Geometric separability of mesoscale patterns in embedding representation and visualization of multidimensional data and complex networks
Fig 6
Hard separability problems in complex data science.
Examples of hard separability problems, the term ‘hard’ indicates difficulty to detect the presence of separability. (a-e) refer to an example of a linearly separable dataset called Parallel lines. (f-j) refer to an example of a nonlinearly separable dataset called Circles. (k-o) refer to an example of a nonlinearly separable dataset called Spirals. (a, f, k) Centroid projection separability (CPS): the two black dots indicate the centroids (median estimator) of the two groups of samples, the black line indicates the projection line, the vertical blue dashed lines indicate the projections of the samples. (b, g, l) Linear discriminant projection separability (LDPS): the black line indicates the first component projection vector of the linear discriminant analysis (LDA), the other graphics are as for (a). (c, h, m) Travelling salesman projection separability (TSPS): the travelling salesman path across the samples is indicated by the black solid lines. (d, i, n) Geometrical separability index (GSI): the black solid lines indicate the first neighbor sample matching. (e, j, o) separability of each measure in the respective dataset: (e) Parallel lines, (j) Circles and (o) Spirals. (p, q) mean and minimum separability of each measure across the three datasets. In (e, j, o) the values of the indices with a significant (p-value < 0.01) geometric separability are marked with a star, which means that these values are very unlikely to be obtained by chance.