User legend - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:19:35Z http://mathoverflow.net/feeds/user/3560 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77085/how-are-epidemic-models-simulated-in-case-of-mobility How are epidemic models simulated in case of mobility? Legend 2011-10-03T23:24:48Z 2011-10-04T00:18:01Z <p>I am not a mathematician but out of curiosity I am trying to implement the <a href="http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIS_model" rel="nofollow">SIS epidemic model</a> when the nodes have mobility to understand how it will change the results. I understand how to perform this simulation in an analytical fashion. However, things get rather confusing when nodes are mobile. To give a background, all models that I have come across do not consider that nodes are mobile.</p> <p>The model assumes that each node can infect <strong>any</strong> node and hence the equations (differential or difference) are valid. But when nodes are mobile, each node is <strong>not able</strong> to infect every other node (the other node might not be within the range) and has to explicity send a message to a node that is susceptible in order to infect it. In that case, give an infection rate <code>B</code>, how do I simulate this when the nodes are mobile? </p> <p>Currently, the way I am doing this is in the following way:</p> <pre><code>def Controller(): for i in range(1,100): randNum = getRand() if (randNum &lt;= InfectionRate): neighbors = getNeighbors(i) ScheduleTransmission(getCurrentTime(), i, neighbors) Schedule(getCurrentTime() + 1, Controller) </code></pre> <p>My problem is that I am not understanding if the infection rate can now be captured through a single value (which was previously <code>B</code>). If not, how does one analyze this scenario? Do I set the <code>InfectionRate</code> as <code>B/numNodes</code> so that the overall probability will be <code>B</code>? Any suggestions?</p> <p><strong>UPDATE: Making it more realistic</strong></p> <pre><code>def Controller(): for i in range(1,100): neighbors = getNeighbors(i) for j in neighbors: randNum = getRand() if (randNum &lt;= InfectionRate): ScheduleTransmission(getCurrentTime(), i, j) Schedule(getCurrentTime() + 1, Controller) </code></pre> http://mathoverflow.net/questions/72668/how-to-compute-kl-divergence-when-pmf-contains-0s How to compute KL-divergence when PMF contains 0s? Legend 2011-08-11T08:39:38Z 2011-08-11T11:11:52Z <p>From the <a href="http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence" rel="nofollow">Wikipedia page</a> on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula:</p> <p><img src="http://i.stack.imgur.com/DOJFR.png" alt="enter image description here"></p> <p>The way I understand this is to compute the PMFs of two given sample sets and then use the above formula to compute the KL-divergence. I am not quite sure I am getting this though. Let us say I have the following set of values:</p> <pre><code>0 0 0 0 0 1 1 0 0 0 0 0 1 2 0 0 0 0 0 20 0 0 0 </code></pre> <p>I am trying to compute the KL-divergence for a sliding window size of 7. So, I start of by choosing the bin size for my histogram as <code>1</code> and (max, min) values as <code>(20, 0)</code> based on the entire dataset. So that when I have the sliding window as follows:</p> <pre><code>0 0 0 0 0 1 1 0 0 0 0 0 1 2 0 0 0 0 0 20 0 0 0 |-----------| |-----------| </code></pre> <p>I compute the PMFs across these two sliding windows as:</p> <pre><code>0 0 0 0 0 1: 0.81818182, 0.18181818 0 0 ... 0 0 0 0 0 1 2: 0.66666667, 0.16666667, 0.16666667 0 0 ... 0 </code></pre> <p>Both these PMFs satisfy the first condition from the Wiki page:</p> <blockquote> <p>The K-L divergence is only defined if P and Q both sum to 1 and if Q(i) > 0 for any i such that P(i) > 0.</p> </blockquote> <p>but failing with the second condition. Can someone please tell me where I am going wrong? </p> http://mathoverflow.net/questions/44015/how-do-i-approach-optimal-control How do I approach Optimal Control? Legend 2010-10-28T20:34:20Z 2011-04-09T21:51:14Z <p>Other than learning basic calculus, I don't really have an advanced background. I was curious to learn about Optimal Control (the theory that involves, bang-bang, Potryagin's Maximum Principle etc.) but any article that I start off with, mentions the following: "Consider a control system of the form..." and then goes on to defining partial differential equations. In short, I am lost. </p> <p>Can someone suggest me a path I should take to learn more about Optimal Control from the very basics?