# A Class of New Metrics forn-Dimensional Unit Hypercube

Authors: Guoxiang Lu
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

We study the metric on then-dimensional unit hypercube. We introduce a class of new metrics for the space, which is information theoretically motivated and has close relation to Jensen-Shannon divergence. These metrics are obtained by discussing a function FDα(P,Q) with the parameter α. We come to the conclusion that the sufficient and necessary condition of the function being a metric is 0<α≤1/2. Finally, by computing basic examples of codons, we show some numerical comparison of the new metrics to the former metric.


---

## Body

## 1. Introduction

As early as 1971, Zadeh introduced a geometric interpretation of fuzzy sets by stating that they can be represented as points in unit hypercube [1]. Many years later, his idea was taken up by Kosko, who built a promising fuzzy-theoretical framework and geometry thereon [2, 3]. This geometry of fuzzy sets was used in [4] to develop the fuzzy polynucleotide space. He demonstrated a polynucleotide molecule as a point in an n-dimensional unit hypercube. This approach enabled us to make quantitative studies such as the measurement of distances, similarities, and dissimilarities between polynucleotide sequences. The n-dimensional unit hypercube enriched by a metricdis named fuzzy polynucleotide space(In,d)withI=[0,1]⊂ℝwhich is a metric space. Torres and Nieto [5, 6] considered the frequencies of the nucleotides at the three base sites of a codon in the coding sequence as fuzzy sets to give an example onI12. Later, Dress, Lokot, and Pustyl’nikov have pointed out that the metric is under theL1-norm and showed the metric properties [7].Because the fuzzy sets which come from the polynucleotide molecules reflect the information of those sequences, we may introduce the related concept in information theory to measure the differences between polynucleotide sequences. In information theory, the relative entropy (also called Kullback-Leibler divergence) is the most common measure to show two probability distributions. But it is not a metric for it does not satisfy symmetric and triangle inequality [8]. In the past time, many pieces of research [9–15] were made to improve the relative entropy. From those references, Jensen-Shannon divergence as an improvement of relative entropy received much attention. In this paper, a class of new metrics inspired by the Jensen-Shannon divergence are introduced in the n-dimensional unit hypercube. These metrics with information-theoretical property of logarithm can replace the former metricdin the fuzzy polynucleotide space.


## 2. Preliminaries

LetX={x1,x2,…,xn}be a fixed set; a fuzzy set inXis defined by
(1)A={(x,μA(x))∣x∈X},
where
(2)μA:X⟶I=[0,1],x↦μA(x).
The number μA(x) denotes the membership degree of the elementxin the fuzzy setA. We can also use the unit hypercube In=[0,1]n to describe all the fuzzy sets inX, because a fuzzy setAdetermines a point P=(μA(x1),μA(x2),…,μA(xn)). Reciprocally, any point P=(a1,a2,…,an)∈In generates a fuzzy set A defined by μA(xi)=ai, i=1,2,…,n. A polynucleotide is representable as such an ordered fuzzy set in [4–6]. Given two fuzzy sets P=(p1,p2,…,pn), Q=(q1,q2,…,qn)∈In, the metricdis defined by
(3)d(P,Q)=∑i=1n|pi-qi|∑i=1nmax{pi,qi}.
With the metricddefined, the fuzzy polynucleotide space is constructed.LetY be a discrete random variable with alphabet 𝒴={y1,y2,…,yn}.P, Qare two probability distributions of Y. Then, the relative entropy between P and Q is defined as
(4)D(P∥Q)=∑i=1n‍pilnpiqi.
Here,lndenotes the natural logarithm for convenience. Furthermore, the Jensen-Shannon divergence is defined by
(5)JSD(P,Q)=12[D(P∥R)+D(Q∥R)],
where R=(1/2)(P+Q).Jensen-Shannon divergence is obviously nonnegative, symmetric and vanishes forP=Q, but it does not fulfill the triangle inequality. And a point in the n-dimensional In is not a probability distribution. In view of the foregoing, the concept of Jensen-Shannon divergence should be generalized. If P,Q are two points in In, this function FDα(P,Q) is studied:
(6)FDα(P,Q)=(2JSD(P,Q))α=[D(P∥R)+D(Q∥R)]α=(∑i=1n‍(piln2pipi+qi+qiln2qipi+qi))α,
where α∈ℝ. In the following sections, we discuss the function to all α∈ℝ and obtain the class of new metrics.


