Graphical Degree Sequences

Stefka Antonova
Faculty Advisor: Dr. Jim DeFranza

For every natural number n ≥ 2, there exists a unique graphical degree sequence of length n with no gaps, which contains exactly one degree repeated twice. Here we investigate the graphical properties of degree sequences with no gaps and of length n that contain one or more blocks of one degree repeated three, four, or more times. In this case the graphical degree sequences are no longer unique. However, when certain conditions are met we can determine whether or not such degree sequences are graphical.

Network Vulnerability and Countermeasures

Matthew Brender
Faculty Advisor: Rich Sharp

Real world threats to network security are as prevalent as they are misunderstood in today’s Internet-infatuated society. Hackers, in order to access private resources, implement a broad, though standardized, range of attacks. I have focused on one powerful method to compromise server authentication processes called a stack-based buffer overflow vulnerability which allows hackers to force malicious code to the runtime stack. With the use of freely distributed tools, including multiple distributions of Linux and VMware Server, I explore the means by which hackers attack as well as some methods of protection individuals can take to safeguard their private information.

Fiscal Policy and Growth: The Case of Puerto Rico

Jasper Gardener Burch I
Saint Lawrence University
Canton, New York

In this presentation we will statistically examine the impact that US federal transfer payments (Medicare, Welfare, Social Security etc.) have on the economic development of Puerto Rico. The economy of Puerto Rico is substantially different from that of the mainland United States. Being a commonwealth however Puerto Rico is subject to US federal laws and policies. This leads us to question the severity of inefficiencies of US federal laws and policies in Puerto Rico.

Canonical Projections

Jasper Gardener Burch I
Faculty Advisor: Duncan Melville

Canonical Projection is a powerful method for constructing a tiling in any dimension. It is used to understand the structure of crystals and can create non periodic tilings such as Penrose. In this presentation I will explain tiling terminology, describe the method of Canonical Projection and touch on some basic results.  

Joe Cleary
Faculty Advisor: Robin Lock

Stein estimators can be used to predict future performance of individuals when we have small samples of previous performance for many individuals. For example, suppose we have data on save percentages for a number of NHL goalies through ten games of the season. Because we have small sample sizes the estimates for save percentages should be quite variable. Goalies with the highest percentages are likely to move towards a more typical save percentage as the season progresses; the same will tend to happen for goalies with low initial percentages. Stein estimators provide a method for adjusting estimates based on the distribution of the collection of goalies. We examine properties and implementation of Stein estimators theoretically, through simulation, and with application to real data.

Chance vs. Skill: Assessing Shootouts in the NHL

Laura Daley
Faculty Advisor: Michael Schuckers

The shootout was adopted by the NHL in the 2005-06 season. It is used in the event of a tie after five minutes of overtime, in which case three players are named for the shootout. If after the three shooters are done a tie still remains, the game goes into a ‘sudden death’ where the game will not end until each team has taken the same amount of shots. With this change, ties are eliminated from NHL competition. This investigation looks at the shootout in the 2005-06, 2006-07, and 2007-08 NHL seasons. We analyze results for shooters and goalies to determine the probability of scoring and whether it differs significantly from player to player.

Analyzing Queueing Models and an Issue of Independence

Brent Davis
Faculty Advisor: Dr. Robin Lock

Queueing Theory is the study of waiting in line at to get service, for example, in a checkout line at a grocery store, a toll booth or a ticket counter. We discuss standard queueing models and a method for simulating customer service times according to these models. A key question is the relationship between queue length and expected waiting time. Based on simulations we find an interesting scenario where the common statistical assumption of independence may not be valid and examine methods for dealing with this situation.

The Price is Right: Using Monte Carlo Simulation to Price Stock Options

Kristen Ehringer
Faculty Advisor: Dr. Robin Lock

Simulations are used to imitate real-life situations when other analyses are too mathematically complex or too difficult to compute. The Monte Carlo method of simulation generates randomly selected values for uncertain variables over and over again to simulate a model. The Monte Carlo simulation can be applied to many financial applications, such as the pricing of options. After discussing stock options, we will use the Monte Carlo simulation to price European options and then compare the results to those of the Black-Scholes equation.

Estimating Atypical Baseball Career Trajectories

Andrew Ford
Faculty Advisor: Robin Lock

It is common in any sport for a player to continue improving throughout their career up until a certain point, at which their production will begin to fall. For example, the number of home runs a player hits generally rises until around age thirty, after which their number of home runs begins to fall. In this investigation we apply different curve fitting methods, such as loess and quadratic regression, to career statistics for individual baseball players to predict a typical career trajectory.

Another question that has come up recently is whether or not the career trajectory of a certain baseball player matches that of the typical player. For example, the number of home runs a player hits could be affected by a number of different variables; ranging from being traded to a team that plays in a park with different field dimensions, to intense training in the off-season, medical enhancements, or even random chance. In our study we use statistical techniques to identify when a certain player’s career trajectory differs significantly from that of the typical player.

