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

Authors: Amirul Aizad Ahmad Fuad; Tahir Ahmad
Journal: Journal of Mathematics
 (2021)

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## Abstract

Recorded electroencephalography (EEG) signals can be represented as square matrices, which have been extensively analyzed using mathematical methods to extract invaluable information concerning brain functions in terms of observed electrical potentials; such information is critical for diagnosing brain disorders. Several studies have revealed that certain such square matrices—in particular, those related to so-called “elementary EEG signals”—exhibit properties similar to those of prime numbers in which every square EEG matrix can be regarded as a composite of these signals. A new approach to ordering square matrices is pivotal to extending the idea of square matrices as composite numbers. In this paper, several ordering concepts are investigated and a new technique for ordering matrices is introduced. Finally, some properties of this matrix order are presented, and the potential applications of this technique to analyzing EEG signals are discussed.


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## Body

## 1. Introduction and Motivation

Epilepsy is a common neurological disease that affects 1% of the world’s population [1]. People of all ages can be affected by this chronic brain disorder [2], which lowers sufferers’ quality of life with the possibility of a seizure occurring at any time. Early diagnosis can not only improve quality of life but also prevent patients from experiencing accidents. The diagnosis of epilepsy is often not straightforward, and misdiagnosis occasionally does occur [3]. A detailed and reliable eyewitness account of the event is the most crucial piece of information for indicative assessment, but this may not be accessible [4]. In most cases, electroencephalography (EEG) is an essential diagnostic test for assessing patients with possible epilepsy. Besides providing diagnostic support [5], EEG can also assist in classifying the underlying epileptic syndrome [6].Mathematical analysis of EEG signals offers medical professionals vital information regarding brain activity during a seizure, thus increasing the understanding of complex brain function [7, 8]. In general, relevant information is extracted from EEG signals via two main methods: linear and nonlinear. Linear analysis (e.g., Fourier and wavelet transforms) has been successfully executed and has produced several good results [9–15]. However, since linear methods disregard the underlying nonlinear EEG dynamics, the results can only provide a limited amount of information concerning the brain’s electrical activity. By contrast, it is commonly accepted that the brain is a chaotic dynamical system; therefore, EEG signals generated by the brain are considered to be chaotic in another sense [16–18]. Additional information can be extracted from EEG signals by progressively incorporating a nonlinear analysis that reveals features that cannot be measured via linear methods [19].Crucial to diagnosing the disorder is to solve the neuromagnetic inverse problem to identify the location of epileptic foci [20]. Therefore, fuzzy topographic topological mapping (FTTM) was introduced in [21] to determine the epileptic foci. More recently, FTTM has been extensively utilized to study the features of the recorded EEG signals of seizure patients (see [22–27]). Most notably, Yun in [28] claimed that one of the components of FTTM—namely, the magnetic contour (MC)—obeys the associative law, which is also satisfied in turn by events in time [29]. The author concluded by stating that the MC is a plane containing information. This prompted Binjadhnan in [25, 30] to perform Krohn–Rhodes decomposition on a set of square matrices of EEG signals, MCnℝ. The scholar found a remarkable result, namely, the EEG signals taken during an epileptic seizure (henceforth called EEG-signal square matrices for the remainder of this paper) are not chaotic, but rather exhibit ordered patterns in the form of simple algebraic structures, as expressed by Theorem 1.Theorem 1 (see [30]).
Any invertible square matrix of EEG-signal readings during an epileptic seizure at timet can be written as a product of elementary EEG signals during an epileptic seizure in one and only one way.Theorem1 states that the elementary EEG signals (i.e., unipotent and diagonal EEG signals) constitute the building blocks of all EEG signals. This theorem, to a certain extent, is similar to the fundamental theorem of arithmetic, which holds that prime numbers are the multiplicative building blocks of the integers [31]. Equally significant are the results that indicate that MCnℝ has properties resembling those of prime numbers via the Jordan–Chevalley decomposition [32]. The well-ordering property of positive integers is vital in producing one of the most beautiful results in the study of prime numbers, namely, the infinitude of prime numbers. Therefore, a technique of ordering matrices is required to extend the work of viewing the elementary EEG signals as prime numbers. The analogy of elementary EEG signals as prime numbers is of vital importance since the pattern of EEG signals can be investigated in terms of the pattern of prime numbers. Hence, the goal of this paper is to introduce a technique of ordering transformed EEG signals (in terms of square matrices).The remainder of this paper is organized as follows. In Section2, a brief review of a few concepts and techniques for ordering matrices is presented, along with their viability for ordering transformed EEG signals. In Section 3, a new technique for ordering matrices, namely, the precede operator, is introduced, allowing several ordering properties to be obtained. Next, this binary relationship is shown to fulfill the partial-order properties; beyond that, it is shown to be totally ordered in Section 4. In Section 5, several results are obtained when the order is applied to symmetric matrices. Then, the implementation of the precede operator to the real data of EEG signals is presented in Section 6. The interpretation of the results and their connection with the prime numbers are discussed in Section 7. Finally, we bring the paper to a close with concluding remarks concerning the need for such a partial order. Throughout the following sections, every matrix is considered to be a square matrix unless otherwise stated.


