**3. Results and discussion**

*Deep Learning Applications*

predicted value is generated.

**2.4 ANN's parameters**

remaining 25%, to validate the model (20) [13].

*RMSE*

with 16 GB of RAM was used to develop the models [13].

**2.6 Computer equipment and software**

from which could be overtrained.

**2.5 Adjustments parameters**

deviations (IPD) is also used.

methodology is that it based on the trial-error procedure [13] to find the optimal combination of parameters for prediction. Once the database is presented to the input layer, the training can start, the data are propagated to the first intermediate layer and the information is treated by the propagation function (Eq. (2)) to obtain a single value (*S*i), being: *x*f the input data (in the input neuron *f*), *w*fi the weight (importance among the neurons *f* and *i*) and *b*i the bias value associated with the intermediate neuron *i*. [13]. The single response is processed by the activation function (Eq. (3)) being: *S*res the output value of the neuron [13]. This process is repeated throughout all the neurons in the intermediate and output layer where the

> *N i fi f i f S wx b* = = + ∑1

> > *<sup>i</sup> res <sup>S</sup> S e* <sup>−</sup> <sup>=</sup> <sup>+</sup> 1

The authors [13] used a total of 80 cases to develop different prediction models

The learning rate and momentum values were set at 0.7 and 0.8, respectively. The models were developed at different training cycles in order to locate the point

The results were analyzed by the authors [13] using different statistics to deter-

( )

*pred experimental <sup>j</sup> y y*

*n*

*y y*

<sup>=</sup>

The input variables, necessary to determine the desired variables, were obtained from the Sigma Aldrich and Chemdraw Professional 15 trial (PerkinElmer) [13]. Microsoft Excel Professional Plus 2013 (Microsoft) was used for RS modeling, and the software EasyNN plus v14.0d (Neural Planner Software Ltd.) was used to ANN modeling [13]. A computer server with an Intel® processor Core™ i7 processor

<sup>1</sup> ·100 *<sup>n</sup> pred experimental*

<sup>−</sup>

*experimental*

*n*

2

<sup>=</sup> <sup>−</sup> <sup>=</sup> <sup>∑</sup> (4)

mean square error (RMSE) (Eq. (4)) or the average absolute percentage deviation (AAPD) (Eq. (5)) for the training and the validation phases. Individual percentage

> 1 *n*

mine the adjustment power, such as the coefficient of determination (R2

*j*

=

*<sup>y</sup> AAPD*

∑

(RS and ANN). In this case, the database was divided into two groups. A first group, with 75% of the cases (60), to train the model and a second group, with the

<sup>1</sup> (3)

(2)

), the root

(5)

**90**

The adjustments for the models developed [13] are shown in **Table 1**. It can be seen heterogeneous results for the surface and neural models.

Response surface models present good determination coefficients in the training phase, varying between the value obtained for the density model (0.994) and the value obtained for the kinematic viscosity model (0.906). These good values contrast with the value obtained for the surface tension model which reports a low determination coefficient value (0.505).

For the first three models (density, speed of sound and kinematic viscosity) the values of determination coefficient obtained in the validation phase are similar (with a minimal descent to the obtained R2 values in the training phase) varying between the value obtained for the density model (0.985) and the R2 value obtained for the kinematic viscosity model (0.885). The response surface model, with the worst-performing behavior for the training phase, the model developed to predict surface tension, showed, for the validation phase, a determination coefficient of 0.503, similar to that obtained in the training phase (0.505).

Regarding the root mean square error values obtained by the response surface models developed by Astray & Mejuto [13], it can be seen that the density model present an RMSE value around 0.001 g·cm−3, in both phases, the speed of sound models around 7.1837 m·s−1 and 6.3226 m·s−1 in training and validation phase, respectively. The model developed to predict kinematic viscosity presents an RMSE value around 0.1003 mm<sup>2</sup> ·s−1 for the training phase and 0.0569 mm2 ·s−1 for validation phase, and the worst model developed, the surface tension model, 8.3304 mN·m−1 and 8.1307 mN·m−1, for training and validation phase, respectively. The size of these errors can best be understood if they are given in terms of average absolute percentage deviation. The AAPD values reported for each phase are very similar. In this case, the errors obtained for each model (for all data) were: 0.08%, 0.31%,


