A novel sequence space related to \(\mathcal{L} {p}\) defined by Orlicz function with application in pattern recognition

R E S E A R C H

Open Access

A novel sequence space related to

L

_p

defined by Orlicz function with application in

pattern recognition

Mohd Shoaib Khan and QM Danish Lohani

*_{Correspondence:}

danishlohani@cs.sau.ac.in Department of Mathematics, South Asian University, New Delhi, 110021, India

Abstract

In the field of pattern recognition, clustering groups the data into different clusters on the basis of similarity among them. Many a time, the similarity level between data points is derived through a distance measure; so, a number of clustering techniques reliant on such a measure are developed. Clustering algorithms are modified by employing an appropriate distance measure due to the high versatility of a data set. The distance measure becomes appropriate in clustering algorithm if weights assigned at the components of the distance measure are in concurrence to the problem. In this paper, we propose a new sequence spaceM(φ,p,F) related toLp using an Orlicz function. Many interesting properties of the sequence space

M(φ,p,F) are established by the help of a distance measure, which is also used to modify thek-means clustering algorithm. To show the efficacy of the modified

k-means clustering algorithm over the standardk-means clustering algorithm, we have implemented them for two real-world data set, viz. a two-moon data set and a path-based data set (borrowed from the UCI repository). The clustering accuracy obtained by our proposed clustering algoritm outperformes the standardk-means clustering algorithm.

MSC: 40H05; 46A45

Keywords: clustering; double sequence;k-means clustering; Orlicz function

1 Introduction

Clustering is the process of separating a data set into different groups (clusters) such that objects in the same cluster should be similar to one another but dissimilar in another cluster [–]. It is a procedure to handle unsupervised learning problems appearing in pattern recognition. The major contribution in the field of clustering came due to the pioneering work of MacQueen [] and Bazdek []. Thek-means clustering algorithm was introduced by MacQueen [], which is based on the minimum distance of the points from the center. The variants ofk-means clustering algorithms were proposed to solve different types of pattern recognitions problems (see [–]). The clustering results ofk-means or its variant can be further enhanced by choosing an appropriate distance measure. Therefore, the distance measure has a vital role in the clustering.

Clustering process is usually carried out through theldistance measure [], but, due to

its trajectory, sometimes it fails to offer good results. Suppose that two pointsxandyare

Figure 1 Geometry oflpnorm.

selected on the boundary of the square (casep= ) and letzbe the center (Figure ). Then

lwill fail to distinguishxandy, but these points may be distinguished byl. If the pointsx

andyare on the circumference of the circle, thenlwill fail to distinguish them. Moreover,

thelp(p≥) distance measures are not flexible, so they cannot be modified as per the need of the clustering problem. Hence, clustering results derived through distance-dependent algorithms basically depend upon two properties of a distance measure: () trajectory and () flexibility. Till now, we have not come across to any distance measure that offers a guaranteed good result for every clustering problems. Clustering is carried out by using other variants of thelpdistance measure. The distance measure of the sequence spacelp,q_,

≤p,q≤ ∞, introduced by Kellogg [] and further studied by Jovanovic and Rakocevic [], Oscar and Carme [], and Ivana et al. [] offers more flexibility in comparison tolp

due to involvement of additional parameterq. Sargent [] introduced another interest-ing sequence spacesm(ϕ) andn(ϕ) closely related tolp. Some useful extensions ofm(ϕ)

andn(ϕ) sequence spaces were proposed by Tripathy and Sen [], Mursaleen [, ], and Vakeel []. Malkowsky et al. [] defined a matrix mapping into the strong Cesàro sequence space [] and studied the modulus function. Recently, for first time, Khan et al. [] defined a distance measure of the double sequence ofM(φ) andN(φ) to cluster the objects. Moreover, Khan et al. in [, ] defined some more similarity measures by using distance measures of the double sequences in the uncertain environment. Mohi-uddine and Alotaibi applied measures of noncompactness to solve an infinite system of second-order differential equations inp spaces [, ]. The double sequence space is

further studied by Mursaleen and Mohiuddine [], Altay and Başar [, ], Başar and Şever [], and Esi and Hazarika []. Moreover, an Orlicz function and a fuzzy set are also used to define other types of double sequence spaces [, –]. The convergence of difference sequence spaces is discussed in [, , ].