## 3. Auxiliary Result Associated withFDα(P,Q)

Definition 1.
Let the functionJ(p,q):[0,+∞)×[0,+∞)→ℝ be defined by
(7)J(p,q)=pln2pp+q+qln2qp+q.In the above definition, we use convention based on continuity that0ln(0/x)=0,x∈[0,+∞).To allα∈ℝ, we wonder whether the function (J(p,q))α can be a metric on the space I.Lemma 2.
J
(
p
,
q
)
≥
0, with equality only for p=q.Proof.
From (7) we can get
(8)J(p,q)=(p+q)(pp+qlnp/(p+q)1/2+qp+qlnq/(p+q)1/2).
The formula above expresses that J(p,q) is the relative entropy of the probability distributions (p/(p+q),q/(p+q)) and (1/2,1/2). With the nonnegativity of the relative entropy [8] the lemma holds.Lemma 3.
If the functionf:[0,+∞)→ℝ is defined by f(x)=J(a,x)-a(ln(2a/(a+x)))2 with a>0, then f is convex function.Proof.
Straightforward derivative shows(9)f′(x)=ln2xa+x+2aa+xln2aa+x,f′′(x)=a(a-x-2xln(2a/(a+x)))x(a+x)2.
Using the standard inequality
(10)lna≥1-1a,lna≤a-1,
we find
(11)a-x-2xln2aa+x≥a-x-2x(2aa+x-1)=(a-x)2a+x≥0.
The equality holds if and only if x=a. So f′′(x)≥0, fis convex function, and the function f(x) gets the minimum0when x=a for f′(a)=0.As a consequence of Lemma3, when x≠a,
(12)1a>(ln(2a/(a+x)))2J(a,x).Lemma 4.
If the functiong:[0,+∞)→ℝ is defined by g(x)=ln(2x/(a+x))/J(a,x) with a>0, then
(13)limx→a+g(x)=1a,limx→a-g(x)=-1a.Proof.
Asg2(x)=(ln(2x/(a+x)))2/J(a,x), using l’Hôspital’s rule we can obtain
(14)limx→ag2(x)=limx→a2aln(2x/(a+x))/x(a+x)ln(2x/(a+x))=1a.
And g(x)<0 in the case x<a, g(x)>0 in the case x>a. Thus, the lemma holds.Assuming0<p<q, we introduce the function h:[0,+∞)→[0,+∞) defined by
(15)h(r)=J(p,r)+J(q,r).Lemma 5.
The functionh(r) has two minima, one at r=p and the other at r=q.Proof.
The derivative of the functionh is
(16)h′(r)=12(ln(2r/(p+r))J(p,r)+ln(2r/(p+r))J(q,r)).
So h′(r)<0 for r∈[0,p) and h′(r)>0 for r∈(q,+∞). It showshis monotonic decreasing in [0,p)and monotonic increasing in [q,+∞).
Next, consider the monotonicity ofhin the open interval (p,q).
From (12), we have
(17)1p>|ln(2p/(p+q))|J(p,q),1q>|ln(2q/(p+q))|J(p,q).
From Lemma 4, we have
(18)limr→p+|ln(2r/(p+r))|J(p,r)=1p,limr→q-|ln(2r/(q+r))|J(q,r)=-1q.
Using (17) and (18), we obtain
(19)limr→p+h′(r)=12(limr→p+ln(2r/(p+r))J(p,r)+ln(2p/(p+q))J(p,q))=12(1p+ln(2p/(p+q))J(p,q))>0,limr→q-h′(r)=12(ln(2q/(p+q))J(p,q)+limr→q-ln(2r/(q+r))J(q,r))=12(ln(2q/(p+q))J(p,q)-1q)<0.
Let(20)A(y,r)=ln(2r/(y+r))J(y,r)=ln(2r/(y+r))rJ(y/r,1)=1rB(y,r);
then
(21)∂B(y,r)∂r=(y(2rln2ry+r+ln2yy+r(2y+(y+r)ln2ry+r)))×(2r(y+r)(J(y,r))3/2)-1≜yC(r)2r(y+r)(J(y,r))3/2.
We have C(y)=0, C′(r)=ln(2r/(y+r))+(1/r)ln(2y/(y+r))(y+rln(2r/(y+r))), C′(y)=0, and C′′(r)=-(1/(r2(y+r)))(y2ln(2y/(y+r))+r2ln(2r/(y+r))). From (10),
(22)y2ln2yy+r+r2ln2ry+r≥y2(1-y+r2y)+r2(1-y+r2r)=(y-r)22≥0.
The equality holds if and only if r=y. This means C′′(r)≤0, and the equality holds if and only if r=y. From the above, C(r)≤0 is easily obtained, and C(r)=0 if and only if r=y. So with respect to variable r in the open interval (p,q), B(p,r) and B(q,r) are both monotonic decreasing, and B(p,r)+B(q,r) is also monotonic decreasing. As
