Adapting the Parameters of RBF Networks Using Grammatical Evolution
Abstract
:1. Introduction
 The element $\overrightarrow{x}$ represents the input pattern from the dataset describing the problem. For the rest of this paper, the notation d will be used to represent the number of elements in $\overrightarrow{x}$.
 The parameter k denotes the number of weights used to train the RBF network, and the associated vector of weights is denoted as $\overrightarrow{w}$.
 The vectors $\overrightarrow{{c}_{i}},\phantom{\rule{4pt}{0ex}}i=1,\dots ,k$ stand for the centers of the model.
 The value $y\left(\overrightarrow{x}\right)$ represents the value of the network for the given pattern $\overrightarrow{x}$.
 They have a simpler structure than other models used in machine learning, such as multilayer perceptron neural networks (MLPs) [11], since they have only one processing layer and therefore have faster training techniques, as well as faster response times.
 They can be used to efficiently approximate any continuous function [12].
 The first phase of the procedure seeks to locate a range of values for the parameters while also reducing the error of the network on the training dataset.
 The rules grammatical evolution uses in the first phase are simple and can be generalized to any dataset for data classification or fitting.
 The determination of the value interval is conducted in such a way that it is faster and more efficient to train the parameters with an optimization method during the second phase.
 After identifying a promising value interval from the first phase, any global optimization method can be used on that value interval to effectively minimize the network training error.
2. Method Description
2.1. Grammatical Evolution
 N is a set of the nonterminal symbols. A series of production rules is associated with every nonterminal symbol. The application of these production rules produces series of terminal symbols.
 T stands for the set of terminal symbols.
 S denotes the start symbol of the grammar and $S\in N$.
 P defines the set of production rules. These are rules that follow the following notations: $A\to a$ or $A\to aB,\phantom{\rule{4pt}{0ex}}A,B\in N,\phantom{\rule{4pt}{0ex}}a\in T$.
 Denote with V the next element form of the current chromosome.
 The next production rule is calculated as: Rule = V mod R. The number R stands for the total number of production rules for the nonterminal symbol that is currently under processing.
 A series of vectors $\overrightarrow{{c}_{i}},\phantom{\rule{4pt}{0ex}}i=1,\dots ,k$ that stand for the centers of the model.
 For every Gaussian unit, an additional parameter ${\sigma}_{i}$ is required.
 The output weight vector $\overrightarrow{w}$.
Algorithm 1 The BNF grammar used in the proposed method to produce intervals for the RBF parameters. By using this grammar in the first phase of the current work, the optimal interval of values for the parameters can be identified. 
S::=<expr> (0) <expr> ::= (<xlist> , <digit>,<digit>) (0) <expr>,<expr> (1) <xlist>::=x1 (0)  x2 (1) .........  xn (n) <digit> ::= 0 (0)  1 (1) 
 For each center $\overrightarrow{{c}_{i}},\phantom{\rule{4pt}{0ex}}i=1,\dots ,k$, there are d variables. As a consequence, every center requires $d\times k$ parameters.
 Every Gaussian unit requires an additional parameter: ${\sigma}_{i},\phantom{\rule{4pt}{0ex}}i=1,\dots ,k$, which means k more parameters.
 The weight vector $\overrightarrow{w}$ used in the output has k parameters.
 The variable for which its original interval will be partitioned, for example, ${x}_{7}$.
 An integer number with values 0 and 1 at the left margin of the interval. If this value is 1, then the left margin of the corresponding variable’s value field will be divided by two; otherwise, no change will be made.
 An integer number with values 0 and 1 at the right end of the range of values of the variable. If this value is 1, then the right end of the corresponding variable’s value field will be divided by two; otherwise, no change will be made.
2.2. The First Phase of the Proposed Algorithm
Algorithm 2 The kmeans algorithm. 