In this paper, we define a new double sequence spaceM(φ,p,F) related toLpusing the following Orlicz function:

M(φ,p,F)

=

x={xmn} ∈:sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F

_|

xmn|

ρ

p

<∞, for someρ> 

.

Obviously,M(φ,p,F) is a norm space, and hence the induced distance measure is repre-sented as

dM(φ,p,F)(x,y) =

sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F

|xmn–ymn|

ρ

p_p

.

clustering problem. Besides, defining the distance measure ofM(φ,p,F), we have also studied some of its mathematically established properties. Finally, the distance measure ofM(φ,p,F) is used in thek-means clustering algorithm, which clusters real-world data sets such as two-moon data set and path-based data set. The clustering results obtained by the modified clustering algorithm is compared with thek-means clustering algorithm to show its efficacy.

2 Preliminaries

Throughout the paper,l∞,c, and cdenote the Banach spaces of bounded, convergent,

and null sequences; ω,N, andRdenote the sets of real (ordinary or single) sequences, natural numbers, and real numbers, respectively.

2.1 Orlicz function [34]

A functionF: [,∞)→[,∞) is called an Orlicz function if (i) F() = ,F(x) > forx> , andF(x)→ ∞asx→ ∞; (ii) Fis convex;

(iii) Fis nondecreasing; and

(iv) Fis continuous from the right of .

An Orlicz functionF is said to satisfy -condition for all values ofxif there exists a

constantK>  such thatF(x)≤KF(x) for allx≥. The -condition is equivalent to

F(Lx)≤KF(x) for all values ofx>  and forL> . An Orlicz functionF can always be represented in the following integral form:

F(x) =

x



η(t)dt,

whereη, known as the kernel ofF, is right-differentiable fort≥,η() = ,η(t) >  for

t> ,ηis nondecreasing, andη(t)→ ∞ast→ ∞.

LetCbe the space of finite sets of distinct positive integers. Given any elementσofC. Let

c(σ) be the sequence{cn(σ)}such thatcn(σ) =  ifn∈σandcn(σ) =  otherwise. Further, let

Cs=

σ∈C: ∞

n=

cn(σ)≤s (cf. [])

be the set of thoseσ whose support has cardinality at mosts, and

=

φ={φn} ∈ω:φ> , φn≥ and

φn n

≤ (n= , , . . .)

,

where ϕn=ϕn–ϕn–.

Forϕ∈, the sequence space, introduced by Sargent [] and known as Sargent’s se-quence space, is defined as follows:

m(φ) =

x={xn} ∈ω:sup

s≥

sup

σ∈Cs

 φs

n∈σ

|xn|

<∞

.

is said to beboundedifx∞=sup_m,n|xmn|<∞. We denote the space of all bounded dou-ble sequences byL∞. Consider a sequencex={xmn} ∈. If for everyε> , there exist n◦=n◦(ε)∈Nand∈Rsuch that

|xmn–|<ε

for allm,n>n◦then we say that the double sequencexisconvergentin thePringheim sense

to the limitand writeP-limxmn=. ByCpwe denote the space of all convergent double sequences in the Pringsheim sense. It is well known that there are such sequences in the spaceCpbut not in the spaceL∞. So, we can consider the spaceCbpof double sequences that are both convergent in the Pringsheim sense and bounded, that is,Cbp=Cp∩L∞. A double sequencex={xmn}is said toconverge regularly to(shortly,r-convergentto) ifxisP-convergent toand the limitsxm:=limnxm,n(m∈N) andxn:=limmxm,n(n∈N)

exist. Note that, in this case, the limitslimmlimnxm,nandlimnlimmxm,nexist and are equal

to theP-limit ofx. Therefore,is called ther-limit ofx.

