Unsupervised clustering of multivariate circular data.

In this paper, we study an unsupervised clustering problem. The originality of this problem lies in the data, which consist of the positions of five separate X-ray beams on a circle. Radiation therapists positioned the five X-ray beam 'projectors' around each patient on a predefined circle. However, similarities exist in positioning for certain groups of patients, and we aim to describe these similarities with the goal of creating pre-adjustment settings that could help save time during X-ray positioning. We therefore performed unsupervised clustering of observed X-ray positions. Because the data for each patient consist of five angle measurements, Euclidean distances are not appropriated. Furthermore, we cannot perform k-means algorithm, usually used for minimizing corresponding distortion because we cannot calculate centers of clusters. We present here solutions to these problems. First, we define a suitable distance on the circle. Then, we adapt an algorithm based on simulated annealing to minimize distortion. This algorithm is shown to be theoretically convergent. Finally, we present simulations on simulated and real data.

Milk acidification by Lactococcus lactis is improved by decreasing the level of dissolved oxygen rather than decreasing redox potential in the milk prior to inoculation.

Although redox potential is very rarely taken into account in food fermentation it could be as influential as pH on bacterial activities. Lactococcus lactis is already known to exhibit a powerful reducing activity in milk but its reduction activity was shown to occur prior to its acidification activity with a potential interaction between these two lactococcal activities. Therefore, acidification lag-type phase could be shortened by decreasing the redox potential of milk before inoculation. As the redox potential is highly dependent on the dissolved oxygen level, our objective was to study their separate and combined influences on acidification and growth kinetics of pure L. lactis strains in milk. Results showed that high level of dissolved oxygen is significantly more influential on growth, and even more on acidification kinetics, than initial decreased redox potential of milk. Reduction of milk was drastic and mostly due to bacterial activity. The redox potential of milk only dropped when dissolved oxygen was entirely consumed. When there was no dissolved oxygen from the beginning, L. lactis immediately decreased the redox potential of milk and acidified afterwards. When the level of dissolved oxygen was initially high, acidification and reduction of milk occurred at the same time. Acidification kinetics was then biphasic with a slower rate during the aerobic stage and a faster rate during the anaerobic stage. The seven strains tested demonstrated diversity in both their acidification kinetics and their adaptation to high level of dissolved oxygen, independent of their growth kinetics. To conclude, we have shown that the level of dissolved oxygen in milk has a dramatic influence on acidification kinetics and could be used to control acidification kinetics in dairy industries.

Robust Bayesian decision theory applied to optimal dosage.

We give a model for constructing an utility function u(theta,d) in a dose prescription problem. theta and d denote respectively the patient state of health and the dose. The construction of u is based on the conditional probabilities of several variables. These probabilities are described by logistic models. Obviously, u is only an approximation of the true utility function and that is why we investigate the sensitivity of the final decision with respect to the utility function. We construct a class of utility functions from u and approximate the set of all Bayes actions associated to that class. Then, we measure the sensitivity as the greatest difference between the expected utilities of two Bayes actions. Finally, we apply these results to weighing up a chemotherapy treatment of lung cancer. This application emphasizes the importance of measuring robustness through the utility of decisions rather than the decisions themselves.