**3.1 Comparison of response surface and neural models**

Once the models have been analyzed separately, it is necessary to make a comparison between them.

As previously stated, the models to predict density are the best models according to the adjustments. This means that this model is useful to predict physical properties of surfactants aqueous solutions (at least with the surfactants studied).

On the one hand, although in general, the AAPD in the RSρ model is around 0.08%, according to the authors [13], some cases present a bigger IPD value (0.25–0.49%). Even so, these values are very low. Despite the good performance of this RS model, the ANN model seems to work a little better, improving each adjustment parameter (see **Table 1**). In fact, the AAPD values in the case of the ANNρ are below to the value obtained by the RSρ model. This improvement is observable in terms of RMSE being, for both phases together (with all the data), very significate (0.0012 g·cm−3 vs. 0.0004 g·cm−3) which represents an important improvement. For both models, it seems clear that the most important variable to determine the density is the concentration with an importance value around 59.00% for the ANN<sup>ρ</sup> model and around 89.25% for ANNρ model [13].

The second-best models, based on their adjustments, are the models to predict the speed of sound. The RSu and ANNu model developed by Astray & Mejuto [13], present good results, in fact, the RSu model presents, for all data, an R<sup>2</sup> value of 0.974 (with some cases presenting an IPD >1%), while for the ANNu presents a better value of determination coefficient (0.998), representing a slight improvement of 2.46%. The same behavior occurs regarding the RMSE, where the ANNu model improved this parameter by around 73.00%. The authors [13] reported that the ANN model has an AAPD value of around 0.10% and a highest IPD value around 0.45%. In both cases, very similar values are obtained for the training and validation phases. Concerning the importance of the variables, in the same way, that the models developed to predict the density, the most important variable to determine the speed of sound is the concentration with an importance value of 89.31% for the RSu model and 63.30% for the ANNu model [13].

The third-best model according to its results is the model to predict the kinematic viscosity. In this case, the behavior of the RSν model is slightly different from the one presented by the ANNν model. Thus, it is observed that the RSν model cannot predict with accuracy the kinematic viscosity and showed a slight dispersion of the data (predicted vs. experimental) that can be seen in the figure presented by Astray & Mejuto [13]. This dispersion is reflected in the adjustment parameters of the RSν model that presents, for all cases, a determination coefficient of 0.903 and an AAPD value of 5.18%). It is noteworthy that, according to Astray & Mejuto [13], there are more than 30 cases with an IPD value in the range of 5.34% to 26.51%. On the other hand, the model based on ANN, predicts with accuracy for the training and the validation phase, showed an R<sup>2</sup> upper than 0.993. According to the AAPD values provided, for all data, the AAPD value obtained by the ANN<sup>ν</sup> model (0.62%) compared to the AAPD value of the RSν model (5.18%) represents a decrease around 88.05% [13]. Regarding the input variables, in the same way,

that the models developed to predict the density and the speed of sound, the most important variable to determine the kinematic viscosity is the concentration [13].

Finally, from all models developed, both surface tension models were the worst models according to their adjustments. These models are the models with the highest dispersion (RSσ model can be seen in Figure 3.G [13]). The adjustment parameters for all data are low, in fact, the determination coefficient for the RSσ model is around 0.503 (being even lower in the case of ANNσ model −0.451-). Taking into account the AAPD values for all data, it can be seen how neither the RSσ model (14.73%) and the ANNσ model (18.13%) are capable to predict with accuracy the surface tension value. Regarding the input variables, once again, the most important variable to determine the surface tension is the concentration [13].

