Two-way interaction of input variables in the sensitivity analysis of neural network models

The major shortcoming of artificial neural networks (ANN) is the difficulty of interpreting the knowledge gained by “black-box” type models. Several methods, commonly called sensitivity analysis, have been proposed to overcome this disadvantage. The relative importance of each input variable on the output can then be determined. One of the existing methods uses partial derivatives (PaD) to visualise the contribution of single variables. However, in ecology, relationships are the result of multivariate and non-linear conditions; phenomena are rarely due to a simple cause or to a unique perturbation. For these reasons, a modification of the PaD (PaD2) was implemented to analyse the contribution of all possible pair-wise combinations of input variables, taking into account the two-way interactions between variables.In the present study, the PaD2 was applied firstly on synthetically generated data in order to test the relevance of the results, with the value and the sign of the contribution being known. Secondly, the method was used on ecological data to predict the density of brown trout spawning redds using 10 habitat characteristics: wetted width, area with suitable spawning gravel for trout per linear meter of river, surface velocity, water gradient, flow/width, mean depth, standard deviation of depth, bottom velocity, standard deviation of bottom velocity and mean speed/mean depth. For both data matrices, a multilayer feedforward neural network with a backpropagation algorithm was used and the PaD2 was then applied to study the two-way interactions of input variables.The results from applying the PaD and PaD2 methods with the generated data were as expected. Using PaD2 on the ecological data, the most important contributions were found to be (1) the relationships between the area with suitable spawning gravel and the water gradient, (2) the relationships between the area with suitable spawning gravel and the bottom velocity and (3) the relationships between the area with suitable spawning gravel and the surface velocity. The contribution profiles of the most contributed paired variables were analysed. The patterns resulting from these interactions are clearly visualised with a 3D representation. From the contribution profile patterns, it was seen that the predicted density of redds closely corresponded to ecological reality. Moreover, the contributions of the variables, which were not significantly differentiated with the PaD were revealed with the PaD2.