From 3c405190e5a17fbf28fb48960eee3902af0f9973 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Wed, 15 May 2024 20:13:58 +0100
Subject: [PATCH 01/32] first draft

---
 papers/sam_morley/banner.png  | Bin 0 -> 506825 bytes
 papers/sam_morley/myst.yml    |  53 +++
 papers/sam_morley/roughpy.bib | 135 +++++++
 papers/sam_morley/roughpy.md  | 697 ++++++++++++++++++++++++++++++++++
 4 files changed, 885 insertions(+)
 create mode 100644 papers/sam_morley/banner.png
 create mode 100644 papers/sam_morley/myst.yml
 create mode 100644 papers/sam_morley/roughpy.bib
 create mode 100644 papers/sam_morley/roughpy.md

diff --git a/papers/sam_morley/banner.png b/papers/sam_morley/banner.png
new file mode 100644
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diff --git a/papers/sam_morley/myst.yml b/papers/sam_morley/myst.yml
new file mode 100644
index 0000000000..95edf882e1
--- /dev/null
+++ b/papers/sam_morley/myst.yml
@@ -0,0 +1,53 @@
+# See docs at: https://mystmd.org/guide/frontmatter
+version: 1
+project:
+  id: scipy-2024-sam_morley
+  title: RoughPy
+  subtitle: Streaming data is rarely smooth
+  short_title: RoughPy
+
+  authors: 
+    - name: Sam Morley
+      email: sam.morley@maths.ox.ac.uk
+      orcid: 0000-0001-5971-7418
+      affiliations: 
+        - University Of Oxford
+
+  # description:
+  keywords: 
+    - time series 
+    - rough paths
+    - signatures 
+    - data science 
+    - machine learning
+  github: datasig-ac-uk/roughpy
+  bibliography: 
+    - roughpy.bib
+  # Add the abbreviations that you use in your paper here
+  abbreviations:
+    RFI: Radio frequency interference
+    NLP: Natural Language Processing
+    ODE: Ordinary Differential Equation
+    CDE: Controlled differential equation
+    PDE: Partial differential equation
+
+  # It is possible to explicitly ignore the `doi-exists` check for certain citation keys
+  error_rules:
+    - rule: doi-exists
+      severity: ignore
+      keys:
+        - Atr03
+        - terradesert
+        - jupyter
+        - sklearn1
+        - sklearn2
+  # A banner will be generated for you on publication, this is a placeholder
+  banner: banner.png
+  # The rest of the information shouldn't be modified
+  subject: Research Article
+  open_access: true
+  license: CC-BY-4.0
+  venue: Scipy 2024
+  date: 2024-07-10
+site:
+  template: article-theme
diff --git a/papers/sam_morley/roughpy.bib b/papers/sam_morley/roughpy.bib
new file mode 100644
index 0000000000..d226364441
--- /dev/null
+++ b/papers/sam_morley/roughpy.bib
@@ -0,0 +1,135 @@
+@inproceedings{tseriotou_etal_2024_sig,
+    title = "Sig-Networks Toolkit: Signature Networks for Longitudinal Language Modelling",
+    author = "Tseriotou, Talia  and
+      Chan, Ryan  and
+      Tsakalidis, Adam  and
+      Bilal, Iman Munire  and
+      Kochkina, Elena  and
+      Lyons, Terry  and
+      Liakata, Maria",
+    editor = "Aletras, Nikolaos  and
+      De Clercq, Orphee",
+    booktitle = "Proceedings of the 18th Conference of the European Chapter of the Association for Computational Linguistics: System Demonstrations",
+    month = mar,
+    year = "2024",
+    address = "St. Julians, Malta",
+    publisher = "Association for Computational Linguistics",
+    url = "https://aclanthology.org/2024.eacl-demo.24",
+    pages = "223--237",
+    abstract = "We present an open-source, pip installable toolkit, Sig-Networks, the first of its kind for longitudinal language modelling. A central focus is the incorporation of Signature-based Neural Network models, which have recently shown success in temporal tasks. We apply and extend published research providing a full suite of signature-based models. Their components can be used as PyTorch building blocks in future architectures. Sig-Networks enables task-agnostic dataset plug-in, seamless preprocessing for sequential data, parameter flexibility, automated tuning across a range of models. We examine signature networks under three different NLP tasks of varying temporal granularity: counselling conversations, rumour stance switch and mood changes in social media threads, showing SOTA performance in all three, and provide guidance for future tasks. We release the Toolkit as a PyTorch package with an introductory video, Git repositories for preprocessing and modelling including sample notebooks on the modeled NLP tasks.",
+}
+
+@inproceedings{NEURIPS2020_4a5876b4,
+ author = {Kidger, Patrick and Morrill, James and Foster, James and Lyons, Terry},
+ booktitle = {Advances in Neural Information Processing Systems},
+ editor = {H. Larochelle and M. Ranzato and R. Hadsell and M.F. Balcan and H. Lin},
+ pages = {6696--6707},
+ publisher = {Curran Associates, Inc.},
+ title = {Neural Controlled Differential Equations for Irregular Time Series},
+ url = {https://proceedings.neurips.cc/paper_files/paper/2020/file/4a5876b450b45371f6cfe5047ac8cd45-Paper.pdf},
+ volume = {33},
+ year = {2020}
+}
+
+@article{Lyons1998,
+abstract = {This paper aims to provide a systematic approach to the treatment of differential equations of the typedyt = Σi fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process.However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed. The results here are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. (For a simple example see [10], [1]).We hope our results will have an influence on infinite dimensional analysis on path spaces, loop groups, etc. as well as in more applied situations. Variable step size algorithms for the numerical integration of stochastic differential equations [8] have been constructed as a consequence of these results.},
+author = {Lyons, Terry J.},
+journal = {Revista Matemática Iberoamericana},
+keywords = {Ecuaciones diferenciales estocásticas; Proceso de difusión; Movimiento browniano; stochastic differential equations; rough paths; Brown motion; Gauss and Markov processes; Lie algebras},
+language = {eng},
+number = {2},
+pages = {215-310},
+title = {Differential equations driven by rough signals.},
+url = {http://eudml.org/doc/39555},
+volume = {14},
+year = {1998},
+}
+
+@inproceedings{liao2021a,
+  edition = {},
+  number = {},
+  journal = {},
+  pages = {},
+  publisher = {British Machine Vision Association},
+  school = {},
+  title = {Logsig-RNN: a novel network for robust and efficient skeleton-based action recognition},
+  volume = {},
+  author = {Liao, S and Lyons, TJ and Yang, W and Schlegel, K and Ni, H},
+  editor = {},
+  year = {2021},
+  organizer = {32nd British Machine Vision Conference (BMVC 2021)},
+ series = {}
+}
+
+@inproceedings{NEURIPS2021_18a9042b,
+ author = {Fermanian, Adeline and Marion, Pierre and Vert, Jean-Philippe and Biau, G\'{e}rard},
+ booktitle = {Advances in Neural Information Processing Systems},
+ editor = {M. Ranzato and A. Beygelzimer and Y. Dauphin and P.S. Liang and J. Wortman Vaughan},
+ pages = {3121--3134},
+ publisher = {Curran Associates, Inc.},
+ title = {Framing RNN as a kernel method: A neural ODE approach},
+ url = {https://proceedings.neurips.cc/paper_files/paper/2021/file/18a9042b3fc5b02fe3d57fea87d6992f-Paper.pdf},
+ volume = {34},
+ year = {2021}
+}
+
+@article{JMLR_v20_16_314,
+  author  = {Franz J. Kiraly and Harald Oberhauser},
+  title   = {Kernels for Sequentially Ordered Data},
+  journal = {Journal of Machine Learning Research},
+  year    = {2019},
+  volume  = {20},
+  number  = {31},
+  pages   = {1--45},
+  url     = {http://jmlr.org/papers/v20/16-314.html}
+}
+
+@misc{pybind11,
+   author = {Wenzel Jakob and Jason Rhinelander and Dean Moldovan},
+   year = {2017},
+   url = {https://github.com/pybind/pybind11},
+   title = {pybind11 -- Seamless operability between C++11 and Python}
+}
+
+@misc{esig,
+  author = {Terry Lyons and David Maxwell},
+  year = 2017,
+  url = {https://github.com/datasig-ac-uk/esig},
+  title = {esig}
+}
+
+@misc{coropa_project,
+  author = {Stephen Buckley and Djalil Chafai and Greg Gyurk\`{o} and Arend
+    Janssen and Christophe Ladroue and Christian Litterer and Terry Lyons 
+    and ChangLiang Xu},
+  year = 2006,
+  url = {https://coropa.sourceforge.io},
+  title = {CoRoPa -- Computational Rough Paths project}
+}
+
+@inproceedings{NEURIPS2019_d2cdf047,
+ author = {Kidger, Patrick and Bonnier, Patric and Perez Arribas, Imanol and Salvi, Cristopher and Lyons, Terry},
+ booktitle = {Advances in Neural Information Processing Systems},
+ editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
+ pages = {},
+ publisher = {Curran Associates, Inc.},
+ title = {Deep Signature Transforms},
+ url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/d2cdf047a6674cef251d56544a3cf029-Paper.pdf},
+ volume = {32},
+ year = {2019}
+}
+
+@misc{dlpack, 
+  author = {DLPack},
+  year = 2023,
+  title = {Open In Memory Tensor structure},
+  url = {https://dmlc.github.io/dlpack/latest/}
+}
+
+@Manual{Granlund12,
+  title = "{GNU MP}: {T}he {GNU} {M}ultiple {P}recision {A}rithmetic {L}ibrary",
+  author = "Torbjörn Granlund and {the GMP development team}",
+  edition = "5.0.5",
+  year = 	 2012,
+  url = "http://gmplib.org/"
+}
diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
new file mode 100644
index 0000000000..6630d36678
--- /dev/null
+++ b/papers/sam_morley/roughpy.md
@@ -0,0 +1,697 @@
+---
+title: RoughPy
+subtitle: Streaming data is rarely smooth 
+exports:
+  - format: pdf
+    template: arxiv_two_column
+    article_type: Article
+bibliography:
+    - roughpy.bib
+abstract: |
+  Rough path theory is a branch of mathematics arising out of stochastic
+  analysis. The principal tool of rough path analysis is the signature, which
+  captures the evolution of a path including the order in which events occurred.
+  This turns out to be a useful tool in data science applications involving
+  sequential data. RoughPy is our new Python package that aims change the way we
+  think about sequential streamed data, by viewing it through the lens of rough
+  paths. In RoughPy, data is wrapped in a stream objects which can be composed
+  and queried to obtain signatures that can be used in analysis. It also
+  provides a platform for further exploration of the connections between rough
+  path theory and data science.
+
+acknowledgements: |
+  This work was supported in part by EPSRC (NSFC) under Grant EP/S026347/1, in
+  part by The Alan Turing Institute under the EPSRC grant EP/N510129/1, the Data
+  Centric Engineering Programme (under the Lloyd’s Register Foundation grant
+  G0095), and the Defence and Security Programme (funded by the UK Government).
+  For the purpose of Open Access, the author has applied a CC-BY public
+  copyright licence to any Author Accepted Manuscript version arising from this
+  submission.
+---
+
+## Introduction
+Sequential data appears everywhere in the modern world: text, finance, health 
+records, radio (and other electromagnetic spectra), sound (and speech), etc.
+These data are problematic for many traditional methods in data science.
+Many of these difficulties arise because of computational constraints, where the 
+complexity of the model is intrinsically linked to the number of observations, 
+and others arise because the events are not observed at regular intervals.
+In this paper, we introduce a new package *RoughPy* for analysing sequential
+data through the lens of *rough path theory*, which provides useful techniques
+for addressing some of the issues with these data.
+
+Rough paths arise in the study of *controlled differential equations* (CDEs),
+which generalise ordinary differential equations (ODEs) and stochastic
+differential equations [@Lyons1998;@http://dx.doi.org/10.1007/978-3-540-71285-5].
+These are equations of the form $\mathrm{d}Y_t = f(Y_t, \mathrm{d}X_t)$, subject 
+to an initial condition $Y_0 = y_0$, that model a non-linear system driven by a 
+input path $X$.
+On simple CDE turns out to be critical to the theory:
+```{math}
+\mathrm{d}S_t = S_t \otimes \mathrm{d}X_t \qquad S_0 = \mathbf{1}.
+```
+The solution of this equation is called the *signature* of $X$.
+It plays the same role as the exponential function for ordinary differential 
+equations (ODEs).
+When the path $X$ is sufficiently regular, the signature can be computed 
+directly as a sequence of iterated integrals.
+In other cases, we can still solve CDEs if we are given higher-order data that 
+can be used in place of the iterated integrals.
+A path equipped with this higher order data is called a *rough path*.
+
+The signature turns out to be a useful summary of sequential data.
+It captures the order of events but not the necessarily the speed at which 
+these events occur.
+The signature is robust to irregular sampling and provides a fixed-size view of
+the data, regardless of how many observations are used to compute it.
+This means the signature can be a useful feature map to be used in machine
+learning for sequential data.
+There are numerous examples of using signatures for analysing sequential data
+outlined in @applications-sec.
+
+Besides signatures, there are two other rough path-based methods that have found
+their way into data science in recent years.
+These are the signature kernel and neural CDEs.
+Both of these enjoy the same robustness of the signature, but can be useful in
+some contexts where using the signature transform might not be optimal.
+We give a short overview of these methods in @rp-in-ds-sec.
+
+There are several Python packages for computing signatures of sequential data, 
+including `esig` [@esig], `iisignature` [@doi.org/10.1145/3371237], and 
+`signatory` [@https://doi.org/10.48550/arXiv.2001.00706].
+These packages simply provide functions for computing signatures from raw, 
+structured data presented in an $n\times d$ array, where $d$ is the dimension 
+of the stream and $n$ is the number of samples.
+This means the user is responsible for interpreting the data as a path and 
+arranging the computations that need to be done
+
+RoughPy is a new package for working with sequential data and rough paths.
+The design philosophy for this package is to shift the emphasis from simply
+computing signatures on data to instead construct streams.
+A stream is a view of some data as if it were a rough path, that can be queried
+over intervals to obtain a signature.
+The actual form of the data is abstracted away in favour of stream objects that
+closely resemble the mathematics.
+The aim is to change the way that users think about sequential data and advance
+the understanding of path-like data analysis.
+
+On top of the streams, RoughPy also provides concrete implementations for
+elements of the various algebras associated with rough path analysis, such as
+free tensor algebras, shuffle tensor algebras, and free Lie algebras [see
+@math-bgd-sec].
+This allows the user to easily manipulate signatures, and other objects, in a
+more natural manner, and quickly develop methods following the mathematics.
+
+
+The paper is organised as follows. In the remainder of this section, we give a
+brief overview of the mathematics associated with rough path theory, and provide
+some additional detail for the signature kernel and neural CDEs. In
+@applications-sec we list several recent applications of signatures and rough
+path-based methods in data science applications. These applications should serve
+to motivate the development of RoughPy. Finally, in @roughpy-sec we give a more
+detailed overview of the RoughPy library, the types and functions it contains,
+and give an example of how it can be used.
+
+
+(math-bgd-sec)=
+### Mathematical background
+In this section we give a very short introduction to signatures and rough path 
+theory that should be sufficient to inform the discussion in the sequel.
+For a far more comprehensive and rigorous treatment, we refer the reader to the 
+recent survey [@https://doi.org/10.48550/arXiv.2206.14674].
+In this section, we write $V$ for the vector space
+$\mathbb{R}^d$, where $d \geq 1$.
+
+A *path* in $V$ is a continuous function $X:[a, b] \to V$, where $a < b$ are
+real numbers. 
+For the purposes of this discussion, we shall further impose the condition 
+that all paths are of bounded variation.
+The value of a path $X$ at some parameter $t\in[a, b]$ is denoted $X_t$. 
+<!-- For each $i\in \{1,\dots, d\}$, the $i$th component of $X$ is denoted  -->
+<!-- $X^{(i)}$; each $X^{(i)}:[a, b] \to \mathbb R$ is a path of bounded variation. -->
+
+The siganture of $X$ is an element of the *(free) tensor algebra*.
+For $n > 0$, the $n$th tensor power of $V$ is defined as $V^{\otimes n} = V
+\otimes\dots\otimes V$, where $n$ copies of $V$ appear on the right-hand side.
+By convention, we take $V^{\otimes 0} = \mathbb{R}$.
+For example, $V^{\otimes 1} = V$, $V^{\otimes 2}$ is the space of $d\times d$ 
+matrices, and $V^{\otimes 3}$ is the space of $d\times d\times d$ tensors.
+The *tensor algebra over $V$* is the space
+```{math}
+\mathrm{T}((V)) = \{\mathbf{x} = (x_0, x_1, \dots) : x_j \in V^{\otimes j}
+\,\forall j \geq 0\}
+```
+equipped with the tensor product $\otimes$ as multiplication.
+The tensor algebra is a *Hopf algebra*, and comes equipped with an
+antipode operation $\alpha_V:\mathrm{T}((V)) \to \mathrm{T}((V))$.
+It contains a group $\mathrm{G}(V)$ of elements under tensor multiplication and
+the antipode.
+The members of $\mathrm{G}(V)$ are called *group-like* elements.
+For each $n \geq 0$, we write $\mathrm{T}^n(V)$ for the *truncated tensor
+algebra* of degree $n$, which is the space of all $\mathbf{x} = (x_0, x_1,
+\dots)$ such that $x_j = \mathbf{0}$ whenever $j > n$.
+Similarly, we write $\mathrm{T}^{>n}(V)$ for the subspace of elements 
+$\mathbf{x} = (x_0, x_1,\dots)$ where $x_j = \mathbf{0}$ whenever $j \leq n$, 
+which is an ideal in $\mathrm{T}((V))$ and 
+$\mathrm{T}^n(V) = \mathrm{T}((V)) / \mathrm{T}^{>n}(V)$.
+The truncated tensor algebra is an algebra, when given the *truncated tensor
+product*, obtained by truncating the full tensor product. 
+
+The signature $\mathrm{S}(X)_{s, t}$ of a path $X:[a,b] \to V$ over a
+subinterval $[s, t)\subseteq [a, b]$ is 
+$\mathrm{S}(X)_{s,t} = (1, \mathrm{S}_1(X)_{s,t}, \dots)\in \mathrm{G}(V)$ 
+where for each $m\geq 1$, $\mathrm{S}_m(X)_{s, t}$ is given by the iterated 
+(Riemann-Stieltjes) integral
+```{math}
+\mathrm{S}_m(X)_{s, t} = \underset{s < u_1 < u_2 < \dots < u_m < t}
+{\int \dots \int}
+\mathrm{d}X_{u_1}\otimes \mathrm{d}X_{u_2}\otimes\dots\otimes \mathrm{d}X_{u_m}.
+```
+The signature respects concatenation of paths, meaning 
+$\mathrm{S}(X)_{s, t} = \mathrm{S}(X)_{s, u} \otimes \mathrm{S}(X)_{u, t}$ 
+for any $s < u < t$.
+Two paths have the same signature if and only if they differ by a 
+*tree-like path* [@http://dx.doi.org/10.4007/annals.2010.171.109].
+The signature is translation invariant, and it is invariant under 
+reparametrisation.
+
+In some sense, the signature is a *unversal non-linearity* on paths.
+Continuous functions on the path can be approximated (uniformly) by linear
+functionals ($\mathrm{G}(V)\to \mathbb{R}$) on the signature.
+This is a consequence of the Stone-Weierstrass theorem. 
+The *dual* of $\mathrm{T}((V))$ is $\mathrm{Sh}((V)):=\mathrm{T}((V^{\ast}))
+\cong\mathrm{T}((V))$ 
+(as a vector space) is the space of these linear functionals on
+$\mathrm{T}((V))$; we call this the *shuffle algebra*.
+The multiplication on $\mathrm{Sh}((V))$ is the *shuffle product*, which
+corresponds to point-wise multiplication of functions on the path.
+
+There are several *Lie algebras* associated to $\mathrm{T}((V))$.
+First we define the *Lie bracket* operation $[\mathbf{x}, \mathbf{y}] =
+\mathbf{x} \otimes \mathbf{y} - \mathbf{y}\otimes \mathbf{x}$, for
+$\mathbf{x},\mathbf{y}\in \mathrm{T}((V))$.
+We define subspaces $L_m$ of $\mathrm{T}((V))$ for each $m\geq 0$ inductively as
+follows: $L_0 = \{\mathbf{0}\}$, $L_1 = V$, and, for $m \geq 1$, 
+```{math}
+L_{m+1} = \mathrm{span}\{[\mathbf{x}, \mathbf{y}] : \mathbf{x}\in V, \mathbf{y}
+\in L_m\}
+```
+The space of Lie series $\mathcal{L}((V))$ over $V$ is the subspace of 
+$\mathrm{T}((V))$ containing sequences of the form $(\ell_0, \ell_1, \cdots)$, 
+where $\ell_j\in L_j$ for each $j\geq 0$.
+Note that $\mathcal{L}((V))\subseteq \mathrm{T}^{>0}(V)$.
+For any $\mathbf{x} \in \mathrm{T}(V)$ we define 
+```{math}
+\exp(\mathbf{x}) = \sum_{n=0}^\infty \frac{\mathbf{x}^{\otimes n}}{n!}
+\quad\text{and}\quad
+\log(\mathbf{1} + \mathbf{x}) = \sum_{n=1}^\infty
+\frac{(-1)^{n-1}}{n}\mathbf{x}^{\otimes n}.
+```
+For any path $X$, we have $\log(\mathrm{S}(X)_{s, t})\in \mathcal{L}((V))$, and,
+more generally we have $\exp(\mathcal{L}((V))) = \mathrm{G}(V)$.
+We call the quantity $\mathrm{LogSig}(X)_{s, t} := \log(\mathrm{S}(X)_{s,t})$
+the *log-signature* of $X$ over $[s, t)$.
+
+(rp-in-ds-sec)=
+### Rough paths in data science 
+Now we turn to the applications of rough path theory to data science.
+Our first task is to form a bridge between sequential data and paths.
+Consider a finite, ordered sequence $\{(t_1, \mathbf{x}_1,\dots,
+t_N,\mathbf{x}_N)\}$ of observations, where $t_j\in \mathbb{R}$, and
+$\mathbf{x}_j\in V$.
+(More generally, we might consider $\mathbf{x}_j\in\mathcal{L}((V))$ instead.
+That is, data that already contains higher-order information.)
+We can find numerous paths that interpolate these observations; a path 
+$X:[t_0, t_N]\to V$ such that, for each $j$, $X_{t_j} = \mathbf{x}_j$.
+The simplest interpolation is to take the path that is linear between adjacent
+observations.
+
+Once we have a path, we need to be able to compute signatures.
+For practical purposes, we truncate all signatures (and log-signatures) to a
+particular degree $M$, which we typically call the *depth*.
+The dimension of the ambient space $d$ is usually called the *width*.
+Using the linear interpolation, we can replace the iterated integrals with a
+tensor exponential of the difference of successive terms:
+```{math}
+\mathrm{Sig}^M([t_j, t_{j+1})) = \exp_M(\mathbf{x}_{j+1} - \mathbf{x}_j) :=
+\sum_{j=0}^M \frac{1}{j!}(\mathbf{x}_{j+1} - \mathbf{x}_j)^{\otimes j}.
+```
+Here, and in the remainder of the paper, we shall denote the empirical signature
+over an interval $I$ by $\mathrm{Sig}(I)$ and the log-signature as
+$\mathrm{LogSig}(I)$.
+We can compute the signature over arbitrary intervals by taking the product of
+the these terms, using the multiplicative property of the signature.
+
+#### The signature transform
+Most of the early applications of rough paths in data science, the (truncated)
+signature was used as a feature map [@NEURIPS2019_d2cdf047].
+This provides a summary of the path that is independent of the parameterisation
+and the number of observations.
+Unfortunately, the signature is a high-dimensional summary.