Algorithm 3 The proposed algorithm used to locate the vectors $\overrightarrow{L},\phantom{\rule{4pt}{0ex}}\overrightarrow{R}$ 

 1.
 Define as ${N}_{c}$ the number of chromosomes that will participate in the the grammatical evolution procedure.
 2.
 Define as k the number of processing nodes of the used RBF model.
 3.
 Define as ${N}_{g}$ the number of allowed generations.
 4.
 Define as ${p}_{s}$ the used selection rate, with ${p}_{s}\le 1$.
 5.
 Define as ${p}_{m}$ the used mutation rate, with ${p}_{m}\le 1$.
 6.
 Define as ${N}_{s}$ the total number of RBF networks that will be created randomly in every fitness calculation.
 7.
 Initialize ${N}_{c}$ chromosomes as sets of random numbers.
 8.
 Set ${f}^{*}=\left[\infty ,\infty \right]$ as the fitness of the best chromosome. The fitness function ${f}_{g}$ of any provided chromosome g is considered an interval ${f}_{g}=\left[{f}_{g,\mathbf{l}\mathbf{o}\mathbf{w}},{f}_{g,\mathbf{u}\mathbf{p}\mathbf{p}\mathbf{e}\mathbf{r}}\right]$
 9.
 Set iter = 0.
 10.
 For $i=1,\dots ,{N}_{c}$ do
 (a)
 (b)
 Produce the bounds $\left[\overrightarrow{{L}_{{p}_{i}}},\overrightarrow{{R}_{{p}_{i}}}\right]$ for the partition program ${p}_{i}$.
 (c)
 Set ${E}_{min}=\infty ,\phantom{\rule{4pt}{0ex}}{E}_{max}=\infty $
 (d)
 For $j=1,\dots ,{N}_{S}$ do
 i.
 Create randomly a set of parameters $\overrightarrow{{g}_{j}}\in $$\left[\overrightarrow{{L}_{{p}_{i}}},\overrightarrow{{R}_{{p}_{i}}}\right]$
 ii.
 Calculate the error ${E}_{\overrightarrow{{g}_{j}}}={\sum}_{k=1}^{M}{\left(y\left(\overrightarrow{{x}_{k}},\overrightarrow{{g}_{j}}\right){t}_{k}\right)}^{2}$
 iii.
 If ${E}_{\overrightarrow{{g}_{j}}}\le {E}_{min}$, then ${E}_{min}={E}_{\overrightarrow{{g}_{j}}}$
 iv.
 If ${E}_{\overrightarrow{{g}_{j}}}\ge {E}_{max}$, then ${E}_{max}={E}_{\overrightarrow{{g}_{j}}}$
 (e)
 EndFor
 (f)
 Set the fitness ${f}_{i}=\left[{E}_{min},{E}_{max}\right]$
 11.
 EndFor
 12.
 Perform the procedure of selection. Initially, the chromosomes of the population are sorted according to their fitness values. Since the fitness values are intervals, the ${L}^{*}$ operator is defined as$$\begin{array}{ccc}\hfill {L}^{*}\left({f}_{a},{f}_{b}\right)& =& \left\{\begin{array}{cc}\mathrm{T}\mathrm{R}\mathrm{U}\mathrm{E},\hfill & {a}_{1}<{b}_{1},\mathrm{O}\mathrm{R}\phantom{\rule{4pt}{0ex}}\left({a}_{1}={b}_{1}\phantom{\rule{4pt}{0ex}}\mathrm{A}\mathrm{N}\mathrm{D}\phantom{\rule{4pt}{0ex}}{a}_{2}<{b}_{2}\right)\hfill \\ \mathrm{F}\mathrm{A}\mathrm{L}\mathrm{S}\mathrm{E},\hfill & \mathrm{O}\mathrm{T}\mathrm{H}\mathrm{E}\mathrm{R}\mathrm{W}\mathrm{I}\mathrm{S}\mathrm{E}\hfill \end{array}\right.\hfill \end{array}$$As a consequence, the fitness value ${f}_{a}$ is considered smaller than ${f}_{b}$ if ${L}^{*}\left({f}_{a},{f}_{b}\right)=\mathrm{T}\mathrm{R}\mathrm{U}\mathrm{E}$. The first $\left(1{p}_{s}\right)\times {N}_{c}$ chromosomes with smaller fitness values are copied without changes to the next generation of the algorithm. The rest of the chromosomes are replaced by chromosomes created in the crossover procedure.
 13.
 Perform the crossover procedure. The crossover procedure will create new ${p}_{s}\times {N}_{c}$ chromosomes. For every pair of created offspring, two parents $(z,w)$ are selected from the current population using the tournament selection method. These parent will produce the offspring $\tilde{z}$ and $\tilde{w}$ using the onepoint crossover method shown in Figure 1.
 14.
 Perform the mutation procedure. In this process, a random number $r\in \left[0,1\right]$ is drawn for every element of each chromosome. The corresponding element is changed randomly if $r\le {p}_{m}$.
 15.
 Set iter = iter + 1
 16.
 If $\mathrm{iter}\le {N}_{g}$, go to step 10.
2.3. The Second Phase of the Proposed Algorithm
 Initialization Step
 (a)
