Recent Developments in Complex and Spatially Correlated Functional Data

As high-dimensional and high-frequency data are being collected on a large scale, the development of new statistical models is being pushed forward. Functional data analysis provides the required statistical methods to deal with large-scale and complex data by assuming that data are continuous functions, e.g., a realization of a continuous process (curves) or continuous random fields (surfaces), and that each curve or surface is considered as a single observation. Here, we provide an overview of functional data analysis when data are complex and spatially correlated. We provide definitions and estimators of the first and second moments of the corresponding functional random variable. We present two main approaches: The first assumes that data are realizations of a functional random field, i.e., each observation is a curve with a spatial component. We call them 'spatial functional data'. The second approach assumes that data are continuous deterministic fields observed over time. In this case, one observation is a surface or manifold, and we call them 'surface time series'. For the two approaches, we describe software available for the statistical analysis. We also present a data illustration, using a high-resolution wind speed simulated dataset, as an example of the two approaches. The functional data approach offers a new paradigm of data analysis, where the continuous processes or random fields are considered as a single entity. We consider this approach to be very valuable in the context of big data.Comment: Some typos fixed and new references adde

Variograms for kriging and clustering of spatial functional data with phase variation

Spatial, amplitude and phase variations in spatial functional data are confounded. Conclusions from the popular functional trace-variogram, which quantifies spatial variation, can be misleading when analyzing misaligned functional data with phase variation. To remedy this, we describe a framework that extends amplitude-phase separation methods in functional data to the spatial setting, with a view towards performing clustering and spatial prediction. We propose a decomposition of the trace-variogram into amplitude and phase components, and quantify how spatial correlations between functional observations manifest in their respective amplitude and phase. This enables us to generate separate amplitude and phase clustering methods for spatial functional data, and develop a novel spatial functional interpolant at unobserved locations based on combining separate amplitude and phase predictions. Through simulations and real data analyses, we demonstrate advantages of our approach when compared to standard ones that ignore phase variation, through more accurate predictions and more interpretable clustering results

Spatial variability clustering for spatially dependent functional data

This paper introduces a method for clustering spatially dependent functional data. The idea is to consider the contribution of each curve to the spatial variability. Thus, we define a spatial dispersion function associated to each curve and perform a k-means like clustering algorithm. The algorithm is based on the optimization of a fitting criterion between the spatial dispersion functions associated to each curve and the representative of the clusters. The performance of the proposed method is illustrated by an application on real data and a simulation study