{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "hide_input": true, "internals": {}, "slideshow": { "slide_type": "skip" }, "tags": [ "hide-input" ] }, "outputs": [], "source": [ "# for QR codes use inline\n", "%matplotlib inline\n", "qr_setting = 'url'\n", "#\n", "# for lecture use notebook\n", "# %matplotlib notebook\n", "# qr_setting = None\n", "#\n", "%config InlineBackend.figure_format='retina'\n", "# import libraries\n", "import numpy as np\n", "import matplotlib as mp\n", "import pandas as pd\n", "import matplotlib.pyplot as plt\n", "import laUtilities as ut\n", "import slideUtilities as sl\n", "import demoUtilities as dm\n", "import pandas as pd\n", "from importlib import reload\n", "from datetime import datetime\n", "from IPython.display import Image\n", "from IPython.display import display_html\n", "from IPython.display import display\n", "from IPython.display import Math\n", "from IPython.display import Latex\n", "from IPython.display import HTML;" ] }, { "cell_type": "markdown", "metadata": { "internals": { "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Markov Chains" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "hide_input": false, "internals": { "frag_number": 3 }, "slideshow": { "slide_type": "-" }, "tags": [ "hide-cell" ] }, "outputs": [ { "data": { "image/jpeg": 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\n", "text/plain": [ "" ] }, "metadata": { "image/jpeg": { "width": 250 } }, "output_type": "display_data" }, { "data": { "text/html": [ "Andrei Markov, 1856 - 1922, St Petersburg." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Image credit: http://en.wikipedia.org/wiki/Andrey_Markov#mediaviewer/File:AAMarkov.jpg\n", "display(Image(\"images/AAMarkov.jpg\", width=250))\n", "display(HTML(\"Andrei Markov, 1856 - 1922, St Petersburg.\"))" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "hide_input": true, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "text/html": [ "AAMarkov by unknown; originally uploaded to en-wikipedia by en:User:Mhym - From the official web site of the Russian Academy of Sciences: http://www.ras.ru/win/db/show_per.asp?P=.id-53175.ln-en.dl-.pr-inf.uk-0. Licensed under Public Domain via Wikimedia Commons." ], "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "HTML('AAMarkov by unknown; originally uploaded to en-wikipedia by en:User:Mhym - From the official web site of the Russian Academy of Sciences: http://www.ras.ru/win/db/show_per.asp?P=.id-53175.ln-en.dl-.pr-inf.uk-0. Licensed under Public Domain via Wikimedia Commons.')" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 3, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "-" } }, "source": [ "Markov was part of the great tradition of mathematics in Russia.\n", "\n", "Markov started out working in number theory but then got interested in probability. He enjoyed poetry and the great Russian poet Pushkin. Markov studied the sequence of letters found in the text of _Eugene Onegin,_ in particular the sequence of consonants and vowels. \n", "\n", "He sought a way to describe the patterns in the text. This eventually led to the idea of a system in which one transitions between states, and the probability of going to another state depends only on the current state.\n", "\n", "Hence, Markov pioneered the study of systems in which the future state of the system depends only on the present state in a random fashion. Classic examples in modern life include the movement of stock prices and the dynamics of animal populations. \n", "\n", "These have since been termed \"Markov Chains.\" \n", "\n", "Markov chains are essential tools in understanding, explaining, and predicting phenomena in computer science, physics, biology, economics, and finance." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 3, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "Today we will study an application of linear algebra. You will see how the concepts we use, such as vectors and matrices, get applied to a particular problem." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 3, "slide_type": "subslide" }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Many applications in computing are concerned with how a system behaves over time. \n", "\n", "Think of a Web server that is processing requests for Web pages, or network that is moving packets from place to place. \n", "\n", "We would like to describe how systems like these operate, and analyze them to understand their performance limits." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 6 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "The way we model this is: \n", "\n", "* we define some vector that describes the state of the system, and \n", "* we formulate a rule that tells us how to compute the next state of the system based on the current state of the system." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 7 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "So we would say that the state of the system at time $k$ is a vector ${\\bf x_k} \\in \\mathbb{R}^n,$ and\n", "\n", "$${\\bf x_{k+1}} = T({\\bf x_k}),\\;\\;\\;\\mbox{for time}\\;k=0,1,2...$$\n", "\n", "where $T: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n.$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 8 }, "slideshow": { "slide_type": "skip" } }, "source": [ "This situation is so common that it goes by many names:\n", "\n", "* In physics, this is called a __dynamical system__. \n", " * Here, ${\\bf x_k}$ might represent the position and velocity of a particle.\n", " \n", "* When studying algorithms, this is called a __recurrence relation.__ \n", " * Here, ${\\bf x_k}$ might represent the number of steps needed to solve a problem of size $k$.\n", " \n", "* Most commonly, this is called a __difference equation.__ \n", " * The reason for this terminology is that it is a discrete analog of a differential equation in $k$." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 9 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "The vector ${\\bf x_k}$ is called the _state vector._" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 10, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "Of course, we are going to be particularly interested in the case where $T$ is a linear transformation. \n", "\n", "Then we know that we can write the difference equation as:\n", "\n", "$$ {\\bf x_{k+1}} = A{\\bf x_k},$$\n", "\n", "where $A \\in \\mathbb{R}^{n\\times n}.$ \n", "\n", "This is a _linear difference equation._" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 10, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "__Example.__ \n", "\n", "Here is our warm-up problem.\n", "\n", "We are interested in the population of two regions, say the city and the suburbs. \n", "\n", "Fix an initial year (say 2000) and let \n", "\n", "$$ {\\bf x_0} = \\left[\\begin{array}{cc}\\mbox{population of the city in 2000}\\\\\\mbox{population of the suburbs in 2000}\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 12 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "\n", "Then\n", "\n", "$$ {\\bf x_1} = \\left[\\begin{array}{cc}\\mbox{population of the city in 2001}\\\\\\mbox{population of the suburbs in 2001}\\end{array}\\right],$$\n", "\n", "$${\\bf x_2} = \\left[\\begin{array}{cc}\\mbox{population of the city in 2002}\\\\\\mbox{population of the suburbs in 2002}\\end{array}\\right],$$\n", "\n", "$$\\dots \\mbox{etc.}$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 13 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "We only concern ourselves with movements of people between the two regions. \n", "* no immigration, emigration, birth, death, etc." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 14 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "We assume that measurements have shown the following pattern: \n", "\n", "in any given year, \n", "\n", "* 5% of the people in the city move to the suburbs, and \n", "* 3% of the people in the suburbs move to the city." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 15 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "You can think of this as:\n", "\n", "$$\\begin{array}{rcc}&\\mbox{From City}&\\mbox{From Suburbs}\\\\\\mbox{To City}& .95&.03\\\\\\mbox{To Suburbs}&.05&.97\\end{array}$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 16 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Then we can capture this update rule as a matrix:\n", "\n", "$$A = \\left[\\begin{array}{rr}.95&.03\\\\.05&.97\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 17, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "We can see that this is correct by verifying that:\n", "\n", "$$\\left[\\begin{array}{cc}\\mbox{city pop. in 2001}\\\\\\mbox{suburb pop. in 2001}\\end{array}\\right] =\\left[\\begin{array}{rr}.95&.03\\\\.05&.97\\end{array}\\right] \\left[\\begin{array}{cc}\\mbox{city pop. in 2000}\\\\\\mbox{suburb pop. in 2000}\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 17, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "## Markov Chains" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 19 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Let's look at $A$ again:\n", "\n", "$$A = \\left[\\begin{array}{rr}.95&.03\\\\.05&.97\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 20 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "We note that $A$ has a special property: each of its columns adds up to 1. \n", "\n", "Also, it would not make sense to have negative entries in $A$." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 21 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "The reason that columns sum to 1 is that the total number of people in the system is not changing over time. \n", "\n", "This leads to three definitions:" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 21 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Definition.