{ "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" ] }, "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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BLw/GdBL4zUH7zcAXgbPAnwNvmHVfVzG2O4HH18N4Bv3/yuDr1KVc6PJ7b9D/24ATg/ffZ4E3jTomP0ErSQ3ochlHkjQkw16SGmDYS1IDDHtJaoBhL0kNMOwlqQGGvSQ1wLCXpAb8H2IXgjRaubfqAAAAAElFTkSuQmCC\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 1005 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 0.33063.\n" ] } ], "source": [ "# a function to integrate using the Monte Carlo approach\n", "def to_integr(x):\n", " return x**2\n", " #return np.exp(np.sin(x))\n", "\n", "# Monte Carlo integrator\n", "def mc_integr(func, a, b, fmax, N):\n", " # func -- function to integrate\n", " # a,b -- the x range of the integration\n", " # fmax -- the maximum value of the integrand\n", " # N -- number of random samples used for the integration\n", " # return -- a real number, estimate of the area\n", " N = int(N) # just making sure that the number of dart throws is integer\n", " x = a+(b-a)*nprnd.random(N) # generating random x coordinated\n", " y = fmax*nprnd.random(N) # generating random y coordinate\n", " counts_under_curve = np.sum(y <= func(x)) # how many dots under the curve?\n", " frac_under_curve = counts_under_curve/N # what is the fraction of dots under the curve?\n", " area = (b-a)*fmax*frac_under_curve # estimate of the area\n", " return area\n", "\n", "# calculating area under the function to_integr using MC integrator\n", "area = mc_integr(to_integr, 0, 1, 3, 100000)\n", "print('Estimate of the area is ', area, '.', sep='')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Above we integrated the function $x^2$ between 0 and 1. The integral should be $1/3$ -- and hopefully after running the cell above, you got a value very close to $1/3$. \n", "\n", ">### Your work\n", "Harden the function `mc_integr()` to be able to handle various user errors. First, expand the function to allow areas under the curve to be negative (this will require you to also have an `fmin` argument, in addition to `fmax`). Second, check if any values of `to_integr(x)` generated in the process of integration fall outside of the range $[f_{\\rm min}, f_{\\rm max}]$. If they do, then expand the range appropriately. Return not just the estimate of the area, but the estimates of $f_{\\rm min}$ and $f_{\\rm max}$ as well.\n", "\n", "## What is the error of MC methods?\n", "The result of the area calculation above came out to be slightly different from $1/3$. Herewithin lies the problem of MC approaches: they are probabilistic, and every time we perform them, the results are different. But how different? We can verify, first of all, that our probabilistic estimate of the area converges to to true one as the number of random samples increases, $N\\to\\infty$ (at least when the true value of the area is known, e.g., when we integrate a function $x^2$). In fact, one can can convince onself that this should be the case rather easily if the numbers are truly uniformley distributed in the