统计与数据科学学院知识库

随着互联网技术的迅猛发展，产生了大量复杂连续数据。然而，传统的多元统计分析方法在处理这些数据时存在一定的局限性。因此，函数型数据分析方法应运而生。本文旨在拓展函数型数据的微分方程分析方法的应用范围，提出函数型聚类方法以及构建函数型预测模型，并基于2002-2022年黄河干流径流量数据开展系统的函数型数据分析方法研究，以期从函数型视角研究径流量数据，全面分析黄河流域径流演变机理，为流域生态环境保护、洪涝灾害防治以及气候变化等研究提供新的理论方法和科学依据。

首先，通过研究径流量曲线、均值曲线和方差曲线，探索了径流量的统计特征，并利用相平面图和主微分分析方法揭示了其季节变动特征。研究结果显示2002-2022年黄河干流径流量呈增加趋势，周期性显著，在2004、2008、2013、2017以及2021年发生了突变，且季节特征明显。

其次，基于非负矩阵分解思想，引入函数型主成分和主微分，构建基于主成分和主微分的函数型聚类方法(PCPDFCM)，探究黄河干流径流量的时空分布特征。利用函数型主成分分析发现径流量主要在2013年前后和2019-2021年显著增大。从空间分布来看，黄河干流径流量整体呈自上而下逐渐减小的趋势，空间差异性明显。此外，采用PCPDFCM方法将黄河干流12个水文站聚为三类，且三类水文站的水文特征差异显著。进一步，利用ArcGIS软件进行可视化展示，直观呈现12个水文站径流量的差异性特征。

最后，结合函数型主成分分析与主微分分析的思想原理，构建基于函数型主微分与主成分的岭回归模型(FPDPCRR)，预测黄河干流径流量。具体地，在利用累积量斜率变化分析法和多元函数型回归模型探究黄河干流径流量影响因素的基础上，采用FPDPCRR方法预测径流量，结果表明所提方法的预测效果较好，与实际情况较为相符，从而为流域水资源综合高效调配和防洪减灾调度提供一定的理论依据。

As the rapid development of Internet technology, a large number of complex and continuous data are produced. However, the traditional multivariate statistical analysis method has some limitations when dealing with these data. Therefore, functional data analysis method came into being. Thesis aims to expand the application range of the differential equation analysis method of functional data, put forward functional clustering method and construct a functional prediction model, the functional data analysis method of the system is studied based on the runoff data of the main stream of the Yellow River from 2002 to 2022, and the runoff data is studied from the functional point of view, comprehensively analyze the runoff evolution mechanism of the Yellow River basin, and provide new theoretical methods and scientific basis for the study of ecological environment protection, flood prevention and climate change in the basin.

Firstly, by studying the runoff curve, mean curve and variance curve, the statistical characteristics of runoff are explored, and its seasonal variation characteristics are revealed by using phase plan and principal differential analysis. The results show that the runoff of the main stream of the Yellow River shows an increasing trend from 2002 to 2022, with obvious periodicity and abrupt changes in 2004, 2008, 2013, 2017 and 2021, with obvious seasonal characteristics.

Secondly, based on the idea of non-negative matrix decomposition, functional principal component and principal differential are introduced, and a functional clustering method (PCPDFCM) combining principal component and principal differential is constructed to explore the temporal and spatial distribution characteristics of runoff in the main stream of the Yellow River. Using functional principal component analysis, it is found that the runoff mainly increased significantly around 2013 and from 2019 to 2021. From the perspective of spatial distribution, the overall runoff of the main stream of the Yellow River is gradually decreasing from top to bottom, with obvious spatial differences. In addition, 12 hydrological stations in the main stream of the Yellow River are grouped into three categories by PCPDFCM method, and the hydrological characteristics of the three types of hydrological stations are significantly different. Furthermore, ArcGIS software is used for visual display, and the difference characteristics of runoff of 12 hydrological stations are presented intuitively.

Finally, combining the principle of functional principal component analysis and principal differential analysis, a ridge regression model (FPDPCRR) based on functional principal differential and principal component analysis is constructed to predict the runoff of the main stream of the Yellow River. Specifically, on the basis of exploring the influencing factors of runoff in the main stream of the Yellow River by using cumulant slope change analysis method and multivariate functional regression model, FPDPCRR method is used to predict runoff. The results show that the prediction effect of the proposed method is good, which is consistent with the actual situation, thus providing a certain theoretical basis for comprehensive and efficient allocation of water resources and flood control and disaster reduction dispatching in the basin.