Lifetime_decay_fit.ipynb

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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Lifetime decay fit\n",
      "\n",
      "**Author:** Antonino Ingargiola  -  **Date:** 07/20/2014"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Abstract\n",
      "In this notebook we fit a [TCSPC](http://en.wikipedia.org/wiki/Ultrafast_laser_spectroscopy#Time-correlated_single_photon_counting) histogram with a model functions that includes an exponential decay, an instrument response function (IRF) and uncorrelated background. The IRF is considered fixed (i.e. measured) and is used to deconvolve the data. This kind of fit is commonly performed on fluorescence lifetime experimental curves.\n",
      "\n",
      "For illustrative purposes, the TCSPC data is contextually simulated by simply extracting random exponential waiting time, adding a random number from the IRF distribution and adding a fraction of uncorrelated events (background). These events are binned in order to obtain a realistic TCSPC histogram. This procedure assures a proper simulation of the Poisson statistics in each bin.\n",
      "\n",
      "The fitting procedure can use both a non-linear least square and a Maximum Likelihood minimization. In the latter case the log-likelihood function obtained from Poisson statistics is maximized.\n",
      "\n",
      "> **Note:** This notebook uses the fitting packages [lmfit](http://lmfit.github.io/lmfit-py/) that allows to easily set constrains on the parameters (i.e. fixed, bounded, etc ...). It also performs rigorous error analysis and allows to easily compare different minimization algorithms (other than least squares).\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Library imports"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import scipy\n",
      "import scipy.stats\n",
      "from scipy.optimize import minimize\n",
      "import matplotlib.pyplot as plt\n",
      "from matplotlib.pyplot import hist, plot\n",
      "\n",
      "import lmfit\n",
      "from lmfit import Parameters, report_fit"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Data simulation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##TCSPC Parameters"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "time_step_s = (60e-9/4096)               # time step in seconds (S.I.)\n",
      "time_step_ns = time_step_s*1e9           # time step in nano-seconds\n",
      "time_nbins = 2048                        # number of time bins\n",
      "\n",
      "time_idx = np.arange(time_nbins)         # time axis in index units\n",
      "time_ns = time_idx*time_step_ns          # time axis in nano-seconds"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Model IRF"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let define the [Exponentially modified Gaussian distribution](http://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution)\n",
      "representing the convolution of a gaussian and an exponential. This function is used to model the IRF.\n",
      "\n",
      "$$f(x;\\mu,\\sigma,\\lambda) = \\frac{\\lambda}{2} e^{\\frac{\\lambda}{2} (2 \\mu + \\lambda \\sigma^2 - 2 x)} \\operatorname{erfc} (\\frac{\\mu + \\lambda \\sigma^2 - x}{ \\sqrt{2} \\sigma})$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def exgauss(x, mu, sig, tau):\n",
      "    lam = 1./tau\n",
      "    return 0.5*lam * np.exp(0.5*lam * (2*mu + lam*(sig**2) - 2*x)) *\\\n",
      "           scipy.special.erfc((mu + lam*(sig**2) - x)/(np.sqrt(2)*sig))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We are going to compute the IRF, using `exgauss` as a PDF. The following parameters have been fitted from an experimental IRF:\n",
      "\n",
      "$$\\sigma = 33\\,\\textrm{ps} \\qquad \\lambda = 100\\,\\textrm{ps}$$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "irf_sig = 0.033\n",
      "irf_mu = 5*irf_sig\n",
      "irf_lam = 0.1\n",
      "x_irf = np.arange(0, (irf_lam*15 + irf_mu)/time_step_ns)\n",
      "p_irf = exgauss(x_irf, irf_mu/time_step_ns, irf_sig/time_step_ns,\n",
      "                irf_lam/time_step_ns)\n",
      "irf = scipy.stats.rv_discrete(name='irf', values=(x_irf, p_irf))\n",
      "x_irf.size"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "114"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(x_irf, p_irf)\n",
      "plt.yscale('log')\n",
