The antibody candidate set generated by the clonal selection algorithm has only a small number of antibodies with high antigen affinity to obtain high-frequency mutations. Among other antibodies, some low-affinity antibodies are replaced by new antibodies to participate in the next clonal selection. A large number of antibodies with high affinity make it difficult to participate in clonal selection and exist in antibody concentration for a long time. This part of inactive antibody forms a “black hole” of the antibody set, which is difficult to remove and update in a timely manner, thus affecting the speed at which the algorithm approximates the optimal solution. Inspired by the mechanism of biological forgetting, an improved clonal selection algorithm is proposed to solve this problem. It aims to use the abstract mechanism of biological forgetting to eliminate antibodies that cannot actively participate in high-frequency mutations in the antibody candidate set and to improve the problem of insufficient diversity of antibodies in the clonal selection algorithm, which is prone to fall into the local optimal. Compared with the existing clonal selection and genetic algorithms, the experiment and time complexity analysis show that the algorithm has good optimization efficiency and stability.
As a heuristic algorithm to solve complex problems with a high mutation rate, the clonal selection algorithm has two important characteristics: it has efficient optimization performance [
The classical clonal selection algorithm has problems of algorithm efficiency, convergence rate, and lack of sufficient theoretical support [
It can be seen from the above literature that the current clonal selection algorithm flow mainly includes steps of selection, cloning, mutation, reelection, and replacement. At present, a large number of improved algorithms are intended to improve the steps of selection, cloning, and mutation, without considering the improvement and optimization of the replacement and update steps of antibody-concentrated antibodies by clonal selection algorithm. In order to improve the update efficiency of the antibody set, this paper tries to find a new scheme to replace the replacement steps used in the current clonal selection algorithm, so as to improve the overall search accuracy and convergence stability of the algorithm.
The antibody candidate set is a set of antibodies produced by the clonal selection algorithm during initialization. The clonal selection algorithm sorts the affinity of the antibodies in the set and selects the first
This kind of antibody that stays in the candidate set of antibodies for a long time can neither converge nor update quickly, jumping out of the local optimal interval. Therefore, there is a phenomenon in which antibodies cannot be selected and update iterations in a short period of time, which is vividly called “black hole” in this paper.
In this paper, an improved clonal selection algorithm based on the mechanism of biological forgetting (FCSA) is proposed. Aiming at the black hole formed by antibodies whose affinity does not meet the mutation and update conditions, the forgetting mechanism is applied to the antibody candidate set update process of the algorithm. The aim is to update these inactive and long-lasting antibodies in the candidate antibody set and enhance the diversity of antibodies in the antibody set, thereby increasing the convergence speed of the algorithm.
In 1885, Ebbinghaus [
Although both attenuation theory and interference theory are supported by experimental results, they lack an objective mechanism to explain why attenuation theory is only applicable to long-term memory, while interference theory is more applicable to short-term memory.
Shuai et al. [
Therefore, Shuai et al. [
Liu et al.’s [
In the complex biological system, the process of forgetting often occurs. This paper argues that forgetting is a process of information loss, and information loss is meaningful under certain circumstances. In reference [
At the same time, the idea of attenuation theory of biological forgetting is introduced into the replacement process of the clonal selection algorithm. The number of iterations of antibodies in the antibody candidate set is taken as the time length, and whether the antibody participates in high-frequency mutation in a certain iteration is taken as the basis of whether the antibody is remembered at the point in time, so as to realize the purpose of replacing the antibody with weak memory degree when the time span is large. Moreover, since the current replacement mechanism of clonal selection is still to replace the
In 2002, De Castro and Von Zuben [
It can be seen that the cloning selection algorithm selects the first
In order to more intuitively explain the state of the candidate antibodies in a certain round of iteration, the affinity between the antigen and the antibody is expressed in the form of distance. According to the previous section, after all antibodies are sorted according to their affinity, the clonal selection algorithm selects the
Antigen and antibody distribution structure.
