Statistical analysis of complex and spatially dependent data: A review of Object Oriented Spatial Statistics

Abstract

We review recent advances in Object Oriented Spatial Statistics, a system of ideas, algorithms and methods that allows the analysis of high dimensional and complex data when their spatial dependence is an important issue. At the intersection of different disciplines – including mathematics, statistics, computer science and engineering – Object Oriented Spatial Statistics provides the right perspective to address key problems in varied contexts, from Earth and life sciences to urban planning. We illustrate a few paradigmatic methods applied to problems of prediction, classification and smoothing, giving emphasis to the key ideas Object Oriented Spatial Statistics relies upon

Exploring Patterns of Demand in Bike Sharing Systems Via Replicated Point Process Models

“…Delicado et al . () and Menafoglio and Secchi ()), estimation of R requires assumptions such as isotropy, i.e. that for some g , to pool data across neighbouring sites.…”

“…(), Secchi et al . () and Menafoglio and Secchi (), among others, but again in the context of a single datum per site, which does not enable direct estimation of spatial correlations and requires assumptions such as isotropy; in our application, the availability of many replications per site enables us to estimate spatial correlations directly and without isotropy assumptions, which, in fact, we show does not hold for the bike sharing network.…”

“…Spatiotemporal processes have been widely studied in the literature, but mostly in the singlereplication context (see for example Li and Guan (2014), Shirota and Gelfand (2017), Diggle (2013) and references therein), not in the many-replication context of this paper. Finally, we mention that clustering methods for spatial functional data have been proposed by Delicado et al (2010), Romano et al (2010), Secchi et al (2013) and Menafoglio and Secchi (2017), among others, but again in the context of a single datum per site, which does not enable direct estimation of spatial correlations and requires assumptions such as isotropy; in our application, the availability of many replications per site enables us to estimate spatial correlations directly and without isotropy assumptions, which, in fact, we show does not hold for the bike sharing network.…”

Cokriging for multivariate Hilbert space valued random fields: application to multi-fidelity computer code emulation

“…In the following developments, we will always consider as ambient space for the analysis a Hilbert space. The use of a Hilbert-space embedding for functional geostatistics is well-documented in the literature (see e.g., [Menafoglio and Secchi, 2017] for a review). Its mathematical and application-oriented convenience is twofold: (i) it allows working by analogy with the scalar setting, providing strong intuitions and interpretations to the concepts involved (e.g., for the concepts of variogram); and (ii) it allows working in a very general setting, which may even involve functional constrained data (e.g., PDFs, [Menafoglio et al, 2014]).…”

Model-free two-sample test for network-valued data

“…It aims at conducting statistical analyses of complex data that cannot be embedded in the standard Euclidean framework (see Marron and Alonso, 2014, with discus-5 sion), by contrast with more traditional data sets composed of numbers or vectors of numbers that naturally lie in a Euclidean space in which standard statistical methods can be applied. Shapes (Dryden and Mardia, 1998), images (Locantore et al, 1999;Wei et al, 2016), manifold-valued data such as directional data (Mardia, 1972), trees (Wang and Marron, 2007), covariance 10 matrices and operators (Dryden et al, 2009;Pigoli et al, 2014), density functions (Menafoglio and Secchi, 2017) are examples of so-called object data. Investigating the relationships between these complex objects requires the development of appropriate statistical tools that can be either generalizations of existing Euclidean methods or novel non-standard approaches (see 15 Sangalli et al, 2014).…”