This paper discusses the genetic diversity of real-coded populations processed by an evolutionary algorithm (EA). Diversity is expressed as a variance or a covariance matrix of individuals contained in the population, in one- or multi-dimensional cases, respectively. We focus on the exploration stage of the optimization, therefore, the fitness function is modeled as noise. We prove that the expected value of genetic diversity achieves a level proportional to the mutation covariance matrix. The proportionality coefficient depends solely on the EA parameters. Formulas are derived to predict the diversity for fitness proportionate, tournament, and truncation selection, with and without arithmetic crossover and with Gaussian mutation. Experimental validation of the multidimensional case shows that prediction accuracy is satisfactory in a broad spectrum of settings of EA parameters.

中文翻译：

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探索阶段非精英进化算法的种群多样性

本文讨论了由进化算法（EA）处理的实编码种群的遗传多样性。多样性分别表示为一维或多维情况下种群中包含的个体的方差或协方差矩阵。我们专注于优化的探索阶段，因此，适应度函数被建模为噪声。我们证明遗传多样性的期望值达到了与突变协方差矩阵成正比的水平。比例系数仅取决于 EA 参数。推导出公式来预测适合度比例、锦标赛和截断选择的多样性，有和没有算术交叉和高斯突变。