From 5f424fd54f554413fdf144b0cae63cda59a12265 Mon Sep 17 00:00:00 2001 From: Adrian Liu Date: Tue, 6 Jun 2017 18:15:23 -0700 Subject: [PATCH] Created Intro to Radio Astronomy folder and added iPython notebook on noise statistics --- .../NoiseStatistics.ipynb | 452 ++++++++++++++++++ 1 file changed, 452 insertions(+) create mode 100644 Lesson3_RadioAstronomyIntro/NoiseStatistics.ipynb diff --git a/Lesson3_RadioAstronomyIntro/NoiseStatistics.ipynb b/Lesson3_RadioAstronomyIntro/NoiseStatistics.ipynb new file mode 100644 index 0000000..3ba2722 --- /dev/null +++ b/Lesson3_RadioAstronomyIntro/NoiseStatistics.ipynb @@ -0,0 +1,452 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Populating the interactive namespace from numpy and matplotlib\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "%pylab inline" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Noise Statistics" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Noise is inherently *random*, so we can't predict its exact value. This is why it's a problem. Imagine, for example, that some experiment measures a signal plus some noise: $s + n$. If we knew the exact value of $n$, it wouldn't be a problem, for we could just subtract it from our final measurement. But if $n$ is random (which it generally is), we can't do that. We'll see soon, though, that we can mitigate the effects of noise by making *repeated measurements*." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Describing Noise" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To understand noise, we need to understand its behavior. Even though we might not know the exact noise contribution for any given measurement (again, it's random!), we can understand its rough behavior. For example, coin flips might be random, so for any individual coin flip I might not know whether to expect heads or tails. But I do know something about the nature of coin flips! I know that there are two possibilities (heads/tails), and they're rougnly equally likely." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The properties of random quantities like noise are often described using probability distribution functions. A probability distribution function tells us how likely different outcomes of a random process are. For the coin-flipping example, the probability distribution function might look something like this:" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[, ]" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "f, ax = plt.subplots(figsize=(9,6))\n", + "ax.bar(np.array([-0.25,0.25]), np.array([0.5, 0.5]),np.array([0.2, 0.2]))\n", + "ax.set_ylim(0,1)\n", + "ax.set_xticks(np.array([-0.15,0.35]))\n", + "ax.set_xticklabels(('Heads', 'Tails'))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here's another example:" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def gauss(x,b,s):\n", + " return np.exp(-0.5 * (x-b)**2 / s**2) /np.sqrt(2.*np.pi*s**2)" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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vhv32y+spLVkC0eUuf6mhWRMjKQ/nTQkOOcQEph7ss0+ebPDRR+GRR4qORmpY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\nUp1znhip3pxzTk5gZs0ygVH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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "f, ax = plt.subplots(figsize=(9,6))\n", + "mean = 0.0\n", + "std = 1.5\n", + "xVals = np.linspace(-10.,10.,1000)\n", + "ax.plot(xVals, gauss(xVals, mean, std),lw=2,color='red')\n", + "ax.set_xlabel(r\"$x$\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This is known