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Markov Chainsby Leah W Berman, Christian Hellings, Lynne L Dotyrestart: with(LinearAlgebra): with(plots): with(plottools):
<Text-field style="Heading 1" layout="Heading 1">A small population example</Text-field>Suppose the movement of a population between a city and its suburbs may be modelled by the following: each year, 5% of the population moves from the city to the suburbs (and 95% of the city population stays in the city) and 3% of the population moves from the suburbs to the city (with the remaining 97% of the suburban population staying in the suburbs). The situation can be modelled with a diagram: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cVRGW1NIU1NGZ1NHdXRQXVNGX0NFXWNHWnBeX2tuYHBmZVVHYGw+YGZGYHA6TE1QQE1VOmtDRV1jR2pTPE1QQD1KU2xsUjpMTVBAPWpUbGxSVG1VcG9nQD12YGpmZVVHQGpUcG9nOlxcbWhob1JacmZgV1dFWnNmZVVHQFJIX1NRcVRGWnBmZVU7X0NGXVNRcVRGVE1QRE1STG1oaG9SOmtTUXE8RmBqZmVVO15gV1dFWnBmZVU7RE1ScG9nOkRtaGg/aUNFX2NGYVNRcVRGWm9mZVU7PmFXV0VqU0BtaGg/alNwb2c6NDpcIlx7XH0=cVRGYUNHU1NGZ1NIdXRQXXNGZUNFXWNHZVNRcVRGVE1QQG1UWG1oaG9SRmFmRmBuPmBXV2VrXmBsXl9rTmBxZmVVa1NsbFJATVNwb2dAPT5gamZlVTttY0d1dFBdOzxtaGg/PmFvZmVVS1M+YW5mZVVHQFJGdXRQXTtpU0h1dFBGYGZ2YHJmZVVHQGJHdXRQPmBmdmBxZmVVS1RsTFZAPXZgZj5hbWZlVUdAbUNFa0NHUkdhQ0VrO2ZgZj5hbzppQ0VrY0d1dFBUbVJsTFZUPWZgZj5hcWZlVWtVbExWWD1qVkRtaGg/ckd1dFBuYGZGYGpOYFdXRWpSVG1oaD9yRmVTUXFcXG9SRmNDR3V0UFpwTmBXV0VqVUhtaGg/alVwP0pWUG1oOkZhV1dFalZAbWhoP2pTcG9nOlBtaGg/alVwP2JGW0NGdTxKU3A/VF1sPmBwZmVVOz5hV1dFalI8bWhoP2lLU0BtU3BvZzpYbWhoP2JGXVNRcVxccl5fbE5gbGZlVTtmYFdXRVpwZjZFXCJce1x9cVRGZ3NHU1NGX3NHdXRQXWNHbUNFXWNGalVITVBATVM6Z0NFXWNGWnJeX2tOQEJIXUNFXWNGa1NRcVRGSE1QQF1zZmVVR0BlQ0VdY0ZacV5fazo+YWZGYG0+YFdXZWs6RG1oaG9SdkBIbVNwb2dAPXJHdXRQbmBmRmBuPmBXV0VacGZlVUdAUkdfU1FxPEJIdXRQWnBOYFdXRVpxZmVVO1htUnBvZ0BdcWpVTG1oaG9SPkFYbVVwb2c6a0NGdXRQOl9TUXE8Ykd1dFBac0ZgV1dFWnBmZVU7X0NGX1NRcTxiR3V0UFprPmBXV0VabWZlVTt2YFdXRVpzZmVVO0RtUnBvZzpIbWhoP2pUcG9nQD1CSHU8ckZdU1FxPDY6XCJce1x9cXRGW1NIU1NGaUNHdXRQYUNGalZwb2dIXXJeX2s+YW5mZVU7YE1TcG9nSD1ETVI8bWhob1N2YGZOYGpeYFdXRVpyZmVVV0BSRmFTUXF0RlpwZmVVV0BqVnBvZzpfY0Z1dFA6ZVNRcXRGWnFmZVVXQGpWcG9nOmFTRnV0UGE7VmBXV0VUTVBEbVM0OlwiXHtcfQ==cXRGW0NIU1NGaVNHdXRQYVNGXUNFXXNHWm1eX2t2QEpWbGxSOl9TRlNTRjphQ0VdO2JHU1NGaTtIbVJsbFI6TE1QQG1VOmlDRV1zR0pUQE1QQD1qU2xsUjpQTVBAbVU6a0NFXTtmYGpeX2t2QEpTbGxSWD1CR1NTRjpnQ0VdO0JIU1NGWnA+YGZGQFprXl9rYkd1dFBhS1NsbFJcXE1ScG9nSD1CR3V0UGE7PmFXV2VtckZTU0ZtQ0d1dFA6a1NRcTxETVJMbWhoP2pVcG9nSD1GYGpmZVU7TmBXV2VtOlBtaGg/alVwb2c6YG1oaG9TWmxGYFdXZW06SG1oaG9TOmVTUXFcXGxiRl9zR3V0UDptU1FxPFZgbGZlVTteYFdXRVpwZmVVS1JOYG12YFdXRWVzR1NjRmE7VmBmTmBtOkBNUERtU1puZkBfc0Zaa2JGWm1uYGZOYG06SE1QRD1KUmxMUzpfQ0hTY0ZhO25AXztyRkRdcDpdS1M6XVtsUkd1dFBuYGZOQFpuXl9sVkBKU2xMUzo8TVBEPUJGTkBacV5fbDY6XCJce1x9cXRGX0NIU2NGYUNIdXRQYWNGSlVwb2dIPUJHdXRQYTtOYFdXZW06PG1oaG9TWmw+YVdXRVpwZmVVO1ZgV1dlbTpAbWhob1Naa0ZhV1dlbTpYbWhob1M6ZVNRcXRGWm1mZVU7PmBXV0VKUlxcbWhoP0pVcG9nOkxtaGhfbUZgZk5gajpITVBETVI6ZUNFX0NGWnFeX2w+QGpWbExTPD1eYGteX2w6VmBmTkBab15fbDo6ZztacjpSRjpyRjpmQDpYPTptO0pTOmNDRV9jRl9TUXE8Qkd1dFA6ZztacjpyRjpyRnV0UDplU1FxPHJHdXRQOm1TUXE8XmBrZmVVOzU6XCJce1x9cXRGY1NGU2NGX0NGdXRQYUNHOjE6XCJce1x9cXRGXVNGU1NGaVNHdXRQYVNGalNwb2dIXWxeX2t2YGtmZVVXQGFDRV1jR2tTUXF0RkxNUEBNVVhtaGhvU2ZgZkZgcG5gV1dlbWJHU1NGZ0NHdXRQYWtVbGxSVG1ScG9nQkhTU0ZnQ0Z1dFBha1ZsbFJQTVZwb2c6VG1oaF9sPmBmRmBvZmBXV0VATVBAbVREbWhob1NOYGZGYG8+YFdXRUhNUEBNVGBtaGhvU15gZkZgbj5hV1dFUE1QQE1UVG1oaD9nQ0VdQ0dlU1FxXFxyXl9rXmBuZmVVa1ZsbFJMTVNwb2dWYGpeX2teYGtmZVVrUmxsUkxNUnBvZ0JHU1NGYVNIdXRQZmBmRmBtPmFXV0VacGZlVUtVbGxSSG1UcG9nckdTU0ZhQ0d1dFA+YWZGYG1mQF5gal5fa1ZgcWZlVUtUbGxSTG1UWnFeX2tmQEpWbGxSUG1TcG9nZmBrXl9rZmByOl9DRV1jR19TUXFcXG1eX2tuYG9mZVVLVGxsUlRNVlpvXl9rdkBKVWxsUlhNU3BvZ3JHU1NGaUNHdXRQPmFmRmBxZkBuYGpeX2t2QGpSbGxSWE1VcG9nYkZTU0Zpc0d1dFBWYGZGYHE1OlwiXHtcfQ==cWRHZ0NFXWNHZ1NRcWRHY0NFXWNHalJsbFJUPVBNVmxsUlQ9XmBmRkBKU2xsUlRtVXBvZ0xtVmxsUlQ9dmBmRmBwPmFXV2VuZmBmRmBwRmFXV2VuTmBmRmBxRmBXV2VuPmBmRmBxTmBXV2VtbmBmRmBxbmBXV2VtVmBmRmBxRmFXV2VtRmBmRmByTmBXV0VbQ0VdQ0hjU1FxZEZtQ0VdQ0hnU1FxZEZqVnBvZ0RNVmxsUmBtU3BvZ0Q9dmBXV2VsdmBmTmBqPmBXV2VsOmNTUXE8dmBXV0VqVnBvZ1prTmBXV2VsPkFfU0ZjU1FxdEZOYGtmYFdXZW1GQF9TRmdTUXF0RmVDRV9TRmlTUXF0RmlDRV9LVnBvZ0pTRE1ScG9nTE1UbExTRD1mQF9jRl1TUXFsVU5gbDptS1NEPVBtUmxMU0Q9VmBmTmBsOmVDRV87dmBmTkBbW28+QV9rVnBvZ1BtVmxMU0BNVlJHOjE6XCJce1x9cVRHXUNFX1NGX1NRcVRHYUNFX1NGalRsTFNAPW5gZk5ga1ZgV1dlbz5hZk5ga1pwPmBmTmBrOl1DRV9LVHBvZ1RNU2xMU0BtVHBvZ1RtU2xMU25gV1dlcF5gZk5ga3ZgV1dlcFZARmFXV2VwTkBETVNwb2dUTVJiRmNTUXFUR21DRV9jRmdTUXFUR2tbbHZgV1dlbzprU1ExOlwiXHtcfQ==cXRGZWNHU1NGY3NHdXRQYVNHaUNFXUNHZVNRcXRGWm1mZVVXQGtDRV1DR19TUXF0RmBNUEBNVEBtaGhvU1RtUmxsUkxNUnBvZ0hdbV5fa15AalRsbFJMPXJHU1NGOm1DRV07dmBrXl9rOlZgZkZAWnBeX2s6PmFmRkBKVkBNUEBdazphQ0VdS1Nab15fa3JGSlVsbFJeYFdXZW1yR1NTRlA9PmFmRkBnU1FxdEZgTVBAXXE6alZwb2dIbVY8TVBAbVQ8bWhoP11DRV1TR11TUXFcXGxeX2tmYG1mZVU7ZmBXV0Vac2ZlVTtUTVNwb2dSRm5gbWZlVTtuYFdXRTw9Z3NHdXRQXFw9VG1WcG9nQkh2YGpmZVVLVXJHYVNRcVxcb2pVTG1oaD9jQ0Vdc0dnU1FxXFxsXl9rdmByZmVVV0BbQ0Vdc0dqVVpzZmVVa1VsbFJcXE1ScG9nUkdTU0ZrO05gZkZgcjpnO2s7dkA6OjI6XCJce1x9cXRGaWNGU1NGaXNGdXRQYXNHXUNFXXNHZVNRcXRGPE1QQG1VVG1oaG9TOmtTUXF0RmdTSFNTRmlTSHV0UGFLVmxsUlxcbVJwb2dIXXNKVkRtaGhvU1htUkJIYVNRcVxcbEpWTG1oaG9TXmBmRmByZmBXV0VQTVBATVZUbWhoP2dDRV1DSGlTUXF0RlhNUEBNVlxcbWhoP2tDRV1rVnBvZzo0OlwiXHtcfQ==cWRGX2NGU1NGaVNIdXRQX2NGY0NFXXNHWnFeX2s+YWpmZVVPYG1OYGZGYHJGYFdXZWxyR1NTRlpuRmBmRmByTmBXV2VsYkdTU0ZrO1BtUmxsUlZgV1dlbG5gal5faz5hbmZlVU9AZUNFXUNIZVNRcWRGXFxNUEBNVlpxRmBmRkBabl5faz5BSlVsbFI6XFxNUEA9QkhdQ0VdO3JGU1NGaztmYGZGQFpxXl9rOj5hZkZAMzpcIlx7XH0=cWRGa0NGU1NGaVNIdXRQX0NIYUNFXXNHWm5eX2s+YWpmZVVPQGdDRV1DSF1TUXFkRlhNUEBNVkRtaGhPUz5hZkZgclZgV1dlbEZhal5faz5hbmZlVU9AXUNFXUNIZVNRcWRGRE1QQE1WVG1oaE9TRkA+YVdXZWw+YXNeX2s+YXNmZVVLVlJIW1NRcTxiRnV0UHZAYG1TcG9nYkdGYW5mZVU7NTpcIlx7XH0=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cXRGW0NFXUNHYVNRcWRGa0NFXUNHalRsbFJMTVRwb2dEPW5gV1dlbDprU1FxZEZnQ0VdQ0dtU1FxZEZrW28+YFdXZW0+QFA9TmBmRmBvOmNDRV1TR2pUbGxSWm1eQExNVnJGYUNFXUNHaVNRcXRGX1tubkBIbVJsbFJMbVRwb2dITVJCR2M7QkdqVWxsUkxtUzo1OlwiXHtcfQ==cXRGYUNFXUNHbVNRcXRGYVtvRmBXV2VtOmNTUXF0RkpVcG9nSD0+YVdXZW1acD5gV1dlbTpfU1FxdEZqU3A1P1wiXHtcfQ==cURHXUNFXUNHYVNRcURHW0NFXUNHZVNRcXRGbUNFXUNHaVNRcXRGUkdbU1FxdEZrQ0VdU0ddU1FxdEZpQ0VdU0dlU1FxPHZgV1dFalZwb2dIbVY6TG1SUkdqU2xsUlA9XmBmRmBwPmBXV2VubmBmRmBwRmBXV2VudmBmRmBwTmBXV2VuPmFmRmBwVmBXV0VtQ0VdY0djU1FxREdKVXBvZ0xNVmpVcG9nTE1VSlZwb2dMTVQ6RkBUPUg9Oms7aVttOjM6XCJce1x9cURHXUNFXUNHYVNRcURHXTtlU1FxREdhQ0VdQ0dnU1FxREdlQ0VdQ0dqVWxsUkw9RmFmRmBuWm9GYGZGQGpTbGxSOjoxOlwiXHtcfQ==cWRGYVNIU1NGZ3NHdXRQX3NGOjE6XCJce1x9cWRGZWNHU1NGZVNIdXRQX1NHY0NFXVNHWmxeX2tmQEpSbGxSUD06Z1NGdXRQX0NHbUNFXWNHYVNRcWRGWm9mZVVPQGpVcG9nRD1SSHV0UFpxRmBXV0VlU0ZTU0ZpO05AWE1TcG9nRF1ualU6Z1txOlhNUEBtVTxtaGhPU1pwPmFXV2VsOlRtaGhPUzpjU1FxZEZUPWdzRmpUbGxSVE1TcG9nQkduYGs6X1twOjoyOlwiXHtcfQ==cWRGZ3NHU1NGZ1NGdXRQX2NHZUNFXWNHWm1eX2tuQGpSbGxSVD06SG1oaE9TOmdTUXFkRkRNUEBNVVhtaGhPU15gZkZgcDpQPVQ9dkA6YE1QQE1VWD1YTVJsbFJUTVZwb2dEPXZgamZlVU9ASlNwb2dEPUJHdXRQX2NHckdlU1FxZEZYPWljR3V0UGZAWD1CR3ZgcWZlVUtTckdaal5fazo6MzpcIlx7XH0=cWRGY1NGU2NGY1NHdXRQX0NHOjE6XCJce1x9cWRGZVNHU2NGYXNHdXRQX1NHYUNFX3NGWmxeX2xWYHJmZVVPQEJHW1NRcWRGWmxmZVVPQEpUcG9nRF1tSlRQbWhoT1NeYGZOYG52YFdXRVA9Y0NIdXRQbmBmTmBuRmFXV2VsckdTY0ZlQ0Z1dFA+YWZOYG9GYFdXZWxSSFNjRmVDRjpMTVY6Ykd1dFBfO15AOkQ9Ols7XFw9YVNIdXRQdkBITVZacGpTWD1mQEhNVXBvZzo0OlwiXHtcfQ==cWRGZ0NHU2NGYWNHdXRQX2NHZ0NFX3NGWnJeX2xWQHJHW0NFX3NGWmteX2xWYHFmZVVPQGFDRV9zRm1TUXFkRkpUQG1oaE9TTmBmTmBuTmBXV2VsQkZeQGJHQkdqVTpKUzpCR1NjRmM7ZmBmTmBuVmBXV2VsckdTY0ZjOz5hZk5AY1NRcWRGYE1QRE1UUG1oaE9TOmlTUXFcXHJacmZlVU9AZ0NFX0NHbVNRcVxcb2pUPG1oaE9TVkBQPUJGZkA6NTpcIlx7XH0=cXRGZ0NGU1NGYUNGdXRQYWNHOjE6XCJce1x9cXRGaUNGU1NGXUNIdXRQYWNHa0NFXVNGWnFeX2tGYHNmZVVXQGVDRV1jRltTUXF0RkxNUEBNU0BtaGhvU1ZgZkZgbE5gV1dlbWJGU1NGX3NGdXRQYTtmYFdXZW1CRzp2QEQ9UkhTU0ZfO1g9RE1UcG9nSF1rXl9rTmBtOmpSOkZAOkQ9Okg9PkBuYFdXZW06XFxtaGg/ckZbU1FxPGJGdXRQOmNTUXE8Ykd1dFA6MzpcIlx7XH0=cXRGaXNHU1NGXUNIdXRQYXNHbUNFXVNGSlZATVBAbVI6YUNFXVNGWm9eX2tGQEpUbGxSRE1ScG9nSF1tSlNAbWhob1NOYGZGYGxOYFdXZW1SRk5gbmZlVVdAW0NFXWNGZVNRcXRGaTtfc0d1dFBhO0ZhV1dlbUJIU1NGYUNGdXRQdkBITVNwb2c6SG1oMjpcIlx7XH0=or a table:
We can construct a matrix with the above data; this is called a stochastic matrix. (Formally, a stochastic matrix is one in which the entries in each column sum to 1.)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 in the year 2005, 60% of people live in the city and 40% of people live in the suburbs. We can make a vector representing this population distribution: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 We can use this to estimate the population distribution in 2006: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 in the year 2006, we expect approximately 58% of people to live in the city and 42% to live in the suburbs.What happens to the population over time: do most people stay in the city? Do most people move to the suburbs? One way is to calculate a sequence of vectors: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Notice that b2 = (A.b1) = A.(A.b) = A2.b and b3 = A.(b2) = A.(A2.b) = A3.b. In general, bk = Ak.b. So if we want to look at where people live a long time in the future, we could simply multiply by some appropriate power of A.The sequence of vectors b, b1 = A.b, b2 = 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.b, b3 = 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.b, ... is called a Markov chain.Let's look at what happens over the next hundred years: the collection li is the list of population distributions from 2006 to 2106.li:=[seq(A^k.b, k=1..100)]:
pointplot(li,color=blue,view=[0..1,0..1]);We can see the evolution over time in the following animation.p:=[seq( pointplot( [seq(li[i], i=1..5*t)], color=blue, view=[0..1,0..1] ), t=0..20 )]:
display( p, insequence=true );
Click on the plot and use the VCR controls in the toolbar. Setting the frames per second to 1 may be helpful.
