| 总结: | <p>In this thesis we consider the application of tools from stochastic analysis and algebra to statistics and machine learning. Most of these tools are different forms of what has become known as signature methods. The signature has been discovered and rediscovered in a few different areas of mathematics in the last 70 years. In short, it maps a path evolving in a vector space to a group enveloping that same space. The reason it is so useful for statistics is twofold: One, its set of invariances is highly desirable in many applications, and two, its image group is highly structured making it particularly amenable to mathematical study using algebraic tools. </p>
<p>The primary aim here is to study how one may use the signature to express statistical properties of a given path, and how these properties can be applied to machine learning. This aim manifests itself as - among other things:</p>
<p> - a new type of cumulants for signatures that have unique combinatorial properties and can be used to characterise independence of paths,</p>
<p> - cumulants on reproducing kernel Hilbert spaces which are related to the signature cumulants, even though signatures are not used explicitly,</p>
<p> - a generalisation of the signature to other types of feature maps into non-commutative algebras,</p>
<p> - a feature map with an initial topology that captures properties of the filtration of stochastic processes, </p>
<p> - and a family of scoring rules with associated divergences, entropies and mutual informations for paths that respect their group structure.</p>
<p>These are divided into separate, self contained chapters that can be read independently of one another.</p>
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