Ordering of Transformed Recorded Electroencephalography (EEG) Signals by a Novel Precede Operator

Table 1

Summary of the techniques of ordering matrices available in the literature.

Ordering technique	Real-world applications	Advantages	Limitations
Multivariate majorization	(i) Measuring income inequalities and comparing the contents of experiments [36, 40]; (ii) Comparing the information of classical or quantum physical states [56]; (iii) Network flow theory [55]; (iv) Measuring income inequalities [57–59]; (v) Measuring experimental designs and survey sampling [60]	(i) More than one attribute of a system, such as income inequality, can be compared; (ii) Comparison between matrices that have different dimensions	Requires the existence of a doubly stochastic matrix
Quantum majorization	(i) Comparing and ranking correlation matrices to assess portfolio risk in a unified framework [41]; (ii) Comparing quantum processes in which a complete set of entropic conditions for state transformations in resources theories of asymmetry and quantum thermodynamics is derived [49]	(i) It is a generalization of matrix majorization; (ii) The technique can be applied to all quantum states, whereas the previous results are limited to a restricted family of states; (iii) Quantum majorization is preferred mainly for two reasons: (a) verification in the data can be easily done and (b) the axiomatic approach commonly used in financial and actuarial mathematics is satisfied	(i) Requires the existence of completely positive and trace-preserving (CPTP) maps; (ii) Additional tools are required for the case of approximate transformations
Loewner’s ordering	Multivariate analysis [61]	Generalization of univariate statistical analysis	Limited to symmetric matrices of the same order
	Matrix ordering of special C-matrices for statistical analysis [62]	Facilitate the comparison of information matrices between corresponding block designs and dispersion of two multinomial distributions	Limited to the special case of a C-matrix in experimental design theory
	Image processing [63–65]	(i) Fundamental concepts of mathematical morphology could be transferred to matrix-valued data; (ii) The ordering technique can be applied to higher-dimensional tensor data	Limited to the set of symmetric matrices
Partial order induced by affine-invariant geometry	Information geometry to perform statistical analysis [66]	(i) The ordering technique is critical to study the monocity of functions; (ii) The ordering technique can be applied to study dynamical systems and convergence analysis of algorithms defined on matrices	Limited to the set of positive definite matrices of dimension derived from the affine-invariant geometry
Sharp partial order	Autonomous linear systems [67, 68]	Enables a comparison between two autonomous systems, and extraction of much more information	Requires the existence of group inverses
Minus partial order	Compartmental control systems [69]	(i) The compartmental control system models’ performance and efficiency, such as infectious disease evolution, are improved; (ii) A reachable successor system can be obtained from a nonreachable one	Requires the existence of generalized inverses