Pairwise Comparisons Matrix Decomposition into Approximation and Orthogonal Component Using Lie Theory
PPAIRWISE COMPARISONS MATRIX DECOMPOSITIONINTO APPROXIMATION AND ORTHOGONALCOMPONENT USING LIE THEORY
W.W. KOCZKODAJ, V.W. MAREK, AND Y. YAYLI
Abstract.
This paper examines the use of Lie group and Lie Alge-bra theory to construct the geometry of pairwise comparisons matrices.The Hadamard product (also known as coordinatewise, coordinate-wise,elementwise, or element-wise product) is analyzed in the context of in-consistency and inaccuracy by the decomposition method.The two designed components are the approximation and orthogonalcomponents. The decomposition constitutes the theoretical foundationfor the multiplicative pairwise comparisons.
Keywords:
AI, approximate reasoning, subjectivity, inconsistency, pair-wise comparison, pairwise comparisons matrix, matrix Lie group, Liealgebra, approximation, orthogonality, decomposition. Introduction
Pairwise comparisons (PCs) take place when we somehow compare twoentities (objects or abstract concepts). According to [9], Raymond Llull iscredited for the first documented use of pairwise comparisons in “A systemfor the election of persons” (
Artifitium electionis personarum ) before 1283and in “An electoral system” (
De arte eleccionis ) in 1299. Both manuscriptswere handwritten (there was no scientific publication process establishedyet) for deciding the winner of elections. a r X i v : . [ m a t h . G M ] J a n KOCZKODAJ, MAREK, AND YAYLI
There are two variants of pairwise comparisons: multiplicative and ad-ditive. The multiplicative PCs variant reflects a relationship:
A is x times [ comparison operator ] than B The additive type expresses: “by how much (the percentage is often used)one entity is [comparison operator] than another entity” . The comparisonoperator could be: bigger, better, more important , or similar comparisons.The multiplicative pairwise comparison is determined by the ratio of twoentities. For instance, the constant π , is one of the most recognized ratiosin mathematics. It is defined as the ratio of a circle circumference to itsdiameter.Entities may be physical objects or abstract concepts (e.g., software de-pendability or software safety). In both cases, we can provide an estimatebased on expert assessment. However, physical measurements (e.g., area orweight) should be considered if they are available.In practice, multiplicative (or ratio) PCs are more popular than addi-tive PCs. However, they are more mathematically challenging than additivecomparisons. The additive PCs can be produced from the multiplicativeform by a logarithmic mapping (introduced in [8]). The logarithmic map-ping is used to provide the proof of convergence and for the interpretationof the limit of convergence in [11, 20, 21].Pairwise comparisons are usually represented by a PC matrix . In thecase of multiplicative PCs, it is a matrix of ratios of entities with 1s on themain diagonal (for the entity being compared to itself) and reciprocal ( x and 1 /x ) values in upper/lower triangles as it is also reasonable to assumethat the ratio of B/A is the reciprocal value of
A/B . C MATRIX DECOMPOSITION 3
For entities which are abstract concepts (e.g., software quality and soft-ware safety), the division operation is undefined but using the ratio stillmakes sense (e.g., by stating: “three times more important”). For this rea-son, ratios are often given by expert assessments.The main goal of PCs is to split 1 into n real values assigned to n entities E i , i = 1 , . . . , n . We call them weights .1.1. Problem outline.
In [16] (a follow-up to [19]), Smarzewski has ob-served that PC matrices form a group under Hadamard product. In thispaper, we will show that it is, in fact, a Lie group.1.2.
Structure of this paper.
A brief introduction to the pairwise com-parisons method is provided in Section 2. Section 2 also includes a simpleexample of using the PC method for a generic exam. Section 3 provides rea-soning for the necessity of theorems and propositions. Section 4 shows theconstruction of Lie groups and Lie algebra for PC matrices. Section 5 intro-duces the exponential transformation and its properties. Section 6 presentsthe main theorem. Section 7 outlines the generalization of our results.2.
