Communication P Systems on Simplicial Complexes with Applications in Cluster Analysis

Abstract

The purpose of this paper is to propose a new kind of P systems on simplicial complexes. We present the basic discrete Morse structure, membrane structures on complexes, and communication rules. A new grid-based clustering technique is described based on this kind of new P systems. Examples are given to show the effect of the algorithm. The new P systems provide an alternative for traditional membrane computing.

1. Introduction

Membrane computing is a new branch of natural computing which is initiated by Păun et al. at the end of 1998, as an attempt to formulate models from the functioning of living cells [1], just like DNA computing coming from genes [2–4]. The advantage of these methods lies in its huge inherent parallelism which has drawn great attention from the scientific community so far. In recent years, many different models of P systems have been proposed, such as cell-like P systems, tissue-like P systems, and spiking neural P systems [5–9]. The obtained computing systems prove to be so powerful that they are equivalent with Turing machines [10] even when using restricted combinations of features and also computationally efficient. Up to now a number of applications were reported in several areas such as biology, biomedicine, linguistics, computer graphics, economics, approximate optimization, cryptography, and so forth.

Traditionally, Morse theory is the subject of differential topology and differential geometry where the topological spaces in question are smooth manifolds. When we want to study discrete problems, we will use combinatorial complexes rather than manifolds. Along this line, discrete Morse theory has been developed [11, 12]. Recently, discrete Morse theory has attracted many researchers because it has found applications in triangulations and graphics. In fact, simplicial complex, the basic data structure in discrete Morse theory, will prove to be an important data structure besides trees and graphs.

Spatial cluster analysis is a traditional problem in knowledge discovery from databases [13]. It has wide applications since increasingly large amounts of data obtained from satellite images, X-ray crystallography, or other automatic equipment are stored in spatial databases. The most classical spatial clustering technique is due to Han and Kamber [13] who developed a variant PAM algorithm called CLARANS, while new techniques are proposed continuously in the literature aiming to reduce the time complexity or to fit for more complicated cluster shapes. Other clustering-like problems include impulsive cluster anticonsensus of discrete multiagent linear dynamic systems [14] and driving general complex networks into prescribed cluster synchronization patterns by using pinning control [15]. In another research, the authors introduce a cooperative article bee colony algorithm for solving clustering problems [16]. Also the authors propose a DNA-based clustering method by the Adleman-Lipton model [17]. For related research, one can also refer to [18–20].

In medical analysis there often appear data clustering problems of various type. The Wisconsin Breast Cancer Showhouse was founded in 1998 as an all-volunteer 501(c)(3) charitable organization by Nance Kinney, a breast cancer survivor (http://www.breastcancershowhouse.org/wbcs2012/index.html). Its mission is to support breast cancer and prostate cancer research at the Medical College of Wisconsin. This breast cancer databases were obtained from the University of Wisconsin Hospitals, Madison from Dr. William H. Wolberg. Many computing methods have been applied to study this data case and, in fact, the Wisconsin Breast Cancer data set is becoming an important testing benchmark for soft computing.

Inspired by the above research, this paper focuses on the joint study of discrete Morse theory with membrane computing. Our purpose is to propose a P system on simplicial complex. Up to our knowledge, this is the first paper to extend membrane computing to complexes. Then we use membrane computing in cluster analysis, providing a new approach to data mining. We first propose a discrete Morse structure for a candidate of a class of new P systems. Then we described for the first time a communication P system on simplices. Then we propose a new method for cluster analysis by simplicial P systems. Finally, we present the Wisconsin Breast Cancer analysis.

2. Discrete Morse Structure

In this section we present some general discrete models which will form the basis of membrane structures. The main idea comes from [11, 21]. In order to do this, we need to present some basic topological concepts. For simplicity we always assume that we are working in an Euclidean space 𝑅^𝑛.

2.1. Simplex without Orientation

A 𝑘-simplex (cell) 𝜎 is the convex hull of 𝑘+1 affinely independent points. More precisely, suppose 𝑎₀,𝑎₁,…,𝑎_𝑘 are affinely independent, that is, 𝑎₁−𝑎₀,…,𝑎_𝑘−𝑎₀ are linearly independent. Then 𝜎 is defined as the set of points in the form 𝑥=∑^𝑘_𝑖=0𝜆_𝑖𝑎^𝑖, where ∑^𝑘_𝑖=0𝜆_𝑖=0 and 𝜆₀,𝜆₁,…,𝜆_𝑘≥0. We will call 𝑘 the dimension of a simplex and write dim(𝜎)=𝑘, while 𝑎₀,𝑎₁,…,𝑎_𝑘 are called vertices of the simplex. A simplex is uniquely indicated by its vertices and hence is expressed as 𝜎=[𝑎₀,𝑎₁,…,𝑎_𝑘], or simply 𝜎=𝑎₀𝑎₁⋯𝑎_𝑘, and will be called a cell in this paper.

