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Sequence spaces \(M(\phi)\) and \(N(\phi)\) with application in clustering
Journal of Inequalities and Applications volume 2017, Article number: 63 (2017)
Abstract
Distance measures play a central role in evolving the clustering technique. Due to the rich mathematical background and natural implementation of \(l_{p}\) distance measures, researchers were motivated to use them in almost every clustering process. Beside \(l_{p}\) distance measures, there exist several distance measures. Sargent introduced a special type of distance measures \(m(\phi)\) and \(n(\phi)\) which is closely related to \(l_{p}\). In this paper, we generalized the Sargent sequence spaces through introduction of \(M(\phi)\) and \(N(\phi)\) sequence spaces. Moreover, it is shown that both spaces are BKspaces, and one is a dual of another. Further, we have clustered the twomoon dataset by using an induced \(M(\phi)\)distance measure (induced by the Sargent sequence space \(M(\phi)\)) in the kmeans clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the \(M(\phi)\)distance measure in the kmeans algorithm.
Introduction
Clustering is a wellknown 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 [1]. The main contribution in the field of clustering analysis was the pioneering work of MacQueen [1] and Bezdek [2]. They had introduced highly significant clustering algorithms such as kmeans [1] and fuzzy cmeans [2]. Among all clustering algorithms, kmeans 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 [3–6] (for more information about the kmeans clustering algorithm, see Section 5). Kmeans 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 [7], but many times it failed to offer good results. In this paper, we define \(M(\phi)\) and \(N(\phi)\)distance measure. Further, \(M(\phi )\)distance is used to cluster twomoon dataset. The output result is compared with the result of Euclidean distance measure to show the efficacy of \(M(\phi)\)distance over the Euclidean distance measure. \(M(\phi )\) and \(N(\phi)\)distance measures are the generalization of \(m(\phi)\) and \(n(\phi)\)distance measures introduced by Sargent [8] and further studied by Mursaleen [9, 10] (to know more about \(m(\phi )\) and \(n(\phi)\), refer to [8–10]). The \(M(\phi)\) and \(N(\phi)\) spaces are closely related to \(l_{p}\) distance measures. \(l_{p}\) measures and its variance are mostly used to solve the problems evolving in the fields of Market prediction [11], Machine Learning [12], Pattern Recognition [13], Clustering [20] etc.
