Khan et al. Journal of Inequalities and Applications (2017) 2017:63 DOI 10.1186/s13660-017-1333-z
RESEARCH
Open Access
Sequence spaces M(φ) and N(φ) with application in clustering Mohd Shoaib Khan1 , Badriah AS Alamri2 , M Mursaleen2,3* and QM Danish Lohani1 *
Correspondence: [email protected] 2 Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia 3 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India Full list of author information is available at the end of the article
Abstract Distance measures play a central role in evolving the clustering technique. Due to the rich mathematical background and natural implementation of lp distance measures, researchers were motivated to use them in almost every clustering process. Beside lp distance measures, there exist several distance measures. Sargent introduced a special type of distance measures m(φ ) and n(φ ) which is closely related to lp . In this paper, we generalized the Sargent sequence spaces through introduction of M(φ ) and N(φ ) sequence spaces. Moreover, it is shown that both spaces are BK-spaces, and one is a dual of another. Further, we have clustered the two-moon dataset by using an induced M(φ )-distance measure (induced by the Sargent sequence space M(φ )) in the k-means clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the M(φ )-distance measure in the k-means algorithm. MSC: 40H05; 46A45 Keywords: clustering; double sequence; k-means clustering; two-moon dataset
1 Introduction Clustering is a well-known procedure to deal with an unsupervised learning problem appearing in pattern recognition. Clustering is a process of organizing data into groups called clusters so that objects in the same cluster are similar to one another, but are dissimilar to objects in other clusters []. The main contribution in the field of clustering analysis was the pioneering work of MacQueen [] and Bezdek []. They had introduced highly significant clustering algorithms such as k-means [] and fuzzy c-means []. Among all clustering algorithms, k-means is the simplest unsupervised clustering algorithm that makes use of a minimum distance from the center, and it has many applications in scientific and industrial research [–] (for more information about the k-means clustering algorithm, see Section ). K-means algorithm is distance dependent, so its outputs vary with changing distance measures. Among all distance measures, a clustering process was usually carried out through the Euclidean distance measure [], but many times it failed to offer good results. In this paper, we define M(φ)- and N(φ)-distance measure. Further, M(φ)-distance is used to cluster two-moon dataset. The output result is compared with the result of Euclidean distance measure to show the efficacy of M(φ)-distance over the Euclidean distance measure. M(φ) and N(φ)-distance measures are the generalization of m(φ)- and n(φ)-distance measures introduced by Sargent [] and further studied by Mursaleen [, © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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] (to know more about m(φ) and n(φ), refer to [–]). The M(φ) and N(φ) spaces are closely related to lp distance measures. lp measures and its variance are mostly used to solve the problems evolving in the fields of Market prediction [], Machine Learning [], Pattern Recognition [], Clustering [] etc. Throughout the paper, by ω we denote the set of all real or complex sequences. Moreover, by l∞ , c and c we denote the Banach spaces of bounded, convergent and null sequences, respectively; and let lp be the Banach space of absolutely p-summable sequences with p-norm · p . For the following notions, we refer to [, ]. A double sequence x = (xjk ) of real or complex numbers is said to be bounded if x∞ < ∞, the space of all bounded double sequences is denoted by L∞ . A double sequence x = (xjk ) is said to converge to the limit L in Pringsheim’s sense (shortly, convergent to L) if for every ε > , there exists an integer N such that |xjk – L| < ε whenever j, k > N . In this case L is called the p-limit of x. If in addition x ∈ L∞ , then x is said to be boundedly convergent to L in Pringsheim’s sense (shortly, bp-convergent to L). A double sequence x = (xjk ) is said to converge regularly to L (shortly, r-convergent to L) if x is p-convergent and the limits xj := limk xjk (j ∈ N) and xk := limj xjk (k ∈ N) exist. Note that in this case the limits limj limk xjk and limk limj xjk exist and are equal to the p-limit of x. In general, for any notion of convergence ν, the space of all ν-convergent double sequences will be denoted by Cν and the limit of a ν-convergent double sequence x by ν- limj,k xjk , where ν ∈ {p, bp, r}. Let denote a vector space of all double sequences with the vector space operations defined coordinate-wise. Vector subspaces of are called double sequence spaces. Let us consider a double sequence x = {xmn } and define the sequence s = {smn } via x by
smn :=
m,n
xij
(m, n ∈ N).
i,j
Then the pair (x, s) and the sequence s = {smn } are called a double series and a sequence of partial sums of the double series, respectively. Let λ be the space of double sequences converging with respect to some linear convergence rule μ- lim : λ → R. The sum of a ∞,∞ double series ∞,∞ i,j= xij with respect to this rule is defined by μi,j= xij := μ- lim smn . Başar and Şever introduced the space Lp in [] |xmn |p < ∞ Lp := {xmn } ∈ :
( ≤ p < ∞)
m,n
corresponding to the space lp for p ≥ and examined some of its properties. Altay and Başar [] have generalized the spaces of double sequences L∞ , Cp and Cbp to L∞ (t) = {xmn } ∈ : sup |xmn |tmn < ∞ , m,n∈N
Cp (t) = {xmn } ∈ : p- lim |xmn – |tmn = , m,n→∞
and Cbp (t) = Cp (t) ∩ L∞ (t),
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respectively, where t = {tmn } is the sequence of strictly positive reals tmn . In the case tmn = , for all m, n ∈ N, L∞ (t), Cp (t) and Cbp (t) reduce to the sets L∞ , Cp and Cbp , respectively. Further, let C be the space whose elements are finite sets of distinct positive integers. Given any element σ of C, we denote by c(σ ) the sequence {cn (σ )} which is such that cn (σ ) = if n ∈ σ , cn (σ ) = otherwise. Further, let Cs = σ ∈ C :
∞
cn (σ ) ≤ s
n=
be the set of those σ whose support has cardinality at most s, and
φn ≤ (n = , , . . .) ,
= φ = {φn } ∈ ω : φ > , φn ≥ and n where φn = φn – φn– and φ = . For φ ∈ , the following sequence spaces were introduced and studied in [] by Sargent and further studied by Mursaleen in [, ]:
|xn | < ∞ , m(φ) = x = {xn } ∈ ω : sup sup s≥ σ ∈Cs φs n∈σ and
n(φ) = x = {xn } ∈ ω : sup
∞,∞
|un | φn < ∞ .