      "irf_mu/time_step_ns"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "11.264000000000001"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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OJ59sNQGHH26Dwe+953REIumVieR/I/BMKk70ww+2e5Om6Ek6VKsGjz5qtQDnn2+1AaoJ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\nIiIiIiIikr3+H5Kc9LHIWTAcAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x145e9a58>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "assert np.allclose(p_irf.sum(), 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally we need to draw random numbers from the IRF's PDF. To do so, we create a discrete random distribution with **scipy**:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "irf = scipy.stats.rv_discrete(name='irf', values=(x_irf, p_irf))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = plt.subplots(1, 2, figsize=(12, 4))\n",
      "ax[0].plot(x_irf, irf.pmf(x_irf));\n",
      "ax[0].set_title('IRF PDF')\n",
      "ax[1].hist(irf.rvs(size=200), bins=x_irf, histtype='step')\n",
      "ax[1].set_title('Random samples from the IRF distribution');\n",
      "ax[0].set_yscale('log')\n",
      "#ax[1].set_yscale('log')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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tz38O/f4PPhiuvTbqaCQlSalJ81TgYWAooY75aGA58C3gTkJbuWnA8z0JIp1O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\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x14b4c2e8>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "assert (p_irf == irf.pmf(x_irf)).all()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Generate the nanotimes"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.random.seed(2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "num_samples = 5e5\n",
      "tau = 2e-9\n",
      "baseline_fraction = 0.03\n",
      "offset = 2e-9\n",
      "\n",
      "sample_decay = np.random.exponential(scale=tau/time_step_s,\n",
      "                                     size=num_samples) + offset/time_step_s\n",
      "sample_decay += irf.rvs(size=num_samples)\n",
      "sample_baseline = np.random.randint(low=0, high=time_nbins,\n",
      "                                    size=baseline_fraction*num_samples)\n",
      "sample_tot = np.hstack((sample_decay, sample_baseline))\n",
      "decay_hist, bins = np.histogram(sample_tot, bins=np.arange(time_nbins + 1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "baseline_true = baseline_fraction*num_samples / time_nbins\n",
      "baseline_true"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "7.32421875"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hist_sim_decay, _, _ = hist(sample_tot, bins=np.arange(time_nbins+1), histtype='step');\n",
      "plt.yscale('log')\n",
      "(hist_sim_decay > 0).all()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "False"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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iDz6Y2FJy8ODE1mxdu5r0ESPsln3Dhxe5SZ2UvXQtfB98sO73ffhhYqvSCRNi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\nRERERILv/wE1Ul/NDqMbCQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x149d3748>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Model function\n",
      "\n",
      "We use a single exponential model convoluted with the model-IRF and with the addition of a **baseline**:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def model(x, tau, ampl, baseline, offset, irf_pdf=irf.pmf(x_irf)):\n",
      "    \"\"\"\n",
      "    Model function used to fit the data.\n",
      "    \"\"\"\n",
      "    y = np.zeros(x.size)\n",
      "    pos_range = (x - offset) >= 0\n",
      "    y[pos_range] = ampl*np.exp(-(x[pos_range] - offset)/tau)\n",
      "    z = np.convolve(y, irf_pdf, mode='same')\n",
      "    z += baseline\n",
      "    return z\n",
      "\n",
      "def residuals(params, x, y, weights):\n",
      "    \"\"\"\n",
      "    Returns the array of residuals for the current parameters.\n",
      "    \"\"\"\n",
      "    tau = params['tau'].value\n",
      "    baseline = params['baseline'].value\n",
      "    offset = params['offset'].value\n",
      "    ampl = params['ampl'].value\n",
      "    return (y - model(x, tau, ampl, baseline, offset))*weights\n",
      "\n",
      "def plot_fit(time, ydata, params, yscale='log', zoom_origin=False):\n",
      "    \"\"\"\n",
      "    Function to plot data and model function.\n",
      "    \"\"\"\n",
      "    plot(time, ydata, marker='.')\n",
      "    plot(time, model(time, **{k: v.value for k, v in params.items()}),\n",
      "         color='r', lw=2.5, alpha=0.7)\n",
      "    if yscale == 'log':\n",
      "        plt.yscale('log')\n",
      "        plt.ylim(1)\n",
      "    if zoom_origin:\n",
      "        dt = time[1] - time[0]\n",
      "        t0 = params['offset'] - 50*dt\n",