A face 𝜏 of a simplex 𝜎 is defined as a simplex generated by a nonempty subset of its vertices. We write 𝜏<𝜎. A face 𝜏 is called a hyperface of 𝜎 if dim𝜏=dim𝜎−1 and is denoted by 𝜏≺𝜎. In this case, 𝜎 is called the parent of 𝜏. Two cells 𝜏₁ and 𝜏₂ are called incident if 𝜏₁,𝜏₂≺𝜎, and 𝜎 is called the coface of 𝜏₁ and 𝜏₂. Two cells 𝜎₁,𝜎₂ are called neighbors if they share a common hyperface. The cone from a vertex 𝑥 to a 𝑘-simplex 𝜎 is the convex hull of 𝑥 and 𝜎 which yields a (𝑘+1)-simplex 𝑥𝜎 provided 𝑥 is not an affine combination of the vertices of 𝜎. A simplicial complex 𝐾 is a finite collection of nonempty simplices for which 𝜎∈𝐾 and 𝜏≺𝜎 implies 𝜏∈𝐾 and 𝜎₁,𝜎₂∈𝐾 implies that 𝜎₁∩𝜎₂ is either empty or a face of both. The underlying space of 𝐾 is the union of simplices: |𝐾|=⋃_𝜎∈𝐾𝜎. 𝐾_𝑝 is a subset of 𝐾 containing simplices of dimension 𝑝.

For a 𝑘-simplex 𝜎, define 𝐾 to be the collection of 𝜎 and all its faces. Then it is clear that 𝐾 is a simplicial complex. We will call this simplicial complex a simple complex or, simply, a complex.

Now we consider some properties of incident and neighborhood relations as described above. First suppose 𝜏₁,𝜏₂≺𝜎 are incident and dim𝜎≥2. Then 𝜏₁,𝜏₂,𝜎 are faces of a simplex. Define 𝜓=𝜏₁∩𝜏₂ and we will show that 𝜓≠∅. This is evident because if 𝜎=𝑣_𝑖0𝑣_𝑖1⋯𝑣_𝑖𝑝, then 𝑝≥2. By removing one vertex we obtain 𝜏₁ and 𝜏₂ with 𝑝 vertices remained. Since 𝑝≥2 we know that there exists at least one common vertex among 𝜏₁, 𝜏₂. By the definition of 𝐾 we get 𝜓∈𝐾 and consequently 𝜏₁, 𝜏₂ are neighbors.

Conversely, if 𝜏₁,𝜏₂≻𝜓 are neighbors, they need not be incident as shown in Figure 1. In the case when they are in the same simplex, however, this is true. In fact, suppose 𝜓=𝑣_𝑖1⋯𝑣_𝑖𝑝. Then all the vertices 𝑣_𝑖1⋯𝑣_𝑖𝑝 belong to 𝜏₁ and 𝜏₂. Then we can assume that 𝜏₁=𝑣_𝑖1⋯𝑣_𝑖𝑝𝑣_𝑗1 and 𝜏₂=𝑣_𝑖1⋯𝑣_𝑖𝑝𝑣_𝑗2. Since both cells are in the same simplex, define 𝜎=𝑣_𝑖1⋯𝑣_𝑖𝑝𝑣_𝑗1𝑣_𝑗2. Then 𝜎∈𝐾. Consequently 𝜏₁, 𝜏₂ are incident. Putting everything together we get a theorem.

(a) incident relation

(b) neighborhood relation

Figure 1 

An attractor model on simplicial complex.

Theorem 2.1. Two incident cells 𝜏₁, 𝜏₂ are neighbors if they are at least one-dimensional. Two neighboring cells are incident provided they are located in the same simplex.

By the definition of neighborhood, if 𝜎₁, 𝜎₂ are neighbors and their common hyperface is 𝜏, then clearly 𝜏=𝜎₁∩𝜎₂ and, consequently, this common hyperface is unique. However, there exist cells with nonempty intersection but they are not neighbors. Now we consider incident cells 𝜏₁,𝜏₂≺𝜎. We will show that their coface is also unique. First if dim𝜏₁=dim𝜏₂=0, then dim𝜎=1. Therefore 𝜎 is the edge joining the two vertices 𝜏₁, 𝜏₂ and hence is unique. Next if dim𝜏₁=dim𝜏₂≥1, then as described previously we define 𝜓=𝜏₁∩𝜏₂. Then 𝜏₁=𝑣₁𝜓,𝜏₂=𝑣₂𝜓 and 𝑣₁≠𝑣₂. Therefore 𝜎=𝑣₁𝑣₂𝜓 and is unique.

Theorem 2.2. Neighboring cells can only share a unique common hyperface. Incident cells have a unique coface (parent).

2.2. Simplex with Orientation

Simplicial complexes with orientation are important tools in the study of topological properties of discrete data structure. Concepts about discrete Morse functions are listed as follows (Robin [11]). For a simplex 𝜎=[𝑎₀,𝑎₁,…,𝑎_𝑘], there are two orientations and the opposite orientation is denoted by −𝜎. Figure 2 shows the orientation of a three-dimensional complex.

Figure 2 

A simplicial complex with orientation in 𝑅3.

