{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "[![Py4Life](https://raw.githubusercontent.com/Py4Life/TAU2015/gh-pages/img/Py4Life-logo-small.png)](http://py4life.github.io/TAU2015/)\n", "\n", "## Lecture 8 - 13.5.2015\n", "### Last update: 14.5.2015\n", "### Tel-Aviv University / 0411-3122 / Spring 2015" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Previously\n", "\n", "- Data analysis\n", "- Summary statistics\n", "- Data visualization\n", "- Packages: [NumPy](http://www.numpy.org/), [Matplotlib](http://matplotlib.org/), [Pandas](pandas.pydata.org), [Seaborn](http://stanford.edu/~mwaskom/software/seaborn/)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Today\n", "\n", "- Sochastic processes\n", "- Probability distributions\n", "- Random numbers\n", "- Simulations\n", "- the [SciPy](http://www.scipy.org/scipylib/index.html) package: provides many user-friendly and efficient numerical integration and optimization, probability distributions, and special mathematical functions." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "from scipy import stats \n", "import seaborn as sns\n", "sns.set_style('white')\n", "sns.set_context('talk', font_scale=1.2)\n", "sns.set_palette(sns.color_palette('Set1'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Stochastic processes\n", "\n", "Stochastic processes are models that describe change through time in the state of a system subject to random events and noise.\n", "The randomness takes into account the variability observed in real-world phenomena. \n", "The simplest example is [Polya's urn](http://en.wikipedia.org/wiki/P%C3%B3lya_urn_model):\n", "\n", "## Polya's Urn\n", "\n", "Imagine an urn with two balls, half of them are blue and half of them are red. \n", "At each turn of the game, a random ball is drawn from the urn, and then returned together with an additional ball of the same color.\n", "The system is the urn, the state of the system is the number of red and blue balls.\n", "The randomness is given by the random draw of balls, and the system evolves from state to state by chance.\n", "Denoting the number of red balls after $t$ draws by $n_t$, we can formulate the model by (the total number of balls after $t$ draws is $t+2$):\n", "\n", "$\n", "n_{t+1} = \\left\\{\\begin{matrix}\n", "n_t + 1, & \\frac{n_t}{t+2}\\\\ \n", "n_t, & 1-\\frac{n_t}{t+2}\n", "\\end{matrix}\\right.\n", "$\n", "\n", "In this process one of the colors will take over after some time:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Z7jOTQf1HH1Pyx9N7vPxkslBiLiK9XuPy5ZTMOo2GTz4N92VNm+ZiRNJf1L8X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\njigy/Otzy7f+mz336dKKZy893OWIRJpAxRxzTc8SEY9TYi6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UItKUNJNFRLxOiblUUpaZ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "time = 1000\n", "randoms = np.random.rand(time)\n", "red = np.ones(time)\n", "blue = np.ones(time)\n", "\n", "for t in range(1,time):\n", " red_freq = red[t-1]/(red[t-1] + blue[t-1])\n", " draw_red = (randoms[t-1] < red_freq)\n", " draw_blue = ~draw_red\n", " red[t] = red[t-1] + draw_red\n", " blue[t] = blue[t-1] + draw_blue\n", "\n", "fig,ax = plt.subplots(1, 2, sharex=True, sharey=False)\n", "ax[0].plot(red)\n", "ax[0].plot(blue)\n", "ax[0].set_xlabel('# draws')\n", "ax[0].set_ylabel('# balls')\n", "ax[1].plot(red/(red + blue))\n", "ax[1].set_xlabel('# draws')\n", "ax[1].set_ylabel('% red')\n", "sns.despine()\n", "fig.tight_layout()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notably, even stochastic processes that *seems* simple (easy to explain) have complex dynamics (hard to predict).