Throughout the paper, by ω we denote the set of all real or complex sequences. Moreover, by \(l_{\infty}\), c and \(c_{0}\) we denote the Banach spaces of bounded, convergent and null sequences, respectively; and let \(l_{p}\) be the Banach space of absolutely psummable sequences with pnorm \({\Vert \cdot \Vert } _{p}\). For the following notions, we refer to [14, 15]. A double sequence \(x = (x_{jk})\) of real or complex numbers is said to be bounded if \(\Vert x \Vert _{\infty} < \infty\), the space of all bounded double sequences is denoted by \(\mathcal{L}_{\infty}\). A double sequence \(x = (x_{jk})\) is said to converge to the limit L in Pringsheim’s sense (shortly, convergent to L) if for every \(\varepsilon> 0\), there exists an integer N such that \(\vert x_{jk}  L\vert < \varepsilon\) whenever \(j,k > N\). In this case L is called the plimit of x. If in addition \(x \in\mathcal{L}_{\infty}\), then x is said to be boundedly convergent to L in Pringsheim’s sense (shortly, bpconvergent to L). A double sequence \(x = (x_{jk})\) is said to converge regularly to L (shortly, rconvergent to L) if x is pconvergent and the limits \(x_{j}: = \lim_{k}x_{jk}\) (\(j \in\mathbb {N} \)) and \(x^{k}: = \lim_{j}x_{jk}\) (\(k \in\mathbb{N} \)) exist. Note that in this case the limits \(\lim_{j}\lim_{k}x_{jk}\) and \(\lim_{k}\lim_{j}x_{jk}\) exist and are equal to the plimit of x. In general, for any notion of convergence ν, the space of all νconvergent double sequences will be denoted by \(\mathcal{C}_{\nu}\) and the limit of a νconvergent double sequence x by \(\nu\textrm{} \lim_{j,k}x_{jk}\), where \(\nu\in\{ p,\mathit{bp},r\}\).
Let Ω denote a vector space of all double sequences with the vector space operations defined coordinatewise. Vector subspaces of Ω are called double sequence spaces. Let us consider a double sequence \(x = \{ x_{mn}\}\) and define the sequence \(s = \{ s_{mn}\}\) via x by
Then the pair \((x,s)\) and the sequence \(s = \{ s_{mn}\}\) 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 \(\mu\textrm{} \lim:\lambda\to \mathbb{R}\). The sum of a double series \(\sum_{i,j = 1}^{\infty,\infty} x_{ij}\) with respect to this rule is defined by \(\mu\textrm{} \sum_{i,j = 1}^{\infty,\infty} x_{ij}: = \mu\textrm{} \lim s_{mn}\). Başar and Şever introduced the space \(L_{p}\) in [16]
corresponding to the space \(l_{p}\) for \(p \ge1\) and examined some of its properties. Altay and Başar [17] have generalized the spaces of double sequences \(L_{\infty}\), \(C_{p}\) and \(C_{\mathit{bp}}\) to
and
respectively, where \(t = \{ t_{mn}\}\) is the sequence of strictly positive reals \(t_{mn}\). In the case \(t_{mn} = 1\), for all \(m,n \in\mathbb{N}\), \(L_{\infty} (t)\), \(C_{p}(t)\) and \(C_{\mathit{bp}}(t)\) reduce to the sets \(L_{\infty}\), \(C_{p}\) and \(C_{\mathit{bp}}\), 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 \(c(\sigma)\) the sequence \(\{ c_{n}(\sigma)\}\) which is such that \(c_{n}(\sigma) = 1\) if \(n \in \sigma\), \(c_{n}(\sigma) = 0\) otherwise. Further, let
be the set of those σ whose support has cardinality at most s, and
where \(\Delta\phi_{n} = \phi_{n}  \phi_{n  1}\) and \(\phi _{0} = 0\).
For \(\phi\in\Phi\), the following sequence spaces were introduced and studied in [8] by Sargent and further studied by Mursaleen in [9, 10]:
and
Remark 1.1