Since the total number
Inspired by the forgetting mechanism, for each antibody in the candidate antibody set, calculate the number of times it is selected as the top
As shown in Figure
Antigen-antibody distribution structure affected by Rac1.
For each antibody in the candidate set, the Rac1 activity of the antibody in layer I is significantly lower than that in layer II. When the Rac1 activity of the layer II antibody exceeds the activity threshold, the antibody is replaced and the entire antibody candidate set is updated. In this way, the clonal selection algorithm avoids the antibody “black hole” formed by the partially unmigrated original antibodies in layer II.
The following definitions relate to the forgetting mechanism: Antibody survival time is the number of iterations that antibodies have participated in the antibody candidate set. Time Appropriate memory is the attribute of each candidate antibody. In an iteration, if the antibody belongs to the best Appropriate memory strength is the appropriate memory accumulated by candidate antibodies during Rac1 protein activity is the index affecting antibody forgetting, determined by the survival time of antibody in candidate antibody concentration and the strength of appropriate memory. Rac1 protein activity is proportional to the survival time of antibodies and inversely proportional to the degree of appropriate memory.
In this paper, an improved clonal selection algorithm (FCSA) inspired by the forgetting mechanism is proposed. Its core implementation idea is to replace the receptor editing mechanism [
The specific implementation method is as follows: In each iteration of the algorithm, the appropriate memory strength and survival time of each antibody candidate set are recorded. After several iterations of the algorithm, antibody forgetting was determined based on whether Rac1 protein activity reached the threshold.
To simplify the calculation, the target test function of antibody affinity to antigen is the function value, which can be expressed as
According to the affinity corresponding to the antibody and antigen, the cloning method performs the cloning operation on the antibody. The higher the affinity, the greater the number of antibodies that will be cloned. The specific cloning formula is
The mutation method aims at the
The specific variation formula is
The method of forgetting determines the necessity of antibody forgetting based on the survival time of the antibody, the appropriate memory intensity, and the activity of the Rac1 protein.
The specific forgetting formula is
The flow of the improved algorithm proposed in this paper is shown in Algorithm
FCSA Input: Output: the best antibody Begin Randomly generate
Calculate the affinity Sort the antibodies in the candidate set according to their affinity, and put the best Update the value of the appropriate memory of antibody See VARIATION METHOD, according to the degree of variation The Select the See FORGETTING METHOD, calculate the Rac1 protein activity of each antibody in forget the antibody Choose the best antibody as the final output
The suspension conditions of the algorithms in Algorithm
In the algorithm, Rac1 protein activity is an inherent property of each candidate antibody, which is calculated based on antibody survival time and appropriate memory strength when the antibody is first selected into the candidate set. And it changes dynamically with the execution of the algorithm. When the property value of the antibody reaches the threshold, it means that the antibody has not mutated in a better direction within the time we expect, and it is not sufficiently competitive with other candidate antibodies. So in the algorithm, the antibody that meets the threshold value will perform the forgetting operation.
Thirteen kinds of testing were done to select CEC test function optimization algorithm functions as experimental test functions, respectively, using the test function of the presented algorithm (FCSA) proposed in [
First, initialize various parameters of the algorithm. The termination criterion in this experiment is to run the GA, BCSA, ECSA, and FCSA until the number of function evaluations reaches the maximum value of 350,000.
Second, find the optimal solution of the test function. Three algorithms were executed to obtain the optimal solution generated by each execution of the algorithm. The average optimal solution, maximum optimal solution, and minimum optimal solution after 100 executions were analyzed.
The purpose of the experiment in this paper is to analyze the effectiveness of the forgetting mechanism applied to the clonal selection algorithm. The performance of the algorithm is mainly evaluated by the quality of the results obtained when the suspension conditions are consistent. This article counts the mean and standard deviation of the results of multiple runs of the algorithm to evaluate the quality of the results. These two indicators reflect the concentration trend and the degree of dispersion of the experimental data, respectively. Therefore, this paper uses these two indicators to verify the effectiveness of the improved algorithm.
