/1qP42e78Ffw8ekHOyRn4H2AYKMKbH399AAAAAElFTkSu\n",
       "QmCC\n"
      ],
      "text/latex": [
       "$$- \\frac{2 \\left(y_{k} - 2 \\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}\\right) e^{logf_{k}} e^{logphi_{i}}}{\\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}}$$"
      ],
      "text/plain": [
       "   ⎛           _____________________________________⎞                  \n",
       "   ⎜          ╱          logb_sy    logf_k  logphiᵢ ⎟  logf_k  logphiᵢ \n",
       "-2⋅⎝y_k - 2⋅╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ        ⎠⋅ℯ      ⋅ℯ        \n",
       "───────────────────────────────────────────────────────────────────────\n",
       "                   _____________________________________               \n",
       "                  ╱          logb_sy    logf_k  logphiᵢ                \n",
       "                ╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ                       "
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.diff(rss_u,logphi_i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-2*(y_k - 2*sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i)))*exp(logf_k)*exp(logphi_i)/sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i))\n"
     ]
    }
   ],
   "source": [
    "print(sy.diff(rss_u,logphi_i))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over **a single pixel** $(i,j)$ at all\n",
    "times $k$. We get the first derivative with respect to `logf_k` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "QmCC\n"
      ],
      "text/latex": [
       "$$- \\frac{2 \\left(y_{k} - 2 \\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}\\right) e^{logf_{k}} e^{logphi_{i}}}{\\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}}$$"
      ],
      "text/plain": [
       "   ⎛           _____________________________________⎞                  \n",
       "   ⎜          ╱          logb_sy    logf_k  logphiᵢ ⎟  logf_k  logphiᵢ \n",
       "-2⋅⎝y_k - 2⋅╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ        ⎠⋅ℯ      ⋅ℯ        \n",
       "───────────────────────────────────────────────────────────────────────\n",
       "                   _____________________________________               \n",
       "                  ╱          logb_sy    logf_k  logphiᵢ                \n",
       "                ╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ                       "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.diff(rss_u,logf_k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-2*(y_k - 2*sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i)))*exp(logf_k)*exp(logphi_i)/sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i))\n"
     ]
    }
   ],
   "source": [
    "print(sy.diff(rss_u,logf_k))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over **all the pixels** $(i,j)$ at **a\n",
    "single time** $k$.\n",
    "\n",
    "### Symbolic Hessian\n",
    "\n",
    "We get the second derivative with respect to `logb_sy` with:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "kFGSRARQsFu2rgVbaQjVnDkYRwukLTW6uV6Hz79+/XTJInNpOpvcmMV3MROUJHFdjq4FhSG11XMq\n",
       "xey/YqXjospgQrLiyq/vCKlLliSLhQdqm2xmgcoZrL6SEa2cu7kVY97ozb+1zeIeG2UQCaFb3Xnl\n",
       "anfLnarepPjulvPQK53JA0YN5V049H9fhvgQHSOd8wAAAABJRU5ErkJggg==\n"
      ],
      "text/latex": [
       "$$\\frac{y_{k} e^{2 logb_{sy}}}{\\left(S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}\\right)^{\\frac{3}{2}}} + \\left(- \\frac{2 y_{k}}{\\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}} + 4\\right) e^{logb_{sy}}$$"
      ],
      "text/plain": [
       "                  2⋅logb_sy                                                   \n",
       "             y_k⋅ℯ                         ⎛                   2⋅y_k          \n",