+If $d > 1$, then the dimension of $\mathrm{T}^M(V)$ is
+```{math}
+\sum_{m=0}^M d^m = \frac{d^{M+1} - 1}{d - 1}
+```
+This generally is a reflection of the complexity of the data, where data with a
+higher complexity generally needs a higher truncation level and thus a larger
+signature.
+
+For some applications, it might be possible to replace the signature with the
+log-signature.
+The log-signature is smaller than the signature, but we lose the universal
+non-linearity property of the signature.
+Alternatively, we might turn to other techniques that don't require a full
+calculation of the signature (such as those mentioned below).
+As the connection between rough paths and data science becomes more
+mathematically mature, we will likely find new ways to use the signature
+without requiring its full size.
+
+(sigker-sec)=
+#### Signature kernels
+Kernel methods are useful tools for learning with sequential data.
+Mathematically, a *kernel* on a set $W$ is a positive-definite function
+$k:W\times W\to \mathbb{R}$. Kernels are often quite easy to evaluate because of
+the kernel trick, which involves embedding the data in a inner product space,
+with a feature map, in which the
+kernel can be evaluated by simply taking an inner product.
+Informally, kernels measure the similarity between two points.
+They are used in a variety of machine learning tasks such as classification.
+
+The *signature kernel* is a kernel induced on the space of paths by combining
+the signature with a secondary kernel defined on the tensor algebra
+[@JMLR_v20_16_314].
+The theory surrounding the signature kernel has been expanded several times
+since their introduction [@NEURIPS2021_18a9042b;@doi:10.1214/23-AAP1973].
+Typically, the kernel on $\mathrm{T}((V))$ will itself by derived from an inner
+product on $V$, extended to the tensor algebra.
+
+
+Unfortunately the tensor algebra is infinite-dimensional, so we can't simply
+evaluate this as an inner product of signatures.
+However, it can be approximated by taking inner products of truncated
+signatures.
+Even better, it turns out that, in certain cases, the signature kernel 
+can be realised as  the solution to a partial differential equation (PDE) of 
+Goursat type.
+This means the full signature kernel can be computed from raw data without 
+needing to compute full signatures [@https://doi.org/10.1137/20M1366794].
+
+In fact, in recent preprint, it has been shown that there are
+higher order solvers for signature kernels by rewriting the kernel 
+solution of a system of PDEs of Goursat type 
+[@https://doi.org/10.48550/arXiv.2404.02926].
+A critical part of their method involves the adjoint of both left and right
+free tensor multiplication, which are not available in any current package
+for computing signatures.
+
+(ncde-sec)=
+#### Neural controlled differential equations
+Neural CDEs are a method for modelling irregular time series
+[@NEURIPS2020_4a5876b4].
+We consider CDEs of the form
+```{math}
+\mathrm{d}Y_t = f_\theta(Y_t)\mathrm{d}X_t
+```
+where $f_\theta$ is a neural network.
+We can treat the path $Y$ as "hidden state" that we can tune using data to
+understand the relationship between the driving path $X_t$ and some response.
+Neural CDEs can be regarded as a continuous-time analogue of a recurrent neural
+network.
+
+Neural CDEs initially showed some promising results on several benchmarks but
+now lag behind current state-of-the-art approaches to time series modelling.
+The latest iteration of neural CDEs are the recently introduced Log-neural
+controlled differential equations [@https://doi.org/10.48550/arXiv.2402.18512],
+which make use of the *Log-ODE* method for solving rough differential equations
+in order to boost the performance of neural CDEs.
+
+(applications-sec)=
+## Current applications of rough paths
+In this section we enumerate several applications where rough paths 
+have been used to develop or improve methods for solving particular problems.
+This list presented here is certainly not exhaustive.
+In addition to the literature cited below, there are numerous worked examples in
+the form of Jupyter notebooks available on the DataSig website 
+(<https://datasig.ac.uk/examples>).
+
+### Detecting interference in radio astronomy data 
+Radio frequency interference (RFI) is a substantial problem in the field of
+radio astronomy.
+Even small amounts of RFI can obscure the faint signals generated by distant
+stellar objects and events.
+The problem of identifying RFI in a signal falls into a class of
+semi-supervised learning tasks called novelty (or anomaly) detection.
+In a recent preprint, @doi:10.48550/arXiv.2402.14892 present a novelty detection 
+framework based on rough path methods to detect RFI in radio astronomy data from 
+several radio telescopes.
+Their result show that their framework is effective at detecting even faint RFI
+within the test data.
+The methodology presented in the paper follows from previous work by 
+@https://doi.org/10.48550/arXiv.2006.03487.
+
+A similar problem of detecting malware by inspecting the streaming tree of
+processes on a computer system is explored in
+@doi:10.1109/CSR51186.2021.9527933.
+Their method uses the signature kernel (see @sigker-sec) combined with a support
+vector machine classifier to identify processes that are malicious compared to
+"normal" behaviour learned via training on labelled data.
+
+### Tracking mood via natural langauge processing
+One application of rough paths in NLP has been in the domain of mental health 
+[@http://dx.doi.org/10.18653/v1/2023.findings-acl.310; @tseriotou_etal_2024_sig].
+In this work, the authors present a model for identifying changes in a person's 
+mood based on their online textual content. 
+Many mental health conditions have symptoms that manifest in the (textual)
+expression, so this could be a powerful tool for mental health professionals to
+identify changes in patients and intervene before the state develops.
+Their model achieves state-of-the-art performance vs existing models on two
+datasets.
+
+### Predicting battery cell degradation
+Another recent application of signatures is to predict the degredation of
+lithium-ion cells [@http://doi.org/10.1016/j.apenergy.2023.121974].
+They use signature features to train a model that can accuractely predict the
+end of life of a cell using relatively low-frequency sampling compared to
+existing models.
+They also observed that the performance at higher frequency was comparable to
+other models.
+
+### Prediction of sepsis in intensive care data
+One of the first effective demonstrations of the utility of signatures and rough
+paths based methods in healthcare was in the 2019 PhysioNet challenge.
+Here teams were invited to develop models to predict sepsis in patients from
+intensive care unit data.
+In this challenge, a team utilising signatures to enhance predictive power
+placed 
+first in the official phase of the challenge 
+[@https://doi.org/10.1097/CCM.0000000000004510].
+Since then, signatures and other rough path based approaches have been used in
+several other clinical contexts 
+[@https://doi.org/10.1038/s41598-024-51989-6; @doi:10.1109/MEDAI59581.2023.00008;
+@tseriotou_etal_2024_sig].
+Clinical data is often irregularly sampled and often exhibits missing data, but
+it can also be very high-frequency and dense.
+Rough path based methods can handle these data in an elegant way, and retain the
+structure of long and short term dependencies within the data.
+
+### Human action recognition
+The task of identifying a specific action performed by a person from a short
+video clip is very challenging. 
+Signatures derived from landmark data extracted from the video has been
+used to train classication models that acheived state-of-the-art performance
+compared with contemporary models [@https://doi.org/10.1007/978-3-030-98519-6_18].
+This work has been continued in numerous additional papers such as 
+[@doi:10.1109/tmm.2023.3318242;@liao2021a]
+(See also preprint papers [@https://doi.org/10.48550/arXiv.2308.12840;
+@https://doi.org/10.48550/arXiv.2403.15212].) 
+Also in the domain of computer vision, signatures have been used to produce
+lightweight models for image classification
+[@https://doi.org/10.1109/CVPRW56347.2022.00409] and in handwriting recognition
+tasks [@https://doi.org/10.1109/TPAMI.2017.2732978].
+
+(roughpy-sec)=
+## RoughPy
+RoughPy is a new library that aims to support the development of connections
+between rough path theory and data science.
+It represents a shift in philosophy from simple computations of signatures for
+sequential data, to a representation of these data as a rough path.
+The design objectives for RoughPy are as follows:
+1. provide a class that presents a rough path view of some source of data as a
+   rough path, exposing methods for querying the data over intervals to get a
+   signature or log-signature;
+2. provide classes and functions that allow the users to interact with the
+   signatures and other algebraic objects in a natural, mathematical manner;
+3. all operations should be differentiable and objects should be interoperable
+   with objects from machine learning, such as TensorFlow (JAX) and PyTorch.
+
+The first two objectives are simple design and implementation problems.
+The final objective presents the most difficulty, especially interoperability
+between RoughPy and common machine learning libraries.
+There are array interchange formats for NumPy-like arrays, such as the Python
+Array API standard [@https://doi.org/10.25080/gerudo-f2bc6f59-001] and the 
+DLPack protocol [@dlpack].
+These provide part of the picture, but in order for them to be fully supported,
+RoughPy must support a variety of compute backends such as CUDA (NVidia), 
+ROCm/HIP (AMD), and Metal (Apple).
+
+RoughPy is a substantial library with numerous components, mostly written in 
+C++ with a Python interface defined using Pybind11 [@pybind11].
+The original design of the library closely followed the C++ template libraries
+libRDE and libalgebra [@coropa_project], although it has seen many iterations 
+since.
+
+In the remainder of this section, we discuss some of the core components of
+RoughPy, give an example of using RoughPy, and discuss the future of RoughPy.
+
+### Free tensors, shuffle tensors, and Lie objects
+In order to property support rough path based methods and allow users to write
+code based on mathematical concepts, we provide realisations of several
+algebra types.
+The algebras provided in RoughPy are `FreeTensor`, `ShuffleTensor`, and `Lie`,
+which define elements of a particular free tensor algebra, shuffle tensor
+algebra, or free Lie algebra respectively.
+Each of these algebras is initialized with a width, depth, and scalar
+coefficient type, encapsulated in a `Context` object.
+
+In addition to the algebra classes, RoughPy provides a number of supporting
+functions, including half-shuffle products for `FreeTensor`/`ShuffleTensor`
+objects, and adjoint operators for left free tensor multiplication.
+These are operations that are frequently used in the theory of rough paths, and
+will likely be necessary in developing new applications later (as in the
+signature kernels).
+
+RoughPy algebras are designed around a flexible scalar ring system that allows
+users to perform calculations with different accuracy, or derive expressions by
+using polynomial coefficients.
+For most applications, floating point numbers will provide a good balance
+between performance and accuracy.
+When more precision is required, rational coefficients can be used
+instead.
+These are backed by GMPLib rationals for fast, arbitrary precision rational
+arithmetic [@Granlund12].
+Polynomial coefficients can be used to derive formulae by performing
+calculations.
+This is a powerful technique for understanding the terms that appear in the
+result, particularly whilst testing and debugging.
+
+
+### Intervals 
+RoughPy is very careful in the way it handles intervals.
+All intervals in RoughPy are half-open, meaning that they include one end point
+but not the other; they are either *clopen* $[a, b) = \{t: a\leq t < b\}$ or 
+*opencl* $(a, b] = \{t : a < t \leq b\}$.
+Besides the type (clopen or opencl), all intervals must provide methods for
+retrieving the infimum ($a$ in the above notation) and the supremum ($b$ above)
+of the interval as double precision floats.
+This is enforced by means of an abstract base class `Interval`.
+The main concrete interval types are `RealInterval`, an interval with arbitrary 
+real endpoints, and `DyadicInterval`, as described below.
+For brevity, in the sequel we shall only consider clopen intervals.
+
+A *dyadic interval* is an interval of the form $D_k^n := [k/2^n, (k+1)/2^n)$, 
+where $k$, $n$ are integers.
+The number $n$ is often described as the *resolution* of the interval.
+The family of dyadic intervals of a fixed resolution $n$ partition the real line 
+so that every real number $t$ belongs to a unique dyadic interval $D_n^k$.
+Moreover, the family of all dyadic intervals have the property that two
+dyadic intervals are either disjoint or one contains the other (including the
+possibility that they are equal).
+
+In many cases, RoughPy will granularise an interval into a dyadic intervals.
+The *dyadic granularisation* of $[a, b)$ with resolution $n$ is 
+$[k_1/2^n, k_2/2^n)$ where $k_1 = \max\{k: k/2^n \leq a\}$ and 
+$k_2 = \max\{k: k/2^n \leq b\}$.
+In effect, the dyadic granularisation is the result of "rounding" each end 
+point to the included end of the unique dyadic interval that contain it.
+
+### Streams
+Streams are central to RoughPy.
+A RoughPy `Stream` is a rough path view of some underlying data.
+It provides two key methods to query the object over intervals to retrieve
+either a signature or log-signature.
+Importantly, once constructed, the underlying data is inaccessible except by
+querying via these methods.
+`Stream`s are designed to be composed in various ways, such as by concatenation,
+in order to build up more complex streams.
+A `Stream` is actually a (type-erased) wrapper around a more minimal
+`StreamInterface` abstract class.
+
+We construct streams by a factory function associated with each different
+`StreamInterface`, which might perform some compression of the underlying data.
+For example, a basic `StreamInterface` is the `LieIncrementStream`, which can be
+constructed using the associated `from_increments` factory function (a static
+method of the class), which accepts an $n \times d$ array of *increment data*. 
+These data will typically be the differences between successive values of the 
+data (but could also include higher-order Lie terms).
+This is similar to the way that libraries such as `esig`, `iisignature`, and
+`signatory` consume data.
+
+RoughPy streams cache the result of log-signature queries over dyadic intervals
+so they can be reused in later calculations.
+To compute the log-signature over any interval $I$, we granularise at a fixed 
+stream resolution $n$ to obtain the interval $\tilde I = [k_1/2^n, k_2/2^n)$, 
+and then compute
+:::{math}
+\mathrm{LogSig}(\tilde I) = \log\biggl(\prod_{k=k_1}^{k_2-1} 
+\exp(\mathrm{LogSig}(D_k^n))\biggr).
+:::
+The $\mathrm{LogSig}(D_k^n)$ terms on the right-hand-side are either retrieved 
+from the cache, or computed from the underlying source.
+This is essentially the Campbell-Baker-Hausdorff formula applied to the 
+log-signatures at the finest level.
+In practice, we can actually reduce the number of terms in the product, by
+merging complementary dyadic intervals that appear in the granularisation.
+We further optimise by using a fused multiply-exponential ($A\exp(B)$)
+operation.
+
+Signatures are always computed by first computing the log-signature and then
+exponentiating.
+Directly computing the signature as a product of exponentials of (cached)
+log-signatures might accumulate enough numerical errors to drift slightly from a
+group-like tensor; that is, the result might not actually be a true
+signature.
+Taking the logarithm and then exponentiating back to obtain the signature has
+the effect of correcting this numerical drift from a true signature.
+
+Aside from the basic `LieIncrementStream`, there are several other
+implementations of the `StreamInterface` currently available in RoughPy.
+The `BrownianStream` approximates Brownian motion by generating normal
+distributed increments over dyadic intervals of arbitrary resolution on demand,
+forming a reasonable approximation of true Brownian motion.
+The `ExternalDataStream` is an interface for loading data from various external
+sources, such as from a database or specialised data format.
+Currently, only sound files are supported but we plan to extend support for
+other sources as the need arises.
+This will certainly include "online" data sources such as computer peripheral
+devices such as microphones.
+
+The other main `StreamInterface` implementation is the `PiecewiseAbelianStream`,
+which is an important construction from the theory of rough paths.
+A piecewise Abelian path, or log-linear path, is an example of a *smooth rough
+path*, which generalises piecewise linear approximations of an arbitrary stream.
+Formally, an *Abelian path* $Y$ is a pair $([a, b), \mathbf{y})$ where $a < b$ and
+$\mathbf{x}\in\mathcal{L}((V))$.
+The log-signature over an arbitrary interval $[u, v) \subseteq [a, b)$ is given
+by
+```{math}
+\mathrm{LogSig}(Y)_{u, v} = \frac{v - u}{b - a}\mathbf{y}.
+```
+A *piecewise Abelian path* is the concatenation of finitely many Abelian paths
+with adjacent intervals.
+For any rough path $X$ and partition $\{a = t_0 < t_1 < \dots < t_N = b\}$ there
+is a piecewise Abelian approximation for this path given by
+```{math}
+\{([t_{j-1}, t_j), \mathrm{LogSig}(X)_{t_{j-1}, t_j}): j=1, \dots, N\}.
+```
+This construction turns out to be vital for computing signature kernels in
+[@https://doi.org/10.25080/gerudo-f2bc6f59-001] and for solving CDEs
+[@http://dx.doi.org/10.1007/978-3-540-71285-5;
+@https://doi.org/10.48550/arXiv.2402.18512].
+In particular, this construction can be used to compress data at some degree,
+which can the be used in computations at a higher degree.
+
+### Example
+In this section we show a very simple example of how to use RoughPy to construct
+a stream and compute a signature.
+This example is similar to the first few steps of the tutorial found in the
+RoughPy documentation.[^roughpydocs]
+We refer the reader to this documentation for much more detail.
+We will construct a stream in $\mathbb{R}^{26}$ by taking each letter in a word,
+"scipy" in this example, as the increments of a path:
+```python
+import numpy as np
+
+text = "scipy"
+increments = np.zeros((5, 26), dtype="int8")
+for i, c in enumerate(text):
+    increments[i, ord(c) - 97] = 1
+```
+Now we import RoughPy and construct a `Stream` using the factory mentioned
+above.
+One other critical ingredient is the algebra `Context`, which is used to set up
+a consistent set of algebra objects with the desired width (26), truncation
+level (2), and coefficient type (`Rational`).
+```python
+import roughpy as rp
+
+ctx = rp.get_context(width=26, 
+                     depth=2, 
+                     coeffs=rp.Rational)
+stream = rp.LieIncrementStream.from_increments(
+  increments, ctx=ctx)
+```
+Now we can compute the signature of the stream over the whole domain of the
+stream $[0, 4]$ by omitting the interval argument:
+```python
+sig = stream.signature()
+print(sig);
+# { 1() 1(3) 1(9) 1(16) 1(19) 1(25) 1/2(3,3) 
+#   1(3,9) 1(3,16) 1(3,25) 1/2(9,9) 1(9,16) 
+#   1(9,25) 1/2(16,16) 1(16,25) 1(19,3) 1(19,9) 
+#   1(19,16) 1/2(19,19) 1(19,25) 1/2(25,25) }
+```
+The first term of the signature is always 1, and the empty parentheses 
+indicate the empty tensor word.
+The next five terms correspond to the counts of each unique letter that 
+appears, the number in parentheses indicates the letter (with `a` being 1).
+The final terms indicate the order in which each pair of letters appear in
+the word.
+For instance, the term `1(3,9)` indicates that a `c` appears before an `i`.
+
+:::{prf:remark} 
+It turns out that most words in the English language can be distinguished 
+using only their level 2 signatures.
+The first level signatures groups words into anagrams.
+The second level signature counts the occurrences of each ordered pair of
+letters.
+There are relatively few words that require level 3 data.
+From the standard Linux dictionary, containing around 80,000 words, there are
+two pairs of words that require the level 3 terms: "toot" and "otto", and "naan"
+and "anna".
+This is shown in the RoughPy documentation.
+Similar patterns can be observed in other languages too, including French,
+Spanish, German, Russian, and Lithuanian.
+:::
+[^roughpydocs]: <https://roughpy.org/user/absolute_beginners.html>
+
+
+### The future of RoughPy
+RoughPy is continuously evolving.
+At time of writing, the current version uses libalgebra and libalgebra-lite
+(libalgebra with fewer templates) for computations.
+Unfortunately, this made it difficult to achieve the differentiability and 
+computation device support that we want.
+We are currently changing the way we implement vectors and algebras to provide
+the support for on-device computation that we want.
+Making the operations differentiable is crucial for machine learning, and will
+be the biggest challenge.
+
+Long term, we need to expand support for signature kernels and CDEs. 
+As applications of these tools grow in data science, we will need to devise
+new methods for computing kernels, or solving CDEs.
+
+
+## Conclusions
+The use of rough path theory in data science is rapidly expanding and provides a
+different way to view sequential data.
+Signatures, and other methods arising from rough path theory, are already used
+in a wide variety of applications, with great effect.
+The next steps in overcoming the difficulty in modeling sequential data will
+require a change of perspective.
+Viewing these data through the lens of rough path theory might provide this
+change.
+
+RoughPy is a new Python library for working with streamed data using rough path
+methods.
+It is designed to abstract away the form and source of data so that analysis can
+be performed by querying path-like objects.
+This approach is much closer to the mathematics.
+It also allows users to interact with the various algebras associated with rough
+paths (free tensor algebra, shuffle tensor algebra, free Lie algebra) in a
+natural way.
+RoughPy is under active development, and a long list of improvements and
+extensions are planned.
+
+
+
+<!-- vim: set tw=80 wrap cc=80 -->