In general, for any notion of convergenceν, the space of allν-convergent double se-quences will be denoted by Cν, and the limit of a ν-convergent double sequence xby

ν-limm,nxmn, whereν∈ {P,bp,r}.

Başar and Sever [] have introduced the spaceLp ofp-summable double sequences corresponding to the spacelp(p≥) of single sequences as

Lp:=

{xmn} ∈:

m,n

|xmn|p<∞

(≤p<∞)

and examined some properties of the space. Altay and Başar [] have generalized the set of double sequences L∞, Cp, and Cbp etc. by defining L∞(t) = ({xmn} ∈ : sup_m,n∈N|xmn|tmn _<∞_), Cp_{(t) = (}{_xmn} ∈_:P_-lim

m,n→∞|xmn–|tmn<∞), and Cbp(t) =

Cp∩L∞, respectively, wheret={tmn}is a sequence of strictly positive realstmn. In the

casetmn=  for allm,n∈N,L∞(t),Cp(t), andCbp(t) reduce to the setsL∞,Cp andCbp, respectively.

Now just to have a better idea about other convergences, especially the linear conver-gence, we first consider the isomorphism defined by Zelster [] as

T:→ω,

x→z= (zi) := (xχ–_(i)),

()

whereχ:N×N→Nis the bijection defined by

χ(, )= , χ(, )= , χ(, )= , χ(, )= 

.. .

χ(,n)= (n– )+ , χ(,n)= (n– )+ , . . . ,

χ(n,n)= (n– )+n, χ(n,n– )=n–n+ , . . . , χ(n, )=n, ..

Let us consider a double sequencex={xmn}and define the sequences={smn}viaxby

smn:=

m,n

i,j

xij (m,n∈N).

For brevity, here and in what follows, we abbreviate the summations ∞_k=∞_l= and

m k=

n l=by

∞,∞

i,j= and

m,n

i,j=, respectively. Then the pair (x,s) and the sequences={smn}

are called a double series and the sequence of partial sums of a double series, respectively. Letλbe the space of double sequences, converging with respect to some linear conver-gence ruleμ-lim:λ→R. The sum of a double series_i∞_,j=,∞xijwith respect to this rule is defined byμ-_i∞_,j=,∞xij:=μ-limsmn.

In this paper, we define an analogoue of Sargent’s sequence in the double sequence space. For this, we first suppose thatU is the space whose elements are finite sets of distinct elements ofN×Nobtained byσ×ς, whereσ∈Csandς∈Ctfor eachs,t≥. Therefore any elementζofUmeans (j,k);j∈σ&k∈ςhaving cardinality atmostst, where

sis the cardinality with respect tom, andtis the cardinality with respect ton. Here, the product saycofstmay be same for differnt sets of positive integersk,l, but in that case,Ukl is different fromUst. Given any elementζ ofU, we denote byc(ζ) the sequence{cmn(ζ)} such that

cmn(ζ) =

⎧ ⎨ ⎩

 if (m,n)∈ζ,

 otherwise.

Further, let

Ust=

ζ∈U: ∞,∞

m,n=

cmn(ζ)≤st

be the set of thoseζ whose support has cardinality at mostst, and let

=

φ={φmn} ∈:φ> , φmn, φmn, φmn≥ and



φmn mn

, 

φmn mn

, 

φmn mn

≤ (m,n= , , . . .)

,

where ϕmn=ϕmn–ϕm–n, ϕmn=ϕmn–ϕmn–, ϕmn=ϕmn–ϕm–n–.

Forϕ∈, we define the sequence space

M(φ,F) =

x={xmn} ∈:sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ F

_|

xmn|

ρ

<∞, for someρ> 

.