In the following we will use [𝑎₀,𝑎₁,…,𝑎_𝑖,…,𝑎_𝑘] to denote a face of 𝜎, where the vertex 𝑎_𝑖 is eliminated. The following chain is defined as the boundary of the complex:𝜕𝜎=^𝑘_𝑖=0(−1)^𝑖𝑎₀,𝑎₁,…,𝑎_𝑖,…,𝑎_𝑘. (2.1)

If 𝑔_𝑖 are integers, then ∑_𝑖𝑔_𝑖𝜎_𝑖 is called a chain, where dim𝜎_𝑖 remains the same within a chain. Boundary operator extends to chains naturally. An important property of boundary operator is that 𝜕∘𝜕=0. For a (𝑘−1)-dimensional simplex 𝜏^𝑘−1 and a 𝑘-dimensional simplex 𝜎_𝑘, define its relationship operator as follows:𝜎^𝑘∶𝜏^𝑘−1=0,𝜏⊀𝜎1,𝜏≺𝜎withsameorientation−1,𝜏≺𝜎withoppositeorientation. (2.2)

In simplicial complex we can define Morse functions which is a tool for optimization.

Definition 2.3. Let 𝐾 be a simplicial complex. A function 𝑓∶𝐾→𝑅 is a discrete Morse function if for every 𝜎^𝑝∈𝐾_𝑝 the following two statements are true:(1)#{𝜏^𝑝+1>𝜎∣𝑓(𝜏)≤𝑓(𝜎)}≤1; #{𝑣^𝑝−1<𝜎∣𝑓(𝑣)≥𝑓(𝜎)}≤1.A simplex 𝜎^𝑝 is critical with index 𝑝 if(2)#{𝜏^𝑝+1>𝜎∣𝑓(𝜏)≤𝑓(𝜎)}=0; #{𝑣^𝑝−1<𝜎∣𝑓(𝑣)≥𝑓(𝜎)}=0.

Discrete gradient can also be defined on the complex 𝐾. If 𝑣 is a critical point, then define 𝑉(𝑣)=0. Otherwise if 𝑣∈𝐾₀ is not critical and the edge 𝑒 satisfies 𝑒>𝑣, 𝑓(𝑒)≤𝑓(𝑣), then define 𝑉(𝑣)=±𝑒 where the sign is determined so that ⟨𝜕(𝑉(𝑣)),𝑣⟩=−1. Here the inner product ⟨⋅,⋅⟩ is the obvious inner product on oriented chains with respect to which the oriented simplices are orthonormal. It is easy to see that, if the edge 𝑒=𝑣𝑢, then 𝑉(𝑣)=−𝑒. Generally speaking, discrete gradient is a mapping𝑉∶𝐶_𝑝(𝐾,𝑍)⟶𝐶_𝑝+1(𝐾,𝑍) (2.3)

A natural gradient flow is Φ(𝜎)=𝜎+𝜕𝑉(𝜎)+𝑉(𝜕𝜎).

3. Communication P Systems on Simplices

3.1. Traditional P Systems

Membrane is a structure serving as a protected reactor. We will identify a membrane 𝑚 with its delimited space. When we say inclusion for membranes, it is always strict inclusion. Now we list some elementary concepts concerning the basic operations of membranes:(i)𝑚,𝑚^′ are vicinal, if 𝑚^′⊂𝑚 and there is no 𝑚^′′ such that 𝑚^′⊂𝑚^′′⊂𝑚,(ii)elementary membrane: with no lower vicinal membranes, skin membrane: with no upper vicinal membranes, we assume there is always a unique skin,(iii)degree: number of membranes,(iv)sibling membranes 𝑚, 𝑚^′: if there is a 𝑚^′′ which is upper vicinal for both 𝑚 and 𝑚^′.

Parentheses expression is often used to describe membrane structures. For example, the membrane structure as shown in Figure 3 has a parentheses expression for membranes as follows: []₂[]₇[]₅[]₈[]₉₆₄[]₃₁. (3.1)

Figure 3 

Traditional membrane structures.

For a set 𝑈, a multiset over 𝑈 is a mapping 𝑀∶𝑈→𝑁, where 𝑁 is the set of nonnegative integers. For 𝑎∈𝑈, 𝑀(𝑎) is the multiplicity of 𝑎 in 𝑀. Suppose the set of objects is 𝑂 with a subset 𝐸 such that objects from 𝐸 are available in the environment in arbitrary multiplicities, that is, its multiset is 𝑀∶𝐸→{∞}. A P system with symport/antiport rules of degree 𝑚≥1 is a constructΠ=𝑂,𝑇,𝐸,𝜇,𝜔₁,…,𝜔_𝑚,𝑅₁,…,𝑅_𝑚,𝑖₀, (3.2) where 𝑂 is the alphabet, 𝑇⊂𝑂 is the alphabet of terminal objects, 𝜇 is a membrane structure of degree 𝑚, 𝜔₁,…,𝜔_𝑚 are the multisets of objects associated with the 𝑚 regions of 𝜇, and 𝑅₁,…,𝑅_𝑚 are finite sets of symport and antiport rules associated with the 𝑚 membranes of 𝜇, and 𝑖₀ is the input/output region.