Example of using pairwise comparisons in practice
A Monte Carlo experiment for pairwise comparisons accuracy improve-ment was presented in [14, 15]. It provided statistical evidence that theaccuracy gain was substantial by using pairwise comparisons. For simplic-ity, let us assume that we have three entities to compare:
A, B, and C . Thethree comparisons are: A to B , A to C , and B to C . We assume the reci-procity of PC matrix M : m ji = 1 /m ij which is reasonable (when comparing B to A , we expect to get the inverse of the comparison A to B ). The examis hence represented by the following PC matrix M : KOCZKODAJ, MAREK, AND YAYLI (2.1) M = [ m ij ] = A/B A/CB/A B/CC/A C/B A/B reads “the ratio between A and B” and may not necessarily be aresult of the division (in the case of the exam problem, the use of divisionoperation makes no mathematical sense but using the ratio is still valid).Ratios of three entities create a triad ( A/B, A/C, B/C ). This triad is saidto be consistent provided
A/B ∗ B/C = A/C . It is illustrated by Fig. 1.Random numbers of dots are hard to count but can be compared in pairsas two random clouds. [
A/B ] reflects the assessed ratio of dots by expertopinions.A large enough number of dots represents the concept of ( numerosity ).They may, for example, represent votes of experts. In an emergency situation(e.g., mine rescue), it is impossible to count votes in a short period of time.The exact number of votes is there but all we need is to assess the numerosityof votes .Symbolically, in a PC matrix M , each triad (or a cycle) is defined by( m ik , m ij , m kj ). Such triad is consistent providing ( m ik ∗ m kj = m ij ). Whenall triads are consistent (known as the consistency condition or transitivitycondition ), the entire PC matrix is considered consistent .Looking at the above exam grading case, we have discovered a pairwisecomparisons method which can be used to construct a PC matrix. The solution to the PC matrix is a vector of weights which are geometric meansof rows. We usually normalize it to 1 as the sum. The justification for theuse of the vector w = [ v i ] of geometric means (GM) of rows is not trivial and C MATRIX DECOMPOSITION 5
Figure 1.
An inconsistency indicator cycleit is the subject of this paper. The exact reconstruction of the PC matrix(say M ) via M = [ v i /v j ] is guaranteed only for the consistent matrices.In our example, the computed weights (as normalized geometric meansof rows) are approximated to: [0 . , . , . A is the most difficult with the weight 0.58. Theeasiest problem is C with the weight 0 . KOCZKODAJ, MAREK, AND YAYLI The problem formulation
This paper examines the use of group theory to construct the geometryof pairwise comparisons matrices. The Hadamard product (also known ascoordinatewise, coordinate-wise, elementwise, or element-wise product) isexamined in the context of inconsistency and inaccuracy. To achieve thisgoal, we provide a proof that PC matrices are represented by a Lie group.Subsequently, a Lie algebra of the Lie group of PC matrices is constructed.A decomposition method of PC matrices is introduced for the Hadamardproduct. One of the components is an approximation PC matrix and theother orthogonal component is interpreted as the approximation inaccuracy.The importance of selecting PC matrix components is also provided in thispaper. Subgroups of the PC matrix Lie group have been identified andpresented as an internal direct product.4.
Lie groups and Lie algebras of PC matrices
The monograph [27] stipulates that, “Intuitively, a manifold is a gener-alization of curves and surfaces to higher dimensions. It is locally Euclideanin that every point has a neighborhood, called a chart, homeomorphic toan open subset of R n ”. We find the above stipulation to be sufficient to befollowed by computer science researchers.A group that is also a differentiable (or smooth) manifold is called Liegroup (after its proponent Sophus Lie). According to [3], a Lie group is anabstract group G with a smooth structure, that is: Definition 4.1. (1) G is a group,(2) G is a smooth manifold, C MATRIX DECOMPOSITION 7 (3) the operation
G × G −→ G , ( x, y ) −→ xy − is smooth.Matrix Lie group operates on matrices. Definition 4.2.
The Lie algebra of a Lie group G is the vector space T e G equipped with the Lie bracket operation [ , ] of vector fields.The bracket operation [ , ] is assumed to be bilinear, antisymmetric, andsatisfies the Jacobi identity: Cyclic([ X, [ Y, Z ]]) = 0 for all
X, Y, Z belongingto this algebra.Lie group and Lie algebra have been analyzed in [25, 2, 12, 4].
Proposition 4.3.