rectangle around the curve. \n", "\n", "Can we additionally understand how quickly the convergence happens? For this, we can explore the distribution of the estimates of the area when we perform the calculation with a varying number $N$ of dart throws. That is, we can generate many different estimates with different realizations of $N$ dart throws and see how different they are from each other. The cell below does precisely this. Hopefully you will see that the larger $N$ is, the more concentrated the estimates are. We quantify this decrease of the spread by calculating the standard deviation $\\sigma$ of the estimates calculated at the same value of $N$. We then compare the standard deviations to the $\\sigma = 1/\\sqrt{N}$ line." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "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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jYJfQxgsLC2MiRJs/wJpddVw5aahnc1f01JWThnC4uZ3N+w97XYrpIxYWxkSIjXsPc6zVzxUTI39U18sm5OBLEF4vtX6LeGFhYUyEWL2rlsQEYVZ+ttelhDWwXxLT8zJ5fVed16WYPmJhYUyEWLOrlgtGD2JAiqvTzPSauQXZvFt5hCPNbV6XYvqAq2EhIgtEpFREykTkvm62f05E3hORzSLyhogUhmz7pnNcqYhc62adxnit7lgrWw80cllB5J9VdJpbkIMqvFFmZxfxwLWwcObQfgS4DigEbgsNA8eTqjpFVacDPwJ+4hxbSHDO7vOABcAvO+fkNiYWrXU+cC+Lolnopo0cSEZqImt2WljEAzfPLGYCZapaoaptwBJgUegOqtoYspgGdF6HtwhYoqqtqrobKHOez5iY9PrOWgb1T+K8EQO9LqXHEn0JzM7PZvWuWhuFNg64GRa5wP6Q5Upn3QeIyBdEpJzgmcWXzvDYu0WkWESKa2vtqgwTnVSVNbvqmJ2fjS9CJjrqqbkFORw82kJ57TGvSzEuczMsuvupP+nPD1V9RFXHA98Avn2Gxz6mqkWqWpSTEz2n78aEKq1uorapNaqaoDrNdfpYVltTVMxzMywqgbyQ5ZFA1Wn2XwLcdJbHGhO1Vu8MnhXPjaLO7U55g/szLjuN1TZOVMxzMyw2AAUiMlZEkgl2WC8N3UFECkIWbwB2OY+XAotFJEVExgIFwNsu1mqMZ9bsqmPC0AEMH9jP61LOymUTclhfUU+rv8PrUoyLXAsLVfUD9wDLge3A06paIiIPichCZ7d7RKRERDYD9wJ3OseWAE8D24C/AV9QVftJNDHnRFsHb+1uYG5B9DVBdZpbkE1Le4DiPTb0Ryxz9e4fVV0GLOuy7oGQx18+zbHfA77nXnXGeG/Dngba/AHmRGETVKdLxmWR5JP3O+lNbLI7uI3x0NryOpJ8wsVjB3tdyllLS0lkRt4g1pVbJ3css7AwxkNry+qYMWoQ/ZOjY4iPU5mVn8V7B47a0B8xzMLCGI8cPt5GSVUjs8dHf9PN7PxsVGF9Rb3XpRiXWFgY45E3K+pRhTkFWV6Xcs6m52WSluyzcaJimIWFMR5ZW1ZHWrKPqSMzvS7lnCX5Epg5djDryuzMIlZZWBjjkXXl9Vw8LoskX2z8Gs7Oz6ai7jhVR054XYpxQWz8lBoTZQ4cOcHuuuMxdalp53tZa01RMcnCwhgPdH6gzs6P/v6KThOHppOVlsy6cmuKikUWFsZ4YF1ZHdkDkpk4NN3rUnpNQoJw6fgs1pbV2ZDlMcjCwpg+pqqsLa9n1vhsRKJrSPJw5uRnU9PUSlmNDVkeaywsjOlju2qOUdvUGlNNUJ2s3yJ2WVgY08fWOR+ks2LgZryu8gb3J29wP+u3iEEWFsb0sXXl9eQN7kfe4P5el+KKWeOyWV9RT0fA+i1iiYWFMX2oI6Csr6hn1rjYO6voNCs/i8YWP9uqGr0uxfQiCwtj+tC2qkYaW/zMisH+ik6Xjgu+NxuFNrZYWBjThzo/QDs/UGPRkIxU8ocMsH6LGONqWIjIAhEpFZEyEbmvm+33isg2EXlXRF4VkdEh2zpEZLPztbTrscZEozcr6skfMoAhGalel+KqWeOz2LCngfaOgNelmF7iWliIiA94BLgOKARuE5HCLru9AxSp6lTgz8CPQradUNXpztdCjIly7R0B3t7dENNnFZ0uHZdFc1sH71Ye8boU00vcPLOYCZSpaoWqtgFLgEWhO6jqSlVtdhbXAyNdrMcYT71beYTmtg5mjY/9sLiks9/CRqGNGW6GRS6wP2S50ll3KncBL4Usp4pIsYisF5GbujtARO529imura0994qNcVHnB+clcXBmMSgtmcLhGdZvEUPcDIvuxjHo9sJrEfkEUAQ8HLJ6lKoWAbcDPxOR8Sc9mepjqlqkqkU5OTm9UbMxrllXXk/h8AwGpSV7XUqfmDU+i437DtPS3uF1KaYXuBkWlUBeyPJIoKrrTiIyH7gfWKiqrZ3rVbXK+bcCWAXMcLFWY1zV0t7Bxn2H46IJqtOs/Cza/AE27T3sdSmmF7gZFhuAAhEZKyLJwGLgA1c1icgM4FGCQVETsn6QiKQ4j7OB2cA2F2s1xlWb9h2mzR+I6fsrurpozGB8CWJNUTEi0a0nVlW/iNwDLAd8wOOqWiIiDwHFqrqUYLPTAOAZZ/TNfc6VT5OBR0UkQDDQ/l1VLSxM1HqzvB5fgnDRmMFel9Jn0lOTmDpyIG9WWFjEAtfCAkBVlwHLuqx7IOTx/FMctw6Y4mZtxvSldeX1TB05kPTUJK9L6VOzxmfx6OsVHGv1MyDF1Y8b4zK7g9sYlx1r9bNl/5G4uL+iq0vHZeMPKBv2NHhdijlHFhbGuGzDngb8AY3JIcnDuXD0IJJ9Cbxp/RZRz8LCGJe9WV5Psi+BC0cP8rqUPtcv2ceMUZk2qGAMsLAwxmXryuuYMSqTfsk+r0vxxKzx2ZRUNXKkuc3rUsw5sLAwxkVHm9spqWqMyyaoTrPys1CFt3Zbv0U0s7AwxkXrd9ejSlzdX9HVtJGZ9EvyWb9FlLOwMMZFb5bX0y/Jx7SRmV6X4pnkxASKxgyyfosoZ2FhjIvWlddRNGYQyYnx/as2a3w2O6uPUdvUGn5nE5Hi+yfYGBfVNrWys/pYXPdXdOocE8vu5o5eFhbGuKTzgzGeBg88lfNGZJCemsi6MmuKilYWFsa4ZF1ZHempiZw3IsPrUjyX6Evg4rFZNqhgFLOwMMYla8vruGRcFok++zUDmJ2fxb6GZvY3NIff2UQc+yk2xgX7G5rZ33CC2dYE9b7Z+cG+G7sqKjr1KCxEZKSIfF1EXhCRDSKyWkR+KSI3iIgFjjFdrHXa5js/IA0UDBlATnoKa21e7qgUdsxgEfkfgnNn/xX4IVADpAITgAXA/SJyn6qudrNQY6LJ2vJ6hqSnkD9kgNelRAwRYfb4LN4oq0NVceawMVGiJwPM/1hVt3azfivwnDML3qjeLcuY6KWqvFlex5z8bPtA7GJWfjbPb66itLqJScOs4z+ahG1C6gwKERnSdZuITFTVNlUt6+5YEVkgIqUiUiYi93Wz/V4R2SYi74rIqyIyOmTbnSKyy/m688zeljHeKa1uou5YG7OsCeoknc1y1hQVfc6kv2GNiHysc0FEvgb85VQ7i4gPeAS4DigEbhORwi67vQMUqepU4M/Aj5xjBwMPAhcDM4EHRST+xnc2Uanzg9D6K06Wm9mPMVn97X6LKHQmYTEPuENEnhGR1QT7LGaeZv+ZQJmqVqhqG7AEWBS6g6quVNXO6+jWAyOdx9cCK1S1QVUPAysI9o8YE/HWldUxNjuN3Mx+XpcSkWblZ/PW7gb8HQGvSzFnoMdhoaoHgb8BlwJjgP9V1WOnOSQX2B+yXOmsO5W7gJfO5FgRuVtEikWkuLa2Nux7MMZt7R0B1lfU213bpzF7fHZwqtnKo16XYs5Aj8NCRFYQbBY6H7ge+KmI/MfpDulmnZ7iuT8BFAEPn8mxqvqYqhapalFOTs7pyjemT7xbeYTjbR3WBHUal47PQuTvlxeb6HAmzVCPqOonVfWI0+k9CzjdnwaVQF7I8kigqutOIjIfuB9YqKqtZ3KsMZFmbVk9InDpODuzOJXBackUDs+wsIgyYcNCnGv/VPX50PWq6lfV74bu08UGoEBExjqX1y4GlnZ57hnAowSDoiZk03LgGhEZ5HRsX+OsMyaivbGrjvNHDGRQWrLXpUS0OQXZbNp3mOOtfq9LMT3UkzOLlSLyRRH5wL0UIpIsIleKyO+Bky5tVVU/cA/BD/ntwNOqWiIiD4nIQme3h4EBwDMisllEljrHNgDfJRg4G4CHnHXGRKxjrX427TvMnAJrggpnbn4O7R3KW7vtEtpo0ZOb8hYAnwaeEpGxwBGCd3D7gJeBn6rq5u4OVNVlwLIu6x4IeTz/VC+qqo8Dj/egPmMiwvryevwBZa6FRVhFYwaRkpjAml11XDlpqNflmB4IGxaq2gL8EviliCQB2cAJVT3idnHGRJM1u2rpl+TjwtF2S1A4qUk+Zo4dzJpd1m8RLXrSZ5EqIl8RkV8A/wDUWlAYc7I1u+q4eNxgUhJ9XpcSFS4ryKGs5hgHj57wuhTTAz3ps/g9wcta3yN4yeyPXa3ImChUebiZirrjzC2wS7h7qrNvx84uokNPwqJQVT+hqo8CHwXmulyTMVHnDecD7zLrr+ixScPSyR6Q8v73zkS2noRFe+cD5wonY0wXa3bVMTTDhiQ/EyLC3IJs3iirIxDo9n5dE0F6EhbTRKTR+WoCpnY+FpFGtws0JtJ1BJS15XXMLcixIcnP0NyCbBqOt7HtoH2URLqeXA1lvXXGnMbWA0c50txul8yehTn5f++3OD93oMfVmNOxKVGNOUdv2BSqZ21IRiqThqWzZpcNBBrpLCyMOUerSms4b0QG2QNSvC4lKl02IYfiPTb0R6SzsDDmHBxtbmfTviNcMfGkiSRND82bkENbR4B15Tb0RySzsDDmHKwpq6UjoFwxye6vOFtFYwaTluxjZWlN+J2NZywsjDkHq0prGdgviel5NsTH2UpOTGB2fjavl9aiapfQRioLC2POUiCgvL6zlrkF2fgS7JLZczFv4hAOHDlBWc3pJt80XrKwMOYsbTvYSG1Tq/VX9IJ5E4PNeNYUFbksLIw5S6ucD7bLJlh/xbkakdmPiUPTWVVql9BGKlfDQkQWiEipiJSJyH3dbL9MRDaJiF9EPtplW4czIdL7kyIZE0lWltYydeRActLtktneMG9SDhv2NHDMLqGNSK6FhYj4gEeA64BC4DYRKeyy2z7gU8CT3TzFCVWd7nwt7Ga7MZ450tzGO/sOM8/OKnrNvAlDaO9Qm5s7Qrl5ZjETKFPVClVtA5YAi0J3UNU9qvouEHCxDmN63ZpddQQULrf+il5TNGYQA1ISrSkqQrkZFrnA/pDlSmddT6WKSLGIrBeRm3q3NGPOzcrSGjL7JzE9L9PrUmJGki+BOfnZrCqtsUtoI5CbYdHdtYRn8hMwSlWLgNuBn4nI+JNeQORuJ1CKa2vtrxHTN/wdAVbuqOGKiUPsktleduXkIRw82kJJlY1CG2ncDItKIC9keSRQ1dODVbXK+bcCWAXM6Gafx1S1SFWLcnKs7dj0jY17D3O4uZ2rC4d6XUrMuWrSEERgxbZqr0sxXbgZFhuAAhEZKyLJwGKgR1c1icggEUlxHmcDs4FtrlVqzBl4ZXs1yb4Eu2TWBVkDUrhw1CBe2W5hEWlcCwtnVr17gOXAduBpVS0RkYdEZCGAiFwkIpXALcCjIlLiHD4ZKBaRLcBK4N9V1cLCeE5VWbGtmkvHZzEgJex0MOYsXF04lJKqRg4cOeF1KSaEqz/tqroMWNZl3QMhjzcQbJ7qetw6YIqbtRlzNspqjrGnvpm75o7zupSYNb9wKD94aQevbKvmzlljvC7HOOwObmPOwAqneeTqydZf4ZbxOQMYl5NmTVERxsLCmDOwYls1U0cOZNjAVK9LiWlXFw5lfUU9jS3tXpdiHBYWxvRQTVMLm/cfYb6dVbju6slDae9Qu0EvglhYGNNDr22vQRW7ZLYPzBg1iKy0ZF6xS2gjhoWFMT308rZqcjP7MWlYutelxDxfgnDlpCGsLK2hzW+jAUUCCwtjeuDoiXbe2FXHdecPQ8Tu2u4L100ZRlOL3wYWjBAWFsb0wCvbqmnrCHD91OFelxI35uTnkJ6ayF/fPeh1KQYLC2N6ZNl7BxkxMJUZNnBgn0lOTODqwqGs2HbImqIigIWFMWEcPdHOml11XD9luDVB9bEbpw6n0ZqiIoKFhTFhWBOUd6wpKnJYWBgTxrL3DpKb2c+aoDyQnJjANYXDrCkqAlhYGHManU1QdhWUd26YOsyaoiKAhYUxp9HZBHWDNUF5xpqiIoOFhTGn0dkEZdOneqezKerlbYdo9Xd4XU7csrAw5hTqj7Xy+s5abpxqV0F5beH0ETS1+Fm5w8aK8oqFhTGn8OKWKvwB5eYLTppyxfSx2eOzGJKewnObKr0uJW65GhYiskBESkWkTETu62b7ZSKySUT8IvLRLtvuFJFdztedbtZpTHeIZz1WAAARtklEQVSe3XSA83MzmGhjQXku0ZfATTNyWVlaQ8PxNq/LiUuuhYWI+IBHgOuAQuA2ESnssts+4FPAk12OHQw8CFwMzAQeFJFBbtVqTFc7q5t478BRbp5hZxWR4uYLcmnvUF7cUuV1KXHJzTOLmUCZqlaoahuwBFgUuoOq7lHVd4GuF1BfC6xQ1QZVPQysABa4WKsxH/DcpgP4EoSF00d4XYpxTBqWQeHwDGuK8oibYZEL7A9ZrnTW9dqxInK3iBSLSHFtrXV8md7REVCef+cA8ybkkD0gxetyTIibL8hlS+VRymqavC4l7rgZFt1dPqK9eayqPqaqRapalJOTc0bFGXMqb5bXc6ixxTq2I9DC6SPwJQjPbTrgdSlxx82wqATyQpZHAj1tbDyXY405J89tqiQ9NZGrJg/xuhTTxZD0VC4ryOYv7xwgEOjp356mN7gZFhuAAhEZKyLJwGJgaQ+PXQ5cIyKDnI7ta5x1xrjq6Il2lm09yI1TR5Ca5PO6HNONj1w4koNHW1i9y5qe+5JrYaGqfuAegh/y24GnVbVERB4SkYUAInKRiFQCtwCPikiJc2wD8F2CgbMBeMhZZ4yrnt1YSUt7gI9fPMrrUswpXFM4jOwByTyxfp/XpcSVRDefXFWXAcu6rHsg5PEGgk1M3R37OPC4m/UZE0pVeeKtvUzPy+T83IFel2NOITkxgY8V5fGr18s5cOQEuZn9vC4pLtgd3MY43iyvp6L2OHdcMtrrUkwYt188CgWeesvOLvqKhYUxjife2ktm/yQbYTYKjBzUnysnDmHJhv02z0UfsbAwBqhpbOHlkmpuuXCkdWxHiU9cMpq6Y628vO2Q16XEBQsLY4AlG/bjDyi3X2xNUNHisgk5jBzUjyfW7/W6lLhgYWHiXntHgKfe3sfcgmzGZqd5XY7pIV+CcPvFo1hf0cDOaruj220WFibuLd1cxcGjLfzD7DFel2LO0OKLRtEvycejr1d4XUrMs7AwcS0QUB5dXc7EoelcMdHu2I42g9OSufWiPF7YfICqIye8LiemWViYuLaytIad1cf47OXjbDa8KHXXnLEo8Ns3dntdSkyzsDBx7Vevl5Ob2Y8PTbOhyKNV3uD+fGjqcJ56ex9Hmm1iJLdYWJi4tXFvAxv2HOauOWNJ8tmvQjT77OXjaW7r4A9v2pVRbrHfEBO3/ntVBZn9k1g8My/8ziaiTR6ewRUTc/jduj20tHd4XU5MsrAwcWnrgaO8sr2aOy8dQ/9kV4dIM33kn+blU3+8ze67cImFhYlLP1peSmb/JO6aO9brUkwvmTl2MHMLsnlkZRmNLe1elxNzLCxM3FlXXsfqnbV8YV4+GalJXpdjetE3FkzicHM7v15t9130NgsLE1dUlR/+rZThA1O541Ib2iPWnJ87kBunDuc3a3ZT09TidTkxxcLCxJW/bT3Elv1H+Or8CTZgYIz62jUTae8I8F+vlnldSkxxNSxEZIGIlIpImYjc1832FBH5k7P9LREZ46wfIyInRGSz8/UrN+s08aG9I8DDL5eSP2QAN1+Q63U5xiVjs9O49aI8nnp7H3vqjntdTsxwLSxExAc8AlwHFAK3iUhhl93uAg6raj7wU+CHIdvKVXW68/U5t+o08eN/1u6movY431gwiUS7ryKmffmqAlISE/jOiyWoqtflxAQ3f2NmAmWqWqGqbcASYFGXfRYBv3ce/xm4SmzMBeOCA0dO8NMVu5g/eQjzJ9sYULFuSEYq914zkVWltby01ea76A1uhkUusD9kudJZ1+0+quoHjgJZzraxIvKOiLwuInO7ewERuVtEikWkuLa2tnerNzHlwRdKAPjOwvNsDKg4ceeloykcnsG/vlhCk11Ke87cDIvufiO7ng+eap+DwChVnQHcCzwpIhkn7aj6mKoWqWpRTk7OORdsYtPykkO8sr2ar8wvYOSg/l6XY/pIoi+B7988hZqmVn788k6vy4l6boZFJRA6jsJIoOpU+4hIIjAQaFDVVlWtB1DVjUA5MMHFWk2Mampp5ztLS5g0LJ1Pz7Eb8OLN9LxM7rhkNL9/cw9b9h/xupyo5mZYbAAKRGSsiCQDi4GlXfZZCtzpPP4o8JqqqojkOB3kiMg4