**HRUMC 2012**

Was held on April 21st at Western New England University in Springfiled, MA

**Nassir Abou Ziki***A Bayesian Approach for Assessing the Performance of a Biometric Authentication Device*

When no successes are observed in *n* independent trials the usual statistical methods won't work to estimate the probability of success, *p*. Several techniques have been devised to give a reasonable upper bound for the probability *p* of a success. These include the Rule of Three which and the Bayesian Rule of Three. A similar problem arises when estimating the matching performance of a biometric authentication device (e.g. a fingerprint scanner). However, in this case, the encountered problem becomes even more complicated since *p* can vary from individual to individual and we have *n _{i} *observations on an individual

*i*. Our work focuses on developing other Bayesian approaches to the latter types of problems. We use a Beta-Binomial distribution to find a better reasonable upper bound on the probability

*p*. We also test our models by estimating the probability of a failure of a biometric authentication device for simulated data.

**James Curro***Ranking NHL players based on all on-ice events*

Abstract: Ratings of individual hockey players are difficult due to the fluid nature of the game. Unlike baseball and football there is no start and stop at plays making it hard to know what you are looking for when analyzing. Hockey is relatively low scoring compared to other similar games such as basketball which further adds to the difficulties for evaluating players. Recently there have been some attempts to get at the value of National Hockey League (NHL) skaters based upon scoring rates (Macdonald (2011)) or at the value of certain events that happen on the ice (Corsi & Fenwick) or the impact of other players that appear on the ice with a given player QualComp. We present a new comprehensive rating that accounts for each of these elements as well as the impact of non-shooting events such as turnovers and hits that occur when a given player is on the ice. The impact of each play is determined by net expected probability leading to a goal for a player’s team (or their opponent) in the subsequent 20 seconds. One desired outcome of this work is to provide a methodology that can quantify the impact of individual hockey players across positions (forwards and defenseman).

**Cassidy Griffin***Obtainability of Strong Orientations: Creating an Efficient Network of One-Way Streets*

Abstract: A strong orientation of a graph is a way to orient the edges so that the resulting directed graph is strongly connected, meaning it is possible to get from any vertex to any other vertex while following the arrows. In his book *Graph Theory and Its Applications to Problems of Society*, Fred Roberts describes a depth-first search algorithm (which we call the Roberts Algorithm) for putting a strong orientation on a graph. We call a strong orientation *obtainable* if it is possible to arrive at that orientation using some application of the Roberts Algorithm. We present results about obtainable and unobtainable orientations, including classes of graphs which have unobtainable orientations and classes of graphs for which all orientations are obtainable. In addition, we discuss relative efficiency of orientations which are obtainable or unobtainable, using four different ways to measure efficiency or optimality.

**Daniel Look (SLU faculty)***Fractal Images Generated from Circles*

As far as mathematical objects go, the circle is rarely considered \sexy".

However, this simple and well-studied object is the basis for many stunning fractal images and, surprisingly, current research. We will discuss how some of these pictures are created; stopping briefly in the fields of group theory and complex dynamics.

**Tom Pasquali***How important are face-offs in the NHL?*

Abstract: Hockey is a very fast paced game, which makes it hard to analyze stats. Unlike baseball and football there is no start and stop at plays making it hard to know what you are looking for when analyzing. However hockey is known to be a game of possession and when there is a stoppage in play there is then a faceoff where there is a 50-50 chance of who will win the faceoff and gain possession. Or is there? The purpose of my project was to analyze face-offs in hockey for all players in the NHL and try to effectively rank the players in the NHL. By doing this and calculating the probability that a faceoff leads to a goal it can then be calculated the expected number of goals a 48% faceoff leader will give you when compared to a 53% faceoff winner. I will also take a look at how location, number of players on the ice and whether a team is home or away affect the outcome of a faceoff.

**Robert Romeo***Kick’em While They’re Up, Kick’em While They’re Down*

Abstract: Imagine that you are an NFL field goal kicker and your team is down by 2 points and it is now 4^{th} down and short. IF you make the field goal, your team wins, but if you miss, your team will lose the game. In the NFL AFC championship game, the Baltimore Raven’s kicker Billy Cundiff was in such a situation; he missed a 32 yard field goal that would have tied the game and taken them into overtime. Everyone always blames the field goal kicker if they miss a field goal, even though they are under a lot of pressure. In this project we investigate the role pressure has in terms of affecting how likely it is that a person will make or miss a field goal. We will be using a Bayesian hierarchical logistic regression model to determine how much of a role pressure has when attempting a field goal in the NFL. We are using kick by kick data from the 2011-2012 NFL regularseason, and some of the potential predictors include the length of the field goal, the point differential at the time prior to the attempt, the quarter and down the kick was attempted.

**Lauren Stemler***Paradigms in Path Counting*

Abstract: Path counting can be described as _nding the number of ways to get between two points A and B in a network. In this talk, we will focus on a “bow-tie" network and several variations to count the number of possible paths and explain why these values arise. Some of

the networks lead interesting results. For example, in one network the number of paths in a network of size *n* satisfies the recursion** ** . . From this intriguing recursion the method of generating functions was used to find a closed-form expression

for the number of paths through the network. A similar process was employed for other networks, most notably including a “Pascal's Bow-Tie" network.