\n", "For example: What is the average time at least 99% of the urn is of the same color? What is the probability to have $k$ reds after $t$ draws? To those interested, read about [Martingales](http://en.wikipedia.org/wiki/Martingale_%28probability_theory%29)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Branching process\n", "\n", "One of the simplest stochastic process is the Branching process of the [Galton-Wason process](http://en.wikipedia.org/wiki/Galton%E2%80%93Watson_process). \n", "\n", "There was concern amongst the Victorians that aristocratic surnames were becoming extinct. \n", "\n", "Francis Galton originally posed the question regarding the probability of such an event in the _Educational Times_ of 1873, and the Reverend Henry William Watson replied with a solution (Watson & Galton, 1875). \n", "\n", "The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). \n", "\n", "This is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups, and is also of use in understanding other processes (as described below); but its application to actual extinction of family names is fraught. In practice, family names change for many other reasons, and dying out of name line is only one factor.\n", "\n", "To make matters a bit easier we will assume that if _N_ people have the same surname than it is protected from extinction. \n", "We further simplify the model by assuming that each male has $n$ offspring and each offspring has a probability $p$ to be male and reach repductive age." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "sns.set_palette(sns.color_palette('muted'))\n", "N = 1000\n", "n = 3\n", "p = 0.35 # probability for a reproductive son" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We simulate a Branching process on the finite space ${0, 1, ..., N}$. \n", "Each state represents the population size - the number of males with the same surname. \n", "\n", "We assume that at each time step $t$ each male has $n$ offspring.\n", "Each one of the offspring has a probability $p$ to become a reproductive male.\n", "So $1-p$ includes female offspring and male offspring that do not reach reproductive age.\n", "\n", "So the number of reproducrive boys per male has a Binomial distribution $B(n,p)$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise - drawing random numbers from the Binomial distribution\n", "\n", "We can draw numbers from a Binomial distribution using `numpy.random.binomial`:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Help on built-in function binomial:\n", "\n", "binomial(...) method of mtrand.RandomState instance\n", " binomial(n, p, size=None)\n", " \n", " Draw samples from a binomial distribution.\n", " \n", " Samples are drawn from a Binomial distribution with specified\n", " parameters, n trials and p probability of success where\n", " n an integer >= 0 and p is in the interval [0,1]. (n may be\n", " input as a float, but it is truncated to an integer in use)\n", " \n", " Parameters\n", " ----------\n", " n : float (but truncated to an integer)\n", " parameter, >= 0.\n", " p : float\n", " parameter, >= 0 and <=1.\n", " size : int or tuple of ints, optional\n", " Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n", " ``m * n * k`` samples are drawn. Default is None, in which case a\n", " single value is returned.\n", " \n", " Returns\n", " -------\n", " samples : {ndarray, scalar}\n", " where the values are all integers in [0, n].\n", " \n", " See Also\n", " --------\n", " scipy.stats.distributions.binom : probability density function,\n", " distribution or cumulative density function, etc.\n", " \n", " Notes\n", " -----\n", " The probability density for the Binomial distribution is\n", " \n", " .. math:: P(N) = \\binom{n}{N}p^N(1-p)^{n-N},\n", " \n", " where :math:`n` is the number of trials, :math:`p` is the probability\n", " of success, and :math:`N` is the number of successes.\n", " \n", " When estimating the standard error of a proportion in a population by\n", " using a random sample, the normal distribution works well unless the\n", " product p*n <=5, where p = population proportion estimate, and n =\n", " number of samples, in which case the binomial distribution is used\n", " instead. For example, a sample of 15 people shows 4 who are left\n", " handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,\n", " so the binomial distribution should be used in this case.\n", " \n", " References\n", " ----------\n", " .. [1] Dalgaard, Peter, \"Introductory Statistics with R\",\n", " Springer-Verlag, 2002.\n", " .. [2] Glantz, Stanton A. \"Primer of Biostatistics.\", McGraw-Hill,\n", " Fifth Edition, 2002.\n", " .. [3] Lentner, Marvin, \"Elementary Applied Statistics\", Bogden\n", " and Quigley, 1972.\n", " .. [4] Weisstein, Eric W. \"Binomial Distribution.\" From MathWorld--A\n", " Wolfram Web Resource.