       "──────────────────────────────────────── + ⎜- ────────────────────────────────\n",
       "                                     3/2   ⎜     _____________________________\n",
       "⎛         logb_sy    logf_k  logphiᵢ⎞      ⎜    ╱          logb_sy    logf_k  \n",
       "⎝S_2_L + ℯ        + ℯ      ⋅ℯ       ⎠      ⎝  ╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ\n",
       "\n",
       "                      \n",
       "            ⎞  logb_sy\n",
       "──────── + 4⎟⋅ℯ       \n",
       "________    ⎟         \n",
       "logphiᵢ     ⎟         \n",
       "            ⎠         "
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.collect(sy.simplify(sy.diff(rss_u,logb_sy,logb_sy)),sy.exp(logb_sy))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y_k*exp(2*logb_sy)/(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i))**(3/2) + (-2*y_k/sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i)) + 4)*exp(logb_sy)\n"
     ]
    }
   ],
   "source": [
    "print(sy.collect(sy.simplify(sy.diff(rss_u,logb_sy,logb_sy)),sy.exp(logb_sy)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over **all the pixels** $(i,j)$ at\n",
    "**all times** $k$. We get the second derivative with respect to\n",
    "`logb_sy` and `logphi_i` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "RK5CYII=\n"
      ],
      "text/latex": [
       "$$\\frac{y_{k} e^{logb_{sy} + logf_{k} + logphi_{i}}}{\\left(S_{2 L} + e^{logb_{sy}} + e^{logf_{k} + logphi_{i}}\\right)^{\\frac{3}{2}}}$$"
      ],
      "text/plain": [
       "          logb_sy + logf_k + logphiᵢ     \n",
       "     y_k⋅ℯ                               \n",
       "─────────────────────────────────────────\n",
       "                                      3/2\n",
       "⎛         logb_sy    logf_k + logphiᵢ⎞   \n",
       "⎝S_2_L + ℯ        + ℯ                ⎠   "
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.collect(sy.simplify(sy.diff(rss_u,logb_sy,logphi_i)),sy.exp(logf_k)*sy.exp(logphi_i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y_k*exp(logb_sy + logf_k + logphi_i)/(S2_L + exp(logb_sy) + exp(logf_k + logphi_i))**(3/2)\n"
     ]
    }
   ],
   "source": [
    "print(sy.collect(sy.simplify(sy.diff(rss_u,logb_sy,logphi_i)),sy.exp(logf_k)*sy.exp(logphi_i)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over **a single pixel** $(i,j)$ at\n",
    "**all times** $k$. We get the second derivative with respect to\n",
    "`logb_sy` and `logf_k` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "03GIfoRNyRjxWZbb5EvycpuMqHSEnmfivuEzyf6VWqLBFtEP/Y9RFT5p3F+iUp3AgDl1olH6hF1P\n",
       "vfwGbKZS7BDfWCop+rkDypKSs5ziBKrTIBrpX2tCaKt8kaMnM2T/SH+vAV/ZkOhHXTwsYsuPInAa\n",
       "RCN/Nezs7LywJWJGCrgMWLYpyjho5aHmwIop/CMuxM9Tbh544cWEPgbVskElbY//e605sGImzZCg\n",
       "94T+xPg4nv1z+fm8cDRsnCVX4t/HZEfj9dKWsNI1fV+9rj3a31d5if8AS+BNLWILZxwAAAAASUVO\n",
       "RK5CYII=\n"
      ],
      "text/latex": [
       "$$\\frac{y_{k} e^{logb_{sy} + logf_{k} + logphi_{i}}}{\\left(S_{2 L} + e^{logb_{sy}} + e^{logf_{k} + logphi_{i}}\\right)^{\\frac{3}{2}}}$$"
      ],
      "text/plain": [
       "          logb_sy + logf_k + logphiᵢ     \n",
       "     y_k⋅ℯ                               \n",
       "─────────────────────────────────────────\n",
       "                                      3/2\n",
       "⎛         logb_sy    logf_k + logphiᵢ⎞   \n",
       "⎝S_2_L + ℯ        + ℯ                ⎠   "
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.collect(sy.simplify(sy.diff(rss_u,logb_sy,logf_k)),sy.exp(logf_k)*sy.exp(logphi_i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y_k*exp(logb_sy + logf_k + logphi_i)/(S2_L + exp(logb_sy) + exp(logf_k + logphi_i))**(3/2)\n"
     ]
    }
   ],
   "source": [
    "print(sy.collect(sy.simplify(sy.diff(rss_u,logb_sy,logf_k)),sy.exp(logf_k)*sy.exp(logphi_i)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over **all the pixels** $(i,j)$ at\n",