From 86330b7e68a466929ffd2d07b76f5ef289e3c5b1 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sun, 19 May 2024 15:20:04 +0100
Subject: [PATCH 02/32] fixed a few problems

---
 papers/sam_morley/roughpy.md | 75 +++++++++++++++++++-----------------
 1 file changed, 40 insertions(+), 35 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 6630d36678..9eab2613b0 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -9,15 +9,15 @@ bibliography:
     - roughpy.bib
 abstract: |
   Rough path theory is a branch of mathematics arising out of stochastic
-  analysis. The principal tool of rough path analysis is the signature, which
-  captures the evolution of a path including the order in which events occurred.
-  This turns out to be a useful tool in data science applications involving
-  sequential data. RoughPy is our new Python package that aims change the way we
-  think about sequential streamed data, by viewing it through the lens of rough
-  paths. In RoughPy, data is wrapped in a stream objects which can be composed
-  and queried to obtain signatures that can be used in analysis. It also
-  provides a platform for further exploration of the connections between rough
-  path theory and data science.
+  analysis. One of the main tools of rough path analysis is the signature,
+  which captures the evolution of an unparametrised path including the order in
+  which events occurred. This turns out to be a useful tool in data science
+  applications involving sequential data. RoughPy is our new Python package
+  that aims change the way we think about sequential streamed data, by viewing
+  it through the lens of rough paths. In RoughPy, data is wrapped in a stream
+  objects which can be composed and queried to obtain signatures that can be
+  used in analysis. It also provides a platform for further exploration of the
+  connections between rough path theory and data science.
 
 acknowledgements: |
   This work was supported in part by EPSRC (NSFC) under Grant EP/S026347/1, in
@@ -32,13 +32,17 @@ acknowledgements: |
 ## Introduction
 Sequential data appears everywhere in the modern world: text, finance, health 
 records, radio (and other electromagnetic spectra), sound (and speech), etc.
-These data are problematic for many traditional methods in data science.
-Many of these difficulties arise because of computational constraints, where the 
-complexity of the model is intrinsically linked to the number of observations, 
-and others arise because the events are not observed at regular intervals.
-In this paper, we introduce a new package *RoughPy* for analysing sequential
-data through the lens of *rough path theory*, which provides useful techniques
-for addressing some of the issues with these data.
+Traditionally, these data are tricky to work with because of the exponential
+complexity and different scales of the underlying process.
+Until recently, with the development of transformers and large language models,
+it has been difficult to capture the long-term pattern whist also capturing the
+short-term fine detail.
+Rough path theory gives us tools to work with sequential, ordered data in a
+mathematically rigorous way, which should provide a means to overcome some of
+the inherent complexity of the data.
+In this paper, we introduce a new package *RoughPy* for working with sequential
+data through the lens of rough path theory, performing rigorous analysis and
+developing new techniques to overcome the difficulties of these data.
 
 Rough paths arise in the study of *controlled differential equations* (CDEs),
 which generalise ordinary differential equations (ODEs) and stochastic
@@ -51,11 +55,12 @@ On simple CDE turns out to be critical to the theory:
 \mathrm{d}S_t = S_t \otimes \mathrm{d}X_t \qquad S_0 = \mathbf{1}.
 ```
 The solution of this equation is called the *signature* of $X$.
-It plays the same role as the exponential function for ordinary differential 
-equations (ODEs).
+It is analogous to the exponential function for ODEs, in that the
+solution of any CDE can be expressed in terms of the signature of the driving
+path.
 When the path $X$ is sufficiently regular, the signature can be computed 
 directly as a sequence of iterated integrals.
-In other cases, we can still solve CDEs if we are given higher-order data that 
+In other cases, we can still solve CDEs if we are given higher order data that 
 can be used in place of the iterated integrals.
 A path equipped with this higher order data is called a *rough path*.
 
@@ -66,14 +71,14 @@ The signature is robust to irregular sampling and provides a fixed-size view of
 the data, regardless of how many observations are used to compute it.
 This means the signature can be a useful feature map to be used in machine
 learning for sequential data.
-There are numerous examples of using signatures for analysing sequential data
+There are numerous examples of using signatures of analysing sequential data
 outlined in @applications-sec.
 
 Besides signatures, there are two other rough path-based methods that have found
 their way into data science in recent years.
 These are the signature kernel and neural CDEs.
-Both of these enjoy the same robustness of the signature, but can be useful in
-some contexts where using the signature transform might not be optimal.
+Both of these enjoy the same robustness of the signature, and expand the range
+of applications of rough path-based methods.
 We give a short overview of these methods in @rp-in-ds-sec.
 
 There are several Python packages for computing signatures of sequential data, 
@@ -88,7 +93,7 @@ arranging the computations that need to be done
 RoughPy is a new package for working with sequential data and rough paths.
 The design philosophy for this package is to shift the emphasis from simply
 computing signatures on data to instead construct streams.
-A stream is a view of some data as if it were a rough path, that can be queried
+A *stream* is a view of some data as if it were a rough path, that can be queried
 over intervals to obtain a signature.
 The actual form of the data is abstracted away in favour of stream objects that
 closely resemble the mathematics.
@@ -96,12 +101,12 @@ The aim is to change the way that users think about sequential data and advance
 the understanding of path-like data analysis.
 
 On top of the streams, RoughPy also provides concrete implementations for
-elements of the various algebras associated with rough path analysis, such as
-free tensor algebras, shuffle tensor algebras, and free Lie algebras [see
-@math-bgd-sec].
+elements of the various algebras associated with rough path analysis.
+These include free tensor algebras, shuffle tensor algebras, and free Lie
+algebras [see @math-bgd-sec].
 This allows the user to easily manipulate signatures, and other objects, in a
-more natural manner, and quickly develop methods following the mathematics.
-
+more natural manner.
+This, in turn, allows us to quickly develop methods following the mathematics.
 
 The paper is organised as follows. In the remainder of this section, we give a
 brief overview of the mathematics associated with rough path theory, and provide
@@ -130,11 +135,11 @@ The value of a path $X$ at some parameter $t\in[a, b]$ is denoted $X_t$.
 <!-- For each $i\in \{1,\dots, d\}$, the $i$th component of $X$ is denoted  -->
 <!-- $X^{(i)}$; each $X^{(i)}:[a, b] \to \mathbb R$ is a path of bounded variation. -->
 
-The siganture of $X$ is an element of the *(free) tensor algebra*.
-For $n > 0$, the $n$th tensor power of $V$ is defined as $V^{\otimes n} = V
-\otimes\dots\otimes V$, where $n$ copies of $V$ appear on the right-hand side.
-By convention, we take $V^{\otimes 0} = \mathbb{R}$.
-For example, $V^{\otimes 1} = V$, $V^{\otimes 2}$ is the space of $d\times d$ 
+The signature of $X$ is an element of the *(free) tensor algebra*.
+For $n \geq 0$, the $n$th tensor power of $V$ is defined recursively by
+$V^{\otimes 0} = \mathbb{R}$, $V^{\otimes 1} = V$, and $V^{\otimes n+1} = V
+\otimes V^{\otimes n}$ for $n > 1$. 
+For example, $V^{\otimes 2}$ is the space of $d\times d$ 
 matrices, and $V^{\otimes 3}$ is the space of $d\times d\times d$ tensors.
 The *tensor algebra over $V$* is the space
 ```{math}
@@ -187,7 +192,7 @@ The multiplication on $\mathrm{Sh}((V))$ is the *shuffle product*, which
 corresponds to point-wise multiplication of functions on the path.
 
