Historia, 50 vs Ahora [Pbm09]

In this section, we present the results for the Iris, Sonar, and XOR. These included the lower-order signatures with varying levels of noise and sizes of N. The results of the con- nection densities can be found in the appendix (see Appendix A.1.1); likewise, the results of the effects of the sizes of N has been moved to the appendix (see Appendix A.1.2).

(a) γ = 0.1 (b) γ = 0.2 (c) γ = 0.3

(d) γ = 0.4 (e) γ = 0.5 (f) γ = 0.6

Figure 5.9: An illustration showing the average transfer function likelihood for the noise levels within the range of γ ∈ {0.1..0.6} on the Iris dataset.

(a) γ = 0.7 (b) γ = 0.8 (c) γ = 0.9

(d) γ = 1.0

Figure 5.10: An illustration showing the average transfer function likelihood for the noise levels within the range of γ ∈ {0.7..1.0} on the Iris dataset.

Fig. 5.9 and Fig. 5.10 shows the average transfer function likelihood for the levels of noise γ ∈ {0.1...1.0}. It was apparent that some of the transfer functions were more likely in comparison to other possibilities. In particular, the combination of the (7, 4) - which according to the indices of the transfer functions refers to the combination of a max activation function (7), and a hyperbolic tangent output function (4). This has been found to work as both a filter mechanism that extracts and normalizes a given feature of the problem as discussed in the earlier chapter.

(a) γ = 0.1 (b) γ = 0.2 (c) γ = 0.3

(d) γ = 0.4 (e) γ = 0.5 (f) γ = 0.6

Figure 5.11: The average transfer function likelihood after thresholding for the noise

(a) γ = 0.7 (b) γ = 0.8 (c) γ = 0.9

(d) γ = 1.0

Figure 5.12: The average transfer function likelihood after thresholding for the noise

levels within the range of γ ∈ {0.7..1.0} on the Iris dataset.

Fig. 5.11 and Fig. 5.12 shows the results after thresholding was applied to the signatures in Fig. 5.9 and Fig. 5.10. After thresholding, more consistent patterns of the transfer function likelihoods were apparent.

(a) γ = 0.1 (b) γ = 0.2 (c) γ = 0.3

(d) γ = 0.3 (e) γ = 0.5 (f) γ = 0.6

Figure 5.13: The average transfer function likelihood for the noise levels within the range of γ ∈ {0.1..0.6} on the Sonar dataset.

(a) γ = 0.7 (b) γ = 0.8 (c) γ = 0.9

(d) γ = 1.0

Figure 5.14: The average transfer function likelihood for the noise levels within the range of γ ∈ {0.7..1.0} on the Sonar dataset.

Similarly, the transfer functions likelihoods for the Sonar dataset shown in Fig.5.13 and Fig. 5.14 have some apparent patterns, however not completely obvious.

(a) γ = 0.1 (b) γ = 0.2 (c) γ = 0.3

(d) γ = 0.4 (e) γ = 0.5 (f) γ = 0.6

Figure 5.15: The average transfer function likelihood after thresholding for the noise

(a) γ = 0.7 (b) γ = 0.8 (c) γ = 0.9

(d) γ = 1.0

Figure 5.16: The average transfer function likelihood after thresholding for the noise

levels within the range of γ ∈ {0.6..1.0} on the Sonar dataset.

Interestingly, as shown by the more obvious and consistent patterns of likely transfer functions for the Sonar with increasing levels of noise (see Fig. 5.15 & Fig. 5.16); one of the likely transfer functions is the combination of a standard deviation activation function and a hyperbolic tangent output function. This is one of the transfer function that has been found to be used as a mechanism for relaying a normalized average of features as highlighted in the earlier chapter. This was considered to be particularly useful for the diabetes problem as well.

(a) γ = 0.1 (b) γ = 0.2 (c) γ = 0.3

(d) γ = 0.4 (e) γ = 0.5 (f) γ = 0.6

Figure 5.17: The average transfer function likelihood for the noise levels within the range of γ ∈ {0.1..0.6} on the XOR dataset.

(a) γ = 0.7 (b) γ = 0.8 (c) γ = 0.9

(d) γ = 1.0

Figure 5.18: The average transfer function likelihood for the noise levels within the range of γ ∈ {0.6..1.0} on the XOR dataset.

Regarding the XOR dataset, Fig. 5.17 & Fig. 5.18 show its result of transfer function likelihood as the level of noise is increased. It also shows consistent patterns as with the later results of the other datasets.