    "**at a single time** $k$. We get the second derivative with respect to\n",
    "`logphi_i` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/latex": [
       "$$\\frac{y_{k} e^{2 logf_{k}} e^{2 logphi_{i}}}{\\left(S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}\\right)^{\\frac{3}{2}}} + \\left(- \\frac{2 y_{k}}{\\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}} + 4\\right) e^{logf_{k}} e^{logphi_{i}}$$"
      ],
      "text/plain": [
       "             2⋅logf_k  2⋅logphiᵢ                                              \n",
       "        y_k⋅ℯ        ⋅ℯ                    ⎛                   2⋅y_k          \n",
       "──────────────────────────────────────── + ⎜- ────────────────────────────────\n",
       "                                     3/2   ⎜     _____________________________\n",
       "⎛         logb_sy    logf_k  logphiᵢ⎞      ⎜    ╱          logb_sy    logf_k  \n",
       "⎝S_2_L + ℯ        + ℯ      ⋅ℯ       ⎠      ⎝  ╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ\n",
       "\n",
       "                              \n",
       "            ⎞  logf_k  logphiᵢ\n",
       "──────── + 4⎟⋅ℯ      ⋅ℯ       \n",
       "________    ⎟                 \n",
       "logphiᵢ     ⎟                 \n",
       "            ⎠                 "
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.collect(sy.simplify(sy.diff(rss_u,logphi_i,logphi_i)),sy.exp(logf_k)*sy.exp(logphi_i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y_k*exp(2*logf_k)*exp(2*logphi_i)/(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i))**(3/2) + (-2*y_k/sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i)) + 4)*exp(logf_k)*exp(logphi_i)\n"
     ]
    }
   ],
   "source": [
    "print(sy.collect(sy.simplify(sy.diff(rss_u,logphi_i,logphi_i)),sy.exp(logf_k)*sy.exp(logphi_i)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over **a single pixel** $(i,j)$ at\n",
    "**all times** $k$. The second derivative with respect to the $\\phi$ of\n",
    "two different pixels is null. We get the second derivative with respect\n",
    "to `logphi_i` and `logf_k` with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/latex": [
       "$$\\frac{y_{k} e^{2 logf_{k}} e^{2 logphi_{i}}}{\\left(S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}\\right)^{\\frac{3}{2}}} + \\left(- \\frac{2 y_{k}}{\\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}} + 4\\right) e^{logf_{k}} e^{logphi_{i}}$$"
      ],
      "text/plain": [
       "             2⋅logf_k  2⋅logphiᵢ                                              \n",
       "        y_k⋅ℯ        ⋅ℯ                    ⎛                   2⋅y_k          \n",
       "──────────────────────────────────────── + ⎜- ────────────────────────────────\n",
       "                                     3/2   ⎜     _____________________________\n",
       "⎛         logb_sy    logf_k  logphiᵢ⎞      ⎜    ╱          logb_sy    logf_k  \n",
       "⎝S_2_L + ℯ        + ℯ      ⋅ℯ       ⎠      ⎝  ╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ\n",
       "\n",
       "                              \n",
       "            ⎞  logf_k  logphiᵢ\n",
       "──────── + 4⎟⋅ℯ      ⋅ℯ       \n",
       "________    ⎟                 \n",
       "logphiᵢ     ⎟                 \n",
       "            ⎠                 "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.collect(sy.simplify(sy.diff(rss_u,logphi_i,logf_k)),sy.exp(logf_k)*sy.exp(logphi_i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y_k*exp(2*logf_k)*exp(2*logphi_i)/(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i))**(3/2) + (-2*y_k/sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i)) + 4)*exp(logf_k)*exp(logphi_i)\n"
     ]
    }
   ],
   "source": [
