Machine-learning classification with additivity and diverse multifractal pathways in multiplicativity

Madhur Mangalam, Henrik Seckler, Damian G. Kelty-Stephen
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Abstract

Evidence of multifractal structures has spread to a wider set of physiological time series supporting the intricate interplay of biological and psychological functioning. These dynamics manifest as random multiplicative cascades, embodying nonlinear relationships characterized by recurring division, branching, and aggregation processes implicating noise across successive generations. This investigation focuses on how well the diversity of multifractal properties can be specific to the type of cascade relationship between generation (i.e., multiplicative, additive, or a mixture) as well as to the type of noise (i.e., including additive white Gaussian noise, fractional Gaussian noise, and various amalgamations) among 15 distinct types of binomial cascade processes. Cross-correlation analysis of multifractal spectral features confirms that these features capture nuanced aspects of cascading processes with minimal redundancy. Principal component analysis using 13 distinct multifractal spectral features shows that different cascade processes can manifest multifractal evidence of nonlinearity for distinct reasons. This transparency of multifractal spectral features to underlying cascade dynamics becomes less amenable to machine-learning strategies. Fully connected neural networks struggled to classify the 15 distinct types of cascade processes based on the respective multifractal spectral features (45.5% accuracy) yet demonstrated improved accuracy when addressing single categories of cross-generation relationships, that is, additive (91.6%), multiplicative (75.4%), or additomultiplicative (70.6%). While traditional principal component analysis reveals distinct loadings attributed to individual noise processes, multiplicative relationships between generations effectively make the constituent noise processes less discernible to neural networks. Neural networks may lack sufficient hierarchical depth required to effectively distinguish among nonadditive cascading processes, recommending either elaborating multifractal geometry or using alternate architectures for machine-learning classification of cascades with multiplicative relationships.

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机器学习分类的可加性和多元多分形路径的可乘性
多分形结构的证据已扩展到更广泛的生理时间序列,支持生物和心理功能的复杂相互作用。这些动态表现为随机乘法级联,体现了非线性关系,其特点是反复出现的分裂、分支和聚集过程,牵涉到连续几代人之间的噪声。这项研究的重点是,在 15 种不同类型的二叉级联过程中,多分形特性的多样性如何与代际级联关系类型(即乘法、加法或混合)以及噪声类型(即包括加性白高斯噪声、分数高斯噪声和各种混合噪声)相匹配。多分形频谱特征的交叉相关分析证实,这些特征以最小的冗余捕捉到级联过程的细微差别。利用 13 个不同的多分形光谱特征进行的主成分分析表明,不同的级联过程会由于不同的原因表现出非线性的多分形证据。多分形频谱特征对基本级联动力学的这种透明性不太适合机器学习策略。全连接神经网络难以根据各自的多分形光谱特征对 15 种不同类型的级联过程进行分类(准确率为 45.5%),但在处理单一类别的跨代关系(即加法关系(91.6%)、乘法关系(75.4%)或加减乘除关系(70.6%))时,准确率却有所提高。传统的主成分分析显示了单个噪声过程的不同载荷,而代际间的乘法关系则有效地降低了神经网络对组成噪声过程的识别能力。神经网络可能缺乏有效区分非相加级联过程所需的足够层次深度,因此建议对多分形几何进行详细说明,或使用其他架构对具有相乘关系的级联进行机器学习分类。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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CiteScore
8.60
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