__ A _probability vector_ is a vector of nonnegative entries that sums to 1. " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 22 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Definition.__ A _stochastic_ matrix is a square matrix of nonnegative values whose columns each sum to 1." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 23, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "__Definition.__ A _Markov chain_ is a dynamical system whose state is a probability vector and which evolves according to a stochastic matrix. \n", "\n", "That is, it is a probability vector ${\\bf x_0}$ and a stochastic matrix $A \\in \\mathbb{R}^{n\\times n}$ such that \n", "\n", "$${\\bf x_{k+1}} = A{\\bf x_k}\\;\\;\\;\\mbox{for}\\;k = 0,1,2,...$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 23, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "So we think of a probability vector ${\\bf x_0}$ as describing how things are \"distributed\" across various categories -- the fraction of items that are in each category. " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 25, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "And we think of the stochastic matrix $A$ as describing how things \"redistribute\" themselves at each time step." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 25, "slide_type": "subslide" }, "slideshow": { "slide_type": "skip" } }, "source": [ "``` {toggle} \n", "Question Time! Q11.1\n", "```" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 27 }, "slideshow": { "slide_type": "slide" } }, "source": [ "__Example.__ Suppose that in 2000 the population of the city is 600,000 and the population of the suburbs is 400,000. What is the distribution of the population in 2001? In 2002? In 2020?" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 28 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Solution.__ First, we convert the population distribution to a probability vector. This is done by simply normalizing by the sum of the vector elements.\n", "\n", "$$600,000 + 400,000 = 1,000,000.$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 29 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "$$\\frac{1}{1,000,000}\\left[\\begin{array}{rr}600,000\\\\400,000\\end{array}\\right] = \\left[\\begin{array}{rr}0.60\\\\0.40\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 30 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Then the distribution of population in 2001 is:\n", "\n", "$$ {\\bf x_{1}} = A{\\bf x_0} = \\left[\\begin{array}{rr}.95&.03\\\\.05&.97\\end{array}\\right]\\left[\\begin{array}{rr}0.60\\\\0.40\\end{array}\\right] = \\left[\\begin{array}{rr}0.582\\\\0.418\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 31 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "And the distribution of the population in 2002 is:\n", "\n", "$$ {\\bf x_{2}} = A{\\bf x_1} = \\left[\\begin{array}{rr}.95&.03\\\\.05&.97\\end{array}\\right]\\left[\\begin{array}{rr}0.582\\\\0.418\\end{array}\\right] = \\left[\\begin{array}{rr}0.565\\\\0.435\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 32 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Note that another way we could have written this is:\n", "\n", "$$ {\\bf x_{2}} = A{\\bf x_1} = A(A{\\bf x_0}) = A^2 {\\bf x_0}.$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 33 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "To answer the question for 2020, i.e., $k=20,$ we note that \n", "\n", "$${\\bf x_{20}} = \\overbrace{A\\cdots A}^{20} {\\bf x_0} = A^{20}{\\bf x_0}.$$" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 33 }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[0.417456 0.582544]\n" ] } ], "source": [ "# stochastic matrix A\n", "A = np.array(\n", " [[0.95,0.03],\n", " [0.05,0.97]])\n", "#\n", "# initial state vector x_0\n", "x_0 = np.array([0.60,0.40])\n", "#\n", "# compute A^20\n", "A_20 = np.linalg.matrix_power(A, 20)\n", "#\n", "# compute x_20\n", "x_20 = A_20 @ x_0\n", "print(x_20)" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 35, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "So we find that after 20 years, only 42% of the population will remain in the city." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 35, "slide_helper": "subslide_end", "slide_type": "subslide" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "skip" } }, "source": [ "``` {toggle}\n", "Question Time! Q11.2\n", "```" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 35, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "## Predicting the Distant Future" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "We noticed that the population of the city is going down. Will everyone eventually live in the suburbs?" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 38 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "A important question about a Markov Chain is: what will happen in the distant future?\n", "\n", "For example, what happens to the population distribution in our example \"in the long run?\" " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 39 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Rather than answering that question right now, we'll take a more interesting example." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 40 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Suppose we have a system whose state transition is described by the stochastic matrix\n", "\n", "$$P = \\left[\\begin{array}{rrr}.5&.2&.3\\\\.3&.8&.3\\\\.2&0&.4\\end{array}\\right]$$\n", "\n", "and which starts int the state \n", "\n", "$${\\bf x_0} = \\left[\\begin{array}{r}1\\\\0\\\\0\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 41 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Consider the Markov Chain defined by $P$ and ${\\bf x_0}$, that is the chain defined as \n", "\n", "$${\\bf x_{k+1}} = P{\\bf x_k}\\;\\;\\;\\mbox{for}\\;k=0,1,2...$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 42 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "What happens to the system as time passes?\n", "\n", "Let's compute the state vectors ${\\bf x_1},\\dots,{\\bf x_{15}}$ to find out." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 43 }, "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "x(0) = [1 0 0]\n", "x(1) = [0.5 0.3 0.2]\n", "x(2) = [0.37 0.45 0.18]\n", "x(3) = [0.329 0.525 0.146]\n", "x(4) = [0.3133 0.5625 0.1242]\n", "x(5) = [0.30641 0.58125 0.11234]\n", "x(6) = [0.303157 0.590625 0.106218]\n", "x(7) = [0.3015689 0.5953125 0.1031186]\n", "x(8) = [0.30078253 0.59765625 0.10156122]\n", "x(9) = [0.30039088 0.59882813 0.10078099]\n", "x(10) = [0.30019536 0.59941406 0.10039057]\n", "x(11) = [0.30009767 0.59970703 0.1001953 ]\n", "x(12) = [0.30004883 0.59985352 0.10009765]\n", "x(13) = [0.30002441 0.59992676 0.10004883]\n", "x(14) = [0.30001221 0.59996338 0.10002441]\n" ] } ], "source": [ "# stochastic matrix A\n", "A = np.array(\n", " [[.5,.2,.3],\n", " [.3,.8,.3],\n", " [.2, 0,.4]])\n", "#\n", "# initial state vector\n", "x = np.array([1,0,0])\n", "#\n", "# array to hold each future state vector\n", "xs = np.zeros((15,3))\n", "# \n", "# compute future state vectors\n", "for i in range(15):\n", " xs[i] = x\n", " print(f'x({i}) = {x}')\n", " x = A @ x" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 44 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "What is going on here? Let's look at these values graphically." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "hide_input": true, "internals": { "frag_helper": "fragment_end", "frag_number": 44 }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "image/png": { "height": 280, "width": 424 }, "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax = plt.subplot(131)\n", "plt.plot(range(15),xs.T[0],'o-')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_1$',size=24)\n", "ax = plt.subplot(132)\n", "plt.plot(range(15),xs.T[1],'o-')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_2$',size=24)\n", "ax = plt.subplot(133)\n", "plt.plot(range(15),xs.T[2],'o-')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_3$',size=24)\n", "plt.tight_layout()" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 46 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Based on visual inspection, these vectors seem to be approaching\n", "\n", "$${\\bf q} = \\left[\\begin{array}{r}.3\\\\.6\\\\.1\\end{array}\\right].$$\n", "\n", "The components of ${\\bf x_k}$ don't seem to be changing much past about $k = 10.$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 47 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "In fact, we can confirm that the this system would be stable at $\\left[\\begin{array}{r}.3\\\\.6\\\\.1\\end{array}\\right]$ by noting that:\n", "\n", "$$\\left[\\begin{array}{rrr}.5&.2&.3\\\\.3&.8&.3\\\\.2&0&.4\\end{array}\\right]\\left[\\begin{array}{r}.3\\\\.6\\\\.1\\end{array}\\right] = \\left[\\begin{array}{r}.15+.12+.03\\\\.09+.48+.03\\\\.06+0+.04\\end{array}\\right] = \\left[\\begin{array}{r}.3\\\\.6\\\\.1\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 48, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "This calculation is exact. So it seems that:\n", "\n", "* the sequence of vectors is approaching $\\left[\\begin{array}{r}.3\\\\.6\\\\.1\\end{array}\\right]$ as a limit, and \n", "* when and if they get to that point, they will stabilize there." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 48, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "## Steady-State Vectors" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 50 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "This convergence to a \"steady state\" is quite remarkable. Is this a general phenomenon? " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 51 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Definition.__ If $P$ is a stochastic matrix, then a __steady-state vector__ (or __equilibrium vector__) for $P$ is a probability vector $\\bf q$ such that:\n", "\n", "$$P{\\bf q} = {\\bf q}.$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 52 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "It can be shown that __every stochastic matrix has at least one steady-state vector.__ \n", "\n", "(We'll study this more closely in a later lecture.)" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 53 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Example.__\n", "\n", "$\\left[\\begin{array}{r}.3\\\\.6\\\\.1\\end{array}\\right]$ is the steady-state vector for $\\left[\\begin{array}{rrr}.5&.2&.3\\\\.3&.8&.3\\\\.2&0&.4\\end{array}\\right].$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 54 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Example.__\n", "\n", "Recalling the population-movement example above." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 55 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "The probability vector ${\\bf q} = \\left[\\begin{array}{r}.375\\\\.625\\end{array}\\right]$ is a steady-state vector for the population migration matrix $A$, because\n", "\n", "$$A{\\bf q} = \\left[\\begin{array}{rr}.95&.03\\\\.05&.97\\end{array}\\right]\\left[\\begin{array}{r}.375\\\\.625\\end{array}\\right] = \\left[\\begin{array}{r}.35625+.01875\\\\.01875+.60625\\end{array}\\right] = \\left[\\begin{array}{r}.375\\\\.625\\end{array}\\right] = {\\bf q}.$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 56, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "To interpret this: \n", "\n", "* if the total population of the region is 1 million, \n", "* then if there are 375,000 persons in the city and 625,000 persons in the suburbs, \n", "* the populations of both the city and the suburbs would stabilize -- they would __stay the same in all future years.__" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 56, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "## Finding the Steady State" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 56 }, "slideshow": { "slide_type": "-" } }, "source": [ "OK, so it seems that the two Markov Chains we have studied so far each have a steady state. This leads to two questions:" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 59 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "* Can we compute the steady state? \n", " * So far we have guessed what the steady state is, and then checked. Can we compute the steady state directly?" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 60 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "* How do we know if:\n", " * a Markov Chain has a unique steady state, and \n", " * whether it will always converge to that steady state?" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 61 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Let's start by thinking about how to compute the steady-state directly.\n", "\n", "__Example.__ Let $P = \\left[\\begin{array}{rr}.6&.3\\\\.4&.7\\end{array}\\right].$ Find a steady-state vector for $P$." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 62 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Solution.__ Let's simply solve the equation $P{\\bf x} = {\\bf x}.$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 63 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "$$P{\\bf x} = {\\bf x}$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 64 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "$$P{\\bf x} -{\\bf x} = {\\bf 0}$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 65 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "$$P{\\bf x} -I{\\bf x} = {\\bf 0}$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 66 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "$$(P-I){\\bf x} = {\\bf 0}$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 67 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Now, $P-I$ is a matrix, so this is a linear system that we can solve." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 68 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "$$P-I = \\left[\\begin{array}{rr}.6&.3\\\\.4&.7\\end{array}\\right] - \\left[\\begin{array}{rr}1&0\\\\0&1\\end{array}\\right] = \\left[\\begin{array}{rr}-.4&.3\\\\.4&-.3\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 69 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "To find all solutions of $(P-I){\\bf x} = {\\bf 0},$ we row reduce the augmented matrix:\n", "\n", "$$\\left[\\begin{array}{rrr}-.4&.3&0\\\\.4&-.3&0\\end{array}\\right] \\sim \\left[\\begin{array}{rrr}-.4&.3&0\\\\0&0&0\\end{array}\\right] \\sim \\left[\\begin{array}{rrr}1&-3/4&0\\\\0&0&0\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 70 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "So $x_1 = \\frac{3}{4}x_2$ and $x_2$ is free. The general solution is $\\left[\\begin{array}{c}\\frac{3}{4}x_2\\\\x_2\\end{array}\\right].