# Non-nilpotent Leibniz algebras with
one-dimensional derived subalgebra00footnotetext: Keywords: Leibniz algebra,
Lie algebra, Derivation, Biderivation, Coquecigrue problem. 00footnotetext:
2020 Mathematics Subject Classification: 16W25, 17A32, 17B30, 17B40, 20M99,
22A30. 00footnotetext: The authors are supported by University of Palermo and
by the “National Group for Algebraic and Geometric Structures, and their
Applications” (GNSAGA – INdAM).

###### Abstract

In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras
with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with
$\operatorname{char}(\mathbb{F})\neq 2$. We prove that such an algebra is
isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie
Leibniz algebra and an abelian algebra. We denote it by $L_{n}$, where
$n=\dim_{\mathbb{F}}L_{n}$. This generalizes the result found in [11], which
is only valid when $\mathbb{F}=\mathbb{C}$ . Moreover, we find the Lie algebra
of derivations, its Lie group of automorphisms and the Leibniz algebra of
biderivations of $L_{n}$. Eventually, we solve the _coquecigrue problem_ for
$L_{n}$ by integrating it into a Lie rack.

Alfonso Di Bartolo, Gianmarco La Rosa, Manuel Mancini

Dipartimento di Matematica e Informatica
Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
<EMAIL_ADDRESS>ORCID: 0000-0001-5619-2644
<EMAIL_ADDRESS>ORCID: 0000-0003-1047-5993
<EMAIL_ADDRESS>ORCID: 0000-0003-2142-6193

## Introduction

Leibniz algebras were introduced by J.-L. Loday in [19] as a non-skew
symmetric version of Lie algebras. Earlier such algebraic structures were also
considered by A. Blokh, who called them D-algebras [5] for their strict
connection with derivations. Leibniz algebras play a significant role in
different areas of mathematics and physics.

Many results of Lie algebras are still valid for Leibniz algebras. One of them
is the _Levi decomposition_ , which states that any Leibniz algebra over a
field $\mathbb{F}$ of characteristic zero is the semidirect sum of its radical
and a semisimple Lie algebra. This makes clear the importance of the problem
of classification of solvable and nilpotent Lie / Leibniz algebras, which has
been dealt with since the early 20th century (see [2], [3], [4], [9], [10],
[11], [13] and [14], just for giving a few examples).

In [16] and [17] nilpotent Leibniz algebras $L$ with one-dimensional derived
subalgebra $[L,L]$ were studied and classified. It was proved that, up to
isomorphism, there are three classes of _indecomposable_ Leibniz algebras with
these properties, namely the _Heisenberg_ algebras $\mathfrak{l}_{2n+1}^{A}$,
which are parameterized by their dimension $2n+1$ and by a matrix $A$ in
canonical form, the _Kronecker_ algebra $\mathfrak{k}_{n}$ and the _Dieudonné_
algebra $\mathfrak{d}_{n}$, both parameterized by their dimension only. We
want to complete this classification by studying non-nilpotent Leibniz
$\mathbb{F}$-algebras with one-dimensional derived subalgebra, where
$\mathbb{F}$ is a field with $\operatorname{char}(\mathbb{F})\neq 2$. Using
the theory of non-abelian extensions of Leibniz algebras introduced in [18],
we prove that a non-nilpotent non-Lie Leibniz algebra $L$ with
$\dim_{\mathbb{F}}L=n$ and $\dim_{\mathbb{F}}[L,L]=1$ is isomorphic to the
direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra
$S_{2}$, i.e. the algebra with basis $\\{e_{1},e_{2}\\}$ and multiplication
table given by $[e_{2},e_{1}]=e_{1}$, and an abelian algebra of dimension
$n-2$. We denote it by $L_{n}$. This generalizes the result found in Theorem
2.6 of [11], where the authors proved that a _complex_ non-split non-nilpotent
non-Lie Leibniz algebra with one-dimensional derived subalgebra is isomorphic
to $S_{2}$.

We study in detail the properties of the algebra $L_{n}$ and we compute the
Lie algebra of derivations $\operatorname{Der}(L_{n})$, its Lie group of
automorphism $\operatorname{Aut}(L_{n}$) and the Leibniz algebra of
biderivations $\operatorname{Bider}(L_{n})$.

Finally, we solve the _coquecigrue problem_ for the Leibniz algebra $L_{n}$.
We mean the problem, formulated by J.-L. Loday in [19], of finding a
generalization of Lie third theorem to Leibniz algebras. Using M. K. Kinyon’s
results for the class of real _split Leibniz algebras_ (see [15]), we show how
to explicitly integrate $L_{n}$ into a Lie rack defined over the vector space
$\mathbb{R}^{n}$.

##  1 Preliminaries

We assume that $\mathbb{F}$ is a field with
$\operatorname{char}(\mathbb{F})\neq 2$. For the general theory we refer to
[1].

######  Definition 1.1.

A _left Leibniz algebra_ over $\mathbb{F}$ is a vector space $L$ over
$\mathbb{F}$ endowed with a bilinear map (called $commutator$ or $bracket$)
$\left[-,-\right]\colon L\times L\rightarrow L$ which satisfies the _left
Leibniz identity_

$\left[x,\left[y,z\right]\right]=\left[\left[x,y\right],z\right]+\left[y,\left[x,z\right]\right],\;\;\forall
x,y,z\in L.$

In the same way we can define a right Leibniz algebra, using the right Leibniz
identity

$\left[\left[x,y\right],z\right]=\left[[x,z],y\right]+\left[x,\left[y,z\right]\right],\;\;\forall
x,y,z\in L.$

Given a left Leibniz algebra $L$, the multiplication
$[x,y]^{\operatorname{op}}=[y,x]$ defines a right Leibniz algebra structure on
$L$.

A Leibniz algebra that is both left and right is called _symmetric Leibniz
algebra_. From now on we assume that $\dim_{\mathbb{F}}L<\infty$.

We have a full inclusion functor
$i\colon\textbf{LieAlg}_{\mathbb{F}}\rightarrow\textbf{LeibAlg}_{\mathbb{F}}$
that embeds Lie algebras over $\mathbb{F}$ into Leibniz algebras over
$\mathbb{F}$. Its left adjoint is the functor
$\pi\colon\textbf{LeibAlg}_{\mathbb{F}}\rightarrow\textbf{LieAlg}_{\mathbb{F}}$,
which associates to each Leibniz algebra $L$ the quotient
$L/\operatorname{Leib}(L)$, where $\operatorname{Leib}(L)$ is the smallest
bilateral ideal of $L$ such that the quotient $L/\operatorname{Leib}(L)$
becomes a Lie algebra. $\operatorname{Leib}(L)$ is defined as the subalgebra
generated by all elements of the form $\left[x,x\right]$, for any $x\in L$,
and it is called the _Leibniz kernel_ of $L$.