    "print(sy.collect(sy.simplify(sy.diff(rss_u,logphi_i,logf_k)),sy.exp(logf_k)*sy.exp(logphi_i)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over **a single pixel** $(i,j)$ at **a\n",
    "single time** $k$. We get the second derivative with respect to `logf_k`\n",
    "with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/latex": [
       "$$\\frac{y_{k} e^{2 logf_{k}} e^{2 logphi_{i}}}{\\left(S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}\\right)^{\\frac{3}{2}}} + \\left(- \\frac{2 y_{k}}{\\sqrt{S_{2 L} + e^{logb_{sy}} + e^{logf_{k}} e^{logphi_{i}}}} + 4\\right) e^{logf_{k}} e^{logphi_{i}}$$"
      ],
      "text/plain": [
       "             2⋅logf_k  2⋅logphiᵢ                                              \n",
       "        y_k⋅ℯ        ⋅ℯ                    ⎛                   2⋅y_k          \n",
       "──────────────────────────────────────── + ⎜- ────────────────────────────────\n",
       "                                     3/2   ⎜     _____________________________\n",
       "⎛         logb_sy    logf_k  logphiᵢ⎞      ⎜    ╱          logb_sy    logf_k  \n",
       "⎝S_2_L + ℯ        + ℯ      ⋅ℯ       ⎠      ⎝  ╲╱  S_2_L + ℯ        + ℯ      ⋅ℯ\n",
       "\n",
       "                              \n",
       "            ⎞  logf_k  logphiᵢ\n",
       "──────── + 4⎟⋅ℯ      ⋅ℯ       \n",
       "________    ⎟                 \n",
       "logphiᵢ     ⎟                 \n",
       "            ⎠                 "
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.collect(sy.simplify(sy.diff(rss_u,logf_k,logf_k)),sy.exp(logf_k)*sy.exp(logphi_i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y_k*exp(2*logf_k)*exp(2*logphi_i)/(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i))**(3/2) + (-2*y_k/sqrt(S2_L + exp(logb_sy) + exp(logf_k)*exp(logphi_i)) + 4)*exp(logf_k)*exp(logphi_i)\n"
     ]
    }
   ],
   "source": [
    "print(sy.collect(sy.simplify(sy.diff(rss_u,logf_k,logf_k)),sy.exp(logf_k)*sy.exp(logphi_i)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When this \"unitary\" expression is re-injected into the \"full\" $RSS$\n",
    "expression, the summation is made over **all the pixels** $(i,j)$ at **a\n",
    "single time** $k$. The block of the Hessian matrix generated by the\n",
    "second derivatives with respect to the `logf_k` parameters is\n",
    "**diagonal**.\n",
    "\n",
    "### Extracting the data for the fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.stats import chi2\n",
    "threshold = -300\n",
    "good_pix = np.where(np.log(chi2.sf(stack_pomc_stab_rss,df=stack_pomc_stab.shape[2]-1)) < -300)\n",
    "data4fit = stack_pomc_stab[good_pix[0],good_pix[1],:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions definitions\n",
    "\n",
    "We come now to the definition of the three functions ($RSS$, gradient\n",
    "and Hessian). We start in fact with a \"constructor\" function, that is a\n",
    "function taking the data as argument and returning a triplet of\n",
    "functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def mk_rss(data=data4fit,\n",
    "           S2=S2_hat0,\n",
    "           base_length=5):\n",
    "    npix,K = data.shape\n",
    "    f = np.ones((K,))\n",
    "    nabla = np.zeros((K-base_length+npix+1,))\n",
    "    hessian = np.zeros((K-base_length+npix+1,K-base_length+npix+1))\n",
    "    d2phi = np.zeros((npix,))\n",
    "    dphidf = np.zeros((npix,K))\n",
    "    d2f = np.zeros((K,))\n",
    "    pred = np.zeros(data.shape)\n",
    "    def rss(par):\n",
    "        par = np.exp(np.copy(par))\n",
    "        b = par[0]\n",
    "        phi = par[1:(npix+1)]\n",
    "        f[:]=1.\n",
    "        f[base_length:] = par[(npix+1):]\n",
    "        pred[:,:] = 2*np.sqrt(np.outer(phi,f)+b+S2)\n",
    "        return np.sum((data-pred)**2)\n",
    "    def grad(par):\n",