as a Gaussian, or equivalently a normal distribution or a bell curve. Gaussians are characterized by their mean and their standard deviation. In this case, I plotted a Gaussian with mean zero and a standard deviation of $1.5$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "A random number governed by this probability distribution function is much more likely to be somewhere near zero (because that's where the probability distribution peaks). It's possible for this random number to take on a value that's far away from the mean, but it's much less likely (as seen from the way the Gaussian dies off rather quickly away from its peak)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In fact, here's a handy rule to remember. In the figure below, we've shaded the regions that are within one standard deviation of the mean (dark grey), two standard deviations (lighter grey), and three standard deviations (lightest grey). It turns out that $68\\%$ of random numbers drawn from a Gaussian distribution fall within one standard deviation (\"within one sigma\"); $95\\%$ fall within two standard deviations (\"within two sigma\"); and $99.7\\%$ fall within three standard deviations (\"within three sigma\"). So while it is possible for a random number drawn from this distribution to be more than three sigma away from the mean, it's quite unlikely." + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 33, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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EkpzGw8yf6N+pxsWIeEMhRiTJlWqm3nmjcTEi3lKIEUliGX19FLa0MBwM0lZZ\n6XU5Sad5bEtMKORxNSKpRyFGJImVHj2KOUdbZSUjaXGv9ypx6s3P5/SSJWQODFDQ2up1OSIpRyFG\nJIlp0cf5d0JdSiKeUYgRSWKjM/UqxMybZs3cK+IZhRiRZOUcZYcPA3C8utrTUpJZ9NJ1tcSILDyF\nGJEkldfeTk53N705OZwqKvK6nKTVsmwZITOKmpoIDg15XY5ISlGIEUlS5WMXfdTK1fNmKDOTjrIy\ngqEQJZHlHURkYSjEiCSpcnUlLZjRS63VpSSyoBRiRJJUdIyGQsz80+BeEW8oxIgkoeDgIMWNjYTM\ntHL1AjgRGdyr5QdEFpZCjEgSKj12jGAoRHt5OUNZWV6Xk/TaysoYSk+noK2NrJ4er8sRSRkKMSJJ\nqGzsoF6Zdy4YpKWqClBrjMhCUogRSUKjg3oVYhZMNDBG/+5FZP4pxIgkG+d0ZZIHon/XCjEiC0ch\nRiTJLD55ktzTp+nPzqazuNjrclJGNMSUNTRgWtFaZEEoxIgkmehSA80rVkBAH/GF0pOfz+mCAjL7\n+ylsbva6HJGUoJ9wIkkmOlOvxsMsvKaVKwGoUJeSyIJQiBFJMhoP453RcTGHDnlbiEiKUIgRSSLB\n4WGKjx0DzkzAJgunSYN7RRaUQoxIEik+doy0kRHaS0sZyM72upyU01ZRwVBGBgVtbWR3dXldjkjS\nU4gRSSLlWi/JUy4YHF1HSa0xIvNPIUYkiUTHYmimXu9ovhiRhaMQI5IsnBu9KiZ6lYwsPI2LEVk4\nCjEiSSKvo4PcU6foW7SIjpISr8tJWc3RSe+OHiUwPOxtMSJJTiFGJElURLqSjldXa5I7D/Xn5NBR\nUkLa0BDFjY1elyOS1PSTTiRJRENM06pVHlcimvROZGEoxIgkieigXo2H8Z4mvRNZGAoxIkkgs6+P\nouZmhoNBTixb5nU5KS8aYioOHwbnPK1FJJkpxIgkgfLDhzHnaFm2jJH0dK/LSXkdJSX0Z2eTe+oU\ni0+e9LockaSlECOSBNSVlGACAXUpiSwAhRiRJFBZXw8oxCSS0S4lhRiReZMW745mVgPcAzQCFcBX\nnHOvTHHMEuA24Hrn3NYJ9vkocC3QDjjgL51zvfHWJZLqAiMjlDY0AHBcISZhNK5eDZwJmCIy9+IK\nMWa2DHiWcBjZbmbrgBfNbKtzbtxPqJmVAdcTDjHjhhIz+yTwZ8AFzjlnZvcA/x/w7um/FZHUVHzs\nGOlDQ3RGMk4EAAAdrklEQVSUlNCXm+t1ORLRvHw5w8EgRcePk9nby8CiRV6XJJJ04u1Oug9ocM5t\nB3DOHQBeBR6Y6ADnXLNz7kHgufGeN7NC4CvAt5wbHb7/EPCHZnZ9nHWJpLwKjYdJSCPp6ZxYvhxz\nTl1KIvNkyhBjZlnADcD2mKe2A9eZWf4Up+ibYPt7gMXAjugG59xR4ATw4anqEpEwhZjEpS4lkfkV\nT0vMFiALaI3ZfhwIApunOH6iSRIui9yPd96L46hLRMb8lq8Qk3iisydX1tV5XIlIcoonxJRF7ttj\ntndF7me60txk59XqdSJxWNLWRk5XF725uZwsLva6HInRVF2NM6Pk6FHSBge9Lkck6UznEuvYwbnB\nyP1sP5njnVefdpE4RH/Db1y1Csw8rkZiDWZn01pRQTAUouzIEa/LEUk68YSYo5H7gpjteZH7lhm+\n9mTnnek5RVLKaIiJjL2QxNOoLiWReRNPiNlLeHBuecz2KqAf2DnD194VuY89byXw/EQHmdmEt9ra\n2hmWIuJPVZEvxmMKMQlLg3slldXW1k76vT1bU4YY51wX8ATw9pinNgNPOef6Z/jajxHuSho9r5mt\nJNwy86+T1DPhTSFGUkleRwd5nZ30L1pEW3ns7wKSKKKDe8uPHCEwMuJxNSILq7a2dtLv7dmKd0zM\nPcCayCR3mNlGoAa4K/L4DjPbFpn7JVYG40yq55zrBv4auGnM5luBHzrnfhH/WxBJTWeNhwloBZFE\n1bt4MR3FxaQPDlJy7JjX5Ygklbhm7HXO1ZnZtcC9ZlZHuMvnKudcQ2SXEqAayI4eY2a5wPuBK4FS\nM/sL4Mkxx+Cc+xszc2b2T4SvSuoHPjj7tyWS/M4KMZLQGlevprC1lYr6eppXrPC6HJGkEffaSc65\nHcCNEzx3J3BnzLZu4LuR22Tn/dt4axCRMzQexj+aVq3irdu2UVlfz66rrvK6HJGkoTZoER/KOXmS\nJW1tDGRm0lpZ6XU5MoWzBveGQh5XI5I8FGJEfKgqcqVL06pVuGBwir3Fa6cLCjhdUEBWby/Fx497\nXY5I0lCIEfGhyoMHAY2H8Q0zjq1ZA0DVm296XIxI8lCIEfGh6HiYxsgXoyS+o2vXArBMIUZkzijE\niPjMoq4uCltaGMrI4MSyZV6XI3E6GgmclfX1mOaLEZkTCjEiPhPtjmiqriak8TC+0V1QwMmiIjL7\n+ylpbPS6HJGkoBAj4jPR7oij69Z5XIlMV7Q1Rl1KInNDIUbEZ5YfOABAQ2SMhfjHsci/WVVkYLaI\nzI5CjIiP5LW3k9/RQX92Nq1VVV6XI9N0NDJfTEV9PYHhYY+rEfE/hRgRH4l2Qxxbswan9ZJ8pzc/\nn/bSUjIGByltaJj6ABGZlH4KivhINMQ0aDyMb0Xni1mmLiWRWVOIEfEL584M6tV4GN+KDu7VuBiR\n2VOIEfGJpc3N5HR10Z2XR2dJidflyAxFW2IqDh0iODjocTUi/qYQI+ITyyJXJR1duxbMPK5GZqo/\nN5eWykrShoepPHTI63JEfE0hRsQnomMoND+M/x057zwAlu/f73ElIv6mECPiAzYyMjqGQvPD+N+R\n9esBWKEQIzIrCjEiPlB29CiZ/f10FhfTXVDgdTkyS8dXrmQoPZ3ipiYWnTrldTkivqUQI+IDK/bu\nBc50Q4i/jaSljQ7wjc7ALCLTpxAj4gPRbofDkW4I8b+GSCBVl5LIzCnEiCS4rJ4eyhoaGA4GR397\nF/8bHdx74ACEQh5XI+JPCjEiCW75/v2YczStWsVwZqbX5cgc6SgtpSs/n5yuLoqOH/e6HBFfUogR\nSXDV+/YBcLimxuNKZE6ZnelSivwbi8j0KMSIJDLnRr/gjmg8TNI5onExIrOiECOSwIqamsjp6qIr\nP5/2sjKvy5E51rBuHc6Mivp60gYGvC5HxHcUYkQSWPXYVhgtNZB0+nNzObFsGWkjIyyPLO4pIvFT\niBFJYOpKSn71GzcCsPKNNzyuRMR/FGJEElR6fz8V9fWEzGjQeklJ69CGDQCs3LsXnPO4GhF/UYgR\nSVDLDxwgGArRvGIFA4sWeV2OzJPWykq68/PJPXWK4sZGr8sR8RWFGJEEtfr114Ez3Q2SpMw4FLl8\nfuWePR4XI+IvCjEiCchCIaojX2gKMcnvUHRcjEKMyLQoxIgkoLIjR1jU08OpwkI6dGl10mtYu5bh\ntDTKGhrI7uryuhwR31CIEUlAqyJXqtRv3KhLq1PAcGYmx9aswZyjOrJiuYhMTSFGJAGtio6Hectb\nPK5EFkq9upREpk0hRiTB5LW1sfTECQaysmhctcrrcmSBHI5car1i3z4Cw8MeVyPiDwoxIgkm2pV0\nZP16QmlpHlcjC+V0YSGtFRVkDgywTLP3isRFIUYkwejS6tRV99a3ArBm926PKxHxB4UYkQSS3d1N\nZV0dI8Hg6GW3kjoOnn8+EA6yFgp5XI1I4lOIEUkgq197jYBzHF27loHsbK/LkQXWVl7OyaIiFnV3\nU37okNfliCQ8hRiRBBLtRnhz0yaPKxFPmHFQXUoicVOIEUkQmb29LDtwgJCZLq1OYdEupTWvvaYF\nIUWmoBAjkiBWvfEGwVCIY2vW0Jeb63U54pHm5cvpzssjr7OTkmPHvC5HJKEpxIgkiGj3QfQ3cUlR\ngcDoVUprX33V42JEEptCjEgCSO/vZ8W+fbgxYyIkdb15wQUArHvlFXUpiUwi7hBjZjVm9gMz+5qZ\nPW5mF8ZxzOfN7FEze8jMvmx27iIwZnaBmYXG3PrMTCveSUpZ9cYbpA0P01RdTW9+vtfliMcaV62i\nOz+f/I4OyhoavC5HJGHFNR2omS0DngWud85tN7N1wItmttU5Vz/BMfcB5zvn3hV5/Ajwd8BtMbve\nDtwx5vER51zzNN+HiK+t37kTgP2bN3tciSQCFwhwYNMmNv/616zbtYvmFSu8LkkkIcXbEnMf0OCc\n2w7gnDsAvAo8MN7OZraecDj5uzGbHwI+ZWabxuy3CWh0zn11zO3JGbwPEd/K7u5mxf79hAKB0W4E\nkWigXfe732niO5EJTBlizCwLuAHYHvPUduA6Mxuv7fuDkXPvGLPtZSAEfGjMti8Bf2xm/2Bml06n\ncJFksfbVVwmEQhw57zxdlSSjTixfzsmlS8k9fZrK+nEbvEVSXjwtMVuALKA1ZvtxIAiM1/59GeCA\n9ugG51wfcBq4GMDMFgP9wDHgY8ALZva3442bEUlm56krScZjxoELw0MP1+3a5XExIokpnhATHWTb\nHrO9K3JfMsExJ507Z1h9V3R/51yXc+5m59ylwErgR8DngC/GU7hIMljc2UnloUMMpadTpwnuJMb+\nSIhZu3s3geFhj6sRSTzTucS6N+ZxMHI/GOf+0WPO2d851wj8V+BnwGfNTJd+S0qI/oZ9aONGhrKy\nPK5GEk17RQWt5eVk9/Swcu9er8sRSTjxhIWjkfuCmO15kfuWCY6J3T96zHj745wLAXdH9imOoy4R\nf3OODTvCw8b2qStJJrBn61YANmyPHZYoIvGEmL1AH1Aes72K8JiWneMcswtYZGajoxQjY2Bygecn\nea39kdeK7boaZWYT3mpra+N4OyKJobShgaUnTtCTm8vhDRu8LkcS1P4tWwgFAlTv2UN2V9fUB4gk\nkNra2km/t2dryhDjnOsCngDeHvPUZuAp51z/OIc9THhg79hjLgJGgMcneblLgAedcxN2/jrnJrwp\nxIifbHzpJQD2XXQRoWBwir0lVfUuXsyhmhqCoRDrNcBXfKa2tnbS7+3ZinfsyT3Amsgkd5jZRqAG\nuCvy+A4z22ZmhQCRCfD+Ebh5zDk+AXzdObcvcswHzOy7ZlYdeXw+cAvwP2b7pkQSXdrgIOe98gpw\nprtAZCJndSlpGQKRUXHN2OucqzOza4F7zawOqASucs5F58MuAaqB7DGHfQb4ipn9A2DAq865+8Y8\n3w5cDrxuZruAnwK3OOdGZvOGRPxgze7dZPb307x8Oe3lsT21Imc7tGEDfTk5FDc1UdzYSGtVldcl\niSSEuEIMgHNuB3DjBM/dCdwZs20kdlvM878AVsf7+iLJJNqV9MYll3hcifhBKC2NfZs3c+Hzz7Px\npZf4lUKMCKBVrEUWXH5bG8sOHmQ4PX10HhCRqbz+trcBULNjB+kDAx5XI5IYFGJEFthbX3wRgAOb\nNjGYnT3F3iJh7RUVNK5cSebAwOgszyKpTiFGZAEFBwdHu5J2X365x9WI3+z+vd8D4PwXX9QAXxEU\nYkQW1HmvvEJ2by/Ny5bRvHy51+WIzxzctInenBxKGhspP3LE63JEPKcQI7JQnGPTCy8AkVYYrXUq\n0zSSlsaeyGDwt/7mNx5XI+I9hRiRBVLV1ETpsWP0LVrE/gsu8Loc8andl16KM2Pd736nGXwl5SnE\niCyQrZF1kt645BJGMjI8rkb86nRREYdqakgbHmaTWmMkxSnEiCyAzBMneOsbbxAKBDSgV2Zt51VX\nAXD+Cy+QNjTkcTUi3lGIEVkAVU88QTAU4sAFF3C6sNDrcsTnGlevpnnZMhb19HDB7t1elyPiGYUY\nkXmW1tVFxU9+Apz5DVpkVszY9Y53AHDZtm0QCnlbj4hHFGJE5lnFj39MsL+fg6tWac0bmTNvbtrE\n6YICijo6WBqZQFEk1SjEiMyjwOAgVU8+CcALl17qcTWSTFwwyCtXXgnAiu99T5PfSUpSiBGZR+U/\n+QkZnZ10rV1L/cqVXpcjSea1t72N7pwc8vbto3D7dq/LEVlwCjEi8yTQ38/y730PgMMf/rAmt5M5\nN5yZOdrCV/3P/6zWGEk5CjEi86Tixz8ms6ODrnXraI+seSMy13Zs2cJgQYFaYyQlKcSIzINgXx/L\nv/99AA7dcotaYWTeDGVk0HDTTYBaYyT1KMSIzIPKp54i4+RJTtfU0BFZ60ZkvjS9+92jrTFFzz/v\ndTkiC0YhRmSOpZ88yfJ//VcADn30o2qFkXkXys4Oj7sCVj34IKZZfCVFKMSIzLHqf/on0np66Lj4\nYjovusjrciRFHL/uOnqXLWNRYyMVP/qR1+WILAiFGJE5lHPoEBU/+QkuEODgJz/pdTmSQlxaGnV/\n9mcAVD/6KGla4VpSgEKMyFxxjtX/+39joRBN7343vZoXRhZY+6WX0nnBBaSfPs2KRx7xuhyReacQ\nIzJHlv7mNxS+/DLDOTkcvuUWr8uRVGRG3ac+hQsEqPrhD8l9802vKxKZVwoxInMg2NvL2m98Awhf\nUj2Un+9xRZKquteupfGGG7BQiHVf/SqMjHhdksi8UYgRmQMrH36YrNZWTq9fT+N73+t1OZLiDn30\nowwUFZG3bx8VP/6x1+WIzBuFGJFZytuzh8qnnsIFAhy4/XYIBr0uSVLcyKJFvPmZzwCw6jvfIfPE\nCY8rEpkfCjEisxDo62P9l7+MhUIc/aM/onvNGq9LEgGg7fLLab38ctJ6elh///0QCnldksicU4gR\nmYXV3/oWixob6V65UoN5JbGYceBzn2OwoICCV16h6oknvK5IZM4pxIjM0NLf/pbKH/+YUFoae++6\ni1BGhtcliZxlqKCA/X/5lwCs+va3yamv97gikbmlECMyA1nNzaz/yleA8CDKHnUjSYJqv/RSmq67\njsDQEBtrawn29HhdksicUYgRmSYbHGTDX/0V6V1dtF16KUc/8AGvSxKZ1MFPfYrulStZdPQo6++7\nT+NjJGkoxIhMh3Oc99WvkrdvH/2lpez7/OchoI+RJLZQdjZv/PVfM5yTQ/ELL7D8+9/3uiSROaGf\nviLTsPz736fs5z9nJCuL1++5h+G8PK9LEolLX2Ule77wBQBWfuc7FD/3nMcVicyeQoxInEp+8QtW\nPfQQzoy9X/gC3evWeV2SyLR0XHop9R//OOYcNffeS/7vfud1SSKzohAjEoelL7xATWQgb/2tt9J2\n+eUeVyQyMw0330zje95DYGiIt959Nzl1dV6XJDJjCjEiUyh4+WU23nMPFgpx5I//WAN5xd/MePPP\n/5zWK64graeHTbffTs7Bg15XJTIjCjEikyh64QXeetddBIaGOHbDDRz62Me8Lklk9oJB9t59N+2X\nXELGqVNc8LnPkbt/v9dViUybQozIBMp+9jM2fulLBIaGaLz+eg7edhuYeV2WyJwIZWTw+j330HbZ\nZaR3dXHB7bezZOdOr8sSmRaFGJFYoRDVDz/M+vvvx0IhDn/4w7z52c/qUmpJOi4jgzdqa2m58krS\neno4/7//d8p/+lOvyxKJm34qi4wR7O7mLXffTfWjj+ICAd788z8Pr4mkFhhJUi49nT1f/CINN91E\nYGSE8x54gDXf+AY2OOh1aSJTUogRich77TUuuvVWin77W4YWL2b3/ffT+L73eV2WyPwLBKi/9Vb2\n33EHoWCQqqeeYvNtt5F97JjXlYlMSiFGUl5gcJCV3/42F372s2Q3NdG1Zg07v/UtOi+6yOvSRBbU\n8T/8Q175u7+jr7ycxW++yUV/+qdUPvkkjIx4XZrIuBRiJKUVbtvGxbfcworvfQ+c48jNN7Prm9+k\nv6LC69JEPNFVU8PLDz5Iy1VXEezvZ+3f/z2bb7uN3Dff9Lo0kXOkeV2AiBdyDxxg5cMPs/SllwDo\nWbGCA5/7HKfOP9/jykS8N5Kby54vfpGWq69m7de/Tt6+fVz0iU9w4pprOPSxj9FfVuZ1iSLANFpi\nzKzGzH5gZl8zs8fN7MI4jvm8mT1qZg+Z2ZfNzh0dGc8+krpqa2vn7mTOkff662z8n/+Ti269laUv\nvcRwdjYHP/lJXv72txVgEsj27du9LkGAtssvZ/s//zNHb7yRUHo6pb/4BVs/9CHOu/9+Fh05Mqev\nNaefdUkZ5pybeiezZcAO4Hrn3HYzWwe8CGx1ztVPcMx9wPnOuXdFHj8CnHbO3TadfWLO6eKpV5KH\nmTHdf/PHHnuMsjG/KQb7+ih+7jkqf/hDFkeaxEcyMmh673tpuOkmhgoK5rTmiTzzzDMUFhYuyGvF\na9euXeQl4CKW3/zmN/n0pz/tdRnnOH36NJs3b/a6jLN0dHRw9dVXz/vrZDU3s/I736HkmWewUCj8\n2hdfzPE//EPaLrsMl55+1v7Nzc3cdNNNcZ9/Jp918b/Iv/uMGy/i7U66D2hwzm0HcM4dMLNXgQeA\ncy7fMLP1wO3A9WM2PwQ8Z2YPOedejWef6b8dkbDAwAAFO3dS8stfUvTiiwT7+wEYysuj6brraHzf\n+xhcutTjKkX8o7+sjL1f+AKHP/IRqh5/nPJ//3cKd+ygcMcOBvPzaX3HO2i74gpObtqES9NIBVkY\nU/5PM7Ms4Abg4ZintgO3m1m+c+5UzHMfJNxVtWPMtpeBEPAh4NU49xGJz8gIvPEGPPMMV/7TP1Gy\nbx/BMfNcnNqwgePXXUfL1VcTysz0sFARf+urrOTNv/gLDn30o5T+4heUP/00ufX1VP7oR1T+6EcM\n5ebSuWULuatXwwUXwHnnaZ4lmTfxxOUtQBbQGrP9OBAENgPPxjx3GeCA9ugG51yfmZ0GLo5s+r04\n9hE5V18f7N8Pe/bAa6/BSy/Bjh3Q3Q1AeWS3rrVrab3ySlquvpr+8vKJzyci0zacn0/j+99P4/ve\nR+6bb1L