A Class of New Metrics for 𝑛-Dimensional Unit Hypercube

Abstract

We study the metric on the n-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 𝐹𝐷_𝛼(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.

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(𝐼^𝑛,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,…,𝑥_𝑛}be a fixed set; a fuzzy set in𝑋is defined by 𝐴=𝑥,𝜇_𝐴(𝑥)∣𝑥∈𝑋, (1) where 𝜇_𝐴∶𝑋⟶𝐼=[0,1],𝑥↦𝜇_𝐴(𝑥). (2) The number 𝜇_𝐴(𝑥) denotes the membership degree of the element𝑥in the fuzzy set𝐴. We can also use the unit hypercube 𝐼^𝑛=[0,1]^𝑛 to describe all the fuzzy sets in𝑋, because a fuzzy set𝐴determines a point 𝑃=(𝜇_𝐴(𝑥₁),𝜇_𝐴(𝑥₂),…,𝜇_𝐴(𝑥_𝑛)). Reciprocally, any point 𝑃=(𝑎₁,𝑎₂,…,𝑎_𝑛)∈𝐼^𝑛 generates a fuzzy set 𝐴 defined by 𝜇_𝐴(𝑥_𝑖)=𝑎_𝑖, 𝑖=1,2,…,𝑛. A polynucleotide is representable as such an ordered fuzzy set in [4–6]. Given two fuzzy sets 𝑃=(𝑝₁,𝑝₂,…,𝑝_𝑛), 𝑄=(𝑞₁,𝑞₂,…,𝑞_𝑛)∈𝐼^𝑛, the metric𝑑is defined by 𝑑()=∑𝑃,𝑄^𝑛𝑖=1𝑝_𝑖−𝑞_𝑖∑^𝑛_𝑖=1max𝑝_𝑖,𝑞_𝑖. (3) With the metric𝑑defined, the fuzzy polynucleotide space is constructed.

Let 𝑌 be a discrete random variable with alphabet 𝒴={𝑦₁,𝑦₂,…,𝑦_𝑛}.𝑃, 𝑄are two probability distributions of 𝑌. Then, the relative entropy between 𝑃 and 𝑄 is defined as 𝐷(𝑃‖𝑄)=^𝑛_𝑖=1𝑝_𝑖𝑝ln^𝑖𝑞_𝑖. (4) Here,lndenotes the natural logarithm for convenience. Furthermore, the Jensen-Shannon divergence is defined by JSD()=1𝑃,𝑄2[𝐷(𝑃‖𝑅)+𝐷(𝑄‖𝑅)], (5) where 𝑅=(1/2)(𝑃+𝑄).

Jensen-Shannon divergence is obviously nonnegative, symmetric and vanishes for 𝑃=𝑄, but it does not fulfill the triangle inequality. And a point in the n-dimensional 𝐼^𝑛 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 𝐼^𝑛, this function 𝐹𝐷_𝛼(𝑃,𝑄) is studied: 𝐹𝐷_𝛼(𝑃,𝑄)=(2JSD(𝑃,𝑄))^𝛼=[𝐷(𝑃‖𝑅)+𝐷(𝑄‖𝑅)]^𝛼=^𝑛_𝑖=1𝑝_𝑖ln2𝑝_𝑖𝑝_𝑖+𝑞_𝑖+𝑞_𝑖ln2𝑞_𝑖𝑝_𝑖+𝑞_𝑖^𝛼, (6) 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 with 𝐹𝐷_𝛼(𝑃,𝑄)

Definition 1. Let the function 𝐽(𝑝,𝑞)∶[0,+∞)×[0,+∞)→ℝ be defined by 𝐽𝑝,𝑞=𝑝ln2𝑝𝑝+𝑞+𝑞ln2𝑞.𝑝+𝑞 (7)

In the above definition, we use convention based on continuity that 0ln(0/𝑥)=0,𝑥∈[0,+∞).

