  In this document, we present key findings in structured matrix approximation
theory, with applications to the regressive representation of dynamic financial
processes. Initially, we explore a comprehensive approach involving generic
nonlinear time delay embedding for time series data extracted from a financial
or economic system under examination. Subsequently, we employ sparse
least-squares and structured matrix approximation methods to discern
approximate representations of the output coupling matrices. These
representations play a pivotal role in establishing the regressive models
corresponding to the recursive structures inherent in a given financial system.
The document further introduces prototypical algorithms that leverage the
aforementioned techniques. These algorithms are demonstrated through
applications in approximate identification and predictive simulation of dynamic
financial and economic processes, encompassing scenarios that may or may not
exhibit chaotic behavior.