A P system is called stable if, even if some rules are still applicable, their application does not change the string/object content of the membrane structure, nor the membrane structure itself. For a subalphabet 𝑊⊂𝑂, we call a system stable over 𝑊 if the projection over 𝑊 of the string/object remains unchanged, even if some rules are still applicable. If 𝑊={𝑎}, we will say stable over 𝑎. If 𝑅=(𝑅^′₁,…,𝑅^′_𝑚), with 𝑅^′_𝑖⊂𝑅_𝑖 for 1≤𝑖≤𝑚, is a subset of rules, we call a P system stable with respect to the rules 𝑅^′ if the P system with rules 𝑅′ is stable (i.e., applications of rules in 𝑅′ do not change the string/object content of the system’s membranes) [19].

3.2. Membrane Structures on Simplices

Now we describe membrane structures on simplicial complexes. First we assume that the complex is a simple complex, that is, a simplex with all its faces including vertices. In Figure 4 a simple three dimensional example is presented with 15 membranes. In the general case when there is a complex {𝑎⁰,𝑎¹,…,𝑎^𝑛} in 𝑅^𝑛, the number of simplices are1+𝐶^𝑛_𝑛+1+𝐶^𝑛−1_𝑛+1+⋯+𝐶¹_𝑛+1=^𝑛+1_𝑖=0𝐶^𝑖_𝑛+1−1=2^𝑛+1−1. (3.3)

Figure 4 

A simplex membrane structure.

The boundary relations of simplices are shown in Figure 5 where the arrows point to the boundary cells.

Figure 5 

A network model for membrane structure.

Now we consider the general cases where 𝐾 is a simplicial complex in 𝑅^𝑛. For an example of 𝐾 in 𝑅³ as shown in Figure 6, the total number of cells is 6+19+21+9=55. For example, the three dimensional simplices are listed as𝑎⁰𝑎¹𝑎⁴𝑎³,𝑎³𝑎¹𝑎⁴𝑎²,𝑎³𝑎⁴𝑎²𝑎⁷,𝑎¹𝑎⁵𝑎²𝑎⁴,𝑎⁴𝑎⁵𝑎⁶𝑎²,𝑎⁴𝑎⁶𝑎⁷𝑎². (3.4)

Figure 6 

A simplicial complex model with all cells.

Generally speaking, a simplicial complex 𝐾 is denoted by a set of vertices {𝑎⁰,𝑎¹,…,𝑎^𝑘} in 𝑅^𝑛. A simplex 𝜎∈𝐾 is called membrane. A membrane is called a max-simplex if it is not a face of another simplex in 𝐾. A simplex is denoted by its vertices. Evidently, vertices are zero-dimensional cells and hence are elementary membranes. If 𝜏≺𝜎 is a face of 𝜎, then we way that 𝜎 is the parent of 𝜏. Figure 5 shows the parents and neighborhood relations of the complex in Figure 4. For the more general simplicial complex as in Figure 6, the network model is shown in Figure 7.

Figure 7 

A network model for the simplicial complex model in Figure 5.

If 𝜎₁, 𝜎₂ are incident, we say there is an upper link (channel) between 𝜎₁, 𝜎₂. If 𝜎₁, 𝜎₂ are neighbors, we say there is a lower link between them. A upper link is denoted by (𝜎₁,𝜎₂)^𝜏, where 𝜏 is their (common) parent, while lower link is written as (𝜎₁,𝜎₂)_𝜏, where 𝜏 is their common hyperface. Upper link is also written by (𝜎₁,𝜎₂)_𝑈 or, simply, (𝜎₁,𝜎₂), while lower link is denoted by (𝜎₁,𝜎₂)_𝐿. Links have no directions. Thus (𝜎₁,𝜎₂) and (𝜎₂,𝜎₁) are identical. We will specify one from these two links, and only one is allowed.

Definition 3.1. A P system on a simplicial complex 𝐾, called a simplicial P system, with antiport and symport rules is a construct Π=𝑚,𝑂,𝐸,𝜔₁,…,𝜔_𝑚,𝑅₁,…,𝑅_𝑚,ch,𝐹𝑖,𝑗_{(𝑖,𝑗)∈ch},𝑖₀, (3.5) where 𝑚 is the number of cells labeled with 1,2,…,𝑚, 𝑂 is the alphabet, 𝐸 is the set of objects with unlimited multiplicity in the environment, 𝜔₁,…,𝜔_𝑚 are initial strings over 𝑂 of multiset, 𝑅₁,…,𝑅_𝑚 are symport and antiport rules associated with the m membranes, ch⊆{(𝑖,𝑗)_𝑅,(𝑖,𝑗)_𝐿∶𝑖,𝑗∈{1,…,𝑚}} is the set of links, and 𝐹(𝑖,𝑗) is a finite set of antiport and/or symport rules associated with the link (𝑖,𝑗)∈ch.

Ceterchi and Martin-Vide [19] proposed a new type of communication P systems with priority relations. They introduced a promoter for a rule to be active and a inhibitor for it to be inactive. Induced by their idea, we will present an ordered system in this section.