## 2. Concepts for Ordering Matrices

Over the past few decades, mathematicians and applied scientists alike have taken a deep interest in the ordering of matrices. Several order relations for matrix algebra have been produced in connection to a series of applications relevant to different branches of mathematics and its applications. These order relations include minus partial order [33], star partial order [34], sharp partial order [35], and matrix majorization [36]. Mitra et al. [37] wrote a comprehensive monograph in which they presented developments in the field of matrix ordering and shorted operators for finite matrices in a unified way, thus sparking research interest in this topic.Matrix partial ordering has applications in many different areas; for instance, Liu [38] developed applications for comparing linear models. Moreover, in the field of statistics, Baksalary and Puntanen [39] presented the best linear unbiased estimator in a general Gauss-Markov model. At the same time, Dahl et al. [40] characterized a binary relation involving stochastic matrices (namely, matrix majorization), which is very useful for comparisons of statistical experiments. Additionally, in the field of finance, Fontanari et al. [41] proposed a technique called quantum majorization to compare and rank correlation matrices such that portfolio risk can be more significantly assessed.The minus partial order is the fundamental matrix partial order, of which almost all subsequent partial orders (including the star and sharp orders) constitute extensions. Such extensions have been created through the addition of restrictions to the minus partial order. The minus partial order (which was originally called the plus order) was established by Hartwig in [33] and independently by Nambooripad in [42] to generalize conventional partial orders on semigroups. Antezana et al. [43] and Šemrl [44] extended this partial order such that it could be applied in an objective way to operators on infinite-dimensional spaces. Djikić et al. [45] documented a new representation of the minus order on the algebra of bounded linear operators on a Hilbert space. The natural partial order of Vagner on inverse semigroups and the star order of Drazin can be extended through minus order [37]. One key feature to note is that these partial orders are defined via the method of generalized inverses.Another essential ordering concept for matrices is majorization, which has been applied across many fields including economics [46], statistics [47, 48], and, most recently, quantum mechanics [49]. This concept was first introduced in a classical book by Hardy et al. [50]. Later, Marshall et al. [36] extensively treated both the theory and application of majorization. Torgersen in [51, 52] studied the generalization of vector majorization and developed the theory of statistical-experiment comparison. This theory is intended to answer the question “What conditions must be fulfilled in order to say that one statistical experiment provides more information than another?” A simple experiment can be found in [53], in which the conventional notion introduced by the author is closely related to that of vector majorization. However, while these generalizations evolved from statistical studies, they are not regularly discussed in the linear-algebra literature. Dahl [54] introduced and studied the generalization of (vector) majorization as it applies in the notable case of matrices with m rows. The classical concept of majorization between vectors can be generalized via matrix majorization [55].Some of the techniques of ordering matrices found in the literature, along with their real-world applications, advantages of the techniques, and their limitations (with respect to the purpose of ordering transformed EEG signals), are summarized in Table1.Table 1
Summary of the techniques of ordering matrices available in the literature.
Ordering techniqueReal-world applicationsAdvantagesLimitationsMultivariate 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 dimensionsRequires the existence of a doubly stochastic matrixQuantum 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 transformationsLoewner’s orderingMultivariate analysis [61]Generalization of univariate statistical analysisLimited to symmetric matrices of the same orderMatrix ordering of special C-matrices for statistical analysis [62]Facilitate the comparison of information matrices between corresponding block designs and dispersion of two multinomial distributionsLimited to the special case of aC-matrix in experimental design theoryImage 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 dataLimited to the set of symmetric matricesPartial order induced by affine-invariant geometryInformation 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 matricesLimited to the set of positive definite matrices of dimensionn derived from the affine-invariant geometrySharp partial orderAutonomous linear systems [67, 68]Enables a comparison between two autonomous systems, and extraction of much more informationRequires the existence of group inversesMinus partial orderCompartmental 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 oneRequires the existence of generalized inversesThe techniques of ordering matrices summarized in Table1 have been deemed unfit to be used to extend the work of Binjadhnan and Ahmad [25, 30] and Fuad and Ahmad [32] since there are some conditions required to be fulfilled, and some are limited to the special matrices. This offers the possibility of introducing and investigating a new partial order of square matrices as discussed in Section 3.