Using Tournament Seedings to Predict Results

Jared Fostveit
Faculty Advisor: Dr. Robin Lock

Scott Berry wrote an article in the magazine Chance in 2000 in which he used NCAA Men’s Basketball Tournament data from 1986-2000 to determine the likelihood of winning based on seeding, probabilities of each seed reaching each round and expected number of upsets per round. We generalize this approach to other sports with varying tournament formats. For example, we use historical data to create a model that will predict winners of the World Golf Matchplay Championships and the NFL playoffs. The former uses a 64-golfer bracket similar to NCAA basketball. The latter has 12 teams that make the playoffs with the top 2 teams in each conference receiving a first round bye and reseeding after the first round.

An Exploration of Sperner’s Lemma

Rebecca Gilbert

In this expository talk, we will explore the proof of a simple result that has surprisingly important implications. Sperner’s Lemma states that, given a triangulated triangle whose vertices have been tri-colored using a Sperner coloring, at least one inner triangle will have vertices of all three colors. There’s no need to be scared away by the vocabulary, however; this presentation will focus on visual presentation rather than notation, and should be accessible to all students. Although the talk will focus on the proof, some time will be given to further results.

Mathematical Analysis of US Presidential Elections

Eloise Hilarides
Faculty Advisor: Dr. Duncan Melville

Abstract:

I will be applying mathematical voting theory to analyze the results of the US 2008 Presidential primary elections. Using as much real data as possible, I will compare our current voting system to other social choice procedures (i.e. Borda Count, Hare System, etc.) by evaluating the results based on the criteria in Arrow’s Impossibility Theorem and more. Furthermore, I will look for methods of determining social choice that re used in the private sector (i.e. hot-or-not system, etc.) for possible use in the public sector. With my results, I may conclude that one method looks more favorable to me (based on a set of assumptions).

Using Weighting Schemes to Account for Coverage Bias in Internet Surveys

Eloise Hilarides
Faculty Advisor: Dr. Mike Sheard

HRUMC 2008 Abstract

Over the past decade, Internet surveys have become a popular method for collecting data about the general population.  In 2005, the Harris Poll published findings which claimed that 74% of the United States Population had access to the Internet somewhere.  While this number has steadily risen over recent years, bias still may be introduced if the population without Internet access is different from the Internet population in regards to the variables of interest.  In this research we studied whether Internet users that only have access to the Internet outside their home can be useful in reducing bias by assuming that they are more similar to those without Internet access than the Internet population as a whole.

I will outline several weighting adjustment schemes aimed at reducing coverage bias.  Data for this study was taken from the Computer and Internet Use Supplement of October 2003 administered by the Current Population Survey.  I will evaluate the schemes based on overall accuracy by considering the reduction in bias for ten variables of interest and the variability of estimates from the schemes.  I find that several of the proposed schemes are successful in improving accuracy.

Modeling Volatility in Today’s Marketplace

Jebaoh Joerg
Faculty Advisor: Dr. Duncan Melville

If you look at the S&P it may have began the day up 2% but by the end of the day could be down 3%. The following day it could gain back everything it lost. This is not always the case. Eighteen months ago a 2% percent move in the market would have been an anomaly, let alone a 5% swing. So what is the best way to model this Volatility? Is simple variance enough, or do we need more complex models such as the GARCH (generalized autoregressive conditional heteroskedastic) models? We will discuss some different methods of modeling volatility and evaluate their performance using past market data.

Measuring the Effects of 9/11: Intervention Models in Time Series

Victor Kai-Rogers
Faculty Advisor: Robin Lock

Intervention Analysis has been applied to a variety of topics in order to observe the impact of various policy implementations, regime changes and more recently effects of terrorism on time series data. There are several ways to model an intervention function namely, impulse function, gradually changing function, prolonged impulse function and a pure jump function. Specifically, I am interested in knowing if and when an Intervention Model is appropriate by observing its improvement over a univariate analysis. The main event of interest in this project is the September 11 th 2001 terrorist attacks on the US and how they have affected various areas of the economy e.g. the S&P 500 returns in the financial sector.

Who's Really Better

Ryan Kimber
Fall 2007 Independent Study

Abstract

In certain experienced baseball circles, there is a common belief that Latin American players are naturally better than American players. I would imagine that this can be quite frustrating for American players, especially at the professional level. How can we determine whether or not this notion is actually true? I have decided to shed some light on the question at hand by using a multiple binary logistic regression analysis to predict the ethnicity of players (0 for Latin Americans and 1 for Americans) based on numerous career offensive statistics such as career batting average, on base percentage, slugging percentage, and fielding percentage. As a result of my analysis, I found a model that correctly predicted the ethnicity of 73.6% of the Americans and 68.7% of the Latin Americans. Also, I performed a 2-sample t-test on all of the statistics I used and found all of the significant tests gave evidence for American supremacy at least in terms of offensive statistics.

Bowling and the Hot Hand

Royce Lawrence
Faculty Advisor: Dr. Michael Schuckers

Hot hand, or just luck? Ever wonder why streaks come as they do? In this presentation we will discuss the results of a study of the hot hand; the tendency to perform at a higher level for a period of time. For example, a bowler may be more likely to continue to throw strikes after previous strikes. Using frame by frame bowling data and statistical methods, we will determine if the hot hand actually exists.

