弱奇异Volterra积分在电子商务中的应用

The Application of Weakly Singular Volterra Integrals in E-Commerce

摘要: 长记忆性、动态波动性以及弱奇异特性给电子商务场景中时序数据带来了建模难题，为了解决这一难题，本文对变指数弱奇异Volterra积分的构建与应用开展了。基于Volterra积分方程，对变指数弱奇异Volterra积分模型进行构建，为了解决弱奇异核与变指数带来的数值求解挑战，本文引入积分与分段多项式插值的Nyström方法，证明出了模型有唯一性与收敛性的解。为了电商销售预测与用户行为分析，将构建的变指数弱奇异Volterra积分模型应用其中，实验数据集通过京东多品类销售时序数据与用户行为数据来构建，并且对比ARIMA模型、LSTM模型，验证出了变指数弱奇异Volterra积分模型优越性。由实验数据得出，在处理具有长记忆性的电商时序数据时，预测精度较ARIMA模型和LSTM模型是最高的，且对突发因素引发的奇异波动的适应性是最强的。本文研究为电子商务场景的复杂时序数据建模提供了新的数学工具，对电子商务的核心场景的电商销售预测和用户行为分析有重要使用价值。

Abstract: The long memory, dynamic volatility, and weak singularity characteristics present modeling challenges for time series data in e-commerce scenarios. To address these challenges, this paper focuses on the construction and application of the variable-index weakly singular Volterra integral. Based on Volterra integral equations, a model for the variable-index weakly singular Volterra integral is developed. To tackle the numerical solution challenges posed by the weakly singular kernel and the variable index, this paper introduces the Nyström method using product integrals and piecewise polynomial interpolation, proving the existence of a unique and convergent solution for the model. To apply the constructed variable-index weakly singular Volterra integral model to e-commerce sales forecasting and user behavior analysis, experimental datasets are created using multi-category sales time series data and user behavior data from JD.com. The performance of the proposed model is compared with that of the ARIMA model and the LSTM model, demonstrating the superiority of the variable-index weakly singular Volterra integral model. The experimental results indicate that when dealing with time series data in e-commerce characterized by long memory, the prediction accuracy is higher than that of both the ARIMA and LSTM models, and the model also shows the strongest adaptability to singular fluctuations caused by sudden factors. This research provides new mathematical tools for modeling complex time series data in e-commerce scenarios, offering significant practical value for e-commerce sales forecasting and user behavior analysis in core business contexts.