Research on Clone Mind Evolution Algorithm

Gang Xie, Hongbo Guo, Keming Xie, and Wenjing Zhao

College of Information Engineering, Taiyuan University of Technology,030024 Taiyuan, Shanxi, P.R. China

xiegangtut@yahoo.com.cn

Abstract. A new algorithm of evolutionary computing, which combinesclone selective algorithm involved in artificial immunity system theoryand mind evolution algorithm (MEA) proposed in reference [4], is pre-sented in this paper. Based on similartaxis which is the one of MEAoperators, some operators borne by the new algorithm including clonemutation, clone crossover, clone selection, is also introduced. Then theclone mind evolution algorithm (CMEA) is developed by using the di-versity principle of antigen-antibody. The simulating results of the rep-resentative evaluation function show that the problem of degenerationphenomenon existing in GA and MEA can be perfectly solved, and therapidity of convergence is evidently improved by CMEA studied in thepaper. In the example of the solution to the numerical problem, thesearch range of solution is expanded and the possibility of finding theoptimal solution is increased.

1 Introduction

In the research field which modern information science and life science over-lap and interpenetrate to form into, artificial immune system (AIS) is an otherresearch focus subsequently following cranial nerves (e.g. neural network) andevolutionary computing (e.g. GA), which is inspired by the biological immunesystem (BIS). BIS mechanics based research on computing model is concentratedon two main aspects: network model of AIS and immune learning algorithm. Theformer aims to construct various computing model, based on the clone selectivetheory of Bernet [1] and the unique network adjusting theory of Jernet [2], toimitate or explain immune phenomena by simulation experiments. The latter isfocused on computing methods with stronger intentness or implement strategiesbased on existed system models. Clonal selection algorithm [3] that is presentedby Castro, Kim, Du, etc. is one of outstanding achievement. The characteristicsof memory, learning and evolution are utilized to implement the task such asmachine learning or pattern recognition. Mind evolution algorithm (MEA) [4]that is a kind of evolution computing method has been applied in the field ofintelligent control [5]. In this paper, how to utilize practicable clone selectivebehavior to design suitable clone selective optimal method in order to improvingthe optimal result of MEA is studied.

D. Sl ↪ezak et al. (Eds.): RSFDGrC 2005, LNAI 3641, pp. 431–440, 2005.c© Springer-Verlag Berlin Heidelberg 2005

432 G. Xie et al.

2 Philosophy of MEA

The nature evolution of biology depends on inheritance and nature selection.The evolutionary process will experience thousands of years. Comparatively, theevolutionary process of the human’s mind is short. The reason is that humanbeing cannot only adapts actively the change of the nature environment butstudies knowledge and experience from predecessor and other people selfcon-sciously. This phenomenon is called similartaxis. During the recent years, withthe development of the exchange way of the information, the development ofhuman’s mind is accelerating. At the same time, many innovations have beenacquired by human. This phenomenon is called dissimilation. Depending on thesimilartaxis and dissimilation, people develop science and technology.

MEA is a new type of evolutional computing method that simulates evolu-tional process of people’s thought. It uses the concept ’population’ of GA, but isradically different from it. “Similartaxis” and “dissimilation” operators are pre-sented. Since memory function and directional study mechanism are introducedand population optimization replaces the individual optimization, the intelli-gence of the algorithm is improved and the search efficiency is also enhanced.

2.1 Population and Group

The set of all individuals is called a population. The population is divided intoseveral groups, and there are two main classes of groups: the winner groups andthe temporary groups.

2.2 Billboard

The billboards, which record the information of the individuals or the groupsincluding theirs serial number, operation and score, provide the environment ofinformation communication among the individuals or the groups. There are twokinds of billboards: one is the local billboard which is used to record informationof individuals in each group; another is the global billboard which is used torecord information of each group in the whole population.

2.3 Similartaxis and Dissimilation

“Similartaxis” performs local competition inside subpopulations among individ-uals and produces local optimal points. At first N individuals are distributednormally around one “winner” with the variance δ and then the scores of themare computed and the one with highest score is the “winner” that will take partin the global competition delegating the subpopulation in the following dissimi-lation.

