Abstract: Statements such as “death rate curve needs to remain single peaked”; “we expect the second peak of infected cases to be smaller compared to the first”, are, unfortunately, all too familiar nowadays. They represent instances of informal descriptions of shapes of functions we can relate to readily, wherein the notion of shape relates exclusively to ‘y direction’ variation. To formalise this, in this talk, I will consider a group-theoretic description of the shape of a function as that which is left behind once ‘x direction’ variability is ignored: only the number and values of local extrema are relevant to describe its shape and not their timings. Accordingly, the shape of a function is shown to be uniquely encoded by a polynomial, and this connection allows us to study structural aspects of the shape space of functions. 

For statistical analysis of functional data (e.g., densely sampled multiple time series data) the upside to this perspective is that (i) a natural dimension reduction mechanism based on local extrema becomes available when analysing amplitudes that avoids the task of registering (or aligning) the data; and (ii) simple finite-dimensional generative shape models can be defined. On the other hand, geometry of the shape space is complicated, and this makes it difficult to compute descriptive statistics (e.g., average shape) and study models induced on the shape space from the original (simpler) function space.  

This is joint work with Ian Jermyn (Durham).