We define the left and the right center of a Leibniz algebra

$\operatorname{Z}_{l}(L)=\left\\{x\in
L\,|\,\left[x,L\right]=0\right\\},\,\,\,\operatorname{Z}_{r}(L)=\left\\{x\in
L\,|\,\left[L,x\right]=0\right\\}.$

The intersection of the left and right center is called the _center_ of $L$
and it is denoted by $\operatorname{Z}(L)$. In general for a left Leibniz
algebra $L$, the left center $\operatorname{Z}_{l}(L)$ is a bilateral ideal,
meanwhile the right center is not even a subalgebra. Furthermore, one can
check that $\operatorname{Leib}(L)\subseteq\operatorname{Z}_{l}(L)$.

The definition of derivation for a Leibniz algebra is the same as in the case
of Lie algebras.

######  Definition 1.2.

A linear map $d\colon L\rightarrow L$ is a _derivation_ of $L$ if

$d([x,y])=[d(x),y]+[x,d(y)],\;\;\forall x,y\in L.$

An equivalent way to define a left Leibniz algebra $L$ is to saying that the
left adjoint maps $\operatorname{ad}_{x}=[x,-]$ are derivations. Meanwhile the
right adjoint maps $\operatorname{Ad}_{x}=[-,x]$ are not derivations in
general. The set $\operatorname{Der}(L)$ of all derivations of $L$ is a Lie
algebra with the usual bracket $[d,d^{\prime}]=d\circ
d^{\prime}-d^{\prime}\circ d$ and the set $\operatorname{Inn}(L)$ spanned by
the left adjoint maps, which are called _inner derivations_ , is an ideal of
$\operatorname{Der}(L)$. Moreover $\operatorname{Aut}(L)$ is a Lie group and
its Lie algebra is precisely $\operatorname{Der}(L)$.

In [19] J.-L. Loday introduced the notion of anti-derivation and biderivation
for a Leibniz algebra.

######  Definition 1.3.

A linear map $D\colon L\rightarrow L$ is an _anti-derivation_ of $L$ if

$D([x,y])=[x,D(y)]-[y,D(x)],\;\;\forall x,y\in L.$

The space $\operatorname{ADer}(L$) of anti-derivations of $L$ has a
$\operatorname{Der}(L)$-module structure with the extra multiplication $d\cdot
D=d\circ D-D\circ d$, for any derivation $d$ and for any anti-derivation $D$,
and one can check that the right adjoint maps $\operatorname{Ad}_{x}$ are
anti-derivations.

######  Definition 1.4.

A _biderivation_ of $L$ is a pair
$(d,D)\in\operatorname{Der}(L)\times\operatorname{ADer}(L)$ such that

$[d(x)+D(x),y]=0,\;\;\forall x,y\in L.$

The set $\operatorname{Bider}(L)$ of all biderivations of $L$ has a Leibniz
algebra structure with the bracket

$[(d,D),(d^{\prime},D^{\prime})]=([d,d^{\prime}],d\cdot D^{\prime})$

and it is defined a Leibniz algebra homomorphism

$L\rightarrow\operatorname{Bider}(L),\;x\mapsto(\operatorname{ad}_{x},\operatorname{Ad}_{x}).$

The pair $(\operatorname{ad}_{x},\operatorname{Ad}_{x})$ is called the _inner
biderivation_ associated with $x\in L$ and the set of all inner biderivations
of $L$ forms a Leibniz subalgebra of $L$.

We recall the definitions of solvable and nilpotent Leibniz algebras.

######  Definition 1.5.

Let $L$ be a right Leibniz algebra over $\mathbb{F}$ and let

$L^{0}=L,\,\,L^{k+1}=[L^{k},L^{k}],\;\;\forall k\geq 0,$

be the _derived series of $L$_. $L$ is _$n-$ step solvable_ if $L^{n-1}\neq 0$
and $L^{n}=0$.

######  Definition 1.6.

Let $L$ be a left Leibniz algebra over $\mathbb{F}$ and let

$L^{(0)}=L,\,\,L^{(k+1)}=[L,L^{(k)}],\;\;\forall k\geq 0,$

be the _lower central series of $L$_. $L$ is _$n-$ step nilpotent_ if
$L^{(n-1)}\neq 0$ and $L^{(n)}=0$.

When $L$ is two-step nilpotent, it lies in different varieties of non-
associative algebras, such as associative, alternative and Zinbiel algebras.
In this case we refer at $L$ as a _two-step nilpotent algebra_ and we have the
following.

######  Proposition 1.7.

  1. (i)

If $L$ is a two-step nilpotent algebra, then
$L^{(1)}=[L,L]\subseteq\operatorname{Z}(L)$ and $L$ is a symmetric Leibniz
algebra;

  2. (ii)

If $L$ is a left nilpotent Leibniz algebra with $\dim_{\mathbb{F}}[L,L]=1$,
then $L$ is two-step nilpotent.

In [16] the classification of nilpotent Leibniz algebras with one-dimensional
derived subalgebra was established. The classification revealed that, up to
isomorphism, there exist only three classes of indecomposable nilpotent
Leibniz algebras of this type.

######  Definition 1.8.

[16] Let $f(x)\in\mathbb{F}\left[x\right]$ be a monic irreducible polynomial.
Let $k\in\mathbb{N}$ and let $A=(a_{ij})_{i,j}$ be the companion matrix of
$f(x)^{k}$. The _Heisenberg_ algebra $\mathfrak{l}_{2n+1}^{A}$ is the
$(2n+1)$-dimensional Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},z\right\\}$ and the brackets
are given by

$[e_{i},f_{j}]=(\delta_{ij}+a_{ij})z,\;[f_{j},e_{i}]=(-\delta_{ij}+a_{ij})z,\;\;\forall
i,j=1,\ldots,n.$

When $A$ is the zero matrix, then we obtain the $(2n+1)-$dimensional
Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$.

