Appears in collection : Statistical Modeling for Shapes and Imaging

We consider a subarea of functional data analysis, where functions of interest are constrained to have pre-determined shape classes. The notion of shape is quite flexible. It can mean a fixed number of modes in the function, say a bimodal or a trimodal function, or the number of modes plus a vector of function heights at the modes. The locations of these modes are left as variables, in order to fit to the data. The basic idea is to define a set of valid functions (with the desired shape constraints) and to solve optimization problems (such as maximum likelihood estimation) on this set. This set is established using the 'deformable template' theory -- choose a function from the correct class and use an appropriate action of the diffeomorphism group to form its orbits. Orbits define shape classes. The larger picture is to learn shape classes from the training data, and then to impose learnt shape constraints in estimating future functions from sparse, noisy data. We present some examples of this framework. First, we introduce the problem of density estimation under arbitrary multimodal shape constraints. While unimodal density estimation is often studied in the literature, there are no general estimators for the multimodal case. Second, we provide a study involving daily electricity consumption data at household level (in Tallahassee, FL) where certain shapes dominate the data. (Joint work with Sutanoy Dasgupta, Ian Jermyn, and Debdeep Pati).