{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Estimation of rates of random events\n", "\n", "*Antonino Ingargiola* -- tritemio AT gmail.com
\n", "**ORCID** [0000-0002-9348-1397](orcid.org/0000-0002-9348-1397)
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# The Model\n", "\n", "Let's assume that $\\{d_i\\}$ is a set of $n$ samples distributed as $D \\sim\\operatorname{Exp}(\\lambda)$.\n", "\n", "Then, the can construct $m = n + 1$ times of random events $\\{t_i\\}$ as:\n", "\n", "$$t_0 = 0, \\quad t_i = \\sum_{j=0}^{i-1} d_j \\quad\\textrm{for}\\quad i > 1$$\n", "\n", "These events are stochastic, statistically independent and have a fixed rate $\\lambda$.\n", "\n", "# Problem: how to estimate $\\lambda$ from $\\{t_i\\}$?\n", "\n", "Given $\\{t_i\\}$ we can trivially compute $\\{d_i\\}$ as:\n", "\n", "$$d_i = t_{i+1} - t_i$$\n", "\n", "Instead of $\\lambda$, we can directly estimate $\\tau = 1/\\lambda$:\n", "\n", "$$\\hat{\\tau} = \\frac{1}{n}\\sum_{i=0}^{n-1} d_i = \\frac{T_n}{n}$$\n", "\n", "The estimator $\\hat{\\tau} \\sim \\frac{1}{n}\\operatorname{Erlang}(n, \\lambda)$.\n", "The Erlang distribution has mean $n/\\lambda$ and variance $n/\\lambda^2$.\n", "\n", "Note that:\n", "\n", "$$\\hat{\\tau} \\sim \\frac{1}{n}\\operatorname{Erlang}(n, \\lambda) = \\operatorname{Erlang}(n, n\\,\\lambda)$$\n", "\n", "thus, $\\hat{\\tau}$ has mean $1/\\lambda$ and variance $1/(n \\lambda)^2$, i.e.\n", "it is an unbiased estimator of $1/\\lambda$.\n", "\n", "However, the fact the $\\hat{\\tau}$ is an unbiased estimator of $1/\\lambda$ \n", "does not mean that $1/\\hat{\\tau}$ is an unbiased estimator of $\\lambda$.\n", "Indeed, as shown in the next section, the unbiased estimator of the rate is:\n", "\n", "$$\\hat\\lambda_u = \\frac{n-1}{T_n}$$ \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It can be shown that $\\hat{\\lambda}_u$ has also the minimum variance among the\n", "unbiased estimators (i.e. it is a minimum variance unbiased estimator, MVUE)\n", "of the rate (see [Lehmann-Scheffé theorem](https://en.wikipedia.org/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem)\n", "and [Rao–Blackwell theorem](https://en.wikipedia.org/wiki/Rao%E2%80%93Blackwell_theorem))." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## MLE estimator of the rate\n", "\n", "For our sample of $n$ observations from an Exponential distribution, \n", "the likelihood function is\n", "\n", "$$L(\\lambda) = \\prod_{i=1}^n f(d_i;\\lambda) = \\prod_{i=1}^n \\lambda\\, e^{-\\lambda d_i}$$\n", "\n", "$$\\log L(\\lambda) = \\sum_{i=1}^n \\log (\\lambda\\, e^{-\\lambda d_i} ) = \n", "n\\log(\\lambda) - \\lambda \\sum_{i=1}^n d_i = \n", "n\\log(\\lambda) - \\lambda T_n$$\n", "\n", "Taking the first derivative and setting it to 0 we obtain:\n", "\n", "$$\\frac{\\rm d}{{\\rm d}\\lambda} \\log L(\\lambda) = \n", "\\frac{n}{\\lambda} - T_n = 0$$\n", "\n", "$$\\Rightarrow\\quad \\hat{\\lambda}_{MLE} = \\frac{n}{T_n} = \\frac{n}{\\sum_{i=1}^n d_i}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Minimum MSE estimator\n", "\n", "Given an estimator $\\Lambda_n$ (a R.V.), we define the mean square