Throughout the paper,_m,n∈ζ means

m∈σ

The spacesM(φ,F),Lp, andL∞can be extended toM(φ,p,F),Lp(F), andL∞(F) as follows:

M(φ,p,F)

=

x={xmn} ∈:sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F _| xmn| ρ p

<∞, for someρ> 

,

Lp(F) =

{xmn} ∈:

m,n

F

|xmn|

ρ

p

<∞, for someρ> 

(≤p<∞),

L∞(F) =

{xmn} ∈:sup

m,nF

_|

xmn|

ρ

<∞, for someρ> 

.

Now, if we take the cardinalitytwith respect tonas , thenM(φ,F) reduce tom(φ,F), andM(φ,p,F) tom(φ,p,F). Here, without further discussingM(ϕ,F), we immediately defineM(φ,p,F) so as not to deviate from our main goal to show thatM(φ,p,F) is a class of new double sequences lying betweenLp(F) andL∞(F). We then further prove certain conditions under whichM(φ,p,F) is same as that ofLp(F) andL∞(F). We can easily see that all results in Section  hold forM(φ,F), which is a particular case ofM(φ,p,F) withp= .

3 Some interesting results related toM(

φ

Theorem . The sequence spaceM(φ,p,F)is a linear space overR.

Proof Letx,y∈M(φ,p,F) andλ,μ∈R. Then there exists positive numbersρ andρ

such that

sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F _| xmn| ρ p <∞ and sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F _| xmn| ρ p

<∞.

Letρ=max(|λ|ρ, |μ|ρ).

()  <p< . Using the well-known inequality|a+b|p_{≤ |a|}p_+|b|p_{for  <p<  and the}

convexity of Orlicz functions, we have

sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F

_|

λxmn+μymn|

ρ

p

≤sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F

_|

λxmn|

ρ

p

+sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F

_|

μymn|

ρ

p

≤sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F

_|

λxmn|

ρ

p

+sup

s,t≥

sup

ζ∈Ust



Ust

m,n∈ζ

F

_|

μymn|

ρ

p

<∞,

() ≤p< +∞. It is easy to see that for alla,b∈R,|a+b|p_≤p_(|a|p_+|b|p_{) andFis}

convex, so that, for alls,t≥,ζ∈Ust,

sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F

|λxmn+μymn|

ρ

p

≤sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

p

F

_|

λxmn|

ρ

p

+sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

p

F

_|

μymn|

ρ

p

≤sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

p+ F

|xmn|

ρ

p

+sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

p+ F

_|

ymn|

ρ

p

<∞.

This shows thatx,y∈M(φ,p,F)⇒λx+μy∈M(φ,p,F).

Remark . The distance measure between two sequencesxnandyninduced byCesqp(F)

can be represented as

dCesqp(F)(x,y) =

_∞ n=  Qn n i=

qifi|xi–yi|

p_p

.

Theorem . M(φ,p,F)⊆M(ψ,p,F)if and only ifsup_s,t≥(φst

ψst) <∞.

Proof Letx∈M(φ,p,F). Then

sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F _| xmn| ρ p

<∞ for someρ> .

Suppose thatsup_s,t≥(φst

ψst) <∞. Thenϕst≤kψstfor some positive numberkand for all s,t∈N, so 

ψst ≤ k

ϕst for alls,t∈N. Therefore we have

 ψst

m,n∈ζ

F _| xmn| ρ p ≤ k φst

m,n∈ζ

F

|xmn|

ρ

p

for eachs,t∈Nand for someρ> .

Now taking the supremum on both sides we get

sup

s,t≥

sup

ζ∈Ust

 ψst

m,n∈ζ

F _| xmn| ρ p

≤ksup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F _| xmn| ρ p

Therefore we have

sup

s,t≥

sup

ζ∈Ust

 ψst

m,n∈ζ

F

|xmn|

ρ

p

<∞ and for someρ> .

Hencex∈M(ψ,p,F).