For every dimension n > , the following group: G = { M = [ m ij ] n × n | M · M T = I, m ij = 1 m ji > for every i, j = 1 , , . . . , n } is an abelian group of n × n PC matrices with an operation · : G × G (cid:55)−→ G defined by ( M, N ) −→ M · N = [ m ij · n ij ] where ” · ” is the Hadamard product.Proof. To begin, we know that I · I − = I so I ∈ G where I = [ η ij ] n × n isthe identity element of the group and satisfies the condition η ij = 1 every i, j = 1 , . . . , n .Now, observe that if M ∈ G then M · M T = I and M T = M − . Thus M T ( M T ) T = I so M − ∈ G .Let M and N be arbitrary elements of G . Notice that by the propertiesof G : N M ( N M ) T = ( N M )( N T M T ) = N ( M M T ) N T = N IN T = I. KOCZKODAJ, MAREK, AND YAYLI G is closed and commutative under Hadamard product. Consequently, wesee that ( G , · ) is an abelian group. (cid:4) Definition 4.4.
Let G be PC matrix Lie group and M ( t ) be a path through G . We say that M ( t ) is smooth if each entry in M ( t ) is differentiable. Thederivative of M ( t ) at the point t is denoted M (cid:48) ( t ) which is the matrix whose ij th element is the derivative of ij th element of M ( t ). Corollary 4.5.
The abelian group G is a PC matrix Lie group.Proof. We know that the Hadamard product ” · ” and the operation M −→ M − = M T are differentiable for every PC matrix M.Thus, G is a PC matrix Lie Group. (cid:4) We also know that the tangent space of any matrix Lie group at unityis a vector space.The tangent space of any matrix group G at unity I will be denoted by T I ( G ) = g where I is the unit matrix of G . Theorem 4.6.
The tangent space of the PC matrix Lie Group G at unity I consists of all n × n real matrices X that satisfy X + X T = 0 .Proof. Recall that any matrix A ∈ G satisfies the condition A · A T = I . Letus consider a smooth path A ( t ) such that A (0) = I .We know that:(4.1) A ( t ) · A ( t ) T = I for all parameters t . By differentiating the equation 4.1, we get(4.2) ddt ( A ( t )) · A ( t ) T + A ( t ) · ddt ( A ( t ) T ) = 0 C MATRIX DECOMPOSITION 9
Considering that: ddt ( A ( t ) T ) = ( ddt ( A ( t ))) T the equation 4.2 can be rewritten as A (cid:48) ( t ) · A ( t ) T + A ( t ) · A (cid:48) ( t ) T = 0and at the point t = 0, we obtain: A (cid:48) (0) + A (cid:48) (0) T = 0Thus, any tangent vector X = A (cid:48) (0) satisfies X + X T = 0. (cid:4) Corollary 4.7.
The Lie algebra of G , denoted by T I ( G ) , is a Lie algebra of G and T I ( G ) is the space of the skew-symmetric n × n matrices. Notice that: dim ( G ) = dimT I ( G ) = n · ( n − . Exponential map and PC matrices
The exponential map is a map from Lie algebra of a given Lie group tothat group. In this Section, we will introduce the exponential transforma-tion from g (the tangent space to the identity element of PC matrix Liegroup G ) to G .Let G be PC matrix Lie group and g be the Lie algebra of G . Then, theexponential map: exp : g −→ G A = [ a ij ] n × n −→ exp [ A ] = [ e a ij ] can be defined so that the following properties (1)-(6) hold:(1) G = { δ ( t ) = e tA | t ∈ R , A ∈ g } is one parameter subgroup of G .(2) Let A and B be two elements of the Lie algebra g . Then, the followingequality holds: e A + B = e A · e B (3) Given any matrix A ∈ g , the tangent vector of the smooth path γ ( t )is equal to A · e tA , that is, γ (cid:48) ( t ) = ddt e tA = A · e tA (4) For any matrix A ∈ g ,( e A ) − = e − A = ( e A ) T = e A T and ( e A )( e A ) T = e A · e A T = e A + A T = e = 1 . (5) For any matrix A ∈ g , γ ( t ) = e tA is a geodesic curve of the pc matrixLie group G passing through the point γ (0) = 1.(6) For any matrix A ∈ g , we would like to stipulate that det ( e A ) = e T rA where
T r ( A ) is the trace function of A . However, this cannot alwaysbe achieved and a counterexample is presented in the Example 5.1. Example 5.1.
