Anomalies and Breaking Trends
Lecture 14: Simulation—Beyond Data, Beyond Equations
Simulation—Beyond Data, Beyond Equations
Lecture 14
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ometimes, we have too many possibilities to consider, and sometimes, SKHQRPHQD DUH VLPSO\ WRR GLI¿FXOW WR FDSWXUH LQ DQ HTXDWLRQ Simulation is a powerful tool in our world. In many cases, rather than analyzing lots of data, you can produce a simulation and analyze what that says about a physical phenomenon. This tool of data analytics allows us to PDNHEHWWHUPHGLFLQHVDQGIDVWHUFDUVDQGWRH[SORUHQHZUHDOPVRIVFLHQWL¿F study—all with the speed and safety of a computer.Monte Carlo Simulation
x The World Series of Poker in Las Vegas is a tournament that begins ZLWKWKRXVDQGVRIFRPSHWLWRUVDQGQDUURZVGRZQWRWKH¿QDOWZR sitting at a table with hundreds of thousands of viewers tuned in to see the outcome. The winner is considered the World Champion of Poker and receives a multimillion-dollar cash prize. When the game is broadcast on ESPN, a
FDUGLVGHDOWDQGTXLFNO\ the probability of each player winning given the current hand is updated. x How is this done?
Could there be a big database of all possible combinations? In that case, a card is dealt, and someone looks up the probabilities for
that given state of the game. Consider how big such a database would need to be—and consider how fast we want our answer. We possibly could do it that way, but there is another simpler way: use a computer to simulate the game.
Simulation is a great tool that can help you play the game of poker.
x To see how, we turn to another card game and travel to Los Alamos, New Mexico, in the 1940s. Stanislaw Ulam, while working on the Manhattan Project that developed the nuclear bomb during World War II, pondered the probabilities of winning a card game of solitaire. Because the computations of the probabilities are inherently complex, Ulam explored another route. On an early mainframe computer that he programmed to simulate solitaire, he would play the game a large number of times and computed the proportion of times that he won.
x Such an approach became known as Monte Carlo simulation, because the methods often depend on an element of chance, such as what cards will be dealt. Today, an ordinary spreadsheet can generate and insert random numbers for you.
x Such methods can be used to simulate more important real- world phenomena, too. At Los Alamos National Laboratory, Ulam and John von Neumann also used the methods to simulate nuclear reactions. Today, Monte Carlo simulation is used to study applications in such areas as physics, mechanics, and economics. x /HW¶VUHWXUQWRSRNHUDQGVSHFL¿FDOO\WKHJDPH7H[DV+ROG¶HPWR
see how simulation could save us from developing a database of trillions and trillions of probabilities. The rules of the game are as IROORZV7ZRFDUGVDUHGHDOWIDFHGRZQWRHDFKSOD\HU7KHQ¿YH community cards are revealed, face up. Each player takes his or KHUEHVW¿YHFDUGSRNHUKDQGIURPKLVRUKHUWZRGRZQFDUGVDQG WKH¿YHFRPPXQLW\FDUGVDQGWKHSOD\HUZLWKWKHEHVWKDQGZLQV During the process of dealing, there are several rounds of betting, and much of the strategy in Texas Hold’em comes from betting. x Here’s where a simulation can help you play the game. You won’t
know, unlike the TV broadcasters in the World Series of Poker, ZKDW KDQG HYHU\RQH KROGV :H ZDQW WR ¿QG WKH RGGV RI ZLQQLQJ from a given two-card starting hand, assuming that no players
Lecture 14: Simulation—Beyond Data, Beyond Equations
simulation. Like Ulam, we put the current state of a game into the computer. Then, we let the computer play thousands or millions of random games and count the fraction of wins, losses, and draws for each player.
x Monte Carlo simulations must be run many, many times. We need a ORWRIQXPEHUV²DORWRIGDWD²WR¿QGZKDWZH¶UHORRNLQJIRU)URP the law of large numbers in mathematics, as we run more and more tests, we will tend toward the expected value. The issue, which isn’t a huge one for computers, is that we need to run hundreds of thousands of experiments. Then, we begin to see the values that we want and expect to see.
