# Creation of Numerical Constants in Robust Gene Expression Programming

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Robust GEP

#### 2.1. Chromosome Evaluation

Algorithm 1: RGEP chromosome evaluation procedure. |

for Each gene in chromosome do |

if The gene is an operator then |

if Number of elements on stack ≥ arity of the operator then |

Pop necessary operands from stack |

Execute the operator |

Push the result back on stack |

end if |

else //Comment: the gene is an operand |

Push the operand on stack |

end if |

end for |

The topmost element on the stack is the value of the evaluated expression. |

#### 2.2. Genetic Operators

## 3. The Improvements

#### 3.1. Numeric Crossover

#### 3.2. Gene Constraint

Algorithm 2: Generation of a chromosome template. |

Input:N, the depth of the derivation tree |

template = T |

for$i=1$ to N do |

template = template + template + F |

end for |

Algorithm 3: Modified chromosome evaluation procedure. |

for Each gene in chromosome do |

if The gene is an operator then |

Pop 2 operands from stack |

if Arity of the operator $<2$ then |

Discard the 2^{nd} operand |

end if |

Execute the operation |

Push the result back on stack |

else //Comment: the gene is an operand |

Push the operand on stack |

end if |

end for |

The element left on the stack is the value of the evaluated expression. |

## 4. Experimental Setup

## 5. Results

#### 5.1. Two Benchmark Cases

## 6. Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

C-RGEP | Constrained RGEP (i.e., RGEP with gene constraint) |

EA | Evolutionary Algorithm |

ERC | Ephemeral Random Constant |

GA | Genetic Algorithm |

GE | Grammatical Evolution |

GEP | Gene Expression Programming |

GP | Genetic Programming |

NLSE | Normalized Least Squares Error |

RGEP | Robust Gene Expression Programming |

RPN | Reverse Polish Notation |

SR | Symbolic Regression |

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**Figure 5.**Modified rotation operator for use with a gene constraint where operator and operand genes are rotated separately. The operator genes are shaded for clarity.

**Figure 6.**Convergence rate of the three constant creation methods on two standard benchmark functions using C-RGEP.

Parameter | Description |
---|---|

Population Size | 200 |

Generations | 200 |

Function Set | {+, −, *, / (protected), sin, cos, tan, ln (protected), |

e^{x}, $\sqrt{\phantom{x}}$ (protected), pass} | |

Terminal Set | {$0\le {C}_{\mathrm{random}}\le 10$, x} (for single-variable problems) |

{$0\le {C}_{\mathrm{random}}\le 10$, x, y} (for bivariable problems) | |

Constant Selection Probability | ${p}_{\mathrm{Tc}}=0.3{p}_{\mathrm{T}}$ (see Equation (3)) ^{1} |

Operator Probabilities | ${p}_{\mathrm{singlePointCrossover}}={p}_{\mathrm{twoPointCrossover}}=0.35$ |

${p}_{\mathrm{mutation}}=0.3$, ${p}_{\mathrm{rotation}}=0.5$^{2} | |

Constant Creation | Random mutation, creep mutation or numeric crossover |

Numeric Mutation Probability | $0.2$ |

Chromosome Length | 15 genes ^{3} |

Selection | Tournament selection of size 3 |

Replacement Selection | Tournament selection of size 2 |

Elitism | Best individual always survives |

Fitness | NLSE |

^{1}Note that terminal selection probability ${p}_{\mathrm{T}}$ always equals 1 when selecting a terminal in the case of gene constraint encoding. In that case, ${p}_{\mathrm{Tc}}=0.3$;

^{2}Rotation is always performed immediately after crossover, before the children are evaluated. A probability of $0.5$ means that approximately half of the children produced by crossover are rotated;

^{3}In the case of gene constraint encoding, this number is obtained from a derivation tree depth of 3.