To all 𝛼∈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 𝐽=𝑝𝑝,𝑞𝑝+𝑞𝑝+𝑞ln𝑝/𝑝+𝑞+𝑞1/2𝑝+𝑞ln𝑞/𝑝+𝑞.1/2 (8) 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 function 𝑓∶[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 𝑓^′(𝑥)=ln2𝑥+𝑎+𝑥2𝑎𝑎+𝑥ln2𝑎,𝑓𝑎+𝑥_′′(𝑥)=𝑎(𝑎−𝑥−2𝑥ln(2𝑎/(𝑎+𝑥)))𝑥(𝑎+𝑥)². (9) Using the standard inequality 1ln𝑎≥1−𝑎,ln𝑎≤𝑎−1, (10) we find 𝑎−𝑥−2𝑥ln2𝑎𝑎+𝑥≥𝑎−𝑥−2𝑥2𝑎=()𝑎+𝑥−1𝑎−𝑥²𝑎+𝑥≥0. (11) 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 Lemma 3, when 𝑥≠𝑎, 1𝑎>(ln(2𝑎/(𝑎+𝑥)))²𝐽().𝑎,𝑥 (12)

Lemma 4. If the function 𝑔∶[0,+∞)→ℝ is defined by 𝑔(𝑥)=ln(2𝑥/(𝑎+𝑥))/𝐽(𝑎,𝑥) with 𝑎>0, then lim_𝑥→𝑎+𝑔(𝑥)=1𝑎,lim_{𝑥→𝑎−}𝑔(𝑥)=−1𝑎. (13)

Proof. As 𝑔²(𝑥)=(ln(2𝑥/(𝑎+𝑥)))²/𝐽(𝑎,𝑥), using l’Hôspital’s rule we can obtain lim_𝑥→𝑎𝑔²(𝑥)=lim_𝑥→𝑎2𝑎ln(2𝑥/(𝑎+𝑥))/𝑥(𝑎+𝑥)((=1ln2𝑥/𝑎+𝑥))𝑎. (14) And 𝑔(𝑥)<0 in the case 𝑥<𝑎, 𝑔(𝑥)>0 in the case 𝑥>𝑎. Thus, the lemma holds.

Assuming 0<𝑝<𝑞, we introduce the function ℎ∶[0,+∞)→[0,+∞) defined by ℎ(𝑟)=𝐽𝑝,𝑟+𝐽𝑞,𝑟. (15)

Lemma 5. The function ℎ(𝑟) has two minima, one at 𝑟=𝑝 and the other at 𝑟=𝑞.