\n", " http://mathworld.wolfram.com/BinomialDistribution.html\n", " .. [5] Wikipedia, \"Binomial-distribution\",\n", " http://en.wikipedia.org/wiki/Binomial_distribution\n", " \n", " Examples\n", " --------\n", " Draw samples from the distribution:\n", " \n", " >>> n, p = 10, .5 # number of trials, probability of each trial\n", " >>> s = np.random.binomial(n, p, 1000)\n", " # result of flipping a coin 10 times, tested 1000 times.\n", " \n", " A real world example. A company drills 9 wild-cat oil exploration\n", " wells, each with an estimated probability of success of 0.1. All nine\n", " wells fail. What is the probability of that happening?\n", " \n", " Let's do 20,000 trials of the model, and count the number that\n", " generate zero positive results.\n", " \n", " >>> sum(np.random.binomial(9,0.1,20000)==0)/20000.\n", " answer = 0.38885, or 38%.\n", "\n" ] } ], "source": [ "help(np.random.binomial)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Use the `binomial` function in NumPy to draw random numbers.\n", "- Plot the histogram of the numbers you've drawn\n", "- Notice how the histogram changes everytime you run the code.\n", "- Compare the histogram to the [binomial distribution](http://en.wikipedia.org/wiki/Binomial_distribution) probability mass function.\n", "\n", "Histograms are plotted using Matplotlib's `hist` function or you could use `seaborn.distplot` to plot the histogram with a density plot on top (see [Visualizing distributions of data\n", " in Python](http://nbviewer.ipython.org/github/mwaskom/seaborn/blob/master/examples/plotting_distributions.ipynb))." ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# You can change these values\n", "size = 10000\n", "n = 3\n", "p = 0.34\n", "# Your code goes here!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- How large should `size` be?\n", "- Does it depend on the choice of `n` and `p`?\n", "- Try to add the keyword argument `bins` to the `hist` function with a value of `100`." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## First simulation\n", "\n", "Based on the assumptions above, the probability that a male has $k$ boys that will reach reproductive age is:\n", "\n", "$$\n", "{n \\choose k} p^k (1-p)^{n - k}.\n", "$$\n", "\n", "To get the total number of male sin the next generation we need to draw lots of samples from this Binomial distribution and sum all the numbers. But the sum of many identical Binomial distributions is also a Binomial distribution. \n", "So if at generation $t$ we have $x_t$ males, then at generation $t+1$ the probability that the number of males is $k$ is:\n", "\n", "$$\n", "{n x_{t} \\choose k} p^k (1-p)^{n x_{t} - k}\n", "$$\n", "\n", "This only depends on the number of males in generation $t$ - $x_t$ (the Markovian _lack of memory_ trait).\n", "Note that if the number of males reaches _0_ or _N_, the process stops.\n", "\n", "The `num_males` array will contain the **population size at each generation**.\n", "We fill it with zeros just as placeholders and put the initial family size, 10, in the first element of the array.\n", "\n", "We then loop over all the generations and at each generation we randomly draw the number of children in the next generation and update the `num_males` array." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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CdwHLtcemiIiIeNGG8OKgOVNTc3/NiETP2fxz1GsHeDT8+n7gC8Di8PUfhL+i294E/BHA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\n8NjM3i533xJCCNF65K+wEKLZXDe2B6t35pNfUsu7K9J48rd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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "ngenerations = 100\n", "num_males = np.zeros(ngenerations)\n", "num_males[0] = 10\n", "\n", "for t in range(ngenerations - 1):\n", " if 0 < num_males[t] < N:\n", " # Update the number of males in the next generation\n", " num_males[t+1] = np.random.binomial(n * num_males[t], p)\n", " # The process is absorbed in 0 and N\n", " else:\n", " num_males[t+1] = num_males[t]\n", "\n", "plt.plot(num_males)\n", "plt.xlabel('# generations')\n", "plt.ylabel('# males');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that at every time step the number of males can stay stable, increase, or\n", "decrease.