$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 71 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "This means that there are an infinite set of solutions. Which one are we interested in?" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 72 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Remember that our vectors ${\\bf x}$ are _probability vectors._ So we are interested in the solution in which the vector elements are nonnegative and sum to 1. \n", "\n", "The simple way to find this is to take any solution, and divide it by the sum of the entries (so that the sum then adds to 1.) \n", "\n", "Let's choose $x_2 = 1$, so the specific solution is:\n", "\n", "$$\\left[\\begin{array}{r}\\frac{3}{4}\\\\1\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 73 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Normalizing this by the sum of the entries ($\\frac{7}{4}$) we get:\n", "\n", "$${\\bf q} = \\left[\\begin{array}{r}\\frac{3}{7}\\\\\\frac{4}{7}\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 74, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "So, we have found how to solve a Markov Chain for its steady state:\n", "\n", "* Solve the linear system $(P-I){\\bf x} = {\\bf 0}.$ \n", "* The system will have an infinite number of solutions, with one free variable. Obtain a general solution.\n", "* Pick any specific solution (choose any value for the free variable), and normalize it so the entries add up to 1.\n" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 74, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "## Existence of, and Convergence to, Steady State" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 74 }, "slideshow": { "slide_type": "-" } }, "source": [ "Finally: when does a system have a __unique__ solution that is a probability vector, and how do we know it will converge to that vector? " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 77 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Of course, a linear system in general might have no solutions, or it might have a unique solution that is not a probability vector. \n", "\n", "So what we are asking is, when does a system defined by a Markov Chain have an infinite set of solutions, so that we can find one of them that is a probability vector?" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 78 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Definition.__ We say that a stochastic matrix $P$ is _regular_ if some matrix power $P^k$ contains only strictly positive entries. " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 79 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "For \n", "\n", "$$P =\\left[\\begin{array}{rrr}.5&.2&.3\\\\.3&.8&.3\\\\.2&0&.4\\end{array}\\right],$$\n", "\n", "We note that $P$ does not have every entry strictly positive. " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 80, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "However:\n", "\n", "$$P^2 = \\left[\\begin{array}{rrr}.37&.26&.33\\\\.45&.70&.45\\\\.18&.04&.22\\end{array}\\right].$$\n", "\n", "Since every entry in $P^2$ is positive, $P$ is a regular stochastic matrix." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 80, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "__Theorem.__ If $P$ is an $n\\times n$ regular stochastic matrix, then $P$ has a unique steady-state vector ${\\bf q}.$ Further, if ${\\bf x_0}$ is any initial state and ${\\bf x_{k+1}} = P{\\bf x_k}$ for $k = 0, 1, 2,\\dots,$ then the Markov Chain $\\{{\\bf x_k}\\}$ converges to ${\\bf q}$ as $k \\rightarrow \\infty.$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 82, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "Note the phrase \"any initial state.\" \n", "\n", "This is a remarkable property of a Markov Chain: it converges to its steady-state vector __no matter what state the chain starts in.__\n", "\n", "We say that the long-term behavior of the chain has \"no memory of the starting state.\"\n" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 82, "slide_helper": "subslide_end", "slide_type": "subslide" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "skip" } }, "source": [ "``` {toggle}\n", "Question Time! Q11.3\n", "```" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 82, "slide_type": "subslide" }, "slideshow": { "slide_type": "slide" } }, "source": [ "__Example.