“Dissimilation”performs global competition. The“winners”of subpopulationsfrom“similartaxis”compete with each other and those having high score are keptto next round but the others are eliminated and are replaced by new individualsdistributed in the solution space. This makes the evolution of the populationheads toward the optimal point and gets there finally.

Research on Clone Mind Evolution Algorithm 433

Intelligent

MEAOptimizingProgram

integral&

Fuzzycontroller

Ke

S Kd

Ku G0(S)u��

d �

��

�

y

� �

�� �

�

�

eR

+ - �

�

+

Fig. 1. Schematic diagram of an optimal fuzzy controller based MEA

2.4 Convergence

It can be proved by means of Markov chains that population of discrete stateexecuted by similartaxis operator is convergent at the global optimal state withtotal probability. But because of localness of similartaxis, there is little probabil-ity that the local optimal state transfers to the global optimal state. In order tothis transfer probability, it is necessary to introduce dissimilation operator [6].

2.5 The Application of MEA

MEA has been successfully applied to intelligent control. The principium of anoptimal fuzzy controller OFC design method based on MEA is showed in figure1. At first the system and the controller are constructed and the membershipfunctions are built with conventional method. Then the fuzzy rules and quanti-fied factors and proportional factors are optimized by use of MEA. During theoptimizing process, the universe of the parameters (including the fuzzy controlrules and quantified factors and proportional factors) is divided into differentsubspaces according to their own solution range and then MEA searches thebest solutions in each subspace and forms a number of parameter groups andevaluates each group synthetically with one criterion. The optimization is alongthe direction that the criterion value reduces. The criterion is selected by de-signers practically to meet the demand of system performance.

When the parameters that make the criterion value least are found, so is theoptimal operation condition.

The design procedures are shown as following:

1. Decide controller structure.2. Select appropriate membership functions, fuzzy variables and universe.3. Set the solution spaces of parameters.4. Make optimizing program and optimize the parameters.5. End of design.

It is convenient to complete this method by software and easy to generalize it.Once it is completed, the program can be used to design any fuzzy controller andwhat the designer need to do is to reset a number of parameters and membershipfunctions.

434 G. Xie et al.

3 Mechanism of Clone Selection

Nowadays, most of researches on the intelligent systems revolve around the mech-anism of inspiring and learning of person brain. Over the last few years, therehas been an ever increasing interest in the area of artificial immune systems(AIS) and their applications.The ability of the immune system to respond to anantigen exists before it ever encounters that antigen.

The immune system relies on the prior formation of an incredibly diversepopulation of B cells and T cells.When an animal is exposed to an antigen, somesubpopulation of its bone marrow derived cells (B lymphocytes) respond by pro-ducing antibodies (Ab). Each cell secretes only one kind of antibody, which isrelatively specific for the antigen. By binding to these antibodies (receptors),and with a second signal from accessory cells, such as the T-helper cell, theantigen stimulates the B cell to proliferate (divide) and mature into terminal(non-dividing) antibody secreting cells, called plasma cells. The various cell di-visions (mitosis) generate a clone, i.e., a set of cells that are the progeny of asingle cell. While plasma cells are the most active antibody secretors, large Blymphocytes, which divide rapidly, also secrete Ab, albeit at a lower rate. WhileB cells secrete Ab, T cells play a central role in the regulation of the B cellresponse and are preeminent in cell mediated immune responses. Lymphocytes,in addition to proliferating and/or differentiating into plasma cells, can differen-tiate into long-lived B memory cells. Memory cells circulate through the blood,lymph and tissues, and when exposed to a second antigenic stimulus commenceto differentiate into large lymphocytes capable of producing high affinity an-tibodies, pre-selected for the specific antigen that had stimulated the primaryresponse [7]. The clou of reference [8] is that antibody in cell surface as offspringof natural exists in the form of receptor, and can selectively react to antigen. Thereaction, which takes place between antigen and receptor, can cause to clonalbreeding of cell. So the great number of clonal cell owns the identical speci-ficity of antibody. Some of these clonal cells in which some cells differentiate toa generation of antibody cell, and others form immunity memory cell so as toattend the second immunity reaction later. Clone selective theory acts as an im-portant enlightenment role for improving the performance of MEA, because ofthe clone selective course of antibody possesses learning, memory development,diversity of antibody, selfadaptive adjustment and such performance, so as toprevent the phenomenon of “prematurity” well, efficiently improve the rapidityof optimization and advance the quality of optimization result.