Determining Win Probabilities Based on Betting Lines on Sporting Events

Rob LaMere
Faculty Advisor: Dr. Sam Vandervelde

We use betting lines and point spread systems to determine probabilities of winning in various sports. In this study we examine different types of betting for major sports such as football, basketball, baseball and hockey. We analyze data from previous seasons to assess the effectiveness of point spreads and betting lines. The goal is to use the odds-makers information to predict who will win the game and the chance of an upset.

Beyond +/-: A Rating System to Compare NHL Players

Dennis Lock
Faculty Advisor: Dr. Michael Schuckers

While there are many statistics, such as plus/minus, to evaluate a hockey player’s performance, few of these statistics effectively compare the worth of different kinds of players. We propose a new, more comprehensive rating method, extending the concept of plus/minus, which aims to take most aspects of a player’s game into account. Each player will be rated on the same scale, where the value of each play is determined by how that play increases or decreases the likelihood of victory. This creates a method of rating where one can compare the value of a pure goal scorer to a defensive specialist, and everyone in between.

Thinking Outside the Box Score

Peter McGoldrick
Faculty Advisor: Dr. Robin Lock

The article A Starting Point for Analyzing Basketball Statistics is just that – a starting point. Do point totals, turnover totals, and rebound counts really give us the best statistical explanation of basketball player and team performances? This study resulted in the formulation of a number of statistics that aim to give a clearer understanding of the quality of both individual and team play in the National Basketball Association (NBA). Based on a central theme of per possession statistics, we discuss the efficiency of team statistics. We will evaluate a player’s performance more accurately by using per minute statistics. With these variations on our classical basketball box score, we are also able to compare basketball statistics from league to league.

Investment Risk Measurement: Approaches to Risk Measurement

Nikolay Naletov
Faculty Advisor: Duncan Melville

This research presents different ways to measure investment risk. In particular, it shows a comparison between VaRs (Value-at-Risk) based on methods such as variance-covariance, historical and Monte Carlo simulations, and GARCH approaches. The paper compares and explains the different techniques and methodologies by using the historical database of the equity portfolio of Crown Royalties Investments, which is the investment club at Saint Lawrence University.

Linking Medical Disciplines and Theories of the Small-World

Renée Rubin
Faculty Advisor: Dr. Collen Knickerbocker

One of the more famous examples of Small-World theory lies in the hands of Kevin Bacon, an otherwise relatively unknown member of the Hollywood clan. The idea that networks exist such that clustering coefficients are high enough to link a large number of components through a limited number of steps is a mind boggling concept. From Kevin Bacon to the number of handshakes any one person is from the president to the extent of the AIDS pandemic, highly linked networks exist nearly everywhere. It has been determined that the world of published documents is impressively scale-free; this project seeks to investigate the Small-World network effect on medical disciplines as well as how linked or clustered they are and enable conclusions to be drawn based on the connectedness of each discipline analyzed.

The Eternal Vision of Leonhard Euler

Jon Sauer
Dr. Sam Vandervelde

Leonhard Euler has often been claimed as being the single most prolific mathematician in history. His achievements span an incredible range of mathematical fields, leaving a lasting imprint on calculus, number theory, graph theory, geometry, and algebra, just to name a few. This film is intended to give viewers an informative sketch of Leonhard Euler’s life and how it intertwined with his constant mathematical output. A section of the video will be shown during the presentation, which should provide a good idea of what the approximately half-hour long final product will look like.

The Mathematics of Flip-flop

Catherine Sheard
St. Lawrence University

Flip-flop is a game played by N people sitting in a circle. The first player says “one,” the next player says “two,” and so on, except in one of the following situations: 1) A player instead says the word “flip” for each occurrence of a factor of seven and each occurrence of the digit seven in her number, and 2) The player instead says the word “flop” for each occurrence of a factor of eight and each occurrence of the digit eight. For example, “ninety-eight” would be replaced by “flip-flip-flop.” Furthermore, each time the word “flip” is said, play changes direction, and each time the word “flop” is said, the next player is skipped. Finally, each time a player says “flip” or “flop,” she changes whether she is sitting or standing. Flip-flop presents many interesting mathematical questions, some of which will be presented. For instance, at what point will all players be standing? Does this point even exist? Where should a person sit if she really likes to say “flip”? What happens as N→∞? Most of these are surprisingly difficult problems.

Performance vs. Pick: A Study of the NBA Draft

Jamie Wolff
Faculty Advisor: Dr. Michael Schuckers

Millions of dollars are invested in the top draft picks of the National Basketball Association (NBA). A significant amount of deliberation and analysis is put into determining which athlete to draft. When making this decision, what are the most important factors to consider? This investigation will explore any patterns of recent NBA drafts considered valuable and what would be the advantage in drafting one player over another. We will use NBA career statistics to assess draft-day decisions based on player productivity.