{ "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": [ "[0.06310025 0.76796064 0.38083596 0.28222382 0.93839985]\n", "[0.37183791 0.97231506 0.72034409 0.75338188 0.36448396]\n", "[0.32547108 0.22990677 0.87781621 0.3031263 0.4982197 ]\n", "[[0.88162177 0.72201118 0.03806709]\n", " [0.80579008 0.03760795 0.42218703]\n", " [0.79615259 0.54270557 0.98428804]\n", " [0.81345949 0.70230362 0.59468543]\n", " [0.35276941 0.86974698 0.3336642 ]]\n" ] }, { "data": { "image/png": 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\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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\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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\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", "***\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 995 random numbers.\n", "Generated 6 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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\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 0.33084.\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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HXZKI9HE6+iggT696mll/nUVTtAmAjbs2MuuvswA4e8zZAVYGT27axi2rNrK+OcLI4kJuGjOcC4YNCrQmEcmPToWCmZUAM4EjgZLd7e7+rzmqq9ebvWR2MhB2a4o2MXvJ7EBD4clN27jx7bU0xhyAdc0Rbnx7LYCCQaQP6Oz00cPAMOBjwB+BUUB9rorqCzbt2pRVe77csmpjMhB2a4w5t6zaGFBFIpJPnQ2Fw9z968Aud38QOBv4UO7K6v2G9R+WVXu+rG+OZNUuIr1LZ0Nh9ydCrZmNB8qB0TmpqI+4buJ1lBSUpLWVFJRw3cTrAqoobmRxYVbtItK7dDYU7jazgcDXgXnAcuCHOauqDzh7zNnMOmkWw/sPxzCG9x/OrJNmBb6T+aYxwykNWVpbaci4aczwgCoSkXwyd9/3Wt3IpEmTfPHixUGX0avNe+sRWjb8lApqqKWKohHXct7hlwZdloi8D2b2mrtP2td6nRopmNlQM7vPzJ5JPB9nZjPfb5HS/Wzc9BRlm25hINUYzkCqKdt0Cxs3PRV0aSKSB52dPnoAeBYYkXj+D+ALuShIgrXq3duIxdJvvRmLNbLq3dsCqkhE8qmzoVDl7o8BMQB3bwWiOatKAtPUnPnQ047aRaR36Wwo7DKzSsABzOxEoC5nVUlgSooz71DuqF1EepfOhsL1xI86OtTM/gI8BFybs6okMGMOvZFQKP3Wm6FQKWMOvTGgikQkn/Z5mQszCxG/tMXJwAcBA952d53N1AsNHzYdiO9baGreSEnxcMYcemOyXUR6t32GgrvHzOxH7j4ZeDMPNUnA1jz6HK9uO4E6+lPOLkKDnmP4dQoFkb6gs9NHvzOzC8zM9r2q9GQvzb6W57cNoM4OADPq7ACe3zaAl2ZrtlCkL+jspbOvB/oDrWbWRHwKyd19QM4qk0C8vK2YiKVf0iJihby8rZjJAdW0267Xt7Dj2TVEa5spqChmwMdG0/+YIQFXJdK7dCoU3L3MzAYBY0m5dLb0PnX0z6o9X3a9voXa37yDR2IARGubqf3NOwAKBpEu1Nkzmq8kfsns3wKzEv/9Rided4aZvW1mK83sK3tZ75Nm5ma2z1OwJbfK2ZVVe77seHZNMhB280iMHc+uCaYgkV6qs/sUrgOOA95z91OBY4Cavb3AzAqAO4EzgXHAxWY2LsN6ZcB/AK9kUbfkyImDmilsc2BZoUc4cVBzQBXFRWszb7+jdhHZP50NhSZ3bwIws2J3f4v44al7czyw0t1XuXsLMAfIdAjLt4lfcbUpwzLJs8nX/ZQJ/TcxwHeCOwN8JxP6b2LydT8NtK6CiuKs2kVk/3R2R/M6M6sA5gK/N7PtwIZ9vGYksDb1PYATUlcws2OAA939/8ysw7OjzOxq4GqAgw46qJMld3/X3/tT5o+cSH1xP8qaGzh3/RJ+fGWwR/l88/mHeXzw69jQPaMFjxWy5PmH+a+PfDqwugZ8bHTaPgUAKwwx4GOjA6tJpDfq1EjB3T/u7rXuPov4PRXuA87fx8syHb6avE534qS4/wZu6MT273b3Se4+afDgwZ0pudu7/t6f8tjoydSX9Acz6kv689joyVx/b7DfyJ9cfQ8WSp8+slCEJ1ffE1BFcf2PGULFJ8YmRwYFFcVUfGKsdjKLdLHOjhSS3P2PnVx1HXBgyvNRpI8uyoDxwAuJ0x+GAfPM7Dx37/U3TJg/ciKtBend31oQZv7Iifw4oJoAYgXbM6Z5rGB73mtpq3rZz9g5/3FCW6PEKgtoHvkp+h8zK+iyRHqVrEMhC4uAsWZ2CLAemAFcsnuhu9cBVbufm9kLwI19IRAA6ov7ZdWeL6HoQDzcPgBC0YEBVLPHmodnsevWRyloATAKtsbYdeujrAFGf3pWoLUtXbqUhQsXUldXR3l5OdOmTWPChAmB1iSyvzq7ozlrictrX0P8PgwrgMfc/U0z+5aZnZer7fYUZc0NWbXnywWHXIXH0k9e81ghFxxyVUAVxe38+eM8f9RJzPjOT/jIzx5hxnd+wvNHncTOnz8eaF1Lly5l/vz51NXFLxpcV1fH/PnzWbp0aaB1JS19DP57PMyqiP936WNBVyTdXM5CAcDdF7j7B9z9UHf/bqLtG+4+L8O6p/SVUQLAueuXEI62prWFo62cu35JQBXF/ddHPs2nDv4i1joQd7DWgXzq4C8GupMZ4PkxJ3LbpVezuXIwbiE2Vw7mtkuv5vkxJwZa18KFC4lE0vfBRCIRFi5cGFBFKZY+BnP/DerWAh7/79x/UzDIXuU0FKRjP77yWi5c8xJlTbvAnbKmXVy45qXAjz4CqGp+k/KiWsyc8qJaqpqDvw7ivR+fQXNx+uGnzcXF3PvxGQFVFFdXV8erw3fy+KF/4IlDnuTxQ//Aq8N3JkcOgXrmPyHW5mLGsUi8XaQDudynIPvw4yuvbbNTeUpAlexx5zNf4b4t/0cEA4zamHPflv+DZ+Dfz/x+YHVtqajKqj1fFg3bwcpBDeys+DqxgkpC0a1Eap/APLbvF+da47bs2kXQSEHamFOzIBEIe0Qw5tQsCKiiuHBT5tt3dNSeL6srW9gx6LPEwlVgRixcxY5Bn2V1ZUugde1Wt6aUd+YNYcWc4bwzbwh1a0r3/SLp0xQKkqY2lvkbbkft+WL/qIdomxqisXh7gGorpkOozVnVoeJ4e8A2rx3IxkXltDaEAaO1IczGReVsXhvskWTSvWn6SNJUhELUxjxje5AO2rKRyuUNrDisivqSEsqamjhiZQ1btwQ7dx8ryDx91VF7PlUvGcAfJh7HvdNnsGVQJUO2beXKp+Zw6pJFDA24thvuv5/nh45kU0kVw5pq+Mjm9fzos58NuCqomz+fLf99O60bNxIePpwhX/wC5eeeG3RZrH/hGSJ/jFDQWEG0tJbCkwsZecqZOdmWQkHSzKg6K2WfQlwhzoyqswOsCq6oqGHT1lVM3LZntFDgIYZVBHtLjyFNUbaUtv9nNKQpGkA16Z4ffxw/uvTq5A763Uds+SPxs0aDcsP99/PkgUfSVFAEwMbSITx5YAXcf3+gwVA3fz4PPbeM+274AZtLCxjaGGXm737PZyDQYFj/wjPcvLiQvxw7hPp+Icoa+jNl8Xa+yzM5CQZNH0mafz/z+8wccg4VIQOcipAxc8g5ge5kBqjdtYuopU8fRS1G7a5gL+l9zT9aKImmj6xKos41/wh+n8I951+S8Yite86/pINX5MfCoWOSgbBbU0ERC4eOCaiiuPtfeJPbzv0om/qFcTM29Qtz27kf5f4Xgj367quvhXnuqIHU9y+IXxKnfwHPHTWQr76Wm+/0GilIO41bPsSWio9RV1ZBS30tjVv2de3D3NtF5h3KHbXnyzG1MW5e1sSdHyhmc4kxtMn59380c3Rt8EcfVQ8clFV7vmwuyTy666g9X37x0ak0t7n0THNBmF98dCpfCKgmgL9+sD+t4YK0ttZwAX/9YG5ufKVQkDS3PfgItw8fS2th/Jtc3YCB3F7aHx58hBsvvzSwuvpTmDEA+lOYYe38eX3Q20yJbuID/hStvpWwV1Ll0/nLoGEcx78EWtsBTU3sLG1/tNEBTcFepb4y0kBNUfsPtMpIsGfzVxdlvsRMR+35Ut8v8xFjHbW/X5o+kjT3VIxIBsJurYVF3FMxIqCK4sKFLRR4+p9rgYcIFwY7TTMuchsbj3yQ1tKtYNBaupWNRz7IuMhtgdYFcFrN8xR5egAUeROn1TwfUEVxH+ehjHV9nIcCqihuENVZtedLpW/Nqv39UihImrqyiqza86W5+iiG7xhMv1gROPSLFTF8x2Caq48KtK6dh8WwcPpOZQtH2XlY8NNH51Y+ypX8D1W+BTxGlW/hSv6HcysfDbSuk8PPZazr5PBzgdb1iZYnM4bVJ1qeDKiiuAt5OGNdF/JwTran6SNJU15fS92A9sexl9fXBlDNHiU7NlDX8BwFtFKWaKsjTEm/0wKtK3xAa1bt+VRcsosp/Jkp/Dmt3UsCKiihuakfU0rb19XUFOw0zbCVjfzr4XfxROhiaqiiiho+Gfs1g1cGeyTZlKa/Y6X/w2N+abKuC3mEk5r+npPtKRQkzVW1G7i9tH/aFFI40sJVtcHubPamF4G2H7StifZrAqgoLrKziKKy9lNYkZ1FGdbOr4ZoJf3D7acYGqKVAVSzx7rXR3LICasoKNjzYRuNFrDu9ZEQ4JHP27aMYbCt4geH3Ehx8S6am/uzZvXRbNsS7FFRAzd+gikH/5IpBSkhGi2kYuNlOdlenwiFuVdMYcOOA2gMhyltbWXEgJ2c/8Bfgi6rW7rx8kup/p/b+d8RR7GjrIIB9bV8fMMb3Pj/gjz+AmKxzIeedtSeL+veOomDJ77Y/gPurZMy35E8jyrf+STNH7gfL9gTWhYtovKdT8LpwdW1oPRsznrlaUYds57ikgaam/qx7vWRLCg9m88FVxYDY+Op2QLV1XtCwBwqY0Ge1QHHXvk1XrsX6ofPI1qylYKmSso2nsexV34tJ9vr9aEw94oprNlVTrQwvvuksbCQNbvKmXvFFAVDBk+veprnyx6huPY+BidmjJ4vK+HpVWM5e0xwX+P6FbbQEGn/7btfwDua1+86mJZ/RBh9yN/Svl1W7zo40LoADlx7IvURp2bsk7SWbCXcVEnVOxdQtinYy42/eswU6laW8eHf/J4BO+vYcUA5fzrho7x9WLA3Jjr93H/h6cej1JesIlbQTChaTFnTGE7/VLBHkQGJAMhNCLTV60Nhw44DkoGwWzQUYsOOAwKqaI/Hr7+DCaFD6FdQRkO0nqWx1Xzqx8FNhQDMXjKbpmj6Tq2maBOzl8wONBRWT/gQI5csp9X3HK8dtiirJ3wosJoA3GJUV49J+3a5uz1otVbLoE0nUb7ppLT2bbadgwKqCaCyZSdvjT2at8YendZe1bwjoIriPnDCMOBkXnpqFDtrmjlgUDGTP3Voor3v6PVHHzWGM+deR+358vj1d3BceBz9wwMwM/qHB3BceByPX39HoHVt2rUpq/Z8+f34i1l37JH0K4wATr/CCOuOPZLfj7840LoaybzvoKP2fHpj2wu0trmfQmsswhvbXgimoIRvHj2eomh6XUXRCN88OthpGogHw+Xfm8K///wjXP69KX0uEKAPjBRKW1tpLGx/glNpa7BHh0wIHUI4lF5XOFTIhNAhAVUUN6z/MDbu2pixPUj/9o8WvnvMpTRN2nNNppKoc/OyRjgruLoWR0ZyUuF7hFNGBq0eYnFkZHBFJWyufZtFkQImDDyZfuEBNLTuYOn2P7J519uB1nXBsPgZ1bes2sj65ggjiwu5acxByXYJVq8PhREDdsb3KaRc5bMgFmPEgJ0BVgX9Csqyas+X6yZex6y/zkqbQiopKOG6idcFWBWc0xSGDJeTOKcp2D/h5pJBbF+/mlHljTSXllDc2ER1XSnNI4P/gCusKOOftSv4564V7dqDdsGwQQqBbqrXh8L5D/ylWx591BCtp3+4/bVeGqLB3h9g936D2Utms2nXJob1H8Z1E68LdH8CwICPjWbajx9k6oNz8cZtWOkgisafz4DrLw+0rm/982cMemk9JSnTIU0FhWyb/AoQ7CWXT7/sczxz12w8smdUbIVhTr8syGN8pLvr9aEABB4AmSyNrea42Li0KaTWWISlsdV8MMC6ALb/up7v199EFQXUEGX525vh5mBr+udz9xN77WFCsfihn964jcbXHuSfz8U44pjg7jl80JKVtEbT/xmVRCMctGRlQBXtccTUUwF4cc5D1G+toayyiqkzPpNsF8nE3NvfUKU7mzRpki9evDjoMrpEdzz66JffncOJ9SMoSbmfQhPOy2UbuOzmGYHV9frRRxEb30j99CjRQVCwDcqeKiC0rJRj/vZGYHWtOPyIDpcd8daKDpeJ5JuZvebuk/a1Xp8YKXRXbQMg6BECwLj6oWmBAFCCMa4+2Ht1Rcc3suPSKJ64PUC0EuoujTLgkcZA62oqLqSkuf3VW5uKg716q8j+6vWHpEp2qijIqj1f6j6+JxB28+J4e5BWDB1Iq6WHaKsZK4bqPsjSMykUJE0NmT9kO2rPF+/gIq0dtefLzrFj+PuowTQUhnGgoTDM30cNZucfI2kyAAALfklEQVTYYK+XI7K/NH0kaZaXbWZAhn0Ky8s2MzHAumpbQwwsbH+WcG1rsN9rps74DL+7+w42DtpzmGe4qJjTZ3wmwKpE9p9GCpLmsptn8Naud2mIRnF3GqJR3tr1bqA7mQEWrxxHS5tMaInF24N0xNRTOf3qayirGgxmlFUN5vSrr9ERPtJjaaQgaf728wWsbRrOe5EYEP8UDkWH87efL+Dozwd36vCjGy8CYNJhy6kIx6htDbF45Tge3XgR3w+sqrgjpp6qEJBeQ6EgaV5b1ESsMH2iPlZQxGuLajn68wEVBUw9vIA5b17BnLUp1xSyFj58ZLBXSRXpbTR9JGmawuVZtefLQ5ddxIePbMHCdYBj4To+fGQLD112UaB1ifQ2GilImpLWOpoK2x/SU9JaF0A16RQAIrmnkYKkOfa4EkLR9CmZULSFY48L+Ma+IpIXCgVJc/Tnz2LysTFKIrXgTkmklsnHxgLdySwi+aPpI2kndNIodjQupK6ujvLyckInTQu6JBHJE4WCpFm6dCnz588nEolfz6euro758+cDMGFCsPfQFZHcy+n0kZmdYWZvm9lKM/tKhuXXm9lyM1tqZgvNLPi7nfdxCxcuTAbCbpFIhIULFwZUkYjkU85CwcwKgDuBM4FxwMVm1vb009eBSe4+AXgC+GGu6pHOqavLfJRRR+0i0rvkcqRwPLDS3Ve5ewswB5ieuoK7/8HdGxJPXwZG5bCebueuL1/Bi8eN483Dj+DF48Zx15evCLokysszn4/QUbuI9C65DIWRwNqU5+sSbR2ZCTyTaYGZXW1mi81scXV1dReWGJy7vnwFJyx4hap6JwRU1TsnLHgl8GCYNm0ahYXp9wIoLCxk2jTtbBbpC3K5o9kytGW8zZuZXQZMAk7OtNzd7wbuhvid17qqwCCN+8OrFLemtxW3xtuDtHtn8sKFe44+mjZtmnYyi/QRuQyFdcCBKc9HARvarmRmpxG/A/DJ7t6cw3q6lUH1mbOto/Z8Wrq+mgdGH8WW8gqG1NUyeH01ygSRviGX00eLgLFmdoiZFQEzgHmpK5jZMcBdwHnuviWHtXQ728oyDaQ6bs+XXz6zkJtDB7C5YhBuITZXDOLm0AH88hkdfSTSF+QsFNy9FbgGeBZYATzm7m+a2bfM7LzEarcCBwCPm9nfzGxeB2/X6yw/9Xia24zTmsPx9iDd2mQ0F6Xf97K5qJhbm4INKxHJj5yevObuC4AFbdq+kfL4tFxuvzv73A8f4C6uYNwfXmVQvbOtzFh+6vF87ocPBFrXlvLM97fsqF1Eehed0RygtgEwNZgy0gypq2VzxaCM7SLS++mCeJLmSyVOcUv6/v7ilma+VBL8DnARyT2FgqS57MxpXLtpJeU7toM75Tu2c+2mlVx2ps5TEOkLNH0kaVa8+AcKn/sNV6eMFsJFxawYM0L3IRbpAzRSkDQvznmI1jbTR60tzbw456GAKhKRfFIoSJr6rTVZtYtI76JQkDRllVVZtYtI76JQkDRTZ3yGcJuT18JFxUyd8ZmAKhKRfNKOZkmze2fyi3Meon5rDWWVVUyd8RntZBbpIxQK0s4RU09VCIj0UZo+EhGRJIWCiIgkKRRERCRJoSAiIkna0Ryg3z76bT707oMMb6lhY1EVfz/0cs646OtBlyUifZhGCgH57aPf5sNv/5SRLdWEcEa2VPPht3/Kbx/9dtCliUgfplAIyIfefZB+sfRrDPWLNfOhdx8MqCIREYVCYIa3ZL6WUEftIiL5oFAIyMaizNcS6qhdRCQfFAoB+fuhl9MQSr/GUEOomL8fenlAFYmIKBQCc8ZFX+dPH7yW9UWDiWGsLxrMnz54rY4+EpFAmXvPuvfupEmTfPHixUGXISLSo5jZa+4+aV/raaQgIiJJCgUREUlSKIiISJJCQUREkhQKIiKSpFAQEZEkhYKIiCQpFEREJEmhICIiSQoFERFJUiiIiEiSQkFERJIUCiIikhTO5Zub2RnAbKAAuNfdv99meTHwEHAssBW4yN3XdHUdF/3qdpYNHc9WG0Slb2P85mU8eskXunozIiI9Xs5GCmZWANwJnAmMAy42s3FtVpsJbHf3w4D/Bn7Q1XVc9KvbeWnYZLaGqsBCbA1V8dKwyVz0q9u7elMiIj1eLqePjgdWuvsqd28B5gDT26wzHdh9p/ongGlmZl1ZxLKh42mx9DuctVgxy4aO78rNiIj0CrkMhZHA2pTn6xJtGddx91agDqhs+0ZmdrWZLTazxdXV1VkVsdUGZdUuItKX5TIUMn3jb3ubt86sg7vf7e6T3H3S4MGDsyqi0rdl1S4i0pflMhTWAQemPB8FbOhoHTMLA+VAl35aj9+8jCJvTmsr8mbGb17WlZsREekVchkKi4CxZnaImRUBM4B5bdaZB1yeePxJ4Hnv4ptGP3rJF5i86SUqYzXgMSpjNUze9JKOPhIRySBnh6S6e6uZXQM8S/yQ1F+4+5tm9i1gsbvPA+4DHjazlcRHCDNyUUv7ADgtF5sREenxcnqegrsvABa0aftGyuMm4FO5rEFERDpPZzSLiEiSQkFERJIUCiIikqRQEBGRJOviI0BzzsyqgfdSmsqJnwndmedVQE2OSmu73a583d7WyXZZX++vvS1Xf2W3/P32F+Suz9Rf7R3s7vs++9fde/QPcHdnnxM/FDYvdXTl6/a2TrbL+np/7W25+iu//ZXLPlN/7f9Pb5g+mp/l83zV0ZWv29s62S7r6/21t+Xqr+yWq7+yW96d+yupx00fvR9mttjdJwVdR0+h/sqO+it76rPs5KO/esNIIRt3B11AD6P+yo76K3vqs+zkvL/61EhBRET2rq+NFEREZC8UCiIikqRQEBGRpD4dCmY2xszuM7Mngq6lJzCz883sHjN7ysxOD7qe7s7MjjCzn5vZE2b2/4Kupycws/5m9pqZnRN0Ld2dmZ1iZi8m/sZO6ar37XWhYGa/MLMtZrasTfsZZva2ma00s68AuPsqd58ZTKXdQ5b9NdfdrwKuAC4KoNzAZdlfK9z988CFQJ887DKb/kr4T+Cx/FbZfWTZXw7sBEqI38Wya+T67Lh8/wAfBiYCy1LaCoB3gTFAEfAGMC5l+RNB193D+utHwMSga+8J/QWcB/wVuCTo2rt7fxG/+9UM4l86zgm69h7QX6HE8qHAI11VQ68bKbj7n2h/n+fjgZUeHxm0AHOA6XkvrhvKpr8s7gfAM+6+JN+1dgfZ/n25+zx3Pwm4NL+Vdg9Z9tepwInAJcBVZtbrPp/2JZv+cvdYYvl2oLirasjpnde6kZHA2pTn64ATzKwS+C5wjJnd5O63BFJd95Oxv4BriX+bKzezw9z950EU1w119Pd1CvAJ4v9gF2R4XV+Vsb/c/RoAM7sCqEn50OvrOvr7+gTwMaACuKOrNtZXQsEytLm7bwU+n+9ieoCO+usnwE/yXUwP0FF/vQC8kN9SeoSM/ZV84P5A/krpETr6+/oN8Juu3lhfGZ6tAw5MeT4K2BBQLT2B+is76q/sqL+yk9f+6iuhsAgYa2aHmFkR8Z1Z8wKuqTtTf2VH/ZUd9Vd28tpfvS4UzOzXwEvAB81snZnNdPdW4BrgWWAF8Ji7vxlknd2F+is76q/sqL+y0x36SxfEExGRpF43UhARkf2nUBARkSSFgoiIJCkUREQkSaEgIiJJCgUREUlSKIi8T2bmZvajlOc3mtmsAEsS2W8KBZH3rxn4hJlVBV2IyPulUBB5/1qBu4EvBl2IyPulUBDpGncCl5pZedCFiLwfCgWRLuDuO4CHgP8IuhaR90OhINJ1bgdmAv2DLkRkfykURLqIu28jftP5mUHXIrK/FAoiXetHgI5Ckh5Ll84WEZEkjRRERCRJoSAiIkkKBRERSVIoiIhIkkJBRESSFAoiIpKkUBARkSSFgoiIJP1/v+YfUICfgGMAAAAASUVORK5CYII=\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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u2BhBRLyyxaxjO7VMMzObOp9ZbGZWc04EZmY150RgZlZzTgRmZjXnRGBmVnOVTQRVqg65dOnIY2n0dNlUpdJlVfoBqULnxo3pu+KKnTXVzskGRd+mekJZlSoqLlkyfl+WLCk6sqmryudSlX5EuGJn1TFbJ5RJegzwbmBhRDxP0hLgqIj4ZA55Cpj6CWWt6qdUjStddp+yfSYbN45fjLHOFTurZDarj14AfB1oVIT5KfCG6YfWeXVIAmazoVVFXlfsrJd2ziyeHxEXSTodICJ2SurqVW1PT3UqKk5UsbNs5QCqUumyStVHXbHToL0tgt9JehQQAJKOBH7d0ahmqEoVFZcsmVp7N6vK51KVfoArdlpmskEE0uUmv0Na+X+HtGvosHYGIGbrNp3qo1WqDjl2wLiMA8UNVal0WZV+RLhiZ5Uxi4PF84BdwMGAgJuAORGR2/W+plt9tEqqVB2yKn2pSj+sumZzsPi7EbEzIjZHxPUR8UfguzMP0czMukHLwWJJBwCLgD0kHU7aGgDYG+ht9TozMyuXiY4aei7wauBA4ANN7b8F/rmDMZmZWY5aJoKI+DTwaUl/ExGX5BiTmZnlaNLzCCLiEkknAEuB3Zva39nJwMzMLB+TDhZL+ijwcuD1pHGClwJ9HY7LzMxy0s5RQ8+KiFcB90bEO4CjgINmslBJ/yBps6TrJV0oaffJX1VfVaoOWZW+VKUfZtBeInggu98haSHwR+Bx012gpEXAqcCyiDgE6AFeMd33q7rBwXTGaqMmzLZtabqMK56q9KUq/TBraOeEsjOBfwWOBT5CKjXxiYg4c1oLTIlgI3AY8Bvgi8CHIuIbrV5T5xPKWtWCmTcPjjwy93BmpCqVLlt9JmXrh1VfuyeUtTNY/H+zh5dIuhzYPSKmXWsoIm6T9D7gFtLWxjfGSwKSVgIrARbXuAJWqyqQrapGdrOqVLpsFW/Z+mHW0E71USQ9C+hvPF8SEfGZ6SxQ0n7AC0m7l+4DPi/ppIhY2/y8iBgABiBtEUxnWVWweHHrX59lK21QlUqXrT6TsvXDrKGdo4Y+C7wPeA7wjOw26abGBI4DfhERw1m5ii8Az5rB+1ValapDVqUvVemHWUM7WwTLgCUx2WBC+24BjpTUS9o1dCxQzwGANqxYke7POCPteli8OK1wGu1lUpW+NOJ93evS7q6+vnL2w6yhncHizwOnRsQds7ZQ6R2kcxN2AtcCfzdRNdM6DxZb93L1Uet2szZYDMwHtkj6PvCnlXVEnDjd4CLi7cDbp/t6MzObPe0kgrM6HYSZmRWnncNH1+cRiJmZFWOi6xFcHRHPkfRbsusVN2YBERF7dzw6MzPruInKUD8nu98rv3DMzCxvE20R7D/RCyPiV7MfjpmZ5W2iE8o2kY7v3wQMAz8F/jN7vKnzoZl1rypVH129GubOBSndr15ddETTNziYPo85c8r/ueRpol1Dj4M/XY/gsoj4ajb9PNLZwWa11Kr6KJTvpLLVq+G880amd+0amV6zppiYpqvxuezYkabL/LnkrZ0TyjZFxBFj2obaOUlhtviEMusmVaoIu77FMYE9PbBzZ76xzJSrwj7cbJ5Qdo+ktwJrSUcPnQT8cobxmZVWlSrCtrJrV9ERTJ2rwk5fO4nglaSzgC8lJYINWZtZLVWpIuzcueOv9Ht68o9lplwVdvomrD4qqQc4PSJOi4jDI+LpEfEGHzFkdVal6qONfejttnezKn0ueZswEUTELuCIiZ5jVjcrVsDAQNoCkNL9wEA5