(i)
The spaces \(m(\phi)\) and \(n(\phi)\) are BKspaces with their usual norms.

(ii)
If \(\phi_{n} = 1\) (\(n = 1,2,3,\ldots\)), then \(m(\phi) = l_{1}\) [\(n(\phi) = l_{\infty} \)], and if \(\phi_{n} = n\) (\(n = 1,2,3,\ldots\)), then \(m(\phi) = l_{\infty}\) [\(n(\phi) = l_{1} \)].

(iii)
\(l_{1} \subseteq m(\phi) \subseteq l_{\infty} \) [\(l_{1} \subseteq n(\phi) \subseteq l_{\infty} \)] for all \(\phi\in\Phi\).

(iv)
For any \(\phi\in\Phi\), \(m(\phi)\neq l_{p}\) [\(n(\phi)\neq l_{q} \)], \(1 < p < \infty\).
In this paper, we define Sargent’s spaces for double sequences \(x = \{ x_{mn}\}\). For this we first suppose U to be the set whose elements are finite sets of distinct elements of \(\mathbb{N} \times\mathbb{N}\) obtained by \(\sigma\times\varsigma\), where \(\sigma\in C_{s}\) and \(\varsigma\in C_{t}\) for each \(s,t \ge1\). Therefore, any element ζ of U means \((m,n)\); \(m \in\sigma\) and \(n \in\varsigma\) 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 \(c(\zeta)\) the sequence \(\{ c_{mn}(\zeta)\}\) such that
Further, we write
for the set of those ζ whose support has cardinality at most st; and
where \(\Delta_{11}\phi_{mn} = \phi_{mn}  \phi_{m  1,n}  \phi_{m,n  1} + \phi_{m  1,n  1}\) and \(\phi_{00}\), \(\phi_{0m}\), \(\phi_{n0} = 0\), \(\forall m,n \in\mathbb{I}^{ +}\). Throughout the paper, we write \(\sum_{m,n \in\zeta}\) for \(\sum_{m \in \sigma} \sum_{n \in\varsigma}\), and \(S(x)\) is used to denote the set of all double sequences that are rearrangements of \(x = \{ x_{mn}\} \in \Omega\). For \(\phi\in\Theta\), we define the following sequence spaces:
and
Then the distances between \(x = \{ x_{mn}\}\) and \(y = \{ y_{mn}\}\) induced by \(M(\phi)\) and \(N(\phi)\) can be expressed as
and
Remark 1.2
If \(\phi_{st} = 1\) (\(s,t = 1,2,3,\ldots\)), then \(M(\phi) = L_{1}\) [\(N(\phi) = L_{\infty} \)], and if \(\phi_{st} = st\) (\(s,t = 1,2,3,\ldots\)), then \(M(\phi) = L_{\infty}\) [\(N(\phi) = L_{1}\)].
We now state the following known results of [18] for single sequences (series) which can also be proved easily for double sequences (series).
Lemma 1.1
If the series \(\sum u_{n}x_{n}\) is convergent for every x of a BKspace E, then the functional \(\sum_{n = 1}^{\infty} u_{n}x_{n}\) is linear and continuous in E.
Lemma 1.2
If E and F are BKspaces, and if \(E \subseteq F\), then there is a real number K such that, for all x of E,
Properties of the spaces \(M(\phi)\) and \(N(\phi)\)
Theorem 2.1
The space \(M(\phi)\) is a BKspace with the norm.
Proof
It is a routine verification to show that \(M(\phi)\) is a normed space with the given norm (2.1), and so we omit it. Now, we proceed to showing that \(M(\phi)\) is complete. Let \(\{ x^{l}\}\) be a Cauchy sequence in \(M(\phi)\), where \(x^{l} = \{ x_{mn}^{l}\}_{m,n = 1,1}^{\infty,\infty}\) for every fixed \(l \in\mathbb{N}\). Then, for a given \(\varepsilon> 0\), there exists a positive integer \(n_{0}(\varepsilon) > 0\) such that
for all \(l,r > n_{0}(\varepsilon)\), which yields, for each fixed \(s,t \ge 1\) and \(\zeta\in U_{st}\),
Therefore
This means that \(\{ \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert \} _{l \in\mathbb{N}}\) is a Cauchy sequence in \(\mathbb{R}\) for every fixed \(s,t \ge1\) and \(\zeta \in U_{st}\). Since \(\mathbb{R}\) is complete, it converges, say
Since absolute convergence implies convergence in \(\mathbb{R}\), hence
Hence we have
Let \(y^{l} = \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert \). Then \(\{ y^{l}\} \in l_{\infty}\). Therefore
Since \(\sum_{m,n \in\zeta} \vert x_{mn}\vert \le\sum_{m,n \in \zeta} \vert x_{mn}  x_{mn}^{l}\vert + \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert \le\varepsilon \phi_{11} + k\), it follows that \(x = \{ x_{mn}\} \in M(\phi)\). Since \(\{ x^{l}\}_{l \in\mathbb{N}}\) was an arbitrary Cauchy sequence, the space \(M(\phi)\) is complete. Now we prove that \(M(\phi)\) has continuous coordinate projections \(p_{mn}\), where \(p_{mn}:\Omega\to K\) and \(p_{mn}(x) = x_{mn}\). The coordinate projections \(p_{mn}\) are continuous since \(\vert x_{mn}\vert \le\sup_{s,t \ge1}\sup_{\zeta\in U_{st}}\phi_{st}\Vert x \Vert _{M(\phi)}\) for each \(m,n \in \mathbb{N}\). □
Remark 2.1
The space \(N(\phi)\) is a BKspace with the norm
Lemma 2.1

(i)
If \(x \in M(\phi)\) [\(x \in N(\phi) \)] and \(u \in S(x)\), then \(u \in M(\phi)\) [\(u \in N(\phi) \)] and \(\Vert u\Vert = \Vert x\Vert \).