Due to these poor results, the authors [13] proposed an alternative ANN model (ANN'σ) to improve the prediction of surface tension. In this new ANN model, the input variables were increased using the predictions of the ANN models of density, speed of sound and kinematic viscosity, that is, this new ANN model presents an input layer with six variables. The ANN'σ model needs more cycles for its training (12800 cycles) compared to the three input variables ANN model (1200 cycles). It can be shown a notable improvement in the adjustments for the training and the validation phases. The determination coefficient increases to 0.985 for the training phase and 0.940 for the validation phase, while the RMSE values decrease from the 9.6851 mN·m−1 (for all case) to 1.9291 mN·m−1. In the same way, an improvement in the AAPD values is observed, which, in the new model, is below 3.86%. This new model takes the predicted speed of sound as the most important variable, unlike the previous ones, where the most important variable was focused on the concentration. The predicted speed of sound is followed by the predicted kinematic viscosity, the predicted density and the concentration (number of carbons and molecular weight present lower importance).

Given the results obtained by the surface models and the neural models [13], it can be concluded that the models developed to determine density, sound speed and kinematic viscosity are models suitable for their use in the laboratory due to the low APPD values that presented (between 0.02% and 5.18%, for all the data cases). Regarding the models for surface tension prediction, as previously mentioned, these cannot be used for laboratory use, because they present errors upper than 10%. The alternative ANN model developed by the authors [13], appears to offer acceptable results in terms of determination coefficient and AAPD value. This alternative model improves the original RS and ANN model.

All the models developed [13] can be improved in different ways. The response surface models could be improved by adapting the experimental cases to an experimental design before the experimental measurements, allowing on the one hand to save economic costs and time, and on the other, favoring the development of an RS model based on a precise experimental design. It would also be very convenient to develop a response surface model trying to find surfactants that allow the variables to vary constantly in a range. All these improvements could favor the improvement of the models destined to predict the density, the speed of the sound and the kinematic viscosity.

Likewise, and given the ANN model that uses six input variables [13], it would be interesting to develop an RS model that includes the predictions of density, speed of sound and kinematic viscosity as input variables of the model (although it would be necessary to see how to treat the different variation of the values within the range understudy).

Neural network models could be improved by including different input variables that are capable of better identification of the different surfactants. Another interesting approach could by the increase the database for their modeling.

**95**

*Modeling the Behavior of Amphiphilic Aqueous Solutions*

alternative to save money and time in the laboratory.

The development of models based on response surfaces and neural networks to predict different physical properties of surfactants aqueous solutions (i) density, ii) speed of sound, iii) kinematic viscosity and iv) surface tension) can be a good

In general terms, this kind of models can adjust, with accuracy, the density, the kinematic viscosity and the speed of sound with determination coefficient upper than 0.902 and lower APPD values than 5.20% (for all data). In contrast to these good adjustments, surface tension models do not work properly and presented (for all data) low determination coefficients (0.503 and 0.451 for RS and ANN model, respectively) and high APPD values (14.73% and 18.13% for RS and ANN model, respectively). It seems that this problem can be solved, in the case of models based on neural networks, with the inclusion of new variables from the predictions of the previous models. With this modification, the new neural model improves (for all data) each adjustment parameter (0.974 and 2.92% for determination coefficient

In conclusion, RS and ANN models can be powerful prediction tools for the properties (density, speed of sound, kinematic viscosity or surface tension) of surfactants aqueous solutions. These models could therefore facilitate daily laboratory work, saving time and money. However, it would be interesting to improve the models using other development alternatives or, even, improve these model using different approaches such as support vector machines or random forests,

Gonzalo Astray thanks to the University of Vigo for his contract supported by *"Programa de retención de talento investigador da Universidade de Vigo para o 2018"* budget application 0000 131H TAL 641. Cecilia Martínez-Castillo and Manuel Alonso-Ferrer thanks to the University of Vigo for her contract supported by FEADER 2018/002B project (Xunta de Galicia, Consellería de Medio Rural, Project "*Desarrollo de modelos de predicción de origen en vinos de denominaciones de origen* 

*DOI: http://dx.doi.org/10.5772/intechopen.95613*

and AAPD value, respectively).

among others.

*gallegas*").

**Acknowledgements**

**4. Conclusion**
