## Five places you didn’t expect to encounter applied math

You don’t need to step into a classroom to have a run-in with mathematics. Professionals from a range of backgrounds — engineering, economics, physics, biology, computer science — use mathematics every day. To celebrate the publication of the much-anticipated Princeton Companion to Applied Mathematics, edited by Nicholas J. Higham, we’re thinking about all of the unique places and situations where applied mathematics is at work. Here is a list of just a few, compiled with a little help from our numerically inclined friends.

#### On the Golf Course

Golf involves mathematics, and not just when keeping score. The flight of your golf ball is affected by how air interacts with the surface of the ball. Did you know that the dimples in golf balls have a purpose, one with a mathematical explanation? Douglas N. Arnold, professor of Mathematics at the University of Minnesota, tells us more:

In the middle of the nineteenth century, when rubber golf balls were introduced, golfers noticed that old scuffed golf balls traveled farther than new smooth balls, although no one could explain this unintuitive behavior. This eventually gave rise to the modern dimpled golf ball. Along the way a great deal was learned about aerodynamics and its mathematical modeling. Hundreds of different dimple patterns have been devised, marketed, and patented. However, even today the optimal dimple pattern lies beyond our reach, and its discovery remains a tough challenge for applied mathematics and computational science.

Check out Dr. Arnold’s entry, “The Flight of a Golf Ball,” where he explains why golf ball dimples are an important part of your Saturday morning tee time.

#### On Wall Street

Wall Street is all about the numbers. Whether modeling the risk of a single stock or mapping the complex interactions that make up the world’s financial structure, mathematics helps the financial sector to study and evaluate systemic risk.

“The complexity, unpredictability, and evolving nature of financial markets continues to provide an enormous challenge to mathematicians, engineers, and economists in identifying, analyzing, and quantifying the issues and risks they pose,” write Dr. René A. Carmona and Dr. Ronnie Sircar of Princeton University.

In their entry, “Financial Mathematics,” Dr. Carmona and Dr. Sircar discuss how the finance industry uses mathematics. They also examine the role of mathematics in understanding and regulating financial markets in light of the financial crisis of 2008.

#### On Your Phone’s Weather App

Do you check the 10-day forecast during the weekend before a big outdoor event, fingers crossed for clear skies and no rain? There’s math behind that “chance of thunderstorms” prediction. NWP [numerical weather prediction] helps meteorologists to predict weather patterns for more than a week ahead. Better numerical schemes are partially responsible for moving us forward from the weather prediction methods of fifty years ago.

In his article “Numerical Weather Prediction,” Peter Lynch presents the mathematical principles of NWP and illustrates the process by considering some specific models and their application to practical forecasting. Dr. Lynch describes the many conditions that can be better predicted using NWP:

NWP models are used to generate special guidance for the marine community. Predicted winds are used to drive wave models, which predict sea and swell heights and periods. Prediction of road ice is performed by specially designed models that use forecasts of temperature, humidity, precipitation, cloudiness, and other parameters to estimate the conditions on the road surface. Trajectories are easily derived from limited-area models. These are vital for modeling pollution drift, for nuclear fallout, smoke from forest fires, and so on. Aviation benefits significantly from NWP guidance, which provides warnings of hazards such as lightning, icing, and clear-air turbulence.

#### In the Airport Security Line

On your next trip through airport security, take a look at the x-ray machine. Once an object, like your suitcase, is scanned, the image can be viewed from multiple angles by a security officer. Threat detection software can also be used to locate problematic items. There is math at work here too.

W. R. B. Lionheart, professor of Applied Mathematics at the University of Manchester, explains this technology in his entry “Airport Baggage Screening with X-Ray Tomography.”

#### While Researching Your Next Vacation

Getting ready for your first vacation of the fall? Buying tickets, making dinner reservations, researching tourist attractions — what did we do without the internet? Or rather, what did we do before the organized internet?

When the internet was still in its early stages, search engines were not as advanced as they are today, and webpage results were ranked by simple rules. Searching for “New York sightseeing” may have led you to the page where the search term appears the most, instead of a page with the most useful information. Today, search engines use a more advanced method for ranking web pages: grouping pages into authority pages, which have many links to them, and hub pages, which point to many authorities. The catch is that these terms depend on one another. How does this work? In the Princeton Companion to Applied Mathematics, editor Nicholas Higham explains the mathematics behind webpage ranking.

Looking for more examples of math in the world? Check out this video from SIAM, where SIAM conference attendees are asked how they use math in their work. Math really is all around us.

## Washington Post highlights summer reading for students

Soon, school will be out for summer, but here at PUP, our “to read” lists keep growing. The Washington Post recently highlighted a unique summer reading list — one compiled by college admissions officers and counselors.

Every year, Brennan Barnard, director of college counseling at The Derryfield School in Manchester, New Hampshire, asks college admissions deans and high school counselors for book recommendations. These selections include books for students, parents, and general book lovers. This year, Frank Cioffi’s One Day in the Life of the English Language makes the list.

Barndard explains the inspiration behind this take on summer reading recommendations:

At The Derryfield School, summer reading has an interesting twist that would have been much more palatable for me as a high school student. Every faculty member chooses a favorite book and students can pick a title from this diverse list. Some students choose books based on their most adored teacher and some based on the brief summary provided. Then there are likely students (like I would have done) who choose the shortest book on the list regardless of topic. During the first week of school, faculty members gather with students who read their recommendation for an engaging discussion.