######  Definition 1.9.

[16] Let $n\in\mathbb{N}$. The _Kronecker_ algebra $\mathfrak{k}_{n}$ is the
$(2n+1)$-dimensional Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n},z\right\\}$ and the brackets
are given by

$\displaystyle[e_{i},f_{i}]=[f_{i},e_{i}]=z,\;\;\forall i=1,\ldots,n$
$\displaystyle[e_{i},f_{i-1}]=z,[f_{i-1},e_{i}]=-z,\;\;\forall i=2,\ldots,n.$

######  Definition 1.10.

[16] Let $n\in\mathbb{N}$. The _Dieudonné_ algebra $\mathfrak{d}_{n}$ is the
$(2n+2)$-dimensional Leibniz algebra with basis
$\left\\{e_{1},\ldots,e_{2n+1},z\right\\}$ and the brackets are given by

$\displaystyle\left[e_{1},e_{n+2}\right]=z,$
$\displaystyle\left[e_{i},e_{n+i}\right]=\left[e_{i},e_{n+i+1}\right]=z,\;\;\forall
i=2,\ldots,n,$ $\displaystyle\left[e_{n+1},e_{2n+1}\right]=z,$
$\displaystyle\left[e_{i},e_{i-n}\right]=z,\;\;\left[e_{i},e_{i-n-1}\right]=-z,\;\;\forall
i=n+2,\ldots,2n+1.$

We want to extend this classification by studying non-nilpotent Leibniz
algebras with one-dimensional derived subalgebra.

##  2 Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

Let $L$ be a non-nilpotent left Leibniz algebra over $\mathbb{F}$ with
$\dim_{\mathbb{F}}L=n$ and $\dim_{\mathbb{F}}\left[L,L\right]=1$. We observe
that such an algebra is two-step solvable since the derived subalgebra
$\left[L,L\right]$ is abelian.

It is well known that a non-nilpotent Lie algebra with one-dimensional derived
subalgebra is isomorphic to the direct sum of the two-dimensional non-abelian
Lie algebra and an abelian algebra (see Section 3 of [12]). Thus we are
interested in the classification of non-Lie Leibniz algebras with these
properties.

In Theorem 2.6 of [11] the authors prove that a _complex_ non-split non-
nilpotent non-Lie Leibniz algebra with one-dimensional derived subalgebra is
isomorphic to the two-dimensional algebra with basis $\\{e_{1},e_{2}\\}$ and
multiplication table $[e_{2},e_{1}]=[e_{2},e_{2}]=e_{1}$. Here we generalize
this result when $\mathbb{F}$ is a general field with
$\operatorname{char}(\mathbb{F})\neq 2$.

######  Proposition 2.1.

Let $L$ be a non-nilpotent left Leibniz algebra over $\mathbb{F}$ with
$\dim_{\mathbb{F}}\left[L,L\right]=1$. Then $L$ has a two-dimensional
bilateral ideal $S$ which is isomorphic to one of the following Leibniz
algebras:

  * (i)

$S_{1}=\langle e_{1},e_{2}\rangle$ with
$\left[e_{2},e_{1}\right]=-\left[e_{1},e_{2}\right]=e_{1}$;

  * (ii)

$S_{2}=\langle e_{1},e_{2}\rangle$ with
$\left[e_{2},e_{1}\right]=\left[e_{2},e_{2}\right]=e_{1}$.

###### Proof.

Let $\left[L,L\right]=\mathbb{F}z$. $L$ is not nilpotent, then

$[L,\left[L,L\right]]\neq 0,$

i.e. $z\notin\mathrm{Z}_{r}(L)$. Since $\left[L,L\right]$ is an abelian
algebra, there exists a vector $x\in L$, which is linearly independent than
$z$, such that $\left[x,z\right]\neq 0$. Thus

$\left[x,z\right]=\gamma z,$

for some $\gamma\in\mathbb{F}^{\ast}$. The subspace $S=\langle x,z\rangle$ is
an ideal of $L$ and it is not nilpotent. In fact

$0\neq\gamma z=\left[x,z\right]\in\left[S,\left[S,S\right]\right].$

Thus $S$ is a non-nilpotent Leibniz algebra. Using the classification of two-
dimensional Leibniz algebras given by C. Cuvier in [8], $S$ is isomorphic
either to $S_{1}$ or to $S_{2}$. ∎

######  Remark 2.1.

The algebras $S_{1}$ and $S_{2}$ are respectively the Leibniz algebras $L_{2}$
and $L_{4}$ of Section 3.1 in [1]. We observe that $S_{1}$ is a Lie algebra,
meanwhile $S_{2}$ is a non-right left Leibniz algebra.

One can see $L$ as an extension of the abelian algebra
$L_{0}=L/S\cong\mathbb{F}^{n-2}$ by $S$ (see [18])

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(1)

It turns out that there exists an equivalence of Leibniz algebra extensions

${0}$${S}$${L_{0}\ltimes_{\omega}S}$${L_{0}}$${0}$${0}$${S}$${L}$${L_{0}}$${0}$$\scriptstyle{i_{2}}$$\scriptstyle{\operatorname{id}_{S}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\theta}$$\scriptstyle{i_{1}}$$\scriptstyle{\operatorname{id}_{B}}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{\sigma}$

where $L_{0}\ltimes_{\omega}S$ is the Leibniz algebra defined on the direct
sum of vector spaces $L_{0}\oplus S$ with the bilinear operation given by

$[(x,a),(y,b)]_{(l,r,\omega)}=(0,[a,b]+l_{x}(b)+r_{y}(a)+\omega(x,y)),$

where

$\omega(x,y)=[\sigma(x),\sigma(y)]_{L}-\sigma([x,y]_{L_{0}})=[\sigma(x),\sigma(y)]_{L}$

is the Leibniz algebra $2$-cocycle associated with (1) and

$l_{x}(b)=[\sigma(x),i(b)]_{L},\;\;r_{y}(a)=[i(a),\sigma(y)]_{L}$

define the action of $L_{0}$ on $S$; $i_{1},i_{2},\pi_{1}$ are the canonical
injections and projection. The Leibniz algebra isomorphism $\theta$ is defined
by $\theta(x,a)=\sigma(x)+i(a)$, for every $(x,a)\in L_{0}\oplus S$.

By Proposition 4.2 of [18], the $2$-cocycle $\omega\colon L_{0}\times
L_{0}\rightarrow S$ and the linear maps $l,r\colon
L_{0}\rightarrow\operatorname{gl}(S)$ must satisfy the following set of
equations

  1. (L1)

$l_{x}([a,b])=[l_{x}(a),b]+[x,l_{x}(b)]$;

  2. (L2)

$r_{x}([a,b])=[a,r_{x}(b)]-[b,r_{x}(a)]$;

  3. (L3)

$[l_{x}(a)+r_{x}(a),b]=0$;

  4. (L4)

$[l_{x},l_{y}]_{\operatorname{gl}(S)}-l_{[x,y]_{L_{0}}}=\operatorname{ad}_{\omega(x,y)}$;

  5. (L5)

$[l_{x},r_{y}]_{\operatorname{gl}(S)}-r_{[x,y]_{L_{0}}}=\operatorname{Ad}_{\omega(x,y)}$;

  6. (L6)

$r_{y}(r_{x}(a)+l_{x}(a))=0$;

  7. (L7)

$l_{x}(\omega(y,z))-l_{y}(\omega(x,z))-r_{z}(\omega(x,y))=$
$=\omega([x,y]_{L_{0}},z)-\omega(x,[y,z]_{L_{0}})+\omega(y,[x,z]_{L_{0}})$

for any $x,y\in L_{0}$ and for any $a,b\in S$. Notice that these equations
where also studied in [6] in the case of Leibniz algebra _split extensions_.