## 3. Precede Operator

As mentioned in Section2, the set of square matrices of EEG signals during a seizure, MCnℝ, has properties similar to those of prime numbers. Therefore, EEG-signal square matrices can be assumed to be analogs of natural numbers. It can be said that one matrix is “greater” than another matrix, just as any natural number can be either greater than or less than another natural number since ℝ is a complete ordered field and ℕ⊂ℝ. With this in mind, the precede operator, denoted by ≻, is introduced as defined by Definition 1.Definition 1.
LetC and C′ be n×n matrices and C≠C′. Matrix C is said to precede C′, written as C≻C′, whenever the first cij>cij′ exists for some i,j. The comparison must be made in the sequence of rows, i.e., R1,R2,…,Rn, until the first cij>cij′ is discovered and denoted as ≻C,C′=cij. Otherwise, if cij′>cij, then C′≻C. When C=C′, i.e., all the corresponding entries for each matrix are the same, then ≻C,C′=c11.
In other words, let us considerC,C′∈Mn such that(1)C=c11c12⋯c1nc21c22⋯c2n⋮⋮⋱⋮cn1c2n⋯cnn,C′=c11′c12′⋯c1n′c21′c22′⋯c2n′⋮⋮⋱⋮cn1′c2n′⋯cnn′.
DefineRi:Mn×Mn⟶RowiC×RowiC′, such that RowiC=ci1,ci2,…,cin and RowiC′=ci1′,ci2′,…,cin′ denote row i in C and C′, respectively.
Next, defineωij:RowiC×RowiC′⟶Mn such that(2)ωij=cij,whencij≥cij′,cij′,whencij′>cij,fori,j=1,2,3,…,n.
Hence, matrixωij=ω11ω12⋯ω1nω21ω22⋯ω2n⋮⋮⋱⋮ωn1ω2n⋯ωnn.
Now, defineω1j′:Mn⟶M1×n such that(3)ω1j′ωij=Row1ωijRow2ωijRow3ωij⋯Rownωij=ω11ω12ω13⋯ω1nω21ω22ω23⋯ω2n⋯ωn1ωn2ωn3⋯ωnn=ω11′ω12′ω13′⋯ω1n′ω1n+1′ω1n+2′⋯ω12n′⋯ω1n2−n′⋯ω1n2−2′ω1n2−1′ω1n2′.
In other words, rearrange and reidentify the entries ofωij.
Next, defineω∗:M1×n⟶ℝ such that(4)ω∗=ω1j′′,whencij≠cij′,ω1j′+1′,whencij=cij′,fori,j=1,2,3,…,n,j′=1,2,3,…,n2.
Finally,≻C,C′=ω∗.
Definition1 is introduced as a map between a matrix and a real number in ℝ, since ℝ is a complete ordered set.Example 1.
Consider two matricesA and B such that(5)A=0.81470.09750.15760.14190.65570.90580.27850.97060.42180.03570.12700.54690.095720.91570.84910.91340.95750.48540.79220.93400.63240.96490.80030.95950.6787,B=0.81470.09750.15760.14190.65570.90580.03180.69480.38160.44560.39220.27690.31710.76550.64630.65550.04620.95020.79520.70940.17120.09710.03440.18690.7547.
As can be clearly seen, the firstaij>bij is found. In this case, a22>b22 or 0.2785>0.0318. Then, A≻B.Theorem 2.
The mapping≻:Mn×Mn⟶ℝ is well-defined.Proof.
LetA,B=C,D, such that A,B,C,D∈Mn×Mn. Therefore, A=C and B=Dimply that(6)≻A,B=≻C,BsinceA=C=≻C,DsinceB=D.
Hence, the mapping of≻:Mn×Mn⟶ℝ is well-defined.
The execution of this definition can be summarized by Algorithm1.
In short,≻:Mn×Mn⟶ℝ, where ≻ is a composition of mappings such that(7)≻=ω∗⊙ω1j′⊙ωij⊙Ri.
The composition of mappings≻ is best illustrated by Figure 1.Algorithm 1: Precede operator.
Step 1: given matricesC,C′∈Mn and C≠C′, determine row i for C and C′, i.e., RiC=ci1,ci2,…,cin and RiC′=ci1′,ci2′,…,cin′.Step 2: determine matrixωij whereby ωij:RiC×RiC′⟶Mn, such that ωij=cij,whencij>cij′cij′,whencij′>cij for i,j=1,2,…,n.Step 3: determine matrixω1j' whereby ω1j′:Mn⟶M1×n, such that ω1j′ωij = Row1ωijRow2ωijRow3ωij⋯Rownωij=ω11ω12ω13⋯ω1nω21ω22ω23⋯ω2n⋯ωn1ωn2ωn3⋯ωnn= ω11′ω12′ω13′⋯ω1n′ω1n+1′ω1n+2′⋯ω12n′⋯ω1n2−n′⋯ω1n2−2′ω1n2−1′ω1n2′.Step 4: determineω∗:M1×n⟶ℝ such that ω∗=ω1j′′,whencij≠cij′ω1j′+1′,whencij′=cij for i,j=1,2,…,n and j′=1,2,…,n2.Figure 1
Composition of mapping≻.