Inspired by this practice, I solicited summer reading recommendations from colleagues in college counseling and admission from high schools and colleges across the nation.

You can view the entire summer reading list here, courtesy of the Washington Post.

One Day in the Life of the English Language was recommended for students by Jeffrey Durso-Finley, director of college counseling at the Lawrenceville School (NJ). Read more about this anti-handbook below, and check out the introduction for yourself.

Generations of student writers have been subjected to usage handbooks that proclaim, “This is the correct form. Learn it”—books that lay out a grammar, but don’t inspire students to use it. By contrast, this antihandbook handbook, presenting some three hundred sentences drawn from the printed works of a single, typical day in the life of the language—December 29, 2008—tries to persuade readers that good grammar and usage matter.

Using real-world sentences rather than invented ones, One Day in the Life of the English Language gives students the motivation to apply grammatical principles correctly and efficiently. Frank Cioffi argues that proper form undergirds effective communication and ultimately even makes society work more smoothly, while nonstandard English often marginalizes or stigmatizes a writer. He emphasizes the evolving nature of English usage and debunks some cherished but flawed grammar precepts. Is it acceptable to end a sentence with a preposition? It is. Can you start a sentence with a conjunction? You can. OK to split an infinitive? No problem.

## Princeton attends Book Expo America 2015

Photo courtesy of Book Expo America.

This week, we packed our books and headed to NYC to take part in Book Expo America. This year, BEA was held in the Javits Center in Manhattan. The event included three days of book signings, author events, and checking out what’s new in the publishing industry. Publishers, booksellers, librarians, retailers, book industry professionals (and fans!) mingled and talked about books–what could be better than a building filled with book lovers?

If you missed BEA this year, don’t worry! You can visit our “digital booth” and page through a few books that we had on display:

The Original Folk and Fairy Tales of the Brothers Grimm: The Complete First Edition
Jacob & Wilhelm Grimm, Translated and edited by Jack Zipes
Illustrated by Andrea Dezsö

These fairy tales may not be what you remember from bedtime. This is a translation of the original Grimm fairy tales. The tales, from the 1812 and 1815 editions, are unique—they reflect diverse voices, rooted in oral traditions, that are absent from the Grimms’ later, more embellished collections of tales.

This book has had a big year. It was one of the Independent’s Best Books of 2014 and one of South China Morning Post’s Best Books of 2014. Page through Chapter One here.

Amazons—fierce warrior women dwelling on the fringes of the known world—were the mythic archenemies of the ancient Greeks.  But just who were these bold barbarian archers on horseback who gloried in fighting, hunting, and sexual freedom? Were Amazons real? In this deeply researched, wide-ranging, and lavishly illustrated book, National Book Award finalist Adrienne Mayor presents the Amazons as they have never been seen before.

View Chapter One for yourself here.

Life’s Engines: How Microbes Made Earth Habitable
Paul G. Falkowski

Paul Falkowski looks “under the hood” of microbes to find the engines of life, the actual working parts that do the biochemical heavy lifting for every living organism on Earth. With insight and humor, he explains how these miniature engines are built—and how they have been appropriated by and assembled like Lego sets within every creature that walks, swims, or flies.

Flip through Chapter One here.

Be sure to peruse our full digital catalogs. Our Fall 2015 list was recently announced!

## 25th anniversary of the launching of the Hubble Space Telescope

Twenty-five years ago today, the shuttle mission STS-31 saw the space shuttle Discovery launch the Hubble Space telescope successfully into orbit. Since then, it has produced the most stunning images of the cosmos humanity has ever seen. (The beautiful image below is of the grand-design spiral galaxy Messier 74!) The Hubble has transformed our understanding of the universe around us, revealing new information about its age and evolution, the life cycle of stars, and the very existence of black holes, among other startling discoveries.

The grand-design spiral galaxy Messier 74 as photographed by the Hubble Space Telescope.

However, behind the beautiful images taken by the telescope, there is the complex story of how the plans for the telescope came to fruition. But it took an amazing amount of work and perseverance to get the first space telescope up and running.

PUP author Robert Zimmerman’s The Universe in a Mirror tells the story of this telescope and the visionaries responsible for its extraordinary accomplishments. He takes readers behind the scenes of one of the most ambitious scientific instruments ever sent into space. After World War II, astronomer Lyman Spitzer and a handful of scientists waged a fifty-year struggle to build the first space telescope capable of seeing beyond Earth’s atmospheric veil. Zimmerman shows how many of the telescope’s advocates sacrificed careers and family to get it launched, and how others devoted their lives to Hubble only to have their hopes and reputations shattered when its mirror was found to be flawed. This is the story of an idea that would not die–and of the dauntless human spirit. Illustrated with striking color images, The Universe in a Mirror describes the heated battles between scientists and bureaucrats, the perseverance of astronauts to repair and maintain the telescope, and much more. Hubble, and the men and women behind it, opened a rare window onto the universe, dazzling humanity with sights never before seen.