Simulations in Our World
x Simulation can help us understand our world. It can help answer TXHVWLRQVLQSUREDELOLW\WKDWFDQEHGLI¿FXOWWRDQVZHU7KLVLVZKDW Ulam was doing when he simulated solitaire.
x Another example is the Monty Hall problem. It’s based on the game show hosted by Monty Hall, in which you are told that there is a 100-dollar bill behind one of three doors and that there is nothing behind the other two. You choose one of the doors. Then, you are told one of the other doors that does not contain the money. At that point, you may change your guess to the remaining door—the one WKDW\RXGLGQRWFKRRVHWKH¿UVWWLPHDQGWKDW\RXZHUHQRWWROGGLG not contain the 100 dollars.
x ,V LW D EHWWHU VWUDWHJ\ WR VWLFN ZLWK \RXU ¿UVW FKRLFH RU VZLWFK" 7KLV TXHVWLRQ DSSHDUHG LQ WKH ³$VN 0DULO\Q´ FROXPQ RIParade magazine in 1990. It caught wide attention. The problem was stated as having goats and a car behind the doors. In her column, Marilyn vos Savant asserted that switching is the best strategy. She got thousands of letters, and 92 percent of them insisted that she was wrong. She settled the argument with a simulation. She called upon “math classes all across the country” to simulate the probabilities using pennies and paper cups. She was right, and of course, the simulation backed it up.
x What else can simulation do? Have you been in a fast-food drive- WKURXJK DQG QRWLFHG WKDW WKH\ WLPH KRZ ORQJ LW WDNHV WR ¿OO \RXU RUGHU" 6XFK LQIRUPDWLRQ FDQ EH TXLWH LPSRUWDQW DQG KHOSIXO7KLV EUDQFK RI VLPXODWLRQ LV FDOOHG TXHXLQJ WKHRU\ ,I ZH NQRZ WKH rate at which customers arrive and the length of time it takes to ¿OO WKH RUGHU ZH FDQ VLPXODWH TXHXLQJ XS RU OLQLQJ XS XQGHU different scenarios.
x When should you have the cashier take orders only and leave the ¿OOLQJ RI RUGHUV WR VRPHRQH HOVH" 6LPXODWLRQ FDQ KHOS \RX determine the impact of such choices. You can see what happens on average. You can also see the extreme cases, or outliers, and GHWHUPLQHWKHLUIUHTXHQF\DQGLIWKH\DUHDFFHSWDEOH
x Similar concepts allow one to model emergency room intake to reduce waiting WLPHV,QWUDI¿FVWXGLHV\RX can model the difference between a roundabout and an intersection with a stoplight. Simulation is a great tool when you might change parameters in the problem. Sometimes, you need an analytical solution
computed directly from the data, but often, a computed number will do just as well. If so, simulation can save a lot of time—and allow \RXWRTXLFNO\WHVWPDQ\PRUHLGHDV
x While it’s only a model, a simulation can, if carefully constructed, have enough realistic behavior that it will uncover enough characteristic behavior to offer insight. With that, decisions FDQ EH PDGH 6LPXODWLRQ LQ JHQHUDO UHTXLUHV VRPH FRPSXWHU programming—but not too much.
A simulation can help planners decide between a roundabout and a stoplight for a particular intersection.
Lecture 14: Simulation—Beyond Data, Beyond Equations
Simulation in Hollywood
x Simulation can model phenomena in our world. Blockbuster movies often contain stunning special effects—particularly computer-generated images (CGI). Such images often rely heavily on simulation.
x In the 1980 Star Wars ¿OPThe Empire Strikes Back, Yoda was a puppet controlled by Muppeteer Frank Oz, the one behind Fozzie the Bear, Miss Piggy, and Grover. In Episode II, Attack of the Clones, from 2002, Yoda was created using CGI. Frank Oz was still the voice, but he no longer controlled the movement as he did when Yoda was a puppet.