######  Remark 2.2.

The first three equations state that the pair $(l_{x},r_{x})$ is a
biderivation of the Leibniz algebra $S$, for any $x\in L_{0}$. Biderivations
of low-dimensional Leibniz algebras were classified in [20] and it turns out
that

  * •

$\operatorname{Bider}(S_{1})=\left\\{(d,-d)\;|\;d\in\operatorname{Der}(S_{1})\right\\}$
and

$\operatorname{Der}(S_{1})=\left\\{\begin{pmatrix}\alpha&\beta\\\ 0&0\\\
\end{pmatrix}\Bigg{|}\;\alpha,\beta\in\mathbb{F}\right\\};$

  * •

$\operatorname{Bider}(S_{2})=\left\\{\left(\begin{pmatrix}\alpha&\alpha\\\
0&0\\\ \end{pmatrix},\begin{pmatrix}0&\beta\\\ 0&0\\\
\end{pmatrix}\right)\Bigg{|}\;\alpha,\beta\in\mathbb{F}\right\\}$.

We study now in detail the non-abelian extension (1) in both cases that $S$ is
isomorphic either to $S_{1}$ or to $S_{2}$.

###  2.1 $S$ is a Lie algebra

When $S\cong S_{1}$, we have that $r_{y}=-l_{y}$, for any $y\in L_{0}$ and the
bilinear operation of $L_{0}\ltimes_{\omega}S_{1}$ becomes

$[(x,a),(y,b)]_{(l,\omega)}=(0,[a,b]+l_{x}(b)-l_{y}(a)+\omega(x,y)).$

The linear map $l_{x}$ is represented by a $2\times 2$ matrix

$\begin{pmatrix}\alpha_{x}&\beta_{x}\\\ 0&0\\\ \end{pmatrix}$

with $\alpha_{x}$,$\beta_{x}\in\mathbb{F}$. From equations (L4)-(L5) it turns
out that

$\omega(x,y)=(\alpha_{x}\beta_{y}-\alpha_{y}\beta_{x})e_{1},\;\;\forall x,y\in
L_{0}$

and the $2$-cocycle $\omega$ is skew-symmetric. Moreover, equations (L6)-(L7)
are automatically satisfied and the resulting algebra
$L_{0}\ltimes_{\omega}S_{1}\cong L$ is a Lie algebra. We conclude that $L$ is
isomorphic to the direct sum of $S_{1}$ and $L_{0}\cong\mathbb{F}^{n-2}$.

###  2.2 $S$ is not a Lie algebra

With the change of basis $e_{2}\mapsto e_{2}-e_{1}$, $S_{2}$ becomes the
Leibniz algebra with basis $\\{e_{1},e_{2}\\}$ and the only non-trivial
bracket given by $[e_{2},e_{1}]=e_{1}$. Now a biderivation of $S_{1}$ is
represented by a pair of matrices

$\left(\begin{pmatrix}\alpha&0\\\ 0&0\\\
\end{pmatrix},\begin{pmatrix}0&\beta\\\ 0&0\\\ \end{pmatrix}\right)$

with $\alpha,\beta\in\mathbb{F}$ and the pair
$(l_{x},r_{x})\in\operatorname{Bider}(S_{2})$ is defined by
$l_{x}(e_{1})=\alpha_{x}e_{1}$ and $r_{x}(e_{2})=\beta_{x}e_{1}$, for any
$x\in L_{0}$.

Equation (L4) states that
$\left[l_{x},l_{y}\right]_{\operatorname{gl}(S_{2})}=\left[\omega(x,y),-\right]$,
with

$\displaystyle\left[l_{x},l_{y}\right]_{\operatorname{gl}(S_{2})}=l_{x}\circ
l_{y}-l_{y}\circ l_{x}$ $\displaystyle=\begin{pmatrix}\alpha_{x}&0\\\
0&0\end{pmatrix}\begin{pmatrix}\alpha_{y}&0\\\
0&0\end{pmatrix}-\begin{pmatrix}\alpha_{y}&0\\\
0&0\end{pmatrix}\begin{pmatrix}\alpha_{x}&0\\\ 0&0\end{pmatrix}=$
$\displaystyle=\begin{pmatrix}\alpha_{x}\alpha_{y}&0\\\
0&0\end{pmatrix}-\begin{pmatrix}\alpha_{x}\alpha_{y}&0\\\
0&0\end{pmatrix}=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix},$

for any $x,y\in L_{0}$. Thus
$\omega(x,y)\in\mathrm{Z}_{l}(S_{2})=\mathbb{F}e_{1}$.

From equation (L5) we have
$\left[l_{x},r_{y}\right]_{\operatorname{gl}(S_{2})}=\left[-,\omega(x,y)\right]_{S_{2}}$,
with

$\left[l_{x},r_{y}\right]_{\operatorname{gl}(S_{2})}=l_{x}\circ
r_{y}-r_{y}\circ l_{x}=\begin{pmatrix}0&\alpha_{x}\beta_{y}\\\
0&0\end{pmatrix}-\begin{pmatrix}0&0\\\
0&0\end{pmatrix}=\begin{pmatrix}0&\alpha_{x}\beta_{y}\\\ 0&0\end{pmatrix}.$

Thus, for every $a=a_{1}e_{1}+a_{2}e_{2}\in S_{2}$ and for every $x,y\in
L_{0}$, we have

$[a,\omega(x,y)]=\left[l_{x},r_{y}\right](a)=\alpha_{x}\beta_{y}a_{2}e_{1}$

i.e. $\omega(x,y)=\alpha_{x}\beta_{y}e_{1}$. Finally, equations (L6) and (L7)
are identically satisfied.

Summarizing we have

$\begin{cases}l_{x}\equiv\begin{pmatrix}\alpha_{x}&0\\\ 0&0\end{pmatrix}\\\
\\\ r_{y}\equiv\begin{pmatrix}0&\beta_{y}\\\ 0&0\end{pmatrix}\\\ \\\
\omega(x,y)=\alpha_{x}\beta_{y}e_{1}\end{cases}$

for every $x,y\in L_{0}$ and the bilinear operation $[-,-]_{(l,r,\omega)}$
becomes

$[(x,a),(y,b)]_{(l,r,\omega)}=(0,(a_{2}b_{1}+\alpha_{x}b_{1}+\beta_{y}a_{2}+\alpha_{x}\beta_{y})e_{1}),$

for any $x$, $y\in L_{0}$ and for any $a=a_{1}e_{1}+a_{2}e_{2}$,
$b=b_{1}e_{1}+b_{2}e_{2}\in S_{2}$.