error (MSE) as:\n", "\n", "$$\\epsilon(\\Lambda_n) = \\mathrm{E}\\{ (\\Lambda_n - \\lambda)^2 \\} = \n", "\\mathrm{Var}\\{ \\Lambda_n\\} + (\\mathrm{E}\\{\\Lambda_n\\} - \\lambda)^2$$\n", "\n", "In general the minimum MSE (MMSE) estimator is defined as:\n", "\n", "$$\\hat{\\lambda}_{MSE} = \\operatorname*{arg\\,min}_{\\Lambda_n} \\epsilon(\\Lambda_n)$$\n", "\n", "If $\\Lambda_n$ has the form:\n", "\n", "$$\\Lambda_{n,c} = \\frac{n - c}{T_n}$$\n", "\n", "we need to find $c$ that minimizes $\\epsilon(\\Lambda_{n,c})$.\n", "It can be shown that the minimum is achieved for $c=2$\n", "\n", "To derive this result use the expression for $\\mathrm{Var}\\{ \\Lambda_{n,c}\\}$\n", "derived in [Appendix 2](#Appendix-2.-Variance-of-the-rate-estimators)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Family of estimators\n", "\n", "We consider the family of estimators \n", "\n", "$$\\Lambda_{n,c} = \\frac{n - c}{T_n}$$\n", "\n", "where $T_n = \\sum d_i \\sim \\textrm{Erlang}(\\lambda, n)$ and $c < n$. We can write:\n", "\n", "$$\\textrm{Erlang PDF:}\\quad f(x \\:|\\: n, \\lambda) = \n", "\\frac{\\lambda^n x^{n-1} e^{-\\lambda x}}{(n-1)!\\,}$$\n", "\n", "$$\\mathrm{E}\\{\\Lambda_{n,c}\\} = \\int \\frac{n - c}{x} \\, f(x \\:|\\: n, \\lambda)\\, dx\n", "= (n - c) \\int_0^\\infty \\frac{1}{x}\\,f(x \\:|\\: n, \\lambda)\\, dx\n", "= (n - c) \\int_0^\\infty \\frac{\\lambda^n x^{n-2} e^{-\\lambda x}}{(n-1)!\\,}\\, dx \n", "= \\frac{(n - c)\\lambda^n}{(n-1)!} \\int_0^\\infty x^{n-2} e^{-\\lambda x}\\, dx$$\n", "\n", "Knowing that (see [Appendix 1](#Appendix-1.-Computation-of-integral-\"A\")):\n", "\n", "$$\\int_0^{+\\infty} x^{n-2} e^{-\\lambda x}\\, dx = \\frac{(n-2)!}{\\lambda^{n-1}}\\qquad\\qquad \\textrm{(A)}$$\n", "\n", "we find that:\n", "\n", "$$\\mathrm{E}\\{\\Lambda_{n,c}\\} = (n - c)\\frac{\\lambda}{n-1} = \\overline{\\Lambda}_{n,c}$$\n", "\n", "Note that \n", "\n", "- for $c=1$ we obtain the unbiased estimator ($\\overline{\\Lambda}_{n,c} = \\lambda$). \n", "- for $c=0$ we obtain the MLE estimator derived in a previous section. \n", "- for $c=2$ we obtain the minimum MSE (MMSE) estimator introduced in a previous section. \n", "\n", "Furthermore, from simulations (see below), I found that for $c=1/3$ the estimator\n", "has median equal to the rate $\\lambda$. This result should not be difficult\n", "to prove analytically." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Simulation of rate estimators" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pylab as plt\n", "import numpy as np" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "n = 5 # number of delays\n", "num_times = n + 1 # number of timestamps\n", "λ = 1000 # simulated rate \n", "num_iter = 100 * 1000" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "np.random.seed(5)\n", "d1 = np.random.exponential(scale=1/λ, size=(num_iter * n)).reshape(num_iter, n)\n", "t = np.cumsum(d1, axis=1)\n", "\n", "assert np.allclose(np.cumsum(d1, axis=1), t)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We $n$ delays, exponentially distributed with a fixed rate. Their cumulative sum (the times) will not have a sharp cut-off: " ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.hist(t.ravel(), bins=50);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The delays are exponential by construction:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.hist(d1.ravel(), bins=40, histtype='step');\n", "plt.yscale('log')\n", "plt.title('$\\{d_i\\}\\quad$ Mean = %.3e Std.Dev. = %.3e ' % (d1.mean(), d1.std()));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here we compare a set of estimators:\n", "\n", "$$\\hat{\\Lambda}_{n,c} = \\frac{n - c}{T_n}$$" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " c n Mean Std MSE Median % > λ\n", " -1.00 5 1.49712 85.90% 99.247% 1.282 71.34%\n", " 0.00 5 1.24760 71.58% 75.744% 1.068 55.67%\n", " 0.33 5 1.16442 66.81% 68.804% 0.997 49.76%\n", " 1.00 5 0.99808 57.27% 57.267% 0.854 36.93%\n", " 2.00 5 0.74856 42.95% 49.769% 0.641 18.41%\n", " 3.00 5 0.49904 28.63% 57.702% 0.427 5.27%\n", " median 5 1.92114 181.83% 203.832% 1.440 73.49%\n" ] }, { "data": { "image/png": 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7AkU/9VaUX+PisP3jD0O5oa0tURacASCEEEWRIF+KaOexfJrYnFf8/AFwSNuP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\nqYpl8FWx7Wd1tFYBtRUt+EClogQQIcgrQAiEkAd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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "kws = dict(bins=np.arange(0, 4, 0.05), histtype='step', lw=1.5, normed=True)\n", "print('{:>8} {:>6} {:>8} {:>8} {:>8} {:>8} {:>8}'\n", " .format('c', 'n', 'Mean', 'Std', 'MSE', 'Median', '% > λ'))\n", "for c in (-1, 0, 1/3, 1, 2, 3):\n", " r_hc = (n - c) / d1.sum(axis=1) / λ\n", " r_hc_err_rms = np.sqrt(np.mean(((r_hc - 1)**2)))\n", " plt.hist(r_hc, **kws, label = 'c = %.1f' % c);\n", " print('%8.2f %6d %8.5f %8.2f%% %8.3f%% %8.3f %8.2f%%' % \n", " (c, n, r_hc.mean(), r_hc.std()*100, r_hc_err_rms*100, np.median(r_hc),\n", " (r_hc > 1).sum() * 100 / num_iter))\n", " \n", "r_hm = 1/np.median(d1, axis=1) / λ\n", "r_hm_err_rms = np.sqrt(np.mean(((r_hm - 1)**2)))\n", "plt.hist(r_hm, **kws, label = 'median');\n", "print('%8s %6d %8.5f %8.2f%% %8.3f%% %8.3f %8.2f%%' % \n", " ('median', n, r_hm.mean(), r_hm.std()*100, r_hm_err_rms*100, np.median(r_hm),\n", " (r_hm > 1).sum() * 100 / num_iter))\n", "plt.xlabel('Normalized Rate')\n", "plt.axvline(1, color='k');\n", "plt.legend()\n", "plt.text(0.35, 0.75, r'$\\Lambda_{n,c} = \\frac{n - c}{T_n}$',\n", " fontsize=24, transform=plt.gcf().transFigure);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can observe that:\n", "\n", "- with $c = 2$ we obtain the mean square error (MSE) in estimating the rate,\n", "- with $c = 1$ we obtained the unbiased estimator of the rate (but higher MSE error),\n", "- with $c = 0$, we obtain the MLE estimator,\n", "- with $c = 1/3$, the estimator has median equal to the true rate ($\\lambda$)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By contrast, when estimating $\\tau = \\lambda^{-1}$,\n", "the MVUE is the empirical mean of the delays ($\\hat{\\tau}$).