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摘要: 符号回归以构建一个能拟合给定数据集的函数模型为目的, 是对基本函数、运算符、变量等进行组合优化的过程.本文提出了一种求解符号回归问题的粒子群优化算法.算法以语法树对函数模型进行表达, 采用基因表达式将语法树编码为一个粒子, 设计了粒子的飞行方法及$r$-邻域环状拓扑的粒子学习关系.为使粒子具有跳出局部极值的能力和减轻粒子快速趋同对全局寻优造成的不利影响, 分别设计了突变算子和散开算子.此外, 为了得到比较简洁的函数模型, 在粒子的评价函数中以罚函数的方式对编码的有效长度进行控制.仿真实验表明, 提出的算法可以获得拟合精度更高、简洁性更好的函数模型.Abstract: Symbolic regression is to construct a function model that fits a given dataset. It is the process of optimally combining various basic functions, operators, and variables. This paper proposes a particle swarm optimization-based algorithm for symbolic regression. In the proposed algorithm, the functional model to be established is represented as a syntax tree, which is encoded as a particle through gene-expression. A specific implementation of particles flying and the $r$-neighborhood learning mechanism of particle swarm were designed. To make particles be capable of jumping out local extremum and to mitigate the negative influence on global optimization resulted from the fast convergence of the particle swarm, mutation and scatter are respectively introduced into the proposed algorithm as operators. Besides, in order to obtain concise functional model, the valid length of the gene-expression-based coding scheme is controlled in manner of introducing a penalty term to the particle evaluation function. Exhaustive simulation experiments are carried out and the results show that, the proposed algorithm can obtain the functional model with higher fitting precision and better conciseness.
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Key words:
- Gene-expression programming /
- particle swarm optimization /
- symbolic regression /
- evolutionary modeling
1) 本文责任编委 伍洲 -
图 10 具有不同算子的算法迭代曲线比较
Fig. 10 Iterative curve comparison of algorithms with different operators
表 2 粒子$X$、$gbest$、$pbest$以及$X'$的编码
Table 2 The code of particle $X$, $gbest$, $pbest$ and $X'$
Node code 0 1 2 3 4 5 6 7 8 9 10 $X$ + $ x $ $\sin$ $-$ ln $ x $ $a$ $ x $ $ x $ $a$ $ a$ $pbest$ + $ x $ $\cos$ / $-$ $ x $ $a$ $ x $ $a$ $ x $ $a$ $gbest$ $\exp$ + $a$ $-$ ln $a$ $a$ $ x $ $a$ $ x $ $a$ $X'$ + + $\sin$ / ln $ x $ $a$ $ x $ $ x $ $a$ $a$ 
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表 3 粒子$X$和$X'$的编码
Table 3 The code of particle $X$ and $X'$
Node code 0 1 2 3 4 5 6 7 8 9 10 $X$ + + sin / ln $ x $ $a$ $ x $ $ x $ $a$ $a$ $X'$ + $\times$ sin cos ln a $a$ $ x $ $ x $ $a$ $a$
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表 4 算法参数
Table 4 Algorithm parameters
参数 参数取值 节点评价个数 $15\, 000\, 000$ 种群数量 50 首段长度 10 约束长度 18 罚因子 0.0001 环状结构邻域大小 $2r+1, r=2$ 终止符 $x, 1$ (单变量); $x, y$ (双变量) 函数及运算符 $+, -, \times, /$, sin, cos, e$^x$, ln 测试次数 100
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表 5 测试函数
Table 5 Benchmark functions