Let us consider the following matrix A = − − C MATRIX DECOMPOSITION 11 A is the element of g , hence the exponential map of A is: e A = e ee e The determinant of e A is: det ( e A ) = e + e − − e A is: T r ( A ) = Σ i =1 a i,i = 0Consequently, det ( e A ) is not equal to e T r ( A ) for the matrix A .6. Internal direct product of Lie group of × PC Matrices
The aim of this section is to provide both geometric and algebraic per-spectives on PC matrices. Our presentation is based on the work [1],[2] and[4]. However, a modified approach is used in this Section. Let us recall thatwe consider only 3 × n >
3. Let us introduce the definition of the internal direct product.We use the notation I n = { , . . . n } . Definition 6.1.
Let G be a group and let { N i | i ∈ I n } be a family of normalsubgroups of G . Then G = N N · · · N n is called the internal direct product of { N i | i ∈ I n } and N i ∩ ( N · · · N i − N i +1 · · · N n ) = { e } for all i ∈ { , , . . . , n } (see [10]). Theorem 6.2.
Let G be a group and { N i | i ∈ I n } be a family of normalsubgroups of G . Then G is an internal direct product of { N i | i ∈ I n } if and only if for all A ∈ G , A can be uniquely expressed as A = a · a · · · a n where a i ∈ N i , i ∈ { , . . . , n } (see [10] ) Theorem 6.3.
Let G be the internal direct product of a family of normalsubgroups { N i | i ∈ I n } . Then G (cid:39) N × N × · · · × N n . The collection M n = ( M n , · ) of all consistent PC matrices M is a multi-plicative Lie subgroup of the Lie group G .Moreover, let us consider additive consistent matrices represented by thefollowing set: l = C g = { A = [ a i,j ] n × n ∈ g | a i,k + a k,j = a i,j } for every ( i, j, k ) ∈ T n where T ( n ) = { ( i, j, k ) ∈ { , , , . . . , n } : i < j < k } was considered in [1] .If A, B ∈ C g then A + B ∈ C g and kA ∈ C g . Thus C g is a Lie subalgebraof ( g, +) . Let us observe that the following equality holds: g = C g ⊕ ( C g ) ⊥ ( C g ) ⊥ = h. It follows that C G is a Lie subgroup of ( G , · ) ; therefore, the followingequality also holds: G = C G ( C G ) ⊥ Considering the above results, we provide the new geometric and alge-braic interpretation for 3 × C MATRIX DECOMPOSITION 13
Proposition 6.4.
Let h = (cid:40) x − x − x xx − x : x ∈ R + (cid:41) and l = (cid:40) y y + z − y z − y − z − z : y, z ∈ R + (cid:41) be two sets of × PC matrices. Then the following holds:(i) h and l are 1 and 2 dimensional Lie subalgebras of the Lie algebra of g ,(ii) the vector space denoted by the Lie algebra of h is the orthogonal com-plement space of the vector space denoted by the Lie algebra of l .Proof. (i) For the proof, let us use sp for the linear span of a set of vectors. h = sp (cid:40) − − − (cid:41) and l = sp (cid:40) − − , − − (cid:41) Since the space h produces one matrix, dim ( h ) = 1 and the space l produces two linearly independent vectors, dim ( l ) = 2. Moreover, h and l are Lie subalgebras of G .(ii) It is implied by the basic properties of Lie algebras. We will show that every element of the Lie algebra g can be writtenas the sum of one element of the Lie subalgebra h and one element of Liesubalgebra l , that is, for all A ∈ G , A = A h + A l , with A h ∈ h and A l ∈ l . a b − a c − b − c = x − x − x xx − x + y y + z − y z − y − z − z where x = 13 ( a − b + c ) y = 13 (2 a + b − c ) z = 13 (2 c − a + b ) (cid:4) Proposition 6.5.