An antiport rule (𝑢,out;𝑣,in)_𝑖 in 𝑅_𝑖 exchanges the multiset 𝑢 inside 𝑅_𝑖 with 𝑣 outside it. A symport rule (𝑥,out)_𝑖 (or (𝑥,in)_𝑖) sends out (takes in) the multiset 𝑥 with respect to membrane 𝑅_𝑖. For a specific membrane 𝑅_𝑖, rules are totally ordered as 𝑟₁>𝑟₂>⋯>𝑟_𝑛>𝑟_𝑛+1>⋯. The rule 𝑟_𝑛+1 is applicable if and only if the system has reached a stable configuration with respect to rules 𝑟₁>⋯>𝑟_𝑛. We can use a queue structure to represent this process.

3.3. Rules in Simplicial P Systems

Now we describe the communication rules in simplicial P systems. For our purpose in this paper, there are mainly four types of communication rules in a simplicial P system. Each type of rule may have operators such as out,in,up,down.

First suppose (𝜏₁,𝜏₂)_𝜎 is an upper link. A rule like [(𝑥,𝑦),up]^𝑈 means that the multiset 𝑥 and 𝑦 from 𝜏₁ and 𝜏₂ transform into 𝑧 and go up to their parent 𝜎. For two cells 𝜏≺𝜎, the antiport rule [𝑢,out;𝑣,in∣𝜏≺𝜎] in 𝜏 means exchanging multiset 𝑢 inside membrane 𝜏 with the multiset 𝑣 outside it (in 𝜎). The symport rule [𝑢,out∣𝜏≺𝜎] sends the multiset 𝑢 outside 𝜏. Another symport rule [𝑢,in∣𝜏≺𝜎] works similarly.

An upper link rule in 𝑅^𝜎(𝜏₁,𝜏₂) may have the following forms:𝑥,𝑦,up;𝛼,𝛽,in^𝑈⟶𝑧,in;𝛾,down,𝑢,out;𝑣,in𝜏₁≺𝜎,𝑢,out;𝑣,in𝜏₂≺𝜎. (3.6)(3.7)

An lower link rule in 𝑅_𝜎(𝜏₁,𝜏₂) may have the following forms:𝑥,𝑦,down;𝛼,𝛽,in_𝐿⟶𝑧,in;𝛾,up,𝑢,out;𝑣,in∣𝜎≺𝜏₁,𝑢,out;𝑣,in∣𝜎≺𝜏₂. (3.8)(3.9)

Equation (3.6) means moving 𝑥 from cell 𝜏₁ and 𝑦 from 𝜏₂ to 𝑧 to parent 𝜎 and simultaneously moving 𝛾 from 𝜎 to 𝛼 into 𝜏₁ and 𝛽 into 𝜏₂. Equation (3.8) means moving 𝑥 from cell 𝜏₁ and 𝑦 from 𝜏₂ to 𝑧 to hyperface 𝜎 and simultaneously moving 𝛾 from 𝜎 to 𝛼 up to 𝜏₁ and 𝛽 up to 𝜏₂.

3.4. Configuration and Computation

Now we describe the configuration and computation of simplicial P systems. For our purpose, change of membrane structure is not involved. A configuration of a simplicial P system is the state of the system described by specifying the objects and rules associated to each membrane. The initial state is called initial configuration. Therefore, the multisets represented by 𝜔_𝑖,1≤𝑖≤𝑑 in Π, constitute the initial configuration of the system.

The system evolves by applying rules in membranes and this evolution is called computation. The computation starts with the multisets specified by 𝜔₁,…,𝜔_𝑚 in the 𝑚 cells. In each time unit, rules are used in a cell. If no rule is applicable for a cell, then no object changes in it. The system is synchronously evolving for all cells.

When the system has reached a configuration in which no rule is any longer applicable, we say that the computation halts. A configuration is stable if, even if some rules are still applicable, their application does not change the object content of the membranes. The computation is successful if and only if it halts, or it is stable. The result of a halting/stable computation is the number described by the multiplicity of objects present in the cell 𝑖₀ in the halting/stable configuration.

4. Cluster Analysis by Simplicial P Systems

4.1. Problem Setting and Algorithm

Now suppose the data set to be clustered is 𝑋={𝑧₁,𝑧₂,…,𝑧_𝑁}⊂Π^𝑛_𝑖=1[𝑃_𝑖,𝑄_𝑖)⊂𝑅^𝑛. We now construct a uniform grid in 𝑅^𝑛 as follows. Choose integers 𝑛_𝑖>0 and let 𝛿_𝑖=(𝑄_𝑖−𝑃_𝑖)/𝑛_𝑖, and divide the interval [𝑃_𝑖,𝑄_𝑖] into 𝑛_𝑖 subintervals [𝑃_𝑖,𝑃_𝑖+𝛿_𝑖),…,[𝑄_𝑖−𝛿_𝑖,𝑄_𝑖). Therefore 𝑋 is contained in the following grid 𝒢 with 𝑁=𝑛₁⋅𝑛₂⋯𝑛_𝑛−𝑁₀ cells, where 𝑁₀ is the number of cells that do not contain data:^𝑛1_𝑡1=1⋯^𝑛𝑛_𝑡𝑛^𝑛_=1𝑖=1𝑃_𝑖+𝑡_𝑖−1𝛿_𝑖,𝑃_𝑖+𝑡_𝑖𝛿_𝑖. (4.1)