 Define as ${N}_{c}$ the number of chromosomes.
 (b)
 Define as ${N}_{g}$ the total number of generations.
 (c)
 Define as k the number of processing nodes of the used RBF model.
 (d)
 Define as $S=\left[{L}_{\mathrm{best}},{R}_{\mathrm{best}}\right]$ the bestlocated interval of the first stage of the algorithm of Section 2.2.
 (e)
 Produce ${N}_{C}$ random chromosomes in S.
 (f)
 Define as ${p}_{s}$ the used selection rate, with ${p}_{s}\le 1$.
 (g)
 Define as ${p}_{m}$ the used mutation rate, with ${p}_{m}\le 1$.
 (h)
 Set iter = 0.
 Fitness calculation step
 (a)
 For $i=1,\dots ,{N}_{g}$, do
 i.
 Compute the fitness ${f}_{i}$ of each chromosome ${g}_{i}$ as ${f}_{i}={\sum}_{j=1}^{m}{\left(y\left(\overrightarrow{{x}_{j}},\overrightarrow{{g}_{i}}\right){t}_{j}\right)}^{2}$
 (b)
 EndFor
 Genetic operations step
 (a)
 Selection procedure. Initially, the population is sorted according to the fitness values. The first $\left(1{p}_{s}\right)\times {N}_{c}$ chromosomes with the lowest fitness values remain intact. The rest of the chromosomes are replaced by offspring that will be produced during the crossover procedure.
 (b)
 Crossover procedure: For every two new offspring $\left(\tilde{z},\tilde{w}\right)$, there are two parents $(z,w)$ that are selected from the current population with the selection procedure of tournament selection. The offspring are produced through the following process:$$\begin{array}{ccc}\hfill \tilde{{z}_{i}}& =& {a}_{i}{z}_{i}+\left(1{a}_{i}\right){w}_{i}\hfill \\ \hfill \tilde{{w}_{i}}& =& {a}_{i}{w}_{i}+\left(1{a}_{i}\right){z}_{i}\hfill \end{array}$$
 (c)
 Perform the mutation procedure. In this process, a random number $r\in \left[0,1\right]$ is drawn for every element of each chromosome. The corresponding element is changed randomly if $r\le {p}_{m}$.
 Termination Check Step
 (a)
 Set $iter=iter+1$
 (b)
 If $\mathrm{iter}\le {N}_{g}$, go to step 2.
3. Experiments
3.1. Experimental Datasets
 The UCI dataset repository, https://archive.ics.uci.edu/ml/index.php (accessed on 5 December 2023);
 The Keel repository, https://sci2s.ugr.es/keel/datasets.php (accessed on 5 December 2023) [75];
 The Statlib URL http://lib.stat.cmu.edu/datasets/ (accessed on 5 December 2023).
3.2. Experimental Results
 The column NEAT (NeuroEvolution of Augmenting Topologies) [115] denotes the application of the NEAT method for neural network training.
 The RBFKMEANS column denotes the original twophase training method for RBF networks.
 The column GENRBF stands for the RBF training method introduced in [116].
 The column PROPOSED stands for the results obtained using the proposed method.
 In the experimental tables, an additional row was added with the title AVERAGE. This row contains the average classification or regression error for all datasets.
4. Conclusions
 The proposed method could be applied to other variants of artificial neural networks.
 Intelligent learning techniques could be used in place of the kmeans technique to initialize the neural network parameters.
 Techniques could be used to dynamically determine the number of necessary parameters for the neural network. For the time being, the number of parameters is considered constant, but this has the consequence of resulting in overtraining phenomena being observed in various datasets.
 Crossover and mutation techniques that focus more on the existing interval construction technique for model parameters could be implemented.
 Efficient termination techniques for genetic algorithms could be used to obtain the most efficient termination of techniques without wasting computing time on unnecessary iterations.
 Techniques that are based on parallel programming could be used to increase the speed of the method.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Expression  Chromosome  Operation 

