RESUMO
Graphical modeling of multivariate functional data is becoming increasingly important in a wide variety of applications. The changes of graph structure can often be attributed to external variables, such as the diagnosis status or time, the latter of which gives rise to the problem of dynamic graphical modeling. Most existing methods focus on estimating the graph by aggregating samples, but largely ignore the subject-level heterogeneity due to the external variables. In this article, we introduce a conditional graphical model for multivariate random functions, where we treat the external variables as conditioning set, and allow the graph structure to vary with the external variables. Our method is built on two new linear operators, the conditional precision operator and the conditional partial correlation operator, which extend the precision matrix and the partial correlation matrix to both the conditional and functional settings. We show that their nonzero elements can be used to characterize the conditional graphs, and develop the corresponding estimators. We establish the uniform convergence of the proposed estimators and the consistency of the estimated graph, while allowing the graph size to grow with the sample size, and accommodating both completely and partially observed data. We demonstrate the efficacy of the method through both simulations and a study of brain functional connectivity network.
RESUMO
Spatial predictions of drift deposits on soil surface were conducted using eight different spatial interpolation methods i.e. classical approaches like the Thiessen method and kriging, and some advanced methods like spatial vine copulas, Karhunen-Loève expansion and INLA. In order to investigate the impact of the number of locations on the prediction, all spatial predictions were conducted using a set of 39 and 47 locations respectively. The analysis revealed that taking more locations into account increases the accuracy of the prediction and the extreme behavior of the data is better modeled. Leave-one-out cross-validation was used to assess the accuracy of the prediction. The Thiessen method has the highest prediction errors among all tested methods. Linear interpolation methods were able to better reproduce the extreme behavior at the first meters from the sprayed border and exhibited lower prediction errors as the number of data points increased. Especially the spatial copula method exhibited an obvious increase in prediction accuracy. The Karhunen-Loève expansion provided similar results as universal kriging and IDW, although showing a stronger change in the prediction as the number of locations increased. INLA predicted the pesticide dispersion to be smooth over the whole study area. Using Delaunay triangulation of the study area, the total pesticide concentration was estimated to be between 2.06% and 2.97% of the total Uranine applied. This work is a first attempt to completely understand and model the uncertainties of the mass balance, therefore providing a basis for future studies.