Estimating an NBA player's impact on his team's chances of winning

Traditional NBA player evaluation metrics are based on scoring differential or some pace-adjusted linear combination of box score statistics like points, rebounds, assists, etc. These measures treat performances with the outcome of the game still in question (e.g. tie score with five minutes left) in exactly the same way as they treat performances with the outcome virtually decided (e.g. when one team leads by 30 points with one minute left). Because they ignore the context in which players perform, these measures can result in misleading estimates of how players help their teams win. We instead use a win probability framework for evaluating the impact NBA players have on their teams' chances of winning. We propose a Bayesian linear regression model to estimate an individual player's impact, after controlling for the other players on the court. We introduce several posterior summaries to derive rank-orderings of players within their team and across the league. This allows us to identify highly paid players with low impact relative to their teammates, as well as players whose high impact is not captured by existing metrics.Comment: To appear in the Journal of Quantitative Analysis of Spor

Self-sufficiency with vitamins and minerals on organic dairy farms

Self-sufficiency of nutrients is a central element in the organic farming principles. In a project involving five private organic dairy farms, we aimed to achieve self-sufficiency in vitamins and minerals at farm level. All the herds are fed 100% organically grown feed, but so far supplements of minerals and vitamins based on inorganic and synthetic products are imported to all farms. The same level and type of supplement was used for the cows all year round, even though all cows were on grass for at least 150 days during the summer period. The average daily intake from the supplement for a lactating cow was 751 mg E vitamin, 111 mg Cu, and 558 mg Zn. The content of vitamin and minerals in the home-grown feeds was modelled taking into account the effect of choice of crops; conservation method; season, plant development and climate conditions at harvest; quality of the silage production, and duration of storage. The modelled contents of vitamins in the main ingredients in the feed ration were verified by measuring the actual vitamin content in the silage at harvest as well as losses during storage. As an example, at one of the farms, where the feed intake was based on 85% grass clover crops during the summer but only 68% during the winter, the home-grown feed could supply the cows with enough vitamin E according to the requirement (800 mg/day) during the summer feeding but not during the winter period. The Cu requirement (10 mg/kg DM) could not be met from home-grown feed during any season. However, supplements of vitamins and minerals secure that requirement was met. The final outcome of the project will result in strategies for achieving self-sufficiency in vitamins and minerals at individual farms through optimization of the choice of forage crops and management of feed production

Bursts and Shocks in a Continuum Shell Model

We study a "burst" event, i. e. the evolution of an initial condition having support only in a finite interval of k-space, in the continuum shell model due to Parisi. We show that the continuum equation without forcing or dissipation can be explicitly written in characteristic form and that the right and left moving parts can be solved exactly. When this is supplemented by the appropriate shock condition it is possible to find the asymptotic form of the burst.Comment: 15 pages, 2 eps figures included, Latex 2e. Contribution to the proceedings of the conference: Disorder and Chaos, in honour of Giovanni Paladin, September 22-24, 1997, in Rom

Nonuniversal Critical Spreading in Two Dimensions

Continuous phase transitions are studied in a two dimensional nonequilibrium model with an infinite number of absorbing configurations. Spreading from a localized source is characterized by nonuniversal critical exponents, which vary continuously with the density phi in the surrounding region. The exponent delta changes by more than an order of magnitude, and eta changes sign. The location of the critical point also depends on phi, which has important implications for scaling. As expected on the basis of universality, the static critical behavior belongs to the directed percolation class.Comment: 21 pages, REVTeX, figures available upon reques