ByTXrIFVq0ame3rSdNkGimHkc2ko8+eSt3YGi98PPAn4PPC7RntEfKGzoY3wYLFZZ0npftaKzReoSn2ZqdkcLN6fNDh8TFNbkC4oY2ZmJddO0bnX5BGImZkVo51LVR4o6VJJd0u6S9Ilkg7MIzgzM+u8SRMB8CngMmAhsAj4ctZmZmYV0E4iWBARn4qIndntAmBBh+MyM7OctJMI7pF0kqSe7OYzi83MKqSdRPBa4GXAndntJVnbtEnaV9LFkm6UdIOko2byfmZ5O+64dJhi43ZcicswNp+EJT38pKwyaf4cyv655FlJddJEEBG3RMSJEbEgu70oIsY5kXtK/j/wHxHxZOAw4IYZvp9Zbo47Dq66anTbVVeVc6XT2wsPPDC67YEHypkMqvS5NCqpbtuWzodoVFLtVDJo54SyA4F/BZ5NOn/gauC0iNg+rQVKewPXAY+PyRae8Qll1k0aJyyZ5W2qlVTbPaGsiKOGHk+6uM2nJF0r6ROSHjn2SZJWShqSNDQ8PDyDxZmZVUOnKqkWcdTQXODpwHkRcTipbMVbxj4pIgYiYllELFuwwAcpWTlElOvmvnTnra9v/H50qpJqEUcNbQe2R8T3sumLSYnBrBSOPXZq7d1sjz2m1t7NqvS55F1JdapHDd3BDI8aiog7gVslHZw1HQtsme77meXtyisfvnI59tjUXjY7djx8pb/HHiOXeyyTKn0ueVe4nXSwuCMLlZ4GfAJ4BPBz4DURcW+r53uw2KyzXLGzmmZtsFjSpyXt2zS9n6TzZxJcRPwo2/9/aHY4asskYGZmndXOrqFDI+K+xkS20j68cyGZmVme2kkEcyTt15iQtD/tXcfAzMxKoJ0V+vuBayRdDARp4NhXATUzq4h2LkzzGUlDpCuUCfjriPBRPmZmFdHWLp5sxe+Vv5lZBbUzRmBmY+RZGbLTqlSx06bHicBsivKuDNlJVarYadPXTvXRU4DBIo/19wll1k36+9PKf6x58+DII3MPZ0bWr289zyeXld9sVh89APiBpIskHS+5CK/VW6sKkA8+mG8cZrOlnaOG3irpTOC/A68BPizpIuCTEfGzTgdo1m0WLx5/i6CvD9atyz2cGfHPOoM2xwiyC8g0LlW5E9gPuFjSezoYm1lXyrsyZCdVqWKnTV87tYZOlbQJeA/wHeCpEbEKOAL4mw7HZ9Z1GpUh581L052uDNlJVarYadPXznkE80knkY3aGI6IhyQ9vzNhmXW3FSvg4x9Pj8u2O2isK6909dG6a2eM4G0TzPNF583MSs7nEZiZ1ZwTgZlZzTkRmJnVnBOBmVnNORGYmdVcYYlAUo+kayVdXlQMlr+qVO1cvTrV6Vm/HubOTdNltXTpyGNp9HTZVOX/K29FbhGcBvjw0xqpStXO1avhvPNGpnftStNlTAZLl8KWMVca2bKlnMmgKv9fRZi0+mhHFiodCHyadMnLf4yICU9Mc/XRaqhK1c6JKnZWydFHFx3B1GzcOH7hv74+2Lo193C6wmxWH+2Ec4E3AQ+1eoKklZKGJA0NDw/nF5l1jKt2Wie1+j9q9X9nI9q6VOVsyspS3B0RmyQtb/W8iBgABiBtEeQUnnVQVap2zp2bdgeN1dMDO3fmH89MTFR9tEyfCbTe4ly8OPdQSqeILYJnAydK2gp8DjhG0toC4rCcVaVq58qVU2vvZkuWTK29m1Xl/6sQEVHYDVgOXD7Z84444oiwali7NmLevAiI6OtL02W0alVET0/qR09Pmi6rJUtSPxq3JUuKjmj6qvL/NVuAoWhjXVzIYHFDtmvon8KDxbWyfHm6L9uuhyqrUvVR/3+NaHewOPcxgmYRsQ5YV2QMZmZ15zOLzcxqzonAzKzmnAjMzGrOicDMrOacCMzMas6JwMys5pwISqBKpXWrVL65Kpo/g7J/JlX6/8r1e9/OWWdF3+p8ZvHatRG9vaPP/OztLecZk6tWje5H41bms3LLrkqfSZX6Mlvfe8pwZnG76nxmcVVKN0Pr8s1lLNZWFa0K6EH5ylDXoTz4VEtqd3sZamtTHUo3t1oRWef5b18unSqpXWiJCZtcVUo3w8Tlm60YPT2tP5Mq/X+VbYsz75La3iLoclUqrVul8s1VUaXPpEp9yf17385AQtG3Og8WR1SrtG7zgF7ZyzdXRZVKalepL2vXpu+7NP3vPR4srpYqldatUsljs27mwWIzM2uLE4GZWc05EZiZ1ZwTgZlZzTkRmJnVnBOBmVnN5Z4IJB0k6VuSbpC0WdJpecdQNlWqqFilSpdmVVFEiYmdwBsj4oeS9gI2SboiIrYUEEvXW70azjtvZHrXrpHpNWuKiWm6qtQXsyop/IQySV8CPhwRV7R6Tp1PKKtDdcgy1oIxK4NSnFAmqR84HPjeOPNWShqSNDQ8PJx3aF2jDtUh69BHs25WWPVRSXsClwBviIjfjJ0fEQPAAKQtgpzD6xp1qQ5pZsUpZItA0m6kJDAYEV8oIoayqFJFxSr1xaxKct8ikCTgk8ANEfGBvJdfNo1B1Magak9PWnGWcXC1Sn0xq5LcB4slPQf4NvAT4KGs+Z8j4qutXlPnweKGKlXsrFIlVbNu1u5gce5bBBFxNaC8l2tmZuPzmcVmZjXnRGBmVnNOBGZmNedEYGZWc04EZmY150RQAoODI4/7+0dPl83gIGzcmOoOlb0vZlXhRNDlBgdHn3m7bVuaLuMKtNGXBx9M02Xui1mVFF59tB11PqGsvz+tMMfq64OtW/OOZmaq1BezMihF9VGb3C23TK29m1WpL2ZV4kTQ5RYvnlp7N6tSX8yqxImgy519NvT2jm7r7U3tZVOlvphViRNBl1uxAgYG0n50Kd0PDKT2smn0Zd68NF3mvphViQeLLXeuPmqWDw8Wm5lZW5wIzMxqzonAzKzmnAjMzGrOicDMrOacCMzMaq6QRCDpeEk3SbpZ0luKiMGK4eqjZt0n90QgqQf4CPA8YAnwSklL8o7D8ufqo2bdqYgtgmcCN0fEzyPiD8DngBcWEIfl7IwzYMeO0W07dqR2MytOEYlgEXBr0/T2rG0USSslDUkaGh4ezi046xxXHzXrTkUkAo3T9rA6FxExEBHLImLZggULcgjLOs3VR826UxGJYDtwUNP0gcDtBcRhOXP1UbPuVEQi+AHwJEmPk/QI4BXAZQXEYTmrUiVVsyqZm/cCI2KnpFOArwM9wPkRsTnvOKwYK1Z4xW/WbXJPBAAR8VXgq0Us28zMRvOZxWZmNedEYGZWc04EZmY150RgZlZzpbhmsaRhYNs0Xz4fuGcWwymS+9J9qtIPcF+61Uz60hcRk56RW4pEMBOShtq5eHMZuC/dpyr9APelW+XRF+8aMjOrOScCM7Oaq0MiGCg6gFnkvnSfqvQD3Jdu1fG+VH6MwMzMJlaHLQIzM5uAE4GZWc3VIhFIeq+kGyX9WNKlkvYtOqbpkvRSSZslPSSpdIfHSTpe0k2Sbpb0lqLjmS5J50u6W9L1RccyU5IOkvQtSTdk/1unFR3TdEjaXdL3JV2X9eMdRcc0U5J6JF0r6fJOLqcWiQC4AjgkIg4FfgqcXnA8M3E98NfAhqIDmSpJPcBHgOcBS4BXSlpSbFTTdgFwfNFBzJKdwBsj4inAkcD/Lunn8iBwTEQcBjwNOF7SkQXHNFOnATd0eiG1SAQR8Y2I2JlNbiRdFa2UIuKGiLip6Dim6ZnAzRHx84j4A/A54IUFxzQtEbEB+FXRccyGiLgjIn6YPf4tacXzsOuId7tI7s8md8tupT0aRtKBwAnAJzq9rFokgjFeC3yt6CBqahFwa9P0dkq4wqkySf3A4cD3io1kerJdKT8C7gauiIhS9iNzLvAm4KFOL6iQC9N0gqQrgQPGmXVGRHwpe84ZpM3gwTxjm6p2+lJSGqettL/YqkbSnsAlwBsi4jdFxzMdEbELeFo2DnippEMionTjOJKeD9wdEZskLe/08iqTCCLiuInmSzoZeD5wbHT5yROT9aXEtgMHNU0fCNxeUCzWRNJupCQwGBFfKDqemYqI+yStI43jlC4RAM8GTpT0P4Ddgb0lrY2IkzqxsFrsGpJ0PPBm4MSI2FF0PDX2A+BJkh4n6RHAK4DLCo6p9iQJ+CRwQ0R8oOh4pkvSgsYRgZL2AI4Dbiw2qumJiNMj4sCI6Cd9T77ZqSQANUkEwIeBvYArJP1I0keLDmi6JL1Y0nbgKOArkr5edEztygbsTwG+ThqQvCgiNhcb1fRIuhD4LnCwpO2SXld0TDPwbOBvgWOy78ePsl+iZfNY4FuSfkz60XFFRHT0sMuqcIkJM7Oaq8sWgZmZteBEYGZWc04EZmY150RgZlZzTgRmZjVsO3bbAAACjklEQVTnRGA2iyRtlTQ/e3zNDN7n1ZIWzl5kZq05EZhNk6QJz8yPiGfN4O1fDTgRWC6cCKz0JD0ju9bE7pIemdWiP2Sc570qe951kj6btfVJuiprv0rS4knaL5D0AUnfAs6R9ChJ38hqxn+MpnpKku7P7pdLWifp4uy6GIPZ2bxIepukH0i6XtKAkpcAy4DB7OSuPSQdIWm9pE2Svi7psZ3+u1qNRIRvvpX+BrwLeB/pegenjzN/KXATMD+b3j+7/zJwcvb4tcAXJ2m/ALgc6MmmPwS8LXt8AqmIXmMZ92f3y4Ffk2orzSGdkfyc5jiyx58FXpA9Xgcsyx7vBlwDLMimXw6cX/Tf3Lfq3CpTdM5q752ksgK/B04dZ/4xwMURcQ9ARDSuJXAU6UI/kFbE75mkHeDzkapcAvxF43kR8RVJ97aI7/sRsR0gK5PcD1wN/KWkNwG9wP7AZlISanYwcAipRApAD3BHi+WYTZkTgVXF/sCepF/PuwO/GzNftFfyutVzmtvHvnc77/tg0+NdwFxJuwNrSL/8b5V0Fin2sQRsjoij2liO2ZR5jMCqYgA4k3StiXPGmX8V8DJJjwKQtH/Wfg2puiPACtKv9Inax9qQzUfS84D9phBzY6V/T3YtgJc0zfstqVAipF1aCyQdlS1nN0lLp7Acswl5i8BKT9KrgJ0R8W/ZdZGvkXRMRHyz8ZyI2CzpbGC9pF3AtaQjc04Fzpf0f4Bh4DXZS1q1j/UO4EJJPwTWA7e0G3ekmvkfB34CbCXt2mq4APiopAdIu6leAnxI0j6k7+25pN1IZjPm6qNmZjXnXUNmZjXnRGBmVnNOBGZmNedEYGZWc04EZmY150RgZlZzTgRmZjX3X1G3u7sJIEp0AAAAAElFTkSuQmCC\n", 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vf90EyLzEbKA9fXlVSZMRFijuXobsajT8/TapXWnPNw11uee9hMTsnMlD2JQAWT/S3pui7mXoOKF+m9Qu9cX+RI5iluQhbEqArB+hexPqH0Xdy7zsUl/sT/rWUYQEPV+7ry2NsBkScoeGpot6jzwSFr2rKHJ3W+4zxNiY/0WBKlyDuA2rVvmjtScmio2ATmOXj1D5Wt9+k5Im39X4eL2i9PuOJEJG1ae0YnZINPYJfUNDLno2vm6aSOM064bE3SqK3FkI+r5pzhx/e1NELusa+PpNXLBuTpOTxUVA++yKR4DPdG3jU+j/Ic00Pt7+wkNI4A71j6pG6fcKkJgdJq3QV1XKFrnzilROQ5HXII2QW2R0eZ4Cc7fEv17SvgBR1Sj9XkFidgfSCn1VpWxhsQrXq8hrkOZYRV6bsvtBnlShj4k+dRRphb6qUrawWIXrVeQ1SHOsIq9NXtegCve3CjaIPnUUvhTQjYYT+uLRrI1Ge9vISLpI4zTr+hgba6+ulld66jRkIej7mDPH355XVLNPRPW1TU2134fQUEqR0eVTU+3Ceciu0LWNQ7pz6CZWZny8vS0UsR7qH1WN0u83+tJRbNrU/ki7axfwgx+0j6kODABvfnN7qcqHH25/62fuXP9bS296k/uy6Qbffstm7VpX9rO1DOj4uP9tm/gXWSde/OL26zU46L40si4Zuno1cOGFe/rDrl1u3td2ySXt173R8LctX96dXWk48UR3LVqvzUEH+dd9/PFk+zQDLr+8uyC4ww9vbzvzTP+6F1zQ3pcmJ10fE+XTl2J2WhG2ClGrPsoWs33keQ3yON86pQ5PQxV+SPiE6LplH+h1korZfeko0v4Tke01kX0MDBSbhiCpXUWS5zXI43zz+kIt+95UwVEA/id0X/8o+3r1K3rrqQN51RMuWiQvW8z2kadN8+YlD5L04VuvbvcmlHk1nqW1CviurbIP1JQkwRZVn9IG3I2P+4N7QtlQkwb9pN1vtwFoVQxGyiKjbGgf8eC2UFCYLxAvFGSZNAAtTbBaXvcm1L/iWWKrMvkywtYtQ3KvAwXchckr42fa8e7Bwe6Ciao4rpvFr9ksAh/j16YX9KOqPCkkJXQNNmwA1qxx8R8LF7q3trp9KUHMDmkUHUg7jp50/LRb7SMv7aRIqvJlFr82vaAfVeXaJqWK/VNMRxpFB/LKrNmt9pGXdlI30lyHpPes6GvVq/cmDboGvUNpjoLkISS/SvJ7JDeTfHvUPo/kl0jeGf3dL+tj+wKURkb8AXfDw8CSJf6slvGgrCOO8B9vfDxclrKVUHDR+HixmUjTEBeIQxllkwZ6AemCrHxBYWR7NtRVq/xBi2ky4IaCxeKEytx2iy+ADWg/r6xIk/01TlX6p8iIJEJGHhOAgwD8dvR5XwA/ADAG4DwA74za3wng3Jn2NZtSqL7Mnr7ykaEsnGNjyUW9sTF/WUqfgBcvtdkURYvKRJqGNNlUQ0Jst1NSIXd4uD377PBwOvG97NKeoWy9afpimimNeD86Wr3+KWYGdROzSX4GwAXRtMLM7id5EIDrzCzwW92RVSnUXhA8i6TKWUurQNb/WlXI1tuJinyViBTUSqMguRjACwB8HcABZnY/AER/nxHYZoLkLSRv2bZtWyZ2FJ2Fs+5ZP+tuf92ospMQvU3pjoLkPgA+BeD/mdkvk25nZuvMbJmZLZs/f34mtkjwTEfd7a8byqQqyqJUR0FyCM5JbDCzT0fND0RDToj+/rwoe3widyiZX1Jhs7muL1utr/xkFcp9JsWXhTfE+Hg+omvSfTYa7euOjKQTs7Mo7emLrA6VAI33hRUr/MdK0xfTkOYFhLxsEBUhiZCRxwSAAC4DcH6s/e8xXcw+b6Z9zUbMDhEXjbOItg6VeYyLo8PD7eJslaNWk5ZCHRx068ZLyhY5DQ87G3wvMCS5Z6Oj6Uq/dhstPT7eXRnSLKb169v7f7fZC0S1QNXFbJIvAfBfAL4DoBmWcyacTnElgIUA7gbwBjN7qNO+shKzfVRBQKyq6F2Fa5MG33VMk820bufbLWmugUqW1pOkYnaXVRJmj5ndAPdU4SPwxnjxVOGLoaqicRWuTRp81zF0bX3tdTvfbklzDfrt2vQbpYvZVacKAmJVReMsIqiLxHcd02QzrcI5FEmaa9Bv16bfkKOYgVCUcBoRNGkZ0IGB9n84XwR3VQhdm7joG4p6L5JQ6dhQlL5v3TQR490K9+Pj3VdF7IbBwXTXYMWK+ryEIdIjRzEDvnKfk5PA8ccn38f27cnW2727PYlaSRJSIpYv95csfctb2kuWLl/efi6hJHdpnHCaGtA+fGVEQyVWfX0hZOveeyezy0fzmiYd80/zNlbSHy07d7qSwXGWL28/Hgls3Oi0HjP3d2JCzqKXqExkdjfkKWaHKFLYrKqYnUYIrkIUt1J/pyNNKVMfVe23Yg+1isyuI0WKd1UVs9MIwVU4hyrYUCd8fTzNNdT17h3kKGZJkeJdVcXsNEJwFc6hCjbUiTSlTH3oevcOchSzJI2wmYb4UEZIVCyDeKRwKHV3UtE4JNbmoVFklfY6fg1CEclJtQAfg4PhlOI+8tAoAH8f993H4eHwSxi+qPM6ZR8QEUmi8qo+ZRmZnZS8UjtXNerVl1I8lLo7FEkej3qfnGyPTh8YSJeSPHQfxsezT3vtuwahNPTd1g6fnGxPOR+6LmlT3sdtJt2+fentk95HXxr9kL3xPlPl7AO9DqoemZ0lZYjZRYqYVYh6zUPEzFPgzuOaFSnIdyskp6Ub4TkLuyR8l4NqZudM0W+7lH2b0tScTlorOe861lnvu+i62/Fj5Xn8bupbZ2GX6muXg9566iGqEPWah4iZp9iZxzUrUpztVkhOSzf7zsIuCd/VZkZHQfIAkv9K8gvR/BjJN+dvWrUpMq3yxEQ4FXUe+MTGqSl/oJUvdXdS0TgUFZ1GyA3dhzxeNvDZm0XQoI+kQvLISPqU91nXX+/2PjYa1XlhQwSYScQA8AUAJwC4PZofBPCdJAJIUVMZYnYoPXVImOxG2AzVas5D5PYJtiMjYWGyW9E4VAvcVzs8VE881J4HPiHXJw77hPqQ8B1vHxxM/kJAM1V6aN++PpNH/XXfPtO8lFCFFzb6EWQlZpO82cxeSPJbZvaCqO02M3t+rh4sBWVoFGkEvEYjnwC9Kgi2VRDayyR0vbq952nE3bR9saj7lUbH6/d+VBZZahS/Ivl0ABbt+EUAHunSvtqTJuo0ryjuPPabNpq239NLh65Xt9clrwjoqt6vqtolHEkcxV8AuBrAs0hugqtK97ZcraoBacS3vMTopoYQD1zyaQxJg5zSiopVENrLJHS9ur0ueUVA53W/ug2i6/d+VHlmGpsCsBecLrEUwHMBDAHYK8m4VlFTWRqFLwDNV8o0zVhtN9PIiBvrjds1NNRehjQU5BTSKDoFtvUzoevluw++/pFF+ds0GtjYWHHXIFQCWP2oOiChRjHzCsA3k7SVOZXhKMzCwmK8bdGi/JxDfIpHvXaaFi1Kfl6hcwjto58IicNJ+0e34nKa/tVoZHbaszp+2r4o8iWpowiK2SQPBLAAwHoAf4g9ZUufCuAiM3t2Xk85aSlDzE5D0YFaSUkT5BQ6BwVKlU/a/pV1X1TAXX3Jomb2sQBOAXAwgA+1tD8K4MyurOszFi4sNvVDUmEw7Ti47xwUKFU+afpXXoGI3fZv9aNqExSzzexjZvYyAKeY2ctapuPM7NMF2lh7pqbaA9NCrw7G/5HTBqCFAs3i+w1leQ3hO4e0+xD54At4C2WUTRuI6BOp48GfS5b4t1XAXe8w41tPZvYpkq8k+Q6SZzWnIozrJZK+U75iRXtZzjRceKG/Pf6UMZtcVfFterm6W53wlXO97DJ/Cd+1a5Pvd8MG51haS5yefLLrY83+tGsXcO21/u03bkx2nF27/GVXRXVIEnB3EYARAC8D8FEArwfwDTOrTBqPqmsU3QZE5fWFnEVQl7J+9i5lZ8sV+ZNlwN2LzewkAA+b2XsBHAPgkG4N7CeqGhCVRVCXyl32LkXeWwXcVZskjuLx6O8OkqMAngRwaH4m9R5VCIjykUVQl0TI3qXsbLmiOiRxFJ8jORfA3wP4JoCfALgiT6N6jTRCnZkbampOacXspAwMpCtVKTG7//CJ5Gm+0OP9pRN5lRYW2ZBEzH6/mW03s08BWATg2Wb27vxN6x0uuST5uvF3yR9/PCwW+kj6z7l7N/COd7QLkxdeCJx00nQBc2LCiY0Ss/sLn0g+MZG8Rvdhh+VrnyiORBXuSL4YwGK0xF2Y2WX5mZWOqovZvfyFKjG7v8hL4JaYXQ5ZBNw1d/RxAM8CcBuApuRkcMkBRZ8jMbu/yOt+S8yuNjM6CgDLAIxZkkcP0bOEIr4lZvcXeWUZkJhdbZKMNn4XwIF5G9LL5CFGh0gjICYt1zk05Mamsy6hKerH1JR76aGV0NBqmhKtErOrTRJHsT+ALSS/SPLq5pS3Yb3Eqaf6heD4l/qcOW6Kt5m1f6nH12sS/yfuxI4d7b/kSH/b8uXtwua6dU7wFP3Dpk3tWoJvrGFgADjzTH90eLcR46J4kkRm/66v3cyuz8WiWVB1MTuPqObBwWLHdSVaCyBdv1OfqT6ZidlVcgh1JY+o5qLFP4nWAkjX79Rneofg0BPJG6K/j5L8Zcv0KMlfFmdi/ckjqrlo8U+itQDS9Tv1md6hU5rxl0R/9zWzp7ZM+5rZU4szsf74Ily7FYJD4l9Iu/Axdy4wPDy9rRmlHW+bmuq+LrKoP6F+Fw/CGx7Wiw69RHDoieS8Thua2UPZm9ObNAXfNWvc4/jChe6fqBshuCn+rVvnhgMajT3/xKFU43FGR4E775ze5gt62rnTRZffdJMTwIE9EduABO1+4jOf8bfHMwroZfreolMp1B/DBdYRwEIAD0ef5wK428wqkxiw6mJ2kUjkFnmSJsuA+kb16TrNuJkdambPBPBFAK82s/3N7OkAXgVAFe4qikRuURXUN3qHJHEULzSza5ozZvYFAN5XZtNC8mKSPyf53Za2eSS/RPLO6O9+WRyrF/FlfpXILbIkrkulYeHC5NmJRbVJ4igeJPkukotJLiK5BsAvMjr+pQBeEWt7J4BrzewwANdG8yLG6tX+zK9xcboTaYTvUBT3qlXJ9yHqha8UahqefNLfR08+uT07sZxFtUkScDcPwHsAvBROs9gI4H1ZidkkFwP4nJk9N5q/A8AKM7uf5EEArjOzIzrtox81iqK1iBAah+5diiyFqn5UDpkE3JFsADjDzN6emWUzc4CZ3Q8AkbN4RsC2CQATALCwD8c/quAkAI1D9zJF3lv1o2rTcejJzHYBOKogW1JhZuvMbJmZLZs/f37Z5hRO0VpE6Hh96KP7hiLvrfpRtUmiUXwrSgT4xyRf15xytOmBaMgJ0d+f53is2hIKfEqaERZwWW19AXfxZIUjI8oe24/4AkXTEMoeG//RoX5UfZI4inlw4vXLAbw6ml6Vo01XAzg5+nwygECIj/Dx8MPp1o9LVCTwJ3/SniV27Vplj+03fKVQ06QOP+ig9ojtgQH3o0P9qF4kKoWa28HJywGsgEtl/gCcaP4fAK6EC/K7G8AbZhLOJWZni4RFESKLsr7qX9Uhy1KoBwP4CIDlcG893QDg7WZ2b7dGmtmbAosKLPVTT/IUsyUsijxR/6ofSYaeLoEbDhoFsADAZ6M2USJ5itkSFkWeqH/VjySOYr6ZXWJmO6PpUgD995pRxcgie+z4uARqkY40ZX3Vv3qHpJHZf0SyEU1/hOwis8UsWb7cLxTulyLhyeGHS6AW6Tj11GTrDQy4ddW/eoMkkdkLAVwA4JioaROcRlFQzObM9KOYnUXUbKPhTysuRIg0/U6idfXJshTq3QCOy8QqkRlZCIJVie4W9SFNv5No3TvMOPRE8mCSV0VZXh8g+anoTShRIlkIgkVHd4v6k6bfSbTuHfTWU00JlVdNE5kdEsSFCOHrd8PD7T86mqVQfSnFlWa8fuitp5rii5pdtw746U/bo2dHR9v/kRsNJ4gLkQZfv3vzm9tfrDADNm1qT1N+2mlO5Faa8XqRRMz+MlzdiMujpjcBONXMKhMU148J9P3pAAARbUlEQVRidhpCAqTERpEFWbxYob5YDl2XQm3hNAAnAPgZgPsBvD5qEzUhJCpKbBRZkEU/Ul+sNnrrqQ9YuND/i09io8iCUP9qNJK/Wae+WG2SvPX0MZJzW+b3I3lxvmaJLAkJ34qQFbMhLkavWtWern542GkPvjT2IeFbVJckQ0/PM7PtzRkzexjAC/IzSWRNSPhWhKxIi6+O9kc/2v7k0JQ+4xLo7t3t+ywxgbVISBIx+3a4GtYPR/PzAFxvZr9VgH2JkJgtRDHkVUdbYnY5ZBaZDeCDAG4k+Um4NOMnANCDohB9SF6is8TsajPj0JOZXQbg9+EKC20D8Doz+3jehgkhqkca0TlN5L/E7GqTRKOAmW0xswvM7CNmtiVvo4QQxRKKll692gnQpPu7ZEm7QN1otLeF6qwPDfnXlZhdbRI5CiFE7+ITqCcmgJUrgQsv3CNU79oFXHst8OST07cfGHDR2UnqrF9yCXDxxXqxom6UWjM7KyRmCzF7FFndv2QWmU3ydJIpyuEIIeqEIqvFTCQZejoQwM0kryT5CpLM2yghRHFkISRLjO5tkrz19C4AhwH4VwCnALiT5Dkkn5WzbUKIAli1yt8+NJR8H0uWZGOLqCZJ33oyuKSAPwOwE8B+AD5J8rwcbRNCFMA11/jb46J1J667LhNTREWZMeCO5NsAnAzgQQAfBfDXZvYkyQEAdwJ4R74mCiHyRGV1xUwkiczeHy7Ibtp7EWa2m+Sr8jFLCFEUoeyvaVBZ3d4miUZxVtxJtCz7XvYmCSGKZGrKn/01TVndFSuSBeytXq1SqHVEcRRC9DkbNrjypK2axNAQMH8+cN99s9vnyAhwzDEuQC/OwMD0LLIjIwq6K4ukcRRyFEL0OXllhE2DAvbKIctSqEKIHqYKwXJVsEGEkaMQos+pQrBcFWwQYeQohOhzshCz45DA+HiydQcHnQ0SuatLktdjhRA9TlyqNAPmzp29mG0GfC/hO5E7d7qssjfdBOzY4dqaGWwBidxVQGK2EH1OFcTsEBK580VithAiEVUWkqtsWz8hRyFEn1NlIbnKtvUTchRC9CghcXjBAic2N6dHHnF/WyGBsbHujp9GDB8fby+bqhKp1UGOQogeJFTedL/92gXq7dv9Yvb27c7JtDIw0O4A5s712xBqjzMw4CLD42VTFa1dHSRmC9GD5ClQxwXmwcHus8dKtC4HidlC9DF5isDxfWeRYlyidbWRoxCiB8lTBI7vO4sU4xKtq01lHUVUn/sOkltJvrNse4SoE1NTfnE4qW4AOC3Ct49Vq6aL5CtW+LcPieFx3WN42Nm7cuV0kX3lSn+a8n6jEhHrZla5CUADwA8BPBPAMIDbAYyF1j/qqKNMCDGd9evNFi0yI93f9evd5KTq6dPo6PT5sTH/PiYnzUZGpq/baPj3OTbmbyenzw8Nhdf1TZOT5V3Tolm/vv16j4y49iwAcIsl+E6upJhN8hgAZ5vZsdH8GQBgZn/nW19ithDJCIncScXkKkRxNxou7Uc/0O39mom6i9kLANzTMn9v1PY/kJwgeQvJW7Zt21aocULUlZBonFRMroLo3E/1ubu9X1lRVUdBT9u0Rx8zW2dmy8xs2fz58wsyS4h6ExKNk4rJVRCd+6k+d7f3Kyuq6ijuBXBIy/zBAGaZx1II0SQkcieNgPZtH/riTiNmp4kCb2aV