(23)h′(r)=12(A(p,r)+A(q,r))=12r(B(p,r)+B(q,r)),limr→p+B(p,r)+B(q,r)>0, and limr→q-B(p,r)+B(q,r)<0, we can see that B(p,r)+B(q,r) has only one zero point in the open interval (p,q) with respect to variable r. As a consequence, h′(r) has only one zero pointx0in the open interval (p,q) with respect to variable r. This means h′(r)>0 in the interval (p,x0), h′(r)<0 in the interval (x0,q). From this, we know h has only one maximum and no minimum in the open interval (p,q).
As a result, the conclusion in the lemma is obtained.Theorem 6.
The function(J(p,q))1/2 is a metric on the space I.Proof.
From Lemma2, the function J(p,q)≥0 with equality only for p=q is proved. Hence, (J(p,q))1/2≥0 with equality only for p=q. It is easy to see that (J(p,q))1/2=(J(q,p))1/2. Because the formula holds,
(24)(J(p,q))1/2=limr→p(J(p,r))1/2+(J(q,r))1/2=limr→q(J(p,r))1/2+(J(q,r))1/2,
and from Lemma 5, the triangle inequality
(25)(J(p,q))1/2≤(J(p,r))1/2+(J(q,r))1/2
can be easily proved for any numberr∈I.Corollary 7.
If0<α<1/2, then the function (J(p,q))α is a metric on the space I.Proof.
The properties of nonnegativity and symmetry can be proved using the same method in Theorem6.
Leta,b>0 and 0<γ<1, then aγ+bγ>(a+b)γ which follows from the concavity of xγ. Now a γ which satisfies α=(1/2)γ can be found. Thus,
(26)(J(p,r))α+(J(q,r))α=(J(p,r))(1/2)γ+(J(q,r))(1/2)γ≥((J(p,r))1/2+(J(q,r))1/2)γ≥(J(p,q))(1/2)γ=(J(p,q))α.
This is the triangle inequality for the function (J(p,q))α.Theorem 8.
If1/2<α<1, then function (J(p,q))α is not a metric on the space I.Proof.
Assuming0<p<q, let l(r)=(J(p,r))α+(J(q,r))α. Firstly, the formula holds:
(27)(J(p,q))α=limr→p(J(p,r))α+(J(q,r))α=limr→q(J(p,r))α+(J(q,r))α.
The derivative of the function l is
(28)l′(r)=α(ln2rp+r(J(p,r))α-1+ln2rq+r(J(q,r))α-1).
Let(29)m(r)=(ln2rp+r(J(p,r))α-1)1/(1-α)=(ln(2r/(p+r)))1/(1-α)J(p,r).
Using l’Hôspital’s rule,
(30)limr→pm(r)=pr(p+r)(1-α)(ln2rp+r)(2α-1)/(1-α)=0.
So
(31)limr→pl′(r)=αln2pp+q(J(p,q))α-1<0.
According to the definition of derivative, there exists a δ>0 such that, for any s∈(p,p+δ),
(32)(J(p,q))α=limr→p+(J(p,r))α+(J(q,r))α>(J(p,s))α+(J(q,s))α.
This shows that the triangle inequality does not hold.Theorem 9.
Ifα≥1, then function (J(p,q))α is not a metric on the space I.Proof.
Letp,q∈I and p≠q; r=(p+q)/2. Consider the following:
(33)J(p,q)-J(p,r)-J(q,r)=pln2pp+q+qln2qp+q-(pln2pp+r+rln2rp+r)-(rln2rq+r+qln2qq+r).
By substitutingr=(p+q)/2,
(34)J(p,q)-J(p,r)-J(q,r)=3p+q2ln3p+q2p+2q+p+3q2lnp+3q2p+2q=J(3p+q2,p+3q2)>0.
For (3p+q)/2,(p+3q)/2∈I and (3p+q)/2≠(p+3q)/2.
Because if0≤x≤1, then xα≤x and (1-x)α≤1-x, we have xα+(1-x)α≤1. Thus,
(35)(J(p,r)J(p,q))α+(J(q,r)J(p,q))α<(J(p,r)J(p,r)+J(q,r))α+(J(q,r)J(p,r)+J(q,r))α≤J(p,r)J(p,r)+J(q,r)+J(q,r)J(p,r)+J(q,r)=1.
As a consequence,(J(p,q))α>(J(p,r))α+(J(q,r))α. This shows that the triangle inequality does not hold.To sum up the theorems and corollary above, we can obtain the main theorem.Theorem 10.
The function(J(p,q))α is a metric on the space I if and only if 0<α≤1/2.


