Brotherhoods of relative entropy, boolean operators and principle component analysis for a gradient of living communities

It is well known that gradient values of the living communities are the most important and valuable data to examine and explain community-environmental relationships. On this context, the purpose of the present study is to find an answer to the question that is how to generate a gradient of living communities along the PCA (Principal Component Analysis) ordination axes by using relative entropy (D[p(x) | |q(x)]), a measure of the non-symmetric difference between two probability distributions, p and q. A hypothetic data matrix composed of 3 plant communities was subjected to relative entropy. Relative entropy could not be calculated from original data since the hypothetic data matrix includes zero values. The data matrix was, therefore, re-arranged by increasing the marginal entropy of each of the communities. Thus shaded relative entropies (D？[p(x) || q(x)]) among the communities were successfully calculated. Next, the locations of the resulted matrix in which each community shares a cell with itself were analyzed using Boolean Operators and principle component analysis (PCA). Lastly, the last PCA was applied in order to determine the locations of the plant communities along the ordination axes. The results of such approach were compared to the results obtained from the original data by using cluster analysis. The results were found very similar as a result of this comparison and, it was formed an opinion to be able to successfully use the combinations of shaded relative entropy and Boolean operators for a gradient of the living communities along the PCA ordination axes