\n", "\n", "Run the simulation code again. Each time you run it you will get something else!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Multiple simulations\n", "\n", "Now, we will simulate many independent families. \n", "\n", "We could run the previous simulation with a loop, but it would be very slow (two nested for loops). \n", "Instead, we vectorize the simulation by considering all independent families at once. \n", "There is a single loop over time. \n", "\n", "At every time step, we update all families simultaneously with vectorized operations on vectors. \n", "The `num_males` array will now contains the number of males of all families at a particular time. \n", "\n", "We define a function that performs the simulation. \n", "\n", "At every time step, we find the families that have not reached the absorbing states 0 and $N$, and we update the number of males with array operations:" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def simulate(num_males, ngenerations):\n", " for t in range(ngenerations):\n", " # Which families to update? Only ones that are not absorbed yet\n", " update = (0 < num_males) & (num_males < N)\n", " # In which families do male births occur?\n", " boys = np.random.binomial(num_males[update] * n, p)\n", " # We update the population size for the non-absorbed families.\n", " num_males[update] = boys" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, let's look at the histograms of the number of males at different times.\n", "These histograms represent the probability distribution of the Stochastic process over many replicates (many surnames),\n", "estimated with independent families (this is often called the _Monte Carlo method_):" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [], "source": [ "nfamilies = 100\n", "n = 3\n", "p = 0.35\n", "\n", "bins = np.linspace(0, 1.2*N, 100)\n", "ngenerations_list = [10, 100, 1000]\n", "\n", "num_males = np.array([10] * nfamilies)" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "% families extinct: 0.27\n" ] }, { "data": { "image/png": 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eD7yNUpRYDbgHuAA4OTMXVW1uAw6iFDfWAO4DzgBmZeazox6xJPWoiHgrcAiw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\nDEmSJEmS1BMsYkiSJEmSpJ7gEquSpJ4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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(1, 3, figsize=(15,5), sharex=True, sharey=False)\n", "for i, ngenerations in enumerate(ngenerations_list):\n", " simulate(num_males, ngenerations)\n", " ax[i].hist(num_males, bins=bins, color='red')\n", " ax[i].set_title(\"%d generations\" % ngenerations)\n", " if i == 0:\n", " ax[i].set_ylabel(\"# families\")\n", " if i == 1:\n", " ax[i].set_xlabel(\"# males in family\")\n", "sns.despine()\n", "fig.tight_layout()\n", "\n", "print(\"% families extinct:\", (num_males==0).mean())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Initially, the number of males in all families started at 10.\n", "After enough time passed it appears to converge to _0_ or _N_.\n", "This is because the states _0_ and _N_ are absorbing; once reached, the chain cannot leave these states.\n", "\n", "Furthermore, these states can be reached from any other state. \n", "From our crude analysis above we see that 1,000 generations are enough for most surnames to either go extinct or reach a size that protects them from extinction (under our simplistic assumptions).\n", "\n", "Interestingly, the % of extinct families is a good approximation to the probability of extinction (strating from 10 males)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise - random numbers from the Poisson distribution\n", "\n", "The Poisson distribution is a commonly used for estimating number of offsprings.\n", "The distribution is easy to use because it has a single parameter $\\lambda$ with a clear interpretation: the average of the distribution.