 There are several *Lie algebras* associated to $\mathrm{T}((V))$.
-First we define the *Lie bracket* operation $[\mathbf{x}, \mathbf{y}] =
+Define the *Lie bracket* operation $[\mathbf{x}, \mathbf{y}] =
 \mathbf{x} \otimes \mathbf{y} - \mathbf{y}\otimes \mathbf{x}$, for
 $\mathbf{x},\mathbf{y}\in \mathrm{T}((V))$.
 We define subspaces $L_m$ of $\mathrm{T}((V))$ for each $m\geq 0$ inductively as
@@ -196,7 +201,7 @@ follows: $L_0 = \{\mathbf{0}\}$, $L_1 = V$, and, for $m \geq 1$,
 L_{m+1} = \mathrm{span}\{[\mathbf{x}, \mathbf{y}] : \mathbf{x}\in V, \mathbf{y}
 \in L_m\}
 ```
-The space of Lie series $\mathcal{L}((V))$ over $V$ is the subspace of 
+The space of formal Lie series $\mathcal{L}((V))$ over $V$ is the subspace of 
 $\mathrm{T}((V))$ containing sequences of the form $(\ell_0, \ell_1, \cdots)$, 
 where $\ell_j\in L_j$ for each $j\geq 0$.
 Note that $\mathcal{L}((V))\subseteq \mathrm{T}^{>0}(V)$.

From 4750c644499fe36ffc8f36603296c9ec3f4ad6b6 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sun, 19 May 2024 19:59:29 +0100
Subject: [PATCH 03/32] fix a couple more problems

---
 papers/sam_morley/roughpy.md | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 9eab2613b0..67c5b00c1a 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -11,11 +11,11 @@ abstract: |
   Rough path theory is a branch of mathematics arising out of stochastic
   analysis. One of the main tools of rough path analysis is the signature,
   which captures the evolution of an unparametrised path including the order in
-  which events occurred. This turns out to be a useful tool in data science
+  which events occur. This turns out to be a useful tool in data science
   applications involving sequential data. RoughPy is our new Python package
   that aims change the way we think about sequential streamed data, by viewing
   it through the lens of rough paths. In RoughPy, data is wrapped in a stream
-  objects which can be composed and queried to obtain signatures that can be
+  object, which can be composed and queried to obtain signatures that can be
   used in analysis. It also provides a platform for further exploration of the
   connections between rough path theory and data science.
 
@@ -84,7 +84,7 @@ We give a short overview of these methods in @rp-in-ds-sec.
 There are several Python packages for computing signatures of sequential data, 
 including `esig` [@esig], `iisignature` [@doi.org/10.1145/3371237], and 
 `signatory` [@https://doi.org/10.48550/arXiv.2001.00706].
-These packages simply provide functions for computing signatures from raw, 
+These packages provide functions for computing signatures from raw, 
 structured data presented in an $n\times d$ array, where $d$ is the dimension 
 of the stream and $n$ is the number of samples.
 This means the user is responsible for interpreting the data as a path and 
@@ -92,7 +92,7 @@ arranging the computations that need to be done
 
 RoughPy is a new package for working with sequential data and rough paths.
 The design philosophy for this package is to shift the emphasis from simply
-computing signatures on data to instead construct streams.
+computing signatures on data to instead work with streams.
 A *stream* is a view of some data as if it were a rough path, that can be queried
 over intervals to obtain a signature.
 The actual form of the data is abstracted away in favour of stream objects that
@@ -106,7 +106,7 @@ These include free tensor algebras, shuffle tensor algebras, and free Lie
 algebras [see @math-bgd-sec].
 This allows the user to easily manipulate signatures, and other objects, in a
 more natural manner.
-This, in turn, allows us to quickly develop methods following the mathematics.
+This allows us to quickly develop methods by following the mathematics.
 
 The paper is organised as follows. In the remainder of this section, we give a
 brief overview of the mathematics associated with rough path theory, and provide

From f7c41d6031d6ef32bbe868d4c64a103fe4f0504c Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sun, 19 May 2024 20:29:35 +0100
Subject: [PATCH 04/32] change the modeline

---
 papers/sam_morley/roughpy.md | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 67c5b00c1a..f7d3bcc659 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -7,6 +7,9 @@ exports:
     article_type: Article
 bibliography:
     - roughpy.bib
+numbering:
+    heading_2: true
+    heading_3: true
 abstract: |
   Rough path theory is a branch of mathematics arising out of stochastic
   analysis. One of the main tools of rough path analysis is the signature,
@@ -644,6 +647,7 @@ the word.
 For instance, the term `1(3,9)` indicates that a `c` appears before an `i`.
 
 :::{prf:remark} 
+:nonumber:
 It turns out that most words in the English language can be distinguished 
 using only their level 2 signatures.
 The first level signatures groups words into anagrams.
@@ -699,4 +703,4 @@ extensions are planned.
 
 
 
-<!-- vim: set tw=80 wrap cc=80 -->
+<!-- nvim: set tw=80 wrap cc=80 -->

From fde193bb3d8541390ab702de4bb8efe82c257a38 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sun, 19 May 2024 22:38:08 +0100
Subject: [PATCH 05/32] final few corrections

---
 papers/sam_morley/roughpy.md | 100 +++++++++++++++++------------------
 1 file changed, 49 insertions(+), 51 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index f7d3bcc659..056cd5a002 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -1,10 +1,6 @@
 ---
 title: RoughPy
 subtitle: Streaming data is rarely smooth 
-exports:
-  - format: pdf
-    template: arxiv_two_column
-    article_type: Article
 bibliography:
     - roughpy.bib
 numbering:
@@ -286,18 +282,17 @@ Informally, kernels measure the similarity between two points.
 They are used in a variety of machine learning tasks such as classification.
 
 The *signature kernel* is a kernel induced on the space of paths by combining
-the signature with a secondary kernel defined on the tensor algebra
+the signature with an inner product defined on the tensor algebra
 [@JMLR_v20_16_314].
 The theory surrounding the signature kernel has been expanded several times
 since their introduction [@NEURIPS2021_18a9042b;@doi:10.1214/23-AAP1973].
 Typically, the kernel on $\mathrm{T}((V))$ will itself by derived from an inner
 product on $V$, extended to the tensor algebra.
 
-
-Unfortunately the tensor algebra is infinite-dimensional, so we can't simply
-evaluate this as an inner product of signatures.
-However, it can be approximated by taking inner products of truncated
-signatures.
+Signatures are infinite objects, so we can't simply evaluate inner products on
+the tensor algebra.
+Fortunately, we can approximate the signature kernel by taking inner products of
+truncated signatures.
 Even better, it turns out that, in certain cases, the signature kernel 
 can be realised as  the solution to a partial differential equation (PDE) of 
 Goursat type.
@@ -315,16 +310,16 @@ for computing signatures.
 (ncde-sec)=
 #### Neural controlled differential equations
 Neural CDEs are a method for modelling irregular time series
-[@NEURIPS2020_4a5876b4].
+.
 We consider CDEs of the form
 ```{math}
-\mathrm{d}Y_t = f_\theta(Y_t)\mathrm{d}X_t
+\mathrm{d}Y_t = f_\theta(Y_t)\,\mathrm{d}X_t
 ```
 where $f_\theta$ is a neural network.
 We can treat the path $Y$ as "hidden state" that we can tune using data to
 understand the relationship between the driving path $X_t$ and some response.
 Neural CDEs can be regarded as a continuous-time analogue of a recurrent neural
-network.
+network [@NEURIPS2020_4a5876b4].
 
 Neural CDEs initially showed some promising results on several benchmarks but
 now lag behind current state-of-the-art approaches to time series modelling.
@@ -336,7 +331,7 @@ in order to boost the performance of neural CDEs.
 (applications-sec)=
 ## Current applications of rough paths
 In this section we enumerate several applications where rough paths 
-have been used to develop or improve methods for solving particular problems.
+have been used to develop or improve methods.
 This list presented here is certainly not exhaustive.
 In addition to the literature cited below, there are numerous worked examples in
 the form of Jupyter notebooks available on the DataSig website 
@@ -348,25 +343,26 @@ radio astronomy.
 Even small amounts of RFI can obscure the faint signals generated by distant
 stellar objects and events.
 The problem of identifying RFI in a signal falls into a class of
-semi-supervised learning tasks called novelty (or anomaly) detection.
-In a recent preprint, @doi:10.48550/arXiv.2402.14892 present a novelty detection 
-framework based on rough path methods to detect RFI in radio astronomy data from 
-several radio telescopes.
+semi-supervised learning tasks called *novelty* (or *anomaly*) *detection*.
+Rough path methods have been applied to develop a novelty detection framework
+based on rough path methods to detect RFI in radio astronomy data from several
+radio telescopes [@doi:10.48550/arXiv.2402.14892].
 Their result show that their framework is effective at detecting even faint RFI
 within the test data.
-The methodology presented in the paper follows from previous work by 
-@https://doi.org/10.48550/arXiv.2006.03487.
+This work is based on a general novelty detection framework
+[@https://doi.org/10.48550/arXiv.2006.03487].
 
-A similar problem of detecting malware by inspecting the streaming tree of
-processes on a computer system is explored in
+Signatures kernels have also been used for a similar problem of detecting
+malware by inspecting the streaming tree of processes on a computer system 
 @doi:10.1109/CSR51186.2021.9527933.
-Their method uses the signature kernel (see @sigker-sec) combined with a support
-vector machine classifier to identify processes that are malicious compared to
-"normal" behaviour learned via training on labelled data.
+Their method uses a support vector machine classifier to identify processes that
+are malicious compared to "normal" behaviour learned via training on labelled
+data.
 
-### Tracking mood via natural langauge processing
-One application of rough paths in NLP has been in the domain of mental health 
-[@http://dx.doi.org/10.18653/v1/2023.findings-acl.310; @tseriotou_etal_2024_sig].
+### Tracking mood via natural language processing
+One application of rough paths in natural language processing has been in the
+domain of mental health [@http://dx.doi.org/10.18653/v1/2023.findings-acl.310;
+@tseriotou_etal_2024_sig].
 In this work, the authors present a model for identifying changes in a person's 
 mood based on their online textual content. 
 Many mental health conditions have symptoms that manifest in the (textual)
@@ -376,23 +372,22 @@ Their model achieves state-of-the-art performance vs existing models on two
 datasets.
 
 ### Predicting battery cell degradation
-Another recent application of signatures is to predict the degredation of
+Another recent application of signatures is to predict the degradation of
 lithium-ion cells [@http://doi.org/10.1016/j.apenergy.2023.121974].
-They use signature features to train a model that can accuractely predict the
-end of life of a cell using relatively low-frequency sampling compared to
-existing models.
+They use signature features to train a model that can accurately predict the end
+of life of a cell using relatively low-frequency sampling compared to existing
+models.
 They also observed that the performance at higher frequency was comparable to
 other models.
 
 ### Prediction of sepsis in intensive care data
 One of the first effective demonstrations of the utility of signatures and rough
-paths based methods in healthcare was in the 2019 PhysioNet challenge.
-Here teams were invited to develop models to predict sepsis in patients from
-intensive care unit data.
-In this challenge, a team utilising signatures to enhance predictive power
-placed 
-first in the official phase of the challenge 
+paths based methods in healthcare was in the 2019 PhysioNet challenge
 [@https://doi.org/10.1097/CCM.0000000000004510].
+In this contest, teams were invited to develop models to predict sepsis in
+patients from intensive care unit data.
+In this challenge, a team utilising signatures to enhance predictive power
+placed first in the official phase of the challenge.
 Since then, signatures and other rough path based approaches have been used in
 several other clinical contexts 
 [@https://doi.org/10.1038/s41598-024-51989-6; @doi:10.1109/MEDAI59581.2023.00008;
@@ -406,11 +401,11 @@ structure of long and short term dependencies within the data.
 The task of identifying a specific action performed by a person from a short
 video clip is very challenging. 
 Signatures derived from landmark data extracted from the video has been
-used to train classication models that acheived state-of-the-art performance
+used to train classification models that achieved state-of-the-art performance
 compared with contemporary models [@https://doi.org/10.1007/978-3-030-98519-6_18].
 This work has been continued in numerous additional papers such as 
 [@doi:10.1109/tmm.2023.3318242;@liao2021a]
-(See also preprint papers [@https://doi.org/10.48550/arXiv.2308.12840;
+(see also preprint papers [@https://doi.org/10.48550/arXiv.2308.12840;
 @https://doi.org/10.48550/arXiv.2403.15212].) 
 Also in the domain of computer vision, signatures have been used to produce
 lightweight models for image classification
@@ -457,7 +452,7 @@ code based on mathematical concepts, we provide realisations of several
 algebra types.
 The algebras provided in RoughPy are `FreeTensor`, `ShuffleTensor`, and `Lie`,
 which define elements of a particular free tensor algebra, shuffle tensor
-algebra, or free Lie algebra respectively.
+algebra, and free Lie algebra respectively.
 Each of these algebras is initialized with a width, depth, and scalar
 coefficient type, encapsulated in a `Context` object.
 
@@ -486,8 +481,8 @@ result, particularly whilst testing and debugging.
 ### Intervals 
 RoughPy is very careful in the way it handles intervals.
 All intervals in RoughPy are half-open, meaning that they include one end point
-but not the other; they are either *clopen* $[a, b) = \{t: a\leq t < b\}$ or 
-*opencl* $(a, b] = \{t : a < t \leq b\}$.
+but not the other; they are either *clopen* $[a, b) := \{t: a\leq t < b\}$ or 
+*opencl* $(a, b] := \{t : a < t \leq b\}$.
 Besides the type (clopen or opencl), all intervals must provide methods for
 retrieving the infimum ($a$ in the above notation) and the supremum ($b$ above)
 of the interval as double precision floats.
@@ -496,8 +491,8 @@ The main concrete interval types are `RealInterval`, an interval with arbitrary
 real endpoints, and `DyadicInterval`, as described below.
 For brevity, in the sequel we shall only consider clopen intervals.
 
-A *dyadic interval* is an interval of the form $D_k^n := [k/2^n, (k+1)/2^n)$, 
-where $k$, $n$ are integers.
+A *dyadic interval* is an interval $D_k^n := [k/2^n, (k+1)/2^n)$, where $k$, $n$
+are integers.
 The number $n$ is often described as the *resolution* of the interval.
 The family of dyadic intervals of a fixed resolution $n$ partition the real line 
 so that every real number $t$ belongs to a unique dyadic interval $D_n^k$.
@@ -521,7 +516,7 @@ Importantly, once constructed, the underlying data is inaccessible except by
 querying via these methods.
 `Stream`s are designed to be composed in various ways, such as by concatenation,
 in order to build up more complex streams.
-A `Stream` is actually a (type-erased) wrapper around a more minimal
+A `Stream` is actually a (type-erasing) wrapper around a more minimal
 `StreamInterface` abstract class.
 
 We construct streams by a factory function associated with each different
@@ -556,8 +551,8 @@ Signatures are always computed by first computing the log-signature and then
 exponentiating.
 Directly computing the signature as a product of exponentials of (cached)
 log-signatures might accumulate enough numerical errors to drift slightly from a
-group-like tensor; that is, the result might not actually be a true
-signature.
+group-like tensor. 
+That is, the result might not actually be a true signature.
 Taking the logarithm and then exponentiating back to obtain the signature has
 the effect of correcting this numerical drift from a true signature.
 
@@ -574,7 +569,7 @@ This will certainly include "online" data sources such as computer peripheral
 devices such as microphones.
 
 The other main `StreamInterface` implementation is the `PiecewiseAbelianStream`,
-which is an important construction from the theory of rough paths.
+which is an important construction from CDE.
 A piecewise Abelian path, or log-linear path, is an example of a *smooth rough
 path*, which generalises piecewise linear approximations of an arbitrary stream.
 Formally, an *Abelian path* $Y$ is a pair $([a, b), \mathbf{y})$ where $a < b$ and
@@ -678,6 +673,10 @@ be the biggest challenge.
 Long term, we need to expand support for signature kernels and CDEs. 
 As applications of these tools grow in data science, we will need to devise
 new methods for computing kernels, or solving CDEs.
+We will also build a framework for constructing and working with linear maps,
+and homomorphisms.
+For example, one very useful linear map is the extension of the $\log$ function
+to the whole tensor algebra.
 
 
 ## Conclusions
@@ -702,5 +701,4 @@ RoughPy is under active development, and a long list of improvements and
 extensions are planned.
 
 
-
-<!-- nvim: set tw=80 wrap cc=80 -->
+% vim: tw=80 wrap cc=80 

From 135ded1a52e6d20d2723af8522f68d3bb973fc93 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sun, 19 May 2024 22:38:19 +0100
Subject: [PATCH 06/32] change to note, avoid duplication

---
 papers/sam_morley/roughpy.bib | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/papers/sam_morley/roughpy.bib b/papers/sam_morley/roughpy.bib
index d226364441..923a508ecd 100644
--- a/papers/sam_morley/roughpy.bib
+++ b/papers/sam_morley/roughpy.bib
@@ -87,14 +87,14 @@ @article{JMLR_v20_16_314
 @misc{pybind11,
    author = {Wenzel Jakob and Jason Rhinelander and Dean Moldovan},
    year = {2017},
-   url = {https://github.com/pybind/pybind11},
+   note = {https://github.com/pybind/pybind11},
    title = {pybind11 -- Seamless operability between C++11 and Python}
 }
 
 @misc{esig,
   author = {Terry Lyons and David Maxwell},
   year = 2017,
-  url = {https://github.com/datasig-ac-uk/esig},
+  note = {https://github.com/datasig-ac-uk/esig},
   title = {esig}
 }
 
@@ -103,7 +103,7 @@ @misc{coropa_project
     Janssen and Christophe Ladroue and Christian Litterer and Terry Lyons 
     and ChangLiang Xu},
   year = 2006,
-  url = {https://coropa.sourceforge.io},
+  note = {https://coropa.sourceforge.io},
   title = {CoRoPa -- Computational Rough Paths project}
 }
 
@@ -123,7 +123,7 @@ @misc{dlpack
   author = {DLPack},
   year = 2023,
   title = {Open In Memory Tensor structure},
-  url = {https://dmlc.github.io/dlpack/latest/}
+  note = {https://dmlc.github.io/dlpack/latest/}
 }
 
 @Manual{Granlund12,
@@ -131,5 +131,5 @@ @Manual{Granlund12
   author = "Torbjörn Granlund and {the GMP development team}",
   edition = "5.0.5",
   year = 	 2012,
-  url = "http://gmplib.org/"
+  note = "http://gmplib.org/"
 }

From af7c7bef7d9b9a1464189052e3edf88c40cb87b4 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Mon, 20 May 2024 20:44:34 +0100
Subject: [PATCH 07/32] Add more keywords and minor fixes

---
 papers/sam_morley/myst.yml   |  2 ++
 papers/sam_morley/roughpy.md | 46 +++++++++++++++++-------------------
 2 files changed, 24 insertions(+), 24 deletions(-)

diff --git a/papers/sam_morley/myst.yml b/papers/sam_morley/myst.yml
index 95edf882e1..e1cfbc540a 100644
--- a/papers/sam_morley/myst.yml
+++ b/papers/sam_morley/myst.yml
@@ -20,6 +20,8 @@ project:
     - signatures 
     - data science 
     - machine learning
+    - signature kernels
+    - Log-ODE method
   github: datasig-ac-uk/roughpy
   bibliography: 
     - roughpy.bib
diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 056cd5a002..7e93e4f599 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -40,12 +40,13 @@ Rough path theory gives us tools to work with sequential, ordered data in a
 mathematically rigorous way, which should provide a means to overcome some of
 the inherent complexity of the data.
 In this paper, we introduce a new package *RoughPy* for working with sequential
-data through the lens of rough path theory, performing rigorous analysis and
-developing new techniques to overcome the difficulties of these data.
+data through the lens of rough path theory, where we can perform rigourous
+analyses and explore different ways to understand sequential data.
 
 Rough paths arise in the study of *controlled differential equations* (CDEs),
 which generalise ordinary differential equations (ODEs) and stochastic
-differential equations [@Lyons1998;@http://dx.doi.org/10.1007/978-3-540-71285-5].
+differential equations
+[@Lyons1998;@http://dx.doi.org/10.1007/978-3-540-71285-5].
 These are equations of the form $\mathrm{d}Y_t = f(Y_t, \mathrm{d}X_t)$, subject 
 to an initial condition $Y_0 = y_0$, that model a non-linear system driven by a 
 input path $X$.
@@ -92,8 +93,8 @@ arranging the computations that need to be done
 RoughPy is a new package for working with sequential data and rough paths.
 The design philosophy for this package is to shift the emphasis from simply
 computing signatures on data to instead work with streams.
-A *stream* is a view of some data as if it were a rough path, that can be queried
-over intervals to obtain a signature.
+A *stream* is a view of some data as if it were a rough path, that can be
+queried over intervals to obtain a signature.
 The actual form of the data is abstracted away in favour of stream objects that
 closely resemble the mathematics.
 The aim is to change the way that users think about sequential data and advance
@@ -123,7 +124,7 @@ In this section we give a very short introduction to signatures and rough path
 theory that should be sufficient to inform the discussion in the sequel.
 For a far more comprehensive and rigorous treatment, we refer the reader to the 
 recent survey [@https://doi.org/10.48550/arXiv.2206.14674].
-In this section, we write $V$ for the vector space
+In the sequel, we write $V$ for the vector space
 $\mathbb{R}^d$, where $d \geq 1$.
 
 A *path* in $V$ is a continuous function $X:[a, b] \to V$, where $a < b$ are
@@ -131,8 +132,6 @@ real numbers.
 For the purposes of this discussion, we shall further impose the condition 
 that all paths are of bounded variation.
 The value of a path $X$ at some parameter $t\in[a, b]$ is denoted $X_t$. 
-<!-- For each $i\in \{1,\dots, d\}$, the $i$th component of $X$ is denoted  -->
-<!-- $X^{(i)}$; each $X^{(i)}:[a, b] \to \mathbb R$ is a path of bounded variation. -->
 
 The signature of $X$ is an element of the *(free) tensor algebra*.
 For $n \geq 0$, the $n$th tensor power of $V$ is defined recursively by
@@ -153,9 +152,9 @@ the antipode.
 The members of $\mathrm{G}(V)$ are called *group-like* elements.
 For each $n \geq 0$, we write $\mathrm{T}^n(V)$ for the *truncated tensor
 algebra* of degree $n$, which is the space of all $\mathbf{x} = (x_0, x_1,
-\dots)$ such that $x_j = \mathbf{0}$ whenever $j > n$.
+\dots)$ such that $x_j = 0$ whenever $j > n$.
 Similarly, we write $\mathrm{T}^{>n}(V)$ for the subspace of elements 
-$\mathbf{x} = (x_0, x_1,\dots)$ where $x_j = \mathbf{0}$ whenever $j \leq n$, 
+$\mathbf{x} = (x_0, x_1,\dots)$ where $x_j = 0$ whenever $j \leq n$, 
 which is an ideal in $\mathrm{T}((V))$ and 
 $\mathrm{T}^n(V) = \mathrm{T}((V)) / \mathrm{T}^{>n}(V)$.
 The truncated tensor algebra is an algebra, when given the *truncated tensor
@@ -191,7 +190,7 @@ The multiplication on $\mathrm{Sh}((V))$ is the *shuffle product*, which
 corresponds to point-wise multiplication of functions on the path.
 
 There are several *Lie algebras* associated to $\mathrm{T}((V))$.
-Define the *Lie bracket* operation $[\mathbf{x}, \mathbf{y}] =
+Define the *Lie bracket* $[\mathbf{x}, \mathbf{y}] =
 \mathbf{x} \otimes \mathbf{y} - \mathbf{y}\otimes \mathbf{x}$, for
 $\mathbf{x},\mathbf{y}\in \mathrm{T}((V))$.
 We define subspaces $L_m$ of $\mathrm{T}((V))$ for each $m\geq 0$ inductively as
@@ -389,9 +388,8 @@ patients from intensive care unit data.
 In this challenge, a team utilising signatures to enhance predictive power
 placed first in the official phase of the challenge.
 Since then, signatures and other rough path based approaches have been used in
-several other clinical contexts 
-[@https://doi.org/10.1038/s41598-024-51989-6; @doi:10.1109/MEDAI59581.2023.00008;
-@tseriotou_etal_2024_sig].
+several other clinical contexts [@https://doi.org/10.1038/s41598-024-51989-6;
+@doi:10.1109/MEDAI59581.2023.00008; @tseriotou_etal_2024_sig].
 Clinical data is often irregularly sampled and often exhibits missing data, but
 it can also be very high-frequency and dense.
 Rough path based methods can handle these data in an elegant way, and retain the
@@ -400,12 +398,12 @@ structure of long and short term dependencies within the data.
 ### Human action recognition
 The task of identifying a specific action performed by a person from a short
 video clip is very challenging. 
-Signatures derived from landmark data extracted from the video has been
-used to train classification models that achieved state-of-the-art performance
-compared with contemporary models [@https://doi.org/10.1007/978-3-030-98519-6_18].
-This work has been continued in numerous additional papers such as 
-[@doi:10.1109/tmm.2023.3318242;@liao2021a]
-(see also preprint papers [@https://doi.org/10.48550/arXiv.2308.12840;
+Signatures derived from landmark data extracted from the video has been used to
+train classification models that achieved state-of-the-art performance compared
+with contemporary models
+[@https://doi.org/10.1007/978-3-030-98519-6_18; @doi:10.1109/tmm.2023.3318242;
+@liao2021a].
+(See also preprint papers [@https://doi.org/10.48550/arXiv.2308.12840;
 @https://doi.org/10.48550/arXiv.2403.15212].) 
 Also in the domain of computer vision, signatures have been used to produce
 lightweight models for image classification
@@ -470,7 +468,7 @@ For most applications, floating point numbers will provide a good balance
 between performance and accuracy.
 When more precision is required, rational coefficients can be used
 instead.
-These are backed by GMPLib rationals for fast, arbitrary precision rational
+These are backed by GMP rationals for fast, arbitrary precision rational
 arithmetic [@Granlund12].
 Polynomial coefficients can be used to derive formulae by performing
 calculations.
@@ -572,8 +570,8 @@ The other main `StreamInterface` implementation is the `PiecewiseAbelianStream`,
 which is an important construction from CDE.
 A piecewise Abelian path, or log-linear path, is an example of a *smooth rough
 path*, which generalises piecewise linear approximations of an arbitrary stream.
-Formally, an *Abelian path* $Y$ is a pair $([a, b), \mathbf{y})$ where $a < b$ and
-$\mathbf{x}\in\mathcal{L}((V))$.
+Formally, an *Abelian path* $Y$ is a pair $([a, b), \mathbf{y})$ where $a < b$
+and $\mathbf{y}\in\mathcal{L}((V))$.
 The log-signature over an arbitrary interval $[u, v) \subseteq [a, b)$ is given
 by
 ```{math}
@@ -586,7 +584,7 @@ is a piecewise Abelian approximation for this path given by
 ```{math}
 \{([t_{j-1}, t_j), \mathrm{LogSig}(X)_{t_{j-1}, t_j}): j=1, \dots, N\}.
 ```
-This construction turns out to be vital for computing signature kernels in
+This construction turns out to be vital for computing signature kernels 
 [@https://doi.org/10.25080/gerudo-f2bc6f59-001] and for solving CDEs
 [@http://dx.doi.org/10.1007/978-3-540-71285-5;
 @https://doi.org/10.48550/arXiv.2402.18512].