u∈S(x) m,n=,
Remark . (i) The spaces m(φ) and n(φ) are BK -spaces with their usual norms. (ii) If φn = (n = , , , . . .), then m(φ) = l [n(φ) = l∞ ], and if φn = n (n = , , , . . .), then m(φ) = l∞ [n(φ) = l ]. (iii) l ⊆ m(φ) ⊆ l∞ [l ⊆ n(φ) ⊆ l∞ ] for all φ ∈ . (iv) For any φ ∈ , m(φ) = lp [n(φ) = lq ], < p < ∞. In this paper, we define Sargent’s spaces for double sequences x = {xmn }. For this we first suppose U to be the set whose elements are finite sets of distinct elements of N × N obtained by σ × ς , where σ ∈ Cs and ς ∈ Ct for each s, t ≥ . Therefore, any element ζ of U means (m, n); m ∈ σ and n ∈ ς having cardinality at most st, where s is the cardinality with respect to m and t is the cardinality with respect to n. Given any element ζ of U, we denote by c(ζ ) the sequence {cmn (ζ )} such that ⎧ ⎨; if (m, n) ∈ ζ , cmn (ζ ) = ⎩; otherwise. Further, we write Ust = ζ ∈ U :
∞,∞ m,n=
cmn (ζ ) ≤ st
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for the set of those ζ whose support has cardinality at most st; and
φmn ≤ (m, n = , , . . .) , = φ = {φmn } ∈ : φ > , φmn ≥ and mn where φmn = φmn – φm–,n – φm,n– + φm–,n– and φ , φm , φn = , ∀m, n ∈ I+ . Through out the paper, we write m,n∈ζ for m∈σ n∈ς , and S(x) is used to denote the set of all double sequences that are rearrangements of x = {xmn } ∈ . For φ ∈ , we define the following sequence spaces:
M(φ) = x = {xmn } ∈ : xM(φ) = sup sup
s,t≥ ζ ∈Ust
|xmn | < ∞ φst m,n∈ζ
and N(φ) = x = {xmn } ∈ : xN(φ) = sup
∞,∞
|umn | φmn < ∞ .
u∈S(x) m,n=
Then the distances between x = {xmn } and y = {ymn } induced by M(φ) and N(φ) can be expressed as
dM(φ) = sup sup
s,t≥ ζ ∈Ust
|xmn – ymn | φst m,n∈ζ
and dN(φ) = sup
∞,∞
|umn – vmn | φmn .
u,v∈S(x) m,n=
Remark . If φst = (s, t = , , , . . .), then M(φ) = L [N(φ) = L∞ ], and if φst = st (s, t = , , , . . .), then M(φ) = L∞ [N(φ) = L ]. We now state the following known results of [] for single sequences (series) which can also be proved easily for double sequences (series). Lemma . If the series un xn is convergent for every x of a BK -space E, then the func∞ tional n= un xn is linear and continuous in E. Lemma . If E and F are BK -spaces, and if E ⊆ F, then there is a real number K such that, for all x of E, xF ≤ KxE .
2 Properties of the spaces M(φ ) and N(φ ) Theorem . The space M(φ) is a BK -space with the norm. xM(φ) = sup sup
s,t≥ ζ ∈Ust
φst
m,n∈ζ
|xmn | .
(.)
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Proof It is a routine verification to show that M(φ) is a normed space with the given norm (.), and so we omit it. Now, we proceed to showing that M(φ) is complete. Let {xl } be a Cauchy sequence in M(φ), where xl = {xlmn }∞,∞ m,n=, for every fixed l ∈ N. Then, for a given ε > , there exists a positive integer n (ε) > such that l x – xr = sup sup M(φ) s,t≥ ζ ∈Ust φst
l x – xr < ε mn mn
m,n∈ζ
for all l, r > n (ε), which yields, for each fixed s, t ≥ and ζ ∈ Ust , xl – xr ≤ εφ mn mn
for all l, r > n (ε).
(.)
m,n∈ζ
Therefore l r x – x < εφ mn mn m,n∈ζ
for all l, r > n (ε).
(.)
m,n∈ζ
This means that { m,n∈ζ |xlmn |}l∈N is a Cauchy sequence in R for every fixed s, t ≥ and ζ ∈ Ust . Since R is complete, it converges, say xl → |xmn | mn m,n∈ζ
as l → ∞.
m,n∈ζ
Since absolute convergence implies convergence in R, hence
xlmn →
m,n∈ζ
as l → ∞.
xmn
Hence we have l xmn – xmn lim l→∞
Let yl =
m,n∈ζ
sup
(.)
m,n∈ζ
m,n∈ζ
m,n∈ζ
= .
(.)
M(φ)
|xlmn |. Then {yl } ∈ l∞ . Therefore
xl ≤ k. mn
l∈N m,n∈ζ
Since m,n∈ζ |xmn | ≤ m,n∈ζ |xmn – xlmn | + m,n∈ζ |xlmn | ≤ εφ + k, it follows that x = {xmn } ∈ M(φ). Since {xl }l∈N was an arbitrary Cauchy sequence, the space M(φ) is complete. Now we prove that M(φ) has continuous coordinate projections pmn , where pmn : → K and pmn (x) = xmn . The coordinate projections pmn are continuous since |xmn | ≤ sups,t≥ supζ ∈Ust φst xM(φ) for each m, n ∈ N. Remark . The space N(φ) is a BK -space with the norm xN(φ) = sup
∞,∞
u∈S(x) m,n=
|umn | φmn .
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Lemma . (i) If x ∈ M(φ) [x ∈ N(φ)] and u ∈ S(x), then u ∈ M(φ) [u ∈ N(φ)] and u = x. (ii) If x ∈ M(φ) [x ∈ N(φ)] and |umn | ≤ |xmn | for every positive integer m, n, then u ∈ M(φ) [u ∈ N(φ)] and u ≤ x. Proof (i) Let x ∈ M(φ), then sups,t≥ supζ ∈Ust |xmn | < ∞ φst m,n∈ζ
φst
m,n∈ζ
|xmn | < ∞. So, we have
for each ζ ∈ Ust and s, t ≥ .
Since the sum of a finite number of terms remains the same for all the rearrangements, |umn | = |xmn | φst m,n∈ζ φst m,n∈ζ
for each u ∈ S(x) and ζ ∈ Ust , s, t ≥ .
Hence sup sup
s,t≥ ζ ∈Ust
|umn | = sup sup |xmn | < ∞, φst m,n∈ζ s,t≥ ζ ∈Ust φst m,n∈ζ
thus u ∈ M(φ) and u = x. (ii) By using the definition, easy to prove.
Theorem . For arbitrary φ ∈ , we have φ ∈ M(φ) and φM(φ) ≤ . Proof Let s and t be arbitrary positive integers, let σ , ς ∈ Ust , and let τ , τ constitute the element of σ and ς exceed by s and t respectively, also from the definition we have mn φ ≥ and ( φmn ) ≤ . Then n∈σ ,m∈ς
| φmn | ≤
s,t
φmn +
n=,m=
≤ φst +
n∈τ ,m∈τ
≤ φst +
φst st φs+,t (s+)t
+ .
≤ φst + st
φmn
n∈τ ,m∈τ
φm–,n– (m – )(n – ) φ
s,t+ + s(t+)
φ
s+,t+ + (s+)(t+) + .
φst = φst . st
φ
s,t+ + s(t+)
φ
s+,t+ + (s+)(t+) + .