If we fix a basis $\\{f_{3},\ldots,f_{n}\\}$ of $L_{0}$ and we denote by

$\alpha_{i}=\alpha_{f_{i}},\;\;\beta_{i}=\beta_{f_{i}},\;\;\forall
i=3,\ldots,n$

then $L$ is isomorphic to the Leibniz algebra with basis
$\left\\{e_{1},e_{2},f_{3},\ldots,f_{n}\right\\}$ and non-zero brackets

$\displaystyle\left[e_{2},e_{1}\right]=e_{1}$
$\displaystyle\left[e_{2},f_{i}\right]=\beta_{i}e_{1},\quad\forall
i=3,\ldots,n$
$\displaystyle\left[f_{i},e_{1}\right]=\alpha_{i}e_{1},\quad\forall
i=3,\ldots,n$
$\displaystyle\left[f_{i},f_{j}\right]=\alpha_{i}\beta_{j}e_{1},\quad\forall
i,j=3,\ldots,n.$

With the change of basis $f_{i}\mapsto
f_{i}^{\prime}=\dfrac{f_{i}}{\beta_{i}}-e_{1}$, if $\beta_{i}\neq 0$, we
obtain that

$\displaystyle[e_{2},f_{i}^{\prime}]=e_{1}-[e_{2},e_{1}]=0,$
$\displaystyle[f_{i}^{\prime},e_{1}]=\gamma_{i}e_{1},\;\;\text{where
}\gamma_{i}=\frac{\alpha_{i}}{\beta_{i}},$
$\displaystyle[f_{i},f_{j}^{\prime}]=\alpha_{i}e_{1}-[f_{i},e_{1}]=0,$
$\displaystyle[f_{i}^{\prime},f_{j}^{\prime}]=\gamma_{i}e_{1}-\frac{1}{\beta_{i}}[f_{i},e_{1}]=0.$

If we denote again $f_{i}\equiv f_{i}^{\prime}$ and
$\alpha_{i}\equiv\gamma_{i}$ when $\beta_{i}\neq 0$, then $L$ has basis
$\\{e_{1},e_{2},f_{3},\ldots,f_{n}\\}$ and non-trivial brackets

$[e_{2},e_{1}]=e_{1},\;\;[f_{i},e_{1}]=\alpha_{i}e_{1},\;\;\forall
i=3,\ldots,n.$

Finally, when $\alpha_{i}\neq 0$, we can operate the change of basis

$f_{i}\mapsto\dfrac{f_{i}}{\alpha_{i}}-e_{2}.$

One can check that the only non-trivial bracket now is $[e_{2},e_{1}]=e_{1}$
and $L$ is isomorphic to the direct sum of $S_{2}$ and the abelian algebra
$L_{0}\cong\mathbb{F}^{n-2}$. This allows us to conclude with the following.

######  Theorem 2.2.

Let $\mathbb{F}$ be a field with $\operatorname{char}(\mathbb{F})\neq 2$. Let
$L$ be a non-nilpotent non-Lie left Leibniz algebra over $\mathbb{F}$ with
$\dim_{\mathbb{F}}L=n$ and $\dim_{\mathbb{F}}[L,L]=1$. Then $L$ is isomorphic
to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra
$S_{2}$ and an abelian algebra of dimension $n-2$. We denote this algebra by
$L_{n}$. ∎

If we suppose that $L$ is a _non-split_ algebra, i.e. $L$ cannot be written as
the direct sum of two proper ideals, then we obtain the following result, that
is a generalization of Theorem 2.6 of [11] and which is valid over a general
field $\mathbb{F}$ with $\operatorname{char}(\mathbb{F})\neq 2$.

######  Corollary 2.3.

Let $L$ be a non-split non-nilpotent non-Lie left Leibniz algebra over
$\mathbb{F}$ with $\dim_{\mathbb{F}}L=n$ and $\dim_{\mathbb{F}}[L,L]=1$. Then
$n=2$ and $L\cong S_{2}$. ∎

Now we study in detail the algebra $L_{n}=S_{2}\oplus\mathbb{F}^{n-2}$ by
describing the Lie algebra of derivations, its Lie group of automorphisms and
the Leibniz algebra of biderivations. Moreover, when $\mathbb{F}=\mathbb{R}$,
we solve the _coquegigrue problem_ (see [7] and [15]) for $L_{n}$ by
integrating it into a Lie rack.

###  2.3 Derivations, automorphisms and biderivations of $L_{n}$

Let $n\geq 2$ and let $L_{n}=S_{2}\oplus\mathbb{F}^{n-2}$. We fix the basis
$\mathcal{B}_{n}=\\{e_{1},e_{2},f_{3},\ldots,f_{n}\\}$ of $L_{n}$ and we
recall that the only non-trivial commutator is $[e_{2},e_{1}]=e_{1}$. A
straightforward application of the algorithm proposed in [20] for finding
derivations and anti-derivations of a Leibniz algebra as pair of matrices with
respect to a fixed basis produces the following.

######  Theorem 2.4.

  1. (i)

A derivation of $L_{n}$ is represented, with respect to the basis
$\mathcal{B}_{n}$, by a matrix

$\left(\begin{array}[]{c | c}\begin{array}[]{c c}\alpha&0\\\ 0&0\\\ \end{array}&\begin{array}[]{c c c c}0&0&\cdots&0\\\ 0&0&\cdots&0\\\ \end{array}\\\ \hline\cr\begin{array}[]{c c}0&a_{3}\\\ 0&a_{4}\\\ \vdots&\vdots\\\ 0&a_{n}\\\ \end{array}&A\end{array}\right)$ (2)

where $A\in\operatorname{M}_{n-2}(\mathbb{F})$.

  2. (ii)

The group of automorphisms $\operatorname{Aut}(L_{n})$ is the Lie subgroup of
$\operatorname{GL}_{n}(\mathbb{F})$ of matrices of the form

$\left(\begin{array}[]{c | c}\begin{array}[]{c c}\beta&0\\\ 0&1\\\ \end{array}&\begin{array}[]{c c c c}0&0&\cdots&0\\\ 0&0&\cdots&0\\\ \end{array}\\\ \hline\cr\begin{array}[]{c c}0&b_{3}\\\ 0&b_{4}\\\ \vdots&\vdots\\\ 0&b_{n}\\\ \end{array}&B\end{array}\right)$ (3)

where $\beta\neq 0$ and $B\in\operatorname{GL}_{n-2}(\mathbb{F})$.

