使用混合函數和遺傳程式設計之符號迴歸增強器

Abstract

符號迴歸之目的在於獲得最符合給定數據集的數學表達式。符號迴歸問題包含多種解法，包括決定性的與啟發式的演算法。遺傳程式設計使用交配和選擇等演化機制來演進族群中的程式。本篇論文提出了一個稱作符號迴歸增強器的框架，此符號迴歸增強器結合了遺傳程式設計和其他符號迴歸方法。其主要的想法為使用來自其他符號迴歸方法的表達式之語法樹來提高遺傳程式設計演化過程的品質與效率。具體來說，本篇論文研究了不同的語法樹規則、混合函式、選擇機制和交配機制來構建符號迴歸增強器算法。符號迴歸增強器的有效性藉由兩個傳統的以及五個基於遺傳程式設計的符號迴歸方法來展示。在來自符號回歸基準和費曼符號迴歸資料庫的二十八個基準中，統計測試表明，每個藉由符號迴歸增強器增強之方法在最少四個到最多二十四個基準中顯著優於各自的符號迴歸方法，且優於的情況多於劣於的情況。此外，本篇論文深入探究基於遺傳程式設計的符號迴歸方法的最佳切換時刻，以確定何時轉換到符號迴歸增強器框架。結果顯示當族群多樣性下降到給定之閾值時，可以指示從基於遺傳程式設計的方法切換到符號迴歸增強器的適當時機。

Symbolic regression (SR) aims to obtain a mathematical expression that most accurately fits a given dataset. The SR problem encompasses various methods, including deterministic and heuristic algorithms. Genetic programming (GP) uses evolutionary mechanisms, such as crossover and selection, to evolve programs in a population. This thesis proposes a framework called the symbolic regressor enhancer (SRE), which combines GP with other SR methods. The main idea is to use the syntax tree of the expression from other SR methods to enhance both the quality and the efficiency of the GP evolutionary procedure. Specifically, this thesis investigates different rules for syntax trees, hybridization, selection, and crossover to construct the SRE algorithm. The effectiveness of SRE is demonstrated using two traditional and five GP-based SR methods. Out of 28 benchmarks from the SR benchmark and Feynman SR database, the statistical tests show that each SRE-enhanced method significantly outperforms the respective SR method in at least 4 and up to 24 benchmarks, with more instances of outperformance than underperformance. Furthermore, this thesis delves into investigating the optimal switching moment for GP-based SR methods to determine when to transition to the SRE framework. The results indicate that a decrease in population diversity to a given threshold can signal the appropriate moment to switch from GP-based methods to SRE.