## 4. Ordered Matrices

In this section, several results are obtained to show that any square matrices together with the precede operator are ordered matrices.Lemma 1.
IfA≻B, then B⊁A.Proof.
LetA≻B. Then, there exists a first aij and a first bij for some i,j∈ℕ such that aij≥bij. Suppose that B⊁A is false; then, B≻A is true. Therefore, there exists a first bij and a first aij for some i,j∈ℕ such that bij≥aij. This is impossible, since A≠B according to Definition 1, and it also contradicts with the assumption that says there exists a first aij and a first bij for some i,j∈ℕ such that aij≥bij, as noted earlier.Lemma 2.
IfA≻B and B≻A, then ≻A,B=a11.Proof.
SupposeA≻B; then, there exists a first aij and a first bij for some i,j∈ℕ, such that aij≥bij. Similarly, if B≻A, then there exists a first bij and a first aij for some i,j∈ℕ, such that bij≥aij. This is only possible if aij=bij,∀i,j=1,2,…,n. In other words, ≻A,B=a11, by Definition 1.Lemma 3.
IfA≻B and B≻C, then A≻C.Proof.
LetA≻B. Then, there exists a first aij and a first bij for some i,j∈ℕ such that aij≥bij. If B≻C, then there exists a first bij′ and a first cij′ for some i,j∈ℕ such that bij′≥cij′. Suppose that A≻C is false; therefore, C≻A. In other words, there exists a first cij″ and a first aij″ for some i,j∈ℕ, such that cij″≥aij″. There are three cases to consider:(i)
cij″>cij′: if cij″>cij′, then bij′≥cij″>cij′ is a contradiction, since cij′ is no longer the first to be found, such that bij′≥cij′ and B≠C, according to Definition 1→←.(ii)
cij′>cij″: if cij′>cij″, then cij″>cij″≥aij″ is a contradiction, since cij″ is no longer the first cij″ to be found, such that cij″≥aij″ but cij′→←.(iii)
cij′=cij″: if cij′=cij″, then bij′≥cij′=cij″, since B≻C, and cij″≥aij″, since C≻A. Therefore, bij′≥cij″≥aij″, which immediately implies bij′≥aij″. In other words, B≻A, which is a contradiction →←.
All three cases lead to contradictions; thus, ifA≻B and B≻C, then A≻C.
Consequently, the binary relation is a partial order.Theorem 3.
The setMn×Mn,≻ is a partially ordered set.Proof.
The setMn×Mn,≻ is a partially ordered set since(i)
It is reflexive, sinceC≻C′=≻C,C′=c11 by Definition 1(ii)
By Lemma2, ≻ is antisymmetric(iii)
By Lemma3, ≻ is transitive
Of equal importance,≻ is totally ordered as well.Theorem 4.
The setMn×Mn,≻ is totally ordered.Proof.
According to Theorem3, the set Mn×Mn,≻ is partially ordered. Next, A=B or A≠B. Consider the case where A≠B, and a contrapositive for Lemma 2 is applied. In other words, ¬A=B⟶¬A≻BandB≻A,; this means that A≠B⟶¬A≻Bor¬B≻A;A≠B⟶B≻AorA≻B. In summary, A=B or A≻B or B≻A. Hence, Mn×Mn,≻ is totally ordered.