  3. (iii)

The Leibniz algebra of biderivations of $L_{n}$ consists of the pairs $(d,D)$
of linear endomorphisms of $L_{n}$ which are represented by the pair of
matrices

$\left(\left(\begin{array}[]{c | c}\begin{array}[]{c c}\alpha&0\\\ 0&0\\\ \end{array}&\begin{array}[]{c c c c}0&0&\cdots&0\\\ 0&0&\cdots&0\\\ \end{array}\\\ \hline\cr\begin{array}[]{c c}0&a_{3}\\\ 0&a_{4}\\\ \vdots&\vdots\\\ 0&a_{n}\\\ \end{array}&A\end{array}\right),\left(\begin{array}[]{c | c}\begin{array}[]{c c}0&\alpha^{\prime}\\\ 0&0\\\ \end{array}&\begin{array}[]{c c c c}0&0&\cdots&0\\\ 0&0&\cdots&0\\\ \end{array}\\\ \hline\cr\begin{array}[]{c c}0&a_{3}^{\prime}\\\ 0&a_{4}^{\prime}\\\ \vdots&\vdots\\\ 0&a_{n}^{\prime}\\\ \end{array}&A^{\prime}\end{array}\right)\right)$ (4)

where $A$,$A^{\prime}\in\operatorname{M}_{n-2}(\mathbb{F})$.

∎

##  3 The integration of the Leibniz algebra $L_{n}$

The _coquecigrue problem_ is the problem formulated by J.-L. Loday in [19] of
finding a generalization of Lie third theorem to Leibniz algebras. Given a
real Leibniz algebra $L$, one wants to find a manifold endowed with a smooth
map, which plays the role of the adjoint map for Lie groups, such that the
tangent space at a distinguished element, endowed with the differential of
this map, gives a Leibniz algebra isomorphic to $L$. Moreover, when $L$ is a
Lie algebra, we want to obatin the simply connected Lie group associated with
$L$. From now on, we assume that the underlying field of any algebra is
$\mathbb{F}=\mathbb{R}$.

In [15] M. K. Kinyon shows that it is possible to define an algebraic
structure, called _rack_ , whose operation, differentiated twice, defines on
its tangent space at the unit element a Leibniz algebra structure.

######  Definition 3.1.

A _rack_ is a set $X$ with a binary operation $\rhd\colon X\times X\rightarrow
X$ which is left autodistributive

$x\rhd(y\rhd z)=(x\rhd y)\rhd(x\rhd z),\;\;\forall x,y,z\in X$

and such that the left multiplications $x\rhd-$ are bijections.

A rack is _pointed_ if there exists an element $1\in X$ such that $1\rhd x=x$
and $x\rhd 1=1$, for any $x\in X$.

A rack is a _quandle_ if the binary operation $\rhd$ is idempotent.

The first example of a rack is any group $G$ endowed with its conjugation

$x\rhd y=xyx^{-1},\;\;\forall x,y\in G.$

We denote this rack by $\operatorname{Conj}(G)$ and we observe that it is a
quandle.

######  Definition 3.2.

A pointed rack $(X,\rhd,1)$ is said to be a _Lie rack_ if $X$ is a smooth
manifold, $\rhd$ is a smooth map and the left multiplications are
diffeomorphisms.

M. K. Kinyon proved that the tangent space $\operatorname{T}_{1}X$ at the unit
element $1$ of a Lie rack $X$, endowed with the bilinear operation

$[x,y]=\frac{\partial^{2}}{\partial s\partial
t}\bigg{|}_{s,t=0}\gamma_{1}(s)\rhd\gamma_{2}(t)$

where $\gamma_{1},\gamma_{2}\colon[0,1]\rightarrow X$ are smooth paths such
that $\gamma_{1}(0)=\gamma_{2}(0)=1$, $\gamma_{1}^{\prime}(0)=x$ and
$\gamma_{2}^{\prime}(0)=y$, is a Leibniz algebra.

He also solved the coquecigrue problem for the class of _split Leibniz
algebras_. Here a Leibniz algebra is said to be _split_ if there exists an
ideal

$\operatorname{Leib}(L)\subseteq I\subseteq\operatorname{Z}_{l}(L)$

and a Lie subalgebra $M$ of $L$ such that $L\cong(M\oplus I,\\{-,-\\})$, where
the bilinear operation $\\{-,-\\}$ is defined by

$\\{(x,a),(y,b)\\}=([x,y],\rho_{x}(b))$

and $\rho\colon M\times I\rightarrow I$ is the action on the $M$-module $I$.
$L$ is said to be the _demisemidirect product_ of $M$ and $I$. More precisely,
we have the following.

######  Theorem 3.3.

[15] Let $L$ be a split Leibniz algebra. Then a Lie rack integrating $L$ is
$X=(H\oplus I,\rhd)$, where $H$ is the simply connected Lie group integrating
$M$ and the binary operation is defined by

$(g,a)\rhd(h,b)=(ghg^{-1},\phi_{g}(b)),$

where $\phi$ is the exponentiation of the Lie algebra action $\rho$.

Some years later S. Covez generalized M. K. Kinyon’s results proving that
every real Leibniz algebra admits an integration into a _Lie local rack_ (see
[7]). More recently it was showed in [16] that the integration proposed by S.
Covez is global for any nilpotent Leibniz algebra. Moreover, when a Leibniz
algebra $L$ is integrated into a Lie quandle $X$, it turns out that $L$ is a
Lie algebra and $X=\operatorname{Conj}(G)$, where $G$ is the simply connected
Lie group integrating $L$.

Our aim here is to solve the coquecigrue problem for the non-nilpotent Leibniz
algebra $L_{n}=S_{2}\oplus\mathbb{F}^{n-2}$. One can check that $S_{2}$ is a
split Leibniz algebra, in the sense of M. K. Kinyon, with
$I=\operatorname{Z}_{l}(S_{2})\cong\mathbb{R}$ and $M\cong\mathbb{R}$. Thus
$L\cong(\mathbb{R}^{2},\\{-,-\\})$ with the bilinear operation defined by

$\\{(x_{1},x_{2}),(y_{1},y_{2})\\}=(0,\rho_{x_{1}}(y_{2}))$

and $\rho_{x_{1}}(y_{2})=x_{1}y_{2}$, for any $x_{1},y_{2}\in\mathbb{R}$. It
turns out that a Lie rack integrating $S_{2}$ is $(\mathbb{R}^{2},\rhd)$,
where

$(x_{1},x_{2})\rhd(y_{1},y_{2})=(y_{1},y_{2}+e^{x_{1}}y_{2}).$

and the unit element is $(0,0)$. Finally, one can check that the binary
operation

$(x_{1},x_{2},x_{3},\ldots,x_{n})\rhd(y_{1},y_{2},y_{3},\ldots,y_{n})=(y_{1},y_{2}+e^{x_{1}}y_{2},y_{3},\ldots,y_{n})$

defines on $\mathbb{R}^{n}$ a Lie rack structure with unit element
$1=(0,\ldots,0)$, such that $(\operatorname{T}_{1}\mathbb{R}^{n},\rhd)$ is a
Leibniz algebra isomorphic to $L_{n}$. This result, combined with the ones of
Section 4 of [16], completes the classification of Lie racks whose tangent
space at the unit element gives a Leibniz algebra with one-dimensional derived
subalgebra.