Conversely, letM(φ,p,F)⊆M(ψ,p,F) and suppose thatsup_s,t≥(φst

ψst) <∞. Then there

exist increasing sequences (si) and (ti) of natural numbers such thatlim(φst

ψst) =∞. Now for

everyb∈R+_{, the set of positive real numbers, there existi}

◦,j◦∈Nsuch that ψϕsiti_siti >bfor

allsi≥i◦andti≥j◦. Henceψ_siti > b

ϕ_siti, so that, for someρ> ,

 ψsiti

m,n∈ζ

F

|xmn|

ρ

p

> b φsiti

m,n∈ζ

F

|xmn|

ρ

p

for allsi≥i◦andti≥j◦. Now taking the supremum oversi≥i◦,ti≥j◦, andζ ∈Ust, we get

sup

si≥i◦,ti≥j◦

sup

ζ∈Ust

 ψsiti

m,n∈ζ

F _| xmn| ρ p

>b sup

si≥i◦,ti≥j◦

sup

ζ∈Ust

 φsiti

m,n∈ζ

F _| xmn| ρ p . ()

Since () holds for allb∈R+(we may takebsufficiently large), we have

sup

si≥i◦,ti≥j◦

sup

ζ∈Ust

 ψsiti

m,n∈ζ

F

|xmn|

ρ

p

=∞

whenx∈M(ϕ,p,F) with  <sup_s

i≥i◦,ti≥j◦supζ∈Ust



φ_siti

m,n∈ζ(F(

|xmn|

ρ )) p_<∞.

Therefore x ∈/ M(ψ,p,F). This contradicts to M(φ,p,F) ⊆ M(ψ,p,F). Hence sup_s,t≥(φst

ψst) <∞.

Corollary . M(φ,p,F) =M(ψ,p,F)if and only ifsup_s,t≥(ηst) <∞andsups,t≥(ηst–) <

∞,whereηst= (ψφstst)for all s,t∈N.

Corollary . M(φ)⊆M(φ,p,F).

Proof Ifp=  andF(x) =x, thenM(φ) =M(φ,p,F). Also,M(φ)⊆M(φ,p,F).

Theorem . The inclusionsLp(F)⊆M(φ,p,F)⊆L∞(F)M(φ,p,F)hold.

Proof Letx∈Lp(F). Then, for someρ> , we have∞_i,j=,,∞(F(|xmn|

ρ ))

p_{<∞. Since (ϕ} st) is

nondecreasing with respect tos,t≥, for someρ> , we have

 φst

m,n∈ζ

F _| xmn| ρ p ≤  φ

m,n∈ζ

F _| xmn| ρ p ≤  φ ∞,∞

i,j=,

F

_|

xi,j|

ρ

p

<∞.

Hencesup_s,t≥sup_{ζ∈Ust φ}

st

m,n∈ζ(F(| xmn|

ThusLp⊆M(φ,p,F). Now letx∈M(φ,p,F). Then for someρ> , we have

sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F _| xmn| ρ p

<∞,

sup

m,n≥

 φ

m,n∈ζ

F

|xmn|

ρ

p

<∞.

⇒F(|xmn|

ρ )≤(Aφ)



p _{for someA>  and allm,n}∈N_{. Thusx}∈L_∞(F_).

Theorem . LetF,F,Fbe Orlicz functions satisfying -condition.Then

() M(φ,p,F)⊆M(φ,p,F◦F),

() M(φ,p,F)∩M(φ,p,F) =M(φ,p,F+F).

Proof () Letx∈M(φ,p,F). Then there existsρ>  such that

sup

s,t≥

sup

ζ∈Ust

 φst

m,n∈ζ

F

|xmn|

ρ

p

<∞.

Let  <ε<  andδwith  <δ<  be such thatF(t) <ε,  <t≤δ. Puttmn=F(|xmnρ |) and

for anyζ ∈Us, consider

m,n∈ζ

F(tmn)

p

=



F(tmn)

p

+



F(tmn)

p

,

where the first sum is overtmn≤δ, and the second is overtmn>δ. From the remark we

have



F(tmn)

p

≤F()p



tp_mn≤F()p



t_mnp , ()

and fortmn>δ, we use the fact that

tmn<tmn δ <  +

tmn

δ .