Finally, we obtained the results of GA, CSA, and FCSA running at 1000 generations and plotted them as line charts. The purpose is to analyze the accuracy and speed of the algorithm by characterizing the relationship between generations and algorithm results.
Among them, the algorithm parameters are set as shown in Table
Initialization parameters.
Algorithm parameter | GA | CSA | FCSA | ECSA | BCSA |
---|---|---|---|---|---|
Cross rate | 0.5 | — | — | — | — |
Mutation rate | 0.13 | 2 | 2 | 2 | 2 |
Initial clone number | — | 5 | 5 | 5 | 5 |
Rac1 threshold | — | — | 3 | — | — |
Execution environment.
OS | Windows 10 professional edition |
CPU | Intel(R) Core(TM) i3-3217U CPU @ 1.80GHZ |
RAM | 12.0 GB |
Compiler version | Python 3.6 |
The test functions selected in this paper are shown in Table
Test function.
Test function | Expression | Optimum |
---|---|---|
Ackley Function |
|
0 |
Bukin Function |
|
0 |
Cross-in-Tray Function |
|
−2.06261 |
Drop-Wave Function |
|
−1 |
Eggholder Function |
|
−959.6407 |
Griewank Function |
|
0 |
Holder Table Function |
|
−19.2085 |
Levy Function |
|
0 |
Rastrigin Function |
|
0 |
Schaffer Function n. 2 |
|
0 |
Schaffer Function n. 4 |
|
0.5 |
Schwefel Function |
|
0 |
Shubert Function |
|
−186.7309 |
Consider the test functions in Table
The results of our experiment are shown in Table
Results of GA, CSA, and FCSA when
ALGs | GA | CSA | FCSA | |||
---|---|---|---|---|---|---|
Mean | Std | Mean | Std | Mean | Std | |
|
−5.0923 |
5.1699 |
−4.1302 |
3.4025 |
− |
|
|
−7.3939 |
4.7669 |
−1.2512 |
6.3021 |
− |
|
|
|
5.0685 |
2.06255 |
6.1577 |
2.06258 |
|
|
9.2563 |
4.3546 |
9.6047 |
2.3679 |
|
|
|
9.3533 |
|
9.5836 |
1.7805 |
|
6.3550 |
|
− |
2.7008 |
−2.5327 |
1.5556 |
−2.3604 |
|
|
1.8907 |
3.8513–001 | 1.9203 |
6.1897 |
|
|
|
−9.7077 |
2.2512 |
−7.2359 |
7.3241 |
− |
|
|
−1.4031 |
9.6354 |
−1.5528 |
1.5445 |
− |
|
|
− |
|
−2.7670 |
3.2928 |
−1.1855 |
1.6805 |
|
−5.00096 |
3.4944 |
− |
1.7190 |
− |
|
|
− |
|
−4.4234 |
4.5491 |
−3.0112 |
3.1088 |
|
1.6615 |
2.3853 |
1.8632 |
3.9038 |
|
|
Results of BCSA, ECSA, and FCSA when
ALGs | BCSA | ECSA | FCSA | |||
---|---|---|---|---|---|---|
Mean | Std | Mean | Std | Mean | Std | |
|
−1.2838 |
2.3931 |
−2.0213 |
|
− |
2.3649 |
|
−7.5633 |
|
−7.4943 |
3.4107 |
− |
3.6480 |
|
−2.5742 |
1.7774 |
−2.5256 |
1.9350 |
− |
|
|
−6.3444 |
1.9519 |
−6.3492 |
1.9929 |
− |
|
|
− |
3.1504 |
−1.4646 |
3.0211 |
−1.4721 |
|
Results of BCSA, ECSA, and FCSA when
ALGs | BCSA | ECSA | FCSA | |||
---|---|---|---|---|---|---|
Mean | Std | Mean | Std | Mean | Std | |
|
−1.3768 |
1.1996 |
− |
1.2061 |
−1.3697 |
|
|
−1.9237 |
|
−1.9198 |
6.4516 |
− |
6.5405 |
|
−7.1533 |
|
−−7.0166 |
4.0096 |
− |
2.7792 |
|
−1.4416 |
2.6102 |
−1.4370 |
3.2709 |
− |
|
|
− |
4.5766 |
−3.2866 |
3.7953 |
−3.2949 |
|
After replacing the updating operator in BCSA with the forgetting operator, set
Finally, the comparison results of the three algorithms of GA, CSA, and FCSA in the 1000th generation are shown in Figure
Results of GA, CSA, and FCSA when
In the test results of the Ackley Function, the optimal interval of FCSA in the process of 100 executions of the algorithm is (−1, 0), the optimal interval of CSA is (−1.5, 0), and the optimal interval of GA is (−2.5, 0). The convergence degree of the GA is worse than that of CSA and FCSA, and FCSA has the best degree of convergence.