## 5. Precede Ordering for Symmetric Matrices

When precede operator is applied to special matrices (namely, symmetric matrices), several properties are obtained.Proposition 1.
IfA and B are symmetric matrices and A≻B, then AT≻BT (i.e., matrix transposition preserves ≻ for symmetric matrices).Proof.
Suppose thatA and B are symmetric matrices and A≻B. It immediately follows that AT≻BT, since AT=A and BT=B.Proposition 2.
IfA and B are symmetric matrices and A≻B, then −B≻−A.Proof.
Suppose thatA≻B; then, there exists a first aij and a first bij for some i,j∈ℕ. Nevertheless, −bij≥−aij, since aij,bij∈ℝ for some i,j∈ℕ, and aij and bij are the first terms of −A and −B, respectively, that exhibit such conditions. Consequently, −B≻−A.Proposition 3.
IfA≻B, then kA≻kB for k∈ℝ+ (i.e., matrix-positive-scalar multiplication preserves ≻).Proof.
Suppose thatA≻B; then, there exists a first aij and a first bij for some i,j∈ℕ such that aij≥bij. Then, kaij≥kbij, since k∈ℝ+. Hence, kA≻kB.Proposition 4.
IfA and B are skew-symmetric matrices and A≻B, then BT≻AT.Proof.
Suppose thatA and B are skew-symmetric matrices; therefore, AT=−A and BT=−B. Nevertheless, if A≻B, then −B≻−A by Proposition 2. Consequently, BT>AT, since A and B are skew-symmetric matrices.Theorem 5.
Suppose thatA,B,C, and D are positive symmetric matrices, such that A≻B and C≻D; then, A+C≻B+D.Proof.
Suppose thatA=aij,B=bij,C=cij, and D=dij are positive symmetric matrices. A≻B implies that there exists a first aij∗ and a first bij∗ for some i′,j′∈ℕ, such that aij∗≥bij∗. Similarly, C≻D implies that there exists a first cij∗ and a first dij∗ for some i″,j″∈ℕ, such that ci″j″∗ and a first di″j″∗ for some i″,j″∈ℕ, such that ci″j″∗≥di″j″∗. Now, ai′j′∗+ci″j″∗≥bi′j′∗+di″j″∗, since ai′j′∗,ci″j″∗,bi′j′∗,di″j″∗∈ℝ and ℝ are the fields. In short, the first ai′j′∗+ci″j″∗≥ the first bi′j′∗+di″j″∗ for i″,j″,i′,j″∈ℕ. This implies that A+C≻B+D.
Theorem5 is best illustrated by Example 2.Example 2.
Consider the positive symmetric matricesA,B,C, and D such that(8)A=561232122416321623,B=151121113721743,C=41710172110119,D=11631611319.
By Definition1, A≻B and C≻D is obtained. Next,(9)A+C=602942292617421742,B+D=162724274824852.
Similarly, by Definition1, A+C≻B+D is obtained.Corollary 1.
Suppose thatA and B are positive symmetric matrices. If A≻B, then A+AT≻B+BT.Proof.
Suppose thatA and B are positive symmetric matrices and that A≻B. Then, AT≻BT, according to Proposition 1. By replacing C=AT and D=BT in Theorem 5, we obtain A+AT≻B+BT.Corollary 2.
Suppose thatA and B are positive symmetric matrices such that A≻B; then, A−B≻B−A.Proof.
Suppose thatA and B are positive symmetric matrices and A≻B. Then, −B≻−A, by Proposition 2. By replacing C=−B and D=−A in Theorem 5, we obtain A−B≻B−A.Example 3.
Suppose thatA and B are two symmetric matrices, such that(10)A=6317311317310,B=11013105113117.
Then, by Definition1, A≻B. Next,(11)A−B=4−74−76242−7,B−A=−47−47−6−2−4−27,which, by Definition 1, implies that A−B≻B−A.Theorem 6.
C≻−C for every positive symmetric matrix C (i.e., a positive symmetric matrix precedes its negation).Proof.
Suppose thatA and B are positive symmetric matrices and that A≻B. Notice that, A+AT is a positive symmetric matrix since(12)A+ATT=AT+ATT=AT+A=A+AT=A+AT.
Similarly,B+BT is also a positive symmetric matrix. Now, A+AT≻B+BT, according to Corollary 1, and −B+BT≻−A+AT, according to Proposition 2. Consequently, A+AT+−B+BT≻B+BT−−A+AT, by Theorem 5. Equivalently,(13)A+AT−B+BT≻B+BT−A+AT⇒A+AT−B+BT≻−A+AT−B+BT−∗.
Notice that ifF and G are positive symmetric matrices, then F−G is a symmetric matrix, since F−GT=F+−GT=FT+−1GT=F−G. Next, let F=A+AT and G=B+BT; therefore, F−G=A+AT−B+BT is symmetric. Replace C=F−G in ∗; then, C≻−C is obtained as required.Example 4.
LetC be a 3×3 symmetric matrix such that(14)C=731231312321,−C=−7−3−12−3−1−3−12−3−21.
Then, by Definition1, we obtain C≻−C.Theorem 7.
Suppose thatA,B, and C are positive symmetric matrices. If C≻A+B and C≻A, then C≻1/2B.Proof.
Suppose thatA,B, and C are positive symmetric matrices. Assume that C≻A+B and C≻A. According to Theorem 6, A≻−A. Since C≻A and A≻−A imply that C≻−A by Lemma 3, we consequently obtain C≻A+B and C≻−A⇒C+C≻A+B+−A by Theorem 3. In other words, 2C≻B. Therefore, by Proposition 3, 1/22C≻1/2B which implies that C≻1/2B.Example 5.
Suppose thatA,B, and C are symmetric matrices such that(15)A=412312106368,B=31041068482,C=132231221221312114.
Clearly, by Definition1, C≻A. Next,(16)A+B=722722161471410⇒C≻A+B,12B=3252534241⇒C≻12B.