7UXi4v2SJf71Ck/LnkTIKHqCS3/+IwCHYo+YvTS0vsRsIZLjE7m73X5yco+o3Wi4+fXrzQYG2oXseNvQUHtbp2nu3OyvSRWYnEx+DebMyUbkRp3FbAAguQrA+XBvQF1sZsHfPBKzhageeQrfFf3a6ooyItyTitmVLVxkZtcAuKZsO4QQs6MKwnedqHKEe1U1CiFEzclTcK1EEFpCfLb6AgmrHOFe2aGnNGjoSYjq0cxg2yxvCjjh2gx48sk9bSMj09dJQnybkZFqZpv1XYPBQX8cyOhoutKzWVyDusdRCCFqzokntqcOv/hiVx87nk48LXHHsmMHsGZNNnZnyZo17baGggXT1icvMi27niiEEKUTL5w0233s3t39frJkYCA/4T2L/eqJQgjRV1QhGDBOFW2aDXIUQojSSRNwNzbmtI5Whofbs9pu2BAWvX1isi97LeBvTyqmT00BQ0PT20JPT75ysKF1uy1Tm5okwRZVnxRwJ0S9CWW1jWeVHR936w4NTW9vNNrbhofb20ZG3D7SBPclyYAbCnZbv97Z0Wnb1qy48WDGkK1ZZdBF3QPu0iCNQoh6kyZLahUy2Pro1lZfVtxQEF5WGXSlUQghakOaLKlVDeTr1lafQwgF4RWdQVeOQghROmmypFZVIO7WVl/AXSgIr+gMunIUQojSSZPV1rfu8HC7aOxrGxlxonFSQqVj4yJzGltDX/ITE+0ieajMbNEZdOUohBCl4wvOCwWQJQ3kCwX3nXpq8l/k8S/5Ji9/+extnZhw2kMrzfmJCadpmLm/X/mK//g/+EEy+7NCYrYQoq/IQgzvRkzOSoxXwJ0QQuREFmJ4N2JyVcX4TshRCCH6iizE8G7E5NDxq1ziVY5CCNFX+ATmUAT06Ki/vRsxOSTcT0y0i+8hxseLTbUuRyGE6Ct8AvNb3+pf97zzgMnJPb/2Gw03v3Zttsdftw5YvrzdYYUc2P33twvfzbem8kBithCi70kTGV60DWnIqxSqniiEEH1Pmsjwom0oeh8+5CiEEH1Pmsjwom0oeh8+5CiEEH3P1JQ/CG5qyp+SPEQ3ArMvJXmIsbHkkexZIEchhOh7Nm1qD6DbuRM45xzgwgv3xE3s2uXmfc6iWR+7G4E5Ll4PD7fXnhgfBzZvVinU1EjMFkJ0QyiddwhfZHa3gngZgrrEbCGESEjaSGvf+t0K4lUQ1EPIUQgh+p7ZREW3lkddujQsJA8NtZdS9ekeVRDUQ8hRCCH6nlA67zlzkm2/ZUs4BuI3v5k+f+21ft1j7739269alcyGPJGjEEL0PVu3+tsff7w4G7Zs8bdfc01xNoSQoxBC9D1V0AFCVME2OQohRN9TBR0gRBVsk6MQQvQ9vmC3oaF0ZVMHuvw2HR8vNoguDXIUQgiB5JlbQ+y11+yPPTjoSrQWGUSXBgXcCSH6nqzKk3ZDkZlqmyjgTgghElIFwbgKNoSQoxBC9D1VEIyrYEMIOQohRN8TKk+aRsxOGpznoyqidQg5CiFE3xMqT3r44cn3ccop7Y4lnvm1yeRkNUXrEBKzhRAiQJqssnlklM0bidlCCNElabLK5pFRtirIUQghRIA0WWV961Y5I2waSnEUJN9AcjPJ3SSXxZadQXIryTtIHluGfUIIAbgKdd2sGxLJqyxc+yjrieK7AF4HYGNrI8kxAG8EsBTAKwCsJTmLTPFCCNE9y5e3Pyk0Gk60brY3Gk6cXru2ffuQSF5l4drH4MyrZI+ZfQ8A2B4jfzyAK8zsCQA/JrkVwNEAbirWQiGEANasadcedu1yacnjwnWIE0+sn2OIUzWNYgGAe1rm743a2iA5QfIWkrds27atEOOEEP1Fr4jR3ZKboyD5ZZLf9UzHd9rM0+Z9f9fM1pnZMjNbNn/+/GyMFkKIFnpFjO6W3IaezGzlLDa7F8AhLfMHA7gvG4uEECIdU1NOpN6xY09bHcXobqna0NPVAN5Ici+ShwI4DMA3SrZJCNGn9IoY3S2liNkkXwvgIwDmA/g8ydvM7Fgz20zySgBbAOwE8KdmliLkRQghsqUXxOhuKeutp6sAXBVYNgWgzx7shBCiulRt6EkIIUTFkKMQQgjRETkKIYQQHZGjEEII0ZGeqEdBchuAkkujJ2Z/AA+WbUQO6LzqR6+em84rOYvMbMaI5Z5wFHWC5C1JCoXUDZ1X/ejVc9N5ZY+GnoQQQnREjkIIIURH5CiKZ13ZBuSEzqt+9Oq56bwyRhqFEEKIjuiJQgghREfkKIQQQnREjqIASJ5N8qckb4umVS3LziC5leQdJI8t085uIPlXJI3k/tE8SX44Ordvk/ztsm1MA8n3R3bfRvI/SY5G7XU/r78n+f3I9qtIzm1ZVuu+SPINJDeT3E1yWWxZ3c/tFZHtW0m+s3ADzExTzhOAswH8lad9DMDtAPYCcCiAHwJolG3vLM7vEABfhAt63D9qWwXgC3BVC18E4Otl25nynJ7a8vltAC7qkfP6PwAGo8/nAjg3+lz7vgjgOQCOAHAdgGUt7bU+NwCNyOZnAhiOzmWsSBv0RFEuxwO4wsyeMLMfA9gK4OiSbZoN/wjgHZhetvZ4AJeZ42sA5pI8qBTrZoGZ/bJldm/sObe6n9d/mtnOaPZrcFUkgR7oi2b2PTO7w7Oo7ud2NICtZvYjM/sNgCvgzqkw5CiK4/Tocf9ikvtFbQsA3NOyzr1RW20geRyAn5rZ7bFFvXBuUyTvAXAigLOi5tqfVwunwT0dAb11XnHqfm6l219K4aJehOSXARzoWbQGwIUA3g/3q/T9AD4I909Kz/qVe195hnM7E244o20zT1ulzq3TeZnZZ8xsDYA1JM8AcDqA96AHzitaZw1cFckNzc0861fqvIBk5+bbzNNWuXPrQOn2y1FkhJmtTLIeyX8B8Llo9l648f0mBwO4L2PTuiZ0biR/C27M93aSgLP/mySPRg3OLek9A/BvAD4P5yhqf14kTwbwKgDjFg2CowbnBaS6Z63U4tw6ULr9GnoqgNgY9msBfDf6fDWAN5Lci+ShAA4D8I2i7ZstZvYdM3uGmS02s8VwHfq3zexncOd2UvSW0IsAPGJm95dpbxpIHtYyexyA70ef635erwDwNwCOM7MdLYtq3RdnoO7ndjOAw0geSnIYwBvhzqkw9ERRDOeRfD7c4+JPALwFAMxsM8krAWyBGwb4UzPbVZqV2XIN3BtCWwHsAHBqueak5gMkjwCwG+5trrdG7XU/rwvg3v75UvQU+DUze2sv9EWSrwXwEQDzAXye5G1mdmzdz83MdpI8He7NwgaAi81sc5E2KIWHEEKIjmjoSQghREfkKIQQQnREjkIIIURH5CiEEEJ0RI5CCCFER+QohCgQkj9pybB7Yxf7OaWZ0VaIvJGjECInSHaMUzKzF3ex+1MAyFGIQpCjED0PyRdGCRmfQnLvqGbBcz3rnRStdzvJj0dti0heG7VfS3LhDO2XkvwQya8COJfk06N6Ft8i+c9oydtD8rHo7wqS15H8ZFQrYgOjaDiSZ5G8meR3Sa6LIsJfD2AZgA109TLmkDyK5PUkbyX5xTpltBU1oOxc65o0FTEB+FsA/wDgnwCc4Vm+FMAd2FNPY17097MATo4+nwbgP2ZovxQul1cjmv8wgLOiz6+Ei85vHuOx6O8KAI/A5fAZAHATgJe02hF9/jiAV0efr0NUcwHAEIAbAcyP5v8ALnq39OuuqTcmpfAQ/cL74HLm/BquEFGclwP4pJk9CABm9lDUfgyA10WfPw7gvBnaAeDfbU+KiJc21zOzz5N8OGDfN8zsXgAgeRuAxQBuAPAyku8AMAJgHoDNcE6qlSMAPBd70nI0ANQm/5SoPnIUol+YB2AfuF/fTwHwq9hyIlnq5tA6re3xfSfZ7xMtn3cBGCT5FABr4Z4c7iF5NpztcQhgs5kdk+A4QqRGGoXoF9YBeDdc/YVzPcuvBXACyacDAMl5UfuNcNk6AVfA6IYZ2uNsjJaD5O8B2C+wno+mU3iQ5D4AXt+y7FEA+0af7wAwn+Qx0XGGSC5NcRwhOqInCtHzkDwJwE4z+zeSDQA3kny5mX2luY65DKNTAK4nuQvAt+DeLHobgItJ/jWAbdiTLTbUHue9AC4n+U0A1wO4O6ndZrY9ql/yHbiswze3LL4UwEUkH4cbBns9gA+TfBrc//X5cMNUQnSNsscKIYToiIaehBBCdESOQgghREfkKIQQQnREjkIIIURH5CiEEEJ0RI5CCCFER+QohBBCdOS/Ab81eFxUttfSAAAAAElFTkSuQmCC\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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\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 $p$ degrade with the propensity $rp$, where $r$ is the rate. Second, existing proteins activate production of new proteins with the propensity $A_0 + \\frac{A_1p^2}{1+(p/p_0)^2}$, where $A_0$ is the basal expression rate, $A_1$ is the expression amplitude, and $p_0$ is the half-maximal expression protein concentration. Simulate this system using the Gillespie algorithm, discussed in class. 