oACwu2zMGVFV/uX5rVQ3tvC9D0+xwQLj1NevncjQ9FS+8qfNHGv1e11O1HLtt8fpg7gHWA5sB55W1RIReUhEFjq7/RbIEpEygs1NnZfXXga8KyJbCHZ8f05VG9yq1cSm5zYd4PnNVXz5qglcOHqQ1+UYj2SkJvGzxdPZW3/8/b4rc+YkVi4rKyoq0uLiYq/LMBGiovYYN/7XG5yfO5Cn/vESfAnWqR3vfrJiJ//56i5+dut0bpph99l0EpGNqloUbj87Lzcxp9XfwRefeofkxAR+vni6BYUB4EtX5nPRmEHc/5f37Ga9s2BhYWKKqvKNP79LSVUjP/rIVIYP7Od1SSZCJPoS+NniGST6EvjM/xZztNkupz0TFhYmpvxkxU6e31zF/7t2ItecN8zrckyEyc3sx6N3XMje+uN89oliWv02UVJPWViYmPH0hv3812tl3FqUx+fnjfe6HBOhLhmXxcMfncb6igbue/Y9Gw6kh2yKMBMTVmyr5lt/eY+5Bdn824fPt7u0zWndNCOX/Q3N/HjFTnLSU/jmdZPsZyYMCwsT9V7cUsVX/7SZ80Zk8MjHL7D7KUyP3HNlPjVNrTy2uoI2f4AHbiwkwS6GOCULCxPVninezzeefZei0YP57aeKSLeZ70wPiQgPLTqPlMQEfvPGbk60dfD9m6fY1XOnYGFholIgoPz36+U8vLyUuQXZPHZHEf2SbTIjc2ZEhPtvmEz/lET+89VdHDnRxn/cMs3+6OiGna+bqNPY0s5nn9jIw8tLWThtBL/+pAWFOXsiwr1XT+CBGwt5ZXsNNz2ylrKaJq/LijgWFiaqbD1wlEW/WMvKHTU8cGMhP1883aZHNb3i03PG8sRdF3P0RDuLfrGWFzYfsCulQlhYmKhwoq2DH7y0nUWPrOVYq58n//ESPj1nrF3BYnrVpeOzePGLc5g4LJ0vL9nMZ35fTNWRE16XFRFsbCgT0VSVV7bX8N2/bmNfQzO3FuXxresnM7C/tSkb9/g7Avxu3R5+/PJOEgS+Mn8Cd1w6OibPYns6NpSFhYlIqsqqnbX8dMVO3q08yrjsNL734SlcOj4r/MHG9JL9Dc38ywtbWVVay9CMFO65Ip+PXZRHSmLshIaFhYlKx1r9vLD5AH9cv49tBxsZOagfX7qygA9fkGv3TxjPrCuv46crdrJhz2GGpKew+KI8bp05itzM6B97zMLCRI2W9g7W7KpjeckhXnrvIMfbOpg8PINPXjqaj1wwkuRECwnjPVXljbI6Hn9jN6t21iLA3IIcrp8yjPmTh5I1IMXrEs+KhYWJWP6OANsPNvHW7nrWVzSwrryO5rYOMlITufa8Ydx+8Sim52Va57WJWPsbmvnThv08v/kAlYdPkCBw4ehBXDIui4vHZnHB6Ez6J0fHbWwRERYisgD4OeADfqOq/95lewrwv8CFQD1wq6rucbZ9E7gL6AC+pKrLT/daFhaRpyOgVB05we664+ypP86OQ01sq2pkx6FGWtoDAIzJ6s+cgmyuPW8Yl4zLsqYmE1VUlW0HG1leUs2q0hq2HjhKQMGXIIzPSaNweAaThmcwLjuNMdlpjBrcP+I6yT0PC2cO7Z3A1UAlwTm5b1PVbSH7fB6YqqqfE5HFwIdV9VYRKQSeAmYCI4BXgAmqesrxhC0selcgoLQHArR3KK3tHbR1BGhpD3CirYMT7X6a2zpoavFzrMVPY0s7h5vbaDjeTsPxVqobW6lubKGmqZWOwN9/vjJSEykckUHh8IFMyxvIJeOyGJqR6uG7NKZ3NbW0s3HvYTbtPcy2g41sq2qk6mjLB/YZnJbMsIxUhg1MZXBaMllpyQxKSyYjNYkBqYmkpyaSlpxI/2QfqUk++iX7SPYlkJKUQLIvgSRfQq8OSdLTsHDzPGkmUKaqFU5BS4BFwLaQfRYB33Ee/xn4hQTbHhYBS1S1FdjtzNE9E3izt4s80tzGLb/q9ac9yblE8qkCXUMehO6hqijQeZiiBAJ/3xZQCIT82xFQAgGlQxV/ILgc+iHfE74EYVD/ZAanJTE0I5X8IdkMy0glb3A/xmSlMTorjaEZKda0ZGJaemoS8yYOYd7EIe+vO9rczp764Nn13vpmDjW2UH20hUONLWw/2Ej98Tba/IEzep0EgcSEYGgkJgjT8jJ54jMX9/bb+QA3wyIX2B+yXAl0fTfv76OqfhE5CmQ569d3OfakSXNF5G7gboBRo0adVZEJCULB0AFndeyZEs7hg/IUh3auFvngs4sEt3V+OAeXBRHwSfBfEcGXEFxOcH7oOv9Ncv6CSfIJKYk+UhITSE5MoH+yj37JifRL8pHu/BWUnppEekqijdhpTDcG9k9iWv9MpuVldrtdVd8/U29qaaexxc+Jtg6a2/ycaO+gtT1Aq7+DVn+Ato4A7X6lvSOAP6AEVPF3KCMy3T9DdzMsuvvk6Prn6qn26cmxqOpjwGMQbIY60wIBMlKT+OXHLzybQ40x5pyJCGkpiaSlJDJsYOQ2y7rZm1gJ5IUsjwSqTrWPiCQCA4GGHh5rjDGmj7gZFhuAAhEZKyLJwGJgaZd9lgJ3Oo8/CrymwQb6pcBiEUkRkbFAAfC2i7UaY4w5DdeaoZw+iHuA5QQvnX1cVUtE5CGgWFWXAr8F/uB0YDcQDBSc/Z4m2BnuB75wuiuhjDHGuMtuyjPGmDjW00tn7Q4oY4wxYVlYGGOMCcvCwhhjTFgWFsYYY8KKmQ5uEakF9npdx1nIBuq8LsID8fi+4/E9Q3y+72h6z6NVNSfcTjETFtFKRIp7ciVCrInH9x2P7xni833H4nu2ZihjjDFhWVgYY4wJy8LCe495XYBH4vF9x+N7hvh83zH3nq3PwhhjTFh2ZmGMMSYsCwtjjDFhWVhEEBH5uoioiGR7XUtfEJGHRWSHiLwrIn8Rke6nEosBIrJAREpFpExE7vO6HreJSJ6IrBSR7SJSIiJf9rqmviQiPhF5R0T+6nUtvcXCIkKISB5wNbDP61r60ArgfFWdCuwEvulxPa4QER/wCHAdUAjcJiKF3lblOj/wNVWdDFwCfCEO3nOoLwPbvS6iN1lYRI6fAv9MN9PHxipVfVlV/c7ieoIzIsaimUCZqlaoahuwBFjkcU2uUtWDqrrJedxE8IMz19uq+oaIjARuAH7jdS29ycIiAojIQuCAqm7xuhYPfRp4yesiXJIL7A9ZriROPjgBRGQMMAN4y9tK+szPCP7hF/C6kN7k2kx55oNE5BVgWDeb7ge+BVzTtxX1jdO9b1V9wdnnfoLNFn/sy9r6kHSzLi7OIEVkAPAs8BVVbfS6HreJyI1AjapuFJF5XtfTmyws+oiqzu9uvYhMAcYCW0QEgk0xm0Rkpqoe6sMSXXGq991JRO4EbgSu0ti96acSyAtZHglUeVRLnxGRJIJB8UdVfc7revrIbGChiFwPpAIZIvKEqn7C47rOmd2UF2FEZA9QpKrRMmLlWRORBcBPgMtVtdbretwiIokEO/CvAg4AG4DbVbXE08JcJMG/fH4PNKjqV7yuxwvOmcXXVfVGr2vpDdZnYbz0CyAdWCEim0XkV14X5AanE/8eYDnBjt6nYzkoHLOBO4Arnf/bzc5f2yZK2ZmFMcaYsOzMwhhjTFgWFsYYY8KysDDGGBOWhYUxxpiwLCyMMcaEZWFhjDEmLAsLY4wxYVlYGOMSEbnImasjVUTSnHkdzve6LmPOht2UZ4yLROTfCI4R1A+oVNUfeFySMWfFwsIYF4lIMsGxoFqAWara4XFJxpwVa4Yyxl2DgQEEx8BK9bgWY86anVkY4yIRWUpwZryxwHBVvcfjkow5KzafhTEuEZFPAn5VfdKZh3udiFypqq95XZsxZ8rOLIwxxoRlfRbGGGPCsrAwxhgTloWFMcaYsCwsjDHGhGVhYYwxJiwLC2OMMWFZWBhjjAnr/wMCjMgmExXNoAAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sigma = 1.0 # standard deviation\n", "mu = 1.0 # mean\n", "x = np.arange(-5,5,0.1)\n", "y = 1/np.sqrt(2*np.pi*sigma)*np.exp(-(x-mu)**2/(2*sigma**2))\n", "plt.plot(x,y)\n", "plt.xlabel('x')\n", "plt.ylabel('P(x)')\n", "plt.title('Normal distribution')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Why were your instructors so confident that the distribution of errors is Gaussian? To understand this, we will start from afar, from random walks.\n", "\n", "Random walks is probably my favorite subject in the modern physics curriculum. Many of the famous names in the late nineteenth and early twentieth century physics and math have contributed to the field, including Einstein, Polya, Wiener, and many others. A single lecture cannot do justice to the subject, and I hope you will pick some of the books to read instead. The \n", "__[Introduction to Probability](http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/pdf.html)__ by Grinstead and Snell is a good starting point. Another great book is __[Random Walks in Biology](https://www.amazon.com/Random-Walks-Biology-Howard-1993-09-07/dp/B01MQZYDZ4)__ by Howard Berg. \n", "\n", "In general, **random walk** is *a stochastic process that consists of a succession of random steps*. The usual and overused analogy is that of a drunkard, who tries to walk, falls, stands up, and, having forgotten which way he came from, chooses a random direction to go to, before falling again and repeating the cycle. There are many different types of random walks. *Discrete time* (DT) walks mean that every step takes a discrete unit of time. In contrast, in *continuous time* (CT) walks, the duration of each step is a real-values variable, sampled from some distribution. Similarly, one can define *discrete space* (DS) or *continuous space* (CS) random walks by whether the step length is fixed or sampled at random from some probability distribution. As we will see later in this lecture details of these probability distributions matter very little (as long as they have finite variances of their own), and we can learn a lot about many different random