If I do an ordinary spline on (x,y) data, then switch x&y and do another spline, will I get the same curve? Obviously the coefficients will be different because they would be applied to (former) y values, but perhaps the two actual curves in the plane would be the same? It turns out they won't be the same. But can we put bounds on how different they might be? ------------------------------------------------------- Last night my 4-year-old asked why it's warmer in summer than in winter. I think the standard answer is because *the sunlight is more direct (closer to perpendicular to the earth's surface) during summer, thus reducing what the solar panel industry calls "cosine losses". * the earth is closer to the sun during the summer (I know this is wrong, at least for summer in the northern hemisphere; we're actually closer to the sun during our winter/Australia's summer) * There's more hours of daylight during the summer, so the earth gets more heat input each day * Since the sun is closer to overhead, the sunlight travels through fewer miles of air as it enters the atmosphere, so more of it hits the ground. (though during the winter, when more of it is getting absorbed in the atmosphere, that should be heating the atmosphere a little as well) So, here's the modeling project idea: quantify the extent that each of these effects has on causing the seasons. This project would involve some 3-d trig, I think, and solving the heat or radiative-heat-transfer ODE with time-varying heat input. I'm not even sure we could come up with a realistic physical model that would allow us to vary cosine losses and hours of daylight independently. Here's an interesting online book that students could use to get some basic physical background: http://www.powerfromthesun.net/book.htm and also an online book that talks directly about the distance-from-the-sun part: http://mitpress.mit.edu/books/full_pdfs/Street-Fighting_Mathematics.pdf ----------------------------------------------------- How would you come up with a quantitative policy on when to call a snow day, either at EMU or at the K-12 level? It would have to mix all kinds of competing objectives, recent weather history (amount of snow on the ground currently) and future forecasts, etc. ------------------------------------------- Suppose you were the engineer designing the olympics luge/bobsled/skeleton track. How would you predict top speeds on the course? And, how would you quantify the safety (or lack thereof) of the track, _before_ building it? I guess I'd start with a roller-coaster type model to predict top speeds--the line of travel being fixed, not subject to steering. But to think about safety, you have to model the ability of the sled to leave the line you want, due to random error. This means you'd have to have some model of how the athlete reacts in doing their steering. And as long as you're modeling control systems, you might as well build a robot-controlled luge :-) ! ------------------------------------------- http://www.analyticsx.com/ Current Contest - 2010 - Predicting Homicides in Philadelphia Philadelphia is a city with 5.8 million people spread out over 47 zip codes and, like any major city, it has its share of crime. The goal of the Analytics X Competition is to use statistical techniques and any data sets you can find to predict where crime, specifically homicides, will occur in the city. The ability to accurately predict where crime is likely to occur allows us to deploy our limited city resources more effectively. Interesting related data sources at http://www.ccri.com/blog/2010/1/11/philadelphia-data-resources.html ------------------------------------------- When I was watching the olympics, I wondered: some of the skaters can do a quadruple-turn jump. What if they didn't have to land it gracefully? Physically speaking, is it possible to do a 4.5 or 5-turn jump? ------------------------------------------- NPR has a story on how some house deeds, from all across the country, include racial restrictions (the most common being "only whites can buy or live here"). Indeed, some houses in my neighborhood here in Ypsi have these clauses in their deeds. Of course these are invalid these days, since a 1968 (?) court ruling, but it occurred to me that the effects can be long-lasting, due to the slow turnover of house owners. So, it would be an interesting modeling problem (perhaps a bit too simple, though) to model how long it might take for a neighborhood to become racially integrated in the aftereffects of the 1968 ruling. Here's the NPR story: http://www.npr.org/templates/story/story.php?storyId=122484215 ------------------------------------------- Here's a real problem facing Ypsi city schools, that could probably benefit from some modeling: * Right now, the district has a deficit, and so they're looking to consolidate classes and lay off teachers. * If you make class sizes bigger, it is likely that some parents will withdraw their kids and send them to charter schools, private schools, or other school districts. * The resulting loss of kids will make class sizes not as big as they would have been under consolidation originally, but * The resulting loss of kids will take money away (roughly $7000 per student), thereby causing another deficit and more teacher layoffs. So, is there an equilibrium? Or is it a downward spiral of death? A similar (but not at all identical) problem is faced at airlines, when deciding how big of an airplane they should assign to a certain route--once the currently assigned plane fills up, you don't know how many more people would have wanted to get on that plane, because you can't sell them tickets (beyond a certain overbooking level). This leads you to underestimate the true demand for the route, so maybe you assign a smaller plane next month, and then you're censoring your demand even more... There's a famous related quote: "Nobody goes there anymore, it's too crowded."--Yogi Berra We could also ask the question at a county-wide level, where kids leaving the Ypsi city schools aren't just "gone", but they're counted as being part of some other school system, which has its own teachers, class sizes, deficit, etc.--is there a county-wide equilibrium? Of course it would be really hard to get data on the class-size-versus-demand tradeoff curve, but we could make some simple assumptions about it: * it's decreasing: higher class size means lower demand * its asymptote is not zero: some parents aren't able to send their kids elsewhere, even if class sizes get huge. * its value at a class size near zero is not infinite--it's obviously limited by the # of kids in the county, but more directly, not all parents will be able to send their kids to the school. But maybe the shape of the graph near zero isn't very important :-) * All together, I'm thinking of a sigmoid sort of curve, but starting high rather than low like most sigmoids do. ----------------------------------------------------- the company Intellectual Ventures * Monte Carlo simulation of malaria spread, to decide where to deploy nets, etc. * "photonic fence" to zap mosquitos with lasers (subject of a TED talk) http://www.newsweek.com/2010/04/08/short-circuiting-malaria.html ----------------------------------------------------- I have a set of data on students taking math courses at EMU for the years 2006-2010. I want to figure out which courses in the standard prerequisite chain are "bottlenecks"--ones that cause people to quit taking math courses and change their major to one that doesn't require so much math. -----------------------------------------------------