Among the test results of the Bukin Function
In the test results of functions, Cross-in-Tray Function, Holder Table Function, Levy Function, and Shubert Function, the CSA and FCSA algorithms converge stably to the global optimal, among which FCSA has the optimal average search accuracy and stability, while the GA still has deviation points and the convergence is not stable.
According to the test results of Eggholder Function, the convergence stability of the CSA and FCSA is worse than GA, but the optimization accuracy is better than GA.
In the test results of Griewank Function, the optimization accuracy of the CSA and FCSA is better than that of GA, and GA has a few deviation points as well as poor convergence stability. With the improved CSA, when
In the test results of Schaffer Function
As can be seen from the experimental results in Table
Results of the BCSA algorithm combined with the forgetting mechanism when
ALGs |
|
| ||
---|---|---|---|---|
Mean | Std | Mean | Std | |
|
−8.5121 |
1.4299 |
−9.1801 |
9.4149 |
|
−7.5371 |
3.7047 |
−1.9109 |
7.2792 |
|
−2.4475 |
1.9376 |
−6.8859 |
3.1488 |
|
−6.2326 |
1.9837 |
−1.4202 |
3.0524 |
|
−1.4873 |
2.7891 |
−3.3390 |
3.6963 |
On the other hand, it can be seen from Figure
According to Section
Overall, the experimental results show that FCSA has higher optimization accuracy and stability than CSA, and FCSA has higher optimization accuracy and convergence stability than GA in most test functions.
It can be seen from the high-dimensional experiments of BCSA, ECSA, and FCSA that FCSA has more advantages over ECSA and BCSA in terms of convergence stability and accuracy. Due to the characteristics of the test function itself, the higher the dimension, the more complex the function change is, which leads to decreased optimization accuracy and stability of the algorithm.
By applying the forgetting mechanism to BCSA, the number of antibodies to be replaced by the original manual definition is changed to the number of antibodies to be replaced by the affinity attribute of antibodies. The forgetting mechanism has a positive effect on improving the convergence speed and convergence stability of such algorithms.
To solve the problem that the CSA cannot in a timely way eliminate antibodies that are not adapted to the new environment and thus form an antibody black hole, we see that by changing the receptor editing mechanism of the clonal selection algorithm to a new forgetting mechanism, the antibody candidate set can be replaced and updated under the regulation of Rac1 protein. Experiments show that FCSA is an effective improved algorithm compared with ECSA and BCSA in terms of optimization efficiency, optimization accuracy, and convergence stability.
Because FCSA changes the substitution step in the current clonal selection algorithm, it is better than the existing improved clonal selection algorithm. However, from the experimental performance in high-dimensional test function, FCSA still has the problem of low optimization precision. In the future, the FCSA will be combined with the existing improved clonal selection algorithm to further optimize the precision and stability of high-dimensional optimization.
We also note that Luo et al. [
At the same time, as an effective updating mechanism, the forgetting mechanism can also be applied to other heuristic algorithms that need to update the population of algorithms.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (61977021), National Social Science Fund (15CGL074), Intelligent Information Processing and Real Time Industrial System Hubei Provincial Key Laboratory Open Fund Project (znxx2018MS05), and Open project of Hubei Key Laboratory of Applied Mathematics (HBAM201902).