#### **Table 1.**

*Adjustments for training and validation phase for the RS and ANN models selected by the authors [13]. Determination coefficient (R2 ) and root mean square error (RMSE) for the models developed by surface (RS) and neural models (ANN). The subscript ρ corresponds to the variable density, u to the speed of sound, ν to the kinematic viscosity and σ is the surface tension. Table adapted from data reported by Astray & Mejuto [13].*

5.18% and 14.73%, for density, speed of sound, kinematic viscosity and surface tension model, respectively. It can be seen how the AAPD value for the density and speed of sound prediction models are very low, the error of the kinematic viscosity model presents an error of 5.18% that can be considered feasible. In these cases, the error that is not acceptable is the one reported by the surface tension model (14.73%) since it is clearly much higher than the rest, and above the 10% which is considered, in our laboratory, as an acceptable error.

With all this, it can be said that the models designed to determine the density, the speed of sound and the kinematic viscosity are useful models for the prediction of these properties. The model to predict the surface tension should not be used due to its high APPD.

The adjustments for the ANN models developed [13] can be shown in **Table 1**. ANN models were developed based on the trial-error method to obtain the best models for each predict output variable (more than 400 neural networks were developed) [13]. All models developed by the authors [13] presented a different topology: i) 3–7-1 for the density model, ii) 3–5-1 for the speed of sound model, iii) 3–6-1 to the kinematic viscosity model and iv) 3–1-1 for the surface tension model. Thus, each model presents, in the input layer, three variables: concentration, number of carbons and molecular weight and intermediate layer of each model varies from a single neuron, to predict the surface tension, to seven in the density model, in addition to that, each selected model has a different number of training cycles [13].

It can be observed (**Table 1**) that, in general, the ANN provided by the authors [13] adjust, properly, the desired variables, both in the training and in the validation phase. The model to predict the density value is the model with the best adjustments, in fact, and take into account the adjustments in terms of determination coefficient and root mean square error, this model presents values of 0.999 and 0.0004 g·cm−3, respectively, for the training phase and values of 0.999 and 0.0003 g·cm−3, respectively, for validation phase. Once again, as was RS models case, the model to predict the density is the model with the best adjustments, in fact, the AAPD values reported for both phases were 0.02%.

The behavior of the rest of the models follows the pattern of the RS models, that is, the models to predict the speed of sound and the kinematic viscosity are, in this order, the second and the third-best model [13].

The model destined to predict the speed of sound presents adjustments, in terms of coefficient of determination, very close to the model destined to predict density (0.998 in both phases), presenting relatively low RMSE values (1.9393 m·s−1 and 1.7093 m·s−1).

The kinematic viscosity model has slightly lower adjustment than the previous two models. In this sense, and always in terms of the determination coefficient, the value for the training phase remains similar to the two previous models, however, for the validation phase, this value falls slightly to 0.994. Even so, the model seems to be predicting the kinematic viscosity values correctly, especially if it be taking into account the low RMSE values (0.0108 mm2 ·s−1 and 0.0104 mm2 ·s−1, for training and validation, respectively) [13].

Finally, the worst model developed using artificial neural networks is the model designed to determine surface tension [13]. It can be seen in **Table 1** show the values obtained fall significantly, in fact, the determination coefficient value falls to 0.449 and 0.457 for the training and validation phase, respectively. It seems clear that this low value of determination coefficient indicates the impossibility of the model to make correct predictions. This fact is demonstrated with the high RMSE values obtained for the training and validation phase (9.6859 mN·m−1 and 9.6827 mN·m−1, respectively).

**93**

*Modeling the Behavior of Amphiphilic Aqueous Solutions*

0.02%, 0.10%, 0.62% and 18.13%, respectively.

model and around 89.25% for ANNρ model [13].

RSu model and 63.30% for the ANNu model [13].

the training and the validation phase, showed an R<sup>2</sup>

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

As stated above, the size of the errors made by the different ANN models can best be understood in terms of AAPD. In this case, the errors obtained (for all data) by density, speed of sound, kinematic viscosity and surface tension model were:

In the same way that occurs with the surface models, the ANN surface tension model should not be used to predict surface tension (APPD above 10%). The other

Once the models have been analyzed separately, it is necessary to make a

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>

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],

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

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

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,

value of

upper than 0.993. According

present good results, in fact, the RSu model presents, for all data, an R<sup>2</sup>

ties of surfactants aqueous solutions (at least with the surfactants studied).

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 proper-

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

three models can be used for prediction.

comparison between them.

*Deep Learning Applications*

to its high APPD.

training cycles [13].

1.7093 m·s−1).

5.18% and 14.73%, for density, speed of sound, kinematic viscosity and surface tension model, respectively. It can be seen how the AAPD value for the density and speed of sound prediction models are very low, the error of the kinematic viscosity model presents an error of 5.18% that can be considered feasible. In these cases, the error that is not acceptable is the one reported by the surface tension model (14.73%) since it is clearly much higher than the rest, and above the 10% which is

With all this, it can be said that the models designed to determine the density, the speed of sound and the kinematic viscosity are useful models for the prediction of these properties. The model to predict the surface tension should not be used due

The adjustments for the ANN models developed [13] can be shown in **Table 1**. ANN models were developed based on the trial-error method to obtain the best models for each predict output variable (more than 400 neural networks were developed) [13]. All models developed by the authors [13] presented a different topology: i) 3–7-1 for the density model, ii) 3–5-1 for the speed of sound model, iii) 3–6-1 to the kinematic viscosity model and iv) 3–1-1 for the surface tension model. Thus, each model presents, in the input layer, three variables: concentration, number of carbons and molecular weight and intermediate layer of each model varies from a single neuron, to predict the surface tension, to seven in the density model, in addition to that, each selected model has a different number of

It can be observed (**Table 1**) that, in general, the ANN provided by the authors [13] adjust, properly, the desired variables, both in the training and in the validation phase. The model to predict the density value is the model with the best adjustments, in fact, and take into account the adjustments in terms of determination coefficient and root mean square error, this model presents values of 0.999 and 0.0004 g·cm−3, respectively, for the training phase and values of 0.999 and 0.0003 g·cm−3, respectively, for validation phase. Once again, as was RS models case, the model to predict the density is the model with the best adjustments, in

The behavior of the rest of the models follows the pattern of the RS models, that is, the models to predict the speed of sound and the kinematic viscosity are, in this

The model destined to predict the speed of sound presents adjustments, in terms of coefficient of determination, very close to the model destined to predict density (0.998 in both phases), presenting relatively low RMSE values (1.9393 m·s−1 and

The kinematic viscosity model has slightly lower adjustment than the previous two models. In this sense, and always in terms of the determination coefficient, the value for the training phase remains similar to the two previous models, however, for the validation phase, this value falls slightly to 0.994. Even so, the model seems to be predicting the kinematic viscosity values correctly, especially if it be taking

Finally, the worst model developed using artificial neural networks is the model designed to determine surface tension [13]. It can be seen in **Table 1** show the values obtained fall significantly, in fact, the determination coefficient value falls to 0.449 and 0.457 for the training and validation phase, respectively. It seems clear that this low value of determination coefficient indicates the impossibility of the model to make correct predictions. This fact is demonstrated with the high RMSE values obtained for the training and validation phase (9.6859 mN·m−1 and 9.6827 mN·m−1,

·s−1 and 0.0104 mm2

·s−1, for training

fact, the AAPD values reported for both phases were 0.02%.

order, the second and the third-best model [13].

into account the low RMSE values (0.0108 mm2

and validation, respectively) [13].

considered, in our laboratory, as an acceptable error.

**92**

respectively).

As stated above, the size of the errors made by the different ANN models can best be understood in terms of AAPD. In this case, the errors obtained (for all data) by density, speed of sound, kinematic viscosity and surface tension model were: 0.02%, 0.10%, 0.62% and 18.13%, respectively.

In the same way that occurs with the surface models, the ANN surface tension model should not be used to predict surface tension (APPD above 10%). The other three models can be used for prediction.