## 4. Metric Property ofFDα(P,Q)

In this section, we mainly prove the following theorem.Theorem 11.
The functionFDα(P,Q) is a metric on the space In if and only if 0<α≤1/2.Proof.
Whenα≤0, D(P∥R)+D(Q∥R)=0, where P=Q. So the function FDα(P,Q)=[D(P∥R)+D(Q∥R)]α is not a metric. When α>0, From (6), we can get FDα(P,Q)=(∑i=1nJ(pi,qi))α. It is easy to see that FDα(P,Q)≥0with equality only for P=Q, and FDα(P,Q)=FDα(Q,P). So what we are concerned with is whether the triangle inequality
(36)FDα(P,Q)≤FDα(P,H)+FDα(Q,H)
holds for anyP,Q,H∈In.
WhenP=Q,FDα(P,Q)=0, the triangle inequality (36) holds apparently. So we assumeP≠Qin the following.
Next consider the value ofαin three cases, respectively,(i)
0<α≤1/2.
From Theorem10, the inequality(J(pi,qi))α≤(J(pi,hi))α+(J(qi,hi))αholds. Applying Minkowski’s inequality, we have
(37)(∑i=1n‍J(pi,qi))α={∑i=1n‍((J(pi,qi))α)1/α}α≤{∑i=1n‍((J(pi,hi))α+(J(qi,hi))α)1/α}α≤{∑i=1n‍((J(pi,hi))α)1/α}α+{∑i=1n‍((J(qi,hi))α)1/α}α=(∑i=1n‍J(pi,hi))α+(∑i=1n‍J(qi,hi))α.
So the triangle inequality (36) holds:(ii)
α≥1.Let R=(1/2)(P+Q); then using Theorem 9 and noticing P≠Q, we find
(38)FDα(P,R)+FDα(Q,R)FDα(P,Q)=(∑i=1nJ(pi,ri))α+(∑i=1nJ(qi,ri))α(∑i=1nJ(pi,qi))α<(∑i=1nJ(pi,ri))α+(∑i=1nJ(qi,ri))α(∑i=1n(J(pi,ri)+J(qi,ri)))α≤1.
This meansFDα(P,Q)>FDα(P,R)+FDα(Q,R). The inequality is not consistent with triangle inequality (36):(iii)
1/2<α<1.
Let(39)F(x1,…,xn)=F1(x1,…,xn)+F2(x1,…,xn),
where
(40)F1(x1,…,xn)=(∑i=1n‍(piln2pipi+xi+xiln2xipi+xi))α,F2(x1,…,xn)=(∑i=1n‍(xiln2xiqi+xi+qiln2qiqi+xi))α.
Then, F(p1,…,pn)=F(q1,…,qn)=FDα(P,Q).