\n", "The distribution of $\\hat{\\tau}$ is shown below:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " n Mean Std Err.RMS Median % > 1/λ\n", " 5 1.00177 44.74% 44.742% 0.936 44.33%\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "kws = dict(bins=np.arange(0, 4, 0.05), histtype='step', lw=1.5, normed=True)\n", "print('{:>6} {:>8} {:>8} {:>8} {:>8} {:>8}'\n", " .format('n', 'Mean', 'Std', 'Err.RMS', 'Median', '% > 1/λ'))\n", "d_h = np.mean(d1, axis=1) * λ\n", "d_h_err_rms = np.sqrt(np.mean(((d_h - 1)**2)))\n", "plt.hist(d_h, **kws, label = r'$\\hat\\tau$');\n", "print('%6d %8.5f %8.2f%% %8.3f%% %8.3f %8.2f%%' % \n", " (n, d_h.mean(), d_h.std()*100, d_h_err_rms*100, np.median(d_h),\n", " (d_h > 1).sum() * 100 / num_iter))\n", "plt.legend(fontsize=18)\n", "plt.xlabel('Normalized Delays')\n", "plt.axvline(1, color='k');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since both $\\hat\\tau = T_n / n$ and $\\hat\\lambda_u = (n-1)/T_n$\n", "are unbiased estimators their empirical mean \n", "computed on $N$ (num_iter) sets of n delays, will converge\n", "to $\\lambda^{-1}$ and $\\lambda$ respectively. In symbols:\n", "\n", "$$\\textrm{E}[\\hat\\lambda_u] = \\lambda$$\n", "\n", "$$\\textrm{E}[\\hat\\tau] = \\lambda^{-1} \\quad\\Rightarrow\\quad \\frac{1}{\\textrm{E}[\\hat\\tau]} = \\lambda$$\n", "\n", "Below we plot the convergence of the expected value,\n", "approximated by an empirical average with number of \"trials\" $N \\to \\infty$.\n", "\n", "Let's start plotting $1/\\textrm{E}[\\hat\\tau]$ for $N \\to \\infty$ first:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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IwIIFvVcOM7OBpl9cE0n5yi9v8MM0M6sr3/bEzMwaXr8JIi+8AGec0delMDMb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\niJiZWc3+Pwz3xkQoGKAyAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "rate_t = 1 / ((d1.mean(axis=1)).cumsum() / np.arange(1, num_iter+1)) / λ\n", "\n", "plt.plot(rate_t[100:])\n", "plt.ylim(0.98, 1.02)\n", "plt.axhline(1, color='k');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And now, let's plot the same for $\\textrm{E}[\\hat\\lambda_u]$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "rate_t2 = ((n - 1) / d1.sum(axis=1)).cumsum() / np.arange(1, num_iter+1) / λ\n", "\n", "plt.plot(rate_t2[100:])\n", "plt.ylim(0.98, 1.02)\n", "plt.axhline(1, color='k');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In both cases there is convergence to $\\lambda$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Appendix 1. Computation of integral \"A\"\n", "\n", "Here we want to compute the integral:\n", "\n", "$$\\int_0^{\\infty} x^{n-2} e^{-\\lambda x}\\, dx$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The $\\Gamma$ function is defined as:\n", "\n", "$$\\Gamma(s) = \\int_0^{\\infty} t^{s-1}\\,e^{-t}\\,{\\rm d}t$$\n", "\n", "Therefore, with the variable substitution $t = \\lambda x$:\n", "\n", "$${\\rm d}t = \\lambda {\\rm d} x$$\n", "\n", "$$\\frac{1}{\\lambda^{n-2}} \\int_0^\\infty t^{n-2} e^{-t}\\,\\frac{{\\rm d}t}{\\lambda}\n", "= \\frac{1}{\\lambda^{n-1}} \\int_0^\\infty t^{n-2} e^{-t}\\,{\\rm d}t\n", "= \\frac{1}{\\lambda^{n-1}} \\Gamma(n-1) = \\frac{(n-2)!}{\\lambda^{n-1}}$$\n", "\n", "So we obtain:\n", "\n", "$$\\int_0^{\\infty} x^{n-2} e^{-\\lambda x}\\, dx = \\frac{(n-2)!