Label | Formula | Fitness/Test Cases |
---|---|---|

(100 random points) | ||

F01 | ${x}^{2}-1.4981x+5.6517$ | $[-10,10]$ |

F02 | $1.7592{x}^{3}+1.7592$ | $[-10,10]$ |

F03 | $1.6190+x+9.8725/{x}^{2}$ | $[-10,10]$ |

F04 | $1.7245x+8.1542xy$ | $[-10,10]\times [-10,10]$ |

F05 | $9.8146-2.7770x/y$ | $[-10,10]\times [-10,10]$ |

F06 | $1.6192+9.8815\mathrm{ln}\left(x\right)$ | $[0.001,20]$ |

F07 | $-0.7391+3.7855\mathrm{sin}\left(x\right)$ | $[-10,10]$ |

F08 | $0.1192+\mathrm{sin}\left(0.6156x\right)$ | $[-10,10]$ |

F09 | $-7.3914+0.1410{\mathrm{e}}^{x}$ | $[-10,10]$ |

F10 | $-8.5410{\mathrm{e}}^{2.1196x}$ | $[-10,10]$ |

F11 | $9.5419+0.6720\mathrm{tan}\left(x\right)$ | $[-10,10]$ |

F12 | $-1.5193+9.1519\sqrt{x}$ | $[0,20]$ |

F13 | $-3.1871\sqrt{x}$ | $[0,20]$ |

F14 | $6.5619x+2.1059\mathrm{cos}\left(y\right)$ | $[-10,10]\times [-10,10]$ |

F15 | $0.1607/\sqrt{x}$ | $[0,20]$ |

F16 | $2.9163\mathrm{ln}\left(x\right)\mathrm{sin}\left(x\right)$ | $[0.001,20]$ |

F17 | $4.4161x\mathrm{sin}\left(y\right)$ | $[-10,10]\times [-10,10]$ |

F18 | $\mathrm{sin}\left(0.3165{x}^{2}\right)$ | $[-10,10]$ |

F19 | $0.5714x/\mathrm{ln}\left(y\right)$ | $[-10,10]\times [0.001,20]$ |

F20 | $3.1928+\mathrm{tan}\left(x\right)-\mathrm{tan}\left(y\right)$ | $[-10,10]\times [-10,10]$ |

F21 | $3.1817{\mathrm{e}}^{\mathrm{sin}\left(x\right)}$ | $[-10,10]$ |

**Table 3.**Results of the evolution of s-expressions using RGEP with two different numeric mutations and the proposed numeric crossover. Bold values indicate that the improvement is not statistically significant at $p<0.05$.

Formula | RGEP (Random Mutation) | RGEP (Creep Mutation) | RGEP (Numeric Crossover) | p-Value (NLSE $\le {10}^{-5}$) | p-Value (NLSE $\le {10}^{-8}$) |
---|---|---|---|---|---|

Median Test NLSE | |||||

No. of Found Formulas (NLSE $\le {10}^{-5}$, NLSE $\le {10}^{-8}$) | |||||

F01 | $0.00254$ | $0.00258$ | $0.00184$ | $0.0349$ | $0.0000$ |

$(12,1)$ | $(\mathbf{19},8)$ | $(\mathbf{22},20)$ | $\mathbf{0.5419}$ | $0.0076$ | |

F02 | $1.90733\times {10}^{-5}$ | $0.00140$ | $7.13315\times {10}^{-6}$ | $0.0455$ | $0.0001$ |

$(19,1)$ | $(17,5)$ | $(29,16)$ | $0.0160$ | $0.0069$ | |

F03 | $0.00907$ | $0.00095$ | $4.00492\times {10}^{-5}$ | $0.0027$ | $0.0000$ |

$(9,1)$ | $(\mathbf{15},8)$ | $(\mathbf{23},17)$ | $\mathbf{0.0989}$ | $0.0375$ | |

F04 | $4.37925\times {10}^{-5}$ | $4.07221\times {10}^{-5}$ | $3.23353\times {10}^{-5}$ | $\mathbf{0.1707}$ | $0.0000$ |

$(\mathbf{10},0)$ | $(7,3)$ | $(\mathbf{16},14)$ | $0.0324$ | $0.0034$ | |

F05 | $0.01820$ | $0.00974$ | $0.00133$ | $0.0005$ | $0.0004$ |

$(2,0)$ | $(\mathbf{11},3)$ | $(\mathbf{15},11)$ | $\mathbf{0.3628}$ | $0.0209$ | |

F06 | $0.07677$ | $0.02654$ | $0.0052965$ | $0.0251$ | $\mathbf{0.0784}$ |

$(2,\mathbf{0})$ | $(\mathbf{10},\mathbf{0})$ | $(\mathbf{9},\mathbf{3})$ | $\mathbf{0.8026}$ | $\mathbf{0.0784}$ | |