“Hubble 01” by NASA

## Math Drives Careers: Author Oscar Fernandez

We know that mathematics can solve problems in the classroom, but what can it do for your business? Oscar Fernandez, author of Everyday Calculus, takes a look at how knowledge of numbers can help your bottom line.

Why You Should Be Learning Math Even If You Don’t Need It for Your Job

I want to tell you a short story about epic triumph in the midst of adversity. Okay, I’m exaggerating a bit, but hear me out.

A couple of years ago, I approached Boston Scientific—an S&P 500 component—with a crazy idea: let me and a team of students from Wellesley College (a liberal arts college for women) and Babson College (a business school) do consulting work for you. It was a crazy idea because what could I—a mathematician who knew nothing about their business—and some students—who hadn’t even graduated yet—possibly offer the company? Plenty, it turns out, all thanks to our common expertise: mathematics.

Mathematics, often depicted in movies as something pocket-protector-carrying people with less than stellar social skills do, is actually quite ubiquitous. I’d even say that mathematicians are the unsung heroes of the world. Alright, that’s a bit of hyperbole. But think about it. Deep in the catacombs of just about every company, there are mathematicians. They work in low light conditions, hunched over pages of calculations stained with days-old coffee, and think up ways to save the company money, optimize their revenue streams, and make their products more desired. You may never notice their efforts, but you’ll surely notice their effects. That recent change in the cost of your flight? Yep, it was one of us trying to maximize revenue. The reason that UPS truck is now waking you up at 6 a.m.? One of us figured out that the minimum cost route passes through your street.

But we’re do-good people too. We help optimize bus routes to get children to school faster and safer. We’ve spent centuries modelling the spread of disease. More recently, we’ve even reduced crime by understanding how it spreads. That’s why I was confident that my team and I could do something useful for Boston Scientific. Simply put, we knew math.

We spent several weeks pouring over data the company gave us. We tried everything we could think of to raise their revenues from certain products. Collectively, we were trained in mathematics, economics, computer science, and psychology. But nothing worked. It seemed that we—and math—had failed.

Then, with about three weeks left, I chanced upon an article from the MIT Technology Review titled “Turning Math Into Cash.” It describes how IBM’s 200 mathematicians reconfigured their 40,000 salespeople over a period of two years and generated \$1 billion in additional revenue. Wow. The mathematicians analyzed the company’s price-sales data using “high-quantile modeling” to predict the maximum amount each customer was willing to spend, and then compared that to the actual revenue generated by the sales teams. IBM then let these mathematicians shuffle around salespeople to help smaller teams reach the theoretical maximum budget of each customer. Genius, really.

I had never heard of quantile regression before, and neither had my students, but one thing math does well is to train you to make sense of things. So we did some digging. We ran across a common example of quantile modelling: food expenditure vs. household income. There’s clearly a relationship, and in 1857 researchers quantified the relationship for Belgian households. They produced this graph:

That red line is the linear regression line—the “best fit to the data.” It’s useful because the slope of the line predicts a 50 cent increase in food expenditure for a \$1 increase in household income. But what if you want information about the food expenditure of the top 5% of households, or the bottom 20%? Linear regression can’t give you that information, but quantile regression can. Here’s what you get with quantile regression:

The red line is the linear regression line, but now we also have various quantile regression lines. To understand what they mean let’s focus on the top-most dashed line, which is the 95th percentile line. Households above this line are in the 95th percentile (or 0.95 quantile) of food expenditure. Similarly, households below the bottom-most line are in the 5th percentile (or 0.05 quantile) of food expenditure. Now, if we graph the slopes of the lines as a function of the percentile (also called “quantile”), we get:

(The red line is the slope of the linear regression line; it doesn’t depend on the quantile, which is why it’s a straight line.) Notice that the 0.95 quantile (95th percentile) slope is about 0.7, whereas the 0.05 quantile (5th percentile) slope is about 0.35. This means that for every \$1 increase in household income, this analysis predicts that households in the 95th percentile of food expenditure will spend 70 cents more, whereas households in the 5th percentile will spend only 35 cents more.

Clearly quantile regression is powerful stuff. So, my team and I went back and used quantile regression on the Boston Scientific data. We came up with theoretical maximum prices that customers could pay based on the region the product was sold in. As with IBM, we identified lots of potential areas for improvement. When my students presented their findings to Boston Scientific, the company took the work seriously and was very impressed with what a few students and one professor could do. I can’t say we generated \$1 billion in new revenue for Boston Scientific, but what I can say is that we were able to make serious, credible recommendations, all because we understood mathematics. (And we were just a team of 5 working over a period of 12 weeks!)

April is Mathematics Awareness Month, and this year’s theme is “math drives careers.” After my Boston Scientific experience and after reading about IBM’s success, I now have a greater appreciation of this theme. Not only can mathematics be found in just about any career, but if you happen to be the one to find it (and use it), you could quickly be on the fast track to success. So in between celebrating March Madness, Easter, Earth Day, and April 15th (I guess you’d only celebrate if you’re due a tax refund), make some time for math. It just might change your career.

Photo by Richard Howard.

Oscar Fernandez is the author of Everyday Calculus. He is assistant professor of mathematics at Wellesley College.

## March Mathness 2015: The Wrap Up

The champion has been crowned! After an eventful and surprising March Madness tournament, Duke has been named the new NCAA national champion.