(ii)
If \(x \in M(\phi)\) [\(x \in N(\phi)\)] and \(\vert u_{mn}\vert \le \vert x_{mn}\vert \) for every positive integer m, n, then \(u \in M(\phi)\) [\(u \in N(\phi)\)] and \(\Vert u\Vert \le \Vert x\Vert \).
Proof
(i) Let \(x \in M(\phi)\), then \(\sup_{s,t \ge1} \sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn}\vert < \infty\). So, we have
Since the sum of a finite number of terms remains the same for all the rearrangements,
Hence
thus \(u \in M(\phi)\) and \(\Vert u\Vert = \Vert x\Vert \).
(ii) By using the definition, easy to prove. □
Theorem 2.2
For arbitrary \(\phi\in\Theta\), we have \(\Delta_{11}\phi\in M(\phi)\) and \(\Vert \Delta_{11}\phi \Vert _{M(\phi)} \le2\).
Proof
Let s and t be arbitrary positive integers, let \(\sigma,\varsigma\in U_{st}\), and let \(\tau_{1}\), \(\tau_{2}\) constitute the element of σ and ς exceed by s and t respectively, also from the definition we have \(\Delta_{11}\phi\ge0\) and \(\Delta_{11} ( \frac{\phi_{mn}}{mn} ) \le0\). Then
□
Lemma 2.2
If \(x \in M(\phi)\) and \(\{ c_{11},c_{12}, \ldots,c_{1n},c_{21},c_{22}, \ldots,c_{2n}, \ldots,c_{m1},c_{m2}, \ldots,c_{mn}\}\) is a rearrangement of \(\{ b_{11},b_{12}, \ldots,b_{1n},b_{21},b_{22}, \ldots,b_{2n}, \ldots,b_{m1},b_{m2}, \ldots,b_{mn}\}\) such that \(\vert c_{11}\vert \ge \vert c_{12}\vert \ge \cdots\ge \vert c_{1n}\vert \), \(\vert c_{21}\vert \ge \vert c_{22}\vert \ge\cdots\ge \vert c_{2n}\vert ,\dots, \vert c_{m1}\vert \ge \vert c_{m2}\vert \ge\cdots\ge \vert c_{mn}\vert \) and \(\vert c_{11}\vert \ge \vert c_{21}\vert \ge\cdots\ge \vert c_{m1}\vert \), \(\vert c_{12}\vert \ge \vert c_{22}\vert \ge\cdots \ge \vert c_{m2}\vert ,\vert c_{n1}\vert \ge \vert c_{n2}\vert \ge\cdots\ge \vert c_{nm}\vert \), then
Proof
In view of Lemma 2.1(i), it is sufficient to consider the case when \(b_{ij} = c_{ij}\) (\(i = 1,2, \ldots,m\); \(j = 1,2, \ldots,n\)). Then writing \(X_{mn} = \sum_{i,j = 1}^{m,n} \vert x_{ij}\vert \), we get
Hence we have \(\sum_{i,j = 1,1}^{m,n} \vert b_{ij}x_{ij}\vert \le \Vert x\Vert _{M(\phi)}\sum_{i,j = 1,1}^{m,n} \vert c_{ij}\vert \Delta_{11}\phi_{ij}\). □
Theorem 2.3
In order that \(\sum u_{ij}x_{ij}\) be convergent [absolutely convergent] whenever \(x \in M(\phi)\), it is necessary and sufficient that \(u \in N(\phi)\). Further, if \(x \in M(\phi)\) and \(u \in N(\phi)\), then
Proof
Necessity. We now suppose that \(\sum u_{ij}x_{ij}\) is convergent whenever \(x \in M(\phi)\), then from Lemma 1.1 we have
for some real number K and all x of \(M(\phi)\). In view of Lemma 2.1(ii), we may replace \(x_{ij}\) by \(x_{ij} \operatorname{sgn} \{ u_{ij}\}\), obtaining
Let \(v \in S(u)\). Then taking x to be a suitable rearrangement of \(\Delta_{11}\phi\), it follows from Eq. (2.7) and Theorem 2.2 and Lemma 2.1(i) that
and thus \(u \in N(\phi)\).
Sufficiency. If \(x \in M(\phi)\) and \(u \in N(\phi)\), it follows from Lemma 2.2 that for every positive integer m and n,
□
Theorem 2.4
In order that \(\sum u_{mn}x_{mn}\) be convergent [absolutely convergent] whenever \(x \in N(\phi)\), it is necessary and sufficient that \(u \in M(\phi)\).
Proof
Since sufficiency is included in Theorem 2.3, we only consider necessity. We therefore suppose that \(\sum u_{mn}x_{mn}\) is convergent whenever \(x \in N(\phi)\). By arguments similar to those used in Theorem 2.3, we may therefore have that
for some real number K and all x of \(N(\phi)\). Let \(x = c(\zeta)\), where \(\zeta\in U_{st}\). Then \(x \in N(\phi)\), and
from Theorem 2.2 and Eq. (2.8) we have
and thus \(u \in M(\phi)\). □
Inclusion relations for \(M(\phi)\) and \(N(\phi)\)
Lemma 3.1
In order that \(M(\phi) \subseteq M(\psi)\) [\(N(\phi) \supseteq N(\psi)\)], it is necessary and sufficient that
Proof
Since each of the spaces \(M(\phi)\) and \(N(\phi )\) is the dual of the other, by Theorems 2.3 and 2.4, the second version is equivalent to the first. Moreover, sufficiency follows from the definition of an \(M(\phi)\) space. We therefore suppose that \(M(\phi) \subseteq M(\psi)\). Since \(\Delta\phi\in M(\phi)\), it follows that \(\Delta \psi\in M(\psi)\), and hence we find that, for every positive integer \(s,t \ge1\),
□
Theorem 3.1