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We aim to investigate if we can improve predictions of stress caused by OCD symptoms using pre-trained models, and present our statistical analysis plan in this paper. With the methods presented in this plan, we aim to avoid bias from data knowledge and thereby strengthen our hypotheses and findings.

The Wrist Angel study, which this statistical analysis plan concerns, contains data from nine participants, between 8 and 17 years old, diagnosed with obsessive-compulsive disorder (OCD). The data was obtained by an Empatica E4 wristband, which the participants wore during waking hours for 8 weeks. The purpose of the study is to assess the feasibility of predicting the in-the-wild OCD events captured during this period.

In our analysis, we aim to investigate if we can improve predictions of stress caused by OCD symptoms, and to do this we have created a pre-trained model, trained on four open-source data for stress prediction. We intend to apply this pre-trained model to the Wrist Angel data by fine-tuning, thereby utilizing transfer learning.
The pre-trained model is a convolutional neural network that uses blood volume pulse, heart rate, electrodermal activity, and skin temperature as time series windows to predict OCD events.

Furthermore, using accelerometer data, another model filters physical activity to further improve performance, given that physical activity is physiologically similar to stress.

By evaluating various ways of applying our model (fine-tuned, non-fine-tuned, pre-trained, non-pre-trained, and with or without activity classification), we contextualize the problem such that it can be assessed if transfer learning is a viable strategy in this domain.

§ SCOPE OF ANALYSIS

This analysis aims to evaluate whether transfer learning is a viable option for enhancing OCD-event predictability, which could be generalized to other areas of interest. This is especially relevant for new prediction tasks, where data scarcity is problematic.

This is done by assessing if a pre-trained deep-learning model can improve the performance of predicting obsessive-compulsive disorder (OCD) events compared to traditional machine learning techniques that were previously applied [Lønfeldt et al., 2023], using the same data set, from here on referred to as Wrist Angel data.

A proposed daily plan for data analysis on Wrist Angel data can be seen in <ref>.

The primary questions this analysis aims to answer are:

* To what extent does deep learning improve performance compared to traditional machine learning techniques?

* How might the performance of a pre-trained model (with and without fine-tuning) compare to that of a non-pre-trained model?

* What is the influence of using a simple activity model on stress prediction?

How does predicting OCD events at various time intervals, such as 0, 1, 2, 3, 4, and 5 minutes before the event, affect performance? This interval between prediction and event will be referred to as prediction lead time.

* How does the length of the signal sequence influence model performance?

A flow chart describing the steps in signal processing and data analysis. The performance of models will be logged in <ref>. (All figures are PDF).

Abbreviations. BVP: Blood volume pulse, EDA: Electrodermal activity, HR: Heart rate, TEMP: Skin temperature, DL: Deep learning, AC: Activity classification.

$^*$ For a visual example of the optimization steps see <ref> in Appendix. The letters correspond to the sub-tables.

$^5$ The preprocessing pipeline is the same as used in [Lønfeldt et al., 2023].

§ PROPOSED DATA PREPROCESSING

To get an understanding of the data, we will perform an exploratory analysis, computing metrics such as means and standard deviations for continuous variables and frequencies for categorical variables. Furthermore, we will plot histograms for physiological and accelerometer data.
The accelerometer data will be used to evaluate and tune the threshold for a physical activity classification.

The preprocessing pipeline will be the same one used in the previous study and is explained in [Lønfeldt et al., 2023] with the possibility of minor changes as explained below.

The data used in this study comes from the E4 wristband, 4 signals will be used in our model: Blood volume pulse (BVP), heart rate (HR), electrodermal activity (EDA), and skin temperature (TEMP). The tags and the timestamp related to the tag are used to pinpoint the stress episodes and to create the dataset.
Additionally, accelerometer data is used for our physical activity classification.

We will extract signals using the window strategy and test different window sizes of 60 seconds and 5 minutes. There will be a 5-minute buffer after the tag where no data will be extracted. There will be no overlapping signals.

Experiments will be done with different prediction lead times to see how early we can predict an OCD event. Specifically, prediction lead times of 0, 1, 2, 3, 4, and 5 minutes. These are practically gaps between the end of the signal and the OCD event tag.
It is done by extracting signals ending at the same time as the tag and increasing the interval up to 5 minutes with a 1-minute interval, resulting in a total of the above-mentioned 6 different ways of extracting event signals before the tag.

A sixth-order Butterworth low-pass filter with a cut-off frequency of 1Hz will be applied to EDA and TEMP. For BVP a second-order Butterworth filter with a high cut-off frequency of 12 Hz and a low cut-off frequency of 2 Hz will be used, while HR will remain unprocessed.

We plan to re-sample the data to obtain the same number of samples for all signals. Both down-sampled (4Hz) and up-sampled (64Hz) data will be used.
BVP and EDA will be re-sampled using python  which uses a Fourier method. HR and TEMP are re-sampled using linear interpolation.

The final step shall be splitting the data and standardizing the signals. This will be done by first splitting the data into train, validation, and test sets. The selected standardization method is observation-wise, meaning, we will standardize the data one observation at a time, each signal individually in all three splits.

To deal with the unbalanced data, under-sampling is used, by randomly selecting non-event windows from all non-event data. The final distribution of our data will be 1/3 event data and 2/3 non-event data.

§ DATA SETS USED TO PRE-TRAIN OUR MODEL

We have pre-trained our deep learning model on four different open-source data sets.
The datasets used are:

* The DTU dataset

It is collected through an experiment conducted at DTU. A total of 28 participants spread over three cohorts participated in the experiment. All participants wore an Empatica E4 wristband. The experiment consisted of three 5-minute phases. First, a pre-task resting phase, then, the task-solving phase, and finally, a post-task resting phase. The timed puzzle task was categorized as a stress event.

* WESAD [Schmidt et al., 2018]

WESAD is a dataset combining physiological data with stress exposure. This dataset contains data from 15 participants, each wearing an Empatica E4.
The participants went through a stressful scenario, where they were exposed to the Trier Social Stress Test (TSST)[Kirschbaum et al., 1993]. The stress test consists of a 5-minute speech and a mental arithmetic task. The non-stressful category comes from scenarios of resting at a table and watching funny video clips.