Next, we prove(p1,…,pn)and(q1,…,qn)are not the extreme points of the functionF(x1,…,xn). From the symmetry, we only need to prove(p1,…,pn)is not the extreme point.
By partial derivative,(41)∂F∂xi|(x1,…,xn)=(p1,…,pn)=∂F1∂xi|(x1,…,xn)=(p1,…,pn)+∂F2∂xi|(x1,…,xn)=(p1,…,pn).
SinceP≠Q, we might as well assumep1≠q1andp1>0. Consider the following
(42)∂F2∂x1|(x1,…,xn)=(p1,…,pn)=αln2p1p1+q1·(∑i=1n‍(piln2pipi+qi+qiln2qipi+qi))α-1≠0,(43)∂F1∂x1|(x1,…,xn)=(p1,…,pn)=limΔx1→01Δx1(F1(p1+Δx1,…,pn)-F1(p1,…,pn))=limΔx1→01Δx1·(p1ln2p12p1+Δx1+(p1+Δx1)ln2(p1+Δx1)2p1+Δx1)α.
Using (10), we have
(44)ln2p12p1+Δx1≤2p12p1+Δx1-1=-Δx12p1+Δx1,ln2(p1+Δx1)2p1+Δx1≤2(p1+Δx1)2p1+Δx1-1=Δx12p1+Δx1.
IfΔx1is small enough, using (44), we find the inequality
(45)(p1ln2p12p1+Δx1+(p1+Δx1)ln2(p1+Δx1)2p1+Δx1)α≤(p1(-Δx12p1+Δx1)+(p1+Δx1)(Δx12p1+Δx1))α=(Δx1)2α(2p1+Δx1)α.
A straight result of (45) is
(46)0≤|1Δx1|·|(p1ln2p12p1+Δx1+(p1+Δx1)ln2(p1+Δx1)2p1+Δx1)α|≤|Δx1|2α-1(2p1+Δx1)α⟶0,ifΔx1⟶0.
So from (43), (46) can be
(47)∂F1∂x1|(x1,…,xn)=(p1,…,pn)=0.
Then, taking (42) and (43) into (47), we have
(48)∂F∂x1|(x1,…,xn)=(p1,…,pn)≠0.
Therefore,(p1,…,pn)is not the extreme point of the functionF(x1,…,xn). For the same reason,(q1,…,qn)is also not the extreme point.
Using the definition of extreme point, there exists a pointH=(h1,…,hn) such that F(h1,…,hn)<F(p1,…,pn)=FDα(P,Q). As F1(h1,…,hn)=FDα(P,H), F2(h1,…,hn)=FDα(Q,H), then FDα(P,H)+FDα(Q,H)<FDα(P,Q). The inequality is not consistent with (36).
From what has been discussed above, the conclusion in the theorem is obtained.


