Probabilistic Fréchet means for time varying persistence diagrams

© 2015, Institute of Mathematical Statistics. All rights reserved.In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in (Dp, Wp), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards. We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each N a map (Dp)^N→ℙ(Dp). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.

Clinical Insights Into the Biology and Treatment of Pancreatic Cancer.

Pancreatic cancer is a devastating disease with a universally poor prognosis. In 2015, it is estimated that there will be 48,960 new cases of pancreatic cancer and that 40,560 people will die of the disease. The 5-year survival rate is 7.2% for all patients with pancreatic cancer; however, survival depends greatly on the stage at diagnosis. Unfortunately, 53% of patients already have metastatic disease at diagnosis, which corresponds to a 5-year survival rate of 2.4%. Even for the 9% of patients with localized disease confined to the pancreas, the 5-year survival is still modest at only 27.1%. These grim statistics highlight the need for ways to identify cohorts of individuals at highest risk, methods to screen those at highest risk to identify preinvasive pathologic precursors, and development of effective systemic therapies. Recent clinical and translational progress has emphasized the relationship with diabetes, the role of the stroma, and the interplay of each of these with inflammation in the pathobiology of pancreatic cancer. In this article, we will discuss these relationships and how they might translate into novel management strategies for the treatment of this disease.