4 Clone Mind Evolutionary Algorithm (CMEA)

In general, the following steps of CMEA are made up of 6 key steps illustratedin figure 2. It is well known that antigen, antibody, affinity of antigen-antibodyis respectively corresponded to the object function, optimal solution, and matchdegree of solution to the object function.

Research on Clone Mind Evolution Algorithm 435

Antigen Recognition

Antibody Production

Affinity Measures

Groups Construction

Affinity Measures

Clone Opoerator

Terminal Condition

�

�

�

�

�

�

Fig. 2. Diagram of CMEA

1. step1: antigen recognition Choose the target function and various con-straints as the antigen of CMEA, then the immune system confirms that theantigen invades;

2. step2: initial antibody production While iterating at the first time, theantibody is produced at random in the whole solution space, or by meansof activating memory cells. At the same time, the foregone antigen is re-moved, and M individuals from the database including the optimal antibody(optimal solution) are choose to produce initial antibody groups;

3. step3: affinity calculation Separately calculates the affinity between anti-gen and antibody, and the affinity between antibody and antibody;

4. step4: groups’ construction N individuals with supreme affinity are ar-ranged in an order. For every individual with supreme affinity, k-1 individualsare randomly choosed among the remaining individuals, and are constructedto a group (the size of group is k). Thus N groups are produced by theidentical operation to the N individual of supreme affinity;

5. step5: Calculate every individual affinity in each group again;6. step6: According to clone operators, produce new group with the following

steps:

(a) clone: Choose A(m) individuals that own higher antibody-antigen affin-ity and lower antibody- antibody affinity, then regard them as clonedindividuals and add clonal results to new groups.

(b) clone mutation: Carry out clone mutation operation on the groups af-ter completing clone operator. In order to reserve the original informationof antibody population, do not operate mutation to A(m) individuals.

436 G. Xie et al.

(c) clone crossover: Choose A(n) individuals that have higher antibody-antibody affinity and lower antigen-antibody affinity, and uniformly codethem. Then, execute crossover operator.

(d) clone selection: Select M new individuals owning the highest affinityto form a new generation of population in order to keep the numberof individuals. Meanwhile, other rejected individuals are deleted fromgroups.

7. step7: Terminal condition Repeat step5 and step6, until satisfy termi-nation condition (convergence criterion), optimal course end. In this paper,limited iteration times are adopted as termination condition. Choose theoptimal individual as the result of algorithm.

5 Convergence Analysis of CMEA

In generally speaking, we consider maximum problem in this paper, to findsolution for an optimal problem ϕ :

∏mi=1[di, ui → R(di ≺ ui)], where m is

the number of the optimized variables, i.e. X = {x1, x2, . . . , xm} . The antigenϕ : Rm → R is the optimized function. Real number code is adopted in thispaper.

Antibody group A = {A1, A2, . . . , An} is an nth multi group; it is a point inthe Antibody population space Sn[9].

Definition 1. M = {A|max(f(A)) = f∗, ∀A ∈ Sn} is called satisfied popula-tion set, that is, any initial antibody population in M at least contains a bestsolution.

The mathematic model of CMEA can be described as: after real numbercoded, CEMA process is a memorized stochastic walk form one state to anotherstate, which can be described by a Markov Chain process.