Then we can define a new data set Ω={𝑥₁,…,𝑥_𝑁}, where 𝑥_𝑖 is each grid point 𝑥_𝑖=[𝑥¹_𝑖,…,𝑥^𝑛_𝑖]:𝑥^𝑗_𝑖=𝑃_𝑗+𝑡_𝑗−1𝛿_𝑗,1≤𝑗≤𝑛,1≤𝑡_𝑗≤𝑛_𝑗. (4.2)

Define a weight function on Ω by 𝑚(𝑥)=#{𝑧_𝑖∈ the cell corresponding to the data 𝑥}. In this way, the original data set is transformed into a new data set Ω with weights equal to the original number of points. Next we will always work on the data set Ω. For simplicity, we will consider a density-based clustering technique. For two points 𝑥,𝑦∈Ω, the similarity is defined by𝑓=𝑚(𝑥)𝑦𝑥,𝑦min,𝑚𝑑𝑥,𝑦, (4.3) where 𝑑(𝑥,𝑦) is the topological distance of the two points. That is, if 𝑥, 𝑦 are incident, then 𝑑(𝑥,𝑦)=1. Else 𝑑(𝑥,𝑦) is the minimal number of edges which form a path connecting 𝑥 and 𝑦. For two subsets 𝐶₁,𝐶₂⊂𝑋, the similarity is defined by𝑓𝐶₁,𝐶₂=max_{𝑥∈𝐶1,𝑦∈𝐶2}𝑓𝑥,𝑦∣𝑥,𝑦areincident. (4.4)

The clustering is implemented by a hierarchical method as follows. At first, each point of the data set 𝑋 forms a cluster which contains a singleton. Then each data point tries to connect another data point in its neighborhood. After this step, a cluster can be found as connected points. Now we construct a simplicial complex 𝐾 corresponding to the data set Ω. On the basis of the rectangles as in (4.1), we add some hyperplanes to form a triangulation. Then each rectangle is decomposed as several simplices. Hyperplanes can be chosen such that the set of simplices satisfy the definition of simplicial complex. And then 𝐾 is defined as the union of such simplices. In the three dimensional case, this is shown as in Figure 8.

Figure 8 

Simplices constructed for three dimensional grids.

Now we will show that the triangulation as above exists. In fact, we need only to consider an inner cell 𝐶=Π^𝑛_𝑖=1[𝑝_𝑖,𝑞_𝑖), where 𝑟_𝑖<𝑝_𝑖<𝑞_𝑖<𝑠_𝑖. There are 2^𝑛 vertices for this cell𝑥¹,…,𝑥^𝑛∣𝑥^𝑖=𝑝_𝑖,𝑞_𝑖. (4.5)

There are totally 3^𝑛−1 surrounding cells^𝑛_𝑖=1𝑥_𝑖,𝑦_𝑖∣𝑥_𝑖,𝑦_𝑖=𝑟_𝑖,𝑝_𝑖,𝑝_𝑖,𝑞_𝑖,𝑞_𝑖,𝑠_𝑖⧵{𝐶}. (4.6)

First we consider the triangulation of the cell 𝐶. Now we choose one vertex 𝑣₀=[𝑧₁,…,𝑧_𝑛] from the cell 𝐶 where 𝑧_𝑖 can be 𝑝_𝑖 or 𝑞_𝑖. Now we denote𝑦_𝑖=𝑝_𝑖,if𝑦_𝑖=𝑞_𝑖𝑞_𝑖,if𝑦_𝑖=𝑝_𝑖. (4.7)

Consider the vertex set𝑆_𝑗=𝑥¹,…,𝑥^𝑗−1,𝑦^𝑗,𝑥^𝑗+1,…,𝑥^𝑛∣𝑥^𝑖=𝑝_𝑖,𝑞_𝑖. (4.8)

Then #(𝑆_𝑗)=2^𝑛−1. Clearly 𝐶 can be decomposed as disjoint cones 𝐶=∪^𝑛_𝑗=1𝑣₀𝑆_𝑗. Notice that 𝑆_𝑗 is an (𝑛−1)-dimensional rectangle and this means that, for any triangulation of 𝑆_𝑗, we can join them with 𝑣₀ to form a triangulation of 𝑣₀𝐶. Therefore by induction we get the construction of triangulation.

Lemma 4.1. The decomposition 𝐶=⋃^𝑛_𝑗=1𝑣₀𝑆_𝑗 is valid with the property that each pair of Int(𝑣₀𝑆_𝑗) is disjoint, where Int(𝐶) is the set of interior points in 𝑅^𝑛.

