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Dynamic financial processes identification using sparse regressive reservoir computers

Fredy Vides, Idelfonso B. R. Nogueira, Gabriela Lopez Gutierrez, Lendy Banegas, Evelyn Flores

TL;DR

This work addresses identifying nonlinear, interconnected financial dynamics from time-series data by marrying sparse structured matrix approximation with nonlinear regressive reservoir computers (NRRCs). It introduces dilated embeddings $\eth_p$ and a sparse compression operator $R_p(n)$ to produce low-dimensional, non-redundant representations used in a regressive mapping $\mathbf{y}_L(t)=W\eth_p(\mathbf{x}_L(t))$, enabling concise identification of dynamics even with limited data. Theoretical results on $\mathrm{rk}_\delta(A)$, sparse linear LS solvers, and compressed representations underpin algorithms for output coupling matrix identification and structured regressive models. Numerical simulations on chaotic and eventually periodic financial-like systems demonstrate accurate, data-efficient forecasting and the approach's potential utility for finance and regulation, with open-source DyNet tools.

Abstract

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.

Dynamic financial processes identification using sparse regressive reservoir computers

TL;DR

This work addresses identifying nonlinear, interconnected financial dynamics from time-series data by marrying sparse structured matrix approximation with nonlinear regressive reservoir computers (NRRCs). It introduces dilated embeddings and a sparse compression operator to produce low-dimensional, non-redundant representations used in a regressive mapping , enabling concise identification of dynamics even with limited data. Theoretical results on , sparse linear LS solvers, and compressed representations underpin algorithms for output coupling matrix identification and structured regressive models. Numerical simulations on chaotic and eventually periodic financial-like systems demonstrate accurate, data-efficient forecasting and the approach's potential utility for finance and regulation, with open-source DyNet tools.

Abstract

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.
Paper Structure (16 sections, 6 theorems, 43 equations, 8 figures, 3 algorithms)

This paper contains 16 sections, 6 theorems, 43 equations, 8 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Structured dynamic transformation model identification
  4. Low-rank approximation and sparse linear least squares solvers
  5. Sparse structured nonlinear regressive model identification
  6. Sparse structured nonlinear autoregressive model identification
  7. Algorithms
  8. Sparse linear least squares solver and structured assembling matrix identification algorithms
  9. Structured coupling matrix identification algorithm
  10. Numerical Simulations and Applications
  11. Sparse autoregressive reservoir computers for dynamical nonlinear financial system behavior identification
  12. Chaotic behavior identification
  13. Eventually approximately periodic behavior identification
  14. Learning interest rates with sparse regressive reservoir computers.
  15. Conclusions
  16. ...and 1 more sections

Key Result

Lemma 3.2

We will have that $\mathrm{rk}_\delta\left(A^\top\right)=\mathrm{rk}_\delta(A)$ for each $\delta>0$ and each $A\in \mathbb{C}^{m\times n}$.

Figures (8)

  • Figure 1: Training orbits data (left), validation orbits data (right). The green line corresponds to validation data, and the red dotted line corresponds to the model's predictions.
  • Figure 2: Matrix factors $\hat{W}$ (top-left) and $R$ (top-right), output coupling matrix $W =\hat{W}R$ (bottom).
  • Figure 3: Training orbits data (left), validation orbits data (right). The green line corresponds to validation data, and the red dotted line corresponds to the model's predictions.
  • Figure 4: Matrix factors $\hat{W}$ (top-left) and $R$ (top-right), output coupling matrix $W =\hat{W}R$ (bottom).
  • Figure 5: Periodic interest rate identification.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • proof
  • Theorem 3.6
  • ...and 3 more