9, 8, 6, 4, 15, 9, 16, 23, 8  9 mod 2 = 1  
<expr>,<expr>  8, 6, 4, 15, 9, 16, 23, 8  8 mod 2 = 0 
(<xlist>,<digit>,<digit>),<expr>  6, 4, 15, 9, 16, 23, 8  6 mod 8 = 6 
(x7,<digit>,<digit>),<expr>  4, 15, 9, 16, 23, 8  4 mod 2 = 0 
(x7,0,<digit>),<expr>  15, 9, 16, 23, 8  15 mod 2 = 1 
(x7,0,1),<expr>  9, 16, 23, 8  9 mod 2 = 1 
(x7,0,1),(<xlist>,<digit>,<digit>)  16, 23, 8  16 mod 8 = 0 
(x7,0,1),(x1,<digit>,<digit>)  23, 8  23 mod 2 = 1 
(x7,0,1),(x1,1,<digit>)  8  8 mod 2 = 0 
(x7,0,1),(x1,1,0) 
Dataset  Classes  Reference 

APPENDICITIS  2  [76] 
AUSTRALIAN  2  [77] 
BALANCE  3  [78] 
CLEVELAND  5  [79,80] 
DERMATOLOGY  6  [81] 
HAYES ROTH  3  [82] 
HEART  2  [83] 
HOUSEVOTES  2  [84] 
IONOSPHERE  2  [85,86] 
LIVERDISORDER  2  [87] 
MAMMOGRAPHIC  2  [88] 
PARKINSONS  2  [89] 
PIMA  2  [90] 
POPFAILURES  2  [91] 
SPIRAL  2  [92] 
REGIONS2  5  [93] 
SAHEART  2  [94] 
SEGMENT  7  [95] 
WDBC  2  [96] 
WINE  3  [97,98] 
Z_F_S  3  [99] 
ZO_NF_S  3  [99] 
ZONF_S  2  [99] 
ZOO  7  [100] 
Dataset  Reference 

ABALONE  [101] 
AIRFOIL  [102] 
BASEBALL  STATLIB 
BK  [103] 
BL  STATLIB 
CONCRETE  [104] 
DEE  KEEL 
DIABETES  KEEL 
FA  STATLIB 
HOUSING  [105] 
MB  [103] 
MORTGAGE  KEEL 
NT  [106] 
PY  [107] 
QUAKE  [108] 
TREASURY  KEEL 
WANKARA  KEEL 
Parameter  Value 

${N}_{c}$  200 
${N}_{g}$  100 
${N}_{s}$  50 
F  10.0 
B  100.0 
k  10 
${p}_{s}$  0.90 
${p}_{m}$  0.05 
Dataset  Rprop  Adam  Neat  RbfKmeans  Genrbf  Proposed 

Appendicitis  16.30%  16.50%  17.20%  12.23%  16.83%  15.77% 
Australian  36.12%  35.65%  31.98%  34.89%  41.79%  22.40% 
Balance  8.81%  7.87%  23.14%  33.42%  38.02%  15.62% 
Cleveland  61.41%  67.55%  53.44%  67.10%  67.47%  50.37% 
Dermatology  15.12%  26.14%  32.43%  62.34%  61.46%  35.73% 
Hayes Roth  37.46%  59.70%  50.15%  64.36%  63.46%  35.33% 
Heart  30.51%  38.53%  39.27%  31.20%  28.44%  15.91% 
HouseVotes  6.04%  7.48%  10.89%  6.13%  11.99%  3.33% 
Ionosphere  13.65%  16.64%  19.67%  16.22%  19.83%  9.30% 
Liverdisorder  40.26%  41.53%  30.67%  30.84%  36.97%  28.44% 
Mammographic  18.46%  46.25%  22.85%  21.38%  30.41%  17.72% 
Parkinsons  22.28%  24.06%  18.56%  17.41%  33.81%  14.53% 
Pima  34.27%  34.85%  34.51%  25.78%  27.83%  23.33% 
Popfailures  4.81%  5.18%  7.05%  7.04%  7.08%  4.68% 
Regions2  27.53%  29.85%  33.23%  38.29%  39.98%  25.18% 
Saheart  34.90%  34.04%  34.51%  32.19%  33.90%  29.46% 
Segment  52.14%  49.75%  66.72%  59.68%  54.25%  49.22% 
Spiral  46.59%  48.90%  50.22%  44.87%  50.02%  23.58% 
Wdbc  21.57%  35.35%  12.88%  7.27%  8.82%  5.20% 
Wine  30.73%  29.40%  25.43%  31.41%  31.47%  5.63% 
Z_F_S  29.28%  47.81%  38.41%  13.16%  23.37%  3.90% 
ZO_NF_S  6.43%  47.43%  43.75%  9.02%  22.18%  3.99% 
ZONF_S  27.27%  11.99%  5.44%  4.03%  17.41%  1.67% 
ZOO  15.47%  14.13%  20.27%  21.93%  33.50%  9.33% 
AVERAGE  26.56%  32.36%  30.11%  28.84%  33.35%  18.73% 
Dataset  Rprop  Adam  Neat  RbfKmeans  Genrbf  Proposed 