In the antibody population space Sn, antibody group transferred from thestate A(k) = {A1(k), A2(k) . . . An(k)} to a new one A(k+1) = {A1(k+1), A2(k+2) . . . An(k + 1)} after the CMEA operation and this process is expressed by:A(k + 1) = T (A(k)) = T c

d ◦ T cs ◦ T c

r ◦ T cc (A(k))

Where, T cd is clone operator, T c

s is clone selection operator, T cr is clone re-

combination operator, and T cc is clone mutation operator.

Mark A(k) = X ,A(k + 1) = Y then the transition probability pxy(k) =p{A(k + 1) = Y |A(k) = X}

When X �= Y :

pxy(k) =

⎧⎪⎨

⎪⎩

0 i ∈ M, j /∈ Mn∏

j=1

pdpks

(qj−1∑

i=1

(pim)d(X,Y )(1 − pi

m)l−d(X,Y )

)

other(1)

Research on Clone Mind Evolution Algorithm 437

when X = Y :

pxy(k) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 −|M|∑

L�=Y

n∏

j=1

pdpks

(qj−1∑

i=1

(pid(X,Y )

m (1 − pim)l−d(X,Y )

))

i ∈ M, j ∈ M

1 −|Sn|−|M|∑

L�=Y

n∏

j=1

pdpks

(qj−1∑

i=1

(pim)d(X,L)(1 − pi

m)l−d(X,L)

)

i /∈ M, j /∈ M

(2)

Equation (1), (2) are the main model of the CMEA.

Theorem 1. The antibody population series of the CMEA algorithm is {A, k ≥0}, and it is a finite nonhomogeneous reducible Markow chain.

Proof. Any antibody in an antibody population A = {A1, A2 . . . An} is a limitedreal number in a limited population, so its state variable is changing in a limitedstate space Sn, for pk

s has relation with population state in time k, so does pxy(k),so it is nonhomogeneous.

From the definition of M , M is a closed set, because:

1. If X, Y ∈ M , then pxy(k) > 0,pxy(k) > 0 i.e.X ↔ Y2. If X ∈ M and Y �∈ M , then pxy(k) = 0, i.e. X �→ Y

So {A(k), k ≥ 0} is reducible.Thus, the theorem 1 is proved.

Theorem 2. The antibody population series {A(k), k ≥ 0} of the CMEA algo-rithm is convergent to satisfied population set with probably 1. That is, to anyinitial state A0

limk→∞

P{A(k) ∈ M |A(0) = A0} = 1 (3)

Proof. Without loss of generality, suppose f(A) has only one maximum, mark:F (A(k)) = max{f(A(k)i), i = 1, 2, . . . n},

P (k) = p{A(k + 1) = Y |A(k) = X ; X, Y ∈ Sn} = (pxy(k); X, Y ∈ Sn) (4)

Equation 4 is called the state transfer matrix.To selection operator, there exist:

pks =

{0 F (X) > F (Y )1 F (X) ≤ F (Y ) (5)

Then for F (Y ) ≥ F (X), we have: pxy(k) = p{T cd ◦ T c

s ◦ T cr ◦ T c

c = Y } > 0 ifF (Y ) < F (X) we have: pxy(k) = 0,Mark:

P (∞) = limk→∞

P (k) = (P∞(X, Y ); X, Y ∈ Sn) (6)

Then:

P∞(X, Y ) ={

> 0 F (X) ≥ F (Y )= 0 F (Y ) < F (X) (7)

438 G. Xie et al.

Obviously: P (∞) is a stochastic matrix, and {A(k), k ≥ 0 is strong ergodic.To any initial state A0 , we have:

limk→∞

P{A(k) = Y |A(0) = A0} = π∞(Y ) (8)

Andk→∞∑

γ∈M

π∞(Y ) = 1 , so:

limk→∞

P{A(k) = Y |A(0) = A0| =∑

Y ∈M

π∞(Y ) = 1 (9)

This completes the proof of Theorem 2.

6 Research Example

In order to verify the preceding analysis, numerical experimentation employingMEA, CMEA and GA are studied by the following classical testing functions.

1. fit1 =3∑

i=1

x2i xi ∈ [−5, 5]

The function is adopted for testing rapidity of convergence, the global min-imum f(0, 0, 0) = 0 .