⎫ +. . ⎪ ⎪ ⎪ ⎪ ⎬ +. . ⎪ . ⎪ ⎪ ⎪ ⎭ .
max(st)-terms
Lemma . If x ∈ M(φ) and {c , c , . . . , cn , c , c , . . . , cn , . . . , cm , cm , . . . , cmn } is a rearrangement of {b , b , . . . , bn , b , b , . . . , bn , . . . , bm , bm , . . . , bmn } such that |c | ≥ |c | ≥ · · · ≥ |cn |, |c | ≥ |c | ≥ · · · ≥ |cn |, . . . , |cm | ≥ |cm | ≥ · · · ≥ |cmn | and |c | ≥ |c | ≥ · · · ≥ |cm |, |c | ≥ |c | ≥ · · · ≥ |cm |, |cn | ≥ |cn | ≥ · · · ≥ |cnm |, then m,n i,j=,
|bij xij | ≤ xM(φ)
m,n i,j=,
|cij | φij .
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Proof In view of Lemma .(i), it is sufficient to consider the case when bij = cij (i = m,n , , . . . , m; j = , , . . . , n). Then writing Xmn = i,j= |xij |, we get m,n
|bij xij | =
i,j=,
m– n– |cij | – |ci,j+ | – |ci+,j | + |ci+,j+ | Xij + |cmn |Xmn i= j=
≤ xM(φ)
m– n–
|cij | – |ci,j+ | – |ci+,j | + |ci+,j+ | φij + xM(φ) |cmn |Xmn
i= j=
= xM(φ)
m,n
|cij | φij .
i,j=,
Hence we have
m,n
i,j=, |bij xij | ≤ xM(φ)
m,n
i,j=, |cij | φij .
Theorem . In order that uij xij be convergent [absolutely convergent] whenever x ∈ M(φ), it is necessary and sufficient that u ∈ N(φ). Further, if x ∈ M(φ) and u ∈ N(φ), then ∞,∞
|uij xij | ≤ uN(φ) xM(φ) .
(.)
i,j=,
Proof Necessity. We now suppose that from Lemma . we have
uij xij is convergent whenever x ∈ M(φ), then
∞,∞ uij xij ≤ KxM(φ) i,j=,
for some real number K and all x of M(φ). In view of Lemma .(ii), we may replace xij by xij sgn{uij }, obtaining ∞,∞
|uij xij | ≤ KxM(φ) .
(.)
i,j=,
Let v ∈ S(u). Then taking x to be a suitable rearrangement of φ, it follows from Eq. (.) and Theorem . and Lemma .(i) that ∞,∞
|vij | φij ≤ K,
i,j=,
and thus u ∈ N(φ). Sufficiency. If x ∈ M(φ) and u ∈ N(φ), it follows from Lemma . that for every positive integer m and n, ∞,∞ i,j=,
|uij xij | ≤ uN(φ) xM(φ) .
Theorem . In order that umn xmn be convergent [absolutely convergent] whenever x ∈ N(φ), it is necessary and sufficient that u ∈ M(φ).
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Proof Since sufficiency is included in Theorem ., we only consider necessity. We there fore suppose that umn xmn is convergent whenever x ∈ N(φ). By arguments similar to those used in Theorem ., we may therefore have that ∞,∞
|umn xmn | ≤ KxN(φ)
(.)
m,n=,
for some real number K and all x of N(φ). Let x = c(ζ ), where ζ ∈ Ust . Then x ∈ N(φ), and xN(φ) = sup
ξ ∈Ust m,n∈ξ
φmn ≤ φst ,
from Theorem . and Eq. (.) we have
|umn | ≤ Kφst
(ζ ∈ Ust ; s, t = , , , . . .),
m,n∈ζ
and thus u ∈ M(φ).
3 Inclusion relations for M(φ ) and N(φ ) Lemma . In order that M(φ) ⊆ M(ψ) [N(φ) ⊇ N(ψ)], it is necessary and sufficient that
φst sup ψ s,t≥ st
< ∞.
Proof Since each of the spaces M(φ) and N(φ) is the dual of the other, by Theorems . and ., the second version is equivalent to the first. Moreover, sufficiency follows from the definition of an M(φ) space. We therefore suppose that M(φ) ⊆ M(ψ). Since φ ∈ M(φ), it follows that ψ ∈ M(ψ), and hence we find that, for every positive integer s, t ≥ ,
φst =
s,t
φij ≤ ψst φM(ψ) ,
where = .
i,j=,
Theorem . (i) L ⊆ M(φ) ⊆ L∞ [L ⊆ N(φ) ⊆ L∞ ] for all φ of . (ii) M(φ) = L [N(φ) = L∞ ] if and only if bp- lims,t φst < ∞. (iii) M(φ) = L∞ [N(φ) = L ] if and only if bp- lims,t (φst /st) > . Proof We prove here the first version, while the second version follows by Theorems . and .. Since φ ≤ φmn ≤ mnφmn for all φ of , we have by Lemma . that (i) is satisfied. Further, from Lemma ., it follows that M(φ) ⊆ L if and only if sups,t≥ φst < ∞, while L∞ ⊆ M(φ) if and only if sups,t≥ (φst /st) < ∞; since the sequences {φst } and {st/φst } are monotonic, (ii) and (iii) are also satisfied. Theorem . Suppose that < p < ∞ and p + q = . Then (i) Given any φ of , M(φ) = Lp [N(φ) = Lq ]. (ii) In order that Lp ⊂ M(φ) [N(φ) ⊂ Lq ], it is necessary and sufficient that /q ) < ∞. sups,t≥ ( (st) φst
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(iii) In order that M(φ) ⊂ Lp [N(φ) ⊃ Lq ], it is necessary and sufficient that φ ∈ Lp . (iv) φ∈Lp M(φ) = Lp [ φ∈Lp N(φ) = Lq ]. Proof (i) Let us suppose that M(φ) = Lp . Then, by Lemma ., there exist real numbers r and r (r > , r > ) such that, for all x of M(φ), r xLp ≤ xM(φ) ≤ r xLp . Taking x = c(ζ ), where ζ ∈ Ust , we have that
r (st) p ≤
st ≤ r (st) p φst
(s, t = , , , . . .),
and hence that
(st) q r ≤ ≤ r φst
(s, t = , , , . . .).
In view of Lemma ., this implies that M(φ) = M(ψ), where ψ = {(mn) q }. Since ψ ∈ M(ψ) by Theorem ., but ψ ∈/ Lq , this leads to a contradiction. Hence (i) follows. (ii) If Lq ⊂ M(φ), arguments similar to those used in the proof of (i) show that (st)/q ≤ Kφst
(s, t = , , , . . .).
(.)
