## 3 Environmental Applications of the Tool and Results
### Importance and Roles of the Causal Variables (Attribution)  
We now describe an alternative way of understanding the role of the individual causal variables on the target variable, which also allows simulations of the latter's behaviour if the former have different values and trends. These simulations have been carried out for climate attribution studies, and we will use one of them here to exemplify the method; see [6] for further details. Figure 2: Modelling migration rates (black line) from ten countries of Sahel to Italy in the period 1995–2009 by NN models (red lines) and ensemble means (blue line) (from [7]). A very important problem in recent climate research has been to understand what has caused the rapid rise in the global mean temperature over the last 60 years. To do this, dynamical models (global climate models--GCMs) have been used, which have shown that without the increase in CO\({}_{2}\) and other greenhouse gases (GHGs), recent global warming would not have occurred. But, given the complexity of these models, it was important to analyse this problem with very different (data-driven) models, such as neural networks and Granger causality models, to test the robustness of their results (see [21; 22] and references therein). As far as NN models are concerned, the concept with which we applied our tool to attribution studies is very simple. At the end of training, our networks found the best weights to reconstruct the trend of the target (over time) according to the input values that occurred in reality. Once these values have been fixed, we have thus found a 'law' that reconstructs the links between the inputs and the target, a law that we can consider valid in general. We can therefore apply it to values of the inputs other than those observed in historical reality, thus simulating different situations that could have occurred and obtaining the related values of the output variable. For instance, if we input to the network with fixed weights a constant or stationary value for one variable and real observed values for all the others, we will see the effect of real variations in the first variable on the target variable and will thus be able to understand its role in the evolution of the target (see the following example). In the climate attribution example given in [6], the target was the global mean temperature, and the causal variables (expressed in terms of radiative forcings) were external natural influences (Sun and volcanoes) and anthropogenic ones (GHGs, black carbon (BC) and sulphate aerosols). In the latter forcings, GHGs and BC have a warming effect, while sulphates have a cooling one. Each of these causal variables has had a certain trend over time since 1850, and we estimated this from observations. But what would have happened if one of these trends had been different, e.g., constant or stationary? A first answer is in Figure 3, taken from [6]. Figure 3: Warming and cooling roles of the distinct anthropogenic forcings by an NN attribution experiment. Black line = observed T; red line = reconstructed T by all inputs with actual values; blue, light blue and green lines = attribution runs with constant forcing(s) (from [6]). In terms of the ensemble means of our NN runs, Figure 3 shows the very good performance of our NN models in reconstructing the real behaviour of temperature when the real trends of all forcings were considered (see how the black and red lines are close). The light blue line shows the further increase in temperature that would have occurred over the last century if sulphates had remained constant at their 1850 value. The blue and green lines come from runs where constant values for the inputs of GHGs and GHGs + BC radiative forcings, respectively, were propagated along the network with fixed weights. This last result shows that without the increase in anthropogenic warming factors, the recent global warming would not have occurred, so humans are responsible for this warming. This result corroborates the analogous one coming from GCMs.