}{\\lambda^{n-1}}\\qquad\\qquad \\textrm{(A)}$$\n" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "Finally, note that $n$ is the number of delays, while the number of times/events is $m = n+1$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Appendix 2. Variance of the rate estimators\n", "\n", "We want to compute the variance for the family of rate estimators $\\Lambda_{n,c}$ defined as\n", "\n", "$$\\Lambda_{n,c} = \\frac{n-c}{T_n}$$\n", "\n", "where $T_n = \\sum d_i$ is the sum of the $n$ delays, and $c$ is a real parameter with $c < n$.\n", "\n", "We start by writing:\n", "\n", "$$\\mathrm{Var}\\{\\Lambda_{n,c}\\} = \\mathrm{E}\\left\\{(\\Lambda_{n,c} - \\mathrm{E}\\{\\Lambda_{n,c}\\})^2 \\right\\} \n", "= \\int \\left( \\Lambda_{n,c} - \\overline{\\Lambda}_{n,c} \\right)^2 f(x \\:|\\: n, \\lambda)\\, dx\n", "= \\int \\left(\\frac{n - c}{x} - (n - c)\\frac{\\lambda}{n-1} \\right)^2 f(x \\:|\\: n, \\lambda)\\, dx =$$\n", "\n", "$$= (n-c)^2 \\int \\left(\\frac{1}{x} - \\frac{\\lambda}{n-1} \\right)^2 f(x \\:|\\: n, \\lambda)\\, dx$$\n", "\n", "For brevity, let's define:\n", "\n", "$$\\left(\\frac{1}{x} - \\frac{\\lambda}{n-1} \\right)^2 = \\left(\\frac{1}{x^2} + \\left(\\frac{\\lambda}{n-1}\\right)^2 \n", "-2 \\frac{\\lambda}{x(n-1)})\\right)\n", "= \\left(\\frac{1}{x^2} + a + \\frac{b}{x} \\right)$$\n", "\n", "From **term 2**:\n", "\n", "$$(n-c)^2 \\int a \\, f(x \\:|\\: n, \\lambda)\\, dx = a (n-c)^2 = (n-c)^2\\left( \\frac{\\lambda}{n-1} \\right)^2$$\n", "\n", "From **term 1**:\n", "\n", "$$(n-c)^2 \\int \\frac{1}{x^2} \\, f(x \\:|\\: n, \\lambda)\\, dx\n", "= (n - c)^2 \\int \\frac{\\lambda^n x^{n-3} e^{-\\lambda x}}{(n-1)!\\,}\\, dx \n", "= \\frac{(n - c)^2\\lambda^n}{(n-1)!} \\int x^{n-3} e^{-\\lambda x}\\, dx \n", "= \\frac{(n - c)^2\\lambda^n}{(n-1)!} \\frac{(n-3)!}{\\lambda^{n-2}} =$$\n", "\n", "$$= \\frac{(n - c)^2\\lambda^2}{(n-1)(n-2)}$$\n", "\n", "And from **term 3**:\n", "\n", "$$(n-c)^2 \\int \\frac{b}{x} \\, f(x \\:|\\: n, \\lambda)\\, dx\n", "= b(n - c)^2 \\int \\frac{\\lambda^n x^{n-2} e^{-\\lambda x}}{(n-1)!\\,}\\, dx \n", "= \\frac{b(n - c)^2\\lambda^n}{(n-1)!} \\int x^{n-2} e^{-\\lambda x}\\, dx \n", "= \\frac{b(n - c)^2\\lambda^n}{(n-1)!} \\frac{(n-2)!}{\\lambda^{n-1}} =$$\n", "\n", "$$= \\frac{b(n - c)^2\\lambda}{(n-1)} = -\\frac{2\\lambda}{n-1} \\frac{(n - c)^2\\lambda}{(n-1)}\n", "= -2 \\frac{(n - c)^2\\lambda^2}{(n-1)^2}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Summing these 3 terms we obtain the variance:\n", "\n", "$$\\mathrm{Var}\\{\\Lambda_{n,c}\\} = \\lambda^2(n-c)^2\\left[ \\frac{1}{n-1} + \\frac{1}{n-2} - 2 \\frac{1}{n-1}\\right]\n", "= \\lambda^2(n-c)^2\\left[\\frac{1}{n-2} - \\frac{1}{n-1}\\right]$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# References\n", "\n", "- [Random: Chapter 6. Point Estimation](http://www.math.uah.edu/stat/point/index.html)\n", "- http://www.amstat.org/publications/jse/v9n1/elfessi.html\n", "- Any statistics book explaining RV parameter's estimators, MLE, MMSE and MVUE." ] } ], "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": "3.5.1" } }, "nbformat": 4, "nbformat_minor": 0 }