## 6. Implementation

As an example of the implementation of the precede operator on real data, two readings of EEG signals’ square matrices are presented. The EEG signals of epileptic seizure patients could be recorded and composed into a set of square matrices (see Figure2). Firstly, the EEG data is gathered from the hospital by the EEG technologists and the three-dimensional data are transformed into two-dimensional data. The transformation of the EEG data into a lower-dimensional MC is executed via a novel technique called flattening the EEG, where the information can be preserved and conveniently analyzed [70].Figure 2
EEG signals during an epileptic seizure of PatientA at a time t is transformed into a square matrix [30].The coordinate system of EEG signals, depicted in Figure3(a), is defined as(17)CEEG=x,y,z,ep|x,y,z,ep∈ℝandx2+y2+z2=r2,Figure 3
(a) EEG coordinate system and (b) EEG projection [70].
(a)(b)Moreover, a functionSt:CEEG⟶MC (see Figure 3(b)) is defined as(18)Stx,y,z,ep=rx+iryr+z,ep=rxr+z,ryr+zepx,y,z.The mappingSt is an injective mapping of a conformal structure since both CEEG and MC were designed and proven as two manifolds [70]. Hence, the mapping St can keep up the data in a specific angle and orientation of the surface throughout the recorded EEG signals. The technique of flat EEG is executed on three groups of EEG signals recorded from three different epileptic patients [70]. The author digitized the EEG signals during epileptic seizures at 256 samples per second using the Nicolet One EEG software. Next, each APD at every second was stored in a file that contained the position of an electrode on a magnetic-contour (MC) plane. Subsequently, the stored data were used to compose a set of square matrices.Differences in surface potential could be recorded using an array of electrodes appended to the scalp; the computed voltages between pairs of electrodes are then clarified, amplified, and recorded. The most widely used system of electrode placement is the International Ten-Twenty System; this is a recommended standard method for characterizing the locations of electrodes at particular time intervals along with the head for recording scalp EEG [71]. The Ten-Twenty system depends upon the connection between the position of an electrode and the underlying area of the cerebral cortex (the “Ten” and “Twenty” refer to 10% and 20% interelectrode distances, respectively) [72]. The electrode position of this system is shown systematically in Figure 4.Figure 4
The international Ten-Twenty system seen from (a) left and (b) above the head [73].
(a)(b)Figure4(a) illustrates the case where almost all of the electrodes are positioned 40% or below from vertex Cz. On the contrary, Figure 4(b) shows the electrode position from the top view of the head by modeling the head as a sphere. We assume that the hemisphere is 80% from the top of the head [70]. In other words, from the front to the back is from Fpz to Oz and from the left to the right is from T3 to T4. In general, every APD at each second is stored in a file containing the position of electrodes on the MC plane, as tabulated in Table 2.Table 2
Average potential difference (APD) at the sensor on anMC [30].
SensorXYAPDFpzx21y21z21Fp1x19y19z19Fp2x20y20z20F3x15y15z15F4x16y16z16C3x9y9z9C4x10y10z10P3x6y6z6P4x7y7z7O1x2y2z2O2x3y3z3F7x17y17z17F8x18y18z18T3x11y11z11T4x12y12z12T5x4y4z4T6x6y6z6Fzx1y1z1Czx13y13z13Pzx8y8z8Ozx14y14z14Then, the readings in Table2 are rewritten in terms of a 5×5 matrix, as shown below.Letx1≤x2≤⋯≤x21,i,j=1,2,3,4,5 and a function βij be defined as(19)βij=xi−15+j,yi−15+j,fori−15+j≤21,0,fori−15+j>21.The mapping ofβij can be rewritten as the following matrix:(20)β11β12β13β14β15β21β22β23β24β25β31β32β33β34β35β41β42β43β44β45β51β52β53β54β55.Specifically,(21)x1,y1x2,y2x3,y3x4,y4x5,y5x6,y6x7,y7x8,y8x9,y9x10,y10x11,y11x12,y12x13,y13x14y14x15,y15x16,y16x17,y17x18,y18x19,y19x20,y20x21,y210000.The corresponding square matrix is generated by substituting the analogous average potential difference of every element into the above matrix. In particular, every single second of the APD is stored in a square matrix that contains the positions of electrodes on theMC plane.Therefore, theMC plane becomes a set of n×n matrices (EEG signals), which is written as follows:(22)MCnℝ=βijztn×n|i,j∈ℤ+,βijzt∈ℝ,where βijzt is a potential-difference reading for EEG signals from a particular ij sensor at time t. For instance, the recorded EEG signals data during seizures at times t=2 and t=3 are tabulated in Tables 3 and 4.Table 3
