Assessment of Two-Dimensional Separative Systems Using Nearest-Neighbor Distances Approach. Part 1: Orthogonality Aspects

We propose here a new approach to the evaluation of two-dimensional and, more generally, multidimensional separations based on topological methods. We consider the apex plot as a graph, which could further be treated using a topological tool: the measure of distances between the nearest neighbors (NND). Orthogonality can be thus defined as the quality of peak dispersion in normalized separation space, which is characterized by two factors describing the population of distances between nearest neighbors: the lengths (di<sub>(o)</sub>) of distances and the degree of similarity of all lengths. Orthogonality grows with the increase of both factors. The NND values were used to calculate a number of new descriptors. They inform about the extent of peak distribution, like the arithmetic mean (<i>A̅</i><sub>(o)</sub>) of NNDs, as well as about the homogeneity of peak distribution, like the geometric mean (<i>G̅</i><sub>(o)</sub>) and the harmonic mean (<i>H̅</i><sub>(o)</sub>). Our new, NND-based approach was compared with another recently published method of orthogonality evaluation: the fractal dimensionality (<i>D</i><sub>F</sub>). The comparison shows that the geometric mean (<i>G̅</i><sub>(o)</sub>) is the descriptor behaving in the most similar way to dimensionality (<i>D</i><sub>F</sub>) and the harmonic mean (<i>H̅</i><sub>(o)</sub>) displays superior sensitivity to the shortest, critical distances between peaks. The latter descriptor (<i>H̅</i><sub>(o)</sub>) can be considered as sufficient to describe the degree of orthogonality based on NND. The method developed is precise, simple, easy to implement, and possible to use for the description of separations in a true or virtual system of any number of dimensions.