Proof. Suppose 𝐱=[𝑎₁,…,𝑎_𝑛]∈𝐶 and 𝑥≠𝑣₀. Consider the point 𝐱_𝑡=(1−𝑡)𝑣₀+𝑡𝐱=[(1−𝑡)𝑧₁+𝑡𝑎₁,…,(1−𝑡)𝑧_𝑛+𝑡𝑎_𝑛] where 𝑡≥1. Let 𝑡₀ be the first zero point of the following function: 𝜓(𝑡)=min_{1≤𝑗≤𝑛}𝑦_𝑗−(1−𝑡)𝑧_𝑗−𝑡𝑎_𝑗. (4.9)
Clearly 𝜙(1)≥0. Let [𝑦_𝐽−(1−𝑡₀)𝑧_𝐽−𝑡₀𝑎_𝐽]=0. Then 𝐱∈𝑣₀𝑆_𝐽.

By the above discussion, we also have another lemma.

Lemma 4.2. Suppose the triangulation of 𝑆_𝑗 in 𝑅^𝑛−1 is {𝑇^𝑠_𝑗}. Then the set ∪^𝑛_𝑗=1∪_𝑠{𝑣₀𝑇^𝑠_𝑗} forms a triangulation of 𝐶.

Finally we obtain a theorem.

Theorem 4.3. A simplicial complex 𝐾 exists corresponding to the grids 𝒢.

For each node 𝑥∈Ω, define its neighborhood as 𝒩(𝑥)={𝑦∈Ω∣𝑦 is incident with 𝑥 in 𝐾}. For a subset 𝐶⊂Ω, define its neighborhood 𝒩(𝐶) as 𝒩(𝐶)=⋃_𝑥∈𝐶𝒩(𝑥).

Now we propose an algorithm for our clustering problem. This is a self-joining cluster technique. Initially, one data point in Ω forms a cluster of singleton. Then each node searches for its neighborhood. If there exists a neighboring node which is similar enough, then activate a link between the two nodes. The final cluster is linked nodes. To define the meaning of similar enough, we need a parameter 𝛿₀ such that 𝑓(𝑥,𝑦)>𝛿₀. Putting everything together, we get the algorithm as shown in Algorithm 1.

Inputs:  Ω={𝑥1,𝑥2,…,𝑥𝑁}, 𝛿0: similarity threshold value, 𝒩𝑖={𝑦∣𝑦,𝑥𝑖 are incident} Outputs:  𝒞={𝐶1,𝐶2,…,𝐶𝑡}: set of clusters, 𝑡: number of clusters, Ω0: outliers Begin Set Ω1=Ω0=∅, 𝐶1=∅, 𝑡=1. for 𝑖=1  to 𝑁   If 𝒩𝑖=∅, then add 𝑥𝑖 into set Ω0. Otherwise add 𝑥𝑖 into set Ω1. For each 𝑥𝑗∈𝒩𝑖,𝑗>𝑖, calculate   the similarity measure 𝑓(𝑥𝑖,𝑦). If 𝑓(𝑥𝑖,𝑦)>𝛿0, then set the edge 𝜎(𝑥𝑖𝑦) as active. In this case, set   𝐶1={𝑥𝑖} if 𝐶1 is still an empty set. end while  Ω1≠∅  do    (1) For each 𝑥∈Ω1 if there exists an active edge 𝜎(𝑥𝑦) with 𝑥≠𝑦, 𝑦∈𝐶𝑡, then add 𝑥 into 𝐶𝑡 and   remove 𝑥 from Ω1.    (2) 𝑡=𝑡+1. end End

Algorithm 1 

A self-joining clustering algorithm.

4.2. Design of a P System

Now we have already defined the simplicial complex 𝐾 as the membrane structure with 𝑀 total membranes. Next we need to specify the alphabet and rules to be used. First we design a binary coding scheme for the weight function 𝑚(⋅)and the distance function 𝑑(𝑥,𝑦). Suppose the length of the coding is 𝐿. In this way, the weight function and the distance function take binary strings as values. Now we suppose 𝑚(𝑥_𝑖)=𝑚_𝑖1⋯𝑚_𝑖𝐿, where 𝑚_𝑖𝑗=0,1. We also need an integer 𝐻 such that if the coding of 𝛿₀ is 𝑑₁𝑑₂⋯𝑑_𝐿, then^𝐿_𝑖=1𝑑_𝑖2^𝐿−𝑖=𝐻. (4.10)

Suppose 𝜇⊂ch is a subset of links. Then we can define a P system as follows:Π=𝑂,𝜇,𝜔₁,…,𝜔_𝑀,𝑅𝑖,𝑗. (4.11)

The working alphabet is𝑂=𝑎_𝑖𝑠,𝑏_𝑖𝑠∪𝑔∶1≤𝑖≤𝑁,1≤𝑠≤𝐿_𝑖𝑠,ℎ_𝑖𝑠∪∶1≤𝑖≤𝑁,1≤𝑠≤𝐿𝛼,𝛽,𝛽₊,𝛽₋∪𝑠_𝑖𝑗∶1≤𝑖,𝑗≤𝑁. (4.12)

Let 𝑣_𝑖=𝑥_𝑖, 𝑖=1,…,𝑁 be the vertices. We will use 𝜎(𝑣_𝑖0⋯𝑣_𝑖𝑘) to denote a 𝑘-dimensional simplex. Therefore 𝜎(𝑣_𝑖) is a vertex or 𝑣_𝑖 for short. Define 𝑁_𝑖=#(𝒩_𝑖). ch is the set of edges in 𝐾.