\n", "\n", "We can draw a random number from a Poisson distribution using `numpy.random.poisson`. \n", "\n", "Below is a plot of a histogram (bars) and with an estimation of the distribution (line) of the Poisson distribution:" ] }, { "cell_type": "code", "execution_count": 188, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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fBTwae60CKoBfxq5zT597HAVWA7l4Oxh9HW/BU7zT/iJjyt5jrbS0e70fF88t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\n8LM/N7CzNpbqaolkhJR/0ltrtzFAkndr7a+BX/e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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.random.poisson(lam=10,size=1000)\n", "sns.distplot(x)\n", "plt.axvline(x=10, color='k')\n", "plt.xlabel('Value')\n", "plt.ylabel('Frequency')\n", "sns.despine()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the code below (copied from the simulation above), the number of sons is distributed Binomially with parameters $n$ and $p$.\n", "\n", "Change the code so that the number of sons will be Poisson distributed \n", "with paramter $\\lambda$ (you can call it `lam` in the code, as `lambda` is a reserved Python word).\n", "\n", "What is the relationship between $\\lambda$ and $n$ and $p$?" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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kOuef4CKhsZb0HgBD5nBeAvQAbwD+D/GezgiwDngkDMP3pvCevyM+XP8M0E68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\nqSzghmsaPJ5RknKAgVPSmHh6fRtf+Ol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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "ngenerations = 100\n", "num_males = np.zeros(ngenerations)\n", "num_males[0] = 10\n", "\n", "for t in range(ngenerations - 1):\n", " if 0 < num_males[t] < N:\n", " # Update the number of males in the next generation\n", " num_males[t+1] = np.random.binomial(n * num_males[t], p)\n", " # The process is absorbed in 0 and N\n", " else:\n", " num_males[t+1] = num_males[t]\n", "\n", "plt.plot(num_males)\n", "plt.xlabel('# generations')\n", "plt.ylabel('# males');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Extinction probability\n", "\n", "An interesting question regarding Branching processes is: what is the probability that given enough time, a lineage will extinct?\n", "\n", "Let's rewrite our simulation function to start all sample populations to start with the same number of males and to continue until the number of males has reached an absorbing state - 0 or N.\n", "\n", "The result of this function will be the final state of each trial population." ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0 0 0 0 0 0 0 0 1017 0 0 0 0 1016 0\n", " 1053 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n", " 1015 1047 0 0 0 0 0 0 0 0 0 0 0 0 1003\n", " 0 0 0 0 0 0 0 0 0 0 0 0 1047 0 0\n", " 0 0 0 0 0 0 0 0 0 1005 0 1035 0 0 0\n", " 0 0 0 0 0 0 0 0 0 0 0 1021 0 0 0\n", " 0 0 0 0 0 0 0 0 0 0]\n" ] } ], "source": [ "def simulate(init_num_males, n, p, nfamilies=100):\n", " num_males = init_num_males * np.ones(nfamilies, dtype=int)\n", " update = (0 < num_males) & (num_males < N)\n", " while update.any():\n", " # Which trials to update?\n", " update = (0 < num_males) & (num_males < N-1)\n", " # In which trials do births occur?\n", " boys = np.random.binomial(num_males[update] * n, p)\n", " # We update the population size for all trials.\n", " num_males[update] = boys\n", " return num_males\n", "num_males = simulate(1, n, p)\n", "print(num_males)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can estimate the extinction probability from the fraction of the simulations that ended with an extinction.\n", "We can also estimate the standard error of the mean (SEM) as the square root of the sample standard deviation." ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Extinction probability starting from a single male is 0.90 ± 0.0302\n" ] } ], "source": [ "mean_prob = (num_males==0).mean()\n", "sem_prob = (num_males==0).std(ddof=1) / np.sqrt(len(num_males))\n", "print(\"Extinction probability starting from a single male is %.2f ± %.4f\" % (mean_prob, sem_prob))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we will estimate the extinction probability for different initial number of males and make an errorbar plot." ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [], "source": [ "init_num_males_range = range(1,100,5)\n", "prob = np.array([simulate(k, n, p)==0 for k in init_num_males_range])\n", "\n", "mean_prob = prob.mean(axis=1)\n", "sem_prob = prob.std(axis=1) / np.sqrt(prob.shape[0])" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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GHEZjjBsLWIcKZOnialasrOfRZ/bR2NxOZUVp2iVJkiQdktMOTAAhhAnAK4C5\n2aYNwG9ijHV5rEt58pKF4ynJQGtbJw+v3cfLTnJBBEmSNHTktDd9COEaYAtwC3BD9vVjYEsI4QP5\nL09Ha3xVGSfNGwvAiqedVS9JkoaWXLYDvQz4IvA0cBlwevZ1GbAS+FII4S2FKFJH56xF4wH4w+oG\n2to7U65GkiTpebk8pr8O+ANwboyxpVv7YyGEnwLLgQ8BP8hjfcqDpYur+eYd29jf1M5TG/dz6vHj\n0i5JkiQJyO0x/SLgBz2CKADZth8Bi/NVmPJn2sQKjps+GvBRvSRJGlpyCaNNQF9rA00EGo+uHBXK\n0sXP78bU2emjekmSNDTkEkZ/B7w/hHBKzxMhhCXA1dlrNAR1hdHd9a2s2+bfGSRJ0tCQy5jRjwMr\ngD+GEO4CVmXbFwOvBBqy12gIOm56JVNqytlV18oDK+uZP3NM2iVJkiQNvGc0xrgaOAP4OXAucG32\n9TLgp8CZ2Ws0BGUymUO9o79f6bhRSZI0NOS06H2M8RngzSGEUmBytnl3jLE975Up75YuruYXDzzH\nhh1NbN/bzPSJFWmXJEmSRricd2ACyIbPHXmuRQV20tyxjK0sZX9jO79fWc/rXzYl7ZIkSdIId9gw\nGkI4N/v2vhhjZ7fjPsUY781LZcq70tIML1k4nrsfreUBw6gkSRoC+uoZvQfoBCqBluxxfzqB0qOu\nSgWzdHE1dz9ay8qNB6g/0EZ11RF1jkuSJOVFX0nk/OzP1h7HKmIvCuMYVZahpa2TP6yu55Uv6mvp\nWEmSpMI6bBiNMd7T17GK0+hRpZx2wjgeXNXAipWGUUmSlK4BL+0UQvhdCOGCPs6fF0L4bX7KUiF1\nLfH0yNp9NLV0pFyNJEkayXLZgWkZMLWP81OBlx9VNRoUL1lYTUkGmls7eWRtQ9rlSJKkESyXMNqf\nmcDBPH6eCqRmbBmLjq0Ckr3qJUmS0tLnVOoQwuuA13VruiqE8IpeLp0IvAL4Yx5rUwEtXVzN0xsP\n8IfVDbS3d1Jamkm7JEmSNAL1t67PacA7uh2fm331dAB4CHhffspSoS1dVM23f7mNhoPtPP3sAZYc\nNzbtkiRJ0gjUZxiNMX4K+BRACKEDuDzG+P3Cl6VCm3FMBXOnjmbjziZWrKw3jEqSpFTkMmb0OODn\nhSpEg69rVv2KlfV0dnamXI0kSRqJcgmjJUBfSzu9NoQw96gr0qA5KxtGd9a2sGFHU6/XbNvTzIUf\ne4wLP/Zu2ajZAAAgAElEQVQY2/Y0D2Z5kiRpBMgljF4PXNvH+Q9kr1GROGFmJZPGlwPOqpckSenI\nJYy+FLirj/N3AeccXTkaTJlMhrO7PaqXJEkabLmE0WOAPX2crwMmH105GmxLT0zC6LptjeysbUm5\nGkmSNNLkEkZ3AKf2cf5UYPfRlaPBdvK8sVSNTv41+L29o5IkaZDlEkZvBa4IIZzf80R2Ifx3Abfl\nqzANjrLSDGcs8FG9JElKR3+L3nf3T8DFwG9CCL8Fnsy2LwHOA54FPpnf8jQYzj6xmnser+XJjfvZ\nd7CNcWNy+ddCkiTpyA24ZzTGuBt4CXAT8GLgmuzrdOA/gDNjjLsKUaQK60VhHGWlGTo64A+rG9Iu\nR5IkjSC5PKYnxrg7xnglyV7007OvSTHGd2fDqorQmIpSTps/DvBRvSRJGlxH9Dw2xtgB7MxzLUrR\n0sXVPLSmgT/GfTS3dlBRntPfUyRJko5IzmE0hLCIZGvQiUCm5/kY43/moS4NspcsGk/mf6C5tYNH\nn9nHWYuq0y5JkiSNAAMOoyGEOcB/AS/r51LDaBGaOK6chbPHsGrTQX6/st4wKkmSBkUuPaNfB15E\nsiXovUBtQSpSapYurk7C6KoG2js6KS35k47vQbFtTzNXfG4VAP/x4UXMmFSRSh2SJKnwcgmjy4Av\nxBi/XKhilK6li6u56VfbqT/QxqpNBzhp7ti0S5IkScNcLrNUDgLbC1WI0jdr8mhmT056IZ1VL0mS\nBkMuYfTHwGsLVYiGhqWLs7sxPV1PZ2dnytVIkqThLpfH9DcCN4cQfpJ9vxFo73lRjHFTfkpTGpYu\nrubHy3exfW8Lm3Y1UV7mEk+SJKlwcgmjT2V/vgh4w2Gu6QRKj6oipSrMGsPEcWXs3dfGAyvrWbZk\nQtolSZKkYSyXMPqPA7jG57pFrqQkw1mLq/nlg3tY8bRhVJIkFdaAw2iM8VMFrGPAQgjzgC8D5wEt\nwK3AdTHGPpeaCiG8B3gjcBIwFngG+CpwU4zREN3N0kVJGF27tZG9Da1plyNJkoaxohoQGEIYB/wO\nmAa8GXgvyZJTtw7g9o8Dm4D3AZcAvwK+AVxfkGKL2JLjx1JZkfyr8di6fSlXI0mShrPD9oyGEN5O\n8tj9v2KMHd2O+1Tg7UCvIgmiS2OM27N1bgHuDyFcFGO8o497T4sx7ul2fE8IoQb4QAjhEzFGuwCz\nRpWVcMaC8dz7RB2PPrM/7XIkSdIw1tdj+u+QhM8fkTwO/84AP7OQYfRiYHlXEAWIMa4IIWwk6e08\nbBjtEUS7PAK8B5gI7MxvqcVt6eJq7n2ijjWbD6RdiiRJGsb6CqPHAcQYW7ofp2wx8P1e2ldmz+Xq\nXGAPsOtoihqOzlgwnrLSDG3tDqeVJEmFc9gwGmPc2NdxSmqAul7a64C5uXxQCGEZ8JfAJ5zA9Keq\nRpey5LixPLLWMaOSJKlwBjyBKYSwIYRw2B2YQgiXhBDW56esIzLgQBlCOJ5kR6n7gBsKVlGR69qN\nSZIkqVBymU1/LMmSSIczlhx7J49ALUnvaE8TgL0D+YAQwnTgf4HtwOtijB35K294OWvR+LRLkCRJ\nw1w+l3Y6ASj0M91VwIm9tC/OnutTCGEicBdJL+qrY4wN+S1veDmmehTzpo1OuwxJkjSM9bnofXY5\np7d3a/p4COHKXi6dCJwM3JnH2npzO/DpEML0bks7nUXSa3tbXzeGEKqAX5L0or4sxujs+QE47YRx\nbNjRBMBNv9rGhWdM4tTjx1Famkm5MkmSNBz0twPTBF44i34yUNXjmk7gAMmSTh/PX2m9+ibwfuAX\nIYRPZWu5AXig+xqjIYS3ATcB58cY7802/xR4MfDXwNQQwtRun/t0jNGZOr04c8F4fnbfbgDuf6qe\n+5+qp7qqjJedVM2yUyZw4rFVlJQYTCVJ0pHpM4zGGL8EfAkghNABXBtj7G1ppUERY9wXQjifZDvQ\nW4BWkt2Xru1xaSb76u5VJMH56z3aO0m2Fr0X/YnJNaMOvZ83bTQbdjRRf6CNOx7cwx0P7mHS+HLO\nPbmGZafUEGaNIZMxmEqSpIHLZW/6IbF1aIxxPckC931dczNwc4+2IVF/Mfv7t84D4N4n6lj+RC0b\ndzSxp6GVn9+/m5/fv5tpE0exbMkEli2pYe600QZTSZLUrwGH0RDC1THGr/Zxvgr4QozxPXmpTEPS\njEkVXHreVC49byrP7mxk+RN1LH+8lm17Wtixt4Vb7tnJLffsZPaUCl6+ZALnLqlh1mQnQUmSpN4N\nOIwC/xZCuBh4R4xxR/cTIYSlwPdIJhIZRkeIY6dW8rZXVnL5K6axbtvzwXR3fSubdzXzvd/s4Hu/\n2cH8GZUsO2UC55xcw9QJo/r/YEmSNGLkEkbfA3wBeCqE8N4Y409CCGXA/wf8DbAROCf/JWqoy2Qy\nzJ85hvkzx/DOV09n9eaDLH+8lvuerKN2fxvPbGvkmW2N/Med21h8bBXnLqnhnJNrmDiuPO3SJUlS\nynIZM/qtEMLdJLPmfxxC+CGwCDgV+BbJ5KaDhSlTxaKkJMPiY6tYfGwVV108kyc37Gf543X831N1\n7G9sZ+WzB1j57AG+eftWTj5uLMuWTOClJ1YzviqXvxdJkqThIqcEEGNcn93T/f9I9nUH+FCM8Yt5\nr0xFr7Qkw6nHj+PU48fx16+dyaPP7OfeJ2p54Ol6Gls6eHzdfh