(i)
\(L_{1} \subseteq M(\phi) \subseteq L_{\infty}\) [\(L_{1} \subseteq N(\phi) \subseteq L_{\infty} \)] for all ϕ of Θ.

(ii)
\(M(\phi) = L_{1}\) [\(N(\phi) = L_{\infty} \)] if and only if \(\mathit{bp}\textit{}\lim_{s,t}\phi_{st} < \infty\).

(iii)
\(M(\phi) = L_{\infty}\) [\(N(\phi) = L_{1}\)] if and only if \(\mathit{bp}\textit{}\lim_{s,t}(\phi_{st}/st) > 0\).
Proof
We prove here the first version, while the second version follows by Theorems 2.3 and 2.4. Since \(\phi_{11} \le\phi_{mn} \le mn\phi_{mn}\) for all ϕ of Θ, we have by Lemma 3.1 that (i) is satisfied. Further, from Lemma 3.1, it follows that \(M(\phi) \subseteq L_{1}\) if and only if \(\sup_{s,t \ge 1}\phi_{st} < \infty\), while \(L_{\infty} \subseteq M(\phi)\) if and only if \(\sup_{s,t \ge1}(\phi_{st}/st) < \infty\); since the sequences \(\{ \phi_{st}\}\) and \(\{ st/\phi_{st}\}\) are monotonic, (ii) and (iii) are also satisfied. □
Theorem 3.2
Suppose that \(1 < p < \infty\) and \(\frac{1}{p} + \frac{1}{q} = 1\). Then

(i)
Given any ϕ of Θ, \(M(\phi)\neq L_{p}\) [\(N(\phi )\neq L_{q}\)].

(ii)
In order that \(L_{p} \subset M(\phi)\) [\(N(\phi) \subset L_{q}\)], it is necessary and sufficient that \(\sup_{s,t \ge1} ( \frac{(st)^{1/q}}{\phi_{st}} ) < \infty\).

(iii)
In order that \(M(\phi) \subset L_{p}\) [\(N(\phi) \supset L_{q}\)], it is necessary and sufficient that \(\Delta\phi\in L_{p}\).