Proof. The derivative of the function ℎ is ℎ^′(𝑟)=12ln2𝑟/𝑝+𝑟𝐽+𝑝,𝑟ln2𝑟/𝑝+𝑟𝐽.𝑞,𝑟 (16) 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 1𝑝>ln2𝑝/𝑝+𝑞𝐽,𝑝,𝑞1𝑞>ln2𝑞/𝑝+𝑞𝐽.𝑝,𝑞 (17) From Lemma 4, we have lim_𝑟→𝑝+ln2𝑟/𝑝+𝑟𝐽=𝑝,𝑟1𝑝,lim_{𝑟→𝑞−}ln2𝑟/𝑞+𝑟𝐽𝑞,𝑟=−1𝑞. (18)
Using (17) and (18), we obtain lim_𝑟→𝑝+ℎ^′(𝑟)=12lim_𝑟→𝑝+ln2𝑟/𝑝+𝑟𝐽+𝑝,𝑟ln2𝑝/𝑝+𝑞𝐽=1𝑝,𝑞21𝑝+ln2𝑝/𝑝+𝑞𝐽𝑝,𝑞>0,lim_{𝑟→𝑞−}ℎ^′(𝑟)=12ln2𝑞/𝑝+𝑞𝐽𝑝,𝑞+lim_{𝑟→𝑞−}ln2𝑟/𝑞+𝑟𝐽=1𝑞,𝑟2ln2𝑞/𝑝+𝑞𝐽−𝑝,𝑞1𝑞<0. (19)
Let 𝐴𝑦,𝑟=ln2𝑟/𝑦+𝑟𝐽=𝑦,𝑟ln2𝑟/𝑦+𝑟√𝑟𝐽=1𝑦/𝑟,1√𝑟𝐵𝑦,𝑟; (20) then 𝜕𝐵𝑦,𝑟=𝑦𝜕𝑟2𝑟ln2𝑟𝑦+𝑟+ln2𝑦𝑦+𝑟2𝑦+𝑦+𝑟ln2𝑟×2√𝑦+𝑟𝑟𝑦+𝑟𝐽𝑦,𝑟^3/2−1≜𝑦𝐶(𝑟)2√𝑟𝑦+𝑟𝐽𝑦,𝑟^3/2. (21) 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), 𝑦²ln2𝑦𝑦+𝑟+𝑟²ln2𝑟𝑦+𝑟≥𝑦²1−𝑦+𝑟2𝑦+𝑟²1−𝑦+𝑟=2𝑟𝑦−𝑟²2≥0. (22) 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 ℎ^′(𝑟)=12𝐴=1𝑝,𝑟+𝐴𝑞,𝑟2√𝑟𝐵𝑝,𝑟+𝐵𝑞,𝑟, (23)lim_𝑟→𝑝+𝐵(𝑝,𝑟)+𝐵(𝑞,𝑟)>0, and lim_{𝑟→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 Lemma 2, 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, 𝐽𝑝,𝑞^1/2=lim_𝑟→𝑝𝐽𝑝,𝑟^1/2+𝐽𝑞,𝑟^1/2=lim_𝑟→𝑞𝐽𝑝,𝑟^1/2+𝐽𝑞,𝑟^1/2, (24) and from Lemma 5, the triangle inequality 𝐽𝑝,𝑞^1/2≤𝐽𝑝,𝑟^1/2+𝐽𝑞,𝑟^1/2 (25) can be easily proved for any number𝑟∈𝐼.