5ft58bf7GZ008Yz7IlNZzlFqSS\nJI0oOYXREMJM4LvAS0h2Mjod+McQQlOM8Wv5L0/DRXlZCWcuHM+ZC8fT3NrBH9c0sPyJOh5cVU9L\nWycPrWngoTUNlJdlOHleX7vOSpKk4SSX2fSXAjcC5cAVMcbvZBeO/zZwYwjhEuBdPSc3ST1VlJfw\n0pNqeOlJNRxsbufBVQ0sf6KWh+M+Wts6eWTt8/sPbNzRyIxJFSlWK0mSCimXntEfAA8Al8cYNwBk\nt9S8JITwbrKTm4Bj8l6lhq0xFaWcd+oEzjt1Avsa23jg6Xr+9497WfnsAQCu/+GzXPOGDl5x+sSU\nK5UkSYWQy0LwnwDO7Qqi3cUYvwWcAqzOV2EaecZVlvHqF0/iQ2+ac6itrb2Tz//3Jr5+2xba2jtT\nrE6SJBXCgMNojPHTMcaOPs6vx6WdlGcL54wB4BcPPMfHb1pH3f62gvyebXuaufBjj3Hhxx5j257m\ngvwOSZL0p3LaIjOEMCqEcEUI4QchhLtCCKdl2yeEEN4GzCxIlRqxrnvjHN7wsskAPLF+P9fcuIZn\ntrqCmCRJw8WAw2gIoQZYQbKm6GuBC4AJ2dMNwKeBq/NdoEa20pIM775oJh958xxGlWXYVdfKh76+\nlt8+ujft0iRJUh7k0jP6aZJF7i8B5nY/EWNsB34GvDpvlUndnH/aRD7/3hOYUlNOS1snn/3xJr55\n+1baHUcqSVJRyyWM/jlwY4zxjsOcX0ePkCrl0/yZY/jy+xaw5LhkHdKf37+bj39nHfUHCjOOVJIk\nFV4uYXQSsKaP853A6KMrR+pbzdgyPv2u43nd2ckKYo+v288HvrqGddscRypJUjHKJYxuBRb0cf4s\n4JmjK0fqX1lphvdeMosPv2k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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.errorbar(x=init_num_males_range, y=mean_prob, yerr=sem_prob)\n", "plt.xlabel('initial # males in family')\n", "plt.ylabel('extinction probability')\n", "sns.despine()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise - check the theorem\n", "\n", "It can be proven that:\n", "\n", "> ### Theorem on extinction probability\n", "\n", "> If the average number of sons is <= 1 the lineage will go to extinction with probability 1.\n", "If the average number of sons is > 1 then the extinction probability is the stationary point of the moment generating function of the distribution of number of sons per male (see Athreya & Ney., 2011).\n", "\n", "Let's check the theorem. \n", "\n", "Calculate the extinction probability of a single male for different values of $n \\cdot p$ such that you include both values lower than 1 and higher than 1 and check that the extinction probability is 1 only when the average number of offspring is lower then 1." ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [], "source": [ "init_num_males = 1\n", "n = 3\n", "\n", "# Your code here" ] }, { "cell_type": "code", "execution_count": 195, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# More code here" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Stationary distribution\n", "\n", "An interesting fact about some branching processes is that after enough time passsed, \n", "the probability to be at a specific state (_i.e._ for the number of males to be $k$) doesn't change from generation to generations.\n", "\n", "This probability is called a _static distribution_. \n", "\n", "In the figure above you can see that the distribution for 10,000 and 100,000 generations is the same, \n", "so we can find the extinction probability by calculating the probability that there are 0 males after 10,000 generations:\n", "\n", "To answer this question the model can be written in the following way. \n", "We denote by $v(t)=(v_0(t), v_1(t), ..., v_N(t))$ the probabilities that there are 0, 1, ..., $N$ males after $t$ generations.\n", "\n", "We denote by $P_{i,j}$ the probability that there are $i$ males in generation $t+1$ given that there are $j$ males in generation $t$.\n", "$P$ is called the _transition matrix_.\n", "\n", "Let's consider a simpler model than before - in this model the population loses a single individual with probability $a$ and gains an individual with probability $b$. This is sometimes called a __birth-death process__ in which $a$ and $b$ are the death and birth rates, respectively.\n", "\n", "This can be written by:\n", "$$\n", "v_i(t+1) = P_{i,0} v_0(t) + P_{i,1} v_1(t) + ... + P_{i,N} v_N(t) = \\sum_{j=0}^{N}{P_{i,j}v_j(t)}\n", "$$\n", "\n", "of as a matrix equation:\n", "$$\n", "v(t+1) = P \\cdot v(t)\n", "$$\n", "\n", "This matrix equation can be written and calculated using NumPy arrays. \n", "\n", "Let's use a simpler model in which the number of males is increased by 1 with probability $a$ and decreased by one with probability $b$:\n", "\n", "$$\n", "P_{i,j} = \\left\\{\\begin{matrix}\n", "a & i=j+1\\\\ \n", "b & i=j-1\\\\ \n", "1-a-b & i=j \\\\\n", "0 & otherwise\n", "\\end{matrix}\\right.