For sufficiency, we suppose that (.) is satisfied. Then, whenever x ∈ Lp and ζ ∈ Ust , m,n∈ζ
|xmn | ≤
m,n∈ζ
p q |xmn | ≤ xLq (st) q < Kφst xLq , p
m,n∈ζ
and hence x ∈ M(φ). In view of (i), it follows that Lq ⊂ M(φ). (iii) By Theorem ., we have φ ∈ M(φ). For sufficiency, we suppose that φ ∈ Lp and that x ∈ M(φ). Then {umn φmn } ∈ L whenever u ∈ Lq , and it therefore follows from Lemma . that {umn xmn } ∈ L whenever u ∈ Lq . Since Lp is the dual of Lq and since M(φ) = Lp , it follows that M(φ) ⊂ Lq . (iv) By using (iii) we have φ∈Lp Mφ ⊆ Lp . Now, for obtaining the complementary re lation Lp ⊆ φ∈Lp Mφ , let us suppose that x ∈ Lp . Then limm,n→∞ xmn = , and hence there is an element u of S(x) such that {|umn |} is a non-increasing sequence. If we take m,n |uij |}, then it is easy to verify that ψ ∈ and that x ∈ M(φ). Since ψ ∈ Lp , ψ = { i,j=, the complementary relation is satisfied.
4 Application of M(φ ) and N(φ ) in clustering In this section, we implement a k-means clustering algorithm by using M(φ)-distance measure. Further, we apply the k-means algorithm into clustering to cluster two-moon data. The clustering result obtained by the M(φ)-distance measure is compared with the results derived by the existing Euclidean distance measures (l ).
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4.1 Algorithm to compute M(φ ) distance Let x = [x , x , x , . . . , xn ]×n and y = [y , y , y , . . . , yn ]×n be two matrices of size × n, and let φm,n = φ,n = n. () Calculate ai = φ,i |xi – yi |, i = , , , . . . , n. () The M(φ)-distance between x and y is d, where d = max{a , a + a , . . . , a + a + · · · + an }.
4.2 K-means clustering algorithm for M(φ )-distance measure Let X = [x , x , x , . . . , xn ] be the data set. () Randomly/judiciously select k cluster centers (in this paper we choose first k data points as the cluster center y = [x , x , . . . , xk ]). () By using M(φ) or N(φ) distance measure (since both are dual of each other, in application point of view, we only consider M(φ)), compute the distance between each data points and cluster centers. () Put data points into the cluster whose M(φ)-distance with its center is minimum. () Define cluster centers for the new clusters evolved due to steps -, the new cluster k i centers are computed as follows: ci = ki j= xi , where ki denotes the number of points in the ith cluster. () Repeat the above process until the difference between two consecutive cluster centers reaches less than a small number ε. 4.3 Two-moon dataset clustering by using M(φ )-distance measure in k-means algorithm Two-moon dataset is a well-known nonconvex data set. It is an artificially designed two dimensional dataset consisting of data points []. Two-moon dataset is visualized as moon-shaped clusters (see Figure ). By using M(φ)-distance measure in the k-means clustering algorithm, the obtained result is represented in Figure . In Figure , we represent the result obtained by using the Euclidean distance measure in the k-means algorithm (we measure the accuracy of the cluster
Figure 1 Original shape of two-moon dataset.
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Figure 2 Clustering induced by M(φ )-distance measure.
Figure 3 Clustering induced by Euclidean distance l2 .
by using the formula, accuracy = (number of data points in the right cluster/total number of data points)). The experimental result shows that cluster accuracy of M(φ)-distance measure is .% while l -distance measure’s clustering accuracy is .%. Thus, M(φ)distance measure substantially improves the clustering accuracy.
5 Conclusions In this paper, we defined Banach spaces M(φ) and N(φ) with discussion of their mathematical properties. Further, we proved some of their inclusion relation. Furthermore, we applied the distance measure induced by the Banach space M(φ) into clustering to cluster the two-moon data by using the k-means clustering algorithm; the result of the experiment shows that the M(φ)-distance measure extensively improves the clustering accuracy. Competing interests The authors declare that they have no competing interests.
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Authors’ contributions All authors of the manuscript have read and agreed to its content and are accountable for all aspects of the accuracy and integrity of the manuscript. Author details 1 Department of Mathematics, South Asian University, New Delhi, India. 2 Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 3 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India. Acknowledgements The second and third authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia. Received: 14 November 2016 Accepted: 14 February 2017 References 1. MacQueen, J, et al.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281-297 (1967) 2. Bezdek, JC: A review of probabilistic, fuzzy, and neural models for pattern recognition. J. Intell. Fuzzy Syst. 1(1), 1-25 (1993) 3. Jain, AK: Data clustering: 50 years beyond k-means. Pattern Recognit. Lett. 31(8), 651-666 (2010) 4. Cheng, M-Y, Huang, K-Y, Chen, H-M: K-means particle swarm optimization with embedded chaotic search for solving multidimensional problems. Appl. Math. Comput. 219(6), 3091-3099 (2012) 5. Yao, H, Duan, Q, Li, D, Wang, J: An improved k-means clustering algorithm for fish image segmentation. Math. Comput. Model. 58(3), 790-798 (2013) 6. Cap, M, Prez, A, Lozano, JA: An efficient approximation to the k-means clustering for massive data. Knowl.-Based Syst. 117, 56-69 (2016) 7. Güngör, Z, Ünler, A: K-harmonic means data clustering with simulated annealing heuristic. Appl. Math. Comput. 184(2), 199-209 (2007) 8. Sargent, W: Some sequence spaces related to the p spaces. J. Lond. Math. Soc. 1(2), 161-171 (1960) 9. Mursaleen, M: Some geometric properties of a sequence space related to p . Bull. Aust. Math. Soc. 67(2), 343-347 (2003) 10. Mursaleen, M: Application of measure of noncompactness to infinite system of differential equations. Can. Math. Bull. 56, 388-394 (2013) 11. Chen, L, Ng, R: On the marriage of p -norms and edit distance. In: Proceedings of the Thirtieth International Conference on Very Large Data Bases, vol. 30, pp. 792-803 (2004) 12. Cristianini, N, Shawe-Taylor, J: An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, Cambridge (2000) 13. Xu, Z, Chen, J, Wu, J: Clustering algorithm for intuitionistic fuzzy sets. Inf. Sci. 178(19), 3775-3790 (2008) 14. Pringsheim, A: Zur theorie der zweifach unendlichen zahlenfolgen. Math. Ann. 53(3), 289-321 (1900) 15. Mursaleen, M, Mohiuddine, SA: Convergence Methods for Double Sequences and Applications. Springer, Berlin (2014) 16. Ba¸sar, F, S¸ ever, Y: The space Lq of double sequences. Math. J. Okayama Univ. 51, 149-157 (2009) 17. Altay, B, Ba¸sar, F: Some new spaces of double sequences. J. Math. Anal. Appl. 309(1), 70-90 (2005) 18. Wilansky, A: Summability Through Functional Analysis. North Holland Math. Stud, vol. 85 (1984) 19. Jain, AK, Law, MHC: Data clustering: a users dilemma. In: Proceedings of the First International Conference on Pattern Recognition and Machine Intelligence (2005) 20. Khan, MS, Lohani, QMD: A similarity measure for atanassov intuitionistic fuzzy sets and its application to clustering. In: Computational Intelligence (IWCI), International Workshop on. IEEE, Dhaka, Bangladesh (2016)