    "        par = np.exp(np.copy(par))\n",
    "        b = par[0]\n",
    "        phi = par[1:(npix+1)]\n",
    "        f[:]=1.\n",
    "        f[base_length:] = par[(npix+1):]\n",
    "        pred[:,:] = np.sqrt(np.outer(phi,f)+b+S2)\n",
    "        nabla[0] = np.sum(-2*b*(data - 2*pred)/pred)\n",
    "        nabla[1:(npix+1)] = np.sum(-2*f.reshape((1,K))*phi.reshape((npix,1))*(data - 2*pred)/pred,1)\n",
    "        nabla[(npix+1):] = np.sum(-2*f.reshape((1,K))*phi.reshape((npix,1))*(data - 2*pred)/pred,0)[base_length:]\n",
    "        return nabla\n",
    "    def hess(par):\n",
    "        par = np.exp(np.copy(par))\n",
    "        b = par[0]\n",
    "        phi = par[1:(npix+1)]\n",
    "        f[:]=1.\n",
    "        f[base_length:] = par[(npix+1):]\n",
    "        pred[:,:] = np.sqrt(np.outer(phi,f)+b+S2)\n",
    "        hessian[:,:] = 0.\n",
    "        hessian[0,0] = np.sum(data*b**2/pred**3+(4-2*data/pred)*b)\n",
    "        hessian[0,1:(npix+1)] = np.sum(data*b*f.reshape((1,K))*phi.reshape((npix,1))/pred**3,1)\n",
    "        hessian[1:(npix+1),0] = np.sum(data*b*f.reshape((1,K))*phi.reshape((npix,1))/pred**3,1)\n",
    "        hessian[0,(npix+1):] = np.sum(data*b*f.reshape((1,K))*phi.reshape((npix,1))/pred**3,0)[base_length:]\n",
    "        hessian[(npix+1):,0] = np.sum(data*b*f.reshape((1,K))*phi.reshape((npix,1))/pred**3,0)[base_length:]\n",
    "        dphidf[:,:] = data*f.reshape((1,K))**2*phi.reshape((npix,1))**2/pred**3+(4-2*data/pred)*f.reshape((1,K))*phi.reshape((npix,1))\n",
    "        d2phi[:] = np.sum(dphidf,1)\n",
    "        for i in range(npix):\n",
    "            hessian[1+i,1+i] = d2phi[i]\n",
    "            hessian[1+i,(npix+1):] = dphidf[i,base_length:]\n",
    "            hessian[(npix+1):,i+1] = dphidf[i,base_length:]\n",
    "        d2f[:] = np.sum(dphidf,0)\n",
    "        for i in range(K-base_length):\n",
    "            hessian[(npix+1+i),(npix+1+i)] = d2f[i+base_length]\n",
    "        return hessian\n",
    "    return (rss,grad,hess)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then \"instantiate\" the three functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rss0, grad0, hess0 = mk_rss(data4fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Getting the initial guesses\n",
    "\n",
    "We use a classical $\\Delta F / F_0$ to get the initial guesses. We set\n",
    "the (initial guess for the) background fluorescence to an arbitrary\n",
    "small but postive value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "baseline = stack_pomc[good_pix[0],good_pix[1],:5].mean(1)\n",
    "f_0 = np.mean(stack_pomc[good_pix[0],good_pix[1],:]/baseline.reshape((12,1)),0)\n",
    "b_0 = 100\n",
    "phi_0 = np.mean((stack_pomc[good_pix[0],good_pix[1],:]-b_0)/f_0.reshape((1,168)),1)\n",
    "par_0 = np.zeros((1+12+163,))\n",
    "par_0[0] = b_0\n",
    "par_0[1:13] = phi_0\n",
    "par_0[13:] = f_0[5:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Checking the analytical expressions\n",
    "\n",
    "The codes returning the gradient and Hessian matrix are somewhat\n",
    "tedious, meaning that we could have made mistakes typing them. It's\n",
    "therefore a good policy to check their results against slow but less\n",
    "error prone codes doing the gradient and Hessian matrix evaluation\n",
    "numerically. We define first a function returning the gradient using the\n",
    "central difference:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def grad_num(fct=rss0,\n",
    "             par=np.log(np.copy(par_0)),\n",
    "             delta=1e-6):\n",
    "    n_par = len(par)\n",
    "    res = np.zeros((n_par,))\n",
    "    for i in range(n_par):\n",
    "        par[i] += delta\n",
    "        res[i] += fct(par)\n",
    "        par[i] -= 2*delta\n",
