Junoh Kang

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time-series /
statistics

Path Signature: Useful Feature for Timeseries

Study note on path signature in the literature of rough path theory. This post introduces the definition, algebraic structure, and probabilistic interpretation of signatures, bridging rough path theory and modern machine learning, based primarily on *A Primer on the Signature Method in Machine Learning*.

(Motivation) Random Variable: Polynomial Features#

Definition (Words / Multi-Indices)#

A word (or multi-index) is a finite sequence with . The length of is . The set is the alphabet. Denote by the set of all words.

Definition (Polynomials for RV)#

For a vector and a multi-index , define . The collection is called the polynomials of .

Theorem (Stone-Weierstrass, Carleman)#

Under suitable integrability conditions on ,

Paths: Iterated Integrals and Signature#

Definition (Path-space)#

Let denote the space of continuous bounded variation paths

Definition (Iterated integrals)#

For and word , define the iterated integral inductively by

with base case as . In other expression,

Definition (Signature)#

The signature of is the collection of real numbers indexed by words

Definition#

Exercise (Linear path)#

Let and define for . Then sol)

Exercise (1D path)#

For , sol)

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