Data processing and engineering techniques enable people to observe and better understand the natural and human-made systems and processes that generate huge amounts of various data types. Data engineers collect data created in almost all fields and formats, such as images, audio, and text streams, biological and financial signals, sensing and many others. They develop and implement state-of-the art machine learning (ML) and artificial intelligence (AI) algorithms using big data to infer valuable information with social and economic value. Furthermore, ML/AI methodologies lead to automate many decision making processes with real-time applications serving people and businesses. As an example, mathematical tools are engineered for analysis of financial data such as prices, trade volumes, and other economic indicators of instruments including stocks, options and futures in order to automate the generation, implementation and maintenance of investment portfolios.
Among the techniques, subspace framework and methods are fundamental, and they have been successfully employed in widely used technologies and real-time applications spanning from Internet multimedia to electronic trading of financial products. In this dissertation, the eigendecomposition of empirical correlation matrix created from market data (normalized returns) for a basket of US equities plays a central role. Then, the merit of approximating such an empirical matrix by a Toeplitz matrix, where closed form solutions for its eigenvalues and eigenvectors exist, is investigated. More specifically, the exponential correlation model that populates such a Toeplitz matrix is used to approximate pairwise empirical correlations of asset returns in a portfolio. Hence, the analytically derived eigenvectors of such a random vector process are utilized to design its eigenportfolios. The performances of the model based and the traditional eigenportfolios are studied and compared to validate the proposed portfolio design method. It is shown that the model based designs yield eigenportfolios that track the variations of the market statistics closely and deliver comparable or better performance.
The theoretical foundations of information theory and the rate-distortion theory that provide the basis for source coding methods, including transform coding, are revisited in the dissertation. This theoretical inquiry helps to construct the basic question of trade-offs between dimension of the eigensubspace versus the correlation structure of the random vector process it represents. The signal processing literature facilitates developing an efficient subspace partitioning algorithm to design novel portfolios by combining eigenportfolios of partitions for US equities that outperform the existing eigenportfolios (EP), market portfolios (MP), minimum variance portfolios (MVP), and hierarchical risk parity (HRP) portfolios for US equities. Additionally, the pdf-optimized quantizer framework is employed to sparse eigenportfolios in order to reduce the (trading) cost of their maintenance. Then, the concluding remarks are presented in the last section of the Dissertation.
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