Generalizations and variations of the fundamental lemma by Willems et al. are
an active topic of recent research. In this note, we explore and formalize the
links between kernel regression and known nonlinear extensions of the
fundamental lemma. Applying a transformation to the usual linear equation in
Hankel matrices, we arrive at an alternative implicit kernel representation of
the system trajectories while keeping the requirements on persistency of
excitation. We show that this representation is equivalent to the solution of a
specific kernel regression problem. We explore the possible structures of the
underlying kernel as well as the system classes to which they correspond.

通过对 Hankel 矩阵的线性方程进行转换，我们得到了系统轨迹的一种隐式核表示，同时保持了激励持久性的要求，并且证明了该表示与一种特定的核回归问题的解是等价的。

探索基本引理与核回归之间的关联

Exploring the Links between the Fundamental Lemma and Kernel Regression

Identifying causal relations among simultaneously acquired signals is an
important problem in multivariate time series analysis. For linear stochastic
systems Granger proposed a simple procedure called the Granger causality to
detect such relations. In this work we consider nonlinear extensions of
Granger's idea and refer to the result as Extended Granger Causality. A simple
approach implementing the Extended Granger Causality is presented and applied
to multiple chaotic time series and other types of nonlinear signals. In
addition, for situations with three or more time series we propose a
conditional Extended Granger Causality measure that enables us to determine
whether the causal relation between two signals is direct or mediated by
another process.

本文提出了扩展格兰杰因果关系方法，应用于多个混沌时间序列和其他非线性信号，同时在三个或更多时间序列的情况下提出了条件扩展格兰杰因果关系度量。