2. fit2 = 100(x21 − x2)2 + (1 − x1)2 xi ∈ [−5, 5]

The minimum point (0, 0) of this function locates at curved surface witha long and narrow paraboloid, so it is difficult to find the minimum. Thefunction is used to test immaturity convergence.

3. fit3 = 0.5 + [sin2(x21 + x2

2)1/2 − 0.5]/[1 + 0.001(x21 + x2

2)]2

Minimum of this function is fit3(0,0)=0. Within scope of 3.14 around (0, 0),there are many protuberant department that is the global suboptimal points.The function characteristic that is properties of strong oscillation and the globaloptimal point surrounded by the suboptimal global points make it is very difficultto find the global optimal solution. In experiments, let M=200, there kinds ofalgorithms are respectively examined 100 times with evaluation function. Butif the optimal solution is not improved within 10 times, then the calculating isterminal in advance. In every operation cycle, if the value of fitness is smallerthan the threshold 0.0001, the algorithm is regarded as success, otherwise failure.The number of successful optimization is denoted as NTS, and the number offailure is denoted as NTF, where NTS+NTF=100. The sum of all successfuliteration times divided by the successes times is the mean successful iterationtimes denoted as NMIS. Table 1 shows test data.

Showed from experiment results, the searching and optimization ability ofCMEA is greater than of MEA and GA. Especially, when the extreme pointis surrounded by the local subextreme points, more embody the superiorityofCMEA. Function fit3 optimization result shows, in 100 times operations, MEAsucceeds four only, and CMEA have 100% rate of success. The optimization re-sult of CMEA is 10e-9 times more accurate than that of MEA.

Research on Clone Mind Evolution Algorithm 439

Table 1. Optimization Result

Evaluation function Algorithm Threshold NTS NTF NMIS Optimal evaluation value

MEA 100 0 20.24 5.252053e-5fit1 CMEA 100 0 19.27 2.225179e-14

Ga 99 1 69.20 6.628564e-6MEA 84 16 79.37 1.864154e-4

fit2 CMEA 0.0001 100 0 8.11 1.272805e-7GA 100 0 98.75 3.464789e-5

MEA 4 96 77 9.172560e-5fit3 CMEA 100 0 11.86 2.947642e-14

GA 82 18 42.63 1.096471e-6

7 Conclusion

Both CMEA and MEA belong to group search strategy, and emphasize the infor-mation exchanging among the individuals of population. So there are similaritiesbetween CMEA and MEA.

Firstly on the structure, both of them circularly proceed with a course thatis “initial population production → dividing into smaller groups → calculatingevaluation function → exchanging information among individuals of groups →producing a new generation of population”. Population is divided into severalgroups to prevent information exchange among groups. So it is helpful for thepopulation differentiation, for the maintenance of diversity and for the preventionof prematurity, eventually the optimal solution is obtained with greater proba-bility; Secondly on the property, both of them inhere parallelism in essence soas to make it difficult to fall into the local minimum in searching process.

On the other hand, due to the introduced operators such as antigen recog-nition, clone, clone mutation, clone crossover, and clone selection etc., there aresome difference between them as follows:

1. clone mutation operator does not affect on the optimal solution which isheld in memory units. Thus it ensures to converge fast the global optimalsolution;

2. The considerable calculation is caused by affinity measure, including theaffinity of the antibody-antigen and the affinity of antibody-antibody. Butit do not influence the rapidity of convergence;

3. By promoting or restraining the antibody production, the function of self-regulation is achieved, and the diversity of individuals is guaranteed. Con-sidering both the local and global search ability, it is especially suitable tooptimize the multimodal function;

4. Mutation operator of MEA is replaced by clone crossover and clone mutation,thus it is sure to extend the search region and to ensure the convergence tothe global optimal solution.

440 G. Xie et al.

Acknowledgements

This work was financed by Chinese Nation Nature Science Foundation(60374029), and Visiting Scholar Foundation of Shanxi Province, P. R. China(2004-18).

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