From e0b0d0dbbdedf6eacf2d4399f88fe362e25cb92a Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Tue, 21 May 2024 13:07:40 +0100
Subject: [PATCH 08/32] Update roles on submission

---
 papers/sam_morley/myst.yml | 4 ++++
 1 file changed, 4 insertions(+)

diff --git a/papers/sam_morley/myst.yml b/papers/sam_morley/myst.yml
index e1cfbc540a..2c62682cfb 100644
--- a/papers/sam_morley/myst.yml
+++ b/papers/sam_morley/myst.yml
@@ -12,6 +12,10 @@ project:
       orcid: 0000-0001-5971-7418
       affiliations: 
         - University Of Oxford
+      roles:
+        - Conceptualization
+        - Writing - original draft
+        - Software
 
   # description:
   keywords: 

From d55ac642576ab744c3ed40c5d1ffd512a5624a85 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Wed, 22 May 2024 07:44:19 +0100
Subject: [PATCH 09/32] Added TL as an author

---
 papers/sam_morley/myst.yml | 12 ++++++++++++
 1 file changed, 12 insertions(+)

diff --git a/papers/sam_morley/myst.yml b/papers/sam_morley/myst.yml
index 2c62682cfb..1fa0b2648f 100644
--- a/papers/sam_morley/myst.yml
+++ b/papers/sam_morley/myst.yml
@@ -16,9 +16,21 @@ project:
         - Conceptualization
         - Writing - original draft
         - Software
+      corresponding: true
+    - name: Terry Lyons
+      email: terry.lyons@maths.ox.ac.uk
+      orcid: 0000-0002-9972-2809
+      affiliations:
+        - University of Oxford
+      roles:
+        - Conceptualization
+        - Supervision
+        - Writing - review & editing
 
   # description:
   keywords: 
+    - sequential data
+    - unparametrised paths
     - time series 
     - rough paths
     - signatures 

From 6e5a85767ed715a4afcf0de8b30f0b01be856360 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Wed, 22 May 2024 07:44:30 +0100
Subject: [PATCH 10/32] update acknowledgements

---
 papers/sam_morley/roughpy.md | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 7e93e4f599..da73d845c0 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -23,9 +23,10 @@ acknowledgements: |
   part by The Alan Turing Institute under the EPSRC grant EP/N510129/1, the Data
   Centric Engineering Programme (under the Lloyd’s Register Foundation grant
   G0095), and the Defence and Security Programme (funded by the UK Government).
-  For the purpose of Open Access, the author has applied a CC-BY public
-  copyright licence to any Author Accepted Manuscript version arising from this
-  submission.
+  Terry Lyons was additionally supported by the Hong Kong Innovation and
+  Technology Commission (InnoHK project CIMDA). For the purpose of Open Access,
+  the author has applied a CC-BY public copyright licence to any Author Accepted
+  Manuscript version arising from this submission.
 ---
 
 ## Introduction
@@ -180,7 +181,7 @@ reparametrisation.
 
 In some sense, the signature is a *unversal non-linearity* on paths.
 Continuous functions on the path can be approximated (uniformly) by linear
-functionals ($\mathrm{G}(V)\to \mathbb{R}$) on the signature.
+functionals ($\mathrm{G}TL(V)\to \mathbb{R}$) on the signature.
 This is a consequence of the Stone-Weierstrass theorem. 
 The *dual* of $\mathrm{T}((V))$ is $\mathrm{Sh}((V)):=\mathrm{T}((V^{\ast}))
 \cong\mathrm{T}((V))$ 

From f2cbdf44e71fbc21d9c1c17149550847ab5fbc4b Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Wed, 22 May 2024 09:24:05 +0100
Subject: [PATCH 11/32] add dois for those that have them, ignore errors on the
 rest

---
 papers/sam_morley/myst.yml    | 13 ++++++++-----
 papers/sam_morley/roughpy.bib | 15 ++++++++++-----
 2 files changed, 18 insertions(+), 10 deletions(-)

diff --git a/papers/sam_morley/myst.yml b/papers/sam_morley/myst.yml
index 1fa0b2648f..43e0857dbc 100644
--- a/papers/sam_morley/myst.yml
+++ b/papers/sam_morley/myst.yml
@@ -54,11 +54,14 @@ project:
     - rule: doi-exists
       severity: ignore
       keys:
-        - Atr03
-        - terradesert
-        - jupyter
-        - sklearn1
-        - sklearn2
+        - Lyons1998
+        - esig
+        - dlpack
+        - pybind11
+        - coropa_project
+        - Granlund12
+        - NEURIPS2021_18a9042b
+ 
   # A banner will be generated for you on publication, this is a placeholder
   banner: banner.png
   # The rest of the information shouldn't be modified
diff --git a/papers/sam_morley/roughpy.bib b/papers/sam_morley/roughpy.bib
index 923a508ecd..3e9b58c65b 100644
--- a/papers/sam_morley/roughpy.bib
+++ b/papers/sam_morley/roughpy.bib
@@ -17,6 +17,7 @@ @inproceedings{tseriotou_etal_2024_sig
     url = "https://aclanthology.org/2024.eacl-demo.24",
     pages = "223--237",
     abstract = "We present an open-source, pip installable toolkit, Sig-Networks, the first of its kind for longitudinal language modelling. A central focus is the incorporation of Signature-based Neural Network models, which have recently shown success in temporal tasks. We apply and extend published research providing a full suite of signature-based models. Their components can be used as PyTorch building blocks in future architectures. Sig-Networks enables task-agnostic dataset plug-in, seamless preprocessing for sequential data, parameter flexibility, automated tuning across a range of models. We examine signature networks under three different NLP tasks of varying temporal granularity: counselling conversations, rumour stance switch and mood changes in social media threads, showing SOTA performance in all three, and provide guidance for future tasks. We release the Toolkit as a PyTorch package with an introductory video, Git repositories for preprocessing and modelling including sample notebooks on the modeled NLP tasks.",
+    doi = "10.48550/arXiv.2312.03523",
 }
 
 @inproceedings{NEURIPS2020_4a5876b4,
@@ -28,7 +29,8 @@ @inproceedings{NEURIPS2020_4a5876b4
  title = {Neural Controlled Differential Equations for Irregular Time Series},
  url = {https://proceedings.neurips.cc/paper_files/paper/2020/file/4a5876b450b45371f6cfe5047ac8cd45-Paper.pdf},
  volume = {33},
- year = {2020}
+ year = {2020},
+ doi = "10.48550/arXiv.1906.08215",
 }
 
 @article{Lyons1998,
@@ -58,7 +60,8 @@ @inproceedings{liao2021a
   editor = {},
   year = {2021},
   organizer = {32nd British Machine Vision Conference (BMVC 2021)},
- series = {}
+ series = {},
+ doi = "10.48550/arXiv.2110.13008",
 }
 
 @inproceedings{NEURIPS2021_18a9042b,
@@ -70,7 +73,7 @@ @inproceedings{NEURIPS2021_18a9042b
  title = {Framing RNN as a kernel method: A neural ODE approach},
  url = {https://proceedings.neurips.cc/paper_files/paper/2021/file/18a9042b3fc5b02fe3d57fea87d6992f-Paper.pdf},
  volume = {34},
- year = {2021}
+ year = {2021},
 }
 
 @article{JMLR_v20_16_314,
@@ -81,7 +84,8 @@ @article{JMLR_v20_16_314
   volume  = {20},
   number  = {31},
   pages   = {1--45},
-  url     = {http://jmlr.org/papers/v20/16-314.html}
+  url     = {http://jmlr.org/papers/v20/16-314.html},
+  doi = "10.48550/arXiv.2102.03657",
 }
 
 @misc{pybind11,
@@ -116,7 +120,8 @@ @inproceedings{NEURIPS2019_d2cdf047
  title = {Deep Signature Transforms},
  url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/d2cdf047a6674cef251d56544a3cf029-Paper.pdf},
  volume = {32},
- year = {2019}
+ year = {2019},
+ doi = "10.48550/arXiv.1905.08494"
 }
 
 @misc{dlpack, 

From d33e15fbeb153ac90ded72fdb35dc4a856aaadde Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 25 Jul 2024 07:44:33 +0100
Subject: [PATCH 12/32] Fixed typo and added comment about installing roughpy.

---
 papers/sam_morley/roughpy.md | 14 +++++++++++++-
 1 file changed, 13 insertions(+), 1 deletion(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index da73d845c0..d4e5bf818b 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -6,6 +6,13 @@ bibliography:
 numbering:
     heading_2: true
     heading_3: true
+exports:
+  - format: pdf
+    template: arxiv_two_column
+    article_type: Article
+  - format: tex
+    template: arxiv_two_column
+    article_type: Article
 abstract: |
   Rough path theory is a branch of mathematics arising out of stochastic
   analysis. One of the main tools of rough path analysis is the signature,
@@ -35,7 +42,7 @@ records, radio (and other electromagnetic spectra), sound (and speech), etc.
 Traditionally, these data are tricky to work with because of the exponential
 complexity and different scales of the underlying process.
 Until recently, with the development of transformers and large language models,
-it has been difficult to capture the long-term pattern whist also capturing the
+it has been difficult to capture the long-term pattern whilst also capturing the
 short-term fine detail.
 Rough path theory gives us tools to work with sequential, ordered data in a
 mathematically rigorous way, which should provide a means to overcome some of
@@ -597,6 +604,11 @@ In this section we show a very simple example of how to use RoughPy to construct
 a stream and compute a signature.
 This example is similar to the first few steps of the tutorial found in the
 RoughPy documentation.[^roughpydocs]
+RoughPy can be installed using `pip`, where prebuilt wheels are available for
+Windows, Linux, and MacOs:
+```
+pip install roughpy
+```
 We refer the reader to this documentation for much more detail.
 We will construct a stream in $\mathbb{R}^{26}$ by taking each letter in a word,
 "scipy" in this example, as the increments of a path:

From 2b6beee08c00ba22324bad6771345e5c3fa2eeaf Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 25 Jul 2024 07:52:35 +0100
Subject: [PATCH 13/32] fixed notation for shuffle tensor and free tensor

---
 papers/sam_morley/roughpy.md | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index d4e5bf818b..e3b4a039c7 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -188,13 +188,13 @@ reparametrisation.
 
 In some sense, the signature is a *unversal non-linearity* on paths.
 Continuous functions on the path can be approximated (uniformly) by linear
-functionals ($\mathrm{G}TL(V)\to \mathbb{R}$) on the signature.
+functionals ($\mathrm{G}(V)\to \mathbb{R}$) on the signature.
 This is a consequence of the Stone-Weierstrass theorem. 
-The *dual* of $\mathrm{T}((V))$ is $\mathrm{Sh}((V)):=\mathrm{T}((V^{\ast}))
+The *dual* of $\mathrm{T}((V))$ is $\mathrm{Sh}(V):=\mathrm{T}((V^{\ast}))
 \cong\mathrm{T}((V))$ 
 (as a vector space) is the space of these linear functionals on
 $\mathrm{T}((V))$; we call this the *shuffle algebra*.
-The multiplication on $\mathrm{Sh}((V))$ is the *shuffle product*, which
+The multiplication on $\mathrm{Sh}(V)$ is the *shuffle product*, which
 corresponds to point-wise multiplication of functions on the path.
 
 There are several *Lie algebras* associated to $\mathrm{T}((V))$.
@@ -207,10 +207,10 @@ follows: $L_0 = \{\mathbf{0}\}$, $L_1 = V$, and, for $m \geq 1$,
 L_{m+1} = \mathrm{span}\{[\mathbf{x}, \mathbf{y}] : \mathbf{x}\in V, \mathbf{y}
 \in L_m\}
 ```
-The space of formal Lie series $\mathcal{L}((V))$ over $V$ is the subspace of 
+The space of formal Lie series $\mathcal{L}(V)$ over $V$ is the subspace of 
 $\mathrm{T}((V))$ containing sequences of the form $(\ell_0, \ell_1, \cdots)$, 
 where $\ell_j\in L_j$ for each $j\geq 0$.
-Note that $\mathcal{L}((V))\subseteq \mathrm{T}^{>0}(V)$.
+Note that $\mathcal{L}(V)\subseteq \mathrm{T}^{>0}(V)$.
 For any $\mathbf{x} \in \mathrm{T}(V)$ we define 
 ```{math}
 \exp(\mathbf{x}) = \sum_{n=0}^\infty \frac{\mathbf{x}^{\otimes n}}{n!}
@@ -218,8 +218,8 @@ For any $\mathbf{x} \in \mathrm{T}(V)$ we define
 \log(\mathbf{1} + \mathbf{x}) = \sum_{n=1}^\infty
 \frac{(-1)^{n-1}}{n}\mathbf{x}^{\otimes n}.
 ```
-For any path $X$, we have $\log(\mathrm{S}(X)_{s, t})\in \mathcal{L}((V))$, and,
-more generally we have $\exp(\mathcal{L}((V))) = \mathrm{G}(V)$.
+For any path $X$, we have $\log(\mathrm{S}(X)_{s, t})\in \mathcal{L}(V)$, and,
+more generally we have $\exp(\mathcal{L}(V)) = \mathrm{G}(V)$.
 We call the quantity $\mathrm{LogSig}(X)_{s, t} := \log(\mathrm{S}(X)_{s,t})$
 the *log-signature* of $X$ over $[s, t)$.
 
@@ -230,7 +230,7 @@ Our first task is to form a bridge between sequential data and paths.
 Consider a finite, ordered sequence $\{(t_1, \mathbf{x}_1,\dots,
 t_N,\mathbf{x}_N)\}$ of observations, where $t_j\in \mathbb{R}$, and
 $\mathbf{x}_j\in V$.
-(More generally, we might consider $\mathbf{x}_j\in\mathcal{L}((V))$ instead.
+(More generally, we might consider $\mathbf{x}_j\in\mathcal{L}(V)$ instead.
 That is, data that already contains higher-order information.)
 We can find numerous paths that interpolate these observations; a path 
 $X:[t_0, t_N]\to V$ such that, for each $j$, $X_{t_j} = \mathbf{x}_j$.
@@ -579,7 +579,7 @@ which is an important construction from CDE.
 A piecewise Abelian path, or log-linear path, is an example of a *smooth rough
 path*, which generalises piecewise linear approximations of an arbitrary stream.
 Formally, an *Abelian path* $Y$ is a pair $([a, b), \mathbf{y})$ where $a < b$
-and $\mathbf{y}\in\mathcal{L}((V))$.
+and $\mathbf{y}\in\mathcal{L}(V)$.
 The log-signature over an arbitrary interval $[u, v) \subseteq [a, b)$ is given
 by
 ```{math}

From 48b767319acc996bc5d3a494d04357d708d3d519 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 25 Jul 2024 08:00:13 +0100
Subject: [PATCH 14/32] fixed naming of the lie algebra

---
 papers/sam_morley/roughpy.md | 13 ++++++-------
 1 file changed, 6 insertions(+), 7 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index e3b4a039c7..614e95f733 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -110,7 +110,7 @@ the understanding of path-like data analysis.
 
 On top of the streams, RoughPy also provides concrete implementations for
 elements of the various algebras associated with rough path analysis.
-These include free tensor algebras, shuffle tensor algebras, and free Lie
+These include free tensor algebras, shuffle tensor algebras, and Lie
 algebras [see @math-bgd-sec].
 This allows the user to easily manipulate signatures, and other objects, in a
 more natural manner.
@@ -190,10 +190,9 @@ In some sense, the signature is a *unversal non-linearity* on paths.
 Continuous functions on the path can be approximated (uniformly) by linear
 functionals ($\mathrm{G}(V)\to \mathbb{R}$) on the signature.
 This is a consequence of the Stone-Weierstrass theorem. 
-The *dual* of $\mathrm{T}((V))$ is $\mathrm{Sh}(V):=\mathrm{T}((V^{\ast}))
-\cong\mathrm{T}((V))$ 
-(as a vector space) is the space of these linear functionals on
-$\mathrm{T}((V))$; we call this the *shuffle algebra*.
+The *dual* of $\mathrm{T}((V))$ is $\mathrm{Sh}(V)$ is the space of 
+these linear functionals on $\mathrm{T}((V))$; we call this the *shuffle 
+algebra*. 
 The multiplication on $\mathrm{Sh}(V)$ is the *shuffle product*, which
 corresponds to point-wise multiplication of functions on the path.
 
@@ -458,7 +457,7 @@ code based on mathematical concepts, we provide realisations of several
 algebra types.
 The algebras provided in RoughPy are `FreeTensor`, `ShuffleTensor`, and `Lie`,
 which define elements of a particular free tensor algebra, shuffle tensor
-algebra, and free Lie algebra respectively.
+algebra, and Lie algebra respectively.
 Each of these algebras is initialized with a width, depth, and scalar
 coefficient type, encapsulated in a `Context` object.
 
@@ -706,7 +705,7 @@ It is designed to abstract away the form and source of data so that analysis can
 be performed by querying path-like objects.
 This approach is much closer to the mathematics.
 It also allows users to interact with the various algebras associated with rough
-paths (free tensor algebra, shuffle tensor algebra, free Lie algebra) in a
+paths (free tensor algebra, shuffle tensor algebra, Lie algebra) in a
 natural way.
 RoughPy is under active development, and a long list of improvements and
 extensions are planned.

From ad3d3dbff9e7d0811d07a5df48da7f1912b3fc97 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 10:06:49 +0100
Subject: [PATCH 15/32] fixed typo

---
 papers/sam_morley/roughpy.md | 87 +++++++++++++++++++-----------------
 1 file changed, 46 insertions(+), 41 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 614e95f733..0ad543f29f 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -54,7 +54,7 @@ analyses and explore different ways to understand sequential data.
 Rough paths arise in the study of *controlled differential equations* (CDEs),
 which generalise ordinary differential equations (ODEs) and stochastic
 differential equations
-[@Lyons1998;@http://dx.doi.org/10.1007/978-3-540-71285-5].
+[@Lyons1998;@10.1007/978-3-540-71285-5].
 These are equations of the form $\mathrm{d}Y_t = f(Y_t, \mathrm{d}X_t)$, subject 
 to an initial condition $Y_0 = y_0$, that model a non-linear system driven by a 
 input path $X$.
@@ -90,8 +90,8 @@ of applications of rough path-based methods.
 We give a short overview of these methods in @rp-in-ds-sec.
 