      "        plt.xlim(t0 - 50*dt, t0 + 50*dt)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# MLE approach"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Log-likelihood function:\n",
      "\n",
      "$$p(k) = \\frac{\\lambda^k}{\\Gamma(k+1)}\\exp(-\\lambda)$$\n",
      "\n",
      "$$\\log p(k) = k \\log \\lambda - \\log\\Gamma(k+1) -\\lambda$$\n",
      "\n",
      "In this context, for each TCSPC bin $i$, we have a Poisson rate $\\lambda_i$ function of the model parameters:\n",
      "\n",
      "$$\\lambda_i = \\lambda(t_i, \\tau, A, t_0, b)$$\n",
      "\n",
      "and a number of counts $y_i$ that are actual the TCSPC bin counts. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def loglike(params, x, ydata):\n",
      "    tau, baseline, offset, ampl = params\n",
      "    ymodel = model(x, tau, ampl, baseline, offset)\n",
      "    return (ymodel - ydata*np.log(ymodel)).sum()\n",
      "\n",
      "def from_params(params):\n",
      "    return [params[k].value for k in ('tau', 'baseline', 'offset', 'ampl')]\n",
      "\n",
      "def to_params(p, params):\n",
      "    for v, k in zip(p, ('tau', 'baseline', 'offset', 'ampl')):\n",
      "        params[k].value = v\n",
      "    return params"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Poisson weights"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Weights** for proper scaling of the residuals according to the Poisson std.dev."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights = 1./np.sqrt(decay_hist)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights[decay_hist == 0] = 1./np.sqrt(baseline_true)\n",
      "(decay_hist == 0).any()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Lifetime fit"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "params = Parameters()\n",
      "params.add('tau', value=2.8)\n",
      "params.add('baseline', value=5)\n",
      "params.add('offset', value=2.814, vary=False)\n",
      "params.add('ampl', value=700)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Check the model with the intial parameters:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_fit(time_ns, decay_hist, params)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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g40YZgzYejjoK/vtfWdZGXCVVdPiG16aBvHV817TD9G13Ci42dmzw+jqGcE/v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\nGcbHjFKh386J5p2VXNJKk0+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MCHZ5tod0cLoG+/bJtq++ElkRWoZZ9sMPO5cf6xqPGhX/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\nok2cJmsAAAAOSURBVCiKoiiKoijB/H+NLrkewwQ/lwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x2176ac8>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Least squares"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we fit the parameters. To more easily converge to the global minimum we first perform a minimization with the Nelder-Mead method (a.k.a. downhill simplex). After that, we perform a second minimization with non-linear least-squares in order to obtain error estimates."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fit_res = lmfit.minimize(residuals, params, args=(time_ns, decay_hist, weights), method='nelder')\n",
      "report_fit(fit_res.params)\n",
      "print 'Reduced Chi-square: %.3f' % fit_res.redchi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[Variables]]\n",
        "     ampl:         3691.373 initial =  700\n",
        "     baseline:     6.317692 initial =  5\n",
        "     offset:       2.814 (fixed)\n",
        "     tau:          1.988625 initial =  2.8\n",
        "[[Correlations]] (unreported correlations are <  0.100)\n",
        "Reduced Chi-square: 1.177\n"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fit_res = lmfit.minimize(residuals, params, args=(time_ns, decay_hist, weights), method='leastsq')\n",
      "report_fit(fit_res.params)\n",
      "print 'Reduced Chi-square: %.3f' % fit_res.redchi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[Variables]]\n",
        "     ampl:         3691.374 +/- 8.272421 (0.22%) initial =  3691.373\n",
        "     baseline:     6.317692 +/- 0.08213102 (1.30%) initial =  6.317692\n",
        "     offset:       2.814 (fixed)\n",
        "     tau:          1.988625 +/- 0.003367729 (0.17%) initial =  1.988625\n",
        "[[Correlations]] (unreported correlations are <  0.100)\n",
        "    C(ampl, tau)                 = -0.722 \n",
        "    C(baseline, tau)             = -0.236 \n",
        "    C(ampl, baseline)            =  0.109 \n",
