Discrete Expressions for the Water Purification in a River, Based on Neural Network Calculations under Incomplete Data Set

Abstract

We propose four models for water purification in a river that flows in a big city area. All models are discrete expressions. They are named a simple expression-, plus dam's effects-, plus underground penetration-, and pair unknown coefficient-models. Using a neural network, we analyze changes of the water quality in a virtual river defined by the models. The objective is to test the simulation-ability of the discrete expressions, and to discuss the possibility of the inverse prediction of the model. If we could predict the inverse operation, we estimate the water purification of a river on use of a data set only.
The neural network is a useful tool to analyze non-linear phenomena. Discrete data set is required in the analysis, which includes observations and descriptor data. The neural network has ability to emulate the phenomena through learning iterations. Defects in the set suspend the iteration. The defects are classified into three cases. The first is that the existence of defect elements is certain but the value is unknown. The second is loss of whole data for a descriptor. The third is uncertainness for the existence of a descriptor. The first case is called "defect", and it was studied recently. However, the latter two were not. It is necessary to discuss the latter two for researches of environmental problems. At the same time, they are important for significance-tests of outputs of the neural networks. We researched neural-network functions on the latter two cases, which are for multi-regression analysis. The main point is to evaluate limits of the functions. The statistical characters of the error are not clear; therefore, to simplify the research, we consider no-error cases. Thus, we define a virtual river whose data are constructed by the uniform random numbers. The defect part is made in the data set on purpose. The researches show the following; 1. A neural network outputs reasonable water quality in a river, even if there is a defect descriptor. 2. The partial derivatives don't indicate accurate descriptor characters, when the target descriptor is not defect one. 3. The cause is another descriptor makes up for the defect; i.e., there are interactions among descriptors. 4. The largest change of whole partial derivatives indicates the complement descriptor. 5. It is possible to calculate characters of the defect descriptor approximately. 6. The possibility is enabling when outlines of phenomena are known. Thus, the discrete expression of a river makes changes of the water quality calculable in general cases, and by the inverse calculations, we can predict the descriptive equation of phenomena by using observation data only.