统计与数据科学学院知识库

数据重构是从部分已观测数据恢复原始数据的过程，是缺失处理的核心任务。 数据重构的关键在于充分、合理地利用数据的特性。目前，随着函数型数据分析 的发展，函数型矩阵填充方法成为数据重构的主流方法之一。然而，现有的函数 型矩阵填充方法在数据学习过程中未充分利用数据的潜在特征，如样本曲线的相 关性、样本的高阶邻域信息等。为了解决这些问题，本文在函数型数据分析框架 下，通过引入非负约束，借助非负矩阵分解(Non-negative Matrix Factorization， NMF)和类信息，提出融合类信息的函数型矩阵填充方法(Non-negative Functional Matrix Completion Method with Class Information，CNFMC)。同时，利用样本信 息的高阶邻域关系以及各视角之间的多样性估计缺失数据，提出基于图正则化的 多视角函数型矩阵填充方法(Based on Graph Regularization Multi-view Non negative Functional Matrix Completion，GMVNFMC)。本文主要研究内容包括以 下两部分： (1) 提出一种融合类信息的函数型矩阵填充方法(CNFMC)。基于非负矩阵分 解构造函数型矩阵填充方法，在此基础上通过聚类划分引入样本类信息，借助类 内样本相关性插补缺失值，并采用自加权集成学习算法动态赋权重计算得最终插 补值。在公共交通数据集PeMS进行了缺失模拟插补实验，并针对空气质量缺失 数据进行实证应用分析。结果表明：相较于K近邻算法、MICE、PACE等10种 插补方法，CNFMC方法插补精度高、鲁棒性好、适用性较强，且耗时可控，能 够保证插补的有效性和准确性。 (2) 提出一种基于图正则化的多视角函数型矩阵填充方法(GMVNFMC)。通 过引入最优图正则化，充分考虑了各视角内样本信息的高阶邻域关系，减少了信 息损失；同时，利用希尔伯特-施密特独立性准则探索不同视角之间包含的互 补信息，进而提高插补精度。分别对空气污染物数据集进行模拟插补实验和实证 应用，结果表明，相较于其他主流插补方法，GMVNFMC 方法具有更好的插补 效果。

Data reconstruction is the process of restoring original data from partially observed data, and is the core task of missing data handling. The key to data reconstruction lies in fully and reasonably utilizing the characteristics of data. Currently, with the development of functional data analysis, functional matrix completion method has become one of the mainstream methods for data reconstruction. However, existing functional matrix completion methods do not fully utilize the potential features of data in the data learning process, such as the correlation of sample curves and high-order neighborhood information of samples. To address these issues, this thesis proposes a non-negative functional matrix completion method with class information (CNFMC) that by introducing non-negative constraints and utilizing non-negative matrix factorization (NMF) and class information within the framework of functional data analysis. Meanwhile, utilizing the high-order neighborhood relationship of sample information and the diversity between different views to estimate missing data, a multi-view non-negative functional matrix completion (GMVNFMC) method based on graph regularization is proposed. The main research content of this thesis includes the following two parts:

(1) Propose a non-negative functional matrix completion method with class information (CNFMC). Based on the non-negative matrix factorization constructor functional matrix completion method, sample class information is introduced through clustering partitioning, missing values are imputed using intra class sample correlation, and the final imputation value is calculated by dynamically assigning weights using a self-weighted ensemble learning algorithm. We conducted missing simulation imputation experiments on the public transportation dataset PeMS and conducted empirical application analysis on air quality missing data. The results show that compared to 10 interpolation methods such as K-nearest neighbor algorithm, MICE, PACE, etc., the CNFMC method has high interpolation accuracy, good robustness, strong applicability, and controllable time consumption, which can ensure the effectiveness and accuracy of imputation. (2) Propose a multi-view functional matrix completion method based on graph regularization (GMVNFMC). By introducing optimal graph regularization, the high-order neighborhood relationships of sample information within each view are fully considered, reducing information loss; Meanwhile, utilizing the Hilbert-Schmidt independence criterion to explore the complementary information contained between different views,thereby improving interpolation accuracy. Simulated interpolation experiments and empirical applications were conducted on the air pollutant datasets, and the results showed that the GMVNFMC method has better imputation performance compared to other mainstream imputation methods.