F07 | $0.03154$ | $0.01631$ | $3.13108\times {10}^{-6}$ | $0.0000$ | $0.0000$ |

$(7,0)$ | $(12,1)$ | $(27,22)$ | $0.0021$ | $0.0000$ | |

F08 | $0.02996$ | $0.03232$ | $0.02669$ | $0.0017$ | $0.0114$ |

$(0,0)$ | $(\mathbf{6},0)$ | $(\mathbf{9},6)$ | $\mathbf{0.4009}$ | $0.0114$ | |

F09 | $0.00047$ | $0.00028$ | $0.00018$ | $0.0375$ | $0.0000$ |

$(13,1)$ | $(\mathbf{14},5)$ | $(\mathbf{23},19)$ | $\mathbf{0.0629}$ | $0.0010$ | |

F10 | $0.00012$ | $0.00136$ | $0.00036$ | $\mathbf{0.5823}$ | $\mathbf{1.0000}$ |

$(\mathbf{7},\mathbf{0})$ | $(\mathbf{4},\mathbf{0})$ | $(\mathbf{9},\mathbf{0})$ | $\mathbf{0.1362}$ | $\mathbf{1.0000}$ | |

F11 | $0.13058$ | $0.24147$ | $0.027811$ | $0.0001$ | $0.0000$ |

$(3,0)$ | $(10,0)$ | $(19,14)$ | $0.0477$ | $0.0000$ | |

F12 | $0.00595$ | $0.02488$ | $0.00620$ | $\mathbf{0.0784}$ | $\mathbf{1.0000}$ |

$(\mathbf{0},\mathbf{0})$ | $(\mathbf{4},\mathbf{1})$ | $(\mathbf{3},\mathbf{0})$ | $\mathbf{0.6965}$ | $\mathbf{0.3125}$ | |

F13 | $0.05749$ | $0.06368$ | $0.03831$ | $\mathbf{0.1835}$ | $\mathbf{0.0801}$ |

$(\mathbf{3},\mathbf{2})$ | $(1,0)$ | $(\mathbf{7},7)$ | $0.0271$ | $0.0061$ | |

F14 | $0.00073$ | $0.00147$ | $0.00052$ | $\mathbf{0.1336}$ | $\mathbf{1.0000}$ |

$(\mathbf{7},\mathbf{0})$ | $(\mathbf{10},\mathbf{0})$ | $(\mathbf{13},\mathbf{0})$ | $\mathbf{0.4777}$ | $\mathbf{1.0000}$ | |

F15 | $0.12812$ | $0.00109$ | $9.39478\times {10}^{-6}$ | $0.0024$ | $0.0000$ |

$(14,1)$ | $(\mathbf{22},13)$ | $(\mathbf{29},25)$ | $\mathbf{0.1615}$ | $0.0135$ | |

F16 | $0.01403$ | $0.01637$ | $0.01035$ | $\mathbf{0.0836}$ | $0.0251$ |

$(\mathbf{4},2)$ | $(\mathbf{4},\mathbf{3})$ | $(\mathbf{10},\mathbf{9})$ | $\mathbf{0.0836}$ | $\mathbf{0.0643}$ | |

F17 | $2.59423\times {10}^{-6}$ | $0.00010$ | $2.22545\times {10}^{-12}$ | $0.0349$ | $0.0000$ |

$(28,6)$ | $(24,21)$ | $(38,38)$ | $0.0040$ | $0.0005$ | |

F18 | $0.77783$ | $0.84318$ | $0.58024$ | $\mathbf{0.3077}$ | $\mathbf{0.1527}$ |

$(\mathbf{1},\mathbf{0})$ | $(\mathbf{2},\mathbf{0})$ | $(\mathbf{3},\mathbf{2})$ | $\mathbf{0.6455}$ | $\mathbf{0.1527}$ | |

F19 | $0.20179$ | $0.21861$ | $0.01328$ | $0.0324$ | $0.0009$ |

$(7,3)$ | $(5,3)$ | $(16,16)$ | $0.0069$ | $0.0009$ | |

F20 | $2.40286\times {10}^{-6}$ | $0.02511$ | $1.27299\times {10}^{-12}$ | $0.0375$ | $0.0000$ |

$(27,16)$ | $(17,16)$ | $(37,36)$ | $0.0000$ | $0.0000$ | |

F21 | $0.00606$ | $0.02130$ | $0.013325$ | $\mathbf{0.1971}$ | $0.0001$ |

$(\mathbf{13},3)$ | $(\mathbf{17},\mathbf{13})$ | $(\mathbf{19},\mathbf{19})$ | $\mathbf{0.6745}$ | $\mathbf{0.1971}$ |

**Table 4.**Results of the evolution of s-expressions using C-RGEP with two different numeric mutations and the proposed numeric crossover. Bold values indicate that the improvement is not statistically significant at $p<0.05$.