(iv)
\(\bigcup_{\Delta\phi\in L_{p}} M(\phi) = L_{p}\) [\(\bigcap_{\Delta\phi\in L_{p}} N(\phi) = L_{q} \)].
Proof
(i) Let us suppose that \(M(\phi) = L_{p}\).
Then, by Lemma 1.2, there exist real numbers \(r_{1}\) and \(r_{2}\) (\(r_{1} > 0\), \(r_{2} > 0\)) such that, for all x of \(M(\phi)\),
Taking \(x = c(\zeta)\), where \(\zeta\in U_{st}\), we have that
and hence that
In view of Lemma 3.1, this implies that \(M(\phi) = M(\psi)\), where \(\psi= \{ (mn)^{\frac{1}{q}}\}\). Since \(\Delta\psi\in M(\psi)\) by Theorem 2.2, but \(\Delta\psi\notin L_{q}\), this leads to a contradiction. Hence (i) follows.
(ii) If \(L_{q} \subset M(\phi)\), arguments similar to those used in the proof of (i) show that
For sufficiency, we suppose that (3.1) is satisfied. Then, whenever \(x \in L_{p}\) and \(\zeta\in U_{st}\),
and hence \(x \in M(\phi)\). In view of (i), it follows that \(L_{q} \subset M(\phi)\).
(iii) By Theorem 2.2, we have \(\Delta\phi\in M(\phi)\). For sufficiency, we suppose that \(\Delta\phi\in L_{p}\) and that \(x \in M(\phi)\). Then \(\{ u_{mn}\Delta_{11}\phi_{mn}\} \in L_{1}\) whenever \(u \in L_{q}\), and it therefore follows from Lemma 2.2 that \(\{ u_{mn}x_{mn}\} \in L_{1}\) whenever \(u \in L_{q}\). Since \(L_{p}\) is the dual of \(L_{q}\) and since \(M(\phi)\neq L_{p}\), it follows that \(M(\phi) \subset L_{q}\).
(iv) By using (iii) we have \(\bigcup_{\Delta\phi\in L_{p}} M_{\phi} \subseteq L_{p}\). Now, for obtaining the complementary relation \(L_{p} \subseteq\bigcup_{\Delta\phi\in L_{p}} M_{\phi}\), let us suppose that \(x \in L_{p}\). Then \(\lim_{m,n \to\infty} x_{mn} = 0\), and hence there is an element u of \(S(x)\) such that \(\{ \vert u_{mn}\vert \}\) is a nonincreasing sequence. If we take \(\psi= \{ \sum_{i,j = 1,1}^{m,n} \vert u_{ij}\vert \}\), then it is easy to verify that \(\psi\in \Theta\) and that \(x \in M(\phi)\). Since \(\Delta\psi\in L_{p}\), the complementary relation is satisfied. □
Application of \(M(\phi)\) and \(N(\phi)\) in clustering
In this section, we implement a kmeans clustering algorithm by using \(M(\phi)\)distance measure. Further, we apply the kmeans algorithm into clustering to cluster twomoon data. The clustering result obtained by the \(M(\phi)\)distance measure is compared with the results derived by the existing Euclidean distance measures (\(l_{2}\)).
Algorithm to compute \(M(\phi)\) distance
Let \(x = [x_{1},x_{2},x_{3}, \ldots,x_{n}]_{1 \times n}\) and \(y = [y_{1},y_{2},y_{3}, \ldots,y_{n}]_{1 \times n}\) be two matrices of size \(1 \times n\), and let \(\phi_{m,n} = \phi_{1,n} = n\).

(1)
Calculate \(a_{i} = \frac{1}{\phi_{1,i}}\vert x_{i}  y_{i}\vert \), \(i = 1,2,3, \ldots,n\).

(2)
The \(M(\phi)\)distance between x and y is d, where
$$d = \max\{ a_{1},a_{1} + a_{2}, \ldots,a_{1} + a_{2} + \cdots+ a_{n}\}. $$
Kmeans clustering algorithm for \(M(\phi)\)distance measure
Let \(X = [x_{1},x_{2},x_{3},\ldots,x_{n}]\) be the data set.

(1)
Randomly/judiciously select k cluster centers (in this paper we choose first k data points as the cluster center \(y = [x_{1},x_{2},\ldots,x_{k}]\)).

(2)
By using \(M(\phi)\) or \(N(\phi)\) distance measure (since both are dual of each other, in application point of view, we only consider \(M(\phi)\)), compute the distance between each data points and cluster centers.

(3)
Put data points into the cluster whose \(M(\phi)\)distance with its center is minimum.

(4)
Define cluster centers for the new clusters evolved due to steps 13, the new cluster centers are computed as follows: \(c_{i} = \frac{1}{k_{i}}\sum_{j = 1}^{k_{i}} x_{i}\), where \(k_{i}\) denotes the number of points in the ith cluster.

