{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Random numbers\n", "Module 4 of Phys212/Biol212 class\n", "### by Ilya Nemenman, 2016-2019\n", "\n", "## General notes\n", "In this notebook, we explore stochastic models, including generation of (pseudo)-random numbers and a variety of models and calculations that employ them. In general, all methods involving generation of random numbers are called *Monte Carlo* methods, referring to the Monte Carlo casino in Monaco, arguably the most famos casino in the world. The history of this term is rather curious. As most modern computational methods we use, claculations with random numbers trace their history to the Manhattan project: making the nuclear bomb in Los Alamos, New Mexico, during the second world war. People like von Neumann, Ulam, Metropolis, Bethe, Feynman -- whose names are familiar to everyone who knows even a bit of modern physics or advanced math -- worked there at the time and are responsible for the bulk of early ideas in computational physics. However, this was a secret laboratory, and projects, whether they involved production of fissile materials or development of algorithms, required codenames. The story goes that the name *Monte Carlo* was suggested by Metropolis as a code name for various stochastic models that were developed at the time by von Neumann and Ulam. And the name stuck.\n", "\n", "A good introduction to probability theory, one of my favorites, but more on the mathematical side, is in the book \n", "__[Introduction to Probability](http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/pdf.html)__ by CM Grinstead and JL Snell." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Why do we need random numbers?\n", "Hopefully an answer to this question will become clearer as we progress through this notebook and see a bunch of their applications. However, even now it is useful to break down various problems where random numbers are used into a few geenral classes.\n", "\n", "First, some processes are **fundamentally random**. Here I am largely talking about quantum mechanics, where all experimental evidence suggests that the randomness is intrinsic and unavoidable. Second, sometimes we simply do not know enough information about the underlying system, and we choose to model this **absence of knowledge** or as randomness, as well. Third, there are processes that lie somewhere invetween these two extremes. For example, motion of molecules of air or interactions of molecules in chemical processes can be described by deterministic Newton's equations. However, these equations give rise to **chaotic** motion (as we discussed in one of the projects in Module 2), which is also not predictable and can be modeled stochastically. Finally, there are some deterministic calculations that are **easier done** using random numbers than using deterministic approaches (e.g., calculating area of a complex object). We will discuss examples of these in subsequent lectures.\n", "\n", ">### Your turn\n", "Think of various physical, chemical, biological processes that involve randomness (e.g., mutations and recombinations in biology, turbulence in physics, and so on) and decide to which of the classes of uses of random numbers deined above they belong." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Introducing concepts of randomness\n", "Our introduction here is deliberately a bit sloppy, with no hard mathematical definitions and proofs. Those interested can find them in the textbook I linked.\n", "\n", "To define the necessary probabilistic concepts, we need \n", " - To define a space of outcomes that a random variable can take (e.g., head or tails, six sides of a dice, etc.). If a random variable is denoted by a small latin letter, then its space of outcomes is typically denoted by the capital version of the same letter, e.g., $x$ and $X$.\n", " - Then we define a probability of a certain outcome $x$ as a limit of frequencies after many random draws, or events. That is, if after $N$ draws, the outcome happened $n_x$ times, then it's frequency is $f_x=n_x/N$, and the probability is $P(x)=\\lim_{N\\to\\infty}f_x=\\lim_{N\\to\\infty}\\frac{n_x}{N}$.\n", " \n", "Probabilities satisfy the following properties, which follow from their definition of limits of frequencies:\n", " - nonnegativity: $P_i\\ge0$;\n", " - unit normalization: $\\sum_{i=1}^N P_i=1$;\n", " - nesting: if $A\\subset B$ then $P(A)\\le P(B)$;\n", " - additivity (for non-disjoint events): $P(A\\cup B)=P(A)+P(B)-P(A\\cap B)$;\n", " - complementarity $P(not\\, A)=1-P(A)$.\n", "\n", "### What if we are studying more than one random variable?\n", "Bivariate distributions (a special case of multivariate ones) $P(x,y)$ defined the probability of both events happening. It contains all of the information about the variables, including\n", " - The marginal distributions: $P(x)=\\sum_{y\\in Y} P(x,y)$ or $P(y)=\\sum_{x\\in X} P(x,y)$ \n", " - The conditional distributions, which can be defined as $P(y|x)=P(x,y)/P(x)$ or $P(x|y)=P(x,y)/P(y)$, so that the probability of both events is the probability of the first event happening, and then the probability of the second happening given that the first one has happened.\n", "The conditional distributions are related using the Bayes theorem, which says $P(x,y)=P(x|y)P(y)=P(y|x)P(x)$, so that $P(x|y)=\\frac{P(y|x)P(x)}{P(y)}$. \n", "\n", "We can now formalize the intuitive concept of dependence among variables. Two random variables are considered to be statistically independent if and only if $P(x,y)=P(x)P(y)$, or, equivalently, $P(x|y)=P(x)$ or $P(y|x)=P(y)$.\n", "\n", "## Probabilities of continuous variables\n", "**To be written**\n", "\n", "## Random numbers module in Python\n", "**To be written**" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['Lock',\n", " 'RandomState',\n", " '__RandomState_ctor',\n", " '__all__',\n", " '__builtins__',\n", " '__cached__',\n", " '__doc__',\n", " '__file__',\n", " '__loader__',\n", " '__name__',\n", " '__package__',\n", " '__path__',\n", " '__spec__',\n", " 'absolute_import',\n", " 'beta',\n", " 'binomial',\n", " 'bytes',\n", " 'chisquare',\n", " 'choice',\n", " 'dirichlet',\n", " 'division',\n", " 'exponential',\n", " 'f',\n", " 'gamma',\n", " 'geometric',\n", " 'get_state',\n", " 'gumbel',\n", " 'hypergeometric',\n", " 'info',\n", " 'laplace',\n", " 'logistic',\n", " 'lognormal',\n", " 'logseries',\n", " 'mtrand',\n", " 'multinomial',\n", " 'multivariate_normal',\n", " 'negative_binomial',\n", " 'noncentral_chisquare',\n", " 'noncentral_f',\n", " 'normal',\n", " 'np',\n", " 'operator',\n", " 'pareto',\n", " 'permutation',\n", " 'poisson',\n", " 'power',\n", " 'print_function',\n", " 'rand',\n", " 'randint',\n", " 'randn',\n", " 'random',\n", " 'random_integers',\n", " 'random_sample',\n", " 'ranf',\n", " 'rayleigh',\n", " 'sample',\n", " 'seed',\n", " 'set_state',\n", " 'shuffle',\n", " 'standard_cauchy',\n", " 'standard_exponential',\n", " 'standard_gamma',\n", " 'standard_normal',\n", " 'standard_t',\n", " 'test',\n", " 'triangular',\n", " 'uniform',\n", " 'vonmises',\n", " 'wald',\n", " 'warnings',\n", " 'weibull',\n", " 'zipf']" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#Initialization\n", "import numpy as np\n", "import numpy.random as nprnd #note the new module we will be using\n", "import matplotlib.pyplot as plt\n", "\n", "%matplotlib inline \n", "dir(nprnd) # see which new functions exist in this new module\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Build-in routines for pseudo-random numbers generation\n", "We will first experiment with the build-in routines for generating uniform standard random numbers (floating point numbers between 0 and 1).\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[0.83654705 0.61883193 0.58793764 0.28319603 0.71339089]\n", "[0.72467775 0.74989385 0.30990007 0.84100082 0.56137152]\n", "[0.30972678 0.08276444 0.15942945 0.26879311 0.23421895]\n", "[[0.39379855 0.41688528 0.11976077]\n", " [0.96018263 0.39387572 0.672743 ]\n", " [0.78234578 0.71078343 0.33331223]\n", " [0.9149731 0.11802462 0.97579149]\n", " [0.96876007 0.45514935 0.6707342 ]]\n" ] }, { "data": { "image/png": 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XAad3t48ANk267j1s87OBo4CbdnL/ccC/MLiG7OnAdfO5/4Xeg+gzdcca4Nzu9qeBY5O0LtibFXO2uaquqqp7u8VrGVxvMsv6TtHyduCvgf8dZ3Ej0qfNrwI+WFV3AVTVHWOucb71aXMBj+pu/xo7XEc1a6rqGuDHu9hkDfDxGrgW2D/J8vna/0IPiD5Td/xim6raBmwFHj2W6kbjwU5XchqDTyCzbM42J3kycGhVfX6chY1Qn7/z4cDhSb6c5NokzxtbdaPRp81vA05OspnB2ZCvG09pEzPS6Ymm7jTXeTbn1B09t5klvduT5GRgNfC7I61o9HbZ5iQPAd4HnDqugsagz995CYNhpqMZ9BL/LcmTquruEdc2Kn3afBLwsap6T5JnAJ/o2nz/6MubiJG+fy30HsScU3cMb5NkCYNu6a66dNOuT5tJ8nvAWcDxVfWzMdU2KnO1eT/gScDVSTYxGKtdP+MHqvu+ti+qqv+rqluAmxkExqzq0+bTgAsAquqrwN4M5mlaqHr9f99dCz0g+kzdsR44pbv9IuAL1R39mVFztrkbbvkwg3CY9XFpmKPNVbW1qpZW1cqqWsnguMvxVbVhMuXOiz6v7c8xOCGBJEsZDDl9f6xVzq8+bb4VOBYgyRMYBMSdY61yvNYDr+jOZno6sLWqtszXky/oIabaydQdSf4S2FBV64GPMuiGbmTQczhxchXvuZ5tfjfwSODC7nj8rVV1/MSK3kM927yg9GzzZcBzk3wHuA/406r60eSq3jM92/wm4CNJ3shgqOXUWf7Al+Q8BkOES7vjKm8FHgpQVWczOM5yHLARuBd45bzuf4b/7SRJI7TQh5gkSbvJgJAkNRkQkqQmA0KS1GRASJKaDAhpDJJcPeMX5mkRMiCkKddd4S+NnQEhDUmyMsl3k3yk+w6Fy5PsM9wDSLK0m7KDJKcm+VySi5PckuSM7vs2vtFNkHfg0NOfnOQrSW5K8tTu8ft2c/5/rXvMmqHnvTDJxcDlY/5nkAADQmpZxWCa7CcCdwMvnGP7JwEvYzAd9TuAe6vqycBXgVcMbbdvVT2TwXeQnNOtO4vB9C6/zWBajHcn2be77xnAKVX1nHlok/Sg2XWVftUtVXVDd/t6YOUc219VVfcA9yTZClzcrf8W8JtD250Hgzn+kzwqyf7Ac4Hjk7y522ZvYEV3+4qqmuWJIzXjDAjpVw3PbnsfsA+wjQd63HvvYvv7h5bv55f/j+04r00xmK75hVV18/AdSZ4G/PRBVy7NI4eYpH42AU/pbr9oN5/jpQBJfofBrJtbGUw897rt32LYzbQrTQUDQurnb4DTk3yF3f9+gbu6x5/N4HsLYPA1qA8Fbuy+mP7te1ypNE+czVWS1GQPQpLUZEBIkpoMCElSkwEhSWoyICRJTQaEJKnJgJAkNRkQkqSm/wchDBhGAN7e9wAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Examples of random number generation using built-in routines. The r1 variable should be pointing at an \n", "# array of 5 random numbers. We do the same calls multiple times to illustrate that the numbers are, indeed,\n", "# seeminly random (or rather pseudo-random)\n", "r1 = nprnd.random(5)\n", "print(r1)\n", "r1 = nprnd.random(5)\n", "print(r1)\n", "r1 = nprnd.random(5)\n", "print(r1)\n", "\n", "# The following command generates a 5x3 matrix of pseudo-random numbers.\n", "r2 = nprnd.random((5, 3))\n", "print(r2)\n", "\n", "# Finally, the following command generates n_rand pseudo-random numbers and histograms them into 50 bins,\n", "# illustrating that the numbers we generate are, indeed, uniformely distributed.\n", "n_rands = int(1e4) # how many random numbers to generate\n", "x = nprnd.random(n_rands)\n", "plt.hist(x, bins=50)\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Most pseudo-random generators fail not because they do not distribute the numbers uniformely along the interval $[0,1)$, but rather because they produce correlations among the generated numbers. For example, making a scatter plot of two consecutive generated numbers against each other, $x_{i+1}$ vs $x_i$, may show some structure, indicative of dependence. We can plot the pairs of consecutive random numbers from the built-in generator against each other to see that no structure is visible. In reality, this generator passed a lot more stringent tests, but this is a good beginning." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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xKeqps40xBmD7S686f49KCfFWquo9IvKpUs+r6t+4K1Z9rVoyhxsf3GaNWMaYSGvEFVWl6qMpuf/NP+VXFfn6t+6N/dYN0RhTEy8hHD95EvsPZZ2mIm8WZYOCqn419/+zjSuOO12dHXR1dtDTN8Cq7z1tI1mbyHGTEmSHR5zn5m8m06xHXCSkUx6vHhkqfFdx2EVjMXitWFdnB8vfPjPsYpgiR4biFRDAGqmjYOWiWRzIZBmO2c4Zu6DQ0zfAA1ss7YUxpryUl2D9k7ub8srgpp5tTtdfMSiISEJErnZaggbr3thvDc4mElox/1QUCP5gx2atYr7vCbfTC1cMCqo6AnzMaQkazBqaTRSkUx59N1/IuuULSAYZJWfqpjlDwWtcp2YJUn30QxH5tIjMFJHp+T+npXKgp2+AzlsfDbsYxgQymMmyeO2PABixIfmmiOuThCAjmv8i9/+jRY8pEJmEeD19AzZOwUTOwGCGT67fSspLcChr6TKMb9Hp05yuv+qVgqqeVuIvMgEBWq8dIezaBJusrL6Om1T+A1XgUHYEL2FVSMb31K4D4U6yIyLtInKTiNyVu3+miLzPWYkcaLXU2ZMrHEQaYcpx1gBaT0GS5h0/ORa5K00AzTDJzjeAo8A7c/f3ALc5K5EDrZb7KBNyVYIlL2s8G9dgirk80Q0SFM5Q1S8CWQBVzUC0JuVatWSOXX4bY1qGyxPdIEHhqIikyPXUEpEzgCNBVi4iF4lIv4jsEJHVZZa5WkSeFZHtIvJPgUteK4sJpsWlU1at51oznFt6SQl9kp1bgH8FZorIvcBi4LpqLxKRJHAn8Kf4VU5PisgGVX22aJkzgRuBxaq6X0ReX/smVNe9sb9pB6IYUy9Hh1qnM0Uz8hLSFN2Dhx0fy4L0PvohsAw/ENwHLFTVHwdY97nADlXdqapHge8Al49Z5sPAnaq6P/derwQvenA2YM3EgXVbdSs7ojTDueUIhN7QDPAnwAXA+cB/DfiaDqB4PPae3GPF3gy8WUQ2ichmEbmo1IpE5HoR6RWR3r179wZ8+9fYiFBjTCtxeaIbpEvql4GPANuAXwD/XUTuDLDuUkfisXF2EnAm8G5gBfAPIpI+5kWqd6nqQlVdOGPGjABvPdp4hoVbHDHGhCFIu4WAs7EKQa4U/gRYoqrfUNVvAJfgH8Sr2QMU56g+FXipxDL/rKpZVX0e6McPEnXT0zdQcxtzRzrFtefNqmcxjDEmkCCZuhV3VUhBgkI/UHyEnAk8E+B1TwJnishpItIGXANsGLNMD36VFCJyEn510s4A6w6se2N/zQmuBgYz3LN5Vz2LYUxk1OsieUpbsk5rMqW4GqtQNiiIyA9EZAPwOuCXIvJjEXkM+CVQtQ5HVYfwM6xuzL3mu6q6XURuFZGlucU2Ar8TkWeBx4BVqvq7iW3SaK02mtkYV6a1e6xbvqBufeAPHrXeUC65GqtQqUvq/53oylX1EeCRMY/dXHRbgU/l/pw4JZ2y3kcOeUmx7r4t4qTj2yxxZIS4GqtQaY7mnxTfF5ETKy3frFYtmWNzMjuSANqSCbLDdhBpBb9+5WDYRTA16Ooc25mzPoL0PrpeRH6D347QC2zJ/Y+Ers4OJjXDMMQWNIJVERgTljB7H60C5qrqbFU9PYqps8NOIGeMMfUWZu+j54BDTt7dmIhafMZ0OtIpS6llQuOqE02QNoIbgZ+JyBMUJcJT1U84KZED09o9Sz1s6iblJblq4axCne7s1Q+HXCITR2H0Psr7KvAj/BHNkayHueWyuYEbm9Mpj1cPDzmfHNtEVyY7zA3rt3LD+q028t2EIiEh9D4qMqSqzrqMNkL+jK57Yz8DgxmEY/NtgH8GuGbpXD65fmtDyxcn7S0237CdO7iV8pIcNylhEzuNEWTU83gFaVN4LNcD6WQRmZ7/c1ckN7o6OwqT7ZT6PDvSKW5fNo+uzo6Wm6mtmXxh2dmkPDcjXd9wQpuT9ba6Zu2cl0553L5sHmuWznW2z0RZmA3N7yfXroDfHTVSXVKLrdmwnWyAELtqyZyyO2FShJWLZlnm1XHqfXHfqInqK01aX6vfvHq0buuKE5dnnRORn7u6q7ODK85x0yc/ykJraFbV05y8cwgqXYIODGb45Pqt9L64j9u65nF/7y42PbfvmOWGVXlgywCLTp9W8nlT2b2bd426Ugsyab2Jp+IJ6h/Y4qZPfpSF1tAsIh8o9biqfrv+xQmX4h+0AH5W4YCfyQ6zeed+vAS0UPV4QzTpSalpUi8NZvjsD7Zb6o0SwmxofnvR7cn4k+08BUQqKAQd/acQKEPqsDbHLEzGtLL2tqR1Jy/DVZqLINVHHy++LyJTgbudlMahGx8Mku3bGNNMDlkalYYbTyvfIeo8EY5rPX0DTZ3qwtqsjSnNLsZLm9buOVt3kDaFH/Dad5MAzgK+66xEDric5LoeJglkbe83xgR0y2Vzna07SJtC8bwKQ8CLqrrHUXmcaPaJdpr4IsaYltCWFI5WaQRsSwrtbZNiP1AuSJvCT6ot0+xsoh1j4q1aQMgvk41IQOje2B/qfArLROTXInJARH4vIq+KyO+dlMaRVUvm4CWt4t4YU1lUanFd1n4EqT76InCZqv7SWSkcy0fUGyynUcOVyzNljBk/l6l4gvQ++k2UA4IJl4KlBDGmjlJe0tnANQh2pdArIuuBHkbPp/Cgs1I5sGbD9rCLEFuWhtyY+nnbrKnO2hMg2JXCifhjEy4ELsv9vc9ZiRyp1KMgnfIQ/EypxhjTzDY9t4+berY5W3+Q3kcfdPbuTWLrLRfS0zfQ9OMZTHykvERTD7g04brvid3c1jXPybqDVB9FXrWoelPPNh7YMmBJt0zTaERAsE4A0eWySjYWQeG+J3ZXfd7qvU3c2B4fXS47b9RvhpMmVu2AbwHBGBMlK86b6WzdQXIfpYEPALOLl1fVTzgrVZ0lRSoe+O0y2jTKlLYkh7MjdiJixiUh8P7zZjlrT4Bg1UePAJuBbUAkW76qzZJmP0/TCMmE8Pk/m8cnbRClU1PakhxssZTb65YvcNoNtZholTMWEXlKVd/WkNIEsHDhQu3trW2K6MVrf2S5j4wTHekUh44OBZ4IpiOd4uCRodgnXXNp5aJZwLFTv0aZCOQP1dPaPW65bG7NQUJEtqjqwmrLBWlTuFtEPiwiJ4vI9PxfTaUJmQWE+vESbnO5R0k65bFqyZyaZgYbGMxYQHBs/ZO7eejpl1H8quFWUHzuvv9QllXfezrwbJK1ChIUjgLdwOPAltxfoFN1EblIRPpFZIeIrK6w3JUioiJSNYrVqqdvoGV2jGaQHcGmR8wZzGS58UF3g4jM+GSHtRB4W+VKYazssDobVxWkTeFTwJtU9be1rFhEksCdwJ8Ce4AnRWSDqj47ZrkTgE8AT9Sy/qC6N/a37I5hwpUUsbEtJjSuMqUGuVLYjp/molbnAjtUdaeqHgW+A1xeYrnP4WdiPTyO96iq2SfYMdFlPYhMmFxlSg1ypTAMbBWRxxidEK9al9QOoHjU2B7gvOIFRKQTmKmqD4nIp4MVuTY2wU40eEkhG2AilGaRTAjDI9Epr2ktyYQ4y5Qa5EqhB/g88DNea1PYEuB1paryC78iEUkAXwL+quqKRK4XkV4R6d27d2+At37NqiVzSHnJml5jGqvdSzClLVqD6y0g1E8yYa1+tTrhuEnOuqgGSYj3LRFpA96ce6hfVYO0NO4BiofdnQq8VHT/BOCtwI/FH7L9X4ANIrJUVUc1ZKvqXcBd4HdJDfDeBfkPbs2G7dbro0llh5VDWftu4soCbO0OODyWBRnR/G7gW8AL+Gf/M0Xkz1X1p1Ve+iRwpoicBgwA1wDvzz+pqgeAk4re58fAp8cGhHo5MhTJcXexkLWDgqnCssaO5nLmtSDX7HcAF6pqP4CIvBm4Dzin0otUdUhEPgZsBJLA11V1u4jcCvSq6oaJFT247o391kvEmIhK0JissVES9sxrXj4gAKjqf4pIoNFLqvoIfpqM4sduLrPsu4OsczysB5IxjSUCp0xN8dJgBhGYyMWghYPGCtLQ3Csi/ygi7879fY1gDc1Nw+WlljHmWKqwafV7eH7tpbz/vFlhF6fluJwQLEhQ+Ev8sQqfAP4n8CzwEWclcmDVkjk2qtmYBkqn/MqEnr4B7vt55flMTO1cdrMP0vvoCPA3ub9I6urs4AbLTGlMw/z+cJbOWx+1lCiOuJxkp2xQEJFtVEgdoqpnOymRI9XmVKgkMcE60ThqxfTFJrgRtRxZLoU1Hef7cv8/mvt/d+7/tYwv7UWoJvIhWkCoXaWAMK3d49KzT+aezbsaWCJjWkdHGF1SVfVFABFZrKqLi55aLSKbgFudlcqBDkt30TT+cHgo7CIY0xTGU4OR8pJOu6QGaWieIiLvyt8RkXcCU5yVyBGXH6KpTXZEudeuEoypOSAIcMU5HU5nYQsSFD4E3CkiL4jIC8CXgb9wViJHujo7mNJmOZCahdXImahw2ahbKwUeevplp+9RNSio6hZVnQ+cDcxX1QWq+pTTUtVZT98Ai9f+yBo+jTE1W3HeTKd1+LUazGSdzboGwXIfHQdcAcwGJuWS16GqkWhT6Okb4MYHt1maC2PMuDywZaDpjh/dG/vDy5IK/DNwAH8U85EqyzYdy3tkjJmITHbYabd0LyE1J4V0mbonSFA4VVUvclYCxyzvkTFmopx2Sxd/BPiBTJapKY/fH85WfT+XqXuCNDT/TETmOSuBY5b3yBjTzLLDyquHh7h20SyODI1UDQiew1nXIFhQeBewRUT6ReQZEdkmIs84K1GdrVoyB89mdjLGNLFh9btpB6rqdnw4CxIULgbOBC4ELsMf6XyZy0LVU1dnB8dPjtZUj8aY+AlaQ5UdVqdZUoMkxMuPbH49MNlZSRwatBwsBsthZVqHy7bSqlcKIrJURH4NPA/8BH9azn9xViIHrF3BgAUE0zrCbmj+HLAI+E9VPQ24ANjkrEQOrFoyh5Rno5mNMa3h/LfMcLbuIEEhq6q/AxIiklDVx4AFzkrkQFdnB7cvi2wHKmNioZnSSTS7B7YMOBvVHCQoDIrI8cBPgXtF5G+ByKW5dJlAqt6mtQeaAtuYliHAHVfPx0taYAgikx121tgcJChcjj9/wieBfwWeI0K9j/J6+gYiMyXnLZfNjUxZg1h8xnQLdKYiBT7z/W1kh63hJyhXjc1BEuIdVNURVR1S1W8BdwKRG+HcvbE/Upk533nG9LCLUDcv/C7DpWefHHYxTJOzhJW1DUFw1dhcaTrOE/FnXesANgA/zN1fBWwF7nVSIkeiNMHOqvufdjrdXqMNDGZsljVjAgj6q3c50U6lcQp3A/uBx4H/hh8M2oDLVXWrk9I4kq86isphttbkWMaYeHE50U6loHC6qs4DEJF/AH4LzFLVV52UxKGoVR01WpQCpjFR4uq39div9jpYq69Sm0JhGLCqDgPPRzEggGVKrcYCwvilvCSLW6j9p1HSqXh0PHD12wordfZ8Efl97rYAqdx9AVRVT3RWqjpLt3vst1QXps7SKY+jQ8Nsem5f2EWJnMFMvH6PSRFGVEmI1KW9sN3h1MJlrxRUNamqJ+b+TlDVSUW3IxMQevoGOBBiQLAEra3rQCbLoexI2MUwETCiyvNrL+WOq+fXZX0Hjw6HOngt0ro39hPmz9bajFuXfbUmqFPSKXr6Buo64MzV4LWWzylt7QnGmLDNfl2q7nPFhzZ4LeosQ6oxJmybd+6v+1zxUx011jsNCiJyUW7Gth0isrrE858SkWdzM7r9m4i8sd5liOPMa/HaWmOan4vBqK7yBzoLCiKSxE+JcTFwFrBCRM4as1gfsFBVzwa+B3yx3uXo6uyg+6r6NO40uzec0MYLay8NuxjGmAZwNXmYyyuFc4EdqrpTVY8C38FPrlegqo+p6qHc3c3Aqa4K0wrZF6ulFn7l1aP09A0EqjKL/qdhTLxFsfqoA9hddH9P7rFyPkSZGd1E5HoR6RWR3r17ax/J172xvyWyLy46fVrFyYIUf1uDTMAR/U/DmHiLXPURpU9GSx6LRGQlsBDoLvW8qt6lqgtVdeGMGbXPOFSplT4ZofaGTc/tQ9CKZ/kvDWZ4YMsd1MXcAAAQU0lEQVSehpXJNDebuKb+vCbpnhPF6qM9wMyi+6cCL41dSETeC3wGWKqqR1wUpFx1ypS2JMMRG0hwKDtS8Sx/asojYwOqDNCRTrVUtt1m4SVrP2y6CM2uela6DApPAmeKyGki0gZcg5+Cu0BEOoGv4geEV1wVpNQczV5SWjJ/+6GjkZsUzzgSpXTxlTTb/OrjGcXuIjS7Sp3tLCio