Average potential difference (APD) of patientA at the sensor on an MC at time t=2 [30].
SensorXYAPDFpz7.6800Fp17.30413.373333.94984375Fp27.3041−2.373315.32792969F33.36913.369122.99746094F43.3691−3.36910.707890625C303.18127.5621875C40−3.181223.01671875P3−3.36913.369184.15105469P4−3.3691−3.369125.2309375O1−7.30412.3733112.3017578O2−7.3041−2.373370.05695313F74.51426.213331.42472656F84.5142−6.213315.34089844T307.6858.33558594T40−7.6853.67203125T5−4.51426.2133114.2680859T6−4.5142−6.2133102.3164453Fz3.1812010.21890625Cz0012.73027344Pz−3.181203.456171875Oz−7.6800Table 4
Average potential difference (APD) for patientA at the sensor on an MC at time t=3 [30].
SensorXYAPDFpz7.6800Fp17.30413.373365.682265625Fp27.3041−2.373341.90457031F33.36913.36918.524765625F43.3691−3.36910.0625C303.1812.902265625C40−3.181279.02796875P3−3.36913.3691110.2065234P4−3.3691−3.369163.74707031O1−7.30412.3733234.7169922O2−7.3041−2.3733198.7374219F74.51426.213397.15632813F84.5142−6.213376.86242188T307.68138.7585156T40−7.68137.3880078T5−4.51426.2133209.6701172T6−4.5142−6.2133230.6576563Fz3.1812018.03042969Cz0060.58832031Pz−3.1812037.07589844Oz−7.6800The data in Tables3 and 4 are then reordered in ascending order of X values and tabulated in Tables 5 and 6, respectively, through the MATLAB programming developed by Binjadhnan [30]. Typically, the program exhibits twenty-one sensor readings and four readings of zero (i.e., no recorded data from the “ghost” sensors at the specified time t); thus, a 5×5 matrix is obtained from each program.Table 5
Reordering the average potential difference (APD) at the sensor for a patientA at time t=2 [30].
SensorXYAPDFpz−7.6800Fp1−7.30412.3733112.3017578Fp2−7.3041−2.373370.05695313F3−4.51426.2133114.2680859F4−4.5142−6.2133102.3164453C3−3.36913.369184.15105469C4−3.3691−3.369125.2309375P3−3.181203.456171875P403.181227.5621875O10−3.181223.01671875O207.6858.33558594F70−7.6853.67203125F80012.73027344T33.1812010.21890625T43.36913.369122.99746094T53.3691−3.36910.707890625T64.51426.213331.42472656Fz4.5142−6.213315.34089844Cz7.30412.373333.94984375Pz7.3041−2.373315.32792969Oz7.6800Table 6
Reordering the average potential difference (APD) at the sensor for a patientA at time t=3 [30].
SensorXYAPDFpz−7.6800Fp1−7.30412.3733234.7169922Fp2−7.3041−2.3733198.7374219F3−4.51426.2133209.6701172F4−4.5142−6.2133230.6576563C3−3.36913.3691110.2065234C4−3.3691−3.369163.74707031P3−3.1812037.07589844P403.18122.902265625O10−3.181279.02796875O207.68138.7585156F70−7.68137.3880078F80060.58832031T33.1812018.03042969T43.36913.36918.524765625T53.3691−3.36910.0625T64.51426.213397.15632813Fz4.5142−6.213376.86242188Cz7.3041−2.373365.68265625Pz7.30412.373341.90457031Oz7.6800Table5 yields the matrix:(23)A2=0112.301870.05695114.2681102.316484.151125.23093.4561727.562223.016758.335653.67212.730310.218922.99750.70789131.424715.340933.949815.327900000.Table6 yields the matrix:(24)A3=0234.717198.7374209.6701230.6577110.206563.7470737.07592.90226679.02797138.7585137.38860.5883218.030438.5247660.062597.156376.862465.682741.904600000.The two matrices,A2 and A3, can be compared using the precede operator. Again,(25)A2=0112.301870.05695114.2681102.316484.151125.23093.4561727.562223.016758.335653.67212.730310.218922.99750.70789131.424715.340933.949815.327900000,A3=0234.717198.7374209.6701230.6577110.206563.7470737.07592.90226679.02797138.7585137.38860.5883218.030438.5247660.062597.156376.862465.682741.904600000.Then, by Definition1, the first aij that is greater than bij (aij>bij) is a12>b12(i.e., 234.7169922>112.3018; hence, A3≻A2. Consequently, the precede operator can be used in the set of EEG signals’ square matrices, allowing one matrix to be said to precede another. The set of EEG signals’ square matrices, along with the precede operator, enables one to see the set of matrices as analogous to the set of real numbers, as related to each other with the “greater than,” > operator. This analogy is best demonstrated in Figure 5.Figure 5
The analogy between the set of EEG-signal square matrices and the precede operator and the set of real numbers and the greater-than operator.


