从经济物理学视角分析标普500金融数据相关矩阵的主成分
Analysis of the Principal Components of Correlation Matrices of S&P 500 Financial Data from an Econophysics Perspective
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中文总结 AI 辅助
本研究从经济物理学视角，通过k-Means聚类等方法分析标普500相关矩阵的主成分，识别市场状态，发现新冠疫情为非典型状态，成果可用于投资组合设计等。
中文摘要 AI 辅助
本论文从经济物理学视角对标普500的集体动态进行分析，将金融市场视为一个复杂系统。利用430家公司的每日对数收益率，在滑动窗口上构建随时间变化的皮尔逊相关矩阵，并通过主导特征值和特征向量研究其谱结构。对几个谱量应用k-Means聚类来识别市场状态（MS）：最大特征值（被解读为市场状态）、主导特征向量的平方元素（视为各股票的相对参与度），以及由前l个特征对构建并归一化以保持相关矩阵解释性的截断矩阵重构项C^l。评估所得聚类的稳定性，计算转移矩阵及其平稳向量，并使用逆参与率量化参与度在资产间的扩散程度。新冠疫情事件在完整相关矩阵中表现为一种非典型状态；重构该状态需要不止一个主成分，C^2和C^3重构项均可再现该状态，尤其在窗口长度q=40且k=5时效果显著。危机期间，相对参与度分布在更多资产上，而2017-2018年区间对动态有贡献但未呈现为孤立机制。该分析为描述性而非预测性，提供了与投资组合设计和交易策略相关的概念及方法论工具，可自然扩展至其他国际交易所。
英文摘要
This thesis analyzes the collective dynamics of the S&P 500 from an econophysics perspective, treating the financial market as a complex system. Using daily logarithmic returns of 430 companies, time-dependent Pearson correlation matrices are constructed over sliding windows and their spectral structure is studied through the leading eigenvalues and eigenvectors. Market States (MS) are identified by applying k-Means clustering to several spectral quantities: the largest eigenvalue (interpreted as the State of the Market), the squared entries of the dominant eigenvector (read as a relative participation of each stock), and truncated matrix reconstructions C^l built from the l largest eigenpairs and normalized to retain a correlation-matrix interpretation. The stability of the resulting clusters is assessed, transition matrices and their stationary vectors are computed, and the inverse participation ratio is used to quantify how participation spreads across assets. The COVID-19 episode emerges as an atypical state in the full correlation matrices; reconstructing it requires more than one principal component, and the C^2 and C^3 constructions reproduce it, particularly for window length q=40 and k=5. During crisis periods the relative participation distributes over a larger number of assets, while the 2017-2018 interval contributes to the dynamics without appearing as an isolated regime. The analysis is descriptive rather than predictive, and provides conceptual and methodological tools relevant to portfolio design and trading strategies, with natural extensions to other international exchanges.
发表机构
- Faculty of Sciences Universidad Nacional Autónoma de México（墨西哥国立自治大学理学院）
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