6hDwMWAj8Evgu6q6XURuFZGlucW6geOB+0Vkq4hsKLO6Cel9cR9HhkYHgKGIXSEEdbQF2k6MyUuKcPuyeXTYeKNjhJE6e8JU9RHgkTGP3Vx0+70u3x/gpp5tJSd4satqY5rfotOn0dXZQe+L+2yipiIu24paPs3FfU/srr6QMaYpbXpuH2f9n5KdEluaCLxpxhR+/crBks+vOG9mycfroeWDgjW0GRNtccxEqwo7ygSElJfgtq55zt67STpXuWNd8owxUVTudDaTHXGWNhtiEBRcXmYZY0wYbnxwm82nMF63dc1j5aJZYRfDGGPqJpMddjafQssHBfADw7rlCyI1etkYYyqx+RQmqKuzgxXnNm9VkgUsY0wtojiiuek8/MzLYRehrKil22gmSRHWLV/AuuULwi6KqbPFZ0wPuwhNSYjgiOZmtN9RAikTHi8p3HH1fH/eDEd1rCY8m57bF3YRmtK1i2Y5G9Ecm6DgsguXCc9wUVoPm4/bxMXze//gbN0tP3gN/IBw44Pbwi6GcWAEuGH9Vro39pPyErEc6GTix+UVVCyuFLo39td90uxatcLMb81sYDBjAcFERtrRrGn1EIugEHa1QruXYPnbZ9oUmMY0keMmhXf4O3h0iJWLZjVdWnCISVBw1XUrqEPZEe59YpdNgWlME2lvS7Ju+YJQztqzw8pjv9rL7cvmjSsVj8teWbEICq66btWi0Xn5kiKc+fopjX1TE3lxyhWW74148Eg4E1MNDGa4Yf3WcSXtvPfD73BQIl8sgkIcDauWTbtrTDkjqkxrb9767npbs2E72YiNERLc9qZs+aDQ0zfAqvufDrsYZhxWLpplM241WHtbkj8cjs+UroOZ6I1dUnA6Jqflg0L3xv7InQkY3wNbBmhva/ldtKkcPDpsv5cIcNl5puV/ca0yeXkcZbLDVgVmTAlTHTaOt3xQiFPDmTFx0GZjfnB5WGv5oGDTcRrTWoZtjCKDDvO4tXxQaOaGyjj18jCmXoZVYz8Q1OXYq5YPCquWzGnKUYMrF81q+NgFY1pFnH86KS/pdOxVyyfEy6eXvWH91pBLMtoDW/aQsVw9JoISAtZBKTy3L5vnLG02xOBKAfzA0GxVNRYQTBR1pFPsvP1S1i1f0NRVs60qKeI0IEAMrhTAH8BmVTXGTNzAYIbZqx8mKRJaJ46EwHGTkqFnPg7DsCqL1/6IVUvm2CQ745WfSyGKIxeNaVZh9uobUbjinI7YdjcfGMxw44PbnKW6aPmg0AxzKRhj6mv9k7tZdPq02PZCymSHnaW6aPmgEPZcCsbEXUL8OUXqKTusbHpuX6x7Ibk6trV8UAh7LoWJiOnVsWkxI4rNiueAq2Ob06AgIheJSL+I7BCR1SWeP05E1ueef0JEZte7DM06TqEaofFzMBgzVlzr7aPA1VgFZ0FBRJLAncDFwFnAChE5a8xiHwL2q+qbgC8Bf13vcnR1dnD7snl0pFMI0dnJo3yFU05UPnvzGksT05wEItn76Fxgh6ruVNWjwHeAy8cscznwrdzt7wEXiNT/yNHV2cGm1e/h+bWXcsfV8+u9eidmv661gkLKS7LiPJun2ph6mFznNppiLoNCB7C76P6e3GMll1HVIeAA8DqHZWrKgWylbN653/l7NPLM/fZl87itax7XLppVMTC0QtAIc0J4Ew+HHbbRuNx7S/2+x16LBlkGEbleRHpFpHfv3r0TLtgtl81t+MHHSwheDSl/x3vZHnQS8vyZ+9j2FhefS0c6VbjUva1rHl9avqBkYE55Sa7NzbYmudeFMan6RExr9/jrK84u+3y7l6i5javdS9S071TjJSQSJ0aNlE5549r3kyIVR3anvCQrF83CS9T3lxXVhHh7gJlF908FXiq3jIhMAqYC+8auSFXvUtWFqrpwxowZEy5YV2cH1y6aNeH1BJUUofuq+XRfOb9wwKt2kj6es/iOdIo1S+dWPeh0pFOFM/fi9paOdIovjTN9wbR2j3XLFxzz3qWSd3V1dtB384WFVAn5986XKV/Vt2n1ewJtz3ilU15dD44pL8ktl82lq7ODlSX2Ly8pfGHZ2dy+bF6g902KsG75Ap793MWFfacSEaoGj450iu6r5tN384Vl1xdkzyu1jJeUwsE1nfJIjuNAmBpH0KwmnfL3zXXLF5Q8yUh5SdYsnVvyKjblJVm3fEHJ7xNgxXkzy3ZmmdbuFfbp7qvm13SCU+kk0nVCPFFHDUm5g/x/AhcAA8CTwPtVdXvRMh8F5qnqR0TkGmCZql5dab0LFy7U3t7eupSxp2+ANRu2j3u0c8pLMNlLMngoyynpFOe/ZQYPbBkYNVgu5SVLJrDKj7QuNbAu5SW54pyOY9Yl+JdRU9qSHDw6fMxr8u9TbrvKlWWscmWb1u5x6dkns/7J3WSHX9tvvKTQfeX8wnt3b+znpcEMp6RTdRmOX+17EuCdZ0xn6+4DJT+XUp9lpc8r/zmXWs9jv9rLwGCmkOYh/7+jxLZW+yzyz5eaHbDSd1Xq+8kvD/DZH2xn/5h8+17CPzEZ+/6l1pPfzpcGM6TbPf5weGjUFJ3F71Vt+4rLkk55vG/+yYXPcKx8GfPrLf6c0ymPg0eyFNealPodFCv1neTLVa7clZ67qWcb9z2xu/C9rzhvJrd1zav6urFKLVvqsyz1OZTbpiBEZIuqLqy6nKugkCvEJcA6IAl8XVU/LyK3Ar2qukFEJgN3A534VwjXqOrOSuusZ1AYa+zBYVq7xy2XzQUq7/xj11HrsuW+9PHuvOMpSy3b4eLAX0uZKv1IypWt1jI3ehvrXb6g63O9H9Vzm12vp9U1RVBwwWVQMMaYVhU0KFg3CWOMMQUWFIwxxhRYUDDGGFNgQcEYY0yBBQVjjDEFket9JCJ7gRfH+fKTgN/WsThRYNscD7bN8TCRbX6jqlYd/Ru5oDARItIbpEtWK7Ftjgfb5nhoxDZb9ZExxpgCCwrGGGMK4hYU7gq7ACGwbY4H2+Z4cL7NsWpTMMYYU1ncrhSMMcZU0JJBQUQuEpF+EdkhIqtLPH+ciKzPPf+EiMxufCnrK8A2f0pEnhWRZ0Tk30TkjWGUs56qbXPRcleKiIpI5HuqBNlmEbk6911vF5F/anQZ6y3Avj1LRB4Tkb7c/n1JGOWsFxH5uoi8IiK/KPO8iMjf5T6PZ0TkbXUtgKq21B9+mu7ngNOBNuBp4Kwxy/wP4Cu529cA68MudwO2+XygPXf7L+OwzbnlTgB+CmwGFoZd7gZ8z2cCfcC03P3Xh13uBmzzXcBf5m6fBbwQdrknuM1/DLwN+EWZ5y8B/gV/6o9FwBP1fP9WvFI4F9ihqjtV9SjwHeDyMctcDnwrd/t7wAUiDZywuP6qbrOqPqaqh3J3N+PPhBdlQb5ngM8BXwQON7JwjgTZ5g8Dd6rqfgBVfaXBZay3INuswIm521M5dobHSFHVn1JiBsoilwPfVt9mIC0iJ9fr/VsxKHQAu4vu78k9VnIZVR0CDgCva0jp3AiyzcU+hH+mEWVVt1lEOoGZqvpQIwvmUJDv+c3Am0Vkk4hsFpGLGlY6N4Js8xpgpYjsAR4BPt6YooWm1t97TSbVa0VNpNQZ/9guVkGWiZLA2yMiK4GFwJ84LZF7FbdZRBLAl4DrGlWgBgjyPU/Cr0J6N/7V4L+LyFtVddBx2VwJss0rgG+q6h0i8g7g7tw2j5R4bStwevxqxSuFPcDMovuncuzlZGGZ3FzSU6l8udbsgmwzIvJe4DPAUlU90qCyuVJtm08A3gr8WERewK973RDxxuag+/Y/q2pWVZ8H+vGDRFQF2eYPAd8FUNXHgcn4OYJaVaDf+3i1YlB4EjhTRE4TkTb8huQNY5bZAPx57vaVwI8014ITUVW3OVeV8lX8gBD1emaoss2qekBVT1LV2ao6G78dZamqRnku1yD7dg9+pwJE5CT86qSK8543uSDbvAu4AEBE/gg/KOxtaCkbawPwgVwvpEXAAVV9uV4rb7nqI1UdEpGPARvxey58XVW3i8itQK+qbgD+Ef8Scwf+FcI14ZV44gJuczdwPHB/rk19l6ouDa3QExRwm1tKwG3eCFwoIs8Cw8AqVf1deKWemIDb/FfA10Tkk/jVKNdF+SRPRO7Dr/47KddOcgvgAajqV/DbTS4BdgCHgA/W9f0j/NkZY4yps1asPjLGGDNOFhSMMcYUWFAwxhhTYEHBGGNMgQUFY4wxBRYUTFMTkWER2SoivxCRH4hIuk7rnV0uC2UzEpF3i0jd0nWIyEdE5AP1Wp9pHRYUTLPLqOoCVX0r/piSj4ZdoCgSkWTxfVX9iqp+O6zymOZlQcFEyePkEn+JyPG5eSGeEpFtInJ57vHZIvJLEflabj6BR0UklXvuHBF5WkQepyi4iMhkEflGbj19IpIfEXydiPTkrlCeF5GPiT8vRV8u2dz0sQUUkW/mct3/TER2isiVucdHnemLyP8Tketyt18QkS+IyOMi0isibxORjSLynIh8pGj1J4rI98WfK+ErufxOiMiFudc+JSL3i8jxReu9WUT+A7hqTDnXiMinJ/yNmJZjQcFEQu5M9wJeS3FwGPgzVX0bflqHO4rSn5+Jnz56LjAIXJF7/BvAJ1T1HWNW/1EAVZ2Hn1ztWyIyOffcW4H346dw/jxwSFU78QNUueqXk4F3Ae8D1gbcxN25cv078E389CuLgFuLljkXf/TuPOAMYFkulcVNwHtzn0Uv8Kmi1xxW1Xep6ncClsPEXMuluTAtJyUiW4HZwBbgh7nHBfiCiPwxMIJ/BfGG3HPPq+rW3O0twGwRmQqkVfUnucfvBi7O3X4X8PcAqvorEXkRP2cQwGOq+irwqogcAH6Qe3wbcHaZMvfkMnQ+KyJvKLPMWPlgtw04vug9Dxe1o/xcVXdCIRXCu/CD41nAplxMbMMPWHnrA76/MYAFBdP8Mqq6IHdQfwj/rP7vgGuBGcA5qpoVPxNq/uy+OAPsMJDCDyLlcrpUmmCpeF0jRfdHKP/7KX5Nft1DjL4yn8xoxesd+5759xlbfs2t/4equqJMWQ6WedyYkqz6yESCqh4APgF8WkQ8/HTnr+QCwvlAxTmnc/MJHBCRd+Ueurbo6Z/m74vIm4FZ+Cmn6+lF4Czx5wefSi6rZ43OzWULTQDLgf/Az/66WETeBCAi7bltMGZcLCiYyFDVPvw5eq8B7gUWikgv/gH9VwFW8UHgzlxDc6bo8S8DSRHZhl/dcl2955tQ1d34Of+fyZW9bxyreRy/jeIXwPPA91V1L/5EQveJyDP4QeIt9SiziSfLkmqMMabArhSMMcYUWFAwxhhTYEHBGGNMgQUFY4wxBRYUjDHGFFhQMMYYU2BBwRhjTIEFBWOMMQX/H01ykwnY3ZX1AAAAAElFTkSuQmCC\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(x[0:n_rands-1],x[1:n_rands],'o')\n", "plt.xlabel('Random number i')\n", "plt.ylabel('Random number i+1')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear congruential random number generator\n", "The linear congruential generator (LCG) illustrated above, with good choices of the parameters, is the simplest good generator in use today. There are many others, somewhat better ones, in use today, which produce sequences that have fewer correlation properties. However, while they may seem a lot more complicated, usually the core remains the same: we use properties of integer arithmetic, such as remainder of an integer division, to redistribute numbers pseudo-randomly in the segment $[0,1)$. The principal problem with the LCG, and the main reason why more complicated versions exist, is not the quality of the sequences that it generates at good choices of parameters. Rather the problem is that there's a strong dependence of the quality of the sequence on the choice of parameters, and there are very few proofs that guarantee the quality for any specific choice of parameters. \n", "\n", ">### Your work\n", "Experiment with the linear congruential random number generator by using different multipliers, moduli, and increments. What happens if the multiplier has common divisers witht he modulus? Do the numbers still look reasonably random?" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# simple linear congruential random number generator\n", "def lcrng(seed=10, multiplier=7**5, modulus=2**31-1,increment=0,n_rands=1):\n", " # seed,multiplier,modulus, and incrment are the corresponding parameters of the generation\n", " # the default values of parameters are one particularly good generator, similar in quality \n", " # to the built-in one.\n", " # n_rands -- number of random numbers to be generated\n", " rand_flt = np.zeros(n_rands) # allocate memory for our pseudo-random numbers\n", " rand_flt[0] = seed # seed\n", " \n", " for i in np.arange(1, n_rands): # generate the pseudo-random integres\n", " rand_flt[i] = np.mod(rand_flt[i-1] * multiplier + increment, modulus)\n", " \n", " rand_flt = rand_flt / modulus # transform the pseudo-random integers into pseudo-random standard uniform numbers\n", " return rand_flt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Having defined the generator, we can now check that its default version produces correct spread of numbers over the range $[0,1)$ and shows no correlation between the consecutive samples." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n_rands = 1000 # the number of random numbers to generate\n", "nbins = 50 # number of bins to histogram the random numbers\n", "\n", "x = lcrng(n_rands=n_rands)\n", "plt.hist(x, nbins) # show the resulting histogram of random numbers\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.show()\n", "\n", "plt.plot(x[0:n_rands-1],x[1:n_rands],'o')\n", "plt.xlabel('Random number i')\n", "plt.ylabel('Random number i+1')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can explore how choosing parameters of the linear congruential generator affects its performance. Make sure to make your own choices, but here's one that is not too good and has a period much shorter than the modulus. Notice the patterns of correlation among consecutive random numbers." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n_rands = 1000 # number of random numbers to be generated\n", "nbins = 50 # number of bins to histogram the random numbers\n", "\n", "x = lcrng(seed = 10, multiplier = 77, modulus = 8100, increment = 2, n_rands=n_rands)\n", "plt.hist(x, nbins) # show the resulting histogram of random numbers\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.title('Histogram of linear congruential pseudo-random numbers')\n", "plt.show()\n", "\n", "plt.plot(x[0:n_rands-1],x[1:n_rands],'o')\n", "plt.xlabel('Random number i')\n", "plt.ylabel('Random number i+1')\n", "plt.title('Correlations between consecutive pseudo-random numbers')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Examples of using the standard uniform pseudo-random number to generate other types of pseudo-random numbers\n", "Here we will study how to generate different types of random pseudo-numbers using just a standard, uniformly distributed pseudo-random number. From this point on, I will be omitting the prefix *psuedo* everywhere, calling the numbers we generate just *random*. However, you should keep in mind that the only such numbers we can generate on a deterministic digital computer are pseudo-random, and this is always implied.\n", "\n", ">### Your turn \n", "Make sure to run each cell below multiple times to convince yourself that the generated numbers indeed change." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Uniform random numbers\n", "The standard linear congruential generator allows us to generate a uniform pseudo-random floating point number between 0 and 1, called the ''standard'' uniform random number. Variables $x$ taken from this generator are denoted as $x\\sim{\\cal U}(1)$. Then how do we generate a number between, say, $a$ and $b$, to be denoted as $y\\sim {\\cal U}(a,b)$? For this, we only need to scale and shift the standard number. That is, if we say that $y=a + (b-a)x$, where $x\\sim {\\cal U}(1)$, then we get $y\\sim{\\cal U}(a,b)$. The probability density for this number is $P(y) = 1/|b-a|$ when $y\\in [a,b)$ and zero otherwise. The code in the next cell illustrates this." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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4j4jYEVgGTBvg9s3MbAhUaRHcJekqSYdJUtUNR+G5NLpWegVwEHB1mj4TOKo/AZuZ2dCqkgh2AmYA7wMWSvp3STtV2bikMZLuBZYCc4DfAssjYnVa5Elg6xbrniSpS1JXd3d3ld2ZmdkA9JkI0if7ORFxPPAhYCpwt6RbJL25j3Vfjog9gEnAPsBfN1usxbozImJKREyZMGFCnwUxM7OB6fOuIUmbAidQtAieAj4BzKLo978K2L6vbUTEcklzgf2AjSSNTa2CScDiAUdvZmaDVqVr6E5gQ+CoiDg8In4YEasjogu4oNVKkiZI2igNrwscAiwAbgaOSYtNBa4dTAHMzGxwqnyPYOeIaNV9c24v600EZkoaQ5FwroyI2ZJ+DVwu6SzgV8B3+hu0mZkNnSqJ4AZJ74mI5QCSNgYuj4i/722liLgf2LPJ9Ecprhe0xVB/YcRfQDGz0aZK19CEniQAEBHLgM3rC8nMzNqpSiJ4WdK2PSOStqPFnT5mZjbyVOkaOgO4TdItafxA4KT6QjIzs3bqMxFExPWS9qK49VPApyPi6dojMzMbIF/L65+qTx9dB3g2Lb+LJCLi1vrCMjOzdqnyhbJzgWOB+cAraXIATgRmZqNAlRbBURTfJXip7mDMzKz9qiSCRymeHOpEYG3nvl6z+lVJBC8A90q6kVIyiIhP1haVmZm1TZVEMCu9zMxsFKpy++jM9NC4bSPi4TbEZGZmbVTlpyrfCdwLXJ/G95DkFoKZ2ShRpWtoOsVD4uYCRMS9kvr8DQLrHF9gbc3Hxuy1qjxraHVErGiY5mcNmZmNElVaBA9Kei8wRtKOwCeBO+oNy8zM2qVKi+ATwK4Ut45eBqwETq4zKDMza58qdw29QPEE0jPqD8fMzNqtyrOGbqbJNYGIOKiWiMzMrK2qXCP4bGl4HHA0sLqecMzMrN2qdA3Na5h0e+lHaszMbISr0jW0SWn0dcDewJa1RWRmZm1VpWtoHsU1AlF0CT0GTKszKDMza58qXUP+FrGZ2ShWpWvo3b3Nj4gfDl04ZmbWblW6hqYBbwFuSuNvo3ju0AqKLiMnAjOzEaxKIghgl4hYAiBpIvDNiPhArZGZjSJ+2J0NZ1UeMTG5JwkkTwE71RSPmZm1WZUWwVxJP6N4zlAAxwE31xqVmZm1TZW7hj4u6R+AA9OkGRHxo3rDMjOzdqnSIgC4B1gVET+XtJ6k8RGxqs7AzMysPar8VOWHgauB/06TtgauqTMoMzNrnyoXi/8F2J/idwiIiEeAzesMyszM2qdKIngpIv7UMyJpLP6pSjOzUaNKIrhF0unAupLeDlwFXFdvWGZm1i5VEsFpQDfwAPDPwE+Af+trJUnbSLpZ0gJJ8yV9Kk3fRNIcSY+kvxsPpgBmZjY4vd41JGkMMDMiTgAu7Oe2VwOnRMQ9ksYD8yTNAU4EboyIcySdRpFoTu1/6GZmNhR6bRFExMvABElr93fDEbEkIu5Jw6uABRR3HB0JzEyLzQSO6u+2zcxs6FT5HsEiil8lmwU83zMxIr5WdSeSJgN7AncBW/Q8siIilkjyHUhmZh3UskUg6btp8Fhgdlp2fOlViaQNgB8AJ0fEyn6sd5KkLkld3d3dVVczM7N+6q1FsLek7YDHgf8cyMYlrUWRBL5f+t2CpyRNTK2BicDSZutGxAxgBsCUKVN8u6qZWU16SwQXANcD2wNdpemi+B7BDr1tWJKA7wALGrqRZgFTgXPS32v7H7aZmQ2VlokgIr4BfEPStyLiowPY9v7A+4AHJN2bpp1OkQCulDSNorXxngFs28zMhkiVp48OJAkQEbdRtB6aOXgg2zQzs6FX5QtlZmY2ijkRmJllzonAzCxzTgRmZplzIjAzy5wTgZlZ5pwIzMwy50RgZpY5JwIzs8w5EZiZZc6JwMwsc04EZmaZcyIwM8ucE4GZWeacCMzMMudEYGaWOScCM7PMORGYmWXOicDMLHNOBGZmmXMiMDPLnBOBmVnmnAjMzDLnRGBmljknAjOzzDkRmJllzonAzCxzTgRmZplzIjAzy5wTgZlZ5pwIzMwy50RgZpY5JwIzs8w5EZiZZa62RCDpIklLJT1YmraJpDmSHkl/N65r/2ZmVk2dLYKLgUMbpp0G3BgROwI3pnEzM+ug2hJBRNwKPNsw+UhgZhqeCRxV1/7NzKyadl8j2CIilgCkv5u3ef9mZtZg2F4slnSSpC5JXd3d3Z0Ox8xs1Gp3InhK0kSA9HdpqwUjYkZETImIKRMmTGhbgGZmuWl3IpgFTE3DU4Fr27x/MzNrUOfto5cBdwI7S3pS0jTgHODtkh4B3p7Gzcysg8bWteGIOL7FrIPr2qeZmfXfsL1YbGZm7eFEYGaWOScCM7PMORGYmWXOicDMLHNOBGZmmXMiMDPLnBOBmVnmnAjMzDLnRGBmljknAjOzzDkRmJllzonAzCxzTgRmZplzIjAzy5wTgZlZ5pwIzMwy50RgZpY5JwIzs8w5EZiZZc6JwMwsc04EZmaZcyIwM8ucE4GZWeacCMzMMudEYGaWOScCM7PMORGYmWXOicDMLHNOBGZmmXMiMDPLnBOBmVnmnAjMzDLnRGBmljknAjOzzHUkEUg6VNLDkhZKOq0TMZiZWaHtiUDSGOCbwP8GdgGOl7RLu+MwM7NCJ1oE+wALI+LRiPgTcDlwZAfiMDMzYGwH9rk18ERp/Elg38aFJJ0EnJRGn5P08AD3txnw9ADXHW5cluFntJQDXJZhR+cCgyvLdlUW6kQiUJNp8ZoJETOAGYPemdQVEVMGu53hwGUZfkZLOcBlGa7aUZZOdA09CWxTGp8ELO5AHGZmRmcSwS+BHSVtL2lt4DhgVgfiMDMzOtA1FBGrJX0c+BkwBrgoIubXuMtBdy8NIy7L8DNaygEuy3BVe1kU8ZrueTMzy4i/WWxmljknAjOzzI2KRCDpIklLJT3YYr4kfSM90uJ+SXu1O8aqKpTlrZJWSLo3vT7f7hirkLSNpJslLZA0X9KnmiwzIuqlYllGSr2Mk3S3pPtSWc5sssw6kq5I9XKXpMntj7RvFctyoqTuUr18qBOxViFpjKRfSZrdZF69dRIRI/4FHAjsBTzYYv5hwE8pvsOwH3BXp2MeRFneCszudJwVyjER2CsNjwd+A+wyEuulYllGSr0I2CANrwXcBezXsMzHgAvS8HHAFZ2OexBlORE4v9OxVizPZ4BLm51HddfJqGgRRMStwLO9LHIkcEkUfgFsJGlie6LrnwplGREiYklE3JOGVwELKL5VXjYi6qViWUaEdKyfS6NrpVfjHSNHAjPT8NXAwZKafRG0oyqWZUSQNAk4HPh2i0VqrZNRkQgqaPZYixH5Rk7enJrDP5W0a6eD6Utqxu5J8YmtbMTVSy9lgRFSL6kL4l5gKTAnIlrWS0SsBlYAm7Y3ymoqlAXg6NT1eLWkbZrMHw7OA/4VeKXF/FrrJJdEUOmxFiPEPcB2EbE78J/ANR2Op1eSNgB+AJwcESsbZzdZZdjWSx9lGTH1EhEvR8QeFN/q30fS3zQsMmLqpUJZrgMmR8RuwM959VP1sCHpCGBpRMzrbbEm04asTnJJBKPmsRYRsbKnORwRPwHWkrRZh8NqStJaFP84vx8RP2yyyIipl77KMpLqpUdELAfmAoc2zPpLvUgaC7yeYd5d2aosEfFMRLyURi8E9m5zaFXsD7xL0iKKpzEfJOl7DcvUWie5JIJZwPvTXSr7ASsiYkmngxoISVv29A1K2oeiDp/pbFSvlWL8DrAgIr7WYrERUS9VyjKC6mWCpI3S8LrAIcBDDYvNAqam4WOAmyJdpRxOqpSl4ZrTuyiu7wwrEfG5iJgUEZMpLgTfFBEnNCxWa5104umjQ07SZRR3bWwm6UngCxQXjoiIC4CfUNyhshB4AfhAZyLtW4WyHAN8VNJq4EXguOH4JqX4lPM+4IHUhwtwOrAtjLh6qVKWkVIvE4GZKn4g6nXAlRExW9IXga6ImEWR9L4raSHFp87jOhdur6qU5ZOS3gWspijLiR2Ltp/aWSd+xISZWeZy6RoyM7MWnAjMzDLnRGBmljknAjOzzDkRmJllzonAbAhJmitpVPxouuXDicBsmEjfGDVrOycCy5Kkyen3BS5Mz7K/QdK65U/0kjZLX/vvea79NZKuk/SYpI9L+kx6fvwvJG1S2vwJku6Q9GD6ljGS1lfxWxO/TOscWdruVZKuA25o82EwA5wILG87At+MiF2B5cDRfSz/N8B7gX2As4EXImJP4E7g/aXl1o+It1A8Q/6iNO0MiscCvAl4G/B/Ja2f5r0ZmBoRBw1Bmcz6zU1Ry9ljEdHzyIh5wOQ+lr85/R7BKkkrKJ5sCfAAsFtpucug+G0JSRum5+G8g+LBYp9Ny4wjPaKC4vHJw/qhbja6ORFYzl4qDb8MrEvxTJqelvK4XpZ/pTT+Cmu+lxqf2xIUjxE+OiIeLs+QtC/wfL8jNxtC7hoyW9MiXn1U8TED3MaxAJIOoHii6grgZ8AnSk8o3XOQcZoNGScCszV9leIponcAA/09gWVp/QuAaWnalyieInu/pAfTuNmw4KePmpllzi0CM7PMORGYmWXOicDMLHNOBGZmmXMiMDPLnBOBmVnmnAjMzDL3/wFqfZvlbISVggAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# generating numbers U(a,b)\n", "n_rands = 1000 # number of random numbers to generate\n", "nbins = 30 # number of bins in the histogram to show\n", "a = 1 # left edge of the interval\n", "b = 4 # right edge of the interval\n", "randAB = a+(b-a)*nprnd.random(n_rands)\n", "\n", "plt.hist(randAB, nbins)\n", "plt.title('Histogram of non-standard uniform random numbers.')\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Multinomial random number\n", "A dice is an example of a multinomial random number with six equally probable outcomes. In general, a multinomial random variable is a random variable that can have a finite, discrete set of values, with a given probability of each of these values. A binomial random number is a special case of a multinominal random number with only two possible outcomes, such as a coin flip.\n", "\n", "We generate multinominal random numbers by sub-dividing the interval between zero and one into subintervals. The number of the sub-intervals is equal to the number of possible outcomes of the multinomial variable, and the length of each sub-interval is equal to the probability of each outcome. We then generate a uniform floating point number between zero and one and check which of the sub-intervals it fell into. The index of the subinterval is then the generated multinomial number. For example, the cell below produces a multinomial random number with 4 possible outcomes, each with the same probability." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# generating integer randoms 1...4\n", "n_rands = 1000 # number of random numbers to generate\n", "rand4 = np.floor(nprnd.random(n_rands)*4) # we first multiply a standard uniform number by 4, so that it spans \n", " # the range [0,4). Flooring the result produces a number [0,1,2,3] with equal probabilities\n", "plt.hist(rand4)\n", "plt.title('Histogram of multinomial random numbers\\n with 4 equiprobable outcomes')\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As another example, the cell below simulates a 6-sides dice, but the probabilities of coming up with each of the sides are $(0.2, 0.3, 0.1, 0.15, 0.1, 0.15)$ instead of the uniform $1/6$. Make sure you carefully study the code -- especially how we verify, which of the subintervals the generated numbers fall into." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# generating integer 1...6 with nonuniform probabilities \n", "n_rands = 1000 # number of random numbers to generate\n", "p1 = .2 # the following lines specify probabilities for each of the 6 outcomes\n", "p2 = .3\n", "p3 = .1\n", "p4 = .15\n", "p5 = .1\n", "p6 = 1 - p1 - p2 - p3 - p4 - p5 # remember that probabilities must sum up to 1, so the probability of the \n", " # last outcome is defined by the probabilities of the five other ones.\n", "\n", "rand6 = nprnd.random(n_rands) # generating n_rands standard uniform numbers \n", "\n", "# Now we could have iterated over all generated numbers and checked which of the subintervals of [0,1) they \n", "# fall into. However, recall that vectorized calculations are always much faster, so we vectorize these checks.\n", "rand6 = (rand6 < p1)*1 + ((rand6 >= p1) & (rand6 < p1+p2))*2 + ((rand6 >= p1+p2) & (rand6 < p1+p2+p3))*3 \\\n", " + ((rand6 >= p1+p2+p3) & (rand6 < p1+p2+p3+p4))*4 \\\n", " + ((rand6 >= p1+p2+p3+p4) & (rand6 < p1+p2+p3+p4+p5))*5 \\\n", " + (rand6>1-p6)*6\n", "\n", "plt.hist(rand6)\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.title('Histogram of multinomial random numbers\\n with 6 outcomes with different probabilities')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">### Your turn\n", "Look at the histogram above. The bins are not positioned uniformly, and are not centered about the integers $1,\\dots,6$. Figure out how to make the figure look nicer." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exponential random numbers \n", "Real-valued variables that have an exponential distribution are also very common. For example, the time between two subsequent nuclear decays, or the time between two nearby spikes in the simplest models of neuronal dynamics is exponentially distributed. Why is that? Nuclear decay has a finite probability $r \\Delta t$ of happening in any interval of time $\\Delta t$. For the next event to happen at time $T$, it must not happen in all of the $T/\\Delta t$ previous windows of the width $\\Delta t$, and then it should happen in the window $[T,T+\\Delta t)$. Combining these, we get the probability of the next event in the interval $[T,T+\\Delta t)$ being equal to $P_{\\rm next event}(T:T+dt)=(1-r \\Delta t)^{T/\\Delta t}r \\Delta t\\to e^{-r T}r \\Delta t$, so that the probability *density* is $P_{\\rm next event}(T)=r e^{-\\rho T}$ (here we used the, so called, *first remarkable limit*; make sure you can do this calculation). More generally, a random variable $x$ is called exponentially distributed with the *mean* $x_0$ (or the *rate* $r=1/x_0$) if the probability density function for this variable obeys $P(x)=\\frac{1}{x_0} e^{-x/x_0}$. We denote such variables as $x\\sim {\\cal E}(x_0)$. The distribution with $x_0=1/r=1$ is called the *standard* exponential distribution, denoted as $x\\sim{\\cal E}(1)$. \n", "\n", "Numerical generation of exponential random numbers illustrates a few important properties of probability distributions. First of all, if we have an access to a standard exponential random number, then by just multiplying it by $x_0$ we will get the exponentially distributed number with a mean of $x_0$. That is, ${\\cal E}(x_0)=x_0{\\cal E}(1)$. This means that we only need to figure out how to generate standard exponential numbers, and the rest can be obtained by a simple multiplication. \n", "\n", "**Figure is needed here**\n", "In its turn, for generation of standard exponential numbers, let's go back to our definition of the probability densitiy of a continuous random variable as a limit of an ever-finer discretization of probabilities of observing the variable in segments of the real axis. Suppose we have a real valued variable $x$, which is distributed according to $P(x)$. This means that the probability of having a sample land in the interval $[x,x+\\Delta x)$ tends to $P(x)\\Delta x$. Suppose now we have defined a new variable $y=y(x)$. We can invert the functional dependence and write $x=x(y)$ (note that we are now assuming that such inverse function exists!) Of course, $y$ is also a random variable. How is it distributed? We note that the probability of a sample to land in our interval should not depend on whether we marked its end points as $[x,x+\\Delta x)$, or as $[y(x),y(x+\\Delta x))\\approx [y(x),y(x) + dy/dx \\Delta x)$. In other words, $P(x)|\\Delta x|=P(y)|\\Delta y|$, where we added the absolute value signs to ensure that, even if an interval is negative (is read from right to left), the probability is still positive. This gives, in the limit of small $\\Delta x$, $P(y) = P(x(y))\\left|\\frac{dx}{dy}\\right|$.\n", "\n", "How does this help with generating exponential numbers? Suppose that $x$ is a standard uniform number, $x\\sim {\\cal U}(1)$, so that $P(x)=1$. Let's choose $y=-\\log x$, resulting in $x= e^{-y}$. Then $P(y)=P(x(y))\\left|\\frac{d e^{-y}}{dy}\\right|=e^{-y}$. In other words, the variable $y$ is a random standard exponential variable, $y\\sim {\\cal E}(1)! So generating samples of such random variables is easy -- just generate standard uniform random numbers and take the negative log of them. \n", "\n", "Below we implement such generation of standard exponential random numbers and compare with the built-in function for their generation, `nprnd.exponential()`. For this, first, we generate exponential random numbers using the built-in function. Second, we compare the built-in functio to our own generator. We also do the comparison on the semi-log plot to illustrate that the numbers in both cases are, indeed, exponentially distributed. The built-in and log-uniform are indistinguishable. Maybe this should not be surprising since, under the hood, the built-in function generates the random numbers in exactly the same way as our own! " ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n_rands=int(1e5)\n", "# First, we generate exponential random numbers using the built-in function.\n", "plt.hist(nprnd.exponential(size=n_rands), 30)\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.title('Histogram of built-in standard exponential numbers')\n", "plt.show()\n", "\n", "# Second, we compare the built-in function to our own generator. We also do the comparison on the semi-log plot\n", "# to illustrate that the bumbers are, indeed, exponentially distributed. \n", "plt.hist([nprnd.exponential(size=n_rands), -np.log(nprnd.random(n_rands))], 30,log=True)\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.title('Comparing histograms of built-in\\n and our own standard exponential numbers')\n", "plt.legend(('built-in generator', '-log U(1)'))\n", "plt.show()\n", "\n", "plt.hist(-5.0*np.log(nprnd.random(n_rands)), 30,log=True)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A uniform random number in a circle\n", "Let's suppose we need a uniform random number in a circle around $(0,0)$ with a radius of 1. This is not a trivial problem -- and, in fact, some of consulting and data science companies use this during interviews. We will explore multiple different ways of generating such numbers. First read the methods and ask yourself: Which of these will work? Which of these will be the fastest? Then we will look at implementations and see if your guesses were correct.\n", "\n", "**First plausible method**: We will generate a random $x$ coordinate and a random $y$ coordinate. We will combine the two to fine the angle defined by them. We will then generate a uniform radius between 0 and 1, and we will put the point at this radius and at the chosen angle. \n", "\n", "**Second plausible method**: We generate the random radius and the angle directly, and generate $x$ and $y$ pairs directly from the angle and the radius.\n", "\n", "**Third plausible method**: This is similar to the second, except that the random variable we generate is $r^2$, not $r$, and we then calculate the distance to the point by taking a square root of the pseudo-random generated number.\n", "\n", "**Fourth plausible method**: We generate a random $x$ coordinate between $(-1,1)$, and the same range for the $y$ coordinate. We then reject all of the points that fell outside the unit circle. In this case, we do not know how many pairs of points to generate precisely, because we do not know how many will be rejected. So we make an estimate: the area of a square is 4; the area of the unit circle is $\\pi$. So if i want to generate $\\mbox{n_nands}$ points, I should generate instead about $4 \\mbox{n_rands}/pi$. I then check how many actually got generated, and if not enough were, then I request the missing number to be generated additionally. This is *not* the fastest way to implement the rejection method (we will discuss this a bit later), but probably the cutest: the function we write for generating numbers on a circle calls itself *recursively* if it needs to generate the missing numbers.\n", "\n", "Once you have thought about these method, run the cell below. " ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Generated 1004 random numbers.\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n_rands = int(1e3)\n", "x = 2*nprnd.random(n_rands) - 1 # generate random x between -1 and 1\n", "y = 2*nprnd.random(n_rands) - 1 # generate random y between -1 and 1\n", "distance = np.sqrt(x**2+y**2) # calculate the distance the points are away from (0,0)\n", "r = nprnd.random(n_rands) # generating random distance from the center\n", "x = x/distance*r # These two lines keep the angle defined by (x,y) and move the \n", "y = y/distance*r # point to the radius defined by the variable r.\n", "plt.plot(x, y, 'o')\n", "plt.title('Putative random numbers on a circle: Method 1')\n", "plt.show()\n", "\n", "\n", "r = nprnd.random(n_rands) # generate random distance from the center between 0 and 1\n", "phi = 2*np.pi*nprnd.random(n_rands) # generate random angle between 0 and 2*pi\n", "x = r*np.cos(phi) # calculate x\n", "y = r*np.sin(phi) # calculate y\n", "plt.plot(x, y, 'o')\n", "plt.title('Putative random numbers on a circle: Method 2')\n", "plt.show()\n", "\n", "r2 = nprnd.random(n_rands) # generate random distance to the center, squared\n", "phi = 2*np.pi*nprnd.random(n_rands) # generate random angle\n", "r = np.sqrt(r2) # generate r from r^2\n", "x = r*np.cos(phi) # x is the cosine projection of (r,phi)\n", "y = r*np.sin(phi) # y is the sine projection of (r,phi)\n", "plt.plot(x, y, 'o')\n", "plt.title('Putative random numbers on a circle: Method 3')\n", "plt.show()\n", "\n", "# Because generating numbers using Method 4 is a bit complicated, we first \n", "# define a function for this, and will then call the function later for the\n", "# actual generation.\n", "def rnd_circle_rejection(n_rands):\n", " N = int(np.ceil(4/np.pi*n_rands)) # need to generate more points because of possible rejections.\n", " # Note that np.ceil makes its argument a double precision number, which just happens to be\n", " # an integer. We need to actually make it an integer type, hence int()\n", " x = 2*nprnd.random(N) - 1 # generate random x\n", " y = 2*nprnd.random(N) - 1 # generate random y\n", " ind = (x**2+y**2 < 1) # which points fall in the circle?\n", " x = x[ind] # choose x of points inside the circle\n", " y = y[ind] # choose y of points inside the circle\n", " print('Generated ', np.size(x), ' random numbers.',sep='') # we do not need this, and you should feel \n", " # free to remove this line.\n", " # I put it here so that you can see how additional data points get generated if we rejected\n", " # too many numbers initially\n", " if (np.size(x)n_rands): # if too many points were generated originally,\n", " x = x[0:n_rands] # then truncate the arrays to the needed size\n", " y = y[0:n_rands]\n", " \n", " return x, y # return the x and y coordinated\n", "\n", "x,y = rnd_circle_rejection(n_rands) # now let's actually generate the numbers \n", "plt.plot(x, y, 'o')\n", "plt.title('Putative random numbers on a circle: Method 4')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Which one of the generators is correct? \n", "\n", "The first method generates a cross of higher density at $\\pm 45^{\\rm o}$ angle. This is easy to understand: the rays that extend in these directions are longer than in all others, and, in particular, $\\sqrt{2}$ longer than in the horizontal and the vertical directions. Thus we generate more points along these diagonal rays, which, in turn, transforms into a higher density once we pull these numbers first onto the rim of the circle, and then spread them over the whole area of the circle. \n", "\n", "Hopefully you also see that the first two of the tried methods generate too many points at the center of the circle compared to the edges. Those of you who know multivariate calculus should be able to extend the calculation we did for generation of exponential random numbers to the multivariate case and relate the higher density of points near the center of the circle to the Jacobian of the transformation we make when we generate such two-dimensional random numbers from uniform standard numbers by transforming from the $(x,y)$ coordinates to $(r,\\phi)$ ones. One can also reason this without the more advanced math. By generating a uniform random $r$, we are putting the same number of points at $0\\le r<0.5$ as at $0.5\\le r<1$. But the area inside the circle of the radius $0.5$ is $\\pi/4$, while the area of the annulus of the width of $0.5$ and the external radius of $1$ is $3\\pi/4$. In other words, the density of points generated using a uniform random $r$ is 4 times as high in the inner circle as it is in the annulus, and this is what you see in the figures. \n", "\n", "> ### Your work\n", "> Time how long it takes each of the generators to generate $10^6$ points in a circle. Which one is faster? (Focus only on *Methods 3, 4*, because the first two are incorrect. In general, performing complex arithmetic operations (trigonometric functions and roots) takes time, slowing down *Method 3*. On the other hand, memory management in *Method 4*, and specifically recursive calls to the function, as well as creating arrays and then stacking them to produce the final output, is also costly. Which of the methods won in your case? Now write an even faster method, which simply generates a bit too many points to start with, so that it almost never has to do the recursive calls. Suprisingly, even though this is wasteful (we generate and then throw away potentially many numbers), as long as it is not too wasteful, such method will be the fastest. *Note: it only takes a few small changes to the function `rnd_circle_rejection()` to implement this.*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Generating a standard normal number \n", "A Gaussian (or normal) random variable is a variable with the probability density function $P(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp \\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]$. $\\mu" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# generating Gaussian random numbers using built-in function\n", "n_rands = int(1e5) # how many numbers to generate?\n", "nbins = 30 # how many bins to histrogram into?\n", "plt.hist(nprnd.randn(n_rands), nbins) # make sure to explore help(nprnd.randn)\n", "plt.title('Built-in normal random numbers generation')\n", "plt.show()\n", "\n", "# and comparing built-in to our own generator\n", "phi = 2 * np.pi * nprnd.random(n_rands) # generating a random angle\n", "r = np.sqrt(2 * nprnd.exponential(size=n_rands)) # generating a random exponentially distributed r^2, \n", " # multiplying by 2, and taking a square root to get the correctly distributed r\n", "rand_norm = r * np.cos(phi) # the x projection of this (x,y) pair is standard-normal distributed, and we \n", " # discard the y projection for now\n", "plt.hist([nprnd.randn(n_rands), rand_norm], nbins)\n", "plt.title('Comparing built-in in our own normal random numbers')\n", "plt.legend(('built-in normal\\n random numbers', 'our normal\\n random numbers'))\n", "plt.show()\n", "\n", "# Now we show that generating just a single x instead of the (x,y) pair doesn't give us the right distribution\n", "x = np.sqrt(2 * nprnd.exponential(size=n_rands)) # we do the same as above, but generate x as a square root of x^2,\n", " # rather than r as a square root of r^2\n", "si = np.round(nprnd.rand(n_rands))*2 - 1 # this generates a random sign for x -- some will be +1 and some will be -1\n", "x = x*si # this is finally our putative standard normal\n", "plt.hist(x, nbins)\n", "plt.title('Incorrectly generated normal random numbers')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Using random numbers for Monte-Carlo (MC) calculations\n", "We use random numbers to perform computations that are fundamentally random, and we will focus on those in the upcoming weeks. However, as we discussed a few lectures ago, there are computations that are fundamentally deterministic, but may be easier to perform with random numbers. One of these is calculation of areas under the curve (definite integrals), known as Monte Carlo integration.