Rough path theory is focused on capturing and making precise the interactions
between highly oscillatory and non-linear systems. It draws on the analysis of
LC Young and the geometric algebra of KT Chen. The concepts and the uniform
estimates, have widespread application and have simplified proofs of basic
questions from the large deviation theory and extended Ito's theory of SDEs;
the recent applications contribute to (Graham) automated recognition of Chinese
handwriting and (Hairer) formulation of appropriate SPDEs to model randomly
evolving interfaces. At the heart of the mathematics is the challenge of
describing a smooth but potentially highly oscillatory and vector valued path
$x_{t}$ parsimoniously so as to effectively predict the response of a nonlinear
system such as $dy_{t}=f(y_{t})dx_{t}$, $y_{0}=a$. The Signature is a
homomorphism from the monoid of paths into the grouplike elements of a closed
tensor algebra. It provides a graduated summary of the path $x$. Hambly and
Lyons have shown that this non-commutative transform is faithful for paths of
bounded variation up to appropriate null modifications. Among paths of bounded
variation with given Signature there is always a unique shortest
representative. These graduated summaries or features of a path are at the
heart of the definition of a rough path; locally they remove the need to look
at the fine structure of the path. Taylor's theorem explains how any smooth
function can, locally, be expressed as a linear combination of certain special
functions (monomials based at that point). Coordinate iterated integrals form a
more subtle algebra of features that can describe a stream or path in an
analogous way; they allow a definition of rough path and a natural linear
"basis" for functions on streams that can be used for machine learning.

粗糙路径理论通过均匀估计来简化对高度振荡和非线性系统之间交互的描述，该理论具有广泛的应用，并通过坐标迭代积分可描述路径的复杂细节，并为机器学习提供了天然的线性基础。

粗略路径，签名和流函数的建模

Rough paths, Signatures and the modelling of functions on streams

Non-linear systems of differential equations have attracted the interest in
fields like system biology, ecology or biochemistry, due to their flexibility
and their ability to describe dynamical systems. Despite the importance of such
models in many branches of science they have not been the focus of systematic
statistical analysis until recently. In this work we propose a general approach
to estimate the parameters of systems of differential equations measured with
noise. Our methodology is based on the maximization of the penalized likelihood
where the system of differential equations is used as a penalty. To do so, we
use a Reproducing Kernel Hilbert Space approach that allows to formulate the
estimation problem as an unconstrained numeric maximization problem easy to
solve. The proposed method is tested with synthetically simulated data and it
is used to estimate the unobserved transcription factor CdaR in Steptomyes
coelicolor using gene expression data of the genes it regulates.

提出了一种基于惩罚最大似然的一般方法来估计带噪声的微分方程组的参数，并使用再生核希尔伯特空间方法来将估计问题形式化为易于解决的无约束数值最大化问题，并用合成的数据测试了所提出的方法，并使用基因表达数据估计未观察到的转录因子 CdaR 在 Steptomyes coelicolor 中的作用。

基于再生核希尔伯特空间的普通微分方程系统估计

Reproducing kernel Hilbert space based estimation of systems of ordinary  differential equations

The use of directed acyclic graphs (DAGs) to represent conditional
independence relations among random variables has proved fruitful in a variety
of ways. Recursive structural equation models are one kind of DAG model.
However, non-recursive structural equation models of the kinds used to model
economic processes are naturally represented by directed cyclic graphs with
independent errors, a characterization of conditional independence errors, a
characterization of conditional independence constraints is obtained, and it is
shown that the result generalizes in a natural way to systems in which the
error variables or noises are statistically dependent. For non-linear systems
with independent errors a sufficient condition for conditional independence of
variables in associated distributions is obtained.

该研究使用有向无环图 (DAG) 表示随机变量之间的条件独立关系，证明了非递归结构方程模型能够通过有向循环图表示条件独立误差，并推导出满足条件独立约束的充分条件以及非线性系统的变量在相关分布中条件独立的条件。