* AffectiveROAD dataset [Haouij et al., 2018]

This is from an experiment with 14 drives, from 10 different participants wearing two Empatica E4s. All participants drive the same route for three different types of roads, being exposed to stressful and non-stressful conditions. The "stress" metric is created based on a subjective measure from a co-driver, and the driver itself.

* ADARP dataset [Sah et al., 2022]

A dataset was created by Washington State University, to examine if it is possible to detect stress in subjects diagnosed with alcohol use disorder (AUD). The data was collected from 11 participants over an average of 7 days, where they were asked to wear an E4 wristband during their daily life and tag times when they felt stressed.

§ DEEP LEARNING MODELING

The event classification model we wish to apply to the Data from the Wrist Angel Feasibility Study consists of a 1D convolutional auto-encoder, from which we use the encoder and attach it to a convolutional classification network. Details about the model can be seen in <ref>. This model will be used both as a pre-trained model and as a non-pre-trained model.

Furthermore, we intend to include a threshold for standard deviation or Fourier transform frequency to classify physical activity based on accelerometer data, before using the event classification model. To determine which of the two classification methods, standard deviation or Fourier transform frequency, will be used, the in-clinic session with dance and relax data is used to create a baseline for when the children are in physical activity. The method that can most effectively distinguish the activity will then be used as our activity classification.

When using the physical activity classification, all observations classified as physical activity will be removed from the data, before performing any fine-tuning/predicting.
If 25% (based on statistics from [6]) or more of the data is removed due to the physical activity classification, we will re-sample the data to fine-tune and predict data with less physical activity.
When the physical activity classification is not used to filter the data, no observations are removed, but will still be classified, such that we can compare the physical activity classification to the final OCD event classification at a later time.

We intend to apply our event classification model in five different ways:

* Apply our pre-trained model directly to the Wrist Angel data to predict OCD events and assess the performance of the model, without the model knowing anything about the specific dataset.

* Fine-tune our pre-trained model to a subset of the Wrist Angel data, and predict OCD events of the remaining data. Assess performance, to see if fine-tuning helps. We will fine-tune in two different ways:

* Random (<ref>): Randomly splitting the Wrist Angel data in 80% train, 10% validation and 10% test.

* Random + Personalized (<ref>): Two fine-tunes. First, we extract one person as the test subject. The remaining 8 persons will be randomly split into 80% train and 20% validation data for the first fine-tuning. Thereafter, a second fine-tuning will be applied to the test subject, where the first week will be split into 80% training and 20% validation. The remaining 7 weeks will be used for testing. This will then be repeated such that each person has been extracted, giving us a total of 9 tests.

* Not using our pre-trained model, but training with the Wrist Angel data on our uninitialized model (not pre-trained). This will be used to assess if a pre-trained model improves performance. Training/fine-tuning in the same two ways as explained above with the pre-trained model.

Flow for fine-tuning method: Random.

Flow for fine-tuning method: Random + Personalized.

Visualization of the two fine-tuning/training methods.

We will apply our event classification model in the stated five different ways, both with and without doing a physical activity classification beforehand. This classification aims at removing time-series windows where the participant has performed physical activity, to not risk predicting it as an OCD event, since physical activity can have similar effects on physiological signals as stress.

§ EVALUATION METHODS

To evaluate our models we will use the metrics accuracy and F1-score.

\begin{equation}
\label{eq:acc}
Accuracy = \frac{TP+TN}{TP+FP+TN+FN}
\end{equation}

\begin{equation}
\label{eq_f1}
\text{F1-score} = 2 \cdot \frac{Precision \cdot Recall}{Precision + Recall} = \frac{2 \cdot TP}{2 \cdot TP + FP + FN}
\end{equation}

Where $TP$ is true positives, $TN$ is true negatives, $FP$ is false positives, and $FN$ is false negatives. The $F1$-$score$ is the harmonic mean of the two metrics $Precision$ and $Recall$.

Furthermore, we will plot Receiver Operating Characteristic (ROC) curves to investigate different thresholds for the classifications, and evaluate these using the Area Under the Curve (AUC).

We will evaluate our model for each of the 5 different ways as explained in <ref>. Additionally, we will evaluate each of these with and without the physical activity classification. Furthermore, we will do everything for 60-second intervals and 5-minute intervals. Finally, we will do everything on the 6 different extracted signals, namely the different periods between the signal and OCD event tag.
In total, this gives us 120 runs, as can be seen in <ref>.
We might not have time to evaluate our model in all these 120 ways. Hence, we intend to optimize the order by first evaluating everything for 0 min prediction lead time, then continue with the additional prediction lead times for the best result, etc. This strategy is visualized in <ref>

\begin{equation}
\label{eq:N_runs}
\text{Number of runs} = 5 \cdot 2 \cdot 2 \cdot 6 = 120
\end{equation}

1c     6cPrediction lead time

1c-2*5min data Physical activity classification     0min     1min     2min     3min     4min     5min

Pre-trained - Predict directly (non fine-tuned)

Pre-trained - Random fine-tuned

Pre-trained - Random + personalized fine-tuned

Uninitialized - Random train

Uninitialized - Random + personalized train
1c     6cPrediction lead time

1c-2*5min data No physical activity classification     0min     1min     2min     3min     4min     5min

Pre-trained - Predict directly (non fine-tuned)

Pre-trained - Random fine-tuned

Pre-trained - Random + personalized fine-tuned

Uninitialized - Random train

Uninitialized - Random + personalized train
1c     6cPrediction lead time

1c-2*1min data Physical activity classification     0min     1min     2min     3min     4min     5min

Pre-trained - Predict directly (non fine-tuned)

Pre-trained - Random fine-tuned

Pre-trained - Random + personalized fine-tuned

Uninitialized - Random train

Uninitialized - Random + personalized train
1c     6cPrediction lead time

1c-2*1min data No physical activity classification     0min     1min     2min     3min     4min     5min

Pre-trained - Predict directly (non fine-tuned)

Pre-trained - Random fine-tuned

Pre-trained - Random + personalized fine-tuned

Uninitialized - Random train

Uninitialized - Random + personalized train

The maximum of 120 ways we intend to apply our model to the Wrist Angel data. Depending on time restrictions we might not complete everything. We intend to first complete the 0 min column, then continue with the best result based on the F1-score, and complete this row for different prediction lead times. Thereafter, we will complete the full column for the best prediction lead time. Continuing interchanging between completing columns and rows, until everything is complete, or we run out of time.

§ DISCUSSION

Results from our deep learning model will be compared to results from the traditional machine learning model presented in