        "Reduced Chi-square: 1.177\n"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_fit(time_ns, decay_hist, params, zoom_origin=False)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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eT+rtJjEBrrs+Rl7KKGP5cjW/7DKYNg0++EDZ6keOtG9icRPLNK7mIcbxO0rl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\nRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x16ada1d0>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Error estimates"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we compute the confidence interval by samplig the reduced $\\chi^2$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ci = lmfit.conf_interval(fit_res)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lmfit.printfuncs.report_ci(ci)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "            99.70%    95.00%    67.40%     0.00%    67.40%    95.00%    99.70%\n",
        "     tau   1.97862   1.98195   1.98535   1.98863   1.99191   1.99532   1.99873\n",
        "baseline   6.07364   6.15659   6.23696   6.33703   6.39842   6.47878   6.56170\n",
        "    ampl3666.739613675.195673683.351253697.352253699.408873707.588043716.00500\n"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Maximum likelihood\n",
      "\n",
      "To find the ML estimate we maximize the log-likelihood function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "res = minimize(loglike, x0=from_params(params), args=(time_ns, decay_hist),\n",
      "               method='Nelder-Mead')\n",
      "\n",
      "params = to_params(res.x, params)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "report_fit(params)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[Variables]]\n",
        "     ampl:         3650.818 +/- 8.272421 (0.23%) initial =  3691.373\n",
        "     baseline:     7.279041 +/- 0.08213102 (1.13%) initial =  6.317692\n",
        "     offset:       2.80642 (fixed)\n",
        "     tau:          2.005321 +/- 0.003367729 (0.17%) initial =  1.988625\n",
        "[[Correlations]] (unreported correlations are <  0.100)\n",
        "    C(ampl, tau)                 = -0.722 \n",
        "    C(baseline, tau)             = -0.236 \n",
        "    C(ampl, baseline)            =  0.109 \n"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_fit(time_ns, decay_hist, params, zoom_origin=False)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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HrfSeFWIZd5lrKipY0vpCSvfCf7tewV3LLpP4NTHEyJHw9ttQUgJXXqnMNbqR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\nBEEQBEEQBEEQBEEQBEEQBEHw5f8BQEOiL7qrpKIAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x14929c88>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "> Note that the ML method gives a better estimation of the true baseline value (`baseline_true`) but the fitted lifetime is not significantly different (in this case)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Apply the style to the notebook"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.core.display import HTML\n",
      "HTML(open(\"./styles/custom2.css\", \"r\").read())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<style>\n",
        "    @font-face {\n",
        "        font-family: \"Computer Modern\";\n",
        "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
        "    }\n",
        "    div.cell{\n",
        "        width:900px;\n",
        "        margin-left:8% !important;\n",
        "        margin-right:auto;\n",
        "    }\n",
        "    div.text_cell_render{\n",
        "        font-family: Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
        "        line-height: 145%;\n",
        "        font-size: 110%;\n",
        "        width:800px;\n",
        "        margin-left:auto;\n",
        "        margin-right:auto;\n",
        "    }\n",
        "    blockquote  p {\n",
        "\t/*margin: 1em 2em;*/\n",
        "\tline-height: 140%;\n",
        "    }\n",
        "\n",
        "    .rendered_html code {\n",
        "\tbackground-color: #f7f7f7;\n",
        "\tborder: 1px solid #dddddd;\n",
        "\tcolor: black;\n",
        "\tline-height: 1.1em;\n",
        "    \tpadding: 0.15em;\n",
        "\tfont-size: 85%;\n",
        "    }\n",
        "\n",
        "    .CodeMirror{\n",
        "            font-family: \"Source Code Pro\", source-code-pro,Consolas, monospace;\n",
        "    }\n",
        "    /*.prompt{\n",
        "        display: None;\n",
        "    }*/\n",
        "    \n",
        "</style>\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 29,
       "text": [
        "<IPython.core.display.HTML at 0x16bc4320>"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    }
   ],
   "metadata": {}
  }
 ]
}