A year of bragging rights goes to PUP paperbacks manager Larissa Skurka (98.6 percent) and PUP executive math and computer science editor Vickie Kearn (98.4 percent), who took first and second place in our ESPN bracket pool. Congrats to both! Check out all of the results here.

As we wrap up March Mathness, here are two final guest posts from basketball fans who used math and Tim Chartier‘s methods to create their brackets.

##### Swearing by Bracketology

By Jeff Smith

My name is Jeff Smith, and I’ve been using Tim Chartier’s math algorithms to help with my March Madness brackets for several years now. I met Tim when we were traveling the ‘circuit’ together in creative ministries training. You may only know Tim for his math prowess, but I knew him for his creativity before I knew he was a brilliant mathematician. He and his wife, Tanya, are professional mimes, and his creativity is genius too.

Several years ago, he mentioned his method for picking brackets at a conference where we were doing some training together. He promised to send me the home page for his site and I could fill out my brackets using his parameters and formula. I was excited to give it a shot. Mainly, because I am part of a men’s group at our church that participates in March Madness brackets every year. Bragging rights are a big deal…for the whole year. You get the picture.

Also, I have two boys who did get one of my genes: the competitive edge. I sat down and explained the process. Because they did not know Tim, they were a little more skeptical, but I promised it wouldn’t hurt to try. That year, in a pool of 40+ guys, we all finished in the top ten. We were all hooked!

Since then, I have contacted Tim each year and reminded him to send me the link to his site where I could put in our numbers to fill out our brackets. Generally, the three of us each incorporate different parameters because we have different philosophies about the process. It has become a family event, where we sit around the dinner table; almost ceremonially, and we take our output and place them in the brackets. The submission is generally preceded by trash talking, prayer, and fasting. (Well, probably not the fasting, because we fill up with nachos and chips during the process.)

Men of March Mathness: Jeff, Samuel, Ben Smith.

This year, I was in South Africa on a mission trip during the annual ritual. Thank God for video chatting and internet access. Halfway across the world, we were still able to be together and place our brackets into the pool. It was such a wonderful experience. While my boys veered from the path, picking intuitively instead of statistically, I didn’t stray far. (I was strong!) If it wouldn’t have been for Villanova, whom I will never choose again in a bracket, I would be leading the pack. But, I’m still in the top ten of the men’s bracket at my church, with an outside shot of winning. In the Princeton bracket, I’m doing even better because I stayed away from the guessing game a little more.

I do not follow college basketball during the season. I’m from central Pennsylvania, and Penn State doesn’t have a good basketball team. So, I have no passion for the basketball season. Periodically, I’ll watch a game because my boys are watching, but generally, basketball season is the long wait until baseball season. (Go Pirates!) So, March Mathness has saved my reputation. It makes me look like a genius. Other guys in the group are looking at my bracket for answers. My boys and I are sworn to secrecy about the formula. The only reason I write this is because I’m sure none of them read this blog! But I’m thankful for Tim and the formula and the chance to look good in front of friends. I have never won the pool, however, if you factor my finishes over the course of the years I have been using Tim’s formula, I have the best average of all the guys.

##### What Do Coaches Have to Do with It?

By Stephen Gorman, College of Charleston student

It’s that time of year again. The time of year when everyone compares brackets to see who did the best. But if your bracket was busted early, don’t worry — you’re not the only one. In fact, nobody came out of the tournament with a perfect bracket.

The unpredictability of these games is an inescapable fact of March Madness. This tournament is so incredibly unpredictable that some people are willing to give out billions to anyone who can create a perfect bracket; Warren Buffett is one of these people. So is he crazy? Or does he realize your odds of creating a perfect bracket are 52 billion times worse than winning the Powerball. In layman’s terms – if you think playing the lottery is crazy, trying to create the perfect bracket is insane.

However, once you can accept the statistics, predicting March Madness becomes a game of bettering you’re odds – and there are many predictive models that can help you out along the way. Some of these models include rating methods, like the Massey method, which takes into account score differentials and strength of schedule. In addition to this, there are weighting methods that can be applied to rating methods; these take into account the significance of particular games and even individual player statistics. However, I noticed there is one thing missing from these predictive models: a method that quantifies the value of a good coach. In order to take into account the importance of a coach, a fellow researcher (John Sussingham) and I decided to create our own rating system for coaches.

Using data available from SportsReference.com, we made a system of rating that incorporated such factors as the coach’s career win percentage, March Madness appearances, and the record of success in March Madness. But before we implemented it, we wanted to justify that it was, indeed, a good way to quantify the strength of a coach. In order to do this, we tested the coach ratings in two ways. The first way being a comparison between how sports writers ranked the top 10 College Basketball coaches of all time and what our coach ratings said were the best coaches of all time. The second way was to test how the coach ratings did by themselves at predicting March Madness.