\n", "$$\n", "\n", "So this matrix has a main diagonal with the value $1-a-b$, above it a diagonal with $a$ and below it a diagonal with $b$. Everything else is 0:\n", "\n", "$$\n", "P = \\begin{pmatrix}\n", "1-a-b & b & 0 & ... \\\\\n", "a & 1-a-b & b & ... \\\\\n", "0 & a & 1-a-b & ... \\\\\n", "... & ... & ... & ...\n", "\\end{pmatrix}\n", "$$\n", "\n", "We can generate diagonal matrices using NumPy's `diag` function:" ] }, { "cell_type": "code", "execution_count": 196, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0.75 0.3 0. ..., 0. 0. 0. ]\n", " [ 0.25 0.45 0.3 ..., 0. 0. 0. ]\n", " [ 0. 0.25 0.45 ..., 0. 0. 0. ]\n", " ..., \n", " [ 0. 0. 0. ..., 0.45 0.3 0. ]\n", " [ 0. 0. 0. ..., 0.25 0.45 0.3 ]\n", " [ 0. 0. 0. ..., 0. 0.25 0.7 ]]\n" ] } ], "source": [ "N = 1000\n", "a = 0.3\n", "b = 0.25\n", "\n", "main_diag = [1-a-b]*(N+1)\n", "lower_diag = [a]*N\n", "upper_diag = [b]*N\n", "P = np.diag(main_diag) + np.diag(lower_diag, 1) + np.diag(upper_diag, -1)\n", "P[0,0] = 1-b\n", "P[N,N] = 1-a\n", "\n", "print(P)\n", "assert np.allclose(P.sum(axis=0), 1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's assume we start from a lineage of 100 males.\n", "We'll iterate the matrix equation for 1000 times.\n", "Matrix multiplication is done using NumPy's `dot` function.\n", "Remember that order matters in matrix multiplication." ] }, { "cell_type": "code", "execution_count": 197, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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ASWt1IVob9ean5bryJ8d9JKkkSWpON8H05cDDI+KtEXF0RGxcq4tS79QrptU+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\nqVP5kiSpHwymQ6Y+7b7aqfw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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "v = np.zeros(N+1)\n", "v[100] = 1.\n", "for _ in range(100):\n", " v = np.dot(P, v)\n", "plt.plot(v)\n", "plt.xlabel('# males')\n", "plt.ylabel('probability')\n", "sns.despine()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is kind of slow. There is a short cut:\n", "$$\n", "v(t+2) = P \\cdot v(t+1) = P \\cdot P \\cdot v(t) = P^2 \\cdot p(t) \\Rightarrow \\\\\n", "v(t) = P^t \\cdot v(0)\n", "$$\n", "\n", "Matrix power is done using `numpy.linalg.matrix_power`:" ] }, { "cell_type": "code", "execution_count": 198, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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x4sUALFiwgJUrV1JfX8/q1atZsWIFS5YsIRKJ0NLSwtKlS3s8x6t2fqIgtChV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\nwZpH9cWO3QTjQX7V+kveU/c7WS6pEGKukhB6C1m36628+qPTmI4LdF1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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "v = np.zeros(N+1)\n", "v[100] = 1.\n", "for generations in (10,100,1000,10000,100000):\n", " u = np.dot(np.linalg.matrix_power(P, generations), v)\n", " plt.plot(u[:200], label=generations)\n", "plt.xlabel('# males')\n", "plt.ylabel('probability')\n", "plt.legend(title=\"Generations\")\n", "sns.despine()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References\n", "\n", "- Watson, HW & Galton, F. 1875. __On the Probability of the Extinction of Families__. _Journal of the Anthropological Institute of Great Britain_, 4:138–144. [PDF](http://galton.org/essays/1870-1879/galton-1874-jaigi-family-extinction.pdf)\n", "- Athreya, K.B., Ney, P.E., 2011. __T. E. Harris and branching processes__. arXiv: [1103.2011](http://arxiv.org/abs/1103.2011)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Fin\n", "This notebook is part of the _Python Programming for Life Sciences Graduate Students_ course given in Tel-Aviv University, Spring 2015.\n", "\n", "The notebook was written using [Python](http://pytho.org/) 3.4.1 and [IPython](http://ipython.org/) 2.1.0 (download from [PyZo](http://www.pyzo.org/downloads.html)).\n", "\n", "The code is available at https://github.com/Py4Life/TAU2015/blob/master/lecture8.ipynb.\n", "\n", "The notebook can be viewed online at http://nbviewer.ipython.org/github/Py4Life/TAU2015/blob/master/lecture8.ipynb.\n", "\n", "This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.\n", "\n", "![Python logo](https://www.python.org/static/community_logos/python-logo.png)" ] } ], "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": "3.4.1" } }, "nbformat": 4, "nbformat_minor": 0 }