\n", "\n", "Suppose we want to calculate the areas under a curve $y=y(x)$, with the limits $x\\in [a,b]$. We can, of course, do this analytically for some functions $y$ or numerically by subdividing the range of $x$ into small segments $\\Delta x$ and adding up the areas of all small rectangles, $y(x)\\times \\Delta x$. An alternative (and dare I say, more fun?) way is as follows. Suppose we know that $y(x)$ is positive and smaller than $f_{\\rm max}$. Then the area under the curve can be written down as $(b-a)f_{\\rm max}\\alpha$, where $\\alpha$ is the fraction of the rectange $b-a$ by $f_{\\rm max}$ that is actually under the curve. How do we estimate this fraction? Suppose I generate random points (dart throws), uniformly distributed over the rectangle $b-a$ by $f_{\\rm max}$. The fraction of those that end up under the curve (and it's easy to check which ones do!) is an estimate of $\\alpha$. The next cell performs such calculation for an arbitrary $y=y(x)$. Note that our implementation is incomplete: we don't check if the function is ever below 0 or is ever above $f_{\\rm max}$, which a better implementation should do. We will address this later.\n", "\n", "The implementation below first defines the function that will be integrated. You can plug different expression there, to get integrals of different functions. Then we define the MC integrator itself. Finally we integrate the function using the integrator." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Estimate of the area is 0.33513000000000004.\n" ] } ], "source": [ "# a function to integrate using the Monte Carlo approach\n", "def to_integr(x):\n", " return x**2\n", " #return np.exp(np.sin(x))\n", "\n", "# Monte Carlo integrator\n", "def mc_integr(func, a, b, fmax, N):\n", " # func -- function to integrate\n", " # a,b -- the x range of the integration\n", " # fmax -- the maximum value of the integrand\n", " # N -- number of random samples used for the integration\n", " # return -- a real number, estimate of the area\n", " N = int(N) # just making sure that the number of dart throws is integer\n", " x = a+(b-a)*nprnd.random(N) # generating random x coordinated\n", " y = fmax*nprnd.random(N) # generating random y coordinate\n", " counts_under_curve = np.sum(y <= func(x)) # how many dots under the curve?\n", " frac_under_curve = counts_under_curve/N # what is the fraction of dots under the curve?\n", " area = (b-a)*fmax*frac_under_curve # estimate of the area\n", " return area\n", "\n", "# calculating area under the function to_integr using MC integrator\n", "area = mc_integr(to_integr, 0, 1, 3, 100000)\n", "print('Estimate of the area is ', area, '.', sep='')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Above we integrated the function $x^2$ between 0 and 1. The integral should be $1/3$ -- and hopefully after running the cell above, you got a value very close to $1/3$. \n", "\n", ">### Your work\n", "Harden the function `mc_integr()` to be able to handle various user errors. First, expand the function to allow areas under the curve to be negative (this will require you to also have an `fmin` argument, in addition to `fmax`). Second, check if any values of `to_integr(x)` generated in the process of integration fall outside of the range $[f_{\\rm min}, f_{\\rm max}]$. If they do, then expand the range appropriately. Return not just the estimate of the area, but the estimates of $f_{\\rm min}$ and $f_{\\rm max}$ as well.\n", "\n", "## What is the error of MC methods?\n", "The result of the area calculation above came out to be slightly different from $1/3$. Herewithin lies the problem of MC approaches: they are probabilistic, and every time we perform them, the results are different. But how different? We can verify, first of all, that our probabilistic estimate of the area converges to to true one as the number of random samples increases, $N\\to\\infty$ (at least when the true value of the area is known, e.g., when we integrate a function $x^2$). In fact, one can can convince onself that this should be the case rather easily if the numbers are truly uniformley distributed in the rectangle around the curve. \n", "\n", "Can we additionally understand how quickly the convergence happens? For this, we can explore the distribution of the estimates of the area when we perform the calculation with a varying number $N$ of dart throws. That is, we can generate many different estimates with different realizations of $N$ dart throws and see how different they are from each other. The cell below does precisely this. Hopefully you will see that the larger $N$ is, the more concentrated the estimates are. We quantify this decrease of the spread by calculating the standard deviation $\\sigma$ of the estimates calculated at the same value of $N$. We then compare the standard deviations to the $\\sigma = 1/\\sqrt{N}$ line." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# investigating the dependence of the accuracy of the MC integrator\n", "# on the number of samples\n", "n_trials = 30 # number of area estimates for each N\n", "N = np.logspace(1, 5, 9) # different values of N to be used\n", "areaN = np.zeros((N.size, n_trials)) # array to store all of the estimated area values\n", "\n", "# for each N and each trial within a fixed value of N, calculate and store the area\n", "for i in np.arange(N.size):\n", " for j in np.arange(n_trials):\n", " areaN[i, j] = mc_integr(to_integr, 0, 1, 3, N[i])\n", "\n", "\n", "# plotting areas as a function of N\n", "plt.semilogx(N, areaN, 'o')\n", "plt.xlabel('N')\n", "plt.ylabel('area')\n", "plt.title('Estimated of the area vs. N')\n", "plt.show()\n", "\n", "# plotting the standard deviation of the area estimates as a function of N\n", "# and comparing to 1/sqrt(N)\n", "plt.loglog(N, np.std(areaN, 1), 'o')\n", "plt.loglog(N, 1/np.sqrt(N))\n", "plt.xlabel('N')\n", "plt.ylabel('standard deviation of the area')\n", "plt.title('Standard deviation of MC estimates of the area vs. N')\n", "plt.legend(('results of simulations', '1/sqrt(N) line'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice how the convergence of the estimators to their true value (the decrease of the standard deviation) goes as $\\sigma = A/\\sqrt{N}$, where $A$ is some constant. The general claim, which is known as the *Central Limit Theorem* is that the $1/\\sqrt{N}$ law holds true for essentially any function being integrated, and only the constant $A$ changes. Moreover, the same scaling is true for almost all simple MC methods, beyond just area estimation: you need to increase the number of smaples by a factor of 100 to get a 10-fold increase in precision. \n", "\n", ">### Your work\n", "Verify that MC integration of different function still converges as $\\sigma=A/\\sqrt{N}$. For this, integrate different functions you can think about. Think about how $A$ depends on the shape of the function being integrated. Suppose you overestimated $f_{\\rm max}$. How would this change $A$? \n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Random walks and the Central Limit Theorem\n", "Every one of you who has taken any science lab can probably recall some mention of the Gaussian (also known as normal) distribution: in many experiments, the stochastic fluctuations around the mean are suppsed to be distributed according to the familiar bell-shaped curve, defined by two parameters: the mean $\\mu$ and the standard deviation $\\sigma$, namely: $P(x) = \\frac{1}{\\sqrt{2\\pi}}\\exp\\left(-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right)$, shown in the next cell.\n", "\n", ">### Your Turn\n", "Vary the mean and the standard deviation, and explore how the distribution changes. Show many different normal distribution on the same axes." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sigma = 1.0 # standard deviation\n", "mu = 1.0 # mean\n", "x = np.arange(-5,5,0.1)\n", "y = 1/np.sqrt(2*np.pi*sigma)*np.exp(-(x-mu)**2/(2*sigma**2))\n", "plt.plot(x,y)\n", "plt.xlabel('x')\n", "plt.ylabel('P(x)')\n", "plt.title('Normal distribution')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Why were your instructors so confident that the distribution of errors is Gaussian? To understand this, we will start from afar, from random walks.\n", "\n", "Random walks is probably my favorite subject in the modern physics curriculum. Many of the famous names in the late nineteenth and early twentieth century physics and math have contributed to the field, including Einstein, Polya, Wiener, and many others. A single lecture cannot do justice to the subject, and I hope you will pick some of the books to read instead. The \n", "__[Introduction to Probability](http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/pdf.html)__ by Grinstead and Snell is a good starting point. Another great book is __[Random Walks in Biology](https://www.amazon.com/Random-Walks-Biology-Howard-1993-09-07/dp/B01MQZYDZ4)__ by Howard Berg. \n", "\n", "In general, **random walk** is *a stochastic process that consists of a succession of random steps*. The usual and overused analogy is that of a drunkard, who tries to walk, falls, stands up, and, having forgotten which way he came from, chooses a random direction to go to, before falling again and repeating the cycle. There are many different types of random walks. *Discrete time* (DT) walks mean that every step takes a discrete unit of time. In contrast, in *continuous time* (CT) walks, the duration of each step is a real-values variable, sampled from some distribution. Similarly, one can define *discrete space* (DS) or *continuous space* (CS) random walks by whether the step length is fixed or sampled at random from some probability distribution. As we will see later in this lecture details of these probability distributions matter very little (as long as they have finite variances of their own), and we can learn a lot about many different random walks by studying the simplest version of a random walk, the DSDT walk with the duration of a step $\\Delta t=1$ and the step size $a=1$. Such walks can exist in any number of dimensions (with 1-d being, of course, the simplest). However, for illustration purposes, below we will mostly focus on 2-d walks on a square lattice, where the walker at any point in time can move in one of four directions: up, down, left, or right. Let's generate a few such walks of different lengths." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# This function will generate a 2d DSDT random walk\n", "def DSDT_2d_walk(N):\n", " # Input:\n", " # N - duration of the walk\n", " # Return:\n", " # x,y -- arrays of length N, coordinates of the walker on every step\n", " \n", " directions = np.floor(nprnd.random(N)*4) # (0,1,2,3) directions of motion on each time step\n", " # Do you understand why multiplication by 1.0 is needed in the next two lines?\n", " # Check what happens if this multiplication is removed.\n", " dx = (directions == 0) - 1.0*(directions == 1) # directions == 0/1 corresponds to right/left motion\n", " dy = (directions == 2) - 1.0*(directions == 3) # directions == 2/3 corresponds to top/down motion\n", " \n", " x = np.cumsum(np.hstack((0,dx))) # x and y coordinates of the walker are a cumulative sum of steps\n", " y = np.cumsum(np.hstack((0,dy))) # Notice that the initial position of the walker is always 0. \n", " # Also notice that the walk of N steps has N+1 positions in it -- there is the initial and the final\n", " # position, surrounding N steps.\n", " \n", " return x, y\n", "\n", "\n", "N = np.array((100, 1000, 10000)) # we will explore random walks of these different lengths\n", "for i in np.arange(N.size):\n", " x, y = DSDT_2d_walk (N[i])\n", " plt.plot(x, y, 'b-o')\n", " plt.title('Random Walk of length ' + str(N[i]))\n", " plt.xlabel('x coordinate')\n", " plt.ylabel('y coordinate')\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice how a moving particle comes back again and again, and how random walk eventually covers almost the entire 2-d space, with very few holes in the middle. This is a crucial property of random walks, which we will refer to again in one of the projects for this Module.\n", "\n", ">### Verification\n", "To verify that the function generates random walks correctly, generate very short walks, e.g., `N=4...10` and see if the walker makes any unallowed steps visually. \n", "\n", ">### Your turn\n", "Run the cells multiple times and convince yourself that the walks are not biased. Sometimes they move more to one side or the other, but other times it's the opposite.\n", "\n", "We can now see that the details of how the individual steps are generated in a walk matter much less than one might naively think. For this, we define a CSDT walk, where in each individual step, the walked moves in both $x$ and $y$ directions by steps of sizes taken from ${\\cal U}(0,2^{1/4})$. We then compare the DS and the CS walks to each other. \n", "\n", "Note the weird $(3/2)^{1/2}$ multiplier. We do this to ensure that the average steps we take in each of the walks are equal. Indeed, for CSDT walk, the steps, on average, are smaller than 1, but we make steps in both the $x$ and the $y$ direction always. The algrebra is a bit more complex than I want to go over here, but I hope the logic is clear; we will return to this multiplier later." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# This function will generate a 2d CSDT random walk with steps taken from the uniform distribution\n", "def CSDT_2d_walk(N):\n", " # Input:\n", " # N - duration of the walk\n", " # Return:\n", " # x,y -- arrays of length N, coordinates of the walker on every step\n", " \n", " dx = (3/2)**(1/2)*(2*nprnd.random(N) - 1) # displacements along x\n", " dy = (3/2)**(1/2)*(2*nprnd.random(N) - 1) # displacements along y\n", " \n", " x = np.cumsum(np.hstack((0,dx))) # x and y coordinates of the walker are a cumulative sum of steps\n", " y = np.cumsum(np.hstack((0,dy))) # Notice that the initial position of the walker is always 0. \n", " # Also notice that the walk of N steps has N+1 positions in it -- there is the initial and the final\n", " # position, surrounding N steps.\n", "\n", " return x, y# comparing different ways of generating random walks\n", "\n", "\n", "N = np.array((100, 1000, 10000)) # we will explore random walks of these different lengths\n", "for i in np.arange(N.size):\n", " xD, yD = DSDT_2d_walk (N[i]) # discrete walk of this length\n", " xC, yC = CSDT_2d_walk (N[i]) # continuous walk of this length\n", " plt.plot(xD, yD, 'b-')\n", " plt.plot(xC, yC, 'r-')\n", " plt.title('Different types of Random Walks of length ' + str(N[i]))\n", " plt.xlabel('x coordinate')\n", " plt.ylabel('y coordinate')\n", " plt.legend(('DSDT walk', 'CSDT walk'))\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that we can easily see the difference between the two walks when there are only a few steps, and the walks do not move far away from the origin. However, at $N=10000$, when individual steps are invisible, it would be hard to distinguish one walk from the other. \n", "\n", "To formalize this observation, we will look at the means and the variances of displacements in walkers of different types. For this, we will generate many different walks of the same type and the same length and average." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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sYtfXMKon+ezOagWM8sZFagAvquoYEbkVmKiqbwKPA0+LyGxcC2Rw8uoMwzCqMU89BRs2wFln5fSw+bTOmqyqu6jqzqraXVVv9dJv9BQIqrpeVY9V1e1UdXd1sZ8rJTVr1qRnz55069aNHj16cN9991Fc7KJ8rl27ln/84x/stNNOdO/enb322os1a9aUuN+4cePo2bMnPXv2pFGjRuywww707NmTk0+OCgiXHh07dmTJkiX8+uuvdO/evfQdDMPIPhs3wvffB9uFXhThUaPcL8dUuRnrFZX69evzvXfjFy1axN///ndWrlzJLbfcwgMPPMDWW2/Njz/+CMCMGTM2WzmVtN9BB7ngjP379+fee++ld+/eeTgzwzByyqWXwsMPw++/w9ixcO658NZb8OGHsM02ORenQgysVze22morhg8fzoMPPoiqsmDBAtq0abM5f4cddqBu3bql7pcK559/Pm+++SYARx11FKeffjoAjz/+OP/6178AOPLII+nVqxfdunVj+PCSw6DMmTOHXXbZhW+++Sal4xuGkWE++sgtn33WKRCAww5zyz/+yLk41a4lcsklsS3BTNCzJ9x/f9n26dy5M8XFxSxatIjTTz+dAw88kJdffpkBAwZwyimnsP3225e639Zbb13qcfbee28+++wzDj/8cObNm8eCBQsAGD9+PIMHuyGmkSNH0rx5c9atW8duu+3GoEGD2HLLLRPqmjFjBoMHD+aJJ56gZ8+eZTthw6iKXHIJ9OgB7dvDgAG5OeZPP7nltdcm5u24Y25kCGEtkTzityZ69uzJnDlzuPLKK1m2bBm77bYbP/kPSgn7pUK/fv347LPPmDZtGl27dmXrrbdmwYIFfPHFF/z1r38FYOjQofTo0YO+ffsyd+5cZs2alVDP4sWLOeKII3jmmWdMgRgGwD33wAMPwOmnw/775+aYhxxScn4J741sUe1aImVtMWSLOXPmULNmTbbaaisAGjVqxNFHH83RRx9NjRo1GDt2LF26dCl1v9Jo06YNy5cv53//+x977703y5Yt48UXX6RRo0ZsscUWfPzxx7z//vt88cUXNGjQgP79+0fO92jSpAnt2rVjwoQJdOvWrXwnbxhVgauuyk69xcVOKZ1/PhxzTGz6O+8k32/t2uzIUwrWEskDixcv5txzz+XCCy9ERJgwYQLLly8HYOPGjUybNo0OHTqUul+q7LHHHtx///3svffe9OvXj3vvvZd+/foBsHLlSpo1a0aDBg2YPn06X375ZWQdderU4fXXX+epp55i9OjRaZy1YeSZoqLMvWiLihLTNm7MTN1jx7pxj2OPhSefDI7lGd7E8OCDblxkwgSoXz8zxy8j1a4lki/WrVtHz549KSwspFatWpx00klcdtllAPz888+cd955qCrFxcUceuihDBo0qNT9UqVfv368++67bLfddnTo0IFly5ZtViIHH3www4YNY+edd2aHHXagb994R8oBDRs2ZMyYMRxwwAE0bNiQI444Is2rYRh5oFs3mDEDNm2CqVNhu+2gQYP06qoV8eqsWxcyEW7cHyQHOO00N4h7//3g/WcBWLYMmjUr/7EyQJWLsd67d2+ND0r1008/RXYNGZnFrrNRYdm0CXznoM89Byec4NbTff8l6wkoLk6el27dzZo5E9699nLbq1dDo0blO0bkYWWSqpZ5noB1ZxlGVee336CgoOz7ffstdOkCcyrtHN+AsHdpX4FkimefDdZ/+CG6zKefupZQOl1ey5cHCgSyokDKgykRw6hqvPii+5r1fx07Qrt2pe83ZYrrJvE57jiYPh223RbOOCOYGV2V2LTJLTdudMog3DIpKnLnfPbZcPnlQfrKlcF6nz7uOvkks9LaZx+YNi2Y45GMNWtKzq+AmBIxjKrCn386pXH88entv9NOEB4T8+YUATByJIS7iSdMgAhT8JRp2rT83T6p4hmtROIrhCZN4MQT3UA2OGVSqxbUqQMjRsB99wXX49//DvZ/9llX7umn3fbSpfD557HHCCugLbZIlGHkSLjgAhg4EEqbxJvE8CWfmBIxjMrC1KnJ+/A3bChfN4ffyvAVQ7t2iZZM55wTrO+1F/zlL+kda//9g5d3lJVTppk2zS2feSYxb4cd3NI3a/fLRBmYfOXFzJs82S1//tm10iC2i2zPPWMV1333BeszZsTWOXOma+U9/DD873+w334u/YUXos+lAo45mhIxjGxTXOxeEn/+mX4dn3wC3bvDzjtH57/3Xul1lPTCXrEiWN+4MXoMJcrEtKwUFsIHHwTb2ei+2bjRWTj5E+98r7aTJ8PNN8eWXbrULX1rqw8/dMuvv06st4b3uvzf/9yyY8cgr2bN2LILF7rlkiWx6W+/HfshkGyWe9eurtx33wVpRUXQuHF0+TxiSsQwss2wYa674uqrXRdOKi/8ePwv2ClTovN//TV2+6mn4OWXY9PCisKnsBDuvjt2LOSRR0qWJfwSFIHHHy+5fBj/5eozPT6EUAYYM8b9unaNPcZFF8FNNzlluGpVUH7mzGBsBJJ3sx1xRGxejbjXZ7j7b8ECd21btowt88orbr/Ond12MoOHJk3csmdPN1/kww8Tj1dRUNUq9evVq5fGM23atIS0XLNgwQI9/vjjtXPnztqlSxcdOHCgzpgxQ4uKivSf//ynduvWTbt37669e/fWOXPmqKpqhw4dtHv37tq9e3ft0qWLXn/99bp+/XqdPHmy9ujRQ3v06KHNmjXTjh07ao8ePXTAgAEZk3efffbRb775RlVVGzZsmNI+FeE6VziefFLVvXZV27YN1svKsGEl7zt8uMurVSs2fdasYL/p0xP3O/FEl9elS1Au/DvvvNg6rr02ulyqTJyY/r6p8uqrQd1z5gTrRUWx5aLOoyy/KC68MPX9v/kmWL/ooti8lSszf11KARfHqczv3Ly/9DP9q4hKpLi4WPv27auPPPLI5rTvvvtOP/30Ux09erQOGjRIi7wHfO7cubps2TJVdUpk8eLFqqq6evVqPeGEE/Tkk0+OqfuUU07Rl156KeMymxLJEMleIMXFqe2/caPq+vWxSuS335IfZ9asxLz//MflHXBAYp7/8mrZMlHG5s1LP4+yKIIVK1RbtUpv37Lw1FOpyZgNJbJmTXTZQYMS0848M1jftClYHzYs89ckBdJVIhW0fVS1+Oijj6hduzbn+m6bcU4X+/Xrx4IFC2jVqhU1vKZq27ZtaRYxE7VRo0YMGzaM119/nWXLUgszf/fddzN06FAALr30UvbzBu0++OADTjzxRADOO+88evfuTbdu3bjppptKrG/JkiXssccevP322ykd3yiBxYvdWElp1KkD9erFpoUj1y1bBs8/H2z7A71hDj/cLaO60bzng8WLE/O+/bZ0+QAiPD5HcvfdQZfPvvsG6e++m9r+Udx+e+y4AZRsjVUS8eMNV13lZouXxbS5YcPo9IceSkx77LFgPTymEjZgqARUP7cnefAFP2XKFHr16hWZd9xxx7HXXnvx2WefMWDAAE488UR22WWXyLKNGzemU6dOzJo1iz59+pQq1t57782QIUO46KKLmDhxIhs2bKCwsJDx48dvdntyxx130Lx5c4qKihgwYACTJ09m54jB24ULF3L44Ydz++23c8ABB5R67GrL6tUwd27QH58M343/zz9Dhw5uALZRI/cSmjLFmduGPbaGX5Thl278CzyqPz8sy3ffueP85S/R/4Mff3QDx4MHQ9u2QfpLL7m+eZ/mzYNxlKVL4ZZbnIuO9u2jz1c11jT27bcDlyMHHRQ7zuIzerSzXFqxwrkUia/vppvgttvghhti9w/P4yiJL7+MtcKaOxe23x4WLXLbN96YqBQ2bIAjj4SyuB465xx3v+fPh9atE/P9j7LCwoo77lECeZNYRNqJyEci8pOITBWRiyPK9BeRlSLyvfe7MR+yZpO2bdsyY8YM/vOf/1CjRg0GDBjAB2HrlTg06s+WhF69ejFp0iRWr15N3bp12WOPPZg4cSKfffbZZiXy4osvsuuuu7LLLrswdepUpvnmkCEKCwsZMGAAd999d9VVIKpukl55J9Q1buxmJocHaiH5y3Xbbd08gm22cS93EadAwDni83n00WD91FPdcvXq2LpuuaV0+XbdNTDNjXLpv8MOcMUVsQoE3IszjG/V5HPzzdCpU/Ljxj/T9evHDqoPHpxoPXbxxc70dt68xPq++sopEJ+w+e5bbyWWv/32xLQ+fQLT3D593L3zrbMgVoF8/707Zp067r6U5PrdD1H70ktueeCBbtmqlWvxxYew9ltltWpVSiWSt7ELoBWwq7e+BTAT6BpXpj8wpiz1VsQxkffff1/79euXUtl77rlHL7zwQlWNHRNRVV21apU2btx485iJauljIvvuu68+8MADesMNN+hLL72kd9xxh3bs2FGLi4t1zpw5uu22226u75RTTtEnnnhCVWPHRBo0aKAnn3yyXnvttSXKnu/rXC7eeMP1R994Y9n3XbdO9c8/3brfr/37725wFFSPP171++/L3wcPqjVqqPrjYvF5GzYkl/G550qvO8Wxr82UZWzkk08Sy61fH5s2dqxLX7xYddSoIH2nncp2bH97xIhgvbCw7OdVHvzjLVmSmFdcHCt3quNjWYbKNiaiqgtU9VtvfTXwE9Cm5L0qJ/vttx8bNmxgxIgRm9O++eYbPvnkE7799lvmz58PQHFxMZMnT450A79mzRrOP/98jjzyyMgxk2Tsvffe3HvvvZvdwA8bNoyePXsiIqxatYqGDRvSpEkTFi5cyDtJYhWICCNHjmT69OnceeedZTz7SoI/JvDLL8nLFBTAunWJ6fXru6/WcFfS8ccHLYMXXgjGKm69tXxyduvmumuiWqR16iTfb/Dg5P31r73muoSivvhL4uijY7dPOil52XBLynveqVs31mTZb4m0bAmnnBKk//hjMFt+2bLgSz+eJ5905tQ+p5/uWiXFxdFed5Pxyy8waVLq5aPwjxc1XhR+Tk47LXcz97NFOpon0z+gI/A70DguvT+wFPgBeAfolmT/s4GJwMT27dsnaNiK8IU8b948PfbYY7Vz587atWtXPeSQQ3TmzJn6zjvv6K677qrdunXTbt266Wmnnabr1q1T1cDEt1u3btqlSxe97rrrNuf5lNYSef/997VWrVq6Zs0aVVXdfvvtdciQITH777jjjnrIIYfoUUcdFdkS8a2zNmzYoAceeKA+9NBDkceqCNc5be6+O/kX6OTJyb+2f/st+Zd92Gw2zHffpdcKueyykvNLI9l+s2enf93C9e6yi+pXX5VcZtSo2PSCgiCvUSNXRzI577svMW3nndO/HvmkoED10kud9V0Fgcpq4gs0AiYBR0fkNQYaeeuHALNKq68idmdVFyr1dY5/+RQXq77zjptbsOWWsfkffBDsFzXvwf+df75bJnuxFhaqdu2auF///sH6o4/GyhV1nIkTVefPL/0ct9suOy/bqDrDc0tOPjn5sTZuTH79SvsNHuzqaNiw8imRCkilVCJAbWAccFmK5X8FWpRUxpRI/qjU1zn88unRI3a7adPoF9Tatam97EqisDAo16uXWy5cqLpqVfR8kNNOS6zfH48pjeJi1aFD3T7jxunmL/nyUpYXfxTTpqWnRLxJudqmTWLe7beX/7yqGekqkXxaZwnwOPCTqt6XpMw2XjlEZHecNdnSqLKGkTbxfpLiY0JEuQsBZ5rrk26Milq1XB//pZc6D65r1sBWWzlvr1EWXfEWWQsWpB6dTwT++U/3mj3wQLdMFv+iLMS7V0nGnntGp3fpEhvvw2f9+mAeSxT+vI577knMu/761GQyyk3eIhuKyF7AZ8CPgD/r6jqgPYCqDhORC4HzgE3AOlyL5fOI6jaTLLLhjjvuWKa45EbZUFWmT59eOSMbpvNcrFjh3Jn7LF8Or7/uBr5POMHN//DJ5H8sXtZNmxKd/+WDJ590g8QlUZKs8efVuXNwDZPdn40bA+UTXyZP77XKTKWLbKiq41VVVHVnVe3p/caq6jBVHeaVeVBVu6lqD1XtW5oCSUa9evVYunQp+VKYVR1VZenSpdSLn1ld2XjttdTLhhUIuJbDqafCbrvBF18E6TNnZkTBbZm/AAAgAElEQVS0zbQJGTDutlvFUCAARx0Vne5bYkHZZC0pVsmaNc5SLtx6+egjp3jOOiu5k0ojK1SLGett27aloKCAxVGuHYyMUK9ePdrGT1Cr6EycGOv9dt993ezhQw9NLPvqq+5FueeeiUGHpk+PfUH6nlsbNXIzoDPJnDluEt6//hWrUPJNkyZw110uDGzYLU6rVqntv+eeLtDV0qVOSYQn3f3f/7lYKscc4ybwNWyYaK7cv39s68/IGXnrzsoWUd1ZhpFAYWEwr6JrVzeP4803XcvBD1T07LPwj3+4df9/snQptGgR1HPLLc49Rjzr1jnFUtLcjarIqlVw7rnw3HPuxd+1q5vn0bBh4P7cqJBUuu4sw8g4y5YlthKSEY6BMW1a0D3VvLlbNmoEf/+7G/D23ZCAmzwWDiSU7Ou3fv3qp0DADXaPHu2Uru+za6edTIFUYUyJGFWHww5z3SK+/6vCQvjtt+iy550Xu/3pp27ZvDkcd5wLagQutKkfDtUn7PsoWWQ6w6gmmBIxqg5+K+SPP5y31cMOcyFMw2axY8ZEW/v4yqZGDeemZJ99kh9nwwa33HbboLvLMKop1WJg3agGHHxwsN6+ves+mTPHbT/3nPO6+sorLkZEFGGPsKXhz984+uiKYx1lGHnClIhRNRg3LnbbVyCQWpCf0uJ/hDnoIOcWfLfdUt/HMKoopkSMyk8mrPF23z275Q2jimJKxKj8pNMiGDsWevVyYyBz5iQGYTIMIyVsYN2oWvhhZAcOdPMTHn44utyeezofVS1aWKvCMMqBtUSMqsNtt7mQr+EJtN27u+X557uZ6H6QIhsQN4yMYErEqNwsWRKs/+tf0WXOPde1OI480pn3mgIxjIxhSsSo3MyYUXoZETj22OzLYhjVEBsTMSo3/sxywzDygikRo/KyaBHceadbHzw4v7IYRjXFlIhROSkshK23DrZHj86fLIZRjTElYlQ+Vq9O9JBrUSsNIy+YEjEqJjNnws47x1pfgbPA8mNr+6xalTu5DMOIIW9KRETaichHIvKTiEwVkYsjyoiIDBWR2SIyWUR2zYesRh7YYQc3WfCVV2LT77gjsewWW+RGJsMwEshnS2QTcLmqdgH6AheISLwXvIHA9t7vbOCR3Ipo5JwPP4yN3xFWGlFROIcOzb5MhmEkJW/zRFR1AbDAW18tIj8BbYBpoWJHAE+pi+H7pYg0FZFW3r5GVSQ+yNPcucF6vI+sf/8bLrww+zIZhpGUCjHZUEQ6ArsAX8VltQFCbxEKvLQYJSIiZ+NaKrT3Yz0YlY+33y45f9KkYD2qVWIYRs7J+8C6iDQCXgEuUdX4EdIok5uEt4eqDlfV3qrau2XLltkQ08gm8+c766q//S06PxwbBGDEiOzLZBhGSuS1JSIitXEK5FlVfTWiSAHQLrTdFpifC9mMHLFhgwthWxLbbhu7feaZWRPHMIyykU/rLAEeB35S1fuSFHsTONmz0uoLrLTxkCpGz55u4mA8bdsGs9ENw6iw5LMlsidwEvCjiHzvpV0HtAdQ1WHAWOAQYDawFjgtD3Ia2WT69NjtTz+Ffv3cenExXHNNbP6ee+ZGLsMwUiKf1lnjiR7zCJdR4ILcSGTknTlzoFOnYLtGXEN54EBzuGgYFYy8D6wb1ZxWrdzyu+9iFYiPquvyuvxyF9I2XrEYhpFXKoSJr1GNKS6Gs892iiIZfshbwzAqHPZZZ+Sezz6DG2+EoiJYuBCaNcu3RIZhpIloFZu01bt3b504cWK+xTBKIt7jbocO8OuveRHFMAyHiExS1d5l3c9aIkZuifpoOeKI3MthGEZGMCVi5Ja1axPTLrss93IYhpERbGDdyB3PPQcff5yYbq7cDaPSYkrEyB1//3t0evPmuZXDMIyMYd1ZRn748cd8S2AYRgawloiRG+IH1Fu0cCFw69bNjzyGYWQEUyJGbogf92jcGLbZJj+yGIaRMUyJGNljyhSnPEaOhD//jM1r0CA/MhmGkVFMiRjZobAQdtopOu+WW3Iri2EYWcOUiJEdrr46Ov3LL6FPn9zKYhhG1jAlYmSHRx5JTJsyBbp1y70shmFkjVJNfEVkaxF5XETe8ba7isgZ2RfNqNSsX5+YZgrEMKocqcwTeRIYB7T2tmcCl2RLIKOKsN9++ZbAMIwckIoSaaGqLwLFAKq6CSjKqlRG5WDx4uhB8mnT4MMP3XqRPSqGUZVJZUzkTxHZElAAEekLrMzEwUVkJPA3YJGqdo/I7w+8AfziJb2qqrdm4thGBthqK7c86ijYeWcXL71Ll9gyNWq4gfR163Ivn2EYWScVJXIZ8CawrYhMAFoCx2bo+E8CDwJPlVDmM1X9W4aOZ2SSmjVdS2P2bKdE+vWLzT/qKLf88svcy2YYRk5IpTtrKrAP8FfgHKAbMD0TB1fVT4FlmajLyCFr18L11wddVYMGueXZZ8eWe+WV3MplGJUYVVi9OrWyRUUutpv/e+yx7MpWEqkokS9UdZOqTlXVKapaCHyRbcFC7CEiP4jIOyISad4jImeLyEQRmbh48eIcilZNufJK+Pe/E9NHj47djo9gaBhGUm66yXkD+vbb0suOGRO7fdZZMH58duQqjaRKRES2EZFeQH0R2UVEdvV+/YFc+az4Fuigqj2A/wNejyqkqsNVtbeq9m7ZsmWORKvGJBvfsBC3hlEihYWJaaowZAjcdpvbPvLIIE8EjjkmcZ9JkxLTPvkkMzKWlZLGRA4CTgXaAveF0lcD12VRps2o6qrQ+lgReVhEWqjqklwc34hg7lx44onE9PhWx6JFuZHHMCoJ778PBxwAX38Nu+0WpI8bB1dcEWzPnesUy08/ue34XuGaNaG4OLH+776DjRuhTp3My14SSVsiqjpKVfcFTlXVfUO/w1X11VwI57WGxFvf3ZN3aS6ObYRYs8YpiS5doH370surgrUIDSOGQw5xy913hzfeCNKjxkGeeirWZ6mvNG68MVqBgFM28caRuaDUMRFVfUVEDhWRq0TkRv+XiYOLyHO48ZUdRKRARM4QkXNF5FyvyDHAFBH5ARgKDFaND0xhZJ299nLL6SF7ittvD9rhhmGUSrgrK9xl1bhxYtlTT3Xfbj7jxrm/m9/llYw5c8olYlqUauIrIsNwYyD7Ao/hXuxfZ+LgqnpCKfkP4kyAjXzyww+Jab6DxY4dY9Mvuijr4hhGVUAEpk6Fgw8O0kaOhNNPd+thpw+HHAING8buf8st8PnncMcd0Lt39uVNRirzRP6qqjuLyGRVvUVEhgA56c4yKjC1vEfnqKNg8GB4/nk49FB44IH8ymUYFYz58+Hll6Pz/KlU4KyydtklUCLxhLu3VqyAJk0Sy3Ttmr6c6ZKKia9virNWRFoDhUCn7IlkVCpE4Mkn4eKL3dIwjM188gm0aeP+HuC6qcLMnBmst2njlpeU4pmwT59oBQLw3ntpiVkuUlEiY0SkKXAPzuT2V+D5bAplVCB8c96aNYO0JXHGcXXrwv33u7jphmEAbgyjf//YtLffhr33ji7vexF6++0g7d573ZyQgQODtCuvTNx32TL3t2zdOjEv25TanaWq/lDOKyIyBqinqhnxnWVUAvzuqaIiOPdc95RuuWV+ZTKMSsCECYlpTZvCu+86C/hUDB0vv9wtDz00sKKPiizdrFn6cpaXlIJSichfgY5+eRFBVUvyd2VUFZo2dcsTTogONGUYVZDx450x4sknw/LlriG+5ZZlc8Lw7ruJaaNHu4Z7u3bJ9+vTB2bNgtq1Y9M7dYJffkkcYM83qQSlehq4F9gL2M375dEWwMgpH3zglhYX3ahG9OvnXInUrQvbbOOmPY0Ykdq+u+4Kjz+e6Bno6qtdXjyXXx5rmvvII+5YGzbElps1Cz76KHl3WL6Q0qZdiMhPQNfKMj+jd+/eOnHixHyLUTWYMgV22smtL15sYx5GtSFZi8N/C65d68YrrrwS6tcP8gsLE2eMT54M8+bFmvKCa+H8+quzyKoIiMgkVS1zAyGV7qwpwDbAgjJLZVRuwg4VbRzEMDbz0EPOYeLq1XDPPUF6lP+qrl2Db7EwzZrldywjU6QU2RCYJiLjRORN/5dtwYw8U1wM//mPW2/TxjzyGtWCdetKftT9lshVV7nlvfe6bqfeveGrr6LHQcKGjVWRVFoiN2dbCKMC8vjjwfpvv+VPDsPIIaW5Yf/zT3j22di0Pn2cU4e+fRPLH3dc5mSrqKRi4psnB8NGXgkHmKrqn1JGteTWW12r44YbgrTPPit5nzVr4NJLY9OivAL5jBqVvnyVhaQD6yIyXlX3EpHVePHV/SxAVTXCbVj+sYH1NNi0yflRCA+c+236lSujPcQZRiXHf8TDr8BwV9aZZ7qhwP32g99/d9ZakyZBr16p1b/99rEz0is66Q6sl+QKfi9vuYWqNg79tqioCsRIk8MPdzaMjz6amGcKxKiCJLM1Pf/8YP3UU+HOO+HAA+GPP1xaWIGsWBFdx777uvorkwIpDyVFNmxe0i+XQhpZ5p133PLcc0suZxhVhOHDg/V99nH+Q486Ch5+2KW9/jrsuWdQZscdE+tI5r/Kt0epLpQ0JjIJ140lQHtgubfeFPgdc8JYNRg3LjHND3xw1lm5lcUwssCRRzpH0w0buob1PvvEfi99+qn7hTniiNjtqBC14Jwl1q8PH38MX3zh0nwnD9WFpEpEVTvB5ngib6rqWG97ILB/bsQzsk78DChwvhUg2rjdMCoRw4e7KILhSIL33Ze8fKr4jg7/+98gzR9PqW49wKnME9nNVyAAqvoOsE/2RDJyRlFRsH7AAW4k8MYbnbc3gC+/zI9chpEhzjknMe2yy0rep0+f6PT164P1qO4tn1atSperKpGKElkiIv8SkY4i0kFEridDcc5FZKSILBKRKUnyRUSGishsEZksIhGeZ4y0CTtUbNbMOee57TaYPdul3XRTfuQyjBzQr190+l13RafXrRusPxXhfnbFCmfMWN1IRYmcALQEXvN+Lb20TPAkENGfspmBwPbe72zA3Mhmklmz3PKAA+DFFxPzO3fOrTyGkUEWlOKoae5cZ0W1fLkbGrzwQthiCzdmkozZs908XD+AVJgmTapfVxaU4oBRRGoCd6pqRBiUDAkg0hEYo6rdI/IeBT5W1ee87RlAf1VN+njYPJEUmTAB9trLrS9bBs0jDO4qh89Nw0hg/Xo3pOc3qqN47z3Y30Z3N5PxeSIAqloEpDi1Jiu0AeaGtgu8NKO8zJ8frDdtCp9/HpvvR8MxjEpI06YlK5A5c0yBZIpUfGd95zlcfAnYHCpeVV/NmlQBUa7QEj6PReRsXHcX7VMJF2bEOvURgT32sJaHUWWIj8URj70mMkcqSqQ5biB9v1CaArlQIgVAOAZYW2B+fCFVHQ4MB9edlQO5DMOooPgW6j4rViTO3TB3cJkjFQeMp+VCkCS8CVwoIs8DfYCVJY2HGGkwbVq+JTCMjBJvD9KkiRswv+02N382mbsSIz1KVSIi0hb4P2BPXAtkPHCxqhaU9+Ai8hzQH2ghIgXATUBtAFUdBowFDgFmA2uBfCq0qkNB6NZ16ZI/OQwjQzzwgJu7cdBBzrrKDw51xRVuuWpVULZBg9zLV5VJpTvrCWA0cKy3faKXdkB5D66qJZoKeyF5LyjvcYw4rr463xIYRsb47TfnfgQCk11wrkySzQUxMkcq80RaquoTqrrJ+z2JmytiVFZ831j/+Ed+5TCMcrLNNtCxY7B9110upjmYAskVqc5YP1FEanq/E8nQjHUjT7z0kltGuX43jErEwoWx29dckx85qjOpKJHTgeOAP7zfMV6aUdmxzmGjEvLGG86/1Vtv5VsSA1KzzvodODwHshi5ont3qFUrNoybYVQClixxrt3BxVJLRrduuZHHSKElIiJtReQ1z1HiQhF5xbPYMiorU6aYXywj73z0ETz7bGJ6gwZwxhmJ6Z9+6gJwRtE29EZq1arkuOdGZkmlO+sJ3HyN1jiXI295aUZFZPlymDcPLrgg2pW7P9r4ai7mihpGNMXFLnb5iSfGulhXhXXrYORIuP12GDECbr3V5SVzjDh7NowZ48x7CwqcRx+bTJg7UjHxbamqYaXxpIhcki2BjHISdqT45JPw55/B9vDhMH58zkUyjHh8M1xwkQFVE3tXb7ghWO+dxC3gVVfBttu69f/9L7MyGqlh1llVmbVrXVxQcB7nwhF6bNqukUeWLIndHjWq5PJ+nDSfK690scyTxf4wckcqLZHTgQeB/+JmrH+OWWdVTPz5H2FeeMEFl/7ttyBtzz2dLwjDyBPxkQFPPbVs+19wAXTokDFxjHJg1lmVgdWrnfOfkpg3L3Z0MczIkbHbEyZkRi7DSIO1a4P1hg1je1x99twz+jE991zngdcUSMUhFeusUSLSNLTdTERGlrSPkSF+/tn9mxo3hrFjSy773HOp1/v11+WTyzDKQbNmwfobb0SXefDB6PT//heuvTbzMhnpk8qYyM6qurkDXVWXA7tkTyRjM7ffHgSLevjhxPz164PRxCtDwSfvvNOZuET5yFq5EnbbLfOyGkaKbNzolueckzyc7I47OqeKW20VuDE54wyoVy83MhqpU2J4XAAR+QEXkna5t90c+ERVd8qBfGWmSoXHPeIIePPNYPuDD5xdpM/FF8PQoTBpEvTyAlDefjtcf31sPR07BmMiFnjKyDO+FdaKFW5ozt9evNit16yZGP/DyD7phsdNZWB9CPC5iLyMG1g/DrijrAcy0qBGXEPxnXfcsnVr96k2dKjbDttLxisQcN1itVK51YaRXcLfML5tx1dfOYXSokV+ZDLKRyoD60+JyERcZEMBjlZVi2SUbTZsgNdfj00rLoYBA9x6UVGQ7geLbt06uq6aNd2+5ubEyDOLFrllODzt7rvnRxYjM6T0eeopDVMcueSDDxLTloam50RNyY13aRrGFIhRAbj/freMskY3KiepDKwb+SBq7KK0GVnPP58dWQwjA2zY4Gw+oGzGhEbFxpRIRcWf0jtypOuKSsaaNcH6LmY0Z1RcTg9NUe7bN39yGJkllXkiF4pIs9LKpYOIHCwiM0RktogkhJMRkVNFZLGIfO/9zsyGHBWSxYvdctCgkruiGjZ0n3V165pnXqNCs802wXrduvmTw8gsqbREtgG+EZEXvZd+RjrXRaQm8BAwEOgKnCAiXSOKvqCqPb3fY5k4doVl4ULX3i8uDuZ9lDRT3TeaHzzYzRmxcQ+jAuMPpvuBNY2qQalKRFX/BWwPPA6cCswSkX+LyLblPPbuwGxVnaOqG4HngSPKWWflZptt3HTck08O0nzF4Ls0fftt18XVpIlzh2IYeeb554OGc0msWuWWflApo2qQ0piIuhmJfnjcTUAz4GURubscx24DzA1tF3hp8QwSkcki8rKItIuqSETOFpGJIjJxcSpPc0Vk/vxgPSpSz803w7JlcMghcNppzrDe5n4YeeaTT+CEE9zM8ltugU2bkpe98Ua3tMe2apHKmMhFIjIJuBuYAOykqucBvYBB5Th2VN9LvEnSW0BHVd0ZeB+INE9S1eGq2ltVe7dMFvqsohNWIj6XXRas16gR63TIMCoAc0OfgTffDEOGuHmwIvDZZ3kTy8ghqXwTtMBNMPwtnKiqxSLyt3IcuwAItyzaAjFvUlUNxy0ZAVTd6AFRLaghQ3Ivh2GUAX/eh881IfOYV15xAaNmzDCbj6pMqb6zsnZgkVrATGAAMA/4Bvi7qk4NlWmlqgu89aOAq1W1ROPASuk7a/ny2IiEPubnyqhAbNgAxx0HL78MtWu7tNJsOerXd75Aw9hjXTFJ13dW3uaJqOom4EJgHPAT8KKqThWRW0XEj19ykYhM9ZxAXoQb2K9avPderAKZPh26dIF3382fTIYRweGHO3+gdeqkvk+8Avnqq8zKZOSfvA5xqepYYGxc2o2h9WuBqhk9oKgITjwxcZb59tvDNPMwY1Q84ue8luRlJxndu2dGFqPiYDPW88VbbyUqkFWrEj33GkYF4f33g/VFi2InD8Zz2mnR6Q0aZFYmI//YGytfxLfz99mn9BC4hlFB2HrrYP2MM2LziosTIzIbVRez2M4HM2bAddfFpn38cV5EMYxUmDEjed7NNzunCZ98Ehtkas4cF8pm//2hd+9YxWNUHUyJZIOiIqcU/Ngf8ey4Y+x2x47ZlsgwysXatcnzWreGtm2DsDY+nTq5H7goz75Fl1G1sO6sbDB0qPtHDR/uPsvefjt52WnTXHhbw6jAhOOihzn77NSG8erUMdduVRVTItnAd+Pu/+P+FpqT+f33sWW7dImeI2IYFYTx4wPX7eedF5t3hwXKrvaYEskG4dC18YRjfowbl31ZDKMUCgtjg2bG069fsL711m7CoY/FRTdMiWSDuyK8s7RuHdue79sXDjwwdzIZRhLOP98pg6IiZ1klEvyeeiq2bNOmcNRR+ZHTqJjYwHquWLAgdvuTT/Ijh2HE8ZgXpeeXXxK/f045JXa7Xj1niTVmjOuJNQxTIvkiBd8RRUVuPuLgwVCzZg5kMqo1e+1V8iz0sM+rZ57JvjxG5cC6s7LFX/6SPO+DD1Kq4tFHnWeU++6DWbOiyyxcGBtm3TDKgm91Bem5MTEMUyLloaAAevSItbjyXbqffDKMGAH//GfsPqqw334pVw9w1VVOJ0XZ6m+zTWA5YxhlZerU0ssAPP54duUwKi+mRMpDu3YweXKgFD75xIV4A2e2e+aZMHBgUP7ss1OuetUq+O232LQ//ogum+qLwDDiiZqJPngwHH887LxzkHb66bmTyahcmBLJBMuXuz6l/v2DNP/f2apVkHbQQSlXefDBMHp0bNp338VuhwdBU2zcGEYMJ5zgln//e5B2ww1uLO6HH1zD2eJ/GCVhSiRd4mN/7rRT7LbfPOjZM63qv/giMe2ii2K3w1HkPvoorcMYBgD/+U+wbl54jLJgSqSMDB3q7Oc3rogboPj119jtcCfyBRe4ZXiiYRpEhWE3jHQJxwcJu3WvXz/3shiVF1MiUUTNOL/pJhCh/cVHogijDn8ZgPfYP7HsiBHQvn2w/eCDrk/A90ZXCn/+GawffHBsnv/Hj+piqFkTrrwypUMY1RxVePjhYLtOHeja1a2bjyujTKhqlfr16tVLy8Whh7pu4Pfei00PuodjftswPzatuLh8x1fVwYNdVccdp7psWWz1gwYlitOkSey2YZTGiy8Gz8trr7m0devc82ZUT4CJmsY7N68tERE5WERmiMhsEbkmIr+uiLzg5X8lIh2zKw+Bx90UTZ4WEhckIQOfcX7Aw/feg2bNYlsdr7wSe4gHHoB7743dv18/eO45Vy5+orxhQOzj3aiRW9ar5543wygLeVMiIlITeAgYCHQFThCRrnHFzgCWq+p2wH+BCKdUWaJRI1591b2I68qGpMVi9PANN5T7sMuXB+tz55Ze/vzz4bDDYtPGjw+sbd55JzZv/XpnkfPBB07RGNWTcGDNvfbKnxxG5SefLZHdgdmqOkdVNwLPA0fElTkCGOWtvwwMEMlOj+3yZYoSqvrMMxk0CFqwmKc5qcR9R3GyW7n55qRlrr0WXn21ZBkWLoz1Ct+wYbCeLChQrVrOs+rGjfDmm4n5Z5wRzC+ZP98Nmj7/vAt3EjbrNKoX4RZqvXr5k8Oo/ORTibQBwt/aBV5aZBlV3QSsBLaMr0hEzhaRiSIycbE/Y7yMFE6dmZB2AO/yJX05jpdi0v/oeyQAh+He2qcyij67a9LoPEVFcOedMGhQ8uM3bhxrIRNP/fqxA+7x1K6dPET7zz+7sf428VcXWLYseZ1G1eXpp91y113zK4dR+cmnEolqUcTbHKVSBlUdrqq9VbV3y5Yt0xJmq347JKS9y0Fsy5zN239hBvvxAetHv8rQq+cxBtePVKOGaxHUr584BrFqVayhVnz3ks/q1bHbYfNLnwYNgvVx4xJNfv2+7XiuuSb5ZPn4UO9G1Sf8bJkzaaO85FOJFADtQtttgfiZEJvLiEgtoAmQvW/n999n8Z2PcyYjIrNn8Rc+Yj86dhIuurM1W2wB99zjWgGff+7GG1q3jt2nSZPYl/0hh6QmSrJOu7VrYdEiF4okPBke4Pffo/cZPz75ceLrMKo2mzYFhhi1aiX/8DCMVMmnEvkG2F5EOolIHWAwEN+r/ybgRzQ4BvjQM0XLDgMG0PyK03mcM1IqvmoVXHEFbEg+7l4qUYPn772XvHz9+pCssTVggFsOHOgsupIplfDcE+sPr17ceCNcfbVbHzWq5LKGkQp5UyLeGMeFwDjgJ+BFVZ0qIreKyOFesceBLUVkNnAZkGAGnGlq1oTttsvM2H1UlxQ4FyWFhW6gvX17eOih2Pz9I+YvpkKTJk55jB3rtuNDtz/yCEyc6PJ9NxcrV6Z3LKNyEnZvsuee+ZPDqDpINj/s80Hv3r114sSJ5arjl1+gaedmNGNFTLp4wzHxl+yUU2LDiIZDjIZp2zZw737mmTBhAvz0U5DesaM7dqbQuLH+eLm33NJ5bI1XYkbVZM2aWOOLJUvcM2AYACIySVV7l3U/c3sSQadO0L/ulzzCuS5h993pzo9A9PhC/LhCjRqJYUSefRYuvzzYfuyxIJiUr1jefTcDwocIK7GoAf1GjdyLJMysWW6/KBfhRuVl0yb43/9i05o0yY8sRtXCWiJJ+PhjNwD5xhuui2vDBvdr3Dix7M8/Q+/eLj88iSuMH4uqJG+7S5cmdkGVF1+R/PQT7LhjdF74EfjHP5wL+ksvdREVjcrFhg1unGuvvWIdTdev7ww/fKrY397IAOm2RCzGehL6948ND1K3rvtFse22bqb5zz/Ddtsl5vtu3c87r2Qlko0vw8WLXQsnXoFA0I32yy9OeTVpEsQwKSm6r1Fx8Q0lxo+HadNcl+mKFbEK5OKL8yObUTUxJZJBwvNBwvjha485po970sMAAA/VSURBVOT9a9bMrDwALVokn5m+ww5OiXTu7Lbt67Rq0a1bdPqQIbmVw6ja2JhIBqldOzH64CWXBOsizrXJ9OlB2ltvuVaCb1GVSz74IHb7q6+C9fjJj0bVIRsfK0b1xZRIhunZ040/+Fx/fWz+Vlu5FoBvJdOqlSsfDsWeK045JXbbbzFB8njuRsXGJg8aucaUSBbYcUdnrtu3r+tOisL/0s9nFLmSfHnZoHrlQ9X5V4v36uyzfr11WRqZx5RIlvjll+g46T577+2WHTrkRp4o/vY3OOCA5PnlmYlv5JZ165yLf1UXTyaeVauSG4YYRnkwJZInxo2D2bNj3b3nGhFnuTVlSpBWu3awfvfduZcp21x0kbOg27gx35KkjmrgHscPaztvnnNrc8UVzkv0hRfCCy+4MtOnO19uAEcd5Tz2JvPwbBjlxeaJGJtZtsy9pL75JhijqeyPx7vvum7Fxo2dh+Wwg8yioqTe+ysUUc44993XdV19/TXcfjv8619B3tix7v5t2OBip1vMdCMV0p0nYkrEiCRqImJlw1ca3bvDjz+62fjh+S9jxsChh+ZPvlSJUgIiye+N73bHMMqCuT0xMkrTpm55TdZdXmaGsK8yVRg+PHA943fXxTubzKdRQ2k8/XS0/zWfsAJp0cJNeO3b11nVmQIxcokpESOSFZ7vybvucl+2jzwCDzwQRMSrqIjAbbfBOefAK68E6fvvH/go8wm7BalonHxy6mWXLHHeErp1c6GSDSOXWHeWEUn4a/aCC2I9/Va0R+aHH9z8nNJo0sS1Rq65xoUrBqdYosIGZ4u333bxYv7735JbDMny4sc/4qlo98aoPFh3lpFR3gyFB6voruJTUSAQdGeFQwWHu7jWrUtutVVU5CaF3nabe8G//nrZ5fziC2dW/cADsa2kVCgqcjFo4ievhjGvvEY+MCViRHLYYS7mSRSqLjZ3ZTKTDdOpU+DHLOxdoEED543ZZ/FiZxo7ZYq7Fl27usiA4ExnS0PVvfh9/vrXYP3ss4NuKJHkJrgPPQRPPOGsyGp5nu7C0TDnzQvWw+diGLnClIiRlB9+iN32u1geesh5OE5l8tqvv7qJbrkgviunQYPkYzg33+yW/kv+22/d8scf3XlecIFzUbNmDey0Ezz5ZOz+TZsmTsb88stYGWrUCExsX3wxtuzy5S7MsR/vfM0aOPdcmDw5ttz558Opp8amtW3rwgZ8/32syfI220Sfq2FkFVWtUr9evXqpkRmef17VvRZVO3RQHTky2PZ/JVFc7Mp07lx+WVauVC0sTEz/4INYWcKybdzofpdeGqT5j8evv7rtxx+PraOsv9GjVTdtUh0xIkjr3Vv122/Tr9P/nX56atdmyBDV/fcv/zU2qjfARE3jnZuXloiINBeR90RklrdslqRckYh87/3ejCpjZI+DDw7Wv/kmOmBWsjjy4OZhAMyZk74M69a5weQmTWLl8XnrrdjtP/6ABx+El192s+9r13Z+wPzxhHHjXDnfU8Cff8L776cv39//7rqZzjorSJs4EXbdNf06fUaOTK3cZZe5wXrDyAf5iidyDfCBqt4pItd421dHlFunqikOmxqZpkmT2O6ZqHjcNWsmtwgK9/Nv2hT06afK8uXOcsqPFum7rv/9d+dz7LzzgvC+J5zglltv7bqi4qlVyykjn7ASiQrYFcVTT7kusjFjEru3ysK6dRV7jophlIV8jYkcAYzy1kcBR+ZJDqMMRCkRcH3+UWGBw5ZPtWsnjrGUxMaNruUTX29BQTC+8Mgj8NJLbr2sfr7q1XNyr1kDV17p0i69FE4/Pfk+Bx7oPB8/8UQQyKs0Cgpig4J17eqO7c/D8QlHHvSp6FZxhgH5UyJbq+oCAG+5VZJy9URkooh8KSJJFY2InO2Vm7h48eJsyGtQssfhBg2cCevEiXDSSc6CaNmy2DI9ewaOAUsj2aB9u3bBSz9M27ap1evjuw254w5YtMilnXSS67bz6dfP+abyCbv1//lnuDqu7dy0aVCXT5s2zjmijz97Pt4ct27dYHAfYOZMN6huGBWedAZSUvkB7wNTIn5HACviyi5PUkdrb9kZ+BXYtrTj2sB6dvEHfdeuVT3yyNiB4H//O3b7kkuiB4zLcpxUf+U5l3Ado0aVrc5+/YLyDRu6tDVrVLfd1hki+Lz8suqECbH7FhSo7rOP6vr1Qdp336kuWJDe+RhGeSDNgfW8zFgXkRlAf1VdICKtgI9VdYdS9nkSGKOqL5dUzmasZxe/hdGjh/tK79MnyGvRIhijKIm1a0sfEwjP2D7uOJg0yX39JyOdxzh+VrivDnzPvqnW2axZ0D2Vh7+TYWSEyjZj/U3AD856CvBGfAERaSYidb31FsCewLScSWhE0q6dUyAAu+8e292TigKBxG6ueMIv4pkzXZyM2bOTl1+6NLXjloRveSbirJ2OPz71fcsy1mMYVY18KZE7gQNEZBZwgLeNiPQWkce8Ml2AiSLyA/ARcKeqmhKpYCQbbA8zeLBbDhvmlsuXl1zeH0w/4wzYfvvoMkVFwXq6ccXDL/+wAhwyBJ5/PvV62rWDVq2CoFCGUZ3Ii4mvqi4FBkSkTwTO9NY/B3bKsWhGGYlSIp06ufDAPo89BlddFbRA3n/fxfhIxpo1bhnvE2vNGqcwunVzXU4zZrgXd5066ckeHowvj/t0EZg/P/39DaMyY25PjHLRuLHr/glbGx1+uAvRCs6RY8OGsMsugXXTpZcGZVesgAED4A2vQ1M1cIMeH/+jYUP47bcgdv1f/gI33JC+7Om2YAzDCMjXZEOjiiDiun9++QVee82lDRkSPQmxQYPY7T//dIPSAB9+6Mq/+24wq/zHHxOP17595mSvUwf+8Q848cTM1WkY1Q1TIkZG8GeAn3eeUyBRbLedWzZq5AbKw/MiAIYOhYsvDrbLOoEwHZ55JvvHMIyqjAWlMjJC8+ZuwLxOnUTvtmFOP925DwkPjCejij2ahlGhqWwmvkYVw7e4Ku3F//77qSmQ+FC2hmFUTEyJGBlh+HC3nDSp5HKnnZZafbkMWWsYRvqYEjEywllnuVbITqUYZR99dG7kMQwjN5gSMXJKly6JaZs2BesLFgTzRAzDqPiYdZaRU8ITA+vUgeuuc9Zc48Y5c2EL8WoYlQtTIkbOefxx2HZb2GefIO3AA/Mnj2EY6WNKxMg5JQV+MgyjcmFjIoZhGEbamBIxDMMw0saUiGEYhpE2pkQMwzCMtDElYhiGYaSNKRHDMAwjbUyJGIZhGGljSsQwDMNImyoXT0REFgO/laOKFsCSDIlTWahu51zdzhfsnKsL5TnnDqrasqw7VTklUl5EZGI6gVkqM9XtnKvb+YKdc3UhH+ds3VmGYRhG2pgSMQzDMNLGlEgiw/MtQB6obudc3c4X7JyrCzk/ZxsTMQzDMNLGWiKGYRhG2pgSMQzDMNLGlIiHiBwsIjNEZLaIXJNvecqDiLQTkY9E5CcRmSoiF3vpzUXkPRGZ5S2beekiIkO9c58sIruG6jrFKz9LRE7J1zmlgojUFJHvRGSMt91JRL7yZH9BROp46XW97dlefsdQHdd66TNE5KD8nElqiEhTEXlZRKZ793qPanCPL/We6Ski8pyI1Ktq91lERorIIhGZEkrL2H0VkV4i8qO3z1ARkXIJrKrV/gfUBH4GOgN1gB+ArvmWqxzn0wrY1VvfApgJdAXuBq7x0q8B7vLWDwHeAQToC3zlpTcH5njLZt56s3yfXwnnfRkwGhjjbb8IDPbWhwHneevnA8O89cHAC956V+/e1wU6ec9EzXyfVwnnOwo401uvAzStyvcYaAP8AtQP3d9Tq9p9BvYGdgWmhNIydl+Br4E9vH3eAQaWS958X7CK8PMu6LjQ9rXAtfmWK4Pn9wZwADADaOWltQJmeOuPAieEys/w8k8AHg2lx5SrSD+gLfABsB8wxvuDLAFqxd9jYBywh7deyysn8fc9XK6i/YDG3gtV4tKr8j1uA8z1Xoy1vPt8UFW8z0DHOCWSkfvq5U0PpceUS+dn3VkO/+H0KfDSKj1eE34X4Ctga1VdAOAtt/KKJTv/ynRd7geuAoq97S2BFaq6ydsOy775vLz8lV75ynS+nYHFwBNeF95jItKQKnyPVXUecC/wO7AAd98mUbXvs0+m7msbbz0+PW1MiTii+gQrve2ziDQCXgEuUdVVJRWNSNMS0isUIvI3YJGqTgonRxTVUvIqxfl61MJ1eTyiqrsAf+K6OZJR6c/ZGwc4AtcF1RpoCAyMKFqV7nNplPUcM37upkQcBUC70HZbYH6eZMkIIlIbp0CeVdVXveSFItLKy28FLPLSk51/ZbkuewKHi8ivwPO4Lq37gaYiUssrE5Z983l5+U2AZVSe8wUna4GqfuVtv4xTKlX1HgPsD/yiqotVtRB4FfgrVfs++2TqvhZ46/HpaWNKxPENsL1n5VEHNwj3Zp5lShvP2uJx4CfV/2/v/kGruuIAjn9/pZDSzYKLOLQRkULQQEMJpUOh4ODkoFAU/wYXoZuCxalDoVDaqXZwdehgB1EXB+skLh38T9QEQZcuFSQgBMGfwzmvvWRJc97Le/Hx/cAl7517cznnniS/nHt/75z8pbPrMtDL0jhCeVbSKz9cMz1mgZd1yHwN2B0Rm+p/gbtr2YaSmd9l5tbM/JjSd39m5kHgBrCvHrayvb3rsK8en7X8m5rV8wmwnfIQcsPJzL+B5xGxoxZ9DTxkTPu4egbMRsSH9We81+ax7eeOgfRr3bcUEbP1Gh7unKvNqB8gbZSNkuXwmJKpcXbU9emzLV9Shqh3gdt120O5H3wdeFK/flSPD+Bcbfs9YKZzruPAQt2Ojbpt/6PtX/FfdtYk5Y/DAnARmKjlH9T3C3X/ZOf7z9br8Ig+s1aG0NZp4K/az5coWThj3cfA98A8cB+4QMmwGqt+Bn6nPPN5TRk5zA2yX4GZev0WgV9ZkZyx1s1pTyRJzbydJUlqZhCRJDUziEiSmhlEJEnNDCKSpGYGEalPdTbdk/X1loj4Y9R1kobFFF+pT3V+squZOTXiqkhD9/7qh0haxY/Atoi4Tfkw2KeZORURR4G9lKUGpoCfKVO2HwKWgT2Z+SIitlE+MLYZeAWcyMz54TdDWjtvZ0n9OwMsZuY0cHrFvingAPA58APwKsuEibcoU04AnAe+zczPgFPAb0OptTQAjkSk9XUjM5co8xW9BK7U8nvAzjrT8hfAxc4CcxPDr6bUxiAira/lzus3nfdvKL9/71HWw5gedsWkQfB2ltS/JcoyxGuWZZ2XpxGxH/5dM3vXICsnrSeDiNSnzPwHuBkR94GfGk5xEJiLiDvAA8rCS9I7wRRfSVIzRyKSpGYGEUlSM4OIJKmZQUSS1MwgIklqZhCRJDUziEiSmr0F/6FwjY4uk+QAAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "N = 10000 # length of a walk\n", "N_walks = 5000 # number of walks for averaging\n", "xD = np.zeros((N_walks,N+1)) # the x coordinates of the N_walks walks of length N, DSDT walk\n", "yD = np.zeros((N_walks,N+1)) # the y coordinates of the N_walks walks of length N, DSDT walk\n", "xC = np.zeros((N_walks,N+1)) # the x coordinates of the N_walks walks of length N, CSDT walk\n", "yC = np.zeros((N_walks,N+1)) # the y coordinates of the N_walks walks of length N, CSDT walk\n", "\n", "for i in np.arange(N_walks):\n", " xD[i,:], yD[i,:] = DSDT_2d_walk(N) # generate a DSDT walk\n", " xC[i,:], yC[i,:] = CSDT_2d_walk(N) # generate a DSDT walk\n", "\n", "# mean x and y positions of both types of walks\n", "mean_xD = np.mean(xD, 0)\n", "mean_yD = np.mean(yD, 0)\n", "mean_xC = np.mean(xC, 0)\n", "mean_yC = np.mean(yC, 0)\n", "\n", "plt.plot(mean_xD, 'b-')\n", "plt.plot(mean_xC, 'r-')\n", "plt.title('Mean x displacements for different walks of length ' + str(N))\n", "plt.xlabel('time')\n", "plt.ylabel('x coordinate')\n", "plt.legend(('DSDT walk', 'CSDT walk'))\n", "plt.show()\n", "\n", "plt.plot(mean_yD, 'b-')\n", "plt.plot(mean_yC, 'r-')\n", "plt.title('Mean y displacements for different walks of length ' + str(N))\n", "plt.xlabel('time')\n", "plt.ylabel('y coordinate')\n", "plt.legend(('DSDT walk', 'CSDT walk'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that, in the example above, we make $N=10^4$ steps, and the mean $x$ and $y$ displacements for 5000 random walks are on the order of about only 1 even at the very end of the time interval. This is relatively very small, and probably consistent with the displacement actually being zero, on average.\n", "\n", ">### Your turn\n", "Run the cells above with larger $N$ and convince yourself that the mean $x$ and $y$ displacements decrease as $N\\to\\infty$.\n", "\n", "So, both CS and DS walks result in the zero mean displacement. Let's now look at what's known as the *mean squared displacement* (which, for the unbiased walks like we are dealing with is equivalent to the walk variance), ${\\rm MSD}= \\left$, where $\\left<\\dots\\right>$ denotes averaging over realizations. " ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "mean_R2D = np.mean(xD**2 + yD**2, 0) # MSD for DS walk\n", "mean_R2C = np.mean(xC**2 + yC**2, 0) # MSD for CS walk \n", "\n", "plt.plot(mean_R2D, 'b-')\n", "plt.plot(mean_R2C, 'r-')\n", "plt.plot(np.arange(N+1), 'g:')\n", "plt.title('Mean squared displacements for different walks of length ' + str(N))\n", "plt.xlabel('time')\n", "plt.ylabel('MSD')\n", "plt.legend(('DSDT walk', 'CSDT walk', 'MSD=time'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Both of the walks have the same MSDs! And both of the MSDs grow linearly with time. In fact, this is the crucial property of random walks: mean squared displacement grows linearly with time. Irrespective of the type of the walk and the distributions of individual steps (provided those have finite variances), we will always get this scaling. \n", "\n", ">### Verification\n", "In fact, the linear scaling of the walk variance (MSD) can be used for verification of computational models of random walks. Let's denote by $\\sigma^2_1$ the variance of a single step of the walk. Then the variance after $T$ steps is $\\sigma^2_T= T\\sigma^2_1$. If the computational model gives a different result, it is wrong.\n", "\n", ">### Your turn\n", "Derive the factor of $(3/2)^{1/2}$ used in the generation of the CSDT random walk above.\n", "\n", "Processes that have the linear scaling of the MSD with time are called *diffusive* processes. They include different random walks, Brownian motion, diffusion per se, and many others. One way to think about diffusion is a limit of a CTCS random walk with finite variances of durations and lengths of jumps, when the mean duration and length go to zero, and the number of steps goes to infinite in such a proportion that the MSD stays linear with time, $${\\rm MSD}=2dDT,$$ where $d$ is the dimensionality of the process, $T$ is its overall duration, and $D$ is what's called the *diffusion coefficient*, defined by the expression above.\n", "\n", ">### Your turn\n", "Generate either a CSDT random walk where the distribution of step sizes is exponential, or the DSCT walk where distribution of time steps is exponential and show that the same linear scaling of the MSD is observed. \n", "\n", "## Central Limit Theorem\n", "Now we are well positioned to answer the original question we started this lecture from: why are distributions of experimental errors Gaussian? Well, errors usually are contributed to by many independent processes that describe irreproducibility of steps taken by the experimentalist and irreproducibility of the conditions that the system experimented on is in. None of the processes dominates the others, and so each produces small contributions, which collectively result in a big effect. Random walks model this: there are many individual steps, each of which is small, but collectively they result in a big displacement. So what is the distribution of displacements in a random walk? To study this, we plot the distribution of $x$ positionsof our CS and DS walkers." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.hist([xD[:,-1],xC[:,-1]],nbins)\n", "plt.xlabel('x position')\n", "plt.ylabel('frequency')\n", "plt.title('Distribution of x displacements')\n", "plt.legend(('DSDT walk', 'CSDT walk'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Shockingly, both of the distributions are indistinguishable Gaussians! This is a demonstration of what is known as a *Central Limit Theorem*, arguably one of the most important theorems in probability theory: a sum of many independent random variables is itself a random variable that tends to be normally distributed with the variance equal to the sum of variances of contributing variables (provided these variances are all finite). It is precisely because of the Central LImit Theorem why we also call Gaussian distributions normal.\n", "\n", ">### Your turn\n", "Show that the CLT property holds for the walks you defined in the previous Your Turn exercise.\n", "\n", ">### Your turn\n", "Verify that the variance of the $x$ displacement of the walk is, indeed, the sum of variances of the individual displacements (for this, you will need to calculate the latter variance analytically).\n", "\n", "Now we can explain why the error of MC methods scales as $\\sim 1/\\sqrt{N}$. If we are calculating a quantity numerically that is **TO BE CONTINUED**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Projects\n", "\n", "There are three projects available for you to choose from in this module. We will first introduce them, and then we will introduce a few more theoretical ideas and modeling topics that are needed to properly address them\n", "\n", ">### Project 1: \n", "**Diffusion-limited aggregation (physics)** Read about diffusion limited aggregation (DLA) __[here](http://paulbourke.net/fractals/dla/)__. Implement DLA as follows. Start with a large square lattice (at least $100\\times 100$, but generally larger; make this a changeable parameter in the model). Make the central point of the lattice the beginning of the DLA cluster (a stationary dot). Release a walking dot on a discretized circle slightly larger than the size of the stationary cluster. Let it perform a random walk, and at each step check if the walking dot comes in contact with a cluster. If it does, the walking dot becomes stationary itself and joins the cluster. If the dot tries to exit the circle, from which it was released, do not let it. Once the dot becomes stationary, repeat the cycle by releasing the next walking dot. Keep repeating this until the cluster consists of a larger number of stationary dots (e.g., 1000 -- again, make this a changeable parameter. At every attachment event, record the state of the lattice (which points have a stationary dot, and which don't), and combine these frames into a movie, as shown in Chapter 8.2 of the book. In your report, instead of a movie, show a few time-lapsed images of the cluster. Finally, measure the maximum horizontal extent of the cluster at every frame and plot it as a function of time. How does the size grow with time? Is this a power law? Which one? The DLA cluster models the deposition of molecules on surfaces -- think of snowflakes, but also growth of bacteria in a bacterial film, where the movable dot is now a food particle, and the stationary dots are bacteria that divide when the food gets eaten.\n", "\n", ">### Project 2:\n", "**Annihilation in different dimensions (chemistry)** A chemical annihilation reaction, $A+A\\to0$, where $A$ denotes a particle, is one of the simplest reactions one can think of. But it also illustrates how properties of random walks depend strongly on the dimension of the system. First, write down the chemical kinetics equation for the system and solve it analytically. At $t\\to\\infty$, how does the concentration of A decrease? Now let's simulate this problem. Initiate walkers on a lattice of a linear size $L$ in 1, 2, and 3 dimensions. In a cellular automaton simulation, this would mean representing a lattice as an an array of size $L$, $L\\times L$, and $L\\times L\\times L$. In an agent-based simulation, this would mean dropping the agents at random over some part of the space with the linear scale of $\\sim L$. Play with different values of $L$, as long as it is much larger than 1. Release $N$ random walkers onto this lattice at random points (again, explore different *even* values of $N$). As the walkers do their 1, 2, and 3 dimensional walks, you should check if any pair of them ends on top of each other. When this happens, the two walkers annihilate each other and disappear. As time increases, measure the number of remaining walkers (you will need to average them over many repeats when this number is of order 1 or less). Does the number of walkers in different dimensions decrease with time as a power law? What is the power? Does the power depend on the dimension? Does any dimension match what we would expect from the analytical, deterministic solution? This model is used in various material science contexts, but also in population biology scenarios.\n", "\n", ">### Project 3:\n", "**Simulating a bistable gene (biochemistry)** Most cells in our bodies have the same DNA, and yet they have very different phenotypic properties. This is because different genes are active or inactive (express corresponding proteins or not) in different cells. Often this is achieved in cells by positive autoregulation: an expressed protein activates expression of additional copies of itself, so that if a large number of proteins is expressed, then they produce even more of themselves, resulting in a fully active gene. On the contrary, if originally only a small number of proteins is expressed, then their effect is insufficient to produce many more, and the gene is inactive. We model such dynamics as a combined effect of two processes. First, existing proteins degrade with the propensity $rn$, where $r$ is the rate and $n$ is the number of protein molecules. Second, existing proteins activate production of new proteins with the propensity $A_0 + \\frac{A_1n^2}{1+(n/n_0)^2}$, where $A_0$ is the basal expression rate, $A_1$ is the expression amplitude, and $n_0$ is the half-maximal expression protein concentration. Simulate this system by simulating production or degradation of individual molecules, or by tracking dynamics of the probability distribution of having a certain number of molecules expressed. Explore how the dynamics of expression depends on time and on the initial conditions. Find parameter values where, depending on the initial condition, there are two stable states: inactive, with $n\\approx 10$ and active with $n\\approx 25$. Does the system stay in either of the stable states forever? If not, quantify statistics of transitions between the states.\n", "\n", "How do we simulate such complex probabilistic systems? A key question is: what should be the dynamical degrees of freedom that we should use to represent the system? This is not a simple question, and it requires a serious discussion.\n", "\n", "### Cellular Automata vs Agent-based simulations\n", "The same systems (usually stochastic, but often also deterministic) can sometimes be simulated in multiple different ways. Consider, for example, Project 2 for this module: the annihilation reaction, where two diffusing individuals of the same species annihilate each other when they meet. We can model this system by choosing random positions of $N$ individuals on the $L\\times L$ lattice. We would then run a loop over time steps, and in each of the time steps we would loop over the individuals. For each individual, we would first generate a step in a random direction, and then we would check if the individual is at the same point on the lattice as some other. If the latter is true, both individuals would then be removed from the future consideration. \n", "\n", "This is called an *agent-based* simulation. To specify such a simulation, one needs to specify the number of agents, their state (the coordinates on the lattice in our case), and the rules, by which the agents evolve with time. The agents are the primary units, the degrees of freedom of the system. The behavioral rules can include the rules for motion, the rules for creation, destruction, for interaction with other agents, and so on. The state of each agent may be much more complex than just the position, and it can include such attributes as reactivity, age, or whether the agent is mobile or not (which is relevant to Project 1, growth of the DLA cluster). The rules and states do not necessarily have to put the agents on the lattice; instead, they can move through the real space. The rules can be deterministic or probabilistic, or a mixture of both. All of these combinations are possible. Such agent-based simulations are used routinely in many different fields, and are a staple of modern science. We can track motion and interactions of individual molecules, including their creation and destruction in chemical reactions. We can track position of individual animals or humans in behavioral or sociological experiments. \n", "\n", "An alternative way of simulating the same annihilation system is to take not the individual, but rather the space itself (always represented as a lattice on a computer) as the main variable. The lattice consists of many cells, and each cell has a certain number of individuals in it. The simulations proceed by, again, looping over time. However, now in each time step, we loop over the lattice sites, the cells, and not the individuals themselves. Each cell has rules for its change based on its own state, as well as the state of the surrounding cells, with which it interacts. In our annihilation problem, the rules could be the following. First if there are more than two individuals in a cells, then the number of them is decreased by two (two are annihilated). And then a random number of individuals from each cell (distributed according to some probability law) is removed from a cell and added to one of the neighboring cells at random. It's pretty clear that one can match the rules, by which we model the cells, to the rules, by which agents are modeled, so that the results of simulations are the same.