Functions Fitcases $F1=x^3+x^2+x$ 20 random points $\subseteq [-1, 1]$ $F2=x^4+x^3+x^2+x$ 20 random points $\subseteq [-1, 1]$ $F3=x^5+x^4+x^3+x^2+x$ 20 random points $\subseteq [-1, 1]$ $F4=x^6+x^5+x^4+x^3+x^2+x$ 20 random points $\subseteq [-1, 1]$ $F5=\sin(x^2)\cos(x)-1$ 20 random points $\subseteq [-1, 1]$ $F6=\sin(x)+\sin(x+x^2)$ 20 random points $\subseteq [-1, 1]$ $F7={\rm ln}(x+1)+{\rm ln}(x^2+1)$ 20 random points $\subseteq [0, 2]$ $F8=\sqrt x$ 20 random points $\subseteq [0, 4]$ $F9=\sin(x)+\sin(y^2)$ 100 random points $\subseteq [-1, 1]$ $F10=2\sin(x)\cos(y)$ 100 random points $\subseteq [-1, 1]$
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表 6 成功运行百分比(%)
Table 6 Percentage of successful runs (%)
Techniques Functions F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 本文算法 ${\bf 100 }$ ${\bf 82}$ ${\bf 22}$ 3 ${\bf 60}$ 84 11 79 ${\bf 94}$ 72 AFSA 97 80 9 2 47 73 26 ${\bf 98}$ 78 ${\bf 80}$ ABCP 89 50 ${\bf 22}$ ${\bf 12}$ 57 ${\bf 87}$ 58 37 33 21 SC 48 22 7 4 20 35 35 16 7 18 NSM 48 16 4 4 19 36 40 28 4 17 SAC2 53 25 7 4 17 32 25 13 4 4 SAC3 56 19 6 2 21 23 25 12 3 8 SAC4 53 17 11 1 20 23 29 14 3 8 SAC5 53 17 11 1 19 27 30 12 3 8 CAC1 34 19 7 7 12 22 25 9 1 15 CAC2 34 20 7 7 13 23 25 9 2 16 CAC4 35 22 7 8 12 22 26 10 3 16 SBS31 43 15 9 6 31 28 31 17 13 33 SBS32 42 26 7 8 36 27 44 30 17 27 SBS34 51 21 10 9 34 33 46 25 26 33 SBS41 41 22 9 5 31 34 38 25 19 33 SBS42 50 22 17 10 41 32 51 24 24 33 SBS44 40 25 16 9 35 43 42 28 33 34 SSC8 66 28 ${\bf 22}$ 10 48 56 59 21 25 47 SSC12 67 33 14 ${\bf 12}$ 47 47 66 38 37 51 SSC16 55 39 20 11 46 44 ${\bf 67}$ 29 30 59 SSC20 58 27 10 9 52 48 63 26 39 51
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表 7 平均最佳适应值(%)
Table 7 Mean best fitness values (%)
Techniques Functions F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 本文算法 ${\bf 0.00}$ ${\bf 0.03}$ 0.30 0.35 ${\bf 0.02}$ ${\bf 0.01}$ 0.14 0.05 ${\bf 0.01}$ 0.82 AFSA ${\bf 0.00}$ 0.06 0.35 0.33 ${\bf 0.02}$ 0.05 0.13 ${\bf 0.01}$ 0.23 ${\bf 0.69}$ ABCP 0.01 0.05 ${\bf 0.07}$ ${\bf 0.10}$ 0.05 0.02 ${\bf 0.06}$ 0.10 0.47 1.06 SC 0.18 0.26 0.39 0.41 0.21 0.22 0.13 0.26 5.54 2.26 NSM 0.16 0.29 0.34 0.40 0.19 0.17 0.11 0.19 5.44 2.16 SAC2 0.16 0.27 0.42 0.50 0.22 0.23 0.15 0.27 5.99 3.19 SAC3 0.13 0.27 0.41 0.48 0.18 0.23 0.15 0.27 5.77 3.13 SAC4 0.15 0.29 0.40 0.46 0.17 0.22 0.15 0.26 5.77 3.03 SAC5 0.15 0.29 0.51 0.46 0.17 0.21 0.15 0.26 5.77 2.98 CAC1 0.33 0.41 0.52 0.53 0.31 0.42 0.17 0.35 7.83 4.40 CAC2 0.32 0.41 0.53 0.53 0.31 0.42 0.17 0.35 7.38 4.30 CAC4 0.33 0.41 0.33 0.53 0.30 0.42 0.17 0.19 7.80 4.32 SBS31 0.18 0.29 0.30 0.36 0.17 0.30 0.15 0.18 4.78 2.75 SBS32 0.18 0.23 0.29 0.36 0.13 0.28 0.10 0.18 4.47 2.77 SBS34 0.16 0.23 0.31 0.33 0.13 0.21 0.11 0.19 4.17 2.90 SBS41 0.18 0.26 0.27 0.38 0.12 0.20 0.13 0.20 4.40 2.75 SBS42 0.12 0.24 0.29 0.30 0.12 0.18 0.10 0.16 3.95 2.76 SBS44 0.18 0.24 0.33 0.35 0.15 0.16 0.11 0.19 2.85 1.75 SSC8 0.09 0.15 0.19 0.29 0.10 0.09 0.07 0.15 3.91 1.53 SSC12 0.17 0.17 0.18 0.28 0.10 0.12 0.07 0.13 3.54 1.45 SSC16 0.10 0.15 0.23 0.26 0.10 0.10 ${\bf 0.06}$ 0.14 3.11 1.22 SSC20 0.10 0.18 0.23 0.30 0.09 0.10 ${\bf 0.06}$ 0.14 2.64 1.23