(a) γ = 0.1 (b) γ = 0.2 (c) γ = 0.3

(d) γ = 0.4 (e) γ = 0.5 (f) γ = 0.6

Figure 5.19: The average transfer function likelihood after thresholding for the noise

(a) γ = 0.7 (b) γ = 0.8 (c) γ = 0.9

(d) γ = 1.0

Figure 5.20: The average transfer function likelihood after thresholding for the noise

levels within the range of γ ∈ {0.6..1.0} on the XOR dataset.

After thresholding, the figures (i.e. Fig. 5.17 & Fig. 5.18), the resulting transfer func- tion likelihoods are as in Fig. 5.19 & Fig. 5.20. This made the most likely transfer func- tions more obvious, and also showed the consistency of the signatures as the level of noise was increased. One observation, as with the later datasets is that some transfer functions likelihoods faded out from the resulting transfer function likelihoods post-thresholding as the noise level was increased. One example is (4, 3), which appeared at noise levels of γ ∈ {0.2, 0.3, 0.4, 0.5}. Another observation is the that like the other datasets, there was a consistency of a significant number of the most likely transfer functions as the noise level was increased. Additionally, another interesting the most likely transfer functions for the different datasets appeared to be unique. This was one of the motivations for studying the consistency and discriminatory ability of problem signatures.

(a) (b)

(c) (d)

Figure 5.21: The correlation between the connection densities of the problems with and without thresholding. (a) and (b) represent results without thresholding, while (c) and (d) are results with thresholding.

The results of the correlation between the various types of signatures presented were used to answer the question of whether lower-order problem signatures were discriminatory by nature.

In terms of the correlation between the problems, it was apparent that the comparison of the connection densities between problems showed weak correlations for both the cor- relation measures used. The results in Fig. 5.21 (a) and (b) illustrates the correlation of the problems connection densities without thresholding as the level of noise was increased, i.e. γ ∈ {0.1...0.9}. While the Figures Fig. 5.21 (c) and Fig. 5.21 (d) presents the results after thresholding as the noise was increased over the same range. This was intended to explore

the possibilities of the correlations differing in both cases. In general, the results show a weak correlation between problems in terms of their connection densities.

(a) (b)

(c) (d)

Figure 5.22: Correlation between the transfer function likelihoods of the problems both with and without thresholding. (a) and (b) are Pearson correlations coefficients, (d) and (c) are Spearman correlation coefficients.

The results of the correlation between the transfer function likelihoods of the problem both with and without thresholding are highlighted in Fig. 5.22. Interestingly, these showed even weaker correlations of between 0.1 and 0.3, in general. Another observation is that the correlations results for the connection densities seemed to be relatively smoother as the level of noise was increased, compared to the transfer function likelihood.

(a) (b)

(c) (d)

Figure 5.23: Correlation between the association associated error of the problems both with and without thresholding. (a) and (b) are Pearson correlations coefficients, (d) and (c) are Spearman correlation coefficients.

Similar results were also observed for the correlation of the associated error of the trans- fer functions between the problems. It also showed what was within the range that indicates weak or no correlation between the problems associated error. Thus, further suggesting that the lower-order signatures were distinct.

(a) (b)

(c) (d)

Figure 5.24: The correlation between the connection densities of the problems both with and without thresholding as the size of N is increased {1..4}.

The results in Fig. 5.24 shows the correlation between the connection densities of the problem the size of N was increased. As expressed earlier, N represents the number of elite models to be sampled from during the extraction process. Fig. 5.24 (a) and (b) show the Spearman and Pearson correlations of the connection densities prior to thresholding. While Fig. 5.24 (c) and (d) show the Spearman and Pearson correlations of the connection densities after thresholding. Interestingly, relative to the results of increasing the level of noise γ, the results of the correlations showed much weaker correlations. Specifically, while the results of the condition of increasing noise seemed to increase gradually to a correlation of 0.5, which is still regarded as a weak correlation, the results of increasing the size of N had a maximum correlation of 0.3.

(a) (b)

(c) (d)

Figure 5.25: Correlation between the transfer function likelihoods of the problems both with and without thresholding as the size of N is increased {1..4}. (a) and (b) are Pearson correlations coefficients, (d) and (c) are Spearman correlation coefficients.

In terms of the correlation of the transfer function likelihoods, the correlations results between the problems are highlighted in Fig. 5.25. Similarly, the results also portrayed suggest that there are either no correlations or very weak correlations between the prob- lems.

(a) (b)

(c) (d)

Figure 5.26: Correlation between the transfer function association of the problems both with and without thresholding as the size of N increases {1..4}. (a) and (b) are Pearson correlations coefficients, (d) and (c) are Spearman correlation coefficients.

Similarly, the results of the correlations for the associated error of the transfer functions between the problems as shown in Fig. 5.25, also suggested that they were very weakly correlated.