Corollary 7. If 0<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 Theorem 6.
Let 𝑎,𝑏>0 and 0<𝛾<1, then 𝑎^𝛾+𝑏^𝛾>(𝑎+𝑏)^𝛾 which follows from the concavity of 𝑥^𝛾. Now a 𝛾 which satisfies 𝛼=(1/2)𝛾 can be found. Thus, 𝐽𝑝,𝑟^𝛼+𝐽𝑞,𝑟^𝛼=𝐽𝑝,𝑟^(1/2)𝛾+𝐽𝑞,𝑟^(1/2)𝛾≥𝐽𝑝,𝑟^1/2+𝐽𝑞,𝑟^1/2𝛾≥𝐽𝑝,𝑞^(1/2)𝛾=𝐽𝑝,𝑞^𝛼. (26) 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. Assuming 0<𝑝<𝑞, let 𝑙(𝑟)=(𝐽(𝑝,𝑟))^𝛼+(𝐽(𝑞,𝑟))^𝛼. Firstly, the formula holds: 𝐽𝑝,𝑞^𝛼=lim_𝑟→𝑝𝐽𝑝,𝑟^𝛼+𝐽𝑞,𝑟^𝛼=lim_𝑟→𝑞𝐽𝑝,𝑟^𝛼+𝐽𝑞,𝑟^𝛼. (27) The derivative of the function 𝑙 is 𝑙^′(𝑟)=𝛼ln2𝑟𝐽𝑝+𝑟𝑝,𝑟^𝛼−1+ln2𝑟𝐽𝑞+𝑟𝑞,𝑟^𝛼−1. (28)
Let 𝑚(𝑟)=ln2𝑟𝐽𝑝+𝑟𝑝,𝑟^{𝛼−11/(1−𝛼)}=ln2𝑟/𝑝+𝑟^1/(1−𝛼)𝐽.𝑝,𝑟 (29) Using l’Hôspital’s rule, lim_𝑟→𝑝𝑚(𝑟)=𝑝𝑟𝑝+𝑟(1−𝛼)ln2𝑟𝑝+𝑟^{(2𝛼−1)/(1−𝛼)}=0. (30) So lim_𝑟→𝑝𝑙^′(𝑟)=𝛼ln2𝑝𝐽𝑝+𝑞𝑝,𝑞^𝛼−1<0. (31) According to the definition of derivative, there exists a 𝛿>0 such that, for any 𝑠∈(𝑝,𝑝+𝛿), 𝐽𝑝,𝑞^𝛼=lim_𝑟→𝑝+𝐽𝑝,𝑟^𝛼+𝐽𝑞,𝑟^𝛼>𝐽𝑝,𝑠^𝛼+𝐽𝑞,𝑠^𝛼. (32) 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. Let 𝑝,𝑞∈𝐼 and 𝑝≠𝑞; 𝑟=(𝑝+𝑞)/2. Consider the following: 𝐽𝑝,𝑞−𝐽𝑝,𝑟−𝐽𝑞,𝑟=𝑝ln2𝑝𝑝+𝑞+𝑞ln2𝑞−𝑝+𝑞𝑝ln2𝑝𝑝+𝑟+𝑟ln2𝑟−𝑝+𝑟𝑟ln2𝑟𝑞+𝑟+𝑞ln2𝑞.𝑞+𝑟 (33) By substituting𝑟=(𝑝+𝑞)/2, 𝐽=𝑝,𝑞−𝐽𝑝,𝑟−𝐽𝑞,𝑟3𝑝+𝑞2ln3𝑝+𝑞+2𝑝+2𝑞𝑝+3𝑞2ln𝑝+3𝑞2𝑝+2𝑞=𝐽3𝑝+𝑞2,𝑝+3𝑞2>0. (34) For (3𝑝+𝑞)/2,(𝑝+3𝑞)/2∈𝐼 and (3𝑝+𝑞)/2≠(𝑝+3𝑞)/2.
Because if 0≤𝑥≤1, then 𝑥^𝛼≤𝑥 and (1−𝑥)^𝛼≤1−𝑥, we have 𝑥^𝛼+(1−𝑥)^𝛼≤1. Thus, 𝐽𝑝,𝑟𝐽𝑝,𝑞^𝛼+𝐽𝑞,𝑟𝐽𝑝,𝑞^𝛼<𝐽𝑝,𝑟𝐽𝑝,𝑟+𝐽𝑞,𝑟^𝛼+𝐽𝑞,𝑟𝐽𝑝,𝑟+𝐽𝑞,𝑟^𝛼≤𝐽𝑝,𝑟𝐽+𝐽𝑝,𝑟+𝐽𝑞,𝑟𝑞,𝑟𝐽𝑝,𝑟+𝐽𝑞,𝑟=1. (35) 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 of 𝐹𝐷_𝛼(𝑃,𝑄)

In this section, we mainly prove the following theorem.

Theorem 11. The function 𝐹𝐷_𝛼(𝑃,𝑄) is a metric on the space 𝐼^𝑛 if and only if 0<𝛼≤1/2.