## 7. Discussion

ptThe key advantage of using the precede operator over the techniques discussed in Section2 is that it does not require to fulfill the necessary conditions (such as the existence of doubly stochastic matrix and generalized inverses), rather, the precede operator “inherits” the totally-ordered property of real numbers. Thus, a similar result provides a piece of evidence that the elementary EEG signals during seizure contain similar attributes to the distribution of prime numbers among positive integers. The similarity of elementary EEG signals with prime numbers is summarized in Table 7.Table 7
The similarity between elementary EEG signals with the prime numbers.
EEG signalsPositive integersDivisibilityDefinition 2 (see [30]): if At and Bt are EEG signals, we say that At divides Bt, written as At|Bt if there exist EEG signals Mt such that Bt=AtMt, where Bt is an invertible EEG signalDefinition 3 (see [31]): if a and b are integers, we say that a divides b if there is an integer c such that b=acUnique factorization theoremTheorem1 [30]Fundamental theorem of arithmetic [31]Building blocksElementary EEG signalsPrime numbersImportant properties(1) Jordan–Chevalley decomposition of elementary EEG signals (see Theorem 10 and Theorem 11 in [74])(2) Theorem 4 (totally ordered)(1) Twin-prime conjecture [31](2) Well-orderedThe resemblance of elementary EEG signals with the prime numbers in terms of its ordering properties corroborates the assertions made by Binjadhnan [30], Barja [27], and Ahmad Fuad and Ahmad [74] that the EEG signals during seizures contain a similar pattern to that of the prime number distribution among positive integers. The premise of viewing the EEG signals as prime numbers enables one to study the dynamics of EEG signals during seizures in terms of the pattern of prime numbers. More importantly, the deduced pattern of seizures is critical to devise a methodology that is capable of predicting a seizure, which in turn would significantly improve the patients’ quality of life [75]. Conversely, it is instructive to explore the property of elementary EEG signals that could possibly exist in the distribution of prime numbers.


## 8. Conclusions

This study introduces a new technique for ordering square matrices, called precede operator, and this binary relation is proven to exhibit partial ordering. In addition, several results obtained by using this new ordering technique are presented. Furthermore, the newly introduced matrix partial order is applied to the square matrices of EEG signals, giving opportunities to further develop EEG-signal square matrices’ structure by studying those features that resemble some of the rich properties of prime numbers. In particular, it will be of interest to extend the totally ordered property of transformed EEG signals to the well-ordered as well. We are working on this problem and hoping to present our findings in a future paper. Furthermore, as this paper is limited to the precede operator implemented only on the transformed EEG signals, it would be intriguing to investigate the feasibility of the proposed technique to other structures, such as, among others, the measurement of income inequality, statistical experiment, and thermodynamics, as previously summarized in Table1.



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