The initial multiset 𝜔_𝑖 stands for the multiset in the membrane 𝜎(𝑣_𝑖):𝜔_𝑖=𝑎^{2𝐿−𝑠𝑁𝑖}_𝑖𝑠,𝑏^{2𝐿−𝑠𝑁𝑖}_𝑖𝑠∪𝑔∶1≤𝑠≤𝐿_{𝑚𝑖𝑠𝑖𝑠},ℎ^{1−𝑚𝑖𝑠}_𝑖𝑠∶1≤𝑠≤𝐿∪𝛽,𝛽₊. (4.13)

For (𝑖,𝑗)∈𝜇, rules are 𝑅(𝑖,𝑗)={𝐴^𝑠_𝑖𝑗,𝐵_𝑖𝑗∶𝑠=1,…,𝐿} with the order as follows:𝐴¹_𝑖𝑗<𝐴²_𝑖𝑗<⋯<𝐴^𝐿_𝑖𝑗𝐴,𝑖⋅𝑗=1,…,𝑁_𝑠𝑖𝑗=𝛽²𝑔_𝑖𝑠ℎ_𝑗𝑠⟶𝜎𝑣_𝑖𝑣_𝑗,𝛽₊∪𝛽²ℎ_𝑖s𝑔_𝑗𝑠⟶𝜎𝑣_𝑖𝑣_𝑗,𝛽₋∪𝜎𝑣_𝑖𝑣_𝑗,𝛽₊⟶𝜎𝑣_𝑖,𝛽^2𝐿₊∪𝜎𝑣_𝑖𝑣_𝑗,𝛽₋⟶𝜎𝑣_𝑗,𝛽^2𝐿₋𝐵_𝑖𝑗=𝛽₊𝑎_𝑖𝑠⟶𝜎𝑣_𝑖𝑣_𝑗,𝑠_𝑖𝑗∪𝛽,in∶1≤𝑠≤𝐿₋𝑎_𝑗𝑠⟶𝜎𝑣_𝑖𝑣_𝑗,𝑠_𝑖𝑗,in∶1≤𝑠≤𝐿. (4.14)

Rules on edges are𝑅_𝑖𝑗=𝑠^𝐻_𝑖𝑗→𝛼∶1≤𝑠≤𝐿. (4.15)

5. Example and Discussion

The Wisconsin breast cancer data set consists of 699 samples with 9 attributes. The samples are classified as two categories, the malignant and the benign. The nine attributes are Clump-Thickness, Cell-Size-Uniformity, Cell-Shape-Uniformity, Marginal-Adhesion, Single-Epi-Cell-Size, Bare-Nuclei, Bland-Chromatin, Normal-Nucleoli, and Mitoses. The data is shown in Figure 9 with an added noise of [0,0.5].

Figure 9 

Wisconsin breast cancer data with perturbation. Attributes 2 and 6.

Now we choose attributes 2 and 6 and accumulate the data samples and use the grid technique proposed in this paper. Then the data set is shown in the following matrix:𝑀_data=3361384900003314320111021956240111135131303211901101035217201112001171110200121111332004014011110010173157106235. (5.1)

Next we construct a simplicial complex 𝐾 as shown in Figure 10 in 𝑅² where each vertex corresponds to the data points as in (5.1).

Figure 10 

Simplices of the Wisconsin breast cancer data.

We now define 𝐿=9. Then 2^𝐿=512 and 𝑁=100−25=75. Now we choose 𝛿₀=1. Then the final cluster is shown in Figure 11. We find five clusters and 31 outliers.

Figure 11 

Cluster result of the Wisconsin breast cancer data.

Next we choose attributes 1 and 2 and then the data matrix is𝑀_data=131831110000411502010008010831210035673322140263111784455013102402303191224540410034724741141001311214116138637222. (5.2)

Now we analyze the effect of the parameter 𝛿₀. First we choose 𝛿₀=0. Then the clustering result is one cluster with all data. Then the ill clustered points is 699. Next let 𝛿₀=1. Then we get two clusters and some outliers. However, in this case the red data are all ill-clustered and hence the error rate is 241. Now choose 𝛿₀=2. In this case we find four clusters. The number of outliers are 46 while the ill-clustered points are 65+74=139. Then the total number is 185. Next choose 𝛿₀=3. This time we find 6 clusters. Number of outliers is 87. Number of ill-clustered points is 91. Number of other error points is 40 hence the total number is 218. Again we choose 𝛿₀=4. In this time we obtain 4 clusters with a large outlier (outlier number: 157, error clusters: 65, other errors: 13, total number: 235). The error line is shown in Figure 12.

Figure 12 

Error number with respect to the parameter 𝛿0.

As a result we see that the best parameter is 𝛿₀=2.

Acknowledgments

This research is supported by the Natural Science Foundation of China (no. 61170038,71071090,60873058), the Natural Science Foundation of Shandong Province (no. ZR2011FM001), and the Shandong Soft Science Major Project (No. 2010RKMA2005).