The comparison of the rankings are shown in the table below:

 Rank Our Results CBS Sports Results Bleacher Report Results 1 John Wooden John Wooden John Wooden 2 Mike Krzyzewski Mike Krzyzewski Bobby Knight 3 Adolph Rupp Bob Knight Mike Krzyzewski 4 Jim Boeheim Dean Smith Adolph Rupp 5 Dean Smith Adolph Rupp Dean Smith 6 Roy Williams Henry Iba Jim Calhoun 7 Jerry Tarkanian Phog Allen Jim Boeheim 8 Al McGuire Jim Calhoun Lute Olson 9 Bill Self John Thompson Eddie Sutton 10 Jamie Dixon Jim Boeheim Jim Phelan

It is clear from the table above that there are striking similarities between all three rankings. This concluded our first test.

For the second test, we decided to use the coach ratings to predict the last fourteen years of March Madness. The results showed that over the last fourteen years, on average, coach ratings had 68.4 precent prediction accuracy and an ESPN bracket score of 946. As a comparison, the uniform (un-weighted) Massey method of rating (over the same timespan) had an average prediction accuracy of 65.2 precent and an average ESPN bracket score of 1006. Having a higher prediction accuracy, but lower ESPN bracket score essentially means that you have predicted more games correctly in the beginning of the tournament, but struggle in the later rounds. This comes to show that not only are these ratings good at predicting March Madness, but they stand their ground when compared to the effectiveness of very popular methods of rating.

To conclude this article, we decided that, this year, we would combine both the Massey ratings and our Coach ratings to make a bracket for March Madness. Over the last fourteen years, the combination-rating had an average prediction accuracy of 66.33 percent and an average ESPN bracket score of 1024. It’s interesting to note that while the prediction accuracy went down from just using the Coach ratings, the ESPN bracket score went up significantly. Even more interestingly, both the prediction accuracy and the ESPN Bracket score were better than uniform Massey.

This year, the combination-ratings had three out of the four Final Four teams correctly predicted with Kentucky beating Duke in the Championship. However, the undefeated Kentucky lost to Wisconsin in the Final Four. Despite this, the combination-ratings bracket still did well, finishing in the 87.6th percentile on ESPN.

## Mathematics Awareness Month 2015: Math Drives Careers

Internet search, pharmaceuticals, insurance, finance, national security, medicine, ecology. What is the link between these diverse career fields? Students graduating with a mathematical sciences degree can find a professional future in all of these fields, and a wide range of others as well. This year’s Mathematics Awareness Month takes a step out of the classroom to show just where mathematics can lead after graduation.

Mathematics Awareness Month is an annual celebration dedicated to increasing public understanding of and appreciation for mathematics. The event, which started in 1986 as Mathematics Awareness Week, adopts a different theme each year. This year’s theme is “Math Drives Careers,” and PUP is excited to bring you a series of guest posts from our authors. Check back all this month for posts about using math to raise revenues, to understand sports and economics, and to solve complex problems.

The organizers of Mathematics Awareness Month explain the importance of mathematics in today’s workforce:

“Innovation is an increasingly important factor in the growth of world economies. It is especially important in key economic sectors like manufacturing, materials, energy, biotechnology, healthcare, networks, and professional and business services. The advances in and applications of the mathematical sciences have become drivers of innovation as new systems and methodologies have become more complex. As mathematics drives innovation, it also drives careers.”
Check out this official Mathematics Awareness Month poster, which includes career descriptions for 10 individuals who used their love for math to find rewarding careers:

Follow along with @MathAware and take a look at Math Awareness Month on Facebook.

## Davidson student hangs onto 97 percent March Madness ranking

Are you still mourning the loss of your perfect bracket after the multiple upsets this March Madness season? Even before the Villanova and NC State match up on Saturday, 99.3 percent of brackets were busted. As experts deem a perfect March Madness bracket impossible, having a nearly perfect bracket is something to brag about. Today, we hear from David College student Nathan Argueta, who argues that knowing a thing or two about math can help with March Madness strategy.

March Mathness: Calculating the Best Bracket

First and foremost… I am far from a Math Major and, prior to this class, the notion that math and sports going hand in hand seemed much more theoretical than based in reality. Now, 48 games later and a 97.2% ranking percentage on ESPN’s Bracket Contest has me thinking otherwise.

In Finite Math, we have explored the realms of creating rankings for teams based on multiple factors (win percentage, quality wins, etc.). Personally, I also take into account teams’ prior experience in the NCAA Tournament. Coaches with experience in the Sweet 16, Final Four, and Championship Game (like Rick Pitino out of Louisville) also factored into my decisions when deciding close games. Rick Pitino has made the Sweet Sixteen for each of the past four years. With a roster whose minutes are primarily distributed amongst second and third year players (players who have had success in the NCAA tournament in the past couple of years) I found it difficult to picture Louisville losing to either UCI, UNI, or even the upcoming battle against upstart NC State (who have successfully busted the majority of brackets in our class’s circuit by topping off Villanova).

In theory, the quest to picking the best bracket on ESPN begins and ends with establishing rankings for each team in the contest. Sure there are four of each seeding (1’s, 2’s, etc.), yet these rankings are very discombobulating when attempting to decide which team will win between a 5th seed and a 12th seed or a 4th seed and a 13th seed. One particular matchup that I found extremely interesting was the one between 13th seeded Harvard and 4th seeded UNC. Gut reaction call—pick UNC. UNC boasts a higher ranking and has ritual success in the postseason. But hold on—Harvard had a terrific record this year (much better than UNC’s, albeit in an easier conference). The difficult thing about comparing Harvard and UNC, however, became this establishment of difficulty of schedule. I nearly chose Harvard, were it not for the fact that Harvard got beaten by about 40 points against UVA while UNC put up more of a fight and only lost by 10 points.