</p> http://mathoverflow.net/questions/57319/optimal-placement-inside-an-arena-given-spatial-density-distribution Optimal placement inside an arena given Spatial Density Distribution? Legend 2011-03-04T05:05:25Z 2011-03-04T08:45:17Z <p>I am not a mathematician so please correct me if I am making any mistakes while explaining my problem (and please help me set the right tags as well because I am not sure what class of problem I am facing). While trying to solve it, I am also trying to frame the right question. </p> <p>From some mobility data that I collected from an experiment I was conducting, I plotted the following spatial distribution graph for the area that I am analyzing. For about 30 people, this graph gives a distribution of how long each person spent in the area (which is a plane with dimensions 200x200). So for instance, over the period of observation (18 hours), the upper-left is the most visited region. </p> <p><img src="http://i52.tinypic.com/33myia9.jpg" alt="alt text"></p> <p>I am trying to place an experimental delivery service that has a proximity of 40m (circle) in the region is a way that it covers the maximum density region. The idea is that I have only a limited amount of resources and need to make the best use of them i.e. within the 40m proximity I should be able to cover the maximum number of people during the observation period. </p> <p>I am guessing that my problem is not a new one and was hoping if someone could guide me in the right direction of understanding and solving this problem.</p> http://mathoverflow.net/questions/15147/minimizing-a-function-containing-an-integral Minimizing a function containing an integral Legend 2010-02-12T22:54:20Z 2010-07-23T20:06:18Z <p>I am trying to optimize a function of the following form:</p> <p>$L = \int_{t=0}^{T}(AR-x)dt$, where A is a system parameter</p> <p>i.e. I am trying to find the optimum x(t) that minimizes L over all admissible x(t)s. R is related to x using a relation:</p> <p>$\frac{dR}{dt} = axRY - bR$, where a and b are system parameters and $R(0) = R_{0}$</p> <p>$\frac{dY}{dt} = -xRY$</p> <p>I was looking at this problem from mainly a simulation perspective. There is a whole amount of work that went into showing that x can take only two specific values that minimize the function over any given interval. Now, what I was thinking was to convert the integral into its discrete formulation and do the following:</p> <p>For $t=1$, Let $x = x_{min}$ and calculate $L_{10}$ Let $x = x_{max}$ and calculate $L_{11}$ Finally, choose the one that has the min L.</p> <p>And then continue with t=2 and so on, in the same way. If I visualize this problem, it is nothing but finding a minimum cost path in a binary tree i.e. something of the following form:</p> <p>---------------------- $L_{00}$<br /> -----------------------/--\---------<br /> ------------------$L_{10}$----$L_{11}$----<br /> -----------------/-----\----/----\-------<br /> -------------$L_{20}$-----$L_{21}$-$L_{22}$-$L_{23}$--</p> <p>and so on until the last T. I am not sure if my thought process of simulating this is in the right direction. Can someone give me some suggestions? </p> http://mathoverflow.net/questions/21813/which-statistical-test-is-significant-here Which statistical test is significant here? Legend 2010-04-19T04:16:53Z 2010-04-19T04:16:53Z <p>As part of a performance evaluation of a system, I have the following piece of information:</p> <pre><code>year | total jobs submitted | failed jobs </code></pre> <p>I have the same piece of information after introducing a new update to the system as a result of which the number of failed jobs decreased. The data looks again like the following: </p> <pre><code>year | total jobs submitted | failed jobs </code></pre> <p>Are there any useful metrics that I can use to say something good about the update that I used on the system?</p> http://mathoverflow.net/questions/77085/how-are-epidemic-models-simulated-in-case-of-mobility/77090#77090 Comment by Legend Legend 2011-10-04T18:21:45Z 2011-10-04T18:21:45Z Great! Thank you once again for your time. http://mathoverflow.net/questions/77085/how-are-epidemic-models-simulated-in-case-of-mobility/77090#77090 Comment by Legend Legend 2011-10-04T00:19:11Z 2011-10-04T00:19:11Z @Robert Israel: +1 Thank you for your comments. So from what I understand, I posted another snippet of the code. Could you please tell me if I am on the right page? Basically, instead of the all-or-none scenario, what I am now doing is calculating the probability for each neighbor within a node's vicinity. http://mathoverflow.net/questions/72668/how-to-compute-kl-divergence-when-pmf-contains-0s/72672#72672 Comment by Legend Legend 2011-08-17T20:44:34Z 2011-08-17T20:44:34Z Very deep and insightful answer. Thank you so much. I have recently