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 $p\\approx 10$ and active with $p\\approx 25$. Does the system stay in either of the stable states forever?\n", "\n", "How do we simulate such spatially extended 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 of the DLA Project\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 annihilation project\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 = 0; No. of active walkers = 86.0.\n", "Time step = 0; 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 = 1; No. of active walkers = 68.0.\n", "Time step = 2; No. of active walkers = 66.0.\n", "Time step = 2; No. of active walkers = 64.0.\n", "Time step = 3; No. of active walkers = 62.0.\n", "Time step = 4; No. of active walkers = 60.0.\n", "Time step = 4; No. of active walkers = 58.0.\n", "Time step = 4; No. of active walkers = 56.0.\n", "Time step = 4; No. of active walkers = 54.0.\n", "Time step = 5; No. of active walkers = 52.0.\n", "Time step = 5; No. of active walkers = 50.0.\n", "Time step = 7; No. of active walkers = 48.0.\n", "Time step = 7; No. of active walkers = 46.0.\n", "Time step = 7; No. of active walkers = 44.0.\n", "Time step = 9; No. of active walkers = 42.0.\n", "Time step = 9; No. of active walkers = 40.0.\n", "Time step = 10; No. of active walkers = 38.0.\n", "Time step = 11; No. of active walkers = 36.0.\n", "Time step = 15; No. of active walkers = 34.0.\n", "Time step = 15; No. of active walkers = 32.0.\n", "Time step = 16; No. of active walkers = 30.0.\n", "Time step = 19; No. of active walkers = 28.0.\n", "Time step = 21; No. of active walkers = 26.0.\n", "Time step = 36; No. of active walkers = 24.0.\n", "Time step = 43; No. of active walkers = 22.0.\n", "Time step = 87; No. of active walkers = 20.0.\n", "Time step = 148; No. of active walkers = 18.0.\n", "Time step = 159; No. of active walkers = 16.0.\n", "Time step = 188; No. of active walkers = 14.0.\n", "Time step = 256; No. of active walkers = 12.0.\n", "Time step = 262; No. of active walkers = 10.0.\n", "Time step = 292; No. of active walkers = 8.0.\n", "Time step = 536; No. of active walkers = 6.0.\n", "Time step = 591; No. of active walkers = 4.0.\n", "Time step = 825; 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": [ "### Coding examples for project 3, more to come" ] }, { "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": [ "# import additional modules, which\n", "from scipy.integrate import odeint\n", "from mpl_toolkits.mplot3d import Axes3D\n", "from matplotlib import cm\n", "\n", "N_events = 100\n", "N_trials = 10\n", "a = 10.0\n", "r = 1.0\n", "\n", "N_part = np.zeros((N_events, N_trials))\n", "t_events = np.zeros((N_events, N_trials))\n", "\n", "for j in range(N_trials):\n", " for i in range(1, N_events):\n", " prop_birth = a\n", " prop_death = N_part[i-1, j]*r\n", " prop_tot = prop_birth + prop_death\n", " prob_birth = prop_birth / prop_tot\n", " which_event = (nprnd.random(1) < prob_birth)\n", " if which_event:\n", " N_part[i, j] = N_part[i-1, j] + 1\n", " else:\n", " N_part[i, j] = N_part[i-1, j] - 1\n", "\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", "Calculate the mean of the 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?\n", "\n", "## Dynamics of the the probability distribution" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def dpdt(p, t, a, r):\n", " dpdt = 0*p\n", " maxn = p.size\n", " n = np.arange(maxn)\n", " dpdt[1:maxn] = a*p[0:maxn-1]\n", " dpdt[0:maxn-1] = dpdt[0:maxn-1] + r*n[1:maxn]*p[1:maxn]\n", " dpdt = dpdt - a*p\n", " dpdt = dpdt - r*n*p\n", " return dpdt\n", "\n", "maxn = 20\n", "p = np.zeros(maxn)\n", "p[0] = 1\n", " \n", "t = np.arange(10)\n", "n = np.arange(maxn)\n", "pp = odeint(dpdt, p, t, args=(a, r))\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 with the results of individual simulations.\n", "\n", ">### You turn\n", "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 }