In order to pick the perfect bracket (which mind you, will never happen), categorizing and ranking teams based on their wins against common opponents with prior sports knowledge is imperative. My school pride got the better of me when I chose Davidson to advance out of the Round of 64 against Iowa simply because I disregarded factors like momentum, size, and location. Looking back, it is no wonder that Davidson lost by over 30 points in what many pundits were looking to be a potential upset match. While mathematically our team’s chances could have more than competed against Iowa, in reality our season was spiraling downwards out of control since the second round of the Atlantic 10 Tournament in which we hardly beat out a surprising La Salle team and got annihilated by an injury plagued VCU team that we shut-out just nine days before. Moral of the story… brackets will be brackets and while math can certainly guide you towards a higher ranking in your class pool, you can kiss perfection good-bye. This is March Madness.

## Using math for March Madness bracket picks

The countdown to fill out your March Madness brackets is on! Who are you picking to win it all?

Today, we hear from Liana Valentino, a student at the College of Charleston who works with PUP authors Amy Langville and Tim Chartier. Liana discusses how math can be applied to bracket selection.

What are the chances your team makes it to the next round?

The madness has begun! Since the top 64 teams have been released, brackets are being made all over the country. As an avid college basketball fan my entire life, this is always my favorite time of the year. This year, I have taken a new approach to filling out brackets that consist of more than my basketball knowledge, I am using math as well.

To learn more about how the math is used to make predictions, information is available on Dr. Tim Chartier’s March Mathness website, where you can create your own bracket using math as well!

My bracket choices are decided using the Colley and Massey ranking methods; Colley only uses wins and losses, while Massey integrates the scores of the games. Within these methods, there are several different weighting options that will change the ratings produced. My strategy is to generate multiple sets of rankings, then determine the probability that each particular team will make it to a specific round. Using this approach, I am able to combine the results of multiple methods instead of having to decide on one to use for the entire bracket.

Choosing what weighting options to use is a personal decision. I will list the ones I’ve used and the reasoning behind them using my basketball awareness.

(1)

Winning games on the road should be rewarded more than winning games at home. Because of that, I use constant rates of .6 for a winning at home, 1.6 for winning away, and 1 for winning at a neutral location; these are the numbers used by the NCAA when determining RPI. I incorporate home and away weightings when performing other weighting methods as well.

(2)

Margin of victory is another factor, but a “blow out” game is defined differently depending on the person. With that in mind, I ran methods using the margin of victory to be both 15 and 20. This means if the margin of victory if 15, then games with a point differential of 15 or higher are weighed the same. These numbers are mainly from personal experience. If a team wins by 20, I would consider that a blowout, meaning the matchup was simply unfair. If a team loses by 15, which in terms of the game is five possessions, the game wasn’t necessarily a blow out, but the winning team is clearly defined as better than the opposition.

In addition to this, I chose to weight games differently if they were close. I defined a close game as a game within one possession, therefore three points. My reasoning behind this was if a team is blowing out every opponent, it means those games are obviously against mismatched opponents, so that does not say very much about them. On the other hand, a team that constantly wins close games shows character. Also, when it comes tournament time, there aren’t going to be many blow out games, therefore teams that can handle close game situations well will excel compared to those who fold under pressure. Because of this, I weighted close games, within three points, 1.5, “blow out” games, greater than 20 points, .5, and any point differential in between as 1.

(3)

Games played at different points in the season are also weighted differently. Would you say a team is the same in the first game as the last? There are three different methods to weight time, as provided by Dr. Chartier using his March Mathness site, linearly, logarithmically, and using intervals. Linear and logarithmic weights are similar in the fact that both increase the weight of the game as the season progresses. These methods can be used if you believe that games towards the end of the season are more important than games at the beginning.

Interval weighting consists of breaking the season into equal sized intervals and choosing specific weightings for each. In one instance, I weighted the games by splitting the season in half, down weighting the first half using .5, and up weighting the second half using 1.5 and 2. These decisions were made because during the first half of the season, teams are still getting to know themselves, while during the second half of the season, there are fewer excuses the make. Also, the second half of the season is when conference games are played, which are generally considered more important than non-conference games. For the people that argue that non conference play is more important because it is usually more difficult than in conference play, I also created one bracket where I up weight the first half of the season and down weight the second half.

(4)

The last different weighting method used was incorporating if a team was on a winning streak. In this case, we would weight a game higher if one team breaks their opponents winning streak. Personally, I defined a winning streak as having won four or more games in a row.

I used several combinations of these various methods and created 36 different brackets that I have used to obtain the following information. Surprisingly, Kentucky only wins the tournament 75% of the time; Arizona wins about 20%, and the remaining 5% is split between Wisconsin and Villanova. Interestingly enough, the only round Kentucky ever loses in is the Final Four, so each time they do make it to the championship, they win. Duke is the only number 1 seed never predicted to win a championship.

Villanova makes it to the championship game 70% of the time, where the only team that prevents them from doing so is Duke, who makes it 25% of the time. The remaining teams for that side of the bracket that make it are Stephen F. Austin and Virginia, both with a 2.5% chance. Kentucky makes it to the championship game 75% of the time, while Arizona makes it 22%, and Wisconsin makes it 3%. However, if Arizona makes it the championship game, they win it 88% of the time. Furthermore, Wisconsin is predicted to play in the championship game once, which they win.

The two teams Kentucky loses to in the Final Four are Arizona, and Wisconsin. During the final four, Kentucky has Arizona as an opponent 39% of the time, where Arizona wins 50% of those matchups. Kentucky’s only other opponent in the final four is Wisconsin, where Wisconsin wins that game only 5% of the time. On the other side, Villanova makes it to the final four 97% of the time, where the one instance they did not was a loss to Virginia. Villanova’s opponent in the Final Four is made up of Duke 72%, Gonzaga 19%, Stephen F. Austin 6%, Utah at 3%. The only seeds that appear in the Final Four are 1, 2, and one 12 seed, Stephen F. Austin one time.

During the Elite 8, Duke is the only number 1 seed that does not make it 100% of the time, with Utah upsetting them in 17% of their matchups. The other Elite 8 member is Gonzaga 97% of the time. Kentucky’s opponent in this round is Notre Dame 47% and Kansas 53% of the time.

In the Sweet 16, there are eight teams that make it every time: Kentucky, Wisconsin, Villanova, Duke, Arizona, Virginia, Gonzaga, and Notre Dame. Kansas is the only number 2 seed not on the list as Wichita State is predicted to beat them in 8% of their matchups. Kentucky’s opponent in the Sweet 16 is Maryland 39%, West Virginia 36%, Valparaiso 14%, and Buffalo 11%. Valparaiso is the only 13 seed predicted to make it to the Sweet 16. Villanova’s opponent is either Northern Iowa 61% or Louisville 38%. Duke appears to be facing either Utah 67%, Stephen F. Austin 19%, or Georgetown 14%.

Now, for the teams that make it into the third round. I’m not sure how many people consider a 9 seed beating an 8 seed an upset, but the number 9 seeds that are expected to progress are Purdue, Oklahoma State, and St. John’s. In regards to the 10 seed, Davidson is the most likely to continue with a 47% chance to move past Iowa, which is the highest percentage for an upset not including the 8-9 seed matchups. Following them is 11 seed Texas, who have a 42% of defeating Butler. For the 12 seeds, Buffalo is the most likely to continue with a 36% chance of beating Virginia. The 13 seed with the best chance of progressing is Valparaiso with 19% over Maryland. Lastly, the only 14 seeds that move on are Georgia State and Albany, which only happens a mere 8% of the time.

In general, Arizona seems to win the championship when using Massey and linear or interval weighting without home and away. This could be because most of their losses happen during the beginning of the season, while they win important games towards the end. Using the Colley method is when most of the upsets are predicted. For example, Stephen F. Austin making it to the championship game happens using the Colley logarithmic weighting. Davidson beating Iowa in the second round is also found many times using different Colley methods.

Overall, there are various methods that include various factors, but there are still qualitative variables that we don’t include. On the other hand, math can do a lot more than people expect. Considering Kentucky is undefeated, I presumed the math would never show them losing, but there is a lot more in the numbers than you think. Combining the various methods on 36 different brackets, I computed the probabilities of teams making it to specific rounds and decided to make a bracket using the combined data. This makes it so I don’t have to decide on solely one weighting that determines my bracket; instead, I use the results from several methods. Unfortunately, there is always one factor we cannot consider, luck! That is why we can only make estimates and never be certain. From my results, I would predict to see a Final Four of Kentucky, Arizona, Villanova, Duke; a championship game of Kentucky, Villanova; and the 2015 national champion being Kentucky.

## Cinderella stories? A College of Charleston student examines March Madness upsets through math

Drew Passarello, a student at the College of Charleston, takes a closer look at how math relates to upsets and predictability in March Madness.

The Madness is coming. In a way, it is here! With the first round of the March Madness tournament announced, the craziness of filling out the tournament brackets is upon us! Can math help us get a better handle on where we might see upsets in March Madness? In this post, I will detail how math helps us get a handle on what level of madness we expect in the tournament. Said another way, how many upsets do we expect? Will there be a lot? We call that a bad year as that leads to brackets having lower accuracy in their predictions. By the end of the article, you will see how math can earmark teams that might be on the cusp of upsets in the games that will capture national attention.

Where am I learning this math? I am taking a sports analytics class at the College of Charleston under the supervision of Dr. Tim Chartier and Dr. Amy Langville. Part of our work has been researching new results and insights in bracketology. My research uses the Massey and Colley ranking methods. Part of my research deals with the following question: What are good years and bad years in terms of March Madness? In other words, before the tournament begins, what can we infer about how predictable the tournament will be?

One way of answering this question is to see how accurate one is at predicting the winners of the tournaments coupled with how high one’s ESPN score is. However, I also wanted to account for the variability of the level of competition going into the tournament, which is why I also looked at the standard deviation of the ratings of those in March Madness. A higher standard deviation implies the more spread out the playing level is. Ultimately, a good year will have a high tournament accuracy, high ESPN score, and a high standard deviation of ratings for those competing in March Madness. Similarly, a bad year will have low tournament accuracy, low ESPN score, and a low standard deviation of the ratings. This assessment will be relative to the ranking method itself and only defines good years and bad years solely in terms of past March Madness data.

