Random forest and similar Machine Learning techniques are already used to generate spatial predictions, but spatial location of points (geography) is often ignored in the modeling process. Spatial auto-correlation, especially if still existent in the cross-validation residuals, indicates that the predictions are maybe biased, and this is suboptimal. This paper presents a random forest for spatial predictions framework (RFsp) where buffer distances from observation points are used as explanatory variables, thus incorporating geographical proximity effects into the prediction process. The RFsp framework is illustrated with examples that use textbook datasets and apply spatial and spatio-temporal prediction to numeric, binary, categorical, multivariate and spatiotemporal variables. Performance of the RFsp framework is compared with the state-of-the-art kriging techniques using fivefold cross-validation with refitting. The results show that RFsp can obtain equally accurate and unbiased predictions as different versions of kriging. Advantages of using RFsp over kriging are that it needs no rigid statistical assumptions about the distribution and stationarity of the target variable, it is more flexible towards incorporating, combining and extending covariates of different types, and it possibly yields more informative maps characterizing the prediction error. RFsp appears to be especially attractive for building multivariate spatial prediction models that can be used as “knowledge engines” in various geoscience fields. Some disadvantages of RFsp are the exponentially growing computational intensity with increase of calibration data and covariates and the high sensitivity of predictions to input data quality. The key to the success of the RFsp framework might be the training data quality—especially quality of spatial sampling (to minimize extrapolation problems and any type of bias in data), and quality of model validation (to ensure that accuracy is not effected by overfitting). For many data sets, especially those with lower number of points and covariates and close-to-linear relationships, model-based geostatistics can still lead to more accurate predictions than RFsp.

Most of the hydrological and hydraulic studies refer to the notion of a return period to quantify design variables. When dealing with multiple design variables, the well-known univariate statistical analysis is no longer satisfactory, and several issues challenge the practitioner. How should one incorporate the dependence between variables? How should a multivariate return period be defined and applied in order to yield a proper design event? In this study an overview of the state of the art for estimating multivariate design events is given and the different approaches are compared. The construction of multivariate distribution functions is done through the use of copulas, given their practicality in multivariate frequency analyses and their ability to model numerous types of dependence structures in a flexible way. A synthetic case study is used to generate a large data set of simulated discharges that is used for illustrating the effect of different modelling choices on the design events. Based on different uni- and multivariate approaches, the design hydrograph characteristics of a 3-D phenomenon composed of annual maximum peak discharge, its volume, and duration are derived. These approaches are based on regression analysis, bivariate conditional distributions, bivariate joint distributions and Kendall distribution functions, highlighting theoretical and practical issues of multivariate frequency analysis. Also an ensemble-based approach is presented. For a given design return period, the approach chosen clearly affects the calculated design event, and much attention should be given to the choice of the approach used as this depends on the real-world problem at hand.

Combined Global Surface Summary of Day and European Climate Assessment and Dataset daily meteorological data sets (around 9000 stations) were used to build spatio‐temporal geostatistical models and predict daily air temperature at ground resolution of 1 km for the global land mass. Predictions in space and time were made for the mean, maximum, and minimum temperatures using spatio‐temporal regression‐kriging with a time series of Moderate Resolution Imaging Spectroradiometer (MODIS) 8 day images, topographic layers (digital elevation model and topographic wetness index), and a geometric temperature trend as covariates. The accuracy of predicting daily temperatures was assessed using leave‐one‐out cross validation. To account for geographical point clustering of station data and get a more representative cross‐validation accuracy, predicted values were aggregated to blocks of land of size 500×500 km. Results show that the average accuracy for predicting mean, maximum, and minimum daily temperatures is root‐mean‐square error (RMSE) =±2°C for areas densely covered with stations and between ±2°C and ±4°C for areas with lower station density. The lowest prediction accuracy was observed at high altitudes (>1000 m) and in Antarctica with an RMSE around 6°C. The model and predictions were built for the year 2011 only, but the same methodology could be extended for the whole range of the MODIS land surface temperature images (2001 to today), i.e., to produce global archives of daily temperatures (a next‐generation http://WorldClim.org repository) and to feed various global environmental models.

Studying phenomena that follow a skewed distribution and entail an extremal behaviour is important in many disciplines. How to describe and model the dependence of skewed spatial random fields is still a challenging question. Especially when one is interested in interpolating a sample from a spatial random field that exhibits extreme events, classical geostatistical tools like kriging relying on the Gaussian assumption fail in reproducing the extremes. Originating from the multivariate extreme value theory partly driven by financial mathematics, copulas emerged in recent years being capable of describing different kinds of joint tail behaviours beyond the Gaussian realm. In this paper spatial vine copulas are introduced that are parametrized by distance and allow to include extremal behaviour of a spatial random field. The newly introduced distributions are fitted to the widely studied emergency and routine scenario data set from the spatial interpolation comparison 2004 (SIC2004). The presented spatial vine copula ranks within the top 5 approaches and is superior to all approaches in terms of the mean absolute error