found out about the Jensen-Shannon divergence metric and am trying to understand if I could utilize that instead. The paper &quot;Divergence Measures Based on the Shannon Entropy&quot; explains that this metric does not face the same issues as the KL-divergence metric. If you have any suggestions on this, it would be a great addition to your already insightful answer. Thank you for your time. http://mathoverflow.net/questions/72668/how-to-compute-kl-divergence-when-pmf-contains-0s Comment by Legend Legend 2011-08-11T10:08:33Z 2011-08-11T10:08:33Z @Ashok: Yes. You are right. It looks like KL-divergence requires that both the PMFs be absolutely continuous with respect to each other. In my case, the second case of infinity is a problem. In my calculation, the histograms contain empty bins which I am not able to understand how to handle. http://mathoverflow.net/questions/44015/how-do-i-approach-optimal-control Comment by Legend Legend 2010-10-29T03:41:16Z 2010-10-29T03:41:16Z @Gilead: That's great to hear. Could you point me towards some good optimization theory material in that case? I've been Googling but am not sure if I'm starting with the right ones... http://mathoverflow.net/questions/44015/how-do-i-approach-optimal-control/44030#44030 Comment by Legend Legend 2010-10-29T00:58:24Z 2010-10-29T00:58:24Z @mhum: Thank you. I will go through that and get back. http://mathoverflow.net/questions/44015/how-do-i-approach-optimal-control/44037#44037 Comment by Legend Legend 2010-10-29T00:58:09Z 2010-10-29T00:58:09Z @Brian Borchers: I've taken Undergrad level Calculus. I've read about Partial differential equations in the process. I have taken a course on Control Systems but that was more towards Laplace transforms and Z-Transforms. I don't remember hearing about optimal control anywhere. And yes, I've studied feedback as part of Control Systems where most of it was related to transfer functions and root locus determination. Its been almost 8 years since I've studied those so I'm feeling a little lost now when I look at bang-bang structure and maximum principles. http://mathoverflow.net/questions/21813/which-statistical-test-is-significant-here Comment by Legend Legend 2010-04-19T15:34:17Z 2010-04-19T15:34:17Z Thank you for your time. Actually I haven't been in touch with statistics for a long time so that explains the reason behind my bad question. Could you kindly elaborate on the tests that you are mentioning? http://mathoverflow.net/questions/21249/need-some-help-with-defining-average-node-connectivity-in-a-random-graph Comment by Legend Legend 2010-04-14T15:58:46Z 2010-04-14T15:58:46Z Actually, I'm still debating the actual use of node connectivity and am not sure if it is the right metric for my case. I want to evaluate the possibility of massing a message between all nodes by constructing a minimum spanning tree. Because one large MST is prone to stability issues in a mobile environment, I am trying to get the average &quot;connectivity&quot; of all the nodes to explore the feasibility of a solution. Please correct me if I am using the wrong set of terminologies here. http://mathoverflow.net/questions/15147/minimizing-a-function-containing-an-integral Comment by Legend Legend 2010-02-12T23:47:52Z 2010-02-12T23:47:52Z Also, I am sorry. I just realized I missed something from the question. Updated it with a second differential equation (but I'm not sure how relevant it is to the overall methodology that I was following). http://mathoverflow.net/questions/15147/minimizing-a-function-containing-an-integral Comment by Legend Legend 2010-02-12T23:43:50Z 2010-02-12T23:43:50Z @Steve: Yes... You are right (but I hate to accept this, because pretty much everything boils down ultimately to physics... Pardon me if I've mistaken). It is more from a problem I am seeing inside the domain of social networks, more specifically, the propagation of a message amongst a set of connected nodes. http://mathoverflow.net/questions/15147/minimizing-a-function-containing-an-integral Comment by Legend Legend 2010-02-12T23:33:10Z 2010-02-12T23:33:10Z @Steve: Currently looking into it.. Thanks. @Leonid: Yes.. All those are positive. http://mathoverflow.net/questions/15147/minimizing-a-function-containing-an-integral Comment by Legend Legend 2010-02-12T23:07:37Z 2010-02-12T23:07:37Z Sorry... A is another parameter. I will clarify this in the question. And no, I meant (AR-x) inside the integral. http://mathoverflow.net/questions/13052/minimizing-an-objective-function Comment by Legend Legend 2010-01-26T17:06:15Z 2010-01-26T17:06:15Z Sorry... Apparently, it looks like I had an error in my latex as well... I modified the entire question...