8619T9MIL5Bw5Qslzz1Hy3HPw8MNQUgIXXQQXXgibN8Nb3gIrV0JM95PITMQTYqKDC9pj\ntndF7ksmOObkOANYusbsXxrHPpJqenuhrQ3a26GlJbztS1+ChgY4ehQOHQrfxus7X7kSLruM3+bn\nE7rmmgUb6yKS0szoXreO7nXrOPTxj5N97BhLdu0ia9s2VtTXw4kT8PTT4VtUMBj+vK5bB9XVEJ3S\n4Oc/h8pKKCqCggJQq6lMYTodl70xj4OR+4nmpo7dP3rM4DT3OdvLL4fvY7/E9HjhH4+MwNBQ+DY8\nfObP4z0eGoKenvCtu/vMn6O3kyfD4SUyduUs99xz9uNgENauhQ0bYOPG8G95l1wCpaUAHHnsMcoU\nYEQ80VdVRV9VFc1bt7LiAx+Aujp45ZXwbdcu2Lcv/EvJwYPh21h/8AdnP87ODoeZ6C03FxYtCm/P\nzj77z9nZ4dadtLSzb+NtCwTCXVxzeVsoC/VaPukCjCfEHI3cx34rRC9paJngmOpxtucBe6exz7ku\nVk9TUsvMDP8WtnRp+P6ZZ+Duu2H58jO31ashI8PrSkVkKmawZk34duONZ7b394fDzYED4RbWpia4\n/364+mpobAy3xHZ2hruO+/rCz4uMxzk36Q1YDPQAD8Rs/2pke9Y4x/w/wAiQG3OeEPDFePcZ57xO\nN91000033XRLnttUOWSyW7zzxHwXqHHObR2z7VfAMefcn4yz/yrgAOF5ZZ6ObLsK+A/grc65ffHs\nM2VhIiIikrLinbH3HmBNZJI7zGwjUAPcFXl8h5ltM7NCgMgEeP8I3DzmHJ8Avh4NJ/HsIyIiIjKR\nuFpiAMzsYuBOoA6oBL7inNsTee5vgA8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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "f, ax = plt.subplots(figsize=(9,6))\n", + "mean = 0.0\n", + "std = 1.5\n", + "xVals = np.linspace(-10.,10.,1000)\n", + "ax.plot(xVals, gauss(xVals, mean, std),lw=2,color='red')\n", + "ax.axvspan(-3.*std, 3.*std, alpha=0.25, color='black')\n", + "ax.axvspan(-2.*std, 2.*std, alpha=0.25, color='black')\n", + "ax.axvspan(-std, std, alpha=0.25, color='black')\n", + "ax.set_xlabel(r\"$x$\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The language of \"number of sigmas\" is often used to describe how noisy a measurement is, or equivalently, how much we're able to \"see\" the signal above the contaminating noise. Suppose we've made a measurement $m$, which is the sum of the signal that we want $s$ and the noise $n$:\n", + "\n", + "$m = s + n$\n", + "\n", + "Suppose $n$ is Gaussian-distributed with mean zero and a standard deviation of $3$, while the signal has value $15$. We would call this a \"five-sigma measurement\", because the size of the signal is five times the standard deviation of the noise. This means that it's overwhelmingly likely that we've measured the signal rather than just noise. Said differently, it's possible that we got really unlucky and had in fact just measured some noise rather than signal. But to mimic the signal, the random noise would have to five standard deviations away from its mean, which we've just seen is highly unlikely." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Averaging down noise" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "But how do we get to a low-noise measurement in the first place? Ideally, one can design an instrument that has very little noise by design. But this may not always be possible. So what do we do? Again, we can't easily subtract off the noise after the fact because it's random (which means we don't know ahead of time what value the noise is going to take)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The best that we can do is to make repeated measurements and to average them together. This tends to have the effect of averaging down the noise. The reason this works is that noise is often distributed with mean-zero Gaussian distributions like the ones we've plotted above. Notice that these distributions are symmetric about zero, which means that the noise is equally likely to be positive or negative. If we make many repeated measurements of the same signal, each time the noise will be different (otherwise it wouldn't be called random!) with it being positive roughly half the time and negative roughly half the time. If we average all our measurements together, the positive and negative noise contributions cancel each other out, and the average will tend to be less noisy than an individual measurement." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's try this out. I can simulate some noise by asking numpy to draw random numbers from a Gaussian distribution." + ] + }, + { + "cell_type": "code", + "execution_count": 89, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "num_samples = 50000\n", + "sigma = 31.2\n", + "noise_samples = np.random.normal(scale=sigma, size=num_samples)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "I picked a standard deviation of $31.2$, so if we take a peek at the numbers, we see that they're typically of that size:" + ] + }, + { + "cell_type": "code", + "execution_count": 90, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[ 15.06446975 39.04686486 -15.85655802 -51.24384466 -28.06541592\n", + " -77.6472985 26.14207299 3.26870247 -41.33022005 8.78763174\n", + " -50.73409185 -22.73161219 27.25016628 -2.8844408 -10.88717499]\n" + ] + } + ], + "source": [ + "print noise_samples[:15]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "But if I now average (say) the first $1000$ samples, I get a much smaller number because the positive and negative noise samples balance each other:" + ] + }, + { + "cell_type": "code", + "execution_count": 91, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1.15569647093\n" + ] + } + ], + "source": [ + "print np.mean(noise_samples[:1000])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This means that if I make $1000$ measurements of the same thing and average the results, the result will be less noisy. Let's see how quickly this averaging occurs:" + ] + }, + { + "cell_type": "code", + "execution_count": 92, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 92, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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+/fXXnCeEqYCu3XxKKRW9hJqFdyIpzBmovXv3UrlyZUSEHTt2+I1xysrKIi0t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\nAUiaDJwPrM3i6V1EMh7UJyLi8lwjLIj7QJmZmXWPwseBSoNYBtgLeDs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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "averaged = np.zeros_like(noise_samples)\n", + "for i in range(num_samples):\n", + " averaged[i] = np.mean(noise_samples[:i+1])\n", + "\n", + "f, ax = plt.subplots(figsize=(9,6))\n", + "ax.loglog(range(1,num_samples+1),abs(averaged),lw=2,color='black')\n", + "ax.loglog(range(1,num_samples+1),sigma/np.sqrt(range(1,num_samples+1)),lw=2,color='red')\n", + "ax.set_ylabel(r\"Absolute value of noise in average\")\n", + "ax.set_xlabel(r\"Number of samples in average\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here I've plotted the residual noise on the vertical axis against the number of samples that I've averaged together. Actually, I plotted the absolute value of the residual noise, since here we're worried only about how much the noise sways our answer, and not which way. The results (in black) exhibit lots of random fluctuations, but the noise seems to follow the overall trend of the red curve, which shows $\\frac{\\sigma}{\\sqrt{N}}$,\n", + "where $\\sigma$ is the standard deviation of the noise, and $N$ is the number of samples." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In our radio observations, every if we make one measurement every $\\Delta t$ seconds and observe for a total of $t$ seconds, we end up with $N = t / \\Delta t$ samples. This means that the noise in our measurement will tend to be proportional to $1/\\sqrt{t}$. As expected, the longer we observe for, the less noisy our measurement." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we plot $1/\\sqrt{N}$ on a linear scale, we see that in a typical measurement it's very easy to drop to a moderate noise level pretty quickly, but to get to an extremely low noise takes a long time because the curve flattens out. For an instrument like HERA, we'll need to observe for about $1000$ hours to get to the sensitivity levels that we want!" + ] + }, + { + "cell_type": "code", + "execution_count": 97, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 97, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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0aFCFqpUkNanZwFRuBiZ1xpIlS7jjjjv43e9+x1133cWyZct2eH3X\nXXflDW94A0cddRRHH300s2bNoq6urkLVSlLtMjCViYFJ3fHXv/6Vu+++m7vvvpu77rqLZ555ZofX\nI4KDDjqIWbNmceSRRzJr1iymTZtmN54klZmBqUwMTOoJK1eu5L777uPee+/l3nvv5aGHHmqePLPJ\n6NGjmTlzJjNnzuSII47giCOOYPfdd69QxZLUPxmYysTApHLYuHEjf/zjH3nggQe4//77uf/++1mx\n4tV37pkyZQpHHHEEhx9+ePOy2267VaBiSeofDExlYmBSb8hMli1bxoMPPsiCBQtYsGABf/jDH1i/\nfv2rtt1rr7047LDDOPTQQ5uXSZMmUbjTjySpPQamMjEwqVK2b9/O008/zR/+8AceeughHnroIR5+\n+GE2btz4qm3HjRvH61//eg455BAOOeQQDj74YKZPn+6tXSSphIGpTAxMqiZNIeqPf/wjDz/8cPPy\n8ssvv2rbwYMHc+CBB/K6172Ogw46qHmZPHmyV6Mk1SwDU5kYmFTtMpMlS5bw6KOP7rAsWrSo1e1H\njRrFgQceyIwZM5qXAw880G49STXBwFQmBib1VevWreOJJ57gscce4/HHH+exxx7jscceY/Xq1a1u\nX1dXx/Tp03dYDjjgAPbZZx8n3ZTUbxiYysTApP4kM2loaOCJJ55oXh5//HH+/Oc/89JLL7W6z+DB\ng9lvv/044IAD2H///dl///2ZNm0a06ZNY+zYsb38E0jSzjEwlYmBSbWgKUg9+eSTzctTTz3FU089\n9apZy1saM2YM++23X6vLqFGjevEnkKTOMTCViYFJtW7Dhg0sXLiQJ598koULF7Jw4UKefvppFi5c\n2Oq0B03Gjh3La1/7Wvbdd9/m9dSpU5k6dSq77767s5pLqggDU5kYmKTWZSYrVqzgL3/5C3/5y194\n5plndni8adOmNvcdOnQo++yzD1OnTmWfffZ51eLVKUnlYmAqEwOT1HWZycqVK3nmmWdYtGhR83rx\n4sUsXry4zYHnTUaPHs3ee+/dvOy11147LKNHj/YbfZK6xcBUJgYmqee98sorPPvssyxevJhnn332\nVUt7V6cARowYwV577cWee+7ZvEyZMqV5vcceezBkyJBe+mkk9SUGpjIxMEm9KzNZtWoVf/3rX3dY\nli5dypIlS1iyZAnr1q3r8H0mTJjA5MmTmTJlClOmTGHy5Mk7LJMmTWLo0KG98BNJqiYGpjIxMEnV\nJTN5+eWXWbJkCcuWLWPp0qUsXbq0+fGyZctYvnw527dv7/C9xo4dy+TJk9ljjz2al0mTJjFp0qTm\nx7vttpsD1KV+xMBUJgYmqe/Zvn07K1asYNmyZc0B6rnnnmteli1bxooVK9i2bVuH7zVo0CB23333\n5iA1ceLEVpdx48Y5wafUBxiYysTAJPVPjY2NNDQ0sHz58uZA9fzzz/P888+zfPny5nVbE3qWGjBg\nAOPGjWP33XffYZkwYcKr1qNHj/aqlVQhBqYyMTBJtW3z5s2sXLmS559/nhUrVjSvS5fVq1fT2d8V\ngwYNYvz48YwfP54JEyYwYcKE5sdN7U3LuHHjHMAu9SADU5kYmCR1xtatW1m1ahUrV65k5cqVrFix\nghdeeIGVK1c2r5ser127tkvvXVdX1xyexo0bt8PjpmW33XZrXg8fPrxMP6XU9xmYysTAJKmnbd68\nmYaGBhoaGnjhhRd2WDc9XrVqVfPzzgxgb2n48OHstttuzcu4ceMYO3Ysu+22W6vrsWPHMmzYsDL9\ntFJ1MTCViYFJUiU1Njby8ssv09DQwKpVq3ZYGhoaWL16dfPzpsdbtmzp8nGGDx/eHJ7Gjh3LmDFj\nXvV4zJgxOzwePXo0gwcPLsNPLZWPgalMDEyS+pLMZP369axevbp5WbVqFS+++CKrV69uXjc9bnq+\ndevWbh1vxIgRzeGp5brpcculZVtdXR0DBw7s4Z9e6piBqUwMTJL6u6aQ9eKLL7JmzZrmINX0uOW6\n6fFLL73EmjVrutxd2CQiqKuraw5Qr3nNa5rXTUvp86Zl1KhRjBgxwtvjqFtqNjBFxHTgS8ByYBLw\n5cx8uIN9XgP8KzAnM2d2sK2BqcrU19dTX19f6TJU5PmoPr11TjKTdevW7RCimoJUy3Xp45deeolX\nXnllp449cOBARo0a1bw0BanSx+0tw4YN65XQ5WekutRkYIqIKcCDFILPgoiYBtwHzMzMxW3sszsw\nh0LI2piZUzs4hoGpyhT/Y690GSryfFSfvnBOtm/fztq1a5sD1Msvv9y8NLWtXbv2Ve1NbRs3btzp\nGgYNGkRdXR2jRo1qdd3WMnLkyB2eDx8+vN3g1RfORy2p1cB0HbBfy6tEEfE7YG1mvrODfecCRxiY\n+h5/+VQXz0f1qYVzsnXr1ubwtHbt2h0et2xrbXnllVdYu3Ytmzdv7pFaBgwY0ByiWq6bHv/whz/k\nc5/7XHNbe8uIESOcMb7Mai4wRcRQYA3wg8z81xbtXwY+DYzLzDYnO4mIHwLHGpj6nlr4Y9CXeD6q\nj+ekc7Zs2dIcnprWa9euZd26dc1tTe3r1q1rft6ybd26dWzatKlH6xo6dGhzeGq5Lm3r7LLrrrs6\nwL6FcgSmao+4hwNDgVUl7SuAgcBhwB3t7O9vE0mqYbvsskvzXFU7Y9u2bc2BquW66fG5557LJZdc\nskNb0+N169axfv36HZ5v3ryZzZs3s2pV6Z+37hs6dOgOAaq9ddPjzixDhw518D3VH5h2L65fLGlf\nV1yP78VaJEk1atCgQc3f9mvNueeey0UXXdSp98pMNm7cuEOIWr9+ffPS1LZhw4Z225ram9qaQtjq\n1at78kdnwIABDB8+vNUw1dReui5ta+/xsGHD+sR9F6s9MDUpHfnXdN2x6zO3SZJUQRHRHComTJjQ\nI+/Z2NjIpk2b2LBhQ3OIagpSra1LH7fXtmXLluaAVi5Dhw5tDlCtLcOGDWu3bdasWcyYMaNs9UH1\nB6ZlxXVppK8rrht6sRZJkqrSgAEDmkPY+PE92/mybds2NmzYwMaNG3cIUqXPm9qalqa2piDXsr3l\ndhs3bmy+OrZmzZpu1XjllVeWPTBV+6DvkcBK4FuZeUGL9q8BHwbGZmabX4GIiP8LHNeZQd89VLIk\nSaoCNTXoOzPXRcQvgGNLXjoMuLG9sNTF4ziaTZIktan6R1kVJp/ctzhhJRExA5gOfK74/IKIeCAi\nxrSy7y5UeSiUJEnVr+rDRGYuioiTgEsjYhGwB3B8Zi4tbjIe2BsY1rRPRIwA3gUcB0yIiE8CN7TY\nR5IkqdOqegxTuXXnHnXqGcVJSS8BzqQw19bdwKcz89kW23h+KiQivgUckJnHt2jzfFRI8RZR/0ph\nSpUlwC8yc1PxNlBfozBX3UTgu5n5X5WrtH+LiA9SmB9wOYX/eX8qM69u8bqfkTKLiOOAzwLXZeaP\nSl7r8POwU5+ZzKzJBZhCYUD5zOLzacBqYGqla6uFBfg2cA1wOoXgtBl4Fqjz/FT83JwIbAfmt2jz\nfFTufJwB3Fv6bw2MABYCpxefjwFeAI6sdM39caHwP3e3lrT9F/Dh4mM/I+U/B28Bflj8/XR2yWsd\nfh529jPTF8YwlctlwNLMXACQmQuBR4ErKlpVDYiI0cDizDwvM3+emf8Phf973hN4R3Ezz08FREQd\n8HHg/pKXPB8VEBHvpfA/FKfkq282/llgVGb+HCAz1wC/Ab7Zu1XWjPcAi0raHgTeWnzsZ6TMMvNW\n4HKgtS9qdebzsFOfmZoMTMXuoNOABSUvLQDeFhGjer+qmpLA/ylp+zmFD8EYz09FfZXCFyq2NjV4\nPiojIvYDvgd8LDNfbmWTf6TwB7ulBcAhEfG6ctdXgxqAs4rnpckxwG1+RnpVWzf168znYac+MzUZ\nmOjcPepUJpn5cmb+raR5MIUgdTeen4qIiLcBf83MP5W85PmojEsodOnsGRHXRsSCiLg0IgZHxB7A\nXrR+TgI4opdrrQVfpdAVdE9EvCki/jdwX2Z+Cz8jvelVA68783noic9MrQYm71FXfd4G3JKZf8Tz\n0+siYizwIQqXu0t5PnpZ8YrFqcBiCuNmzgEuAi4EfoLnpNdl5jPA8RRuyTUf2C0zLyy+7PmorM78\n++/0OarVwNTEe9RVgYjYBfgohXFMLXl+es8VwIWZ2djONp6P3jOVwlQpN2VxOpTMnAfcBLyT/5lG\nxXPSu8YCd1IYhP++iPh+yeuej8rqzL9/t89R1c/DVCbeo666XAZ8ITP/Wnzu+elFEXE68KfiINXW\neD5638jiel1J+28ojJWZVHzuOeklETGLwpiygyiM8fsP4AMR8QyFK07g+aiUzvyO2unfY7UamJ6k\nMHBsYkn7ZApfb3+o1yuqURHxz8DjmXlbi2bPT+/6KHBccUzGDiJiO4VvzXk+elfTJLvjStpXFNer\nKMz109o5SeD35SutZl1CYdjAJmj+BuMY4F+Ab+BnpGIysyEi2vs83NOJbTr8zNRkl1xmrgPKfo86\ntS8izgSGZuYPWrSNoDCw0vPTez4EHAq8vsXyEPCH4uPr8Hz0qsxcATxA4W4FLe0GrKHwzZ4fAEeX\nvH448EBmLil7kbVnNC26bYrd19+gMAH0evyMVFp7n4elndim489MpSeiquAEWK+l8ItnWvH5DAoT\nWO1Z6dpqYQHOBn4LnNRiOQP4TwpXPj0/lT0/d7DjxJWej94/BzOB9cDhxecB3Ab8U/H5aApXok4o\nPp9A4QrU31W69v64UBhj+TQwuEXb54CLi4/9jPTOedgPaAQ+WNLe4edhZz8ztdolR3Z8jzqVSUSc\nQyHpQ2Hm1pauzsxtgOen8pq/vuvnpfdl5oLiv3l9RCyg8Mt9bhavyGbmSxExG/hfEXECha6Fd2Xm\nHypXdf+Vmd+IiEHA9cXzERTGmH2j+LqfkTIrjiP7IIXfTWdHRENm/gY693nY2c9MTd9LTpIkqTNq\ncgyTJElSVxiYJEmSOmBgkiRJ6oCBSZIkqQMGJkmSpA4YmCRJkjpgYJIkSeqAgUmSJKkDBiZJ/U5E\nzIiIyyJidUQ0RsQX2tjuExHxVHGb6yLibb1dq6S+wZm+JfVbEXEkcC/w18yc2sY2n6dw/69zerU4\nSX2KV5gk9WfjgM3AXsV7SLXmAOA7vVeSpL7IwCSpPzsJuIjCjVI/2MY2B2Xmfb1XkqS+yMAkqT87\nDLgSWAicFhF1LV+MiNcBj1eiMEl9i4FJUr8UEa8FFmdhoOb3gaHAP5ZsdhJwS2/XJqnvMTBJ6q9O\nAm4rPr4W2Maru+VObLGNJLXJwCSpv3ozMA8gM1cBNwGHF7vhiIihwLDMXF25EiX1FQYmSf1ORAwC\nxmbmyhbN32fHwd/HAXf2cmmS+igDk6T+6I1A6TffbgeWAGcVA9VJwK29XZikvsnAJKk/ajl+CYDi\n4O//D9gNeDtwBPBA75cmqS8yMEnqj44B7mml/QdAI/A54Ln0VgeSOsnAJKlfiYiJwKbM3Fr6WmY+\nT6Eb7vXYHSepCwxMkvqb9wLPtPP694prA5OkThtU6QIkqSdExBuBOcD5wMsRsQK4tJVut98Ct2Tm\nC71do6S+K+zClyRJap9dcpIkSR0wMEmSJHXAwCRJktQBA5MkSVIHDEySJEkdMDBJkiR1wMAkSZLU\nAQOTJElSBwxMkiRJHTAwSZIkdcDAJEmS1IH/H0YEGoMdmUvdAAAAAElFTkSuQmCC\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "f, ax = plt.subplots(figsize=(9,6))\n", + "ax.plot(range(1,100), 1/np.sqrt(np.array(range(1,100))), lw=2, color='black')\n", + "ax.set_xlabel(r\"$N$\")\n", + "ax.set_ylabel(r\"$1/\\sqrt{N}$\")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 2", + "language": "python", + "name": "python2" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.13" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +}