Fig 1.
Summarization of the the existing tools and their limitations and a brief introduction to our approach.
Fig 2.
Results given for various combinations of αX and αY in associated datasets were better than Pearson correlation coefficient in the majority of cases.
9 heatmaps are shown in the figure, each representing one combination of αX and αY. Each heatmap, calculates BPA for different combinations of and
IX was kept equal to IY for the dataset to be associated and X was calculated by the addition of IX and NX and similarly Y. The value of BPA obtained is encoded in color, the darker being higher. Further, below every heatmap the corresponding scatterplot for X and Y is shown. The straight black line shows the relationship function. On the top of every heatmap the normalized mutual information content (MI), the Bayesian Probability of Association (BPA, for the correct choice of
and
which is equal to αX and αY respectively) and the Pearson correlation value (PC) has been displayed. On the top of every scatter plot, the values of the Spearman Rank Correlation (SR) and the Kendall Tau Rank Correlation (KT) is also shown. It can be observed that when
and
exceed αX and αY respectively the results start to approach 0.
Fig 3.
Results given for various combinations of αX and αY in unassociated datasets outperformed Pearson correlation coefficient in the majority of cases.
The figure is arranged in the exact same way as Fig 2. IX and IY were calculated independently for the dataset to be unassociated and X was calculated by the addition of IX and NX and similarly Y. The value of BPA obtained is encoded in color, the darker being higher. As in Fig 2, below every heatmap the corresponding scatterplot for X and Y is shown. The straight black line shows the relationship function. On the top of every heatmap the normalized mutual information content (MI), the Bayesian Probability of Association (BPA, for the correct choice of and
which is equal to αX and αY respectively) and the Pearson correlation value (PC) has been displayed. On the top of every scatter plot, the values of the Spearman Rank Correlation (SR) and the Kendall Tau Rank Correlation (KT) is also shown.
Fig 4.
Distribution of various associational measures over the range of 0 to 1 for all the synthetic datasets.
The distribution of results of all the five measures over all the synthetic datasets is shown. For every dataset, only the correct BPA was chosen. Further, absolute values were taken for Pearson Correlation Coefficient, Spearman Rank Correlation Coefficient and Kendall Tau Rank Correlation to bring all the measures on the same scale. Observe how a more negative value also suggests a stronger correlation.
Fig 5.
Association was detected in 19 out of 25 datasets with underlying association in the low Pearson correlation coefficient (<0.35) group: BPA for combinations of various and
are shown as a heatmap for 9 datasets.
Below every heatmap, a scatterplot of the actual data is shown. Also, a black line shows the relationship function that was calculated by curve fitting approaches. On the top left of the heatmap, the unique index of every dataset is presented. On the top of the heatmap Pearson correlation coefficient (PC) along with the p-value in brackets and the normalized mutual information content (MI) for the dataset is given. On the top of the scatterplot, the Spearman rank correlation coefficient (SR) and the Kendall tau rank correlation coefficient (KT) are given.
Fig 6.
Association was detected in 25 out of 25 datasets with underlying association in the moderate Pearson correlation coefficient (0.35–0.63) group: The figure is arranged in exactly the same way as Figs 5 and 9 datasets have been shown.
BPA for combinations of various and
are shown as a heatmap for 9 datasets. Below every heatmap, a scatterplot of the actual data is shown. Also, a black line shows the relationship function that was calculated by curve fitting approaches. On the top left of the heatmap, the unique index of every dataset is presented. On the top of the heatmap Pearson correlation coefficient (PC) along with the p-value in brackets and the normalized mutual information content (MI) for the dataset is given. On the top of the scatterplot, the Spearman rank correlation coefficient (SR) and the Kendall tau rank correlation coefficient (KT) are given.
Fig 7.
Association was detected in 25 out of 25 datasets with underlying association in the high Pearson correlation coefficient (>0.63) group: The figure is arranged in exactly the same way as Fig 5.
BPA for combinations of various and
are shown as a heatmap for 9 datasets. Below every heatmap, a scatterplot of the actual data is shown. Also, a black line shows the relationship function that was calculated by curve fitting approaches. On the top left of the heatmap, the unique index of every dataset is presented. On the top of the heatmap Pearson correlation coefficient (PC) along with the p-value in brackets and the normalized mutual information content (MI) for the dataset is given. On the top of the scatterplot, the Spearman rank correlation coefficient (SR) and the Kendall tau rank correlation coefficient (KT) are given.
Fig 8.
Results for 15 datasets with no underlying association: Data from different EEG recordings was taken and the channel (electrode position) was picked at random.
As two different recordings must not have any underlying association, BPA was checked for different combinations of and
and it was found that the results were pretty low. The figure is arranged in exactly the same way as Figs 5 and 9 datasets have been shown. Below every heatmap, a scatterplot of the actual data is shown. Also, a black line shows the relationship function that was calculated by curve fitting approaches. On the top left of the heatmap, the unique index of every dataset is presented. On the top of the heatmap Pearson correlation coefficient (PC) along with the p-value in brackets and the normalized mutual information content (MI) for the dataset is given. On the top of the scatterplot, the Spearman rank correlation coefficient (SR) and the Kendall tau rank correlation coefficient (KT) are given.
Fig 9.
The new approach could identify associations in cases of non-linearity: A), Five different relationship functions were taken and the results of the new approach were tested for various combinations of actual and estimated relationship functions. It was observed that the new approach could identify associations when the correct relationship function was estimated. B), Results were compared with the Pearson Correlation Coefficient (PC), the Spearman Rank Correlation Coefficient (SR), the Kendall Tau Rank Correlation Coefficient (KT) and the normalized mutual information content (MI). It was observed that while the three correlation coefficients could not identify associations in all the cases, Bayesian Probability of Association (BPA) could. Normalized Mutual Information Content was lower than the suggested approach but as seen it was higher than Pearson Correlation Coefficient.