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Open Access
Sequence spaces M(φ) and N(φ) with application in clustering Mohd Shoaib Khan1 , Badriah AS Alamri2 , M Mursaleen2,3* and QM Danish Lohani1 *
Correspondence: [email protected] 2 Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia 3 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India Full list of author information is available at the end of the article
Abstract Distance measures play a central role in evolving the clustering technique. Due to the rich mathematical background and natural implementation of lp distance measures, researchers were motivated to use them in almost every clustering process. Beside lp distance measures, there exist several distance measures. Sargent introduced a special type of distance measures m(φ ) and n(φ ) which is closely related to lp . In this paper, we generalized the Sargent sequence spaces through introduction of M(φ ) and N(φ ) sequence spaces. Moreover, it is shown that both spaces are BK-spaces, and one is a dual of another. Further, we have clustered the two-moon dataset by using an induced M(φ )-distance measure (induced by the Sargent sequence space M(φ )) in the k-means clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the M(φ )-distance measure in the k-means algorithm. MSC: 40H05; 46A45 Keywords: clustering; double sequence; k-means clustering; two-moon dataset
1 Introduction Clustering is a well-known procedure to deal with an unsupervised learning problem appearing in pattern recognition. Clustering is a process of organizing data into groups called clusters so that objects in the same cluster are similar to one another, but are dissimilar to objects in other clusters []. The main contribution in the field of clustering analysis was the pioneering work of MacQueen [] and Bezdek []. They had introduced highly significant clustering algorithms such as k-means [] and fuzzy c-means []. Among all clustering algorithms, k-means is the simplest unsupervised clustering algorithm that makes use of a minimum distance from the center, and it has many applications in scientific and industrial research [–] (for more information about the k-means clustering algorithm, see Section ). K-means algorithm is distance dependent, so its outputs vary with changing distance measures. Among all distance measures, a clustering process was usually carried out through the Euclidean distance measure [], but many times it failed to offer good results. In this paper, we define M(φ)- and N(φ)-distance measure. Further, M(φ)-distance is used to cluster two-moon dataset. The output result is compared with the result of Euclidean distance measure to show the efficacy of M(φ)-distance over the Euclidean distance measure. M(φ) and N(φ)-distance measures are the generalization of m(φ)- and n(φ)-distance measures introduced by Sargent [] and further studied by Mursaleen [, © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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] (to know more about m(φ) and n(φ), refer to [–]). The M(φ) and N(φ) spaces are closely related to lp distance measures. lp measures and its variance are mostly used to solve the problems evolving in the fields of Market prediction [], Machine Learning [], Pattern Recognition [], Clustering [] etc. Throughout the paper, by ω we denote the set of all real or complex sequences. Moreover, by l∞ , c and c we denote the Banach spaces of bounded, convergent and null sequences, respectively; and let lp be the Banach space of absolutely p-summable sequences with p-norm · p . For the following notions, we refer to [, ]. A double sequence x = (xjk ) of real or complex numbers is said to be bounded if x∞ < ∞, the space of all bounded double sequences is denoted by L∞ . A double sequence x = (xjk ) is said to converge to the limit L in Pringsheim’s sense (shortly, convergent to L) if for every ε > , there exists an integer N such that |xjk – L| < ε whenever j, k > N . In this case L is called the p-limit of x. If in addition x ∈ L∞ , then x is said to be boundedly convergent to L in Pringsheim’s sense (shortly, bp-convergent to L). A double sequence x = (xjk ) is said to converge regularly to L (shortly, r-convergent to L) if x is p-convergent and the limits xj := limk xjk (j ∈ N) and xk := limj xjk (k ∈ N) exist. Note that in this case the limits limj limk xjk and limk limj xjk exist and are equal to the p-limit of x. In general, for any notion of convergence ν, the space of all ν-convergent double sequences will be denoted by Cν and the limit of a ν-convergent double sequence x by ν- limj,k xjk , where ν ∈ {p, bp, r}. Let denote a vector space of all double sequences with the vector space operations defined coordinate-wise. Vector subspaces of are called double sequence spaces. Let us consider a double sequence x = {xmn } and define the sequence s = {smn } via x by
smn :=
m,n
xij
(m, n ∈ N).
i,j
Then the pair (x, s) and the sequence s = {smn } are called a double series and a sequence of partial sums of the double series, respectively. Let λ be the space of double sequences converging with respect to some linear convergence rule μ- lim : λ → R. The sum of a ∞,∞ double series ∞,∞ i,j= xij with respect to this rule is defined by μi,j= xij := μ- lim smn . Başar and Şever introduced the space Lp in [] |xmn |p < ∞ Lp := {xmn } ∈ :
( ≤ p < ∞)
m,n
corresponding to the space lp for p ≥ and examined some of its properties. Altay and Başar [] have generalized the spaces of double sequences L∞ , Cp and Cbp to L∞ (t) = {xmn } ∈ : sup |xmn |tmn < ∞ , m,n∈N
Cp (t) = {xmn } ∈ : p- lim |xmn – |tmn = , m,n→∞
and Cbp (t) = Cp (t) ∩ L∞ (t),
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respectively, where t = {tmn } is the sequence of strictly positive reals tmn . In the case tmn = , for all m, n ∈ N, L∞ (t), Cp (t) and Cbp (t) reduce to the sets L∞ , Cp and Cbp , respectively. Further, let C be the space whose elements are finite sets of distinct positive integers. Given any element σ of C, we denote by c(σ ) the sequence {cn (σ )} which is such that cn (σ ) = if n ∈ σ , cn (σ ) = otherwise. Further, let Cs = σ ∈ C :
∞
cn (σ ) ≤ s
n=
be the set of those σ whose support has cardinality at most s, and
φn ≤ (n = , , . . .) ,
= φ = {φn } ∈ ω : φ > , φn ≥ and n where φn = φn – φn– and φ = . For φ ∈ , the following sequence spaces were introduced and studied in [] by Sargent and further studied by Mursaleen in [, ]:
|xn | < ∞ , m(φ) = x = {xn } ∈ ω : sup sup s≥ σ ∈Cs φs n∈σ and
n(φ) = x = {xn } ∈ ω : sup
∞,∞
|un | φn < ∞ .