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It looks like the population is converging to some value! We can see what the population is in 2106:A^(100).b;So, approximately 37.5% of the people live in the city and 62.5% live in the suburbs in 2106.We can look farther into the future: this time we'll look in the future by hundred year jumps.li2:={seq(A^(100*k).b, k=1..5)};And by thousands:li3:={seq(A^(1000*k).b, k=1..5)};It looks like the lists of vectors are converging---in the far, far future, about 37.5% will live in the city and 62.5% will live in the suburbs.How does the initial distribution of the population (60% city, 40% suburbs) affect the long-range distribution of people in this area? We can start with 30% of the people living in the city and 70% in the suburbs.
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b2 := A.b1;Let's skip ahead to 100 years from now:A^(100).b;It looks like the same long-term distribution as before: 37.5% in the city, 62.5% in suburbs. So at least for this example, the initial distribution of folks doesn't seem to affect the long-term one. Is this always true? More later.bb:=<1,0>;
A^(1000).bb;
<Text-field style="Heading 2" layout="Heading 2">Exercises</Text-field>Suppose that in a different city, it was found that each year, 12% of the population moves from the city to the suburbs (and 88% of the city population stays in the city) and 4% of the population moves from the suburbs to the city (with the remaining 96% of the suburban population staying in the suburbs).1. What is the stochastic matrix representing the movements to and from the city and the suburbs?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. If in 2005, 82% of the population lived in the city and 18% live in the suburbs, what will the population distribution be in 2006? What about 2008?3. In the far future, what do you think the population distribution will be?Do you think the population distribution in the future depends on the initial distribution? Choose a random distribution (but make sure the total population percentages add up to 100%) and guess what the population distribution will be in the far future.
<Text-field style="Heading 1" layout="Heading 1">A more complicated example</Text-field>For this example we assume the population moves among three locations: City, Suburbs, Rural. Here's the table that indicates the movement of people from location to location.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
From CityFrom SuburbsFrom RuralTo City.96.01.015To Suburbs.03.98.005To Rural.01.01.980
Here's the stochastic matrix for this changing population.P := Matrix(<< 0.96 | 0.01 | 0.015 >,
< 0.03 | 0.98 | 0.005 >,
< 0.01 | 0.01 | 0.98 >>);