 There are several Python packages for computing signatures of sequential data, 
-including `esig` [@esig], `iisignature` [@doi.org/10.1145/3371237], and 
-`signatory` [@https://doi.org/10.48550/arXiv.2001.00706].
+including `esig` [@esig], `iisignature` [@10.1145/3371237], and 
+`signatory` [@10.48550/arXiv.2001.00706].
 These packages provide functions for computing signatures from raw, 
 structured data presented in an $n\times d$ array, where $d$ is the dimension 
 of the stream and $n$ is the number of samples.
@@ -111,7 +111,7 @@ the understanding of path-like data analysis.
 On top of the streams, RoughPy also provides concrete implementations for
 elements of the various algebras associated with rough path analysis.
 These include free tensor algebras, shuffle tensor algebras, and Lie
-algebras [see @math-bgd-sec].
+algebras ([see Section %s](#math-bgd-sec)).
 This allows the user to easily manipulate signatures, and other objects, in a
 more natural manner.
 This allows us to quickly develop methods by following the mathematics.
@@ -125,13 +125,15 @@ to motivate the development of RoughPy. Finally, in @roughpy-sec we give a more
 detailed overview of the RoughPy library, the types and functions it contains,
 and give an example of how it can be used.
 
+RoughPy is open source (BSD 3-Clause) and available on GitHub
+[https://github.com/datasig-ac-uk/roughpy](https://github.com/datasig-ac-uk/roughpy).
 
 (math-bgd-sec)=
 ### Mathematical background
 In this section we give a very short introduction to signatures and rough path 
 theory that should be sufficient to inform the discussion in the sequel.
 For a far more comprehensive and rigorous treatment, we refer the reader to the 
-recent survey [@https://doi.org/10.48550/arXiv.2206.14674].
+recent survey [@10.48550/arXiv.2206.14674].
 In the sequel, we write $V$ for the vector space
 $\mathbb{R}^d$, where $d \geq 1$.
 
@@ -182,19 +184,22 @@ The signature respects concatenation of paths, meaning
 $\mathrm{S}(X)_{s, t} = \mathrm{S}(X)_{s, u} \otimes \mathrm{S}(X)_{u, t}$ 
 for any $s < u < t$.
 Two paths have the same signature if and only if they differ by a 
-*tree-like path* [@http://dx.doi.org/10.4007/annals.2010.171.109].
+*tree-like path* [@10.4007/annals.2010.171.109].
 The signature is translation invariant, and it is invariant under 
 reparametrisation.
 
-In some sense, the signature is a *unversal non-linearity* on paths.
-Continuous functions on the path can be approximated (uniformly) by linear
-functionals ($\mathrm{G}(V)\to \mathbb{R}$) on the signature.
-This is a consequence of the Stone-Weierstrass theorem. 
-The *dual* of $\mathrm{T}((V))$ is $\mathrm{Sh}(V)$ is the space of 
-these linear functionals on $\mathrm{T}((V))$; we call this the *shuffle 
-algebra*. 
+The *dual* of $\mathrm{T}((V))$ is the 
+*shuffle algebra* $\mathrm{Sh}(V)$.
+This is the space of linear functionals $\mathrm{T}((V))\to \mathbb{R}$
+and consists of sequences $(\lambda_0, \lambda_1, \dots)$ with
+$\lambda_k\in (V^{\ast})^{\otimes k}$ and where $\lambda_k = 0$ for all $k$
+larger than some $N$.
 The multiplication on $\mathrm{Sh}(V)$ is the *shuffle product*, which
 corresponds to point-wise multiplication of functions on the path.
+Continuous functions on the path can be approximated (uniformly) by shuffle
+tensors acting on $\mathrm{G}(V)$ on the signature.
+This is a consequence of the Stone-Weierstrass theorem. 
+This property is sometimes referred to as *universal non-lineararity*.
 
 There are several *Lie algebras* associated to $\mathrm{T}((V))$.
 Define the *Lie bracket* $[\mathbf{x}, \mathbf{y}] =
@@ -204,7 +209,7 @@ We define subspaces $L_m$ of $\mathrm{T}((V))$ for each $m\geq 0$ inductively as
 follows: $L_0 = \{\mathbf{0}\}$, $L_1 = V$, and, for $m \geq 1$, 
 ```{math}
 L_{m+1} = \mathrm{span}\{[\mathbf{x}, \mathbf{y}] : \mathbf{x}\in V, \mathbf{y}
-\in L_m\}
+\in L_m\}.
 ```
 The space of formal Lie series $\mathcal{L}(V)$ over $V$ is the subspace of 
 $\mathrm{T}((V))$ containing sequences of the form $(\ell_0, \ell_1, \cdots)$, 
@@ -217,10 +222,11 @@ For any $\mathbf{x} \in \mathrm{T}(V)$ we define
 \log(\mathbf{1} + \mathbf{x}) = \sum_{n=1}^\infty
 \frac{(-1)^{n-1}}{n}\mathbf{x}^{\otimes n}.
 ```
-For any path $X$, we have $\log(\mathrm{S}(X)_{s, t})\in \mathcal{L}(V)$, and,
-more generally we have $\exp(\mathcal{L}(V)) = \mathrm{G}(V)$.
-We call the quantity $\mathrm{LogSig}(X)_{s, t} := \log(\mathrm{S}(X)_{s,t})$
-the *log-signature* of $X$ over $[s, t)$.
+For any path $X$, we have 
+$\mathrm{LogSig}(X)_{s, t} := \log(\mathrm{S}(X)_{s, t})\in \mathcal{L}(V)$, 
+and we call this the *log-signature* of $X$ over $[s, t)$.
+This is an alternative representation of the path, but doesn't enjoy the same
+unievrsal non-linearity of the signature.
 
 (rp-in-ds-sec)=
 ### Rough paths in data science 
@@ -240,7 +246,7 @@ Once we have a path, we need to be able to compute signatures.
 For practical purposes, we truncate all signatures (and log-signatures) to a
 particular degree $M$, which we typically call the *depth*.
 The dimension of the ambient space $d$ is usually called the *width*.
-Using the linear interpolation, we can replace the iterated integrals with a
+Using linear interpolation, we can replace the iterated integrals with a
 tensor exponential of the difference of successive terms:
 ```{math}
 \mathrm{Sig}^M([t_j, t_{j+1})) = \exp_M(\mathbf{x}_{j+1} - \mathbf{x}_j) :=
@@ -303,12 +309,12 @@ Even better, it turns out that, in certain cases, the signature kernel
 can be realised as  the solution to a partial differential equation (PDE) of 
 Goursat type.
 This means the full signature kernel can be computed from raw data without 
-needing to compute full signatures [@https://doi.org/10.1137/20M1366794].
+needing to compute full signatures [@10.1137/20M1366794].
 
 In fact, in recent preprint, it has been shown that there are
 higher order solvers for signature kernels by rewriting the kernel 
 solution of a system of PDEs of Goursat type 
-[@https://doi.org/10.48550/arXiv.2404.02926].
+[@10.48550/arXiv.2404.02926].
 A critical part of their method involves the adjoint of both left and right
 free tensor multiplication, which are not available in any current package
 for computing signatures.
@@ -330,7 +336,7 @@ network [@NEURIPS2020_4a5876b4].
 Neural CDEs initially showed some promising results on several benchmarks but
 now lag behind current state-of-the-art approaches to time series modelling.
 The latest iteration of neural CDEs are the recently introduced Log-neural
-controlled differential equations [@https://doi.org/10.48550/arXiv.2402.18512],
+controlled differential equations [@10.48550/arXiv.2402.18512],
 which make use of the *Log-ODE* method for solving rough differential equations
 in order to boost the performance of neural CDEs.
 
@@ -356,7 +362,7 @@ radio telescopes [@doi:10.48550/arXiv.2402.14892].
 Their result show that their framework is effective at detecting even faint RFI
 within the test data.
 This work is based on a general novelty detection framework
-[@https://doi.org/10.48550/arXiv.2006.03487].
+[@10.48550/arXiv.2006.03487].
 
 Signatures kernels have also been used for a similar problem of detecting
 malware by inspecting the streaming tree of processes on a computer system 
@@ -367,7 +373,7 @@ data.
 
 ### Tracking mood via natural language processing
 One application of rough paths in natural language processing has been in the
-domain of mental health [@http://dx.doi.org/10.18653/v1/2023.findings-acl.310;
+domain of mental health [@10.18653/v1/2023.findings-acl.310;
 @tseriotou_etal_2024_sig].
 In this work, the authors present a model for identifying changes in a person's 
 mood based on their online textual content. 
@@ -379,7 +385,7 @@ datasets.
 
 ### Predicting battery cell degradation
 Another recent application of signatures is to predict the degradation of
-lithium-ion cells [@http://doi.org/10.1016/j.apenergy.2023.121974].
+lithium-ion cells [@10.1016/j.apenergy.2023.121974].
 They use signature features to train a model that can accurately predict the end
 of life of a cell using relatively low-frequency sampling compared to existing
 models.
@@ -389,13 +395,13 @@ other models.
 ### Prediction of sepsis in intensive care data
 One of the first effective demonstrations of the utility of signatures and rough
 paths based methods in healthcare was in the 2019 PhysioNet challenge
-[@https://doi.org/10.1097/CCM.0000000000004510].
+[@10.1097/CCM.0000000000004510].
 In this contest, teams were invited to develop models to predict sepsis in
 patients from intensive care unit data.
 In this challenge, a team utilising signatures to enhance predictive power
 placed first in the official phase of the challenge.
 Since then, signatures and other rough path based approaches have been used in
-several other clinical contexts [@https://doi.org/10.1038/s41598-024-51989-6;
+several other clinical contexts [@10.1038/s41598-024-51989-6;
 @doi:10.1109/MEDAI59581.2023.00008; @tseriotou_etal_2024_sig].
 Clinical data is often irregularly sampled and often exhibits missing data, but
 it can also be very high-frequency and dense.
@@ -408,14 +414,14 @@ video clip is very challenging.
 Signatures derived from landmark data extracted from the video has been used to
 train classification models that achieved state-of-the-art performance compared
 with contemporary models
-[@https://doi.org/10.1007/978-3-030-98519-6_18; @doi:10.1109/tmm.2023.3318242;
+[@10.1007/978-3-030-98519-6_18; @doi:10.1109/tmm.2023.3318242;
 @liao2021a].
-(See also preprint papers [@https://doi.org/10.48550/arXiv.2308.12840;
-@https://doi.org/10.48550/arXiv.2403.15212].) 
+(See also preprint papers [@10.48550/arXiv.2308.12840;
+@10.48550/arXiv.2403.15212].) 
 Also in the domain of computer vision, signatures have been used to produce
 lightweight models for image classification
-[@https://doi.org/10.1109/CVPRW56347.2022.00409] and in handwriting recognition
-tasks [@https://doi.org/10.1109/TPAMI.2017.2732978].
+[@10.1109/CVPRW56347.2022.00409] and in handwriting recognition
+tasks [@10.1109/TPAMI.2017.2732978].
 
 (roughpy-sec)=
 ## RoughPy
@@ -436,7 +442,7 @@ The first two objectives are simple design and implementation problems.
 The final objective presents the most difficulty, especially interoperability
 between RoughPy and common machine learning libraries.
 There are array interchange formats for NumPy-like arrays, such as the Python
-Array API standard [@https://doi.org/10.25080/gerudo-f2bc6f59-001] and the 
+Array API standard [@10.25080/gerudo-f2bc6f59-001] and the 
 DLPack protocol [@dlpack].
 These provide part of the picture, but in order for them to be fully supported,
 RoughPy must support a variety of compute backends such as CUDA (NVidia), 
@@ -452,7 +458,7 @@ In the remainder of this section, we discuss some of the core components of
 RoughPy, give an example of using RoughPy, and discuss the future of RoughPy.
 
 ### Free tensors, shuffle tensors, and Lie objects
-In order to property support rough path based methods and allow users to write
+In order to properly support rough path based methods and allow users to write
 code based on mathematical concepts, we provide realisations of several
 algebra types.
 The algebras provided in RoughPy are `FreeTensor`, `ShuffleTensor`, and `Lie`,
@@ -592,9 +598,9 @@ is a piecewise Abelian approximation for this path given by
 \{([t_{j-1}, t_j), \mathrm{LogSig}(X)_{t_{j-1}, t_j}): j=1, \dots, N\}.
 ```
 This construction turns out to be vital for computing signature kernels 
-[@https://doi.org/10.25080/gerudo-f2bc6f59-001] and for solving CDEs
-[@http://dx.doi.org/10.1007/978-3-540-71285-5;
-@https://doi.org/10.48550/arXiv.2402.18512].
+[@10.25080/gerudo-f2bc6f59-001] and for solving CDEs
+[@10.1007/978-3-540-71285-5;
+@10.48550/arXiv.2402.18512].
 In particular, this construction can be used to compress data at some degree,
 which can the be used in computations at a higher degree.
 
@@ -627,11 +633,10 @@ level (2), and coefficient type (`Rational`).
 ```python
 import roughpy as rp
 
-ctx = rp.get_context(width=26, 
-                     depth=2, 
-                     coeffs=rp.Rational)
+ctx = rp.get_context(width=26, depth=2, 
+      coeffs=rp.Rational)
 stream = rp.LieIncrementStream.from_increments(
-  increments, ctx=ctx)
+         increments, ctx=ctx)
 ```
 Now we can compute the signature of the stream over the whole domain of the
 stream $[0, 4]$ by omitting the interval argument:

From bfde2635d390b29d0b23b1828e68e0a183f2cc66 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 10:08:38 +0100
Subject: [PATCH 16/32] added missing period at end of paragraph

---
 papers/sam_morley/roughpy.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 0ad543f29f..1a068ee505 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -96,7 +96,7 @@ These packages provide functions for computing signatures from raw,
 structured data presented in an $n\times d$ array, where $d$ is the dimension 
 of the stream and $n$ is the number of samples.
 This means the user is responsible for interpreting the data as a path and 
-arranging the computations that need to be done
+arranging the computations that need to be done.
 
 RoughPy is a new package for working with sequential data and rough paths.
 The design philosophy for this package is to shift the emphasis from simply

From df1f072dcab60b126fd88cb5e980c57e317c514b Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 10:10:16 +0100
Subject: [PATCH 17/32] fixed typo in 'one simple'

---
 papers/sam_morley/roughpy.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 1a068ee505..dc1026a63c 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -58,7 +58,7 @@ differential equations
 These are equations of the form $\mathrm{d}Y_t = f(Y_t, \mathrm{d}X_t)$, subject 
 to an initial condition $Y_0 = y_0$, that model a non-linear system driven by a 
 input path $X$.
-On simple CDE turns out to be critical to the theory:
+One simple CDE turns out to be critical to the theory:
 ```{math}
 \mathrm{d}S_t = S_t \otimes \mathrm{d}X_t \qquad S_0 = \mathbf{1}.
 ```

From d6e41e9114e2d922adb1ed9090636753aa4a2647 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 10:12:41 +0100
Subject: [PATCH 18/32] fixed linebreak before period

---
 papers/sam_morley/roughpy.md | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index dc1026a63c..dc5fcb4826 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -321,8 +321,7 @@ for computing signatures.
 
 (ncde-sec)=
 #### Neural controlled differential equations
-Neural CDEs are a method for modelling irregular time series
-.
+Neural CDEs are a method for modelling irregular time series.
 We consider CDEs of the form
 ```{math}
 \mathrm{d}Y_t = f_\theta(Y_t)\,\mathrm{d}X_t

From e622a79a319fcccb07b3abe07710f917485d0dda Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 10:14:34 +0100
Subject: [PATCH 19/32] clarify the use of floats and defaults

---
 papers/sam_morley/roughpy.md | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index dc5fcb4826..0c9547131d 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -476,8 +476,9 @@ signature kernels).
 RoughPy algebras are designed around a flexible scalar ring system that allows
 users to perform calculations with different accuracy, or derive expressions by
 using polynomial coefficients.
-For most applications, floating point numbers will provide a good balance
-between performance and accuracy.
+For most applications, single or double precision floating point numbers will 
+provide a good balance between performance and accuracy.
+(Double precision floats are the default.)
 When more precision is required, rational coefficients can be used
 instead.
 These are backed by GMP rationals for fast, arbitrary precision rational

From 4dcfc11b70a496f3cc12c5e2935e058aaa68dbf6 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 10:19:55 +0100
Subject: [PATCH 20/32] change the wording to remove 'in the sequel'

---
 papers/sam_morley/roughpy.md | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 0c9547131d..b409c42f77 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -134,7 +134,7 @@ In this section we give a very short introduction to signatures and rough path
 theory that should be sufficient to inform the discussion in the sequel.
 For a far more comprehensive and rigorous treatment, we refer the reader to the 
 recent survey [@10.48550/arXiv.2206.14674].
-In the sequel, we write $V$ for the vector space
+For the remainder of the paper, we write $V$ for the vector space 
 $\mathbb{R}^d$, where $d \geq 1$.
 
 A *path* in $V$ is a continuous function $X:[a, b] \to V$, where $a < b$ are
@@ -188,8 +188,7 @@ Two paths have the same signature if and only if they differ by a
 The signature is translation invariant, and it is invariant under 
 reparametrisation.
 
-The *dual* of $\mathrm{T}((V))$ is the 
-*shuffle algebra* $\mathrm{Sh}(V)$.
+The *dual* of $\mathrm{T}((V))$ is the *shuffle algebra* $\mathrm{Sh}(V)$.
 This is the space of linear functionals $\mathrm{T}((V))\to \mathbb{R}$
 and consists of sequences $(\lambda_0, \lambda_1, \dots)$ with
 $\lambda_k\in (V^{\ast})^{\otimes k}$ and where $\lambda_k = 0$ for all $k$
@@ -202,8 +201,9 @@ This is a consequence of the Stone-Weierstrass theorem.
 This property is sometimes referred to as *universal non-lineararity*.
 
 There are several *Lie algebras* associated to $\mathrm{T}((V))$.
-Define the *Lie bracket* $[\mathbf{x}, \mathbf{y}] =
-\mathbf{x} \otimes \mathbf{y} - \mathbf{y}\otimes \mathbf{x}$, for
+Define a *Lie bracket* operation on $\mathrm{T}((V))$ by the formula 
+$[\mathbf{x}, \mathbf{y}] = \mathbf{x} \otimes \mathbf{y} - 
+\mathbf{y}\otimes \mathbf{x}$, for
 $\mathbf{x},\mathbf{y}\in \mathrm{T}((V))$.
 We define subspaces $L_m$ of $\mathrm{T}((V))$ for each $m\geq 0$ inductively as
 follows: $L_0 = \{\mathbf{0}\}$, $L_1 = V$, and, for $m \geq 1$, 

From 4216b5778a4c81b02bcd6a51b6062847ef4115d5 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 10:31:04 +0100
Subject: [PATCH 21/32] changed the wording from 'speed' to 'rate'

---
 papers/sam_morley/roughpy.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index b409c42f77..d8ef0c52dc 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -73,7 +73,7 @@ can be used in place of the iterated integrals.
 A path equipped with this higher order data is called a *rough path*.
 