x In order to operate Yoda in a computer as opposed to the hand of a puppeteer, animators create a digital wire frame of the character. Such a model can contains over 50,000 vertices connected by lines. That number of vertices is needed to capture the detail of Yoda. x To move and animate Yoda, animators sometimes simply decide on
VSHFL¿FSODFHVIRU<RGD¶VDUPIRUH[DPSOHWREHLQVSDFHDQGWLPH 7KHQLWLVWKHFRPSXWHUV¶MREWR¿JXUHRXWZKHUHWKDWOLPEZLOOEH in intervening frames.
x Animating Yoda’s hair is even more complicated. Unlike the movement of his body, the movement of his hair may not be VSHFL¿HGH[FHSWLQWKH¿UVWIUDPHRIWKHVFHQH*HQHUDOO\LWLVXS to the computer to determine how his hair would move given the PRYHPHQWRIKLVERG\2IWHQWKHFRPSXWHULVDOVR¿JXULQJRXWWKH movement of his body.
x Simulation is used to determine how his hair will move. A model is built. In particular, animators model hair as springs. You can determine how springy hair is in the model, too. Think of a bed, where some springs are bouncier than others. Then, you let the computer determine this given the force acting on the hair. This allows animators to put digital doubles into scenes. By simulating
the movement of hair, it may not be exact and perfect, but it is close enough that the audience buys into it.
Simulation in Science
x Of course, Hollywood is different from science. In science, a simulation is used to predict or explain behavior. In the movies, a simulation needs only to produce images that give the appearance of reality. But simulations in entertainment and science are becoming FORVHU WRR &*, VLPXODWLRQV IRU VFLHQWL¿F SXUSRVHV FDQ PDNH LW easier to visualize large data sets in motion, further blurring the line EHWZHHQVFLHQWL¿FVLPXODWLRQDQGHQWHUWDLQPHQWTXDOLW\&*, x Simulation not only visualizes imaginary worlds for Hollywood,
EXW LW FDQ DOVR KHOS XV XQGHUVWDQG RXU XQLYHUVH VFLHQWL¿FDOO\7KH Bolshoi simulation is a massive, incredibly detailed model of the universe’s 14-billion-year history. The images it is producing are amazing and being closely studied.
x The simulations create frame after frame of video. They simulate WKHHYROXWLRQRIWKHXQLYHUVH7KH\GRWKLVE\¿UVWH[DPLQLQJWKH data from NASA’s WMAP explorer, which maps out the cosmic microwave background radiation. That radiation is the light that was left over from the big bang. That data can be used as the starting conditions of the universe, and then the supercomputer can simulate how the universe evolved.
x While there are reaches of the universe we have yet to explore, there are many regions that we do know. So, the supercomputer’s results are compared to parts of the universe that we do know. And they match up really, really well.
Gladwell, The Tipping Point.
Neuwirth and Arganbright, The Active Modeler. Suggested Reading
Lecture 14: Simulation—Beyond Data, Beyond Equations
1. You can think about how to simulate many aspects of life. If you can VLPXODWH LW ZKDW TXHVWLRQV PLJKW \RX H[SORUH ZLWK LW" +RZ KHDY\ PXVW WUDI¿F EH IRU D URXQGDERXW WR EH OHVV HIIHFWLYH WKDQ D IRXUZD\ stop? Which board games could be simulated, and could you compare VWUDWHJLHVWKDWSHRSOHSOD\"7KLVLVWKH¿UVWVWHSLQEXLOGLQJDVLPXODWLRQ and knowing why are you creating it.
2. Many video games are a form of simulation. Video games must have D TXLFN UHVSRQVH WR \RXU GHFLVLRQV &DQ \RX GHWHUPLQH ZKDW W\SH RI underlying model the program is using? Sometimes, possibly even often, you may not know. But create your own models and play the game as a data analyst.
3. When watching movies, pay attention to the presence of special effects. What looks real, and where is the special effect less than real? It can be GLI¿FXOWWRFRQFHQWUDWHRQWKLV<RXPLJKWKDYHWRZDWFKWKHPRYLHPRUH than once.