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表 8 "Ⅴ"型函数建模的算法参数
Table 8 Algorithm parameters for "V" function
参数 参数取值 迭代次数 50 000 种群数量 300 首段长度 25 约束长度 35 罚因子 0.05 环状结构邻域大小 $2r+1, r=2$ 终止符 $x, 1, 0.2, 2, 4, 0.7, 0.1, 0.1415, 0.01, 3$ 函数及运算符 $+, -, \times, /, \sqrt{~~}, \sin, \cos, {\rm e}^x, \ln, x^2$ 测试次数 100
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算法 评价指标 F1 F2 F3 F11 TSO Average fitness 0 0.09 0.63 0.26 Standard deviation 0 0.35 0.79 0.36 Perfect hits in % 100 92 57 52 Number of evaluations 1 220 4 660 7 000 9 020 本文算法 Average fitness 0 0 0.04 0.23 Standard deviation 0 0 0.26 0.30 Perfect hits in % 100 100 97 73 Number of evaluations 692 1 589 6 645 8 350
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表 10 观测数据
Table 10 Observation data
$x$ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 $y$ 0.250841, $-$0.373517, $-$1.957016, $-$2.699129, 1.619885 4.684452, 2.686799, $-$0.501882, $-$4.718648, $-$7.787834 0.056755, 9.363730, 6.761007, 0.824883, $-$5.821076 $ -$12.512195, $-$5.349206, 12.234454, 12.324229, 3.656108
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表 11 不同约束长度下的函数模型
Table 11 Function models with different constrained length
约束长度 有效长度 编码 模型表达式 25 24 $\times, \cos, \times, x, /, \cos, x, \sqrt{~~}, \cos, \exp, \cos, \sin, \sin, x, \sqrt{~~}, \cos, $ $\dfrac{x\cdot \cos x}{\sqrt {{\rm e}^{\sin x}}}\cdot \cos(\cos(\cos(\sin(\sqrt{\cos(\dfrac{\sqrt{x}-x}{x^2})}))))$ $/, -, \times, \sqrt{~~}, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x$ 22 21 $\times, \cos, /, x, x, \sqrt{~~}, \exp, +, \sin, \sin, \sin, x, \cos, /, \ln, \cos, \sqrt{~~}, $ $\dfrac{x\cdot \cos x}{\sqrt{{\rm exp}({\sin x}+ \sin(\sin(\cos(\dfrac{\ln{\sqrt {{\rm e}^{\cos x}}}}{\cos x}))))}}$ $x, \exp, \cos, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x$ 20 19 $\times, /, x, \cos, \sqrt{~~}, x, \exp, +, \cos, \sin, \sin, x, \sqrt{~~}, \cos, /, x, +, $ $\dfrac{x\cdot \cos x}{\sqrt{{\rm exp}({\sin x}+ \cos(\sin(\sqrt{\cos(\dfrac{x}{2x})})))}}$ $/x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x$ 18 18 $ /, \cos, /, x, \sqrt{~~}, x, +, \exp, -, \sin, \exp, \ln, x, \sin, /, x, x, $ $\dfrac{x\cdot \cos x}{\sqrt{{\rm e}^{\sin x}+{\rm e}^{\sin x}-\ln{(\dfrac{x}{x})}}}$ $x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x$ 15 13 $/, \cos, /, x, \sqrt{~~}, x, +, \exp, \exp, \sin, \sin, x, x, x, x, x, \times, $ $\dfrac{x\cdot \cos x}{\sqrt{{\rm e}^{\sin x}+{\rm e}^{\sin x}}}$ $x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x$
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