 The signature turns out to be a useful summary of sequential data.
-It captures the order of events but not the necessarily the speed at which 
+It captures the order of events but not the necessarily the rate at which 
 these events occur.
 The signature is robust to irregular sampling and provides a fixed-size view of
 the data, regardless of how many observations are used to compute it.

From a7fffdf6b49f6221da65cf6443943e74c1f85f57 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 10:51:10 +0100
Subject: [PATCH 22/32] remove semicolon

---
 papers/sam_morley/roughpy.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index d8ef0c52dc..20ce462527 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -642,7 +642,7 @@ Now we can compute the signature of the stream over the whole domain of the
 stream $[0, 4]$ by omitting the interval argument:
 ```python
 sig = stream.signature()
-print(sig);
+print(sig)
 # { 1() 1(3) 1(9) 1(16) 1(19) 1(25) 1/2(3,3) 
 #   1(3,9) 1(3,16) 1(3,25) 1/2(9,9) 1(9,16) 
 #   1(9,25) 1/2(16,16) 1(16,25) 1(19,3) 1(19,9) 

From 22c67e0260eade1ecf7099353218c05a4e3a8715 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Thu, 1 Aug 2024 11:08:28 +0100
Subject: [PATCH 23/32] added paragraph about applications on datasig

---
 papers/sam_morley/roughpy.md | 7 +++++++
 1 file changed, 7 insertions(+)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 20ce462527..dc0d37e15d 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -656,6 +656,13 @@ The final terms indicate the order in which each pair of letters appear in
 the word.
 For instance, the term `1(3,9)` indicates that a `c` appears before an `i`.
 
+This is only the beginning of the story. 
+From here, we can use the signatures to compute the similarity between streams,
+via the signature kernel for instance, or used as features in a variety of
+machine learning problems.
+More detailed examples of how to use signatures in data science are given on the
+DataSig website [https://datasig.ac.uk/examples](https://datasig.ac.uk/examples).
+
 :::{prf:remark} 
 :nonumber:
 It turns out that most words in the English language can be distinguished 

From 65ad5e3e4ab4111f26b48bd4b3e2bcedc82a643b Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:09:20 +0100
Subject: [PATCH 24/32] Fixed case of "of"

---
 papers/sam_morley/myst.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/papers/sam_morley/myst.yml b/papers/sam_morley/myst.yml
index 43e0857dbc..5bf779a875 100644
--- a/papers/sam_morley/myst.yml
+++ b/papers/sam_morley/myst.yml
@@ -11,7 +11,7 @@ project:
       email: sam.morley@maths.ox.ac.uk
       orcid: 0000-0001-5971-7418
       affiliations: 
-        - University Of Oxford
+        - University of Oxford
       roles:
         - Conceptualization
         - Writing - original draft

From 47ecd3469185a9e3021e2e57ef121ff1871b5f24 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:11:21 +0100
Subject: [PATCH 25/32] fix reference to later section

---
 papers/sam_morley/roughpy.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index dc0d37e15d..6e469e1ae8 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -80,7 +80,7 @@ the data, regardless of how many observations are used to compute it.
 This means the signature can be a useful feature map to be used in machine
 learning for sequential data.
 There are numerous examples of using signatures of analysing sequential data
-outlined in @applications-sec.
+outlined in [Section %s](#applications-sec).
 
 Besides signatures, there are two other rough path-based methods that have found
 their way into data science in recent years.

From 41b5d0a4323385910b16effb829f749a0fd06144 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:11:52 +0100
Subject: [PATCH 26/32] remove surplus comma

---
 papers/sam_morley/roughpy.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 6e469e1ae8..3970afb593 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -21,7 +21,7 @@ abstract: |
   applications involving sequential data. RoughPy is our new Python package
   that aims change the way we think about sequential streamed data, by viewing
   it through the lens of rough paths. In RoughPy, data is wrapped in a stream
-  object, which can be composed and queried to obtain signatures that can be
+  object which can be composed and queried to obtain signatures that can be
   used in analysis. It also provides a platform for further exploration of the
   connections between rough path theory and data science.
 

From 97fc6ce1003ebda84d7cbe7fea639e4234c9734e Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:19:01 +0100
Subject: [PATCH 27/32] Fix a few issues in mathematical introduction

---
 papers/sam_morley/roughpy.md | 10 ++++++----
 1 file changed, 6 insertions(+), 4 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 3970afb593..4700f593e1 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -163,10 +163,10 @@ The members of $\mathrm{G}(V)$ are called *group-like* elements.
 For each $n \geq 0$, we write $\mathrm{T}^n(V)$ for the *truncated tensor
 algebra* of degree $n$, which is the space of all $\mathbf{x} = (x_0, x_1,
 \dots)$ such that $x_j = 0$ whenever $j > n$.
-Similarly, we write $\mathrm{T}^{>n}(V)$ for the subspace of elements 
+Similarly, we write $\mathrm{T}^{>n}((V))$ for the subspace of elements 
 $\mathbf{x} = (x_0, x_1,\dots)$ where $x_j = 0$ whenever $j \leq n$, 
 which is an ideal in $\mathrm{T}((V))$ and 
-$\mathrm{T}^n(V) = \mathrm{T}((V)) / \mathrm{T}^{>n}(V)$.
+$\mathrm{T}^n(V) = \mathrm{T}((V)) / \mathrm{T}^{>n}((V))$.
 The truncated tensor algebra is an algebra, when given the *truncated tensor
 product*, obtained by truncating the full tensor product. 
 
@@ -183,6 +183,7 @@ where for each $m\geq 1$, $\mathrm{S}_m(X)_{s, t}$ is given by the iterated
 The signature respects concatenation of paths, meaning 
 $\mathrm{S}(X)_{s, t} = \mathrm{S}(X)_{s, u} \otimes \mathrm{S}(X)_{u, t}$ 
 for any $s < u < t$.
+This property is usually called *Chen's relation*.
 Two paths have the same signature if and only if they differ by a 
 *tree-like path* [@10.4007/annals.2010.171.109].
 The signature is translation invariant, and it is invariant under 
@@ -193,7 +194,8 @@ This is the space of linear functionals $\mathrm{T}((V))\to \mathbb{R}$
 and consists of sequences $(\lambda_0, \lambda_1, \dots)$ with
 $\lambda_k\in (V^{\ast})^{\otimes k}$ and where $\lambda_k = 0$ for all $k$
 larger than some $N$.
-The multiplication on $\mathrm{Sh}(V)$ is the *shuffle product*, which
+(Here $V^{\ast}$ denotes the dual space of $V$. In our notation $V^{\ast} \cong
+V$.) The multiplication on $\mathrm{Sh}(V)$ is the *shuffle product*, which
 corresponds to point-wise multiplication of functions on the path.
 Continuous functions on the path can be approximated (uniformly) by shuffle
 tensors acting on $\mathrm{G}(V)$ on the signature.
@@ -201,7 +203,7 @@ This is a consequence of the Stone-Weierstrass theorem.
 This property is sometimes referred to as *universal non-lineararity*.
 
 There are several *Lie algebras* associated to $\mathrm{T}((V))$.
-Define a *Lie bracket* operation on $\mathrm{T}((V))$ by the formula 
+Define a *Lie bracket* on $\mathrm{T}((V))$ by the formula 
 $[\mathbf{x}, \mathbf{y}] = \mathbf{x} \otimes \mathbf{y} - 
 \mathbf{y}\otimes \mathbf{x}$, for
 $\mathbf{x},\mathbf{y}\in \mathrm{T}((V))$.

From ddfac4a018840fb596661720616df60520368c3c Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:27:14 +0100
Subject: [PATCH 28/32] Fixed issues in section 3

---
 papers/sam_morley/roughpy.md | 14 +++++++++-----
 1 file changed, 9 insertions(+), 5 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 4700f593e1..b4186375b5 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -238,7 +238,8 @@ Consider a finite, ordered sequence $\{(t_1, \mathbf{x}_1,\dots,
 t_N,\mathbf{x}_N)\}$ of observations, where $t_j\in \mathbb{R}$, and
 $\mathbf{x}_j\in V$.
 (More generally, we might consider $\mathbf{x}_j\in\mathcal{L}(V)$ instead.
-That is, data that already contains higher-order information.)
+That is, data that already contains higher-order information. In our language,
+it is a genuine rough path.)
 We can find numerous paths that interpolate these observations; a path 
 $X:[t_0, t_N]\to V$ such that, for each $j$, $X_{t_j} = \mathbf{x}_j$.
 The simplest interpolation is to take the path that is linear between adjacent
@@ -248,8 +249,8 @@ Once we have a path, we need to be able to compute signatures.
 For practical purposes, we truncate all signatures (and log-signatures) to a
 particular degree $M$, which we typically call the *depth*.
 The dimension of the ambient space $d$ is usually called the *width*.
-Using linear interpolation, we can replace the iterated integrals with a
-tensor exponential of the difference of successive terms:
+Using linear interpolation, we can compute the iterated integrals explicitly 
+using a free tensor exponential of the difference of successive terms:
 ```{math}
 \mathrm{Sig}^M([t_j, t_{j+1})) = \exp_M(\mathbf{x}_{j+1} - \mathbf{x}_j) :=
 \sum_{j=0}^M \frac{1}{j!}(\mathbf{x}_{j+1} - \mathbf{x}_j)^{\otimes j}.
@@ -265,14 +266,17 @@ Most of the early applications of rough paths in data science, the (truncated)
 signature was used as a feature map [@NEURIPS2019_d2cdf047].
 This provides a summary of the path that is independent of the parameterisation
 and the number of observations.
-Unfortunately, the signature is a high-dimensional summary.
+Unfortunately, the signature grows geometrically with truncation depth.
 If $d > 1$, then the dimension of $\mathrm{T}^M(V)$ is
 ```{math}
 \sum_{m=0}^M d^m = \frac{d^{M+1} - 1}{d - 1}
 ```
 This generally is a reflection of the complexity of the data, where data with a
 higher complexity generally needs a higher truncation level and thus a larger
-signature.
+signature. It is worth noting that this still represents a significant
+compression of stream information in many cases. The truncation depth one needs
+depends on the stream and the (relative) length of intervals over which you need
+to query the stream.
 
 For some applications, it might be possible to replace the signature with the
 log-signature.

From 20d0d2ba9a31daf52af7029fc8cc736494766376 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:33:14 +0100
Subject: [PATCH 29/32] fix wording and reformat pars

---
 papers/sam_morley/roughpy.md | 35 +++++++++++++++--------------------
 1 file changed, 15 insertions(+), 20 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index b4186375b5..99a7d0bfc9 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -263,30 +263,25 @@ the these terms, using the multiplicative property of the signature.
 
 #### The signature transform
 Most of the early applications of rough paths in data science, the (truncated)
-signature was used as a feature map [@NEURIPS2019_d2cdf047].
-This provides a summary of the path that is independent of the parameterisation
-and the number of observations.
-Unfortunately, the signature grows geometrically with truncation depth.
-If $d > 1$, then the dimension of $\mathrm{T}^M(V)$ is
+signature was used as a feature map [@NEURIPS2019_d2cdf047]. This provides a
+summary of the path that is independent of the parameterisation and the number
+of observations. Unfortunately, the signature grows geometrically with
+truncation depth. If $d > 1$, then the dimension of $\mathrm{T}^M(V)$ is
 ```{math}
 \sum_{m=0}^M d^m = \frac{d^{M+1} - 1}{d - 1}
 ```
-This generally is a reflection of the complexity of the data, where data with a
-higher complexity generally needs a higher truncation level and thus a larger
-signature. It is worth noting that this still represents a significant
-compression of stream information in many cases. The truncation depth one needs
-depends on the stream and the (relative) length of intervals over which you need
-to query the stream.
+The size of the signature is a reflection of the complexity of the data, where
+data with a higher complexity generally needs a higher truncation level and thus
+a larger signature. It is worth noting that this still represents a significant
+compression of stream information in many cases. 
 
 For some applications, it might be possible to replace the signature with the
-log-signature.
-The log-signature is smaller than the signature, but we lose the universal
-non-linearity property of the signature.
-Alternatively, we might turn to other techniques that don't require a full
-calculation of the signature (such as those mentioned below).
-As the connection between rough paths and data science becomes more
-mathematically mature, we will likely find new ways to use the signature
-without requiring its full size.
+log-signature. The log-signature is smaller than the signature, but we lose the
+universal non-linearity property of the signature. Alternatively, we might turn
+to other techniques that don't require a full calculation of the signature (such
+as the signature kernel below). As the connection between rough paths and data
+science becomes more mathematically mature, we will likely find new ways to use
+the signature without requiring its full size.
 
 (sigker-sec)=
 #### Signature kernels
@@ -323,7 +318,7 @@ solution of a system of PDEs of Goursat type
 [@10.48550/arXiv.2404.02926].
 A critical part of their method involves the adjoint of both left and right
 free tensor multiplication, which are not available in any current package
-for computing signatures.
+for computing signatures. These functions are provided by RoughPy.
 
 (ncde-sec)=
 #### Neural controlled differential equations

From f44cf593666d40308c9f184d44f1d7aac5fffa8a Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:36:52 +0100
Subject: [PATCH 30/32] fix more wording

---
 papers/sam_morley/roughpy.md | 85 ++++++++++++++++--------------------
 1 file changed, 38 insertions(+), 47 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 99a7d0bfc9..86f1522ff1 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -289,36 +289,32 @@ Kernel methods are useful tools for learning with sequential data.
 Mathematically, a *kernel* on a set $W$ is a positive-definite function
 $k:W\times W\to \mathbb{R}$. Kernels are often quite easy to evaluate because of
 the kernel trick, which involves embedding the data in a inner product space,
-with a feature map, in which the
-kernel can be evaluated by simply taking an inner product.
-Informally, kernels measure the similarity between two points.
+with a feature map, in which the kernel can be evaluated by simply taking an
+inner product. Informally, kernels measure the similarity between two points.
 They are used in a variety of machine learning tasks such as classification.
 
 The *signature kernel* is a kernel induced on the space of paths by combining
 the signature with an inner product defined on the tensor algebra
-[@JMLR_v20_16_314].
-The theory surrounding the signature kernel has been expanded several times
-since their introduction [@NEURIPS2021_18a9042b;@doi:10.1214/23-AAP1973].
-Typically, the kernel on $\mathrm{T}((V))$ will itself by derived from an inner
-product on $V$, extended to the tensor algebra.
+[@JMLR_v20_16_314]. The theory surrounding the signature kernel has been
+expanded several times since their introduction
+[@NEURIPS2021_18a9042b;@doi:10.1214/23-AAP1973]. Typically, the inner product on
+$\mathrm{T}((V))$ will itself by derived from an inner product on $V$, extended
+to the tensor algebra.
 
 Signatures are infinite objects, so we can't simply evaluate inner products on
-the tensor algebra.
-Fortunately, we can approximate the signature kernel by taking inner products of
-truncated signatures.
-Even better, it turns out that, in certain cases, the signature kernel 
-can be realised as  the solution to a partial differential equation (PDE) of 
-Goursat type.
-This means the full signature kernel can be computed from raw data without 
-needing to compute full signatures [@10.1137/20M1366794].
-
-In fact, in recent preprint, it has been shown that there are
-higher order solvers for signature kernels by rewriting the kernel 
-solution of a system of PDEs of Goursat type 
-[@10.48550/arXiv.2404.02926].
-A critical part of their method involves the adjoint of both left and right
-free tensor multiplication, which are not available in any current package
-for computing signatures. These functions are provided by RoughPy.
+the tensor algebra. Fortunately, we can approximate the signature kernel by
+taking inner products of truncated signatures. Even better, it turns out that,
+in certain cases, the signature kernel can be realised as  the solution to a
+partial differential equation (PDE) of Goursat type. This means the full
+signature kernel can be computed from raw data without needing to compute full
+signatures [@10.1137/20M1366794].
+
+In fact, in recent preprint, it has been shown that there are higher order
+solvers for signature kernels by rewriting the kernel solution of a system of
+PDEs of Goursat type [@10.48550/arXiv.2404.02926]. A critical part of their
+method involves the adjoint of both left and right free tensor multiplication,
+which are not available in any current package for computing signatures. These
+functions are provided by RoughPy.
 
 (ncde-sec)=
 #### Neural controlled differential equations
@@ -342,34 +338,29 @@ in order to boost the performance of neural CDEs.
 
 (applications-sec)=
 ## Current applications of rough paths
-In this section we enumerate several applications where rough paths 
-have been used to develop or improve methods.
-This list presented here is certainly not exhaustive.
-In addition to the literature cited below, there are numerous worked examples in
-the form of Jupyter notebooks available on the DataSig website 
-(<https://datasig.ac.uk/examples>).
+In this section we enumerate several applications where rough paths have been
+used to develop or improve methods. This list presented here is certainly not
+exhaustive. In addition to the literature cited below, there are numerous
+additional references and worked examples, in the form of Jupyter notebooks,
+available on the DataSig website (<https://datasig.ac.uk/examples>).
 
 ### Detecting interference in radio astronomy data 
 Radio frequency interference (RFI) is a substantial problem in the field of
-radio astronomy.
-Even small amounts of RFI can obscure the faint signals generated by distant
-stellar objects and events.
-The problem of identifying RFI in a signal falls into a class of
-semi-supervised learning tasks called *novelty* (or *anomaly*) *detection*.
-Rough path methods have been applied to develop a novelty detection framework
-based on rough path methods to detect RFI in radio astronomy data from several
-radio telescopes [@doi:10.48550/arXiv.2402.14892].
-Their result show that their framework is effective at detecting even faint RFI
-within the test data.
-This work is based on a general novelty detection framework
-[@10.48550/arXiv.2006.03487].
+radio astronomy. Even small amounts of RFI can obscure the faint signals
+generated by distant stellar objects and events. The problem of identifying RFI
+in a signal falls into a class of semi-supervised learning tasks called
+*novelty* (or *anomaly*) *detection*. Rough path methods have been applied to
+develop a novelty detection framework based on rough path methods to detect RFI
+in radio astronomy data from several radio telescopes
+[@doi:10.48550/arXiv.2402.14892]. Their result show that their framework is
+effective at detecting even faint RFI within the test data. This work is based
+on a general novelty detection framework [@10.48550/arXiv.2006.03487].
 
 Signatures kernels have also been used for a similar problem of detecting
-malware by inspecting the streaming tree of processes on a computer system 
-@doi:10.1109/CSR51186.2021.9527933.
-Their method uses a support vector machine classifier to identify processes that
-are malicious compared to "normal" behaviour learned via training on labelled
-data.
+malware by inspecting the streaming tree of processes on a computer system
+@doi:10.1109/CSR51186.2021.9527933. Their method uses a support vector machine
+classifier to identify processes that are malicious compared to "normal"
+behaviour learned via training on a corpus of normality.
 
