# Towards Hyper-Dimensional Variography Using the Product-Sum Covariance Model
## 4 Conclusions  
This study presents an easy and practical way of modeling spatial variability in cases where hyper-dimensional covariance (variogram) structures are required to describe observed spatial variability. It could be especially useful in modeling anisotropic conditions with multiple axes of anisotropy, often encountered in environmental and ecological sciences, or in cases where multiple units have to be used to span dimensions of interest (e.g., space and time). The method is simpler than classical approaches to modeling anisotropy (see [PERSON] and [PERSON], (2016), for details) and requires fewer parameters to be estimated from the data. It represents an alternative to approaches that use time series and spatial analysis conjointly to represent variability at unsampled locations (e.g., [PERSON] et al. (2016)).  
We showed that this approach yields admissible hyper-dimensional covariance models using valid lower-dimensional covariance models as basic blocks. It allows a relatively easy modeling of full 3D-anisotropic space evolving in time, a task that so far has been challenging.  
We showed that the original sequential product-sum modeling method can yield results slightly different from the method we propose in this study due to the existence of tolerances and a potentially higher estimated nugget. We simplified the modeling approach and made it applicable to sparse data by proposing an \"all at once\" modeling approach.  
One of the deficiencies of this study is the lack of the specific-case analysis of the criteria for choosing a product-sum model and its higher dimensional derivates instead of other covariance models. While it should be the method of choice in modeling the data characterized by negative non-separability, it might not be suitable for data characterized by positive non-separability and pointwise non-separability (both positive and negative). For more information on this topic, please see [PERSON] et al. (2019), [PERSON] (2017) and [PERSON] et al. (2018).  
Many ecological processes are nonlinear and can result in biased predictions when using coarse-resolution datasets to drive ecological models. We presented a straightforward approach to estimate spatiotemporal variability and demonstrated its application to spatial scaling of ecological processes. This approach relaxes previous assumptions regarding isotropicity of variability for  
Figure 1: Application of kriged shortwave radiation (i.e., PAR) to estimate gross primary productivity (GPP) over a 2.5\({}^{\circ}\) longitude \(\times\) 2.5\({}^{\circ}\) latitude region (\(\sim\)220 km) in central Oklahoma. The black curve illustrates the nonlinear relationship between GPP and PAR from a canopy model. The histogram shows the distribution of PAR values within the region on May 30 of 2006 (16:00–6:30 local time). Due to the saturation of GPP at high PAR, estimates of GPP from the spatially-averaged PAR (red circle) are higher than the actual regional-scale GPP (blue line).  
hyper-dimensional analysis, which enables applications across a wider range of environmental and ecological problems.