{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "hide_input": true, "internals": {}, "slideshow": { "slide_type": "skip" }, "tags": [ "remove-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 matplotlib import animation\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": true, "internals": { "frag_number": 3 }, "slideshow": { "slide_type": "-" }, "tags": [ "remove-cell" ] }, "outputs": [ { "data": { "image/jpeg": 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", "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": "markdown", "metadata": { "hide_input": false, "slideshow": { "slide_type": "skip" } }, "source": [ "```{margin}\n", "AAMarkov by unknown; \n", "originally uploaded to en-wikipedia by \n", "en:User:Mhym \n", " - 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. \n", " Licensed under Public Domain via Wikimedia Commons.\n", "```" ] }, { "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 patterns in sequences, such as text like _Eugene Onegin_. 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. This has turned out to be a terrifically useful idea. For example, it is the starting point for analysis of 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. " ] }, { "cell_type": "markdown", "metadata": { "internals": { "frag_helper": "fragment_end", "frag_number": 21 }, "slideshow": { "slide_type": "slide" } }, "source": [ "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 ${\\bf x_0} \\in \\mathbb{R}^n$ 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": "slide" }, "tags": [ "remove-cell" ] }, "source": [ "``` {toggle} \n", "Question Time! Q11.2\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 counts to a probability vector, in which elements sum to 1. \n", "\n", "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": 4, "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_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": 5, "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": 6, "metadata": { "hide_input": true, "internals": { "frag_helper": "fragment_end", "frag_number": 44 }, "slideshow": { "slide_type": "-" }, "tags": [ "remove-input" ] }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "
" ] }, "metadata": { "image/png": { "height": 469, "width": 629 } }, "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 a steady-state vector for $\\left[\\begin{array}{rrr}.5&.2&.3\\\\.3&.8&.3\\\\.2&0&.4\\end{array}\\right]$\n", "\n", "because\n", "\n", "$$ \\left[\\begin{array}{rrr}.5&.2&.3\\\\.3&.8&.3\\\\.2&0&.4\\end{array}\\right]\\left[\\begin{array}{r}.3\\\\.6\\\\.1\\end{array}\\right] = \\left[\\begin{array}{r}.3\\\\.6\\\\.1\\end{array}\\right] $$" ] }, { "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": "slide" }, "tags": [ "remove-cell" ] }, "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": "markdown", "metadata": { "hide_input": false, "internals": { "frag_helper": "fragment_end", "frag_number": 86 }, "slideshow": { "slide_type": "-" } }, "source": [ "\n", "\n", "
\n", "\n", "\"Figure\"\n", " \n", "
" ] }, { "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 unique 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": 7, "metadata": { "hide_input": true, "internals": { "frag_helper": "fragment_end", "frag_number": 97 }, "slideshow": { "slide_type": "-" }, "tags": [ "remove-input" ] }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "
" ] }, "metadata": { "image/png": { "height": 469, "width": 629 } }, "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": 8, "metadata": { "hide_input": true, "internals": { "frag_helper": "fragment_end", "frag_number": 101 }, "slideshow": { "slide_type": "fragment" }, "tags": [ "remove-input" ] }, "outputs": [ { "data": { "image/png": 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\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "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", "plt.close()\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=20.0)\n", "fig.save()\n", "#\n", "def anim(frame):\n", " fig.ax.view_init(azim = frame, elev = 20)\n", " # fig.canvas.draw()\n", "#\n", "# create and display the animation \n", "HTML(animation.FuncAnimation(fig.fig, anim,\n", " frames = 5 * np.arange(72),\n", " fargs = None,\n", " interval = 100).to_jshtml(default_mode = 'loop'))" ] }, { "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 (ipykernel)", "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": "3.9.18" }, "rise": { "scroll": true, "theme": "beige", "transition": "fade" } }, "nbformat": 4, "nbformat_minor": 1 }