Proof. When 𝛼≤0, 𝐷(𝑃‖𝑅)+𝐷(𝑄‖𝑅)=0, where 𝑃=𝑄. So the function 𝐹𝐷_𝛼(𝑃,𝑄)=[𝐷(𝑃‖𝑅)+𝐷(𝑄‖𝑅)]^𝛼 is not a metric. When 𝛼>0, From (6), we can get 𝐹𝐷_𝛼(𝑃,𝑄)=(∑^𝑛_𝑖=1𝐽(𝑝_𝑖,𝑞_𝑖))^𝛼. It is easy to see that 𝐹𝐷_𝛼(𝑃,𝑄)≥0with equality only for 𝑃=𝑄, and 𝐹𝐷_𝛼(𝑃,𝑄)=𝐹𝐷_𝛼(𝑄,𝑃). So what we are concerned with is whether the triangle inequality 𝐹𝐷_𝛼(𝑃,𝑄)≤𝐹𝐷_𝛼(𝑃,𝐻)+𝐹𝐷_𝛼(𝑄,𝐻) (36) holds for any𝑃,𝑄,𝐻∈𝐼^𝑛.
When𝑃=𝑄,𝐹𝐷_𝛼(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 Theorem 10, the inequality(𝐽(𝑝_𝑖,𝑞_𝑖))^𝛼≤(𝐽(𝑝_𝑖,ℎ_𝑖))^𝛼+(𝐽(𝑞_𝑖,ℎ_𝑖))^𝛼holds. Applying Minkowski’s inequality, we have ^𝑛_𝑖=1𝐽𝑝_𝑖,𝑞_𝑖^𝛼=^𝑛_𝑖=1𝐽𝑝_𝑖,𝑞_𝑖^{𝛼1/𝛼𝛼}≤^𝑛_𝑖=1𝐽𝑝_𝑖,ℎ_𝑖^𝛼+𝐽𝑞_𝑖,ℎ_𝑖^{𝛼1/𝛼𝛼}≤^𝑛_𝑖=1𝐽𝑝_𝑖,ℎ_𝑖^{𝛼1/𝛼𝛼}+^𝑛_𝑖=1𝐽𝑞_𝑖,ℎ_𝑖^{𝛼1/𝛼𝛼}=^𝑛_𝑖=1𝐽𝑝_𝑖,ℎ_𝑖^𝛼+^𝑛_𝑖=1𝐽𝑞_𝑖,ℎ_𝑖^𝛼. (37) 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 𝐹𝐷_𝛼(𝑃,𝑅)+𝐹𝐷_𝛼(𝑄,𝑅)𝐹𝐷_𝛼()=∑𝑃,𝑄_𝑛𝑖=1𝐽𝑝_𝑖,𝑟_𝑖^𝛼+∑^𝑛_𝑖=1𝐽𝑞_𝑖,𝑟_𝑖^𝛼∑^𝑛_𝑖=1𝐽𝑝_𝑖,𝑞_𝑖^𝛼<∑^𝑛_𝑖=1𝐽𝑝_𝑖,𝑟_𝑖^𝛼+∑^𝑛_𝑖=1𝐽𝑞_𝑖,𝑟_𝑖^𝛼∑^𝑛_𝑖=1𝐽𝑝_𝑖,𝑟_𝑖+𝐽𝑞_𝑖,𝑟_𝑖^𝛼≤1. (38) This means𝐹𝐷_𝛼(𝑃,𝑄)>𝐹𝐷_𝛼(𝑃,𝑅)+𝐹𝐷_𝛼(𝑄,𝑅). The inequality is not consistent with triangle inequality (36):(iii)1/2<𝛼<1.
Let 𝐹𝑥₁,…,𝑥_𝑛=𝐹₁𝑥₁,…,𝑥_𝑛+𝐹₂𝑥₁,…,𝑥_𝑛, (39) where 𝐹₁𝑥₁,…,𝑥_𝑛=^𝑛_𝑖=1𝑝_𝑖ln2𝑝_𝑖𝑝_𝑖+𝑥_𝑖+𝑥_𝑖ln2𝑥_𝑖𝑝_𝑖+𝑥_𝑖^𝛼,𝐹₂𝑥₁,…,𝑥_𝑛=^𝑛_𝑖=1𝑥_𝑖ln2𝑥_𝑖𝑞_𝑖+𝑥_𝑖+𝑞_𝑖ln2𝑞_𝑖𝑞_𝑖+𝑥_𝑖^𝛼. (40) Then, 𝐹(𝑝₁,…,𝑝_𝑛)=𝐹(𝑞₁,…,𝑞_𝑛)=𝐹𝐷_𝛼(𝑃,𝑄).
Next, we prove(𝑝₁,…,𝑝_𝑛)and(𝑞₁,…,𝑞_𝑛)are not the extreme points of the function𝐹(𝑥₁,…,𝑥_𝑛). From the symmetry, we only need to prove(𝑝₁,…,𝑝_𝑛)is not the extreme point.
By partial derivative,𝜕𝐹𝜕𝑥_{𝑖(𝑥1,…,𝑥𝑛)=(𝑝1,…,𝑝𝑛)}=𝜕𝐹₁𝜕𝑥_{𝑖(𝑥1,…,𝑥𝑛)=(𝑝1,…,𝑝𝑛)}+𝜕𝐹₂𝜕𝑥_{𝑖(𝑥1,…,𝑥𝑛)=(𝑝1,…,𝑝𝑛)}. (41) Since𝑃≠𝑄, we might as well assume𝑝₁≠𝑞₁and𝑝₁>0. Consider the following 𝜕𝐹₂𝜕𝑥_{1(𝑥1,…,𝑥𝑛)=(𝑝1,…,𝑝𝑛)}=𝛼ln2𝑝₁𝑝₁+𝑞₁⋅^𝑛_𝑖=1𝑝_𝑖ln2𝑝_𝑖𝑝_𝑖+𝑞_𝑖+𝑞_𝑖ln2𝑞_𝑖𝑝_𝑖+𝑞_𝑖^𝛼−1≠0,𝜕𝐹₁𝜕𝑥_{1(𝑥1,…,𝑥𝑛)=(𝑝1,…,𝑝𝑛)}=lim_Δ𝑥1→01Δ𝑥₁𝐹₁𝑝₁+Δ𝑥₁,…,𝑝_𝑛−𝐹₁𝑝₁,…,𝑝_𝑛=lim_Δ𝑥1→01Δ𝑥₁⋅𝑝₁ln2𝑝₁2𝑝₁+Δ𝑥₁+𝑝₁+Δ𝑥₁2𝑝ln¹+Δ𝑥₁2𝑝₁+Δ𝑥₁^𝛼. (42)(43) Using (10), we have ln2𝑝₁2𝑝₁+Δ𝑥₁≤2𝑝₁2𝑝₁+Δ𝑥₁−1=−Δ𝑥₁2𝑝₁+Δ𝑥₁,2𝑝ln¹+Δ𝑥₁2𝑝₁+Δ𝑥₁≤2𝑝₁+Δ𝑥₁2𝑝₁+Δ𝑥₁−1=Δ𝑥₁2𝑝₁+Δ𝑥₁. (44) IfΔ𝑥₁is small enough, using (44), we find the inequality 𝑝₁ln2𝑝₁2𝑝₁+Δ𝑥₁+𝑝₁+Δ𝑥₁2𝑝ln¹+Δ𝑥₁2𝑝₁+Δ𝑥₁^𝛼≤𝑝₁−Δ𝑥₁2𝑝₁+Δ𝑥₁+𝑝₁+Δ𝑥₁Δ𝑥₁2𝑝₁+Δ𝑥₁^𝛼=Δ𝑥₁^2𝛼2𝑝₁+Δ𝑥₁^𝛼. (45) A straight result of (45) is 10≤Δ𝑥₁⋅𝑝₁ln2𝑝₁2𝑝₁+Δ𝑥₁+𝑝₁+Δ𝑥₁2𝑝ln¹+Δ𝑥₁2𝑝₁+Δ𝑥₁^𝛼≤Δ𝑥₁^2𝛼−12𝑝₁+Δ𝑥₁^𝛼⟶0,ifΔ𝑥₁⟶0. (46) So from (43), (46) can be 𝜕𝐹₁𝜕𝑥_{1(𝑥1,…,𝑥𝑛)=(𝑝1,…,𝑝𝑛)}=0. (47) Then, taking (42) and (43) into (47), we have 𝜕𝐹𝜕𝑥_{1(𝑥1,…,𝑥𝑛)=(𝑝1,…,𝑝𝑛)}≠0. (48)
Therefore,(𝑝₁,…,𝑝_𝑛)is not the extreme point of the function𝐹(𝑥₁,…,𝑥_𝑛). For the same reason,(𝑞₁,…,𝑞_𝑛)is also not the extreme point.
Using the definition of extreme point, there exists a point 𝐻=(ℎ₁,…,ℎ_𝑛) such that 𝐹(ℎ₁,…,ℎ_𝑛)<𝐹(𝑝₁,…,𝑝_𝑛)=𝐹𝐷_𝛼(𝑃,𝑄). As 𝐹₁(ℎ₁,…,ℎ_𝑛)=𝐹𝐷_𝛼(𝑃,𝐻), 𝐹₂(ℎ₁,…,ℎ_𝑛)=𝐹𝐷_𝛼(𝑄,𝐻), then 𝐹𝐷_𝛼(𝑃,𝐻)+𝐹𝐷_𝛼(𝑄,𝐻)<𝐹𝐷_𝛼(𝑃,𝑄). The inequality is not consistent with (36).
From what has been discussed above, the conclusion in the theorem is obtained.

5. Comparison between 𝐹𝐷_1/2 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 (0,1,0,0,0,0,0,1,1,0,0,0)∈𝐼¹². (49)

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, (0.3,0.4,0.1,0.2,0,1,0,0,0,0,0,1)∈𝐼¹² (50) 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 𝐹𝐷_1/2, 𝐹𝐷_1/2(histidine,proline)=𝐹𝐷_1/2(CAU,CCG)=1.6651; 𝐹𝐷_1/2(histidine,serine)=𝐹𝐷_1/2(CAU,UCG)=2.0393; 𝐹𝐷_1/2(histidine,arginine)=𝐹𝐷_1/2(CAU,CGU)=1.1774; 𝐹𝐷_1/2(arginine,glutamine)=𝐹𝐷_1/2(CGU,CAG)=1.6651; 𝐹𝐷_1/2(lysine,glycine)=𝐹𝐷_1/2(AAA,GGG)=2.0393.

From the above, the 𝐹𝐷_1/2 is larger than 𝑑. But the value does not change the relationship of the distances between different codons. This shows that the new metric 𝐹𝐷_1/2 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 metric𝐹𝐷_1/2, 𝐹𝐷_1/2(XCG,proline)=𝐹𝐷_1/2(XCG,CCG)=0.7408; 𝐹𝐷_1/2(XCG,serine)=𝐹𝐷_1/2(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)∈𝐼¹² or (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]), 𝑑(𝑀.𝑡𝑢𝑏𝑒𝑟𝑐𝑢𝑙𝑜𝑠𝑖𝑠,𝐸.𝑐𝑜𝑙𝑖)=0.2483. (51) For the metric 𝐹𝐷_1/2, 𝐹𝐷_1/2(𝑀.𝑡𝑢𝑏𝑒𝑟𝑐𝑢𝑙𝑜𝑠𝑖𝑠,𝐸.𝑐𝑜𝑙𝑖)=0.2859. (52)

It is easy to obtain that 𝐹𝐷_𝛼 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: when 𝑃, 𝑄are two points in the n-dimensional unit hypercube 𝐼^𝑛, 𝐹𝐷_𝛼(𝑃,𝑄)is a metric if and only if 0<𝛼≤1/2.

In Section 4, 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 𝐹𝐷_𝛼 to the former metric 𝑑 by computing some basic examples of codons. This shows the improvement is comprehensive.

With 0<𝛼≤1/2, we can also study the metric space (𝐼^𝑛,𝐹𝐷_𝛼) 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.

Acknowledgment

This paper is supported by the Fundamental Research Funds for the Central Universities (FRF-CU) Grant no. 2722013JC082.