基于统计机器学习的高维复杂系统建模研究

Research on Regression Modeling of High-Dimensional Complex Systems Based on Statistical Machine Learning

为了解决"维数灾难"给高维复杂系统数据驱动建模带来的过拟合、计算复杂度高等问题,探究特征提取算法对回归结果的影响。首先使用主成分分析法、非负矩阵分解法、局部线性嵌入法和均匀流形近似与投影法分别进行降维,提取关键特征后在多项式模型与随机响应面模型中进行回归,最后在草炭土土壤和电力系统的数据集上进行了仿真建模分析。仿真结果表明,在草炭土土壤中使用主成分分析法降维后的预测效果最好,在电力系统中使用非负矩阵分解法降维后的概率潮流回归结果最准确。经过分析后可知,主成分分析法适用于具有明显线性相关关系的高维数据集,基于流形的降维算法在本征维数较低的情况下不利于构建相似流形,提取关键特征的效果较差。

In order to solve the problems of over-fitting and high computational complexity brought by “dimensional disaster” to data-driven modeling of high-dimensional complex systems, the influence of feature extraction algorithm on regression results is explored. In this paper, the principal component analysis method, non-negative matrix factorization method, locally linear embedding method, uniform manifold approximation and projection method were used for dimensionality reduction respectively. After extracting the key features, regression was carried out in the polynomial model and stochastic response surface model. Finally, simulation modeling analysis was carried out on data sets of peat soil and power system. The simulation results show that the prediction effect of peat soil is the best after dimension reduction by principal component analysis, and the probability power flow regression is the most accurate in the power system after dimension reduction by nonnegative matrix decomposition. It can be concluded that principal component analysis is suitable for high-dimensional data sets with obvious linear correlation, and the dimensionality reduction algorithm based on manifolds is not conducive to constructing similar manifolds when the intrinsic dimensionality is low, and it is not good at extracting key features on the time.