\n", "\n", "This type of simulations is called *cellular automaton* simulation (that is, there is an automaton that obeys a certain rules in every cell). To specify such simulation, we need to specific the number and the geometric arrangement of the cells, as well as the rules by which the cells evolve in time. As in agent-based approaches, the rules can be deterministic or probabilistic, and the state of the lattice can be either very simple (e.g., occupied or not) or very complex (how many individuals and of which type are in a cell). Cells may be real physical cells, or they can be discretized bins of the continuous space, for example exchanging molecules. These type of approaches are also used routinely in applications. For example, we can predict forest fires by viewing a forest as consisting of lattice sites that have states such as *on fire*, *combustible*, or *barren*, and with rules that define how fires emerge and how they spread to new combustible cells. Spread of epidemics can also be models by cellular automata. And even diffusion of molecules through space can be viewed as cells with rules for exchanging molecules.\n", "\n", "Cellular automata and agent-based simulations are not exclusive. Some problems may require the use of both, as we will illustrate below using the example of the DLA cluster growth. \n", "\n", "### Which representation to use?\n", "If both methods can be used often to achieve the same results for the same systems, which one ''should'' be used? Both methods can be made equally accurate, and both are similarly easy (or hard) to code. So often the choice is left to the scientist, and either choice is reasonable. However, sometimes there is a big distinction: the execution time. And here common sense rules. In both methods, the main loop is over time. But then the first loops over the agents, and the second over the cells. It is clear that the number of operations per each time step that would need to be performed in each cycle of an agent-based simulation would scale at least as $N$, the number of agents, and may require even more time if many pairs of agents interact, e.g., $N^2$. (The time cost of a computation is called the *time complexity* or the *computational complexity*.) Let's generalize this and denote the scaling of the time in the agent-based formulation as $N^a$. In contrast, for cellular automata, this internal cycle would scale with the number of cells, $\\propto L^d$ in a $d$-dimensional system. If the agents and the cells have similarly complex rules, than the best type of simulations is determined by whether $N^a$ is larger than $L^d$ or not -- the smaller of the two should be chosen. For example, if we have only a handful of individuals diffusing on a big lattice and they only rarely interact with each other, then agent-based approaches are the way to go. If, on the other hand, each cell, on average, has $\\gg1$ individuals, then we can move many of them from one cell to another in just a few computer commands, rather than a few commands per individual -- and cellular automata simulations will be faster in this case.\n", "\n", "In my experience reviewing papers for scientific journals, the choice of the representation between cellular automata and agents is probably one of the most common and the most costly mistakes an computational scientist can (and do) make. Choose wrong, and your code will spend idle cycles moving millions of individuals one at a time, or analyzing cells where nothing at all happens." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Analysis for Project 1: DLA\n", "The most important thing we should do is to decide how to represent the data. Let's consider the costs of different choices.\n", "\n", "*Cellular automaton*. There's only one walker at any given time. Thus nothing will be happening over most of the lattice, and running a cellular automaton would be unnecessarily costly. \n", "\n", "*Agent-based simulation*. From the first glance, this seems like a way to go. Since only one walker is moving, let's represent it as an agent. However, at every step of the walker, we need to check if it is touching the DLA cluster. Thus if the cluster is represented by agents that are inactive, and the walker is an active agent, then every step of the walker will require checking if it is touching any of the previous agents. This will require $O(N^2)$ steps, where the first power in $N^2$ comes from having to launch $N$ agents, and the second from having to check every other agent every time. This will be costly too.\n", "\n", "*Mixed simulation*. The best solution here is to represent the walker as an agent, but the cluster as a cellualr automaton, where, say, a 0 on the lattice at a certain point means that it does not belong to the cluster, while 1 sayd that it does. Then to decide whether an agent is touching the cluster, one only needs to check the state of the cell under it, which does not scale with $N$. Thus the time complexity of the problem decreases to $O(N)$ from $O(N^2)$ in this representation.\n", "\n", "The code below implements the simplest DLA cluster using this representation. You should feel free to use this in your solution for Project 1. However, this is not the whole solution -- a lot of modifications to the code are still needed to answer the questions posed by the Project." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Attached walker No. 0\n", "Attached walker No. 100\n", "Attached walker No. 200\n", "Attached walker No. 300\n", "Attached walker No. 400\n", "Attached walker No. 500\n", "Attached walker No. 600\n", "Attached walker No. 700\n", "Attached walker No. 800\n", "Attached walker No. 900\n" ] }, { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# initialization block\n", "N_walkers = 1000 # how many walkers in a cluster\n", "L = 100 # lattice size\n", "lattice = np.zeros((L,L)) # the lattice itself\n", "lattice[int(L/2), int(L/2)] = 1 # seeding the DLA cluster\n", "\n", "# The following function starts the walker somewhere on the lattice.\n", "# For now, I coded the function to start a walker on one of the edges of the lattice.\n", "# However, as the accompanying text describes, starting the walker on the edge is not really ok.\n", "# Indeed, this will break the circular symmetry of the cluster. Further, it will take the walker a \n", "# a very long time to find the claster if it starts far away. You will need to re-write this function\n", "# to start the walker at a circle that is slightly larger than the current cluster size. This is why\n", "# we supply the lattice as the argument to this function.\n", "def start_walker(lattice):\n", " L = lattice.shape[0] # getting the lattice size\n", " side = np.floor(4*nprnd.random()) # which side the walker will be releazed on\n", " start_coord = 1+int(np.floor((L-2)*nprnd.random())) # which site on the chosen side the walker \n", " # will be released on. Notice that we don't let the walkers to be released at edges, but only\n", " # with coordinates [1,L-2]. This makes it easier later to check if the walker is approaching an edge,\n", " # which we block.\n", " if (side==0): # left side\n", " x = 1\n", " y = start_coord\n", " elif (side==1): # top\n", " x = start_coord\n", " y = L-2\n", " elif (side==2): # right\n", " x = L-2\n", " y = start_coord \n", " else: # bottom \n", " x = start_coord\n", " y = 1\n", " \n", " return (x,y) # return the walker initial position\n", "\n", "\n", "# The following function runs a walker till it ends up joining the cluster.\n", "# At this point, the function checks if the walker is trying to leave the finite range\n", "# of the lattice. Instead, to speed things up, you should recode this so that\n", "# the walker is not allowed to leave the circle that is slightly larger than the \n", "# current cluster size.\n", "def run_walker(lattice):\n", " L = lattice.shape[0] # getting the lattice size\n", " x, y = start_walker(lattice) # we initiate the walker position\n", " ii=0\n", " # keep running the walker till it hits the cluster and gets stuck\n", " while (True):\n", " # Keep generating plausible steps until an acceptable one (not hitting a wall)\n", " # is generated.\n", " while(True): \n", " # this code is copied from the DSDT_2d_walk() function above\n", " directions = int(np.floor(nprnd.random(1)*4)) # (0,1,2,3) directions of motion on each time step\n", " dx = (directions == 0) - 1.0*(directions == 1) # directions == 0/1 corresponds to right/left motion\n", " dy = (directions == 2) - 1.0*(directions == 3) # directions == 2/3 corresponds to top/down motion\n", " # we now verify that the walker doesn't approach the edge. If it doesn't, we break the loop.\n", " # If it does, we continue pulling random numbers and trying different steps.\n", " if((int(x+dx)>0)&(int(x+dx)0)&(int(y+dy)0): # the walker is near the cluster\n", " lattice[x,y] = 1 # add the walker to the cluster\n", " break\n", " \n", " return (lattice)\n", "\n", "# release N_walkers walkers and build the DLA cluster on the lattice. \n", "for i in np.arange(N_walkers):\n", " lattice = run_walker(lattice)\n", " if (np.round(i/100)==i/100): # signal attachment of every 100th walker\n", " print('Attached walker No. '+str(i)) # we do this to monitor progress\n", "\n", "# now we show the final cluster lattice\n", "plt.imshow(lattice, cmap='Greys', interpolation='nearest')\n", "plt.title('DLA cluster')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">### Your turn:\n", "Use the code above and make the necessary improvements to it, and then use it to answer questions posed in Project 1." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Analysis for the Project 2: annihilation \n", "First let's analyze this system assuming that it obeys the usual mass-action law. Then since annihilation requires *two* particles to be at the same place at the same time, the propensity of the particle number $n$ (or the concentration) decrease will be proportional to $n^2$ and will obey the following ordinary differential equation\n", "$$\\frac{dn}{dt} = -r n^2.$$\n", "This is easy to solve by separating the variables:\n", "$$\\frac{dn}{n^2} = -rdt,$$\n", "$$-\\left.\\frac{1}{n}\\right|_{n_0}^{n(t)} = -\\left.rt\\right|_0^t,$$\n", "$$ \\frac{1}{n}=\\frac{1}{n_0}+rt,$$\n", "$$n(t) = \\frac{n_0}{1+rtn_0},$$\n", "where $n_0$ is the initial particle number. In the limit of long time, \n", "$$n(t)\\to \\frac{1}{rt},$$\n", "which is a slow, power-law decay. The problem formulation asks us to compare (quantitatively!) this decay to results of simulations in different number of dimensions. (As a side note, such comparison will require many simulation runs for averaging, and probably is better done in log-log axes.)\n", "\n", "Now we are in a good position to think how to code this problem. The first question we need to ask, as always, is how to represent the data and what the dynamical variables should be for this project. In the simplest case, if we have the lattice of size $L$, then each time step of the project will have the time complexity $O(L)$. In the agent-based approach, each time step will require looping over all agents, and then, for each agent, we will have to compare its position with all other agents to verify if the pair is annihilating. The resulting time complexity is $O(N^2)$. So it may seem that going with the cellular automaton formulation will be faster for all but the largest lattices. However, the situation is not that simple. First of all, the lattice doesn't have to be finite -- the particles can, in principle, walk very far from where they were released. Second, when doing loops over agents, we do not have to move and compare positions for agents that are inactive. Checking if an agent is active costs almost nothing, ans so if we are looping only over the active agents, the computational costs will scale as $O(N_{\\rm active}N)$, where the factor $N$ is due to checking if an active agent is at the same position as *any other one of $N$ agents*. Guided by the analytical solution above, we venture that most of the time the system will spend in the regime with very few remaining particles, so that $N_{\\rm active}\\sim 1$. Thus the computational complexity of the problem is closer to $O(N)$ rather than $O(N^2)$. At this point, it becomes clear that simulating the system as a set of agents is going to be faster than as a cellular automaton.\n", "\n", "The code below implements the simplest version of the annihilation problem using this representation. You should feel free to use this in your solution for Project 2. However, this is not the whole solution -- a lot of modifications to the code are still needed to answer the questions posed by the Project." ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Time step = 0; No. of active walkers = 198.0.\n", "Time step = 0; No. of active walkers = 196.0.\n", "Time step = 0; No. of active walkers = 194.0.\n", "Time step = 0; No. of active walkers = 192.0.\n", "Time step = 0; No. of active walkers = 190.0.\n", "Time step = 0; No. of active walkers = 188.0.\n", "Time step = 0; No. of active walkers = 186.0.\n", "Time step = 0; No. of active walkers = 184.0.\n", "Time step = 0; No. of active walkers = 182.0.\n", "Time step = 0; No. of active walkers = 180.0.\n", "Time step = 0; No. of active walkers = 178.0.\n", "Time step = 0; No. of active walkers = 176.0.\n", "Time step = 0; No. of active walkers = 174.0.\n", "Time step = 0; No. of active walkers = 172.0.\n", "Time step = 0; No. of active walkers = 170.0.\n", "Time step = 0; No. of active walkers = 168.0.\n", "Time step = 0; No. of active walkers = 166.0.\n", "Time step = 0; No. of active walkers = 164.0.\n", "Time step = 0; No. of active walkers = 162.0.\n", "Time step = 0; No. of active walkers = 160.0.\n", "Time step = 0; No. of active walkers = 158.0.\n", "Time step = 0; No. of active walkers = 156.0.\n", "Time step = 0; No. of active walkers = 154.0.\n", "Time step = 0; No. of active walkers = 152.0.\n", "Time step = 0; No. of active walkers = 150.0.\n", "Time step = 0; No. of active walkers = 148.0.\n", "Time step = 0; No. of active walkers = 146.0.\n", "Time step = 0; No. of active walkers = 144.0.\n", "Time step = 0; No. of active walkers = 142.0.\n", "Time step = 0; No. of active walkers = 140.0.\n", "Time step = 0; No. of active walkers = 138.0.\n", "Time step = 0; No. of active walkers = 136.0.\n", "Time step = 0; No. of active walkers = 134.0.\n", "Time step = 0; No. of active walkers = 132.0.\n", "Time step = 0; No. of active walkers = 130.0.\n", "Time step = 0; No. of active walkers = 128.0.\n", "Time step = 0; No. of active walkers = 126.0.\n", "Time step = 0; No. of active walkers = 124.0.\n", "Time step = 0; No. of active walkers = 122.0.\n", "Time step = 0; No. of active walkers = 120.0.\n", "Time step = 0; No. of active walkers = 118.0.\n", "Time step = 0; No. of active walkers = 116.0.\n", "Time step = 0; No. of active walkers = 114.0.\n", "Time step = 0; No. of active walkers = 112.0.\n", "Time step = 0; No. of active walkers = 110.0.\n", "Time step = 0; No. of active walkers = 108.0.\n", "Time step = 0; No. of active walkers = 106.0.\n", "Time step = 0; No. of active walkers = 104.0.\n", "Time step = 0; No. of active walkers = 102.0.\n", "Time step = 0; No. of active walkers = 100.0.\n", "Time step = 0; No. of active walkers = 98.0.\n", "Time step = 0; No. of active walkers = 96.0.\n", "Time step = 0; No. of active walkers = 94.0.\n", "Time step = 0; No. of active walkers = 92.0.\n", "Time step = 0; No. of active walkers = 90.0.\n", "Time step = 0; No. of active walkers = 88.0.\n", "Time step = 1; No. of active walkers = 86.0.\n", "Time step = 1; No. of active walkers = 84.0.\n", "Time step = 1; No. of active walkers = 82.0.\n", "Time step = 1; No. of active walkers = 80.0.\n", "Time step = 1; No. of active walkers = 78.0.\n", "Time step = 1; No. of active walkers = 76.0.\n", "Time step = 1; No. of active walkers = 74.0.\n", "Time step = 1; No. of active walkers = 72.0.\n", "Time step = 1; No. of active walkers = 70.0.\n", "Time step = 2; No. of active walkers = 68.0.\n", "Time step = 2; No. of active walkers = 66.0.\n", "Time step = 3; No. of active walkers = 64.0.\n", "Time step = 3; No. of active walkers = 62.0.\n", "Time step = 3; No. of active walkers = 60.0.\n", "Time step = 3; No. of active walkers = 58.0.\n", "Time step = 3; No. of active walkers = 56.0.\n", "Time step = 4; No. of active walkers = 54.0.\n", "Time step = 4; No. of active walkers = 52.0.\n", "Time step = 4; No. of active walkers = 50.0.\n", "Time step = 4; No. of active walkers = 48.0.\n", "Time step = 4; No. of active walkers = 46.0.\n", "Time step = 6; No. of active walkers = 44.0.\n", "Time step = 7; No. of active walkers = 42.0.\n", "Time step = 8; No. of active walkers = 40.0.\n", "Time step = 9; No. of active walkers = 38.0.\n", "Time step = 11; No. of active walkers = 36.0.\n", "Time step = 11; No. of active walkers = 34.0.\n", "Time step = 12; No. of active walkers = 32.0.\n", "Time step = 14; No. of active walkers = 30.0.\n", "Time step = 16; No. of active walkers = 28.0.\n", "Time step = 20; No. of active walkers = 26.0.\n", "Time step = 23; No. of active walkers = 24.0.\n", "Time step = 28; No. of active walkers = 22.0.\n", "Time step = 41; No. of active walkers = 20.0.\n", "Time step = 55; No. of active walkers = 18.0.\n", "Time step = 94; No. of active walkers = 16.0.\n", "Time step = 95; No. of active walkers = 14.0.\n", "Time step = 118; No. of active walkers = 12.0.