Let H = (cid:40) N | N = k k k kk k , where k ∈ R + (cid:41) and L = (cid:40) W | W = y yz y z yz z , where y, z ∈ R + (cid:41) be two sets of × matrices. Then:(i) H and L are normal subgroups of the Lie group of G C MATRIX DECOMPOSITION 15 (ii) The Lie group G is the internal direct product of the normal Lie sub-groups H and L . In particular: G (cid:39) H × L. Proof.
We know that: G = (cid:40) M | M = m m m m m m , m , m , m ∈ R + (cid:41) Let us consider: A = ξ η ξ γ η γ ∈ G Using the logarithmic transformation we get:˜ A = ln ( ξ ) ln ( η ) − ln ( ξ ) 0 ln ( γ ) − ln ( η ) − ln ( γ ) 0 The Proposition 6.5 implies that:˜ A = x − x − x xx − x + y y + z − y z − y − z − z = A h + A l Moreover, we know that: x = ln ( ξ ) − ln ( η ) + ln ( γ )3 = ln ( ξγη ) / y = 2 ln ( ξ ) + ln ( η ) − ln ( γ )3 = ln ( ξ ηγ ) / z = 2 ln ( γ ) − ln ( ξ ) + ln ( η )3 = ln ( γ ηξ ) / Using the exponential transformation for ˜ A = A h + A l we conclude: exp ( ˜ A ) = exp ( A h ) · exp ( A l )which implies: A = A H · A L (cid:4) Observe that the matrix A L is a consistent PC matrix. However, thefollowing proposition needs to be considered for deciding on the classificationof the matrix A H . Proposition 6.6. (i) Lie algebra of the PC matrix Lie group of consis-tent L is l .(ii) Lie algebra of the PC matrix normal Lie subgroup H is h . From the above proposition for additive consistent matrices, we concludethat h is the orthogonal complement space of l . Hence every element of h can be classified as ortho-additive consistent matrix and every element of H is an ortho-consistent PC matrix. C MATRIX DECOMPOSITION 17 Generalization outline
Following [22], we would like to point out that PC matrices of the size n > × j , we need to also delete column j . Fig. 2 demonstrates how to obtainone PC matrix 3 × × Figure 2.
Generatization to n > n < m , A and B are square matrices of degree n and m respectively. We call matrix A a submatrix of B ( A ⊂ B ) if there exist injection σ : { , . . . , n } → { , . . . , m } such that for all n, m ∈ { , . . . , n } a ij = b σ ( i ) σ ( j ) . The PC matrix reconstruction from its 3 × S of geometric means of all corresponding ele-ments in these components. The need for geometric mean use comes fromthe occurrence of the same matrix element ( n − n − n − / S does not need to be consistenteven though all submatrices of S are consistent. However, the reconstructionprocess converges to a consistent PC matrix as partly proven by [11] andcompleted in [20]. The above reconstruction process will be analyzed in theplanned follow-up paper. Conclusions
Using fundamental theorems from [23] and [24], the collaborative re-search effort [17] demonstrated that group generalization for pairwise com-parisons matrices is a challenge. In particular, [17] provided the proof thatelements of a multiplicative PC matrix must be selected from a torsion-freeabelian group.Our paper demonstrates that the multiplicative PC matrices (not theelements of a PC matrix) generate a Lie group for the Hadamard product.Lie algebras of these Lie groups are identified here. It has been shown thatLie algebras form spaces of skew-symmetric matrices. It has also been proven
C MATRIX DECOMPOSITION 19 that the Lie group of PC matrices can be represented as an internal directproduct using the direct summability property of vector spaces.In conclusion, a relatively simple concept of pairwise comparisons turnsout to be related to the theory of Lie groups and Lie algebras (what iscommonly regarded as very sophisticated mathematics). For the first time,the decomposition of a PC matrix into an approximation component andorthogonal component (interpreted as the approximation error) was ob-tained. Without such decomposition, the pairwise comparisons method hasremained incomplete for 722 years from its first scholarly presentation.
Acknowledgments
We thank Tuˇg¸ce C¸ alci for the verification of mathematical formulas andthe terminology associated with them. The authors recognize the efforts ofTiffany Armstrong in proofreading this text and Lillian (Yingli) Song withthe technical editing. We also acknowledge that algebraic terminology andbasic concepts are based on [26].
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