Formula | C-RGEP (Random Mutation) | C-RGEP (Creep Mutation) | C-RGEP (Numeric Crossover) | p-Value (NLSE $\le {10}^{-5}$) | p-Value (NLSE $\le {10}^{-8}$) |
---|---|---|---|---|---|

Median Test NLSE | |||||

No. of Found Formulas (NLSE $\le {10}^{-5}$, NLSE $\le {10}^{-8}$) | |||||

F01 | $2.92910\times {10}^{-6}$ | $2.06778\times {10}^{-8}$ | $8.81672\times {10}^{-12}$ | $0.0007$ | $0.0000$ |

$(29,0)$ | $(33,22)$ | $(44,41)$ | $0.0091$ | $0.0001$ | |

F02 | $8.20371\times {10}^{-6}$ | $7.64020\times {10}^{-6}$ | $7.32903\times {10}^{-6}$ | $\mathbf{0.5353}$ | $0.0076$ |

$(\mathbf{30},1)$ | $(\mathbf{30},\mathbf{6})$ | $(\mathbf{33},\mathbf{9})$ | $\mathbf{0.5353}$ | $\mathbf{0.4009}$ | |

F03 | $4.45163\times {10}^{-5}$ | $0.00649$ | $1.33771\times {10}^{-5}$ | $\mathbf{0.3125}$ | $0.0000$ |

$(\mathbf{20},0)$ | $(14,6)$ | $(\mathbf{25},21)$ | $0.0238$ | $0.0007$ | |

F04 | $4.62953\times {10}^{-5}$ | $3.18061\times {10}^{-5}$ | $1.55923\times {10}^{-10}$ | $0.0050$ | $0.0000$ |

$(16,0)$ | $(\mathbf{23},10)$ | $(\mathbf{30},28)$ | $\mathbf{0.1615}$ | $0.0002$ | |

F05 | $0.00449$ | $2.32946\times {10}^{-6}$ | $1.03801\times {10}^{-10}$ | $0.0000$ | $0.0000$ |

$(3,0)$ | $(\mathbf{26},5)$ | $(\mathbf{32},30)$ | $\mathbf{0.2225}$ | $0.0000$ | |

F06 | $0.00543$ | $0.03705$ | $0.00610$ | $\mathbf{0.0629}$ | $0.0032$ |

$(\mathbf{5},0)$ | $(3,0)$ | $(\mathbf{12},8)$ | $0.0117$ | $0.0032$ | |

F07 | $0.03993$ | $0.00023$ | $5.24175\times {10}^{-9}$ | $0.0000$ | $0.0000$ |

$(8,0)$ | $(19,2)$ | $(29,26)$ | $0.0455$ | $0.0000$ | |

F08 | $0.02794$ | $0.02883$ | $0.02918$ | $0.0220$ | $0.0220$ |

$(0,0)$ | $(\mathbf{2},0)$ | $(\mathbf{5},5)$ | $\mathbf{0.2380}$ | $0.0220$ | |

F09 | $0.00021$ | $0.00017$ | $1.83179\times {10}^{-9}$ | $0.0093$ | $0.0000$ |

$(17,0)$ | $(15,6)$ | $(30,28)$ | $0.0025$ | $0.0000$ | |

F10 | $0.00037$ | $0.00077$ | $0.00014$ | $\mathbf{0.0854}$ | $\mathbf{1.0000}$ |

$(\mathbf{7},\mathbf{0})$ | $(\mathbf{9},\mathbf{0})$ | $(\mathbf{14},\mathbf{0})$ | $\mathbf{0.2340}$ | $\mathbf{1.0000}$ | |

F11 | $0.23872$ | $1.71681\times {10}^{-5}$ | $1.83038\times {10}^{-10}$ | $0.0000$ | $0.0000$ |

$(7,0)$ | $(23,1)$ | $(36,30)$ | $0.0083$ | $0.0000$ | |

F12 | $0.00913$ | $0.01491$ | $0.01200$ | $\mathbf{0.1676}$ | $\mathbf{0.3125}$ |

$(\mathbf{1},\mathbf{0})$ | $(\mathbf{2},\mathbf{0})$ | $(\mathbf{4},\mathbf{1})$ | $\mathbf{0.4009}$ | $\mathbf{0.3125}$ | |