SinceFis nondecreasing and convex, we have

F(tmn)≤F

 +tmn δ

<  F() +

 F

tmn

δ

.

SinceFsatisfies -condition, we have

F(tmn) < k

tmn

δ F() +  k

tmn

δ F() =k

tmn

δ F().

Hence

F(tmn)

p

<

ktmn

δ F()

p

Therefore



F(tmn)

≤max

,

_kF

() δ

pp

m,n∈ζ

t_mnp . ()

By () and () we haveM(φ,p,F)⊆M(φ,p,F◦F).

() The proof follows from the inequality

sup

s,t≥

sup

ζ∈Ust

 φst _m,n∈ζ

(F+F)

|xmn|

ρ

p_p

≤sup

s,t≥

sup

ζ∈Ust

 φst _m,n∈ζ

F

_|

xmn|

ρ

pp

+sup

s,t≥

sup

ζ∈Ust

 φst _m,n∈ζ

F

_|

xmn|

ρ

pp

<∞.

Theorem . TheM(φ,p,F)satisfy the following relations: () M(φ,p,F) =Lp(F)if and only ifsup_s,t≥(ϕst) <∞,

() M(φ,p,F) =L∞(F)if and only ifsups,t≥(φstst) <∞.

Proof () If we takeϕst=  for alls,t∈N, then we haveM(φ,p,F) =Lp(F).

() By Theorem . we easily get that M(φ,p,F) = L∞(F) if and only if sup_s,t≥(_φst

st) <∞.

3.1 k-means algorithm forM(

φ

LetX={x,x, . . . ,xn}be a given data set. Then the proposed clustering algoritm works as

follows.

Step-: Select firstkdata points as the cluster centerxk={x,x, . . . ,xk}(wherekis the

number of clusters).

Step-: Compute the distance between each data point and cluster center through

M(φ,p,F)distance measure.

Step-: Put the data point into that cluster whoseM(φ,p,F)distance with its center is minimal.

Step-: Redefine cluster centers for newly evolved clusters due to the above steps; the new cluster centers are computed asci=_k_i

ki

j=xi, wherekiis the number of

points in theith cluster.

Step-: Repeat Step  to Step  until the difference between two consecutive cluster centers becomes less than a desired small number.

3.2 Clustering by using the inducedM(

φ

Figure 2 Two-moon data set.

(a) Clustering byl

(b) Clustering byM(φ,p,F)

Figure 4 Path-based data set.

(a) Clustering byl

(b) Clustering byM(φ,p,F)

clustering is %(Figure (b)). Moreover, in the case of a path-based data set, we take ϕ=n,∀m,n,p= , andF(x) =|x|. The clustering accuracy of the path-based data set by using thek-mean clustering algorithm is %, whereas by using the proposed modefied

k-means clustering algorithm it is %as shown in Figures (a) and (b), respectively.

4 Conclusions

The parametersφ,p,F involved in the sequence spaceM(φ,p,F) give additional three degrees of freedom to its induced distance measure. Therefore, it is more flexible in com-parison to thelpor weightedlpdistance measure. The flexibility in the distance measure can be judiciously used in the clustering of the real-world data sets. We have proposed only a modifiedk-means clustering algorithm; in the similar fashion, other distance-based clus-tering algorithms can also be modified. So, improvement in many clusclus-tering algorithms is possible due to a distance measure ofM(φ,p,F). We have shown the efficacy of an

M(φ,p,F)-basedk-means clustering algorithm over thel-basedk-means clustering

al-gorithm on the basis of better clustering accuracy obtained for a two-moon data set and path-based data set.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both 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.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 25 May 2017 Accepted: 11 October 2017 References

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