u∈S(x) m,n=,
Remark . (i) The spaces m(φ) and n(φ) are BK -spaces with their usual norms. (ii) If φn = (n = , , , . . .), then m(φ) = l [n(φ) = l∞ ], and if φn = n (n = , , , . . .), then m(φ) = l∞ [n(φ) = l ]. (iii) l ⊆ m(φ) ⊆ l∞ [l ⊆ n(φ) ⊆ l∞ ] for all φ ∈ . (iv) For any φ ∈ , m(φ) = lp [n(φ) = lq ], < p < ∞. In this paper, we define Sargent’s spaces for double sequences x = {xmn }. For this we first suppose U to be the set whose elements are finite sets of distinct elements of N × N obtained by σ × ς , where σ ∈ Cs and ς ∈ Ct for each s, t ≥ . Therefore, any element ζ of U means (m, n); m ∈ σ and n ∈ ς having cardinality at most st, where s is the cardinality with respect to m and t is the cardinality with respect to n. Given any element ζ of U, we denote by c(ζ ) the sequence {cmn (ζ )} such that ⎧ ⎨; if (m, n) ∈ ζ , cmn (ζ ) = ⎩; otherwise. Further, we write Ust = ζ ∈ U :
∞,∞ m,n=
cmn (ζ ) ≤ st
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for the set of those ζ whose support has cardinality at most st; and
φmn ≤ (m, n = , , . . .) , = φ = {φmn } ∈ : φ > , φmn ≥ and mn where φmn = φmn – φm–,n – φm,n– + φm–,n– and φ , φm , φn = , ∀m, n ∈ I+ . Through out the paper, we write m,n∈ζ for m∈σ n∈ς , and S(x) is used to denote the set of all double sequences that are rearrangements of x = {xmn } ∈ . For φ ∈ , we define the following sequence spaces:
M(φ) = x = {xmn } ∈ : xM(φ) = sup sup
s,t≥ ζ ∈Ust
|xmn | < ∞ φst m,n∈ζ
and N(φ) = x = {xmn } ∈ : xN(φ) = sup
∞,∞
|umn | φmn < ∞ .
u∈S(x) m,n=
Then the distances between x = {xmn } and y = {ymn } induced by M(φ) and N(φ) can be expressed as
dM(φ) = sup sup
s,t≥ ζ ∈Ust
|xmn – ymn | φst m,n∈ζ
and dN(φ) = sup
∞,∞
|umn – vmn | φmn .
u,v∈S(x) m,n=
Remark . If φst = (s, t = , , , . . .), then M(φ) = L [N(φ) = L∞ ], and if φst = st (s, t = , , , . . .), then M(φ) = L∞ [N(φ) = L ]. We now state the following known results of [] for single sequences (series) which can also be proved easily for double sequences (series). Lemma . If the series un xn is convergent for every x of a BK -space E, then the func∞ tional n= un xn is linear and continuous in E. Lemma . If E and F are BK -spaces, and if E ⊆ F, then there is a real number K such that, for all x of E, xF ≤ KxE .
2 Properties of the spaces M(φ ) and N(φ ) Theorem . The space M(φ) is a BK -space with the norm. xM(φ) = sup sup
s,t≥ ζ ∈Ust
φst
m,n∈ζ
|xmn | .
(.)
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Proof It is a routine verification to show that M(φ) is a normed space with the given norm (.), and so we omit it. Now, we proceed to showing that M(φ) is complete. Let {xl } be a Cauchy sequence in M(φ), where xl = {xlmn }∞,∞ m,n=, for every fixed l ∈ N. Then, for a given ε > , there exists a positive integer n (ε) > such that l x – xr = sup sup M(φ) s,t≥ ζ ∈Ust φst
l x – xr < ε mn mn
m,n∈ζ
for all l, r > n (ε), which yields, for each fixed s, t ≥ and ζ ∈ Ust , xl – xr ≤ εφ mn mn
for all l, r > n (ε).
(.)
m,n∈ζ
Therefore l r x – x < εφ mn mn m,n∈ζ
for all l, r > n (ε).
(.)
m,n∈ζ
This means that { m,n∈ζ |xlmn |}l∈N is a Cauchy sequence in R for every fixed s, t ≥ and ζ ∈ Ust . Since R is complete, it converges, say xl → |xmn | mn m,n∈ζ
as l → ∞.
m,n∈ζ
Since absolute convergence implies convergence in R, hence
xlmn →
m,n∈ζ
as l → ∞.
xmn
Hence we have l xmn – xmn lim l→∞
Let yl =
m,n∈ζ
sup
(.)
m,n∈ζ
m,n∈ζ
m,n∈ζ
= .
(.)
M(φ)
|xlmn |. Then {yl } ∈ l∞ . Therefore
xl ≤ k. mn
l∈N m,n∈ζ
Since m,n∈ζ |xmn | ≤ m,n∈ζ |xmn – xlmn | + m,n∈ζ |xlmn | ≤ εφ + k, it follows that x = {xmn } ∈ M(φ). Since {xl }l∈N was an arbitrary Cauchy sequence, the space M(φ) is complete. Now we prove that M(φ) has continuous coordinate projections pmn , where pmn : → K and pmn (x) = xmn . The coordinate projections pmn are continuous since |xmn | ≤ sups,t≥ supζ ∈Ust φst xM(φ) for each m, n ∈ N. Remark . The space N(φ) is a BK -space with the norm xN(φ) = sup
∞,∞
u∈S(x) m,n=
|umn | φmn .
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Lemma . (i) If x ∈ M(φ) [x ∈ N(φ)] and u ∈ S(x), then u ∈ M(φ) [u ∈ N(φ)] and u = x. (ii) If x ∈ M(φ) [x ∈ N(φ)] and |umn | ≤ |xmn | for every positive integer m, n, then u ∈ M(φ) [u ∈ N(φ)] and u ≤ x. Proof (i) Let x ∈ M(φ), then sups,t≥ supζ ∈Ust |xmn | < ∞ φst m,n∈ζ
φst
m,n∈ζ
|xmn | < ∞. So, we have
for each ζ ∈ Ust and s, t ≥ .
Since the sum of a finite number of terms remains the same for all the rearrangements, |umn | = |xmn | φst m,n∈ζ φst m,n∈ζ
for each u ∈ S(x) and ζ ∈ Ust , s, t ≥ .
Hence sup sup
s,t≥ ζ ∈Ust
|umn | = sup sup |xmn | < ∞, φst m,n∈ζ s,t≥ ζ ∈Ust φst m,n∈ζ
thus u ∈ M(φ) and u = x. (ii) By using the definition, easy to prove.
Theorem . For arbitrary φ ∈ , we have φ ∈ M(φ) and φM(φ) ≤ . Proof Let s and t be arbitrary positive integers, let σ , ς ∈ Ust , and let τ , τ constitute the element of σ and ς exceed by s and t respectively, also from the definition we have mn φ ≥ and ( φmn ) ≤ . Then n∈σ ,m∈ς
| φmn | ≤
s,t
φmn +
n=,m=
≤ φst +
n∈τ ,m∈τ
≤ φst +
φst st φs+,t (s+)t
+ .
≤ φst + st
φmn
n∈τ ,m∈τ
φm–,n– (m – )(n – ) φ
s,t+ + s(t+)
φ
s+,t+ + (s+)(t+) + .
φst = φst . st
φ
s,t+ + s(t+)
φ
s+,t+ + (s+)(t+) + .