\n", "Time step = 132; No. of active walkers = 10.0.\n", "Time step = 1085; No. of active walkers = 8.0.\n", "Time step = 2692; No. of active walkers = 6.0.\n", "Time step = 3526; No. of active walkers = 4.0.\n", "Time step = 12976; No. of active walkers = 2.0.\n", "Finished execution after 19999 steps; No. of active walkers remaining = 2.0.\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# The following function runs the walkers as agents till either all walkers are annihilated\n", "# or the prescribed number of temporal steps is over. This is only done in 1d. \n", "# To use this for your Project 2 solution, you'll need to extend the code to 2d and 3d.\n", "# This you should do by writing different pieces of code -- do not try to make the same functions\n", "# general enough to deal with all dimensions simultaneously. You may also need to think about\n", "# how to speed up the code, specifically how to make the internal comparison loop (which \n", "# checks if two walkers share the same position) faster. For this, you may want to sort the walkers\n", "# by their activity status, so that one needs to compare the position of the currently analyzed\n", "# walker with the positions of only other active walkers. However, leave this modification \n", "# till the very end -- and first make the code work in different dimensions, and get it to \n", "# answer the questions that the Project poses (that is, average over many runs and output\n", "# the average particle number at a given time)\n", "\n", "N = 2*int(100) # Number of walkers; we need to make sure that the number is even, so that \n", " # the walkers can annigilate each other\n", "W = 100 # the spread along the x axis of where the walkers will be located\n", "T = 20000 # simulation duration\n", "\n", "# initial positions of N walkers are integers pulled out of a Gaussian with std.dev. of W\n", "position = np.round(W*nprnd.randn(N))\n", "activity = np.ones(N) # originally all walkers are active\n", "N_active = N # initial number of active walkers\n", "\n", "N_remaining = N # these will be the output arrays, recording the number of remaining walkers\n", "t_events = 0 # and the times when this number changes\n", "\n", "for i in np.arange(T):\n", " for n1 in np.arange(N):\n", " # if the walker is active\n", " if (activity[n1]>0):\n", " # then move walker\n", " position[n1] = position[n1] + (2*(nprnd.rand(1)>0.5)-1)\n", " # now loop over all other walkers and see if two walkers are on the same site\n", " for n2 in np.arange(N):\n", " # if the second walker is active, is not the same as the first, \n", " # and has the same position as the first\n", " if ((activity[n2]>0)&(position[n1]==position[n2])&(n1!=n2)):\n", " # then deactivate both walkers\n", " activity[n1] = 0\n", " activity[n2] = 0\n", " N_active = np.sum(activity) # how many walkers remain active?\n", " # output progress tracking\n", " print('Time step = '+ str(i)+'; No. of active walkers = ' + str(N_active)+'.')\n", " # append the number of walkers and the time stamp to the output variables\n", " N_remaining = np.hstack((N_remaining, N_active)) \n", " t_events = np.hstack((t_events,i))\n", " break # do not check for more position coincidences if \n", " # one coincidence was found (breaks for n2 loop)\n", "\n", " if (N_active==0): # if no more active walkers\n", " break # stop running the time loop over i; exit the simulation\n", "\n", "print('Finished execution after '+str(i)+' steps; No. of active walkers remaining = ' + str(N_active)+'.')\n", "\n", "plt.loglog(t_events, N_remaining, '-o')\n", "plt.xlabel('time')\n", "plt.ylabel('No. of remaining particles')\n", "plt.title('Dynamics of particle annihilation')\n", "plt.show()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">### Your turn:\n", "Use the code above and make the necessary improvements to it, and then use it to answer questions posed in Project 2." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Analysis for Project 3: biochemical bistability\n", "\n", "Before solving the actual, rather complex, problem in this Project, let's first deal with a simpler, toy problem. \n", "Suppose proteins in a cell (or these could be other types of particles, such as molecules in a simpler chemical reaction, or nuclei of a certain chemical element, or even people with some attribute, such as infected by a disease) are produced at a certain constant rate $r$ randomly. They disappear similarly randomly, at some fixed rate $\\alpha$ per existing particle. This is known as the *birth-death process*. How do we model the number of individuals in the system, or the probabilities of this number? \n", "\n", "If we forgo the stochastic nature of birth and death processes, then the deterministic equation describing the mean number of particles in the system is \n", "$$\\frac{dn}{dt}=\\alpha-rn.$$ How do we generalize this to account for the randomness? There are multiple ways, and we will consider two.\n", "\n", "#### Stochastic Simulations Approach (SSA)\n", "This is a stochastic system, so that every time that we simulate it, the result is likely to be different. The system state is specified by a single number $n$, the number of protein molecules (or other types of particles), which changes with time. There are two types of *reaction* events that can lead to the change: the creation and the destruction of one particle at a time. The creation happens with the propensity of $\\alpha$ per unit time, and the destruction with the propensity $rn$, which is the destruction rate per particle times the current number of particles. The total propensity for *any* reaction to happen is thus $p_{\\rm tot}=\\alpha+rn$. If we are dealing with real chemical reactions in a well-mixed container, then it is known that the distribution of times between two successive reaction events is exponential, with the mean equal to the inverse of the total propensity (for non-chemical systems, the exponential distribution between the events is often a surprisingly good approximation too). This immediately suggests a way of simulating this system. We start at time $t$ with $n$ particles. We generate an exponentially distributed random number $\\Delta t$ with the mean $1/(\\alpha+rn)$, that is, $\\Delta t \\sim {\\cal E}\\left(\\frac{1}{p_{\\rm tot}}\\right)$. This will correspond to the waiting time to the next reaction event happening. We thus increment the time $t\\to t+\\Delta t$. Now we need to decide which of the two reactions will happen at that time? Clearly, the reaction with the higher propensity will have a higher chance to happen. So we generate a binary random number with the probability of 0 given by $\\alpha/(\\alpha+rn)$, and the probability of 1 given by $1-P(0)= rn/(\\alpha+rn)$. If the generated number is 0, it means that the creation event happened, and we change $n\\to n+1$. If the number is 1, then a particle was destroyed, and so we change $n\\to n-1$. We then continue by generating the next reaction event time and the next reaction event type, and so on. \n", "\n", "This method of generating a single random trajectory of a state of the system with time is known as Gillespie or the *stochastic simulation algorithm (SSA)*. One typically generates many equivalent random trajectories and then calculates averages of these many realizations to find means or variances of the number of particles at any time. The code below has an SSA simulation of this simple birth-death system, showing diversity of the random trajectories that can be realized. The algorithm is easy to update for many different particle types -- the state of the system is then described by the set of numbers of particles of each species. It is also easy to account for many types of reactions: the total propensity for all reactions is the sum of all propensities, and the identity of a reaction event is determined by drawing a multinomial random number, with the probability of each outcome proportional to its propensity.\n", "\n", "Importantly for this lecture, SSA is a type of an agent-based simulation. We have a simple agent -- the system -- and the state of the agent, $n$, is changed according to some probabilistic rules. It is easy to extend such algorithms to more complicated scenarios, where the state is given not just by the number, but also by the physical position of the individual particles.\n" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# The following code implements the SSA solution for the simple\n", "# birth-death toy process. To use this code for this Project, you will need to change the\n", "# propensity of birth to be A0 + A1*n**2/(1+(n/n0)**2). You will then need to play with \n", "# parameters of the simulation (A0, A1, n0, and r) until you are able to find parameters that \n", "# create the two resuested stable states. After that you will need to simulate the system\n", "# multiple times and study the statistics of transitions between the stable states, if such\n", "# transitions happen.\n", "#\n", "# You may also need to change the code to execute not for a given number of reaction events,\n", "# but for a certain duration of time.\n", "\n", "N_events = 100 # How many reaction events (birth and death events) to simulate\n", "N_trials = 10 # How many different simulation runs to produce\n", "a = 10.0 # birth rate\n", "r = 1.0 # death rate\n", "\n", "# initializing the output\n", "N_part = np.zeros((N_events, N_trials)) # N_parts[i] will be the number of particles after reaction event i\n", "t_events = np.zeros((N_events, N_trials)) # t_events[i] will be the time of the reaction event i\n", " # by convention, the 0'th event happens at t=0, and we will start with 0 particles\n", "\n", "for j in range(N_trials): # simulate N_trial random trials\n", " for i in range(1, N_events): # with each simulation running for N_events events\n", " prop_birth = a # propenisty of birth\n", " prop_death = N_part[i-1, j]*r # propensity of death is given by r*(# of particles in run j at the\n", " # current, i-1'th, time)\n", " prop_tot = prop_birth + prop_death # total propensity\n", " prob_birth = prop_birth / prop_tot # probability that the next event will be birth\n", " which_event = (nprnd.random(1) < prob_birth) # randomly generate identity of the next event\n", " if which_event: # if birth, \n", " N_part[i, j] = N_part[i-1, j] + 1 # then increase the # of particles\n", " else:\n", " N_part[i, j] = N_part[i-1, j] - 1 # otherwise decrease it \n", "\n", " # now generate the time to the next reaction from the exponential distribution with \n", " # the mean of 1/(total propensity) and increment the time\n", " t_events[i, j] = t_events[i-1, j] + nprnd.exponential(1)*(1/prop_tot)\n", "\n", "plt.plot(t_events, N_part, '-')\n", "plt.xlabel('time')\n", "plt.ylabel('number of particles')\n", "plt.title('Individual trajectories in the birth-death process')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "> Your turn and Verification\n", "To verify the simulation, we can look back to our deterministic chemical kinetics version of this system. Setting $dn/dt$ to zero in the equation above, we get a theoretical prediction for the steady state concentration of the protein is $\\left = \\frac{\\alpha}{r}$. Thus to verify the SSA code, we can calculate the mean of the simulated trajectories at large $t$. Does the mean agree with the theoretical prediction? Remebering our lecture about Random Walks, what should the variance of $n$ be as a function of the mean? Do the simulations agree with this prediction?\n", "\n", ">Your turn\n", "Change the code above to model positive auto-regulation for protein production. Then look for parameters that would achieve the required results. To help you look, consider that the deterministic chemical kinetics version of this system is \n", "$$\\frac{dn}{dt}= A_0 + \\frac{A_1n^2}{1+(n/n_0)^2} - rn.$$\n", "Steady states for this dynamics will be given by $A_0 + \\frac{A_1n^2}{1+(n/n_0)^2} - rn=0$. Sketch the left hand sizde of this algebraic equation to understand under which conditions it will have multiple solutions." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Text Track 2: can be skipped in first reading**\n", "\n", "#### Master Equation and the dynamics of the probability distribution\n", "Another way of studying stochastic chemical kinetics systems relies on our observation that sometimes the same processes can be described as cellular automata or as agent-based processes. We will illustrate this here using the toy birth-death system, but similar analysis can be done for more complicated systems, including the auto-regulating protein. \n", "\n", "While the birth-deatch process is a stochastic system, with different outcomes for every simulation run, there's a very special cellular automaton that allows us to simulate the system *deterministically*. Specifically, let each cell of the cellular automaton represent a different value of $n$, the number of particles in the system. The state of each cell will be denoted by $P(n)$, and it will represent the probability that the system at the time $t$ has exactly $n$ particles. What are the rules by which this cellular automaton evolves?\n", "\n", "These rules for this cellular automaton are defined by what's called the *master equation*. The probability of being in the state $n$ changes with time. There are four ways, four rules, that can change it. First, starting in the state $n$, a new particle may get created, resulting in $n+1$ particles. This has the propensity of $\\alpha$, and leads to a decrease of $P(n)$. Second, starting in the state $n$, a particle may be destroyed with the propensity $rn$, resulting in $n-1$ particles. This also decreases $P(n)$, since the system leaves this state. Since both of these processes can only happen if the system has $n$ particles, it means the total rate of negative change of $P(n)$ is $-(\\alpha+rn)P(n)$. Two other rules increase the probability $P(n)$. Starting with $n-1$ particles, one can create one. This has a positive rate of change of the probability of having $n$ particles, $\\alpha P(n-1)$. Finally, starting with math $n+1$ particles, one can get destroyed. Since this ends in the state $n$, this has a positive effect on $P(n)$ particles, $r(n+1) P(n+1)$. \n", "\n", "Putting this all together, we get: \n", "$$\\frac{dP(n)}{dt}=-P(n)(rn+\\alpha)+P(n+1)r(n+1)+P(n-1)\\alpha,$$\n", "and for $n=0$, this changes to\n", "$$\\frac{dP(0)}{dt}=-P(0)\\alpha+P(n+1)r(n+1)$$\n", "because particles cannot be destroyed there. This is a set of coupled ordinary differential equations that describes the temporal evolution of the set of probabilities $P(0),P(1),P(2),\\dots$. It is collectively known as the *master equation* (even though these are many equations). If we discretize time, and use the usual Euler, RK2, odeint, or any other method of solving these equations, we will transform this system into a deterministic cellular automaton, which solves for probabilities of having $n$ particles at discrete time steps. \n", "\n", "The script below solves the master equation for the evolution of probability in the birth-death system." ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from scipy.integrate import odeint # importing functionf for solving ODEs\n", "from mpl_toolkits.mplot3d import Axes3D # and for suface plotting\n", "from matplotlib import cm\n", "\n", "maxn = 20 # the simulation requires us to choose what will be the maximal n we will track\n", "p = np.zeros(maxn) # initializing p(n)\n", "p[0] = 1 # for consistency with the SSA case above, at time t=0, there is exactly 1 particle,\n", " # so that p[0]=1\n", "t = np.arange(10) # we will simulate for 10 time points, t=0...9\n", "n = np.arange(maxn) # create value of n corresponding to each p[n]\n", "\n", "# this function defines the right hand side of the dP/dt=... equation, as described above\n", "def dpdt(p, t, a, r):\n", " dpdt = 0*p # initializing the derivatives array\n", " maxn = p.size # maximum number of particles read out from the size of p \n", " n = np.arange(maxn) # setting values of n for each cell of p\n", " dpdt[1:maxn] = a*p[0:maxn-1] # increase of p[n] due to creation from n-1\n", " dpdt[0:maxn-1] = dpdt[0:maxn-1] + r*n[1:maxn]*p[1:maxn] # increase of p[n] due to destruction from n+1\n", " dpdt = dpdt - a*p # decrease in p[n] due to creation from n\n", " dpdt = dpdt - r*n*p # decrease in p[n] due to destruction from n\n", " return dpdt\n", "\n", " \n", "pp = odeint(dpdt, p, t, args=(a, r)) # solve for the evolution of probability\n", "\n", "# plotting \n", "T, N = np.meshgrid(t, n)\n", "ax = Axes3D(plt.figure())\n", "ax.plot_surface(T, N, np.transpose(pp),cmap=cm.coolwarm)\n", "plt.title('Evolution of the probability for the birth-death process')\n", "plt.xlabel('time')\n", "plt.ylabel('particle number')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">### Your turn and verification\n", "Compare the mean and the variance of the solved probablity distribution in the cell above with the results of the SSA approach. The two methods must give the same results. Visually, they do: most of the individual trajectories concentrate where the probability is expected to be large. But it's always nice to compare things quantitatively.\n", "\n", ">### Your turn\n", "Change the code above to solve the auto-regulated gene problem isntead of the toy birth-death problem.\n", "\n", ">### You turn\n", "This is the first time we encounter surface plots. Figure out how to change the plot above to make it into a 2-d color intensity plot." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.1" } }, "nbformat": 4, "nbformat_minor": 2 }