###### Abstract
The recycling sites denoted by \(s_{i}(x,y)\) (\(i=1,2,\ldots,q\)) can be rewritten in vector form as \(S(s_{1},s_{2},\cdots,s_{q})\). The return amount of each site, denoted by \((s_{i})\) (\(i=1,2,\ldots,q\)), can be rewritten in vector form as \(Z\big{(}Z(s_{1}),\,Z(s_{2}),\,\cdots,Z\big{(}s_{q}\big{)}\big{)}\) and considered as a random variable. Considering random disturbance, in a certain period, the return amounts of arbitrary \(n\) (\(n<q\)) recycling sites can be obtained by detection and denoted by \(Z(s_{1}),\,Z(s_{2}),\,\cdots,Z(s_{n})\) respectively. There are two challenges for researchers to deal with. _Challenge 1_: It is difficult to obtain a quantitative description for the spatial structure correlation between the given \(Z(s_{i})\) (\(i=1,2,\ldots,n\)) and the unknown \(\hat{Z}(s_{i}),\,(i=n+1,n+2,\ldots,q)\). _Challenge 2_: Even though the quantitative description of the spatial structure correlation is obtained, it is hard to predict the unknown \(\hat{Z}(s_{i})\) (\(i=n+1,n+2,\ldots,q\)) based on the given return amounts of \(n\) recycling sites, which will result in an accurate prediction of the return amounts of all the recycling sites in area \(\mathbf{D}\). Figure 2: A limited two-dimensional area D with q recycling sites. In order to solve the above two challenges, it is necessary to analyze the spatial structure of the return amounts between recycling sites under the conditions of spatial correlation and randomness and establish a spatial correlation random model. Firstly, correlation arises frequently in data observed within certain intervals of space, and the data indeed exhibit a significant amount of positive autocorrelation. The waste electrical and electronic equipment (WEEE) return amounts in every recycling sites are often spatially correlated because they are obtained in similar conditions during the recycling process and related to similar properties. Spatial correlation is different from temporal correlation, which is usually represented via time series models. In fact, spatial correlation models allow one to represent contiguity in space rather than in time. Therefore, it is vital for us to develop a quantitative mathematical description model of the spatial structure correlation between the given return amounts and the unknown return amounts. In the practical application of the space modeling method for spatial correlation, an important assumption is the second-order randomness, which means that the variables can be arbitrary values in a certain period. The second-order randomness means that the properties of the random function satisfied by the variables remain unchanged. The mean value of the random function is a constant. And the covariance between the random recoveries of any two nodes, only depends on the distance between them. The first scientific issue we need to handle is to prove that the spatial structure of the return amount of the recycling network follows the second-order randomness property. If the property is proved to be true, the quantitative mathematical description model of the spatial structure correlation could be developed. Secondly, the second scientific issue is to develop a mathematical prediction model to predict the unknown return amounts based on the given return amounts of n recycling sites. It is difficult for both academia and industry to predict the WEEE return due to the uncertainty of the return amount. The return estimation amounts are not exactly equal to the real amounts. There is a deviation between the estimation and the real amount, which leads to the inaccuracy prediction. The reason for this phenomenon is that predicting the return values only uses known value information on measured recycling sites considering the spatial correlation. Therefore, developing a mathematical prediction model and decrease this deviation to improve the precision of WEEE return amount prediction is another crucial scientific issue.