__ Consider a computer system that consists of a disk, a CPU, and a network interface. " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 85 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "A set of jobs are loaded into the system. Each job makes requests for service from each of the components of the system.\n", "\n", "After receiving service, the job then next requests service from the same or a different component, and so on. " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 86 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Jobs move between system components according to the following diagram. \n", "\n", "For example, after receiving service from the Disk, 70% of jobs return to the disk for another unit of service; 20% request service from the network interface; and 10% request service from the CPU." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "hide_input": true, "internals": { "frag_helper": "fragment_end", "frag_number": 86 }, "slideshow": { "slide_type": "-" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "" ] }, "metadata": { "image/png": { "width": 500 } }, "output_type": "display_data" } ], "source": [ "# Image credit: my own drawing\n", "display(Image(\"images/disk-cpu-net.png\", width=500))" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 86 }, "slideshow": { "slide_type": "-" } }, "source": [ "Assume the system runs for a long time. Determine whether the system will stabilize, and if so, find the fraction of jobs that are using each device once stabilized. Which device is busiest? Least busy?" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 89 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "__Solution.__ From the diagram, the movement of jobs among components is given by:\n", "\n", "$$ \\begin{array}{rccc}&\\mbox{From Disk}&\\mbox{From Net}&\\mbox{From CPU}\\\\\\mbox{To Disk}&0.70&0.10&0.30\\\\\\mbox{To Net}&0.20&0.80&0.30\\\\\\mbox{To CPU}&0.10&0.10&0.40\\end{array}$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 90 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "This corresponds to the stochastic matrix:\n", "\n", "$$P = \\left[\\begin{array}{rrr}0.70&0.10&0.30\\\\0.20&0.80&0.30\\\\0.10&0.10&0.40\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 91 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "First of all, this is a regular matrix because $P$ has all strictly positive entries. So it has a steady-state vector." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 92 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Next, we find the steady state of the system by solving $(P-I){\\bf x} = {\\bf 0}:$\n", "\n", "$$[P-I\\; {\\bf 0}] = \\left[\\begin{array}{rrrr}-0.30&0.10&0.30&0\\\\0.20&-0.20&0.30&0\\\\0.10&0.10&-0.60&0\\end{array}\\right] \\sim \\left[\\begin{array}{rrrr}1&0&-9/4&0\\\\0&1&-15/4&0\\\\0&0&0&0\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 93 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Thus the general solution of $(P-I){\\bf x} = {\\bf 0}$ is \n", "\n", "$$x_1 = \\frac{9}{4} x_3, \\;\\;\\; x_2 = \\frac{15}{4} x_3,\\;\\;\\; x_3\\;\\mbox{free.}$$\n", "\n", "Since $x_3$ is free (there are infinitely many solutions) we can find a solution whose entries sum to 1. This is:\n", "\n", "$${\\bf q} \\approx \\left[\\begin{array}{r}.32\\\\.54\\\\.14\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 94 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "So we see that in the long run, \n", "* about 32% of the jobs will be using the disk, \n", "* about 54% will be using the network interface, and \n", "* about 14% will be using the CPU." ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 95 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Again, the important fact here is that we did not have to concern ourselves with the state in which that the system started. \n", "\n", "The influence of the starting state is eventually lost!" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 96 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Let's demonstrate that computationally. Let's say that at the start, all the jobs happened to be using the CPU. Then:\n", "\n", "$${\\bf x}_0 = \\left[\\begin{array}{r}0\\\\0\\\\1\\end{array}\\right].$$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 97 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Then let's look at how the three elements of the ${\\bf x}$ vector evolve with time, by computing $P{\\bf x}_0$, $P^2{\\bf x}_0$, $P^3{\\bf x}_0$, etc." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "hide_input": false, "internals": { "frag_helper": "fragment_end", "frag_number": 97 }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "image/png": { "height": 280, "width": 424 }, "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#\n", "# system definition\n", "A = np.array(\n", " [[.7,.1,.3],\n", " [.2,.8,.3],\n", " [.1,.1,.4]])\n", "x = np.array([0,0,1])\n", "xs = np.zeros((15,3))\n", "#\n", "# compute 15 steps in the evolution of the system stae\n", "for i in range(15):\n", " xs[i] = x\n", " x = A @ x\n", "#\n", "# plot the results\n", "ax = plt.subplot(131)\n", "plt.plot(range(15),xs.T[0],'o-')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_1$',size=24)\n", "ax = plt.subplot(132)\n", "plt.plot(range(15),xs.T[1],'o-')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_2$',size=24)\n", "ax = plt.subplot(133)\n", "plt.plot(range(15),xs.T[2],'o-')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_3$',size=24)\n", "plt.tight_layout()" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 99 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Notice how the system starts in the state $\\left[\\begin{array}{r}0\\\\0\\\\1\\end{array}\\right]$,\n", "\n", "but quickly (within about 10 steps) reaches the equilibrium state that we predicted: $\\left[\\begin{array}{r}.32\\\\.54\\\\.14\\end{array}\\right].$" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 100 }, "slideshow": { "slide_type": "fragment" } }, "source": [ "Now let's compare what happens if the system starts in a different state, say $\\left[\\begin{array}{r}1\\\\0\\\\0\\end{array}\\right]$:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "hide_input": true, "internals": { "frag_helper": "fragment_end", "frag_number": 101 }, "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "image/png": { "height": 280, "width": 424 }, "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "A = np.array([[.7,.1,.3],[.2,.8,.3],[.1,.1,.4]])\n", "x = np.array([1,0,0])\n", "ys = np.zeros((15,3))\n", "for i in range(15):\n", " ys[i] = x\n", " x = A.dot(x)\n", "ax = plt.subplot(131)\n", "plt.plot(range(15),xs.T[0],'o-',range(15),ys.T[0],'o-r')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_1$',size=24)\n", "ax = plt.subplot(132)\n", "plt.plot(range(15),xs.T[1],'o-',range(15),ys.T[1],'o-r')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_2$',size=24)\n", "ax = plt.subplot(133)\n", "plt.plot(range(15),xs.T[2],'o-',range(15),ys.T[2],'o-r')\n", "ax.set_ylim([0,1])\n", "plt.title(r'$x_3$',size=24)\n", "plt.tight_layout()" ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 102, "slide_helper": "subslide_end" }, "slide_helper": "slide_end", "slideshow": { "slide_type": "fragment" } }, "source": [ "This shows graphically that even though the system started in a very different state, it quickly converges to the steady state regardless of the starting state." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "hide_input": true, "slideshow": { "slide_type": "fragment" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "image/png": { "height": 253, "width": 277 }, "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "A = np.array([[.7,.1,.3],[.2,.8,.3],[.1,.1,.4]])\n", "x = np.array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.], [0.5, 0., 0.5], [0., 0.5, 0.5], [0.5, 0.5, 0.]]).T\n", "colors = ['r', 'b', 'g', 'y', 'c', 'm']\n", "npts = x.shape[1]\n", "fig = ut.three_d_figure((11, 1), fig_desc = 'Convergence to Steady State',\n", " xmin = 0, xmax = 1, ymin = 0, ymax = 1, zmin = 0, zmax = 1, qr = None)\n", "for pt in range(npts):\n", " fig.plotPoint(x[0][pt], x[1][pt], x[2][pt], colors[pt])\n", "for i in range(10):\n", " xnext = A @ x\n", " for pt in range(npts):\n", " # fig.plotPoint(xnext[0][pt], xnext[1][pt], xnext[2][pt], colors[pt])\n", " fig.plotLine(np.array([[x[0][pt], xnext[0][pt]], [x[1][pt], xnext[1][pt]], [x[2][pt], xnext[2][pt]]]).T, \n", " colors[pt])\n", " x = xnext\n", "fig.text(x[0][0]-0.05, x[1][0], x[2][0]+0.05, 'Steady State', 'Steady State', 12)\n", "fig.plotPoint(x[0][0], x[1][0], x[2][0], 'k')\n", "fig.ax.view_init(azim=40.0,elev=45.0)\n", "fig.save()" ] }, { "cell_type": "markdown", "metadata": { "hide_input": false }, "source": [ "Although we have been showing each component converging separately, in fact the entire state vector of the system can be thought of as evolving in space.\n", "\n", "This figure shows the movement of the state vector starting from six different initial conditions, and shows how regardless of the starting state, the eventual state is the same." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Summary\n", "\n", "* Many phenomena can be describe using Markov's idea:\n", " * There are \"states\", and\n", " * Transition between states only depends on the current state.\n", "* Examples: population movement, jobs in a computer, consonants/vowels in a text...\n", "* Such a system can be expressed in terms of a stochastic matrix and a probability vector.\n", "* Evolution of the system in time is described by matrix multiplication.\n", "* Using linear algebraic tools we can predict the steady state of such a system!" ] } ], "metadata": { "celltoolbar": "Slideshow", "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": "3.7.10" } }, "nbformat": 4, "nbformat_minor": 1 }