A comprehensive review of plus-minus ratings for evaluating individual players in team sports

The increasing availability of data from sports events has led to many new directions of research, and sports analytics can play a role in making better decisions both within a club and at the level of an individual player. The ability to objectively evaluate individual players in team sports is one aspect that may enable better decision making, but such evaluations are not straightforward to obtain. One class of ratings for individual players in team sports, known as plus-minus ratings, attempt to distribute credit for the performance of a team onto the players of that team. Such ratings have a long history, going back at least to the 1950s, but in recent years research on advanced versions of plus-minus ratings has increased noticeably. This paper presents a comprehensive review of contributions to plusminus ratings in later years, pointing out some key developments and showing the richness of the mathematical models developed. One conclusion is that the literature on plus-minus ratings is quite fragmented, but that awareness of past contributions to the field should allow researchers to focus on some of the many open research questions related to the evaluation of individual players in team sports. Keywords: rating system, ranking, regression, regularizationpublishedVersio

Offensive and defensive plus-minus player ratings for soccer

Rating systems play an important part in professional sports, for example, as a source of entertainment for fans, by influencing decisions regarding tournament seedings, by acting as qualification criteria, or as decision support for bookmakers and gamblers. Creating good ratings at a team level is challenging, but even more so is the task of creating ratings for individual players of a team. This paper considers a plus–minus rating for individual players in soccer, where a mathematical model is used to distribute credit for the performance of a team as a whole onto the individual players appearing for the team. The main aim of the work is to examine whether the individual ratings obtained can be split into offensive and defensive contributions, thereby addressing the lack of defensive metrics for soccer players. As a result, insights are gained into how elements such as the effect of player age, the effect of player dismissals, and the home field advantage can be broken down into offensive and defensive consequences. View Full-Text Keywords: association football, linear regression, regularization, rankingpublishedVersio

Ordinal versus nominal regression models and the problem of correctly predicting draws in soccer

Ordinal regression models are frequently used in academic literature to model outcomes of soccer matches, and seem to be preferred over nominal models. One reason is that, obviously, there is a natural hierarchy of outcomes, with victory being preferred to a draw and a draw being preferred to a loss. However, the often used ordinal models have an assumption of proportional odds: the influence of an independent variable on the log odds is the same for each outcome. This paper illustrates how ordinal regression models therefore fail to fully utilize independent variables that contain information about the likelihood of matches ending in a draw. However, in practice, this flaw does not seem to have a substantial effect on the predictive accuracy of an ordered logit regression model when compared to a multinomial logistic regression model. Keywords: association football, forecasting, ordered regressionpublishedVersio

Predicting match outcomes in association football using team ratings and player ratings

The main goal of this article is to compare the performance of team ratings and individual player ratings when trying to forecast match outcomes in association football. The well-known Elo rating system is used to calculate team ratings, whereas a variant of plus-minus ratings is used to rate individual players. For prediction purposes, two covariates are introduced. The first represents the pre-match difference in Elo ratings of the two teams competing, while the second is the average difference in individual ratings for the players in the starting line-ups of the two teams. Two different statistical models are used to generate forecasts. The first type is an ordered logit regression (OLR) model that directly outputs probabilities for each of the three possible match outcomes, namely home win, draw and away win. The second type is based on competing risk modelling and involves the estimation of scoring rates for the two competing teams. These scoring rates are used to derive match outcome probabilities using discrete event simulation. Both types of models can be used to generate pre-game forecasts, whereas the competing risk models can also be used for in-game predictions. Computational experiments indicate that there is no statistical difference in the prediction quality for pre-game forecasts between the OLR models and the competing risk models. It is also found that team ratings and player ratings perform about equally well when predicting match outcomes. However, forecasts made when using both team ratings and player ratings as covariates are significantly better than those based on only one of the ratings. Keywords: Elo rating, competing risk, ordered logit regression, plus-minus rating, survival analysis.acceptedVersio

RescUSim and IPython: An environment for offshore emergency preparedness planning

Emergency preparedness is crucial for oil and gas operators. While accidents in this industry are commonly connected to oil spill disasters, helicopter accidents are, in terms of incidence rates, a more grave concern in Norway. A recent helicopter accident near Bergen has brought this subject back into focus. We introduce RescUSim, a simulator for rescue missions after offshore helicopter accidents, which is implemented as an open source library with bindings for the Python language. We discuss the modules in the existing Python ecosystem that are used for data preparation and analysis. We show how RescUSim and the interactive computing environment IPython can join forces to provide a tool for planning rescue preparedness for oil and gas related offshore activities

Maximizing performance with an eye on the finances: a chance-constrained model for football transfer market decisions

Composing a team of professional players is among the most crucial decisions in association football. Nevertheless, transfer market decisions are often based on myopic objectives and are questionable from a financial point of view. This paper introduces a chance-constrained model to provide analytic support to club managers during transfer windows. The model seeks a top-performing team while adapting to different budgets and financial-risk profiles. In addition, it provides a new rating system that is able to numerically reflect the on-field performance of football players and thus contribute to an objective assessment of football players. The model and rating system are tested on a case study based on real market data. The data from the case study are available online for the benefit of future research

On the relationship between +/- ratings and event-level performance statistics

This work considers the challenge of identifying and properly assessing the contribution of a single player towards the performance of a team. In particular, we study the use of advanced plus-minus ratings for individual football players, which involves evaluating a player based on the goals scored and conceded with the player appearing on the pitch, while compensating for the quality of the opponents and the teammates as well as other factors. To increase the understanding of plus-minus ratings, event-based data from matches are first used to explain the observed variance of ratings, and then to improve their ability to predict outcomes of football matches. It is found that event-level performance statistics can explain from 22% to 38% of the variance in plus-minus ratings, depending on player positions, while incorporating the event-level statistics only marginally improves the predictive power of plus-minus ratings.publishedVersio

RescUSim and IPython : an environment for offshore emergency preparedness planning

Emergency preparedness is crucial for oil and gas operators. While accidents in this industry are commonly connected to oil spill disasters, helicopter accidents are, in terms of incidence rates, a more grave concern in Norway. A recent helicopter accident near Bergen has brought this subject back into focus. We introduce RescUSim, a simulator for rescue missions after offshore helicopter accidents, which is implemented as an open source library with bindings for the Python language. We discuss the modules in the existing Python ecosystem that are used for data preparation and analysis. We show how RescUSim and the interactive computing environment IPython can join forces to provide a tool for planning rescue preparedness for oil and gas related offshore activities.publishedVersio

Evaluating the efficiency of the association football transfer market using regression based player ratings

In recent times, the use of quantitative methods to improve decisions within sports has increased. In association football, large amounts of match data has become available. This work first shows how simple match data describing the players on pitch and the time for events such as goals and red cards, can be used to derive an objective player rating. The rating is based on solving a large linear regression model. The resulting player ratings are in turn used as input to a regression model for analyzing transfer fees. It is shown that the performance of players, as reflected in the player ratings, is an important predictor of transfer fees. At the same time, several other important factors that determine the size of transfer fees are identified

Determining departure times in dynamic and stochastic maritime routing and scheduling problem

In maritime transportation, decisions are made in a dynamic setting where many aspects of the future are uncertain. However, most academic literature on maritime transportation considers static and deterministic routing and scheduling problems. This work addresses a gap in the literature on dynamic and stochastic maritime routing and scheduling problems, by focusing on the scheduling of departure times. Five simple strategies for setting departure times are considered, as well as a more advanced strategy which involves solving a mixed integer mathematical programming problem. The latter strategy is significantly better than the other methods, while adding only a small computational effort