In order to get a feel for how accurate the ranking methods will be for this year, I created a regression line based on years 2001 – 2014 that had tournament accuracy as the dependent variable and standard deviation of the ratings of those in March Madness as the independent variable. Massey is predicted to have 65.81% accuracy for predicting winners this year whereas Colley is predicted to have 64.19%accuracy. The standard deviation of the ratings for those expected to be in the tournament was 8.0451 for Massey and 0.1528 for Colley, and these mostly resemble the standard deviation of the ratings of the March Madness teams in 2002 and 2007.

After this assessment, I wanted to figure out what defines an upset relative to the ratings. To answer this, I looked at season data and focused on uniform Massey. Specifically for this year, I used the first half of the season ratings to predict the first week of the second half of the season and then updated the ratings. After this, I would use these to predict the next week and update the ratings again and so on until now. For games incorrectly predicted, the median in the difference of ratings was 2.2727, and the mean was 3.0284. I defined an upset for this year to be those games in which the absolute difference in the ratings is greater than or equal to three. This definition of an upset is relative to this particular year. I then kept track of the upsets for those teams expected to be in the tournament. I looked at the number of upsets each team had and the number of times each team gets upset, along with the score differential and rating differences for these games. From comparing these trends, I determined the following teams to be upset teams to look for in the tournament: Indiana, NC State, Notre Dame, and Georgetown. These teams had a higher ratio of upsets over getting upset when compared to the other teams. Also, these teams had games in which the score differences and rating differences were larger than those from the other teams in March Madness.

I am still working on ways to weight these upset games from the second half of the season, and one of the approaches relies on the score differential of the game. Essentially, teams who upset teams by a lot of points should benefit more in the ratings. Similarly, teams who get upset by a lot of points should be penalized more in the ratings. For a fun and easy bracket, I am going to weight upset games heavily on the week before conference tournament play and a week into conference tournament play. These two weeks gave the best correlation coefficient in terms of accuracy from these weeks and the accuracy from March Madness for both uniform Massey and Colley. Let the madness begin!

## May the odds be in your favor — March Mathness begins

Let the games begin! After the excitement of Selection Sunday, brackets are ready for “the picking.” Have you started making your picks?

Check out the full schedule of teams selected yesterday, and join the fun by submitting a bracket to the official Princeton University Press March Madness tournament pool.

Before you do, we recommend that you brush up on your bracketology by checking out PUP author Tim Chartier’s strategy:

For more on the math behind the madness, head over to Dr. Chartier’s March Mathness video page. Learn three popular sport ranking methods and how to create March Madness brackets with them. Let math make the picks!

Be sure to follow along with our March Mathness coverage on our blog, and comment below with your favorite strategy for making March Madness picks.

## The math behind March Madness

It’s almost that time again. The beginning of the March Madness basketball tournament is a few days away, and here at PUP, we cannot wait!

We’re marking our calendars (find the schedule here) and going over our bracketology, with a little help from PUP author Tim Chartier.

To kick off the countdown, we bring you an article from the Post and Courier, who checked in with Dr. Chartier about how numbers can be the best strategy in bracketology.

College basketball fans seeking to cash in on March Madness need to turn on their calculators and turn off their allegiances.

That was the message Dr. Tim Chartier, a math professor at Davidson and published author, brought to cadets at The Citadel on Monday night.

“The biggest mistake people make in bracketology is they go with their heart no matter what the data says,” said Chartier, who has made studying the mathematics of the NCAA basketball tournament part of his students’ course work at Davidson. “They just can’t let a certain team win or they just have to see their team do well.

“It’s hard not to do that, because that is part of the fun.”

Chartier has made it easier for the average fan to use math in filling out their own brackets at the March Mathness website marchmathness.davidson.edu. The site will get a lot of traffic after the NCAA tournament field is announced on March 15.

Dr. Tim Chartier is a numbers guy, and not only during basketball season. He likes to show students how math can apply outside of the classroom. How can reposting on Twitter kill a movie’s opening weekend? How can you use mathematics to find your celebrity look-alike? What is Homer Simpson’s method for disproving Fermat’s Last Theorem? Dr. Chartier explores these and other questions in his book Math Bytes.

(Photo courtesy of Davidson College)

As Dr. Chartier and others gear up for basketball lovers’ favorite time of year, PUP reminds you to mark your calendars for these key dates.

Check back here soon for more hoop scoop!

• Selection Sunday, March 15, ESPN

• First and Second Rounds, March 20, 22 or March 21, 23

• Greensboro Regional, March 27, 29, Greensboro Coliseum (Greensboro, North Carolina)

• Oklahoma City Regional, March 27, 29, Chesapeake Energy Arena (Oklahoma City, Oklahoma)

• Albany Regional, March 28, 30, Times Union Center (Albany, New York)

• Spokane Regional, March 28, 30, Spokane Veterans Memorial Arena (Spokane, Washington)

• National Semifinals, April 5, Amalie Arena (Tampa Bay, Florida)

• Championship Game, April 7, Amalie Arena (Tampa Bay, Florida)