(5)
Repeat the above process until the difference between two consecutive cluster centers reaches less than a small number ε.
Twomoon dataset clustering by using \(M(\phi)\)distance measure in kmeans algorithm
Twomoon dataset is a wellknown nonconvex data set. It is an artificially designed two dimensional dataset consisting of 373 data points [19]. Twomoon dataset is visualized as moonshaped clusters (see Figure 1).
By using \(M(\phi)\)distance measure in the kmeans clustering algorithm, the obtained result is represented in Figure 2. In Figure 3, we represent the result obtained by using the Euclidean distance measure in the kmeans algorithm (we measure the accuracy of the cluster 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 \(M(\phi)\)distance measure is 84.72% while \(l_{2}\)distance measure’s clustering accuracy is 78.55%. Thus, \(M(\phi)\)distance measure substantially improves the clustering accuracy.
Conclusions
In this paper, we defined Banach spaces \(M(\phi)\) and \(N(\phi)\) 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 \(M(\phi)\) into clustering to cluster the twomoon data by using the kmeans clustering algorithm; the result of the experiment shows that the \(M(\phi)\)distance measure extensively improves the clustering accuracy.
References
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. 281297 (1967)
Bezdek, JC: A review of probabilistic, fuzzy, and neural models for pattern recognition. J. Intell. Fuzzy Syst. 1(1), 125 (1993)
Jain, AK: Data clustering: 50 years beyond kmeans. Pattern Recognit. Lett. 31(8), 651666 (2010)
Cheng, MY, Huang, KY, Chen, HM: Kmeans particle swarm optimization with embedded chaotic search for solving multidimensional problems. Appl. Math. Comput. 219(6), 30913099 (2012)
Yao, H, Duan, Q, Li, D, Wang, J: An improved kmeans clustering algorithm for fish image segmentation. Math. Comput. Model. 58(3), 790798 (2013)
Cap, M, Prez, A, Lozano, JA: An efficient approximation to the kmeans clustering for massive data. Knowl.Based Syst. 117, 5669 (2016)
Güngör, Z, Ünler, A: Kharmonic means data clustering with simulated annealing heuristic. Appl. Math. Comput. 184(2), 199209 (2007)
Sargent, W: Some sequence spaces related to the \(\ell_{p}\) spaces. J. Lond. Math. Soc. 1(2), 161171 (1960)
Mursaleen, M: Some geometric properties of a sequence space related to \(\ell_{p}\). Bull. Aust. Math. Soc. 67(2), 343347 (2003)
Mursaleen, M: Application of measure of noncompactness to infinite system of differential equations. Can. Math. Bull. 56, 388394 (2013)
Chen, L, Ng, R: On the marriage of \(\ell_{p}\)norms and edit distance. In: Proceedings of the Thirtieth International Conference on Very Large Data Bases, vol. 30, pp. 792803 (2004)
Cristianini, N, ShaweTaylor, J: An Introduction to Support Vector Machines and Other KernelBased Learning Methods. Cambridge University Press, Cambridge (2000)
Xu, Z, Chen, J, Wu, J: Clustering algorithm for intuitionistic fuzzy sets. Inf. Sci. 178(19), 37753790 (2008)
Pringsheim, A: Zur theorie der zweifach unendlichen zahlenfolgen. Math. Ann. 53(3), 289321 (1900)
Mursaleen, M, Mohiuddine, SA: Convergence Methods for Double Sequences and Applications. Springer, Berlin (2014)
Başar, F, Şever, Y: The space \(\mathcal{L}_{q}\) of double sequences. Math. J. Okayama Univ. 51, 149157 (2009)
Altay, B, Başar, F: Some new spaces of double sequences. J. Math. Anal. Appl. 309(1), 7090 (2005)
Wilansky, A: Summability Through Functional Analysis. North Holland Math. Stud, vol. 85 (1984)
Jain, AK, Law, MHC: Data clustering: a users dilemma. In: Proceedings of the First International Conference on Pattern Recognition and Machine Intelligence (2005)
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)
Acknowledgements
The second and third authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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Khan, M.S., Alamri, B.A., Mursaleen, M. et al. Sequence spaces \(M(\phi)\) and \(N(\phi)\) with application in clustering. J Inequal Appl 2017, 63 (2017). https://doi.org/10.1186/s136600171333z
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DOI: https://doi.org/10.1186/s136600171333z
MSC
 40H05
 46A45
Keywords
 clustering
 double sequence
 kmeans clustering
 twomoon dataset