# Towards Hyper-Dimensional Variography Using the Product-Sum Covariance Model
## 2 Theory
### Modeling of the Hyper-Dimensional Variogram Based on the Product-Sum Model  
Our extension of the product-sum model is based on three previously unexplored properties of Equation (2) that allow for extrapolating the approach to hyper-dimensional cases encountered in environmental science and ecology. First, we notice that the validity of the product-sum covariance model using basic COV\({}^{(1)}\) class of covariances \(C_{s}\) and \(C_{t}\) is not dependent on the dimensionality of the basic covariance models, and the same applies to corresponding variograms (Equation (3)). The minimum and necessary conditions to assure validity of the product-sum model are: (1) that basic COV\({}^{(1)}\) building blocks are valid models, and (2) that constants \(a_{1}\)-\(a_{3}\) are subjected to constraints mentioned earlier in Equation (2) ([PERSON] et al., 2001). Thus, the product-sum model validity holds for any basic covariance(s) dimensionality (equivalent to \(C_{s}\) and \(C_{t}\) in Equation (2)), as long as they represent a valid model (for validity criteria, see [PERSON] and [PERSON], 2012). Subsequently, a COV\({}^{(2)}\) class of covariance modeled in the first step starting from basic COV\({}^{(1)}\) components can be used as a basic covariance in the subsequent steps, thus increasing the dimensionality of the resulting covariance. Second, Equation (2) is symmetric in terms of \(C_{s}\) and \(C_{t}\) in the sense that if these terms exchange places in the equation, it will still converge to the same expression given that the constants a\({}_{1}\)-a\({}_{3}\) are data-driven, (note that constants a\({}_{1}\)-a\({}_{3}\) are not directly modeled, but rather implicitly through global and partial sills estimated from the data; please see Equations (4)-(8) in [PERSON] et al. (2001) for further clarifications). We will later show that fitting the model to data can be done sequentially, mimicking the approach from the original paper by [PERSON] et al. (2001), or, as we propose, all at once, thus avoiding the need to define tolerances. Third, Equation (4) can be rewritten as:  
\[k\ =\ \frac{sill(s)+sill(t)-sill(s,t)}{sill(s)\ sill(t)} \tag{6}\]  
Based on the argument from analogy (S  
This approach, analogous to spatio-temporal modeling in Equation (2), starts by selecting the first pair of independent one-dimensional variograms to be combined into a product-sum model. In the case of separately modeling variability in horizontal and vertical directions and time, it requires first creating the product-sum model for the first arbitrarily selected pair of dimensions; for example, horizontal and vertical directions:  
\[C_{h,v}(h_{h}J_{v})^{(2)}=a_{1}C_{h}(h_{h})^{(1)}C_{v}(h_{v})^{(1)}+a_{2}C_{h}( h_{h})^{(1)}+a_{3}C_{v}(h_{v})^{(1)} \tag{8}\]  
Second, \(C_{h,v}(h_{h}J_{v})^{(2)}\) and \(C_{t}(h_{t})^{(1)}\) are combined into a final product-sum model:  
\[C_{h,v,t}(h_{h}J_{v})^{(3)}=a_{4}C_{h,v}(h_{h}J_{v})^{(2)}C_{t}(h_{t})^{(1)}+a_ {5}C_{h,v}(h_{h}J_{v})^{(2)}+a_{6}C_{t}(h_{t})^{(1)} \tag{9}\]  
After substituting Equation (8) into Equation (9) it yields:  
\[ \begin{split} C_{h,v,t}(h_{h}J_{v})h_{t})^{(3)}&= \boldsymbol{a_{1}a_{4}C_{h}(h_{h})^{(1)}C_{v}(h_{v})^{(1)}C_{t}(h_{t})^{(1)}+ \boldsymbol{a_{2}a_{4}C_{h}(h_{h})^{(1)}C_{t}(h_{t})^{(1)}+\boldsymbol{a_{3}a _{4}C_{v}(h_{v})^{(1)}C_{t}(h_{t})^{(1)}}+\boldsymbol{a_{3}a_{4}C_{v}(h_{v})^{ (1)}C_{t}(h_{t})^{(1)}}+\boldsymbol{+a_{1}a_{5}C_{h}(h_{h})^{(1)}C_{t}(h_{t}) ^{(1)}}+\boldsymbol{+a_{2}a_{5}C_{h}(h_{h}J_{v})^{(1)}+\boldsymbol{a_{3}a_{5}C _{v}(h_{v})^{(1)}}+\boldsymbol{a_{6}C_{t}(h_{t})^{(1)}}}}\end{split}  \tag{10}\]  
Two important conclusions follow from Equation (10). First, any \(n\)-dimensional variogram (covariance) can be broken down to one-dimensional components following \(n\)\(-\)\(1\) modeling steps, each comprising the modeling of one product-sum variogram of the lower dimensionality. Second, Equation (10) shows that the order of modeling lower-dimensional variograms is not important, as it converges to the same model.