Kriging prediction for manifold-valued random fields

The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications, such as shape analysis, diffusion tensor imaging and the analysis of covariance matrices. In many cases, data are spatially distributed but it is not trivial to take into account spatial dependence in the analysis because of the non linear geometry of the manifold. This work proposes a solution to the problem of spatial prediction for manifold valued data, with a particular focus on the case of positive definite symmetric matrices. Under the hypothesis that the dispersion of the observations on the manifold is not too large, data can be projected on a suitably chosen tangent space, where an additive model can be used to describe the relationship between response variable and covariates. Thus, we generalize classical kriging prediction, dealing with the spatial dependence in this tangent space, where well established Euclidean methods can be used. The proposed kriging prediction is applied to the matrix field of covariances between temperature and precipitation in Quebec, Canada.This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jmva.2015.12.00

A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space

We address the problem of predicting spatially dependent functional data belonging to a Hilbert space, with a Functional Data Analysis approach. Having defined new global measures of spatial variability for functional random processes, we derive a Universal Kriging predictor for functional data. Consistently with the new established theoretical results, we develop a two-step procedure for predicting georeferenced functional data: first model selection and estimation of the spatial mean (drift), then Universal Kriging prediction on the basis of the identified model. The proposed methodology is applied to daily mean temperatures curves recorded in the Maritimes Provinces of Canada

Stochastic simulation of soil particle-size curves in heterogeneous aquifer systems through a Bayes space approach

We address the problem of stochastic simulation of soil particle-size curves (PSCs) in heterogeneous aquifer systems. Unlike traditional approaches that focus solely on a few selected features of PSCs (e.g., selected quantiles), our approach considers the entire particle-size curves and can optionally include conditioning on available data. We rely on our prior work to model PSCs as cumulative distribution functions and interpret their density functions as functional compositions. We thus approximate the latter through an expansion over an appropriate basis of functions. This enables us to (a) effectively deal with the data dimensionality and constraints and (b) to develop a simulation method for PSCs based upon a suitable and well defined projection procedure. The new theoretical framework allows representing and reproducing the complete information content embedded in PSC data. As a first field application, we demonstrate the quality of unconditional and conditional simulations obtained with our methodology by considering a set of particle-size curves collected within a shallow alluvial aquifer in the Neckar river valley, Germany

A Comparison Between Machine Learning and Functional Geostatistics Approaches for Data-Driven Analyses of Sediment Transport in a Pre-Alpine Stream

The problem of providing data-driven models for sediment transport in a pre-Alpine stream in Italy is addressed. This study is based on a large set of measurements collected from real pebbles, traced along the stream through radio-frequency identification tags after precipitation events. Two classes of data-driven models based on machine learning and functional geostatistics approaches are proposed and evaluated to predict the probability of movement of single pebbles within the stream. The first class built upon gradient-boosting decision trees allows one to estimate the probability of movement of a pebble based on the pebbles' geometrical features, river flow rate, location, and subdomain types. The second class is built upon functional kriging, a recent geostatistical technique that allows one to predict a functional profile-that is, the movement probability of a pebble, as a function of the pebbles' geometrical features or the stream's flow rate-at unsampled locations in the study area. Although grounded in different perspectives, both models aim to account for two main sources of uncertainty, namely, (1) the complexity of a river's morphological structure and (2) the highly nonlinear dependence between probability of movement, pebble size and shape, and the stream's flow rate. The performance of the two methods is extensively compared in terms of classification accuracy. The analyses show that despite the different perspectives, the overall performance is adequate and consistent, which suggests that both approaches can provide modeling frameworks for sediment transport. These data-driven approaches are also compared with physics-based ones that are classically used in the hydrological literature. Finally, the use of the developed models in a bottom-up strategy, which starts with the prediction/classification of a single pebble and then integrates the results into a forecast of the grain-size distribution of mobilized sediments, is discussed

Telemedicine for cardiac surgery candidates

Background: Cardiac surgery is generally well or over-represented in many Western countries. Since the southern part of Switzerland relies on 300 km distance centers for cardiac surgery, we started a project of telemedicine for the distant evaluation of cardiac surgery candidates. We report our experience of the results of the diagnosis made by telemedicine and by direct scrutiny of coronary angiograms. Methods: Coronary angiography was performed at the distant hospital by an invasive cardiology team. Teletransmission of images was performed using three Integrated Service Digital Network (ISDN) lines by direct transmission of recent recording. A total of 98 cases were reviewed (87 aorto-coronary bypass candidates, seven valvular and four congenital heart disease). We further performed a prospective blinded comparison of 47 consecutive cases with severe coronary artery disease (CAD) with respect to localization and number of significant coronary lesions, obtained by direct scrutiny of the original angiograms and the evaluation obtained with the teletransmitted images. Results: In 89 cases of the 98 analyzed (91%) correct diagnosis and surgical approach could be established by distant transmission. In nine cases (9%, all aortocoronary bypass candidates) definitive diagnosis and treatment was feasible only by direct scrutiny of the original angiograms. Five critically ill patients were urgently referred to the surgical care center based on the correct distant diagnosis. The blinded comparison of distant diagnosis and direct scrutiny of angiograms in defining 1-2-3 vessel CAD was good: r=0.87, P≪0.01. Conclusion: Initial experience using non-sophisticated telemedical transmission of angiograms of cardiac surgery candidates seems to be a promising facility for distantly located center

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

In this paper we propose Universal trace co-kriging, a novel methodology for interpolation of multivariate Hilbert space valued functional data. Such data commonly arises in multi-fidelity numerical modeling of the subsurface and it is a part of many modern uncertainty quantification studies. Besides theoretical developments we also present methodological evaluation and comparisons with the recently published projection based approach by Bohorquez et al. (Stoch Environ Res Risk Assess 31(1):53–70, 2016. https://doi.org/10.1007/s00477-016-1266-y). Our evaluations and analyses were performed on synthetic (oil reservoir) and real field (uranium contamination) subsurface uncertainty quantification case studies. Monte Carlo analyses were conducted to draw important conclusions and to provide practical guidelines for all future practitioners