Abstract: Modelling a large bundle of curves arises in a broad spectrum of real applications. In this talk, I will give a selective overview of functional data analysis in high dimensions based on my recent research. The first part extended the graphical models concept to model the conditional dependence structure among p random functions. We developed an extension of the glasso criterion (fglasso), which estimates the functional graphical model by imposing a block sparsity constraint on the precision matrix, via a group lasso penalty. We established the concentration inequalities of the estimates, which guarantee the desirable graph support recovery property. In functional data analysis, the classical assumption is that each function has been measured over a densely sampled grid. However, in practice the functions have often been observed, with measurement error, at a relatively small number of points. In the second part, we considered a class of doubly functional graphical models to capture the evolving conditional dependence relationship among a large number of sparsely or densely sampled functions. In the last part, we provided a general theory for large-scale Gaussian curve time series, where the temporal and cross-sectional dependence across multiple curve observations exist and the number of functional variables, p, may be large relative to the number of observations, n. We proposed a novel functional stability measure for multivariate stationary processes based on their spectral properties and use it to establish some useful concentration bounds on the sample covariance matrix function. These concentration bounds serve as a fundamental tool for further theoretical analysis, in particular, for deriving nonasymptotic upper bounds on the errors of the regularized estimates in high dimensional settings.

About the speaker: Shaojun Guo is currently an Associate Professor in the Institute of Statistics and Big Data at Renmin University of China. Before that, he was an Assistant Professor in Academy of Mathematics and Systems Science at Chinese Academy of Sciences since 2008 and also Research Fellow in Department of Statistics at London school of Economics and Political Science since 2014 to 2016.  He completed his Ph.D. in Mathematical Statistics from Academy of Mathematics and Systems Science at Chinese Academy of Sciences in 2008. From 2009 to 2010 he was a Visiting Postdoctoral Research Associate in the Department of Operations Research and Financial Engineering (ORFE) at Princeton University, hosted by Professor Jianqing Fan.

See his personal website in details: sites.google.com/site/guoshaojun20170709/