    "        res[i] -= fct(par)\n",
    "        par[i] += delta\n",
    "    return res/2/delta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define next a function returning a numerical evaluation of the\n",
    "Hessian matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def hess_num(fct=rss0,\n",
    "             par=np.log(np.copy(par_0)),\n",
    "             delta=1e-4):\n",
    "    n_par = len(par)\n",
    "    res = np.zeros((n_par,n_par))\n",
    "    f0 = fct(par)\n",
    "    for i in range(n_par):\n",
    "        for j in range(i,n_par):\n",
    "            par[i] += delta\n",
    "            par[j] += delta\n",
    "            f3 = fct(par)\n",
    "            par[i] -= delta\n",
    "            f2 = fct(par)\n",
    "            par[i] += delta\n",
    "            par[j] -= delta\n",
    "            f1 = fct(par)\n",
    "            par[i] -= delta\n",
    "            res[i,j] = (f3-f2-f1+f0)/delta**2\n",
    "            if j > i:\n",
    "                res[j,i] = (f3-f2-f1+f0)/delta**2\n",
    "    return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now do the tests:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial RSS value: \n",
      " 17547562.449 \n",
      "\n",
      "Maximal relative difference (in absolute value) between analytical and numerical gradient: \n",
      " 5.89580034732e-08 \n",
      "\n",
      "Maximal relative difference (in absolute value) between analytical and numerical Hessian: \n",
      " 0.000847340050286 \n",
      "\n"
     ]
    }
   ],
   "source": [
    "g0 = grad0(np.log(par_0))\n",
    "g0n = grad_num(rss0,np.log(np.copy(par_0)))\n",
    "h0 = hess0(np.log(par_0))\n",
    "h0n = hess_num(rss0,np.log(np.copy(par_0)),delta=1e-3)\n",
    "not_zero = np.abs(h0) > 0.\n",
    "print('Initial RSS value: \\n', rss0(np.log(par_0)), '\\n')\n",
    "print('Maximal relative difference (in absolute value) between analytical and numerical gradient: \\n', np.max(np.abs(g0-g0n)/np.abs(g0)),'\\n')\n",
    "print('Maximal relative difference (in absolute value) between analytical and numerical Hessian: \\n', np.max(np.abs(h0n[not_zero]-h0[not_zero])/np.abs(h0[not_zero])), '\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Doing the fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 1975.518323\n",
      "         Iterations: 178\n",
      "         Function evaluations: 347\n",
      "         Gradient evaluations: 524\n",
      "         Hessian evaluations: 178\n"
     ]
    }
   ],
   "source": [
    "from scipy.optimize import minimize\n",
    "res = minimize(rss0,np.log(par_0),method='Newton-CG',jac=grad0,hess=hess0,options={'disp': True, 'xtol': 1e-8})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see here the clear reduction in the $RSS$, starting at 17547562 and\n",
    "ending at 1975.\n",
    "\n",
    "### Definition of a function returning the prediction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def mk_pred(p_vector,\n",
    "            log_scale=False,\n",
    "            stabilise_var=True,\n",
    "            S2=S2_hat0,\n",
    "            G=G_hat0,\n",
    "            base_length=5):\n",
    "    if log_scale:\n",
    "        p_vector = np.exp(p_vector)\n",
    "    phi = p_vector[1:13]\n",
    "    b = p_vector[0]\n",
    "    f = np.ones((168,))\n",
    "    f[base_length:] = p_vector[13:]\n",
    "    pred = np.outer(phi,f)+b\n",
    "    if stabilise_var:\n",
    "        return 2*np.sqrt(pred+S2)\n",
    "    else:\n",
    "        return G*pred"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pointwise time course estimation: Time course estimate\n",
    "-------------------------------------------------------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 25)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "JKnH/h/Nylh3+kaFQQAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799c109e48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_hat = np.zeros((168,))\n",