walks by studying the simplest version of a random walk, the DSDT walk with the duration of a step $\\Delta t=1$ and the step size $a=1$. Such walks can exist in any number of dimensions (with 1-d being, of course, the simplest). However, for illustration purposes, below we will mostly focus on 2-d walks on a square lattice, where the walker at any point in time can move in one of four directions: up, down, left, or right. Let's generate a few such walks of different lengths." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# This function will generate a 2d DSDT random walk\n", "def DSDT_2d_walk(N):\n", " # Input:\n", " # N - duration of the walk\n", " # Return:\n", " # x,y -- arrays of length N, coordinates of the walker on every step\n", " \n", " directions = np.floor(nprnd.random(N)*4) # (0,1,2,3) directions of motion on each time step\n", " # Do you understand why multiplication by 1.0 is needed in the next two lines?\n", " # Check what happens if this multiplication is removed.\n", " dx = (directions == 0) - 1.0*(directions == 1) # directions == 0/1 corresponds to right/left motion\n", " dy = (directions == 2) - 1.0*(directions == 3) # directions == 2/3 corresponds to top/down motion\n", " \n", " x = np.cumsum(np.hstack((0,dx))) # x and y coordinates of the walker are a cumulative sum of steps\n", " y = np.cumsum(np.hstack((0,dy))) # Notice that the initial position of the walker is always 0. \n", " # Also notice that the walk of N steps has N+1 positions in it -- there is the initial and the final\n", " # position, surrounding N steps.\n", " \n", " return x, y\n", "\n", "\n", "N = np.array((100, 1000, 10000)) # we will explore random walks of these different lengths\n", "for i in np.arange(N.size):\n", " x, y = DSDT_2d_walk (N[i])\n", " plt.plot(x, y, 'b-o')\n", " plt.title('Random Walk of length ' + str(N[i]))\n", " plt.xlabel('x coordinate')\n", " plt.ylabel('y coordinate')\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice how a moving particle comes back again and again, and how random walk eventually covers almost the entire 2-d space, with very few holes in the middle. This is a crucial property of random walks, which we will refer to again in one of the projects for this Module.\n", "\n", ">### Verification\n", "To verify that the function generates random walks correctly, generate very short walks, e.g., `N=4...10` and see if the walker makes any unallowed steps visually. \n", "\n", ">### Your turn\n", "Run the cells multiple times and convince yourself that the walks are not biased. Sometimes they move more to one side or the other, but other times it's the opposite.\n", "\n", "We can now see that the details of how the individual steps are generated in a walk matter much less than one might naively think. For this, we define a CSDT walk, where in each individual step, the walked moves in both $x$ and $y$ directions by steps of sizes taken from ${\\cal U}(0,2^{1/4})$. We then compare the DS and the CS walks to each other. \n", "\n", "Note the weird $(3/2)^{1/2}$ multiplier. We do this to ensure that the average steps we take in each of the walks are equal. Indeed, for CSDT walk, the steps, on average, are smaller than 1, but we make steps in both the $x$ and the $y$ direction always. The algrebra is a bit more complex than I want to go over here, but I hope the logic is clear; we will return to this multiplier later." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# This function will generate a 2d CSDT random walk with steps taken from the uniform distribution\n", "def CSDT_2d_walk(N):\n", " # Input:\n", " # N - duration of the walk\n", " # Return:\n", " # x,y -- arrays of length N, coordinates of the walker on every step\n", " \n", " dx = (3/2)**(1/2)*(2*nprnd.random(N) - 1) # displacements along x\n", " dy = (3/2)**(1/2)*(2*nprnd.random(N) - 1) # displacements along y\n", " \n", " x = np.cumsum(np.hstack((0,dx))) # x and y coordinates of the walker are a cumulative sum of steps\n", " y = np.cumsum(np.hstack((0,dy))) # Notice that the initial position of the walker is always 0. \n", " # Also notice that the walk of N steps has N+1 positions in it -- there is the initial and the final\n", " # position, surrounding N steps.\n", "\n", " return x, y# comparing different ways of generating random walks\n", "\n", "\n", "N = np.array((100, 1000, 10000)) # we will explore random walks of these different lengths\n", "for i in np.arange(N.size):\n", " xD, yD = DSDT_2d_walk (N[i]) # discrete walk of this length\n", " xC, yC = CSDT_2d_walk (N[i]) # continuous walk of this length\n", " plt.plot(xD, yD, 'b-')\n", " plt.plot(xC, yC, 'r-')\n", " plt.title('Different types of Random Walks of length ' + str(N[i]))\n", " plt.xlabel('x coordinate')\n", " plt.ylabel('y coordinate')\n", " plt.legend(('DSDT walk', 'CSDT walk'))\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that we can easily see the difference between the two walks when there are only a few steps, and the walks do not move far away from the origin. However, at $N=10000$, when individual steps are invisible, it would be hard to distinguish one walk from the other. \n", "\n", "To formalize this observation, we will look at the means and the variances of displacements in walkers of different types. For this, we will generate many different walks of the same type and the same length and average." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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" ] }, "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": [ "## Stochastic simulations of biochemical systems\n", "\n", "### Generating system trajectories\n", "Here we simulate the system that has the propensity of particle creation $a$ and the propensity of destruction as $rn$, where $n$ is the number of particles. The mean expected number of particles is $\\left=a/r$." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import numpy.random as nprnd\n", "import matplotlib.pyplot as plt\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": 22, "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 }