⎫ +. . ⎪ ⎪ ⎪ ⎪ ⎬ +. . ⎪ . ⎪ ⎪ ⎪ ⎭ .
max(st)-terms
Lemma . If x ∈ M(φ) and {c , c , . . . , cn , c , c , . . . , cn , . . . , cm , cm , . . . , cmn } is a rearrangement of {b , b , . . . , bn , b , b , . . . , bn , . . . , bm , bm , . . . , bmn } such that |c | ≥ |c | ≥ · · · ≥ |cn |, |c | ≥ |c | ≥ · · · ≥ |cn |, . . . , |cm | ≥ |cm | ≥ · · · ≥ |cmn | and |c | ≥ |c | ≥ · · · ≥ |cm |, |c | ≥ |c | ≥ · · · ≥ |cm |, |cn | ≥ |cn | ≥ · · · ≥ |cnm |, then m,n i,j=,
|bij xij | ≤ xM(φ)
m,n i,j=,
|cij | φij .
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Proof In view of Lemma .(i), it is sufficient to consider the case when bij = cij (i = m,n , , . . . , m; j = , , . . . , n). Then writing Xmn = i,j= |xij |, we get m,n
|bij xij | =
i,j=,
m– n– |cij | – |ci,j+ | – |ci+,j | + |ci+,j+ | Xij + |cmn |Xmn i= j=
≤ xM(φ)
m– n–
|cij | – |ci,j+ | – |ci+,j | + |ci+,j+ | φij + xM(φ) |cmn |Xmn
i= j=
= xM(φ)
m,n
|cij | φij .
i,j=,
Hence we have
m,n
i,j=, |bij xij | ≤ xM(φ)
m,n
i,j=, |cij | φij .
Theorem . In order that uij xij be convergent [absolutely convergent] whenever x ∈ M(φ), it is necessary and sufficient that u ∈ N(φ). Further, if x ∈ M(φ) and u ∈ N(φ), then ∞,∞
|uij xij | ≤ uN(φ) xM(φ) .
(.)
i,j=,
Proof Necessity. We now suppose that from Lemma . we have
uij xij is convergent whenever x ∈ M(φ), then
∞,∞ uij xij ≤ KxM(φ) i,j=,
for some real number K and all x of M(φ). In view of Lemma .(ii), we may replace xij by xij sgn{uij }, obtaining ∞,∞
|uij xij | ≤ KxM(φ) .
(.)
i,j=,
Let v ∈ S(u). Then taking x to be a suitable rearrangement of φ, it follows from Eq. (.) and Theorem . and Lemma .(i) that ∞,∞
|vij | φij ≤ K,
i,j=,
and thus u ∈ N(φ). Sufficiency. If x ∈ M(φ) and u ∈ N(φ), it follows from Lemma . that for every positive integer m and n, ∞,∞ i,j=,
|uij xij | ≤ uN(φ) xM(φ) .
Theorem . In order that umn xmn be convergent [absolutely convergent] whenever x ∈ N(φ), it is necessary and sufficient that u ∈ M(φ).
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Proof Since sufficiency is included in Theorem ., we only consider necessity. We there fore suppose that umn xmn is convergent whenever x ∈ N(φ). By arguments similar to those used in Theorem ., we may therefore have that ∞,∞
|umn xmn | ≤ KxN(φ)
(.)
m,n=,
for some real number K and all x of N(φ). Let x = c(ζ ), where ζ ∈ Ust . Then x ∈ N(φ), and xN(φ) = sup
ξ ∈Ust m,n∈ξ
φmn ≤ φst ,
from Theorem . and Eq. (.) we have
|umn | ≤ Kφst
(ζ ∈ Ust ; s, t = , , , . . .),
m,n∈ζ
and thus u ∈ M(φ).
3 Inclusion relations for M(φ ) and N(φ ) Lemma . In order that M(φ) ⊆ M(ψ) [N(φ) ⊇ N(ψ)], it is necessary and sufficient that
φst sup ψ s,t≥ st
< ∞.
Proof Since each of the spaces M(φ) and N(φ) is the dual of the other, by Theorems . and ., the second version is equivalent to the first. Moreover, sufficiency follows from the definition of an M(φ) space. We therefore suppose that M(φ) ⊆ M(ψ). Since φ ∈ M(φ), it follows that ψ ∈ M(ψ), and hence we find that, for every positive integer s, t ≥ ,
φst =
s,t
φij ≤ ψst φM(ψ) ,
where = .
i,j=,
Theorem . (i) L ⊆ M(φ) ⊆ L∞ [L ⊆ N(φ) ⊆ L∞ ] for all φ of . (ii) M(φ) = L [N(φ) = L∞ ] if and only if bp- lims,t φst < ∞. (iii) M(φ) = L∞ [N(φ) = L ] if and only if bp- lims,t (φst /st) > . Proof We prove here the first version, while the second version follows by Theorems . and .. Since φ ≤ φmn ≤ mnφmn for all φ of , we have by Lemma . that (i) is satisfied. Further, from Lemma ., it follows that M(φ) ⊆ L if and only if sups,t≥ φst < ∞, while L∞ ⊆ M(φ) if and only if sups,t≥ (φst /st) < ∞; since the sequences {φst } and {st/φst } are monotonic, (ii) and (iii) are also satisfied. Theorem . Suppose that < p < ∞ and p + q = . Then (i) Given any φ of , M(φ) = Lp [N(φ) = Lq ]. (ii) In order that Lp ⊂ M(φ) [N(φ) ⊂ Lq ], it is necessary and sufficient that /q ) < ∞. sups,t≥ ( (st) φst
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(iii) In order that M(φ) ⊂ Lp [N(φ) ⊃ Lq ], it is necessary and sufficient that φ ∈ Lp . (iv) φ∈Lp M(φ) = Lp [ φ∈Lp N(φ) = Lq ]. Proof (i) Let us suppose that M(φ) = Lp . Then, by Lemma ., there exist real numbers r and r (r > , r > ) such that, for all x of M(φ), r xLp ≤ xM(φ) ≤ r xLp . Taking x = c(ζ ), where ζ ∈ Ust , we have that
r (st) p ≤
st ≤ r (st) p φst
(s, t = , , , . . .),
and hence that
(st) q r ≤ ≤ r φst
(s, t = , , , . . .).
In view of Lemma ., this implies that M(φ) = M(ψ), where ψ = {(mn) q }. Since ψ ∈ M(ψ) by Theorem ., but ψ ∈/ Lq , this leads to a contradiction. Hence (i) follows. (ii) If Lq ⊂ M(φ), arguments similar to those used in the proof of (i) show that (st)/q ≤ Kφst
(s, t = , , , . . .).
(.)