    "f_hat[5:] = res.x[13:]\n",
    "par_se = np.sqrt(np.diag(scipy.linalg.inv(hess0(res.x))))\n",
    "f_up = np.zeros((168,))\n",
    "f_up[5:] = res.x[13:] + 1.96*par_se[13:]\n",
    "f_low = np.zeros((168,))\n",
    "f_low[5:] = res.x[13:] - 1.96*par_se[13:]\n",
    "fig, ax = plt.subplots(dpi=600,figsize=(10,8))\n",
    "ax.fill_between(np.arange(168)*0.15,np.exp(f_low),np.exp(f_up),facecolor='grey',edgecolor='grey')\n",
    "ax.plot(np.arange(168)*0.15,np.exp(f_hat),lw=1,color='red')\n",
    "plt.grid()\n",
    "plt.xlabel('Time (s)',fontsize=25)\n",
    "plt.ylabel(r'$\\hat{f}_k$',fontsize=25)\n",
    "plt.xlim([0,25])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(138.21082385726555, 182.92690289032578)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "rkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f799c150a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pred = mk_pred(res.x,True,True)\n",
    "the_range = [np.min(data4fit),np.max(data4fit)]\n",
    "plt.figure(dpi=600,figsize=(10,8))\n",
    "plt.subplot(341)\n",
    "plt.plot(np.arange(168)*0.15,pred[0,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[0,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(342)\n",
    "plt.plot(np.arange(168)*0.15,pred[1,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[1,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(343)\n",
    "plt.plot(np.arange(168)*0.15,pred[2,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[2,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(344)\n",
    "plt.plot(np.arange(168)*0.15,pred[3,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[3,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(345)\n",
    "plt.plot(np.arange(168)*0.15,pred[4,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[4,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(346)\n",
    "plt.plot(np.arange(168)*0.15,pred[5,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[5,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(347)\n",
    "plt.plot(np.arange(168)*0.15,pred[6,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[6,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(348)\n",
    "plt.plot(np.arange(168)*0.15,pred[7,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[7,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(349)\n",
    "plt.plot(np.arange(168)*0.15,pred[8,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[8,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(3,4,10)\n",
    "plt.plot(np.arange(168)*0.15,pred[9,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[9,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(3,4,11)\n",
    "plt.plot(np.arange(168)*0.15,pred[10,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[10,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)\n",
    "plt.subplot(3,4,12)\n",
    "plt.plot(np.arange(168)*0.15,pred[11,:],color='red')\n",
    "plt.plot(np.arange(168)*0.15,data4fit[11,:],color='black')\n",
    "plt.xlim([0,25])\n",
    "plt.ylim(the_range)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Data and fit after variance stabilization. The `RSS` is 1976 giving a\n",
    "probability of 0.986 (a bit large).\n",
    "\n",
    "Thanks\n",
    "---------------\n",
    "\n",
    "This work was done in collaboration with:\n",
    "-   Sebastien Joucla\n",
    "-   Romain Franconville\n",
    "-   Andeas Pippow\n",
    "-   Peter Kloppenburg\n",
    "\n",
    "Thank you for your attention!\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "calibration.close()"
   ]
  }
 ],
 "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": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}