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Fig. 2 In-house comparison of Shannon's entropy distribution for the QuBiLS-MAS 2D-Indices considering the non-stochastic, simple stochastic, double-stochastic and mutual probability matrix formalisms.

The mutual probability matrices are obtained dividing each entry between the total sum of their elements, in this way, symmetrical matrices where the total sum is equal to 1 are obtained.

Additionally, the kth total (or local-fragment) mutual probability (MP) graph-theoretical electronic-density matrix (MP-GEDM, (_{(F)} {boldsymbol{mathcal{M}}}_{mp}^{k})) and edge-adjacency matrix (MP-EAM, (_{(F)} {boldsymbol{mathcal{E}}}_{{varvec{mp}}}^{k})) are introduced.

The superior performance of the mutual probability formalism with respect to the other three matrix transformations justifies the theoretical contribution of this scheme in the computation of the QuBiLS-MAS 2D-MDs.

Figure 2 shows similar entropy distributions for the non-, double- and simple-stochastic matrix approaches, while the best behavior is obtained with the mutual probability approach.

No statistical probability matrix would have predicted that to occur.

For the Markov model, we generated a transfer area matrix and transfer probability matrix for 2005 2014 (Supplementary Table 15).

Basic knowledge of probability, matrix algebra and mathematical statistics.

As shown in Fig. 5, the transition probability matrix constructed at 48 ns lag time fulfills the Markovian behavior criterion and this transition probability matrix was chosen for the subsequent analyses.

The eigenvector corresponding to the first relaxation mode of the selected transition probability matrix is depicted in Fig. 6A.

The 48 ns lag-time (highlighted with red line) was used for constructing transition probability matrix and subsequent analysis.