ABALONE  4.55  4.30  9.88  7.37  9.98  5.16 
AIRFOIL  0.002  0.005  0.067  0.27  0.121  0.004 
BASEBALL  92.05  77.90  100.39  93.02  98.91  81.26 
BK  1.60  0.03  0.15  0.02  0.023  0.025 
BL  4.38  0.28  0.05  0.013  0.005  0.0004 
CONCRETE  0.009  0.078  0.081  0.011  0.015  0.006 
DEE  0.608  0.630  1.512  0.17  0.25  0.16 
DIABETES  1.11  3.03  4.25  0.49  2.92  1.74 
HOUSING  74.38  80.20  56.49  57.68  95.69  21.11 
FA  0.14  0.11  0.19  0.015  0.15  0.033 
MB  0.55  0.06  0.061  2.16  0.41  0.19 
MORTGAGE  9.19  9.24  14.11  1.45  1.92  0.014 
NT  0.04  0.12  0.33  8.14  0.02  0.007 
PY  0.039  0.09  0.075  0.012  0.029  0.019 
QUAKE  0.041  0.06  0.298  0.07  0.79  0.034 
TREASURY  10.88  11.16  15.52  2.02  1.89  0.098 
WANKARA  0.0003  0.02  0.005  0.001  0.002  0.003 
AVERAGE  11.71  11.02  11.97  10.17  12.54  6.46 
Dataset  $\mathit{F}=3$  $\mathit{F}=5$  $\mathit{F}=10$ 

Appendicitis  15.57%  16.60%  15.77% 
Australian  24.29%  23.94%  22.40% 
Balance  17.22%  15.39%  15.62% 
Cleveland  52.09%  51.65%  50.37% 
Dermatology  37.23%  36.81%  35.73% 
Hayes Roth  35.72%  32.31%  35.33% 
Heart  16.32%  15.54%  15.91% 
HouseVotes  4.35%  3.90%  3.33% 
Ionosphere  12.50%  11.44%  9.30% 
Liverdisorder  28.08%  28.19%  28.44% 
Mammographic  17.49%  17.15%  17.72% 
Parkinsons  16.25%  15.17%  14.53% 
Pima  23.29%  23.97%  23.33% 
Popfailures  5.31%  5.86%  4.68% 
Regions2  25.97%  26.29%  25.18% 
Saheart  28.52%  28.59%  29.46% 
Segment  44.95%  48.77%  49.22% 
Spiral  15.49%  18.19%  23.58% 
Wdbc  5.43%  5.01%  5.20% 
Wine  7.59%  8.39%  5.63% 
Z_F_S  4.37%  4.26%  3.90% 
ZO_NF_S  3.79%  4.21%  3.99% 
ZONF_S  2.34%  2.26%  1.67% 
ZOO  11.90%  10.50%  9.33% 
AVERAGE  19.03%  18.93%  18.73% 
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Tsoulos, I.G.; Tzallas, A.; Karvounis, E. Adapting the Parameters of RBF Networks Using Grammatical Evolution. AI 2023, 4, 10591078. https://doi.org/10.3390/ai4040054
Tsoulos IG, Tzallas A, Karvounis E. Adapting the Parameters of RBF Networks Using Grammatical Evolution. AI. 2023; 4(4):10591078. https://doi.org/10.3390/ai4040054
Chicago/Turabian StyleTsoulos, Ioannis G., Alexandros Tzallas, and Evangelos Karvounis. 2023. "Adapting the Parameters of RBF Networks Using Grammatical Evolution" AI 4, no. 4: 10591078. https://doi.org/10.3390/ai4040054