## 5. Comparison betweenFD1/2 and d

As [5, 6] mentioned, we focus on the RNA alphabet {U,C,A,G}. Code U as (1,0,0,0): 1 shows that the first letter U is present, 0 shows that the second letter C does not appear, 0 shows that the third letter A does not appear, and 0 shows that the fourth letter G does not appear. Thereby, C is represented as (0,1,0,0), A is represented as (0,0,1,0), G is represented as (0,0,0,1). So any codon can correspond to a fuzzy set as a point in the 12-dimensional fuzzy polynucleotide space I12. For example, the codon CGU would be recorded as
(49)(0,1,0,0,0,0,0,1,1,0,0,0)∈I12.However, there exist  some cases in which there is no sufficient knowledge about the chemical structure of a particular sequence. One therefore may deal with base sequences not necessarily at a corner of the hypercube, and some components of the fuzzy set are not either 0 or 1. For example,(50)(0.3,0.4,0.1,0.2,0,1,0,0,0,0,0,1)∈I12
expresses a codon XCG. In this case, the first letter X is unknown and corresponds to U to an extent of 0.3, C to an extent of 0.4, A to an extent of 0.1, G to an extent of 0.2.(
1
) For the metric d (in [5]),d(histidine,proline)=d(CAU,CCG)=0.8;d(histidine,serine)=d(CAU,UCG)=1;d(histidine,arginine)=d(CAU,CGU)=0.5;d(arginine,glutamine)=d(CGU,CAG)=0.8;d(lysine,glycine)=d(AAA,GGG)=1.(
2
) For the metric FD1/2,FD1/2(histidine,proline)=FD1/2(CAU,CCG)=1.6651;FD1/2(histidine,serine)=FD1/2(CAU,UCG)=2.0393;FD1/2(histidine,arginine)=FD1/2(CAU,CGU)=1.1774;FD1/2(arginine,glutamine)=FD1/2(CGU,CAG)=1.6651;FD1/2(lysine,glycine)=FD1/2(AAA,GGG)=2.0393.From the above, theFD1/2 is larger than d. But the value does not change the relationship of the distances between different codons. This shows that the new metric FD1/2 reflects more information of the difference between codons. Next, the distances between codon XCG mentioned and proline and serine are as follows:(1)
For the metricd(in [5]),d(XCG,proline)=d(XCG,CCG)=0.3333;d(XCG,serine)=d(XCG,UCG)=0.3784;(2)
For the metricFD1/2,FD1/2(XCG,proline)=FD1/2(XCG,CCG)=0.7408;FD1/2(XCG,serine)=FD1/2(XCG,UCG)=0.8271.We apply the comparison to complete genomes. In [5], Torres a.nd Nieto computed the frequencies of the nucleotides A, C, G, and T at the three base sites of a codon in two bacteria M. tuberculosis and E. coli and obtained two points corresponding to either(0.1632, 0.3089, 0.1724, 0.3556, 0.2036, 0.3145, 0.1763, 0.3056, 0.1645, 0.3461, 0.1593, 0.3302)∈I12or(0.1605, 0.2420, 0.2600, 0.3374, 0.3116, 0.2286, 0.2846, 0.1752, 0.2619, 0.2568, 0.1831, 0.2981)∈I12. For the metric d (in [5]),
(51)d(M.tuberculosis,E.coli)=0.2483.
For the metric FD1/2,
(52)FD1/2(M.tuberculosis,E.coli)=0.2859.It is easy to obtain thatFDα is closer to 1 and also larger thandwhen 0<α<1/2.


## 6. Concluding Remarks

By the discussion in the above sections, we come to the main conclusion: whenP, Qare two points in the n-dimensional unit hypercube In, FDα(P,Q)is a metric if and only if 0<α≤1/2.In Section4, the method in case (iii) can also be used to prove that triangle inequality (36) does not hold in case (ii). But the method in case (ii) is intuitive, and we can find one determinate pointRbeyond the existence. So we adopt the method in case (ii) when α≥1.In this paper, we extend the method in [2, 4–6] to discuss the new fuzzy polynucleotide space. By considering all the possible values of parameterα, we obtain the new class of metrics in the space. At last, we numerically compare the new metrics FDα to the former metric d by computing some basic examples of codons. This shows the improvement is comprehensive.With0<α≤1/2, we can also study the metric space (In,FDα) using the theory of metric space, such as the Pythagoras theorem, the isometric property, the isomorphism property, and the limit property in the future. We think the new metrics can interpret more biological significance for the sequences of the polynucleotide and be useful in the bioinformatics.



---

*Source: 942687-2013-10-23.xml*