F13 | $0.00950$ | $0.02490$ | $1.00544\times {10}^{-5}$ | $0.0003$ | $0.0000$ |

$(8,2)$ | $(15,\mathbf{12})$ | $(25,\mathbf{21})$ | $0.0414$ | $\mathbf{0.0561}$ | |

F14 | $0.00039$ | $0.00046$ | $8.19655\times {10}^{-5}$ | $0.0238$ | $0.0011$ |

$(14,1)$ | $(14,2)$ | $(25,12)$ | $0.0238$ | $0.0040$ | |

F15 | $4.91555\times {10}^{-7}$ | $5.36922\times {10}^{-7}$ | $8.49969\times {10}^{-12}$ | $\mathbf{0.5287}$ | $0.0000$ |

$(\mathbf{31},7)$ | $(\mathbf{26},22)$ | $(\mathbf{34},34)$ | $\mathbf{0.1031}$ | $0.0155$ | |

F16 | $0.00867$ | $0.00921$ | $0.00341$ | $\mathbf{0.1471}$ | $0.0018$ |

$(\mathbf{15},7)$ | $(12,10)$ | $(\mathbf{22},21)$ | $0.0349$ | $0.0173$ | |

F17 | $3.97238\times {10}^{-7}$ | $2.09001\times {10}^{-9}$ | $2.45460\times {10}^{-13}$ | $0.0324$ | $0.0000$ |

$(34,7)$ | $(\mathbf{35},28)$ | $(\mathbf{43},42)$ | $\mathbf{0.0536}$ | $0.0022$ | |

F18 | $0.79627$ | $0.98743$ | $0.59888$ | $0.0063$ | $0.0009$ |

$(3,0)$ | $(\mathbf{7},1)$ | $(\mathbf{13},10)$ | $\mathbf{0.1336}$ | $0.0040$ | |

F19 | $0.01666$ | $0.00045$ | $0.00854$ | $\mathbf{0.1416}$ | $0.0004$ |

$(\mathbf{14},4)$ | $(\mathbf{23},\mathbf{19})$ | $(\mathbf{21},\mathbf{19})$ | $\mathbf{0.6892}$ | $\mathbf{1.0000}$ | |

F20 | $5.21087\times {10}^{-8}$ | $7.30382\times {10}^{-12}$ | $2.39878\times {10}^{-15}$ | $0.0001$ | $0.0000$ |

$(31,20)$ | $(38,36)$ | $(47,47)$ | $0.0117$ | $0.0034$ | |

F21 | $4.72883\times {10}^{-7}$ | $4.59902\times {10}^{-8}$ | $1.90733\times {10}^{-11}$ | $\mathbf{0.5093}$ | $0.0000$ |

$(\mathbf{34},10)$ | $(\mathbf{28},24)$ | $(\mathbf{37},37)$ | $\mathbf{0.0588}$ | $0.0076$ |

**Table 5.**Results of the evolution of two benchmark s-expressions using C-RGEP with two different numeric mutations and the proposed numeric crossover.

Constant Creation Method | Keijzer 14 | Korns 2 | |||||
---|---|---|---|---|---|---|---|

Minimum Test NLSE | NLSE ${10}^{-5}$ | NLSE ${10}^{-8}$ | Minimum Test NLSE | NLSE ${10}^{-5}$ | NLSE ${10}^{-8}$ | ||

Random mutation | $3.642\times {10}^{-10}$ | 10 | 2 | $1.242\times {10}^{-9}$ | 11 | 1 | |

Creep mutation | $3.855\times {10}^{-13}$ | 12 | 10 | $5.761\times {10}^{-10}$ | 20 | 6 | |

Numeric crossover | $0.000$ | 16 | 14 | $2.337\times {10}^{-13}$ | 30 | 20 |

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**MDPI and ACS Style**

Fajfar, I.; Tuma, T.
Creation of Numerical Constants in Robust Gene Expression Programming. *Entropy* **2018**, *20*, 756.
https://doi.org/10.3390/e20100756

**AMA Style**

Fajfar I, Tuma T.
Creation of Numerical Constants in Robust Gene Expression Programming. *Entropy*. 2018; 20(10):756.
https://doi.org/10.3390/e20100756

**Chicago/Turabian Style**

Fajfar, Iztok, and Tadej Tuma.
2018. "Creation of Numerical Constants in Robust Gene Expression Programming" *Entropy* 20, no. 10: 756.
https://doi.org/10.3390/e20100756