 ### Tracking mood via natural language processing
 One application of rough paths in natural language processing has been in the

From 7dd660738b5b389e1be16ea1a8d6ed6a3df10e74 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:42:56 +0100
Subject: [PATCH 31/32] fix remaining issues

---
 papers/sam_morley/roughpy.md | 261 ++++++++++++++++-------------------
 1 file changed, 116 insertions(+), 145 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 86f1522ff1..772f4177da 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -365,39 +365,34 @@ behaviour learned via training on a corpus of normality.
 ### Tracking mood via natural language processing
 One application of rough paths in natural language processing has been in the
 domain of mental health [@10.18653/v1/2023.findings-acl.310;
-@tseriotou_etal_2024_sig].
-In this work, the authors present a model for identifying changes in a person's 
-mood based on their online textual content. 
+@tseriotou_etal_2024_sig]. In this work, the authors present a model for
+identifying changes in a person's mood based on their online textual content.
 Many mental health conditions have symptoms that manifest in the (textual)
 expression, so this could be a powerful tool for mental health professionals to
-identify changes in patients and intervene before the state develops.
-Their model achieves state-of-the-art performance vs existing models on two
-datasets.
+identify changes in patients and intervene before the state develops. Their
+model achieves state-of-the-art performance vs existing models on two datasets.
 
 ### Predicting battery cell degradation
 Another recent application of signatures is to predict the degradation of
-lithium-ion cells [@10.1016/j.apenergy.2023.121974].
-They use signature features to train a model that can accurately predict the end
-of life of a cell using relatively low-frequency sampling compared to existing
+lithium-ion cells [@10.1016/j.apenergy.2023.121974]. They use signature features
+to train a model that can accurately predict the end of life of a cell using
+relatively low-frequency sampling compared to existing models. They also
+observed that the performance at higher frequency was comparable to other
 models.
-They also observed that the performance at higher frequency was comparable to
-other models.
 
 ### Prediction of sepsis in intensive care data
 One of the first effective demonstrations of the utility of signatures and rough
 paths based methods in healthcare was in the 2019 PhysioNet challenge
-[@10.1097/CCM.0000000000004510].
-In this contest, teams were invited to develop models to predict sepsis in
-patients from intensive care unit data.
-In this challenge, a team utilising signatures to enhance predictive power
-placed first in the official phase of the challenge.
-Since then, signatures and other rough path based approaches have been used in
-several other clinical contexts [@10.1038/s41598-024-51989-6;
-@doi:10.1109/MEDAI59581.2023.00008; @tseriotou_etal_2024_sig].
-Clinical data is often irregularly sampled and often exhibits missing data, but
-it can also be very high-frequency and dense.
-Rough path based methods can handle these data in an elegant way, and retain the
-structure of long and short term dependencies within the data.
+[@10.1097/CCM.0000000000004510]. In this contest, teams were invited to develop
+models to predict sepsis in patients from intensive care unit data. In this
+challenge, a team utilising signatures to enhance predictive power placed first
+in the official phase of the challenge. Since then, signatures and other rough
+path based approaches have been used in several other clinical contexts
+[@10.1038/s41598-024-51989-6; @doi:10.1109/MEDAI59581.2023.00008;
+@tseriotou_etal_2024_sig]. Clinical data is often irregularly sampled and often
+exhibits a high degree of missingness, but it can also be very high-frequency
+and dense. Rough path based methods can handle these data in an elegant way, and
+retain the structure of long and short term dependencies within the data.
 
 ### Human action recognition
 The task of identifying a specific action performed by a person from a short
@@ -417,10 +412,9 @@ tasks [@10.1109/TPAMI.2017.2732978].
 (roughpy-sec)=
 ## RoughPy
 RoughPy is a new library that aims to support the development of connections
-between rough path theory and data science.
-It represents a shift in philosophy from simple computations of signatures for
-sequential data, to a representation of these data as a rough path.
-The design objectives for RoughPy are as follows:
+between rough path theory and data science. It represents a shift in philosophy
+from simple computations of signatures for sequential data, to a representation
+of these data as a rough path. The design objectives for RoughPy are as follows:
 1. provide a class that presents a rough path view of some source of data as a
    rough path, exposing methods for querying the data over intervals to get a
    signature or log-signature;
@@ -439,170 +433,148 @@ These provide part of the picture, but in order for them to be fully supported,
 RoughPy must support a variety of compute backends such as CUDA (NVidia), 
 ROCm/HIP (AMD), and Metal (Apple).
 
-RoughPy is a substantial library with numerous components, mostly written in 
-C++ with a Python interface defined using Pybind11 [@pybind11].
-The original design of the library closely followed the C++ template libraries
-libRDE and libalgebra [@coropa_project], although it has seen many iterations 
-since.
+RoughPy is a substantial library with numerous components, mostly written in C++
+with a Python interface defined using Pybind11 [@pybind11]. The original design
+of the library closely followed the C++ template libraries libRDE and libalgebra
+[@coropa_project], although it has seen many iterations since.
 
 In the remainder of this section, we discuss some of the core components of
 RoughPy, give an example of using RoughPy, and discuss the future of RoughPy.
 
 ### Free tensors, shuffle tensors, and Lie objects
 In order to properly support rough path based methods and allow users to write
-code based on mathematical concepts, we provide realisations of several
-algebra types.
-The algebras provided in RoughPy are `FreeTensor`, `ShuffleTensor`, and `Lie`,
-which define elements of a particular free tensor algebra, shuffle tensor
-algebra, and Lie algebra respectively.
-Each of these algebras is initialized with a width, depth, and scalar
-coefficient type, encapsulated in a `Context` object.
+code based on mathematical concepts, we provide realisations of several algebra
+types. The algebras provided in RoughPy are `FreeTensor`, `ShuffleTensor`, and
+`Lie`, which define elements of a particular free tensor algebra, shuffle tensor
+algebra, and Lie algebra respectively. Each of these algebras is initialized
+with a width, depth, and scalar coefficient type, encapsulated in a `Context`
+object.
 
 In addition to the algebra classes, RoughPy provides a number of supporting
-functions, including half-shuffle products for `FreeTensor`/`ShuffleTensor`
-objects, and adjoint operators for left free tensor multiplication.
-These are operations that are frequently used in the theory of rough paths, and
-will likely be necessary in developing new applications later (as in the
-signature kernels).
+functions, including antipodes and half-shuffle products for
+`FreeTensor`/`ShuffleTensor` objects, and adjoint operators for left free tensor
+multiplication. These are operations that are frequently used in the theory of
+rough paths, and will likely be necessary in developing new applications later
+(as in the signature kernels).
 
 RoughPy algebras are designed around a flexible scalar ring system that allows
 users to perform calculations with different accuracy, or derive expressions by
-using polynomial coefficients.
-For most applications, single or double precision floating point numbers will 
-provide a good balance between performance and accuracy.
-(Double precision floats are the default.)
-When more precision is required, rational coefficients can be used
-instead.
-These are backed by GMP rationals for fast, arbitrary precision rational
-arithmetic [@Granlund12].
+using polynomial coefficients. For most applications, single or double precision
+floating point numbers will provide a good balance between performance and
+accuracy. (Double precision floats are the default.) When more precision is
+required, rational coefficients can be used instead. These are backed by GMP
+rationals for fast, arbitrary precision rational arithmetic [@Granlund12].
 Polynomial coefficients can be used to derive formulae by performing
-calculations.
-This is a powerful technique for understanding the terms that appear in the
-result, particularly whilst testing and debugging.
-
+calculations. This is a powerful technique for understanding the terms that
+appear in the result, particularly whilst testing and debugging.
 
 ### Intervals 
-RoughPy is very careful in the way it handles intervals.
-All intervals in RoughPy are half-open, meaning that they include one end point
-but not the other; they are either *clopen* $[a, b) := \{t: a\leq t < b\}$ or 
-*opencl* $(a, b] := \{t : a < t \leq b\}$.
-Besides the type (clopen or opencl), all intervals must provide methods for
-retrieving the infimum ($a$ in the above notation) and the supremum ($b$ above)
-of the interval as double precision floats.
-This is enforced by means of an abstract base class `Interval`.
-The main concrete interval types are `RealInterval`, an interval with arbitrary 
-real endpoints, and `DyadicInterval`, as described below.
-For brevity, in the sequel we shall only consider clopen intervals.
+RoughPy is very careful in the way it handles intervals. All intervals in
+RoughPy are half-open, meaning that they include one end point but not the
+other; they are either *clopen* $[a, b) := \{t: a\leq t < b\}$ or *opencl* $(a,
+b] := \{t : a < t \leq b\}$. Besides the type (clopen or opencl), all intervals
+must provide methods for retrieving the infimum ($a$ in the above notation) and
+the supremum ($b$ above) of the interval as double precision floats. This is
+enforced by means of an abstract base class `Interval`. The main concrete
+interval types are `RealInterval`, an interval with arbitrary real endpoints,
+and `DyadicInterval`, as described below. For brevity, we shall only consider
+clopen intervals.
 
 A *dyadic interval* is an interval $D_k^n := [k/2^n, (k+1)/2^n)$, where $k$, $n$
-are integers.
-The number $n$ is often described as the *resolution* of the interval.
-The family of dyadic intervals of a fixed resolution $n$ partition the real line 
-so that every real number $t$ belongs to a unique dyadic interval $D_n^k$.
-Moreover, the family of all dyadic intervals have the property that two
+are integers. The number $n$ is often described as the *resolution* of the
+interval. The family of dyadic intervals of a fixed resolution $n$ partition the
+real line so that every real number $t$ belongs to a unique dyadic interval
+$D_n^k$. Moreover, the family of all dyadic intervals have the property that two
 dyadic intervals are either disjoint or one contains the other (including the
 possibility that they are equal).
 
-In many cases, RoughPy will granularise an interval into a dyadic intervals.
-The *dyadic granularisation* of $[a, b)$ with resolution $n$ is 
-$[k_1/2^n, k_2/2^n)$ where $k_1 = \max\{k: k/2^n \leq a\}$ and 
-$k_2 = \max\{k: k/2^n \leq b\}$.
-In effect, the dyadic granularisation is the result of "rounding" each end 
-point to the included end of the unique dyadic interval that contain it.
+In many cases, RoughPy will granularise an interval into a dyadic intervals. The
+*dyadic granularisation* of $[a, b)$ with resolution $n$ is $[k_1/2^n, k_2/2^n)$
+where $k_1 = \max\{k: k/2^n \leq a\}$ and $k_2 = \max\{k: k/2^n \leq b\}$. In
+effect, the dyadic granularisation is the result of "rounding" each end point to
+the included end of the unique dyadic interval that contain it.
 
 ### Streams
-Streams are central to RoughPy.
-A RoughPy `Stream` is a rough path view of some underlying data.
-It provides two key methods to query the object over intervals to retrieve
-either a signature or log-signature.
-Importantly, once constructed, the underlying data is inaccessible except by
-querying via these methods.
+Streams are central to RoughPy. A RoughPy `Stream` is a rough path view of some
+underlying data. It provides two key methods to query the object over intervals
+to retrieve either a signature or log-signature. Importantly, once constructed,
+the underlying data is inaccessible except by querying via these methods.
 `Stream`s are designed to be composed in various ways, such as by concatenation,
-in order to build up more complex streams.
-A `Stream` is actually a (type-erasing) wrapper around a more minimal
-`StreamInterface` abstract class.
+in order to build up more complex streams. A `Stream` is actually a
+(type-erasing) wrapper around a more minimal `StreamInterface` abstract class.
 
 We construct streams by a factory function associated with each different
 `StreamInterface`, which might perform some compression of the underlying data.
 For example, a basic `StreamInterface` is the `LieIncrementStream`, which can be
 constructed using the associated `from_increments` factory function (a static
-method of the class), which accepts an $n \times d$ array of *increment data*. 
-These data will typically be the differences between successive values of the 
-data (but could also include higher-order Lie terms).
-This is similar to the way that libraries such as `esig`, `iisignature`, and
-`signatory` consume data.
+method of the class), which accepts an $n \times d$ array of *increment data*.
+These data will typically be the differences between successive values of the
+data (but could also include higher-order Lie terms). This is similar to the way
+that libraries such as `esig`, `iisignature`, and `signatory` consume data.
 
 RoughPy streams cache the result of log-signature queries over dyadic intervals
-so they can be reused in later calculations.
-To compute the log-signature over any interval $I$, we granularise at a fixed 
-stream resolution $n$ to obtain the interval $\tilde I = [k_1/2^n, k_2/2^n)$, 
-and then compute
+so they can be reused in later calculations. To compute the log-signature over
+any interval $I$, we granularise at a fixed stream resolution $n$ to obtain the
+interval $\tilde I = [k_1/2^n, k_2/2^n)$, and then compute
 :::{math}
 \mathrm{LogSig}(\tilde I) = \log\biggl(\prod_{k=k_1}^{k_2-1} 
 \exp(\mathrm{LogSig}(D_k^n))\biggr).
 :::
-The $\mathrm{LogSig}(D_k^n)$ terms on the right-hand-side are either retrieved 
-from the cache, or computed from the underlying source.
-This is essentially the Campbell-Baker-Hausdorff formula applied to the 
-log-signatures at the finest level.
-In practice, we can actually reduce the number of terms in the product, by
-merging complementary dyadic intervals that appear in the granularisation.
-We further optimise by using a fused multiply-exponential ($A\exp(B)$)
-operation.
+The $\mathrm{LogSig}(D_k^n)$ terms on the right-hand-side are either retrieved
+from the cache, or computed from the underlying source. This is essentially the
+Campbell-Baker-Hausdorff formula applied to the log-signatures at the finest
+level. In practice, we can actually reduce the number of terms in the product,
+by merging complementary dyadic intervals that appear in the granularisation. We
+further optimise by using a fused multiply-exponential ($A\exp(B)$) operation.
 
 Signatures are always computed by first computing the log-signature and then
-exponentiating.
-Directly computing the signature as a product of exponentials of (cached)
-log-signatures might accumulate enough numerical errors to drift slightly from a
-group-like tensor. 
-That is, the result might not actually be a true signature.
-Taking the logarithm and then exponentiating back to obtain the signature has
-the effect of correcting this numerical drift from a true signature.
+exponentiating. Directly computing the signature as a product of exponentials of
+(cached) log-signatures might accumulate enough numerical errors to drift
+slightly from a group-like tensor. That is, the result might not actually be a
+true signature. Taking the logarithm and then exponentiating back to obtain the
+signature has the effect of correcting this numerical drift from a true
+signature.
 
 Aside from the basic `LieIncrementStream`, there are several other
-implementations of the `StreamInterface` currently available in RoughPy.
-The `BrownianStream` approximates Brownian motion by generating normal
-distributed increments over dyadic intervals of arbitrary resolution on demand,
-forming a reasonable approximation of true Brownian motion.
-The `ExternalDataStream` is an interface for loading data from various external
-sources, such as from a database or specialised data format.
-Currently, only sound files are supported but we plan to extend support for
-other sources as the need arises.
-This will certainly include "online" data sources such as computer peripheral
-devices such as microphones.
+implementations of the `StreamInterface` currently available in RoughPy. The
+`BrownianStream` approximates Brownian motion by generating normal distributed
+increments over dyadic intervals of arbitrary resolution on demand, forming a
+reasonable approximation of true Brownian motion. The `ExternalDataStream` is an
+interface for loading data from various external sources, such as from a
+database or specialised data format. Currently, only sound files are supported
+but we plan to extend support for other sources as the need arises. This will
+certainly include "online" data sources such as computer peripheral devices
+(e.g. microphones).
 
 The other main `StreamInterface` implementation is the `PiecewiseAbelianStream`,
-which is an important construction from CDE.
-A piecewise Abelian path, or log-linear path, is an example of a *smooth rough
-path*, which generalises piecewise linear approximations of an arbitrary stream.
-Formally, an *Abelian path* $Y$ is a pair $([a, b), \mathbf{y})$ where $a < b$
-and $\mathbf{y}\in\mathcal{L}(V)$.
-The log-signature over an arbitrary interval $[u, v) \subseteq [a, b)$ is given
-by
+which is an important construction from CDE. A piecewise Abelian path, or
+log-linear path, is an example of a *smooth rough path*, which generalises
+piecewise linear approximations of an arbitrary stream. Formally, an *Abelian
+path* $Y$ is a pair $([a, b), \mathbf{y})$ where $a < b$ and
+$\mathbf{y}\in\mathcal{L}(V)$. The log-signature over an arbitrary interval $[u,
+v) \subseteq [a, b)$ is given by
 ```{math}
 \mathrm{LogSig}(Y)_{u, v} = \frac{v - u}{b - a}\mathbf{y}.
 ```
 A *piecewise Abelian path* is the concatenation of finitely many Abelian paths
-with adjacent intervals.
-For any rough path $X$ and partition $\{a = t_0 < t_1 < \dots < t_N = b\}$ there
-is a piecewise Abelian approximation for this path given by
+with adjacent intervals. For any rough path $X$ and partition $\{a = t_0 < t_1 <
+\dots < t_N = b\}$ there is a piecewise Abelian approximation for this path
+given by
 ```{math}
 \{([t_{j-1}, t_j), \mathrm{LogSig}(X)_{t_{j-1}, t_j}): j=1, \dots, N\}.
 ```
-This construction turns out to be vital for computing signature kernels 
+This construction turns out to be vital for computing signature kernels
 [@10.25080/gerudo-f2bc6f59-001] and for solving CDEs
-[@10.1007/978-3-540-71285-5;
-@10.48550/arXiv.2402.18512].
-In particular, this construction can be used to compress data at some degree,
-which can the be used in computations at a higher degree.
+[@10.1007/978-3-540-71285-5; @10.48550/arXiv.2402.18512]. In particular, this
+construction can be used to compress data at some degree, which can the be used
+in computations at a higher degree.
 
 ### Example
 In this section we show a very simple example of how to use RoughPy to construct
-a stream and compute a signature.
-This example is similar to the first few steps of the tutorial found in the
-RoughPy documentation.[^roughpydocs]
-RoughPy can be installed using `pip`, where prebuilt wheels are available for
-Windows, Linux, and MacOs:
+a stream and compute a signature. This example is similar to the first few steps
+of the tutorial found in the RoughPy documentation.[^roughpydocs] RoughPy can be
+installed using `pip`, where prebuilt wheels are available for Windows, Linux,
+and MacOs:
 ```
 pip install roughpy
 ```
@@ -618,10 +590,9 @@ for i, c in enumerate(text):
     increments[i, ord(c) - 97] = 1
 ```
 Now we import RoughPy and construct a `Stream` using the factory mentioned
-above.
-One other critical ingredient is the algebra `Context`, which is used to set up
-a consistent set of algebra objects with the desired width (26), truncation
-level (2), and coefficient type (`Rational`).
+above. One other critical ingredient is the algebra `Context`, which is used to
+set up a consistent set of algebra objects with the desired width (26),
+truncation level (2), and coefficient type (`Rational`).
 ```python
 import roughpy as rp
 

From e9f57ca130e3467119ff37fecb13d86e57af1342 Mon Sep 17 00:00:00 2001
From: Sam Morley <admin@inakleinbottle.com>
Date: Sat, 3 Aug 2024 21:43:45 +0100
Subject: [PATCH 32/32] Remove export format

---
 papers/sam_morley/roughpy.md | 7 -------
 1 file changed, 7 deletions(-)

diff --git a/papers/sam_morley/roughpy.md b/papers/sam_morley/roughpy.md
index 772f4177da..e279eac19b 100644
--- a/papers/sam_morley/roughpy.md
+++ b/papers/sam_morley/roughpy.md
@@ -6,13 +6,6 @@ bibliography:
 numbering:
     heading_2: true
     heading_3: true
-exports:
-  - format: pdf
-    template: arxiv_two_column
-    article_type: Article
-  - format: tex
-    template: arxiv_two_column
-    article_type: Article
 abstract: |
   Rough path theory is a branch of mathematics arising out of stochastic
   analysis. One of the main tools of rough path analysis is the signature,