For sufficiency, we suppose that (.) is satisfied. Then, whenever x ∈ Lp and ζ ∈ Ust , m,n∈ζ
|xmn | ≤
m,n∈ζ
p q |xmn | ≤ xLq (st) q < Kφst xLq , p
m,n∈ζ
and hence x ∈ M(φ). In view of (i), it follows that Lq ⊂ M(φ). (iii) By Theorem ., we have φ ∈ M(φ). For sufficiency, we suppose that φ ∈ Lp and that x ∈ M(φ). Then {umn φmn } ∈ L whenever u ∈ Lq , and it therefore follows from Lemma . that {umn xmn } ∈ L whenever u ∈ Lq . Since Lp is the dual of Lq and since M(φ) = Lp , it follows that M(φ) ⊂ Lq . (iv) By using (iii) we have φ∈Lp Mφ ⊆ Lp . Now, for obtaining the complementary re lation Lp ⊆ φ∈Lp Mφ , let us suppose that x ∈ Lp . Then limm,n→∞ xmn = , and hence there is an element u of S(x) such that {|umn |} is a non-increasing sequence. If we take m,n |uij |}, then it is easy to verify that ψ ∈ and that x ∈ M(φ). Since ψ ∈ Lp , ψ = { i,j=, the complementary relation is satisfied.
4 Application of M(φ ) and N(φ ) in clustering In this section, we implement a k-means clustering algorithm by using M(φ)-distance measure. Further, we apply the k-means algorithm into clustering to cluster two-moon data. The clustering result obtained by the M(φ)-distance measure is compared with the results derived by the existing Euclidean distance measures (l ).
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4.1 Algorithm to compute M(φ ) distance Let x = [x , x , x , . . . , xn ]×n and y = [y , y , y , . . . , yn ]×n be two matrices of size × n, and let φm,n = φ,n = n. () Calculate ai = φ,i |xi – yi |, i = , , , . . . , n. () The M(φ)-distance between x and y is d, where d = max{a , a + a , . . . , a + a + · · · + an }.
4.2 K-means clustering algorithm for M(φ )-distance measure Let X = [x , x , x , . . . , xn ] be the data set. () Randomly/judiciously select k cluster centers (in this paper we choose first k data points as the cluster center y = [x , x , . . . , xk ]). () By using M(φ) or N(φ) distance measure (since both are dual of each other, in application point of view, we only consider M(φ)), compute the distance between each data points and cluster centers. () Put data points into the cluster whose M(φ)-distance with its center is minimum. () Define cluster centers for the new clusters evolved due to steps -, the new cluster k i centers are computed as follows: ci = ki j= xi , where ki denotes the number of points in the ith cluster. () Repeat the above process until the difference between two consecutive cluster centers reaches less than a small number ε. 4.3 Two-moon dataset clustering by using M(φ )-distance measure in k-means algorithm Two-moon dataset is a well-known nonconvex data set. It is an artificially designed two dimensional dataset consisting of data points []. Two-moon dataset is visualized as moon-shaped clusters (see Figure ). By using M(φ)-distance measure in the k-means clustering algorithm, the obtained result is represented in Figure . In Figure , we represent the result obtained by using the Euclidean distance measure in the k-means algorithm (we measure the accuracy of the cluster
Figure 1 Original shape of two-moon dataset.
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Figure 2 Clustering induced by M(φ )-distance measure.
Figure 3 Clustering induced by Euclidean distance l2 .
by using the formula, accuracy = (number of data points in the right cluster/total number of data points)). The experimental result shows that cluster accuracy of M(φ)-distance measure is .% while l -distance measure’s clustering accuracy is .%. Thus, M(φ)distance measure substantially improves the clustering accuracy.
5 Conclusions In this paper, we defined Banach spaces M(φ) and N(φ) with discussion of their mathematical properties. Further, we proved some of their inclusion relation. Furthermore, we applied the distance measure induced by the Banach space M(φ) into clustering to cluster the two-moon data by using the k-means clustering algorithm; the result of the experiment shows that the M(φ)-distance measure extensively improves the clustering accuracy. Competing interests The authors declare that they have no competing interests.
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Authors’ contributions All authors of the manuscript have read and agreed to its content and are accountable for all aspects of the accuracy and integrity of the manuscript. Author details 1 Department of Mathematics, South Asian University, New Delhi, India. 2 Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 3 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India. Acknowledgements The second and third authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia. Received: 14 November 2016 Accepted: 14 February 2017 References 1. MacQueen, J, et al.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281-297 (1967) 2. Bezdek, JC: A review of probabilistic, fuzzy, and neural models for pattern recognition. J. Intell. Fuzzy Syst. 1(1), 1-25 (1993) 3. Jain, AK: Data clustering: 50 years beyond k-means. Pattern Recognit. Lett. 31(8), 651-666 (2010) 4. Cheng, M-Y, Huang, K-Y, Chen, H-M: K-means particle swarm optimization with embedded chaotic search for solving multidimensional problems. Appl. Math. Comput. 219(6), 3091-3099 (2012) 5. Yao, H, Duan, Q, Li, D, Wang, J: An improved k-means clustering algorithm for fish image segmentation. Math. Comput. Model. 58(3), 790-798 (2013) 6. Cap, M, Prez, A, Lozano, JA: An efficient approximation to the k-means clustering for massive data. Knowl.-Based Syst. 117, 56-69 (2016) 7. Güngör, Z, Ünler, A: K-harmonic means data clustering with simulated annealing heuristic. Appl. Math. Comput. 184(2), 199-209 (2007) 8. Sargent, W: Some sequence spaces related to the p spaces. J. Lond. Math. Soc. 1(2), 161-171 (1960) 9. Mursaleen, M: Some geometric properties of a sequence space related to p . Bull. Aust. Math. Soc. 67(2), 343-347 (2003) 10. Mursaleen, M: Application of measure of noncompactness to infinite system of differential equations. Can. Math. Bull. 56, 388-394 (2013) 11. Chen, L, Ng, R: On the marriage of p -norms and edit distance. In: Proceedings of the Thirtieth International Conference on Very Large Data Bases, vol. 30, pp. 792-803 (2004) 12. Cristianini, N, Shawe-Taylor, J: An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, Cambridge (2000) 13. Xu, Z, Chen, J, Wu, J: Clustering algorithm for intuitionistic fuzzy sets. Inf. Sci. 178(19), 3775-3790 (2008) 14. Pringsheim, A: Zur theorie der zweifach unendlichen zahlenfolgen. Math. Ann. 53(3), 289-321 (1900) 15. Mursaleen, M, Mohiuddine, SA: Convergence Methods for Double Sequences and Applications. Springer, Berlin (2014) 16. Ba¸sar, F, S¸ ever, Y: The space Lq of double sequences. Math. J. Okayama Univ. 51, 149-157 (2009) 17. Altay, B, Ba¸sar, F: Some new spaces of double sequences. J. Math. Anal. Appl. 309(1), 70-90 (2005) 18. Wilansky, A: Summability Through Functional Analysis. North Holland Math. Stud, vol. 85 (1984) 19. Jain, AK, Law, MHC: Data clustering: a users dilemma. In: Proceedings of the First International Conference on Pattern Recognition and Machine Intelligence (2005) 20. Khan, MS, Lohani, QMD: A similarity measure for atanassov intuitionistic fuzzy sets and its application to clustering. In: Computational Intelligence (IWCI), International Workshop on. IEEE, Dhaka, Bangladesh (2016)
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