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.

Visual imagery in autobiographical memory: The role of repeated retrieval in shifting perspective.

Recent memories are generally recalled from a first-person perspective whereas older memories are often recalled from a third-person perspective. We investigated how repeated retrieval affects the availability of visual information, and whether it could explain the observed shift in perspective with time. In Experiment 1, participants performed mini-events and nominated memories of recent autobiographical events in response to cue words. Next, they described their memory for each event and rated its phenomenological characteristics. Over the following three weeks, they repeatedly retrieved half of the mini-event and cue-word memories. No instructions were given about how to retrieve the memories. In Experiment 2, participants were asked to adopt either a first- or third-person perspective during retrieval. One month later, participants retrieved all of the memories and again provided phenomenology ratings. When first-person visual details from the event were repeatedly retrieved, this information was retained better and the shift in perspective was slowed.