A Class of New Metrics for -Dimensional Unit Hypercube
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 with the parameter α. We come to the conclusion that the sufficient and necessary condition of the function being a metric is . Finally, by computing basic examples of codons, we show some numerical comparison of the new metrics to the former metric.
As early as 1971, Zadeh introduced a geometric interpretation of fuzzy sets by stating that they can be represented as points in unit hypercube . 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  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 metricis named fuzzy polynucleotide spacewithwhich 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 on. Later, Dress, Lokot, and Pustyl’nikov have pointed out that the metric is under the-norm and showed the metric properties .
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 . 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 metricin the fuzzy polynucleotide space.
Letbe a fixed set; a fuzzy set inis defined by where The number denotes the membership degree of the elementin the fuzzy set. We can also use the unit hypercube to describe all the fuzzy sets in, because a fuzzy setdetermines a point . Reciprocally, any point generates a fuzzy set defined by , . A polynucleotide is representable as such an ordered fuzzy set in [4–6]. Given two fuzzy sets , , the metricis defined by With the metricdefined, the fuzzy polynucleotide space is constructed.
Let be a discrete random variable with alphabet ., are two probability distributions of . Then, the relative entropy between and is defined as Here,denotes the natural logarithm for convenience. Furthermore, the Jensen-Shannon divergence is defined by where .
Jensen-Shannon divergence is obviously nonnegative, symmetric and vanishes for , but it does not fulfill the triangle inequality. And a point in the n-dimensional is not a probability distribution. In view of the foregoing, the concept of Jensen-Shannon divergence should be generalized. If , are two points in , this function is studied: where . In the following sections, we discuss the function to all and obtain the class of new metrics.
3. Auxiliary Result Associated with
Definition 1. Let the function be defined by
In the above definition, we use convention based on continuity that .
To all , we wonder whether the function can be a metric on the space .
Lemma 2. , with equality only for .
Lemma 3. If the function is defined by with , then is convex function.
Proof. Straightforward derivative shows Using the standard inequality we find The equality holds if and only if . So , is convex function, and the function gets the minimumwhen for .
As a consequence of Lemma 3, when ,
Lemma 4. If the function is defined by with , then
Proof. As , using l’Hôspital’s rule we can obtain And in the case , in the case . Thus, the lemma holds.
Assuming , we introduce the function defined by
Lemma 5. The function has two minima, one at and the other at .
Proof. The derivative of the function is
So for and for . It showsis monotonic decreasing in and monotonic increasing in .
Next, consider the monotonicity ofin the open interval .
From (12), we have From Lemma 4, we have
Using (17) and (18), we obtain
Let then We have , , , and . From (10), The equality holds if and only if . This means , and the equality holds if and only if . From the above, is easily obtained, and if and only if . So with respect to variable in the open interval , and are both monotonic decreasing, and is also monotonic decreasing. As , and , we can see that has only one zero point in the open interval with respect to variable . As a consequence, has only one zero pointin the open interval with respect to variable . This means in the interval , in the interval . From this, we know has only one maximum and no minimum in the open interval .
As a result, the conclusion in the lemma is obtained.
Theorem 6. The function is a metric on the space .
Proof. From Lemma 2, the function with equality only for is proved. Hence, with equality only for . It is easy to see that . Because the formula holds, and from Lemma 5, the triangle inequality can be easily proved for any number.
Corollary 7. If , then the function is a metric on the space .
Proof. The properties of nonnegativity and symmetry can be proved using the same method in Theorem 6.
Let and , then which follows from the concavity of . Now a which satisfies can be found. Thus, This is the triangle inequality for the function .
Theorem 8. If, then function is not a metric on the space .
Proof. Assuming , let . Firstly, the formula holds:
The derivative of the function is
Let Using l’Hôspital’s rule, So According to the definition of derivative, there exists a such that, for any , This shows that the triangle inequality does not hold.
Theorem 9. If, then function is not a metric on the space .
Proof. Let and ; . Consider the following:
For and .
Because if , then and , we have . Thus, As a consequence,. 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 is a metric on the space if and only if .
4. Metric Property of
In this section, we mainly prove the following theorem.
Theorem 11. The function is a metric on the space if and only if .
Proof. When , , where . So the function is not a metric. When , From (6), we can get . It is easy to see that with equality only for , and . So what we are concerned with is whether the triangle inequality
holds for any.
When,, the triangle inequality (36) holds apparently. So we assumein the following.
Next consider the value ofin three cases, respectively,(i).
From Theorem 10, the inequalityholds. Applying Minkowski’s inequality, we have So the triangle inequality (36) holds:(ii).Let ; then using Theorem 9 and noticing , we find This means. The inequality is not consistent with triangle inequality (36):(iii).
Let where Then, .
Next, we proveandare not the extreme points of the function. From the symmetry, we only need to proveis not the extreme point.
By partial derivative, Since, we might as well assumeand. Consider the following Using (10), we have Ifis small enough, using (44), we find the inequality A straight result of (45) is So from (43), (46) can be Then, taking (42) and (43) into (47), we have
Therefore,is not the extreme point of the function. For the same reason,is also not the extreme point.
Using the definition of extreme point, there exists a point such that . As , , then . The inequality is not consistent with (36).
From what has been discussed above, the conclusion in the theorem is obtained.
5. Comparison between and
As [5, 6] mentioned, we focus on the RNA alphabet . Code U as : 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 , A is represented as , G is represented as . So any codon can correspond to a fuzzy set as a point in the 12-dimensional fuzzy polynucleotide space . For example, the codon CGU would be recorded as
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, 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.
For the metric (in ), ; ; ; ; .
For the metric , ; ; ; ; .
From the above, the is larger than . But the value does not change the relationship of the distances between different codons. This shows that the new metric 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 metric(in ), ; ;(2) For the metric, ; .
We apply the comparison to complete genomes. In , 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 , , , , , , , , , , , or , , , , , , , , , , , . For the metric (in ), For the metric ,
It is easy to obtain that is closer to 1 and also larger thanwhen .
6. Concluding Remarks
By the discussion in the above sections, we come to the main conclusion: when , are two points in the n-dimensional unit hypercube , is a metric if and only if .
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 pointbeyond the existence. So we adopt the method in case (ii) when .
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 parameter, we obtain the new class of metrics in the space. At last, we numerically compare the new metrics to the former metric by computing some basic examples of codons. This shows the improvement is comprehensive.
With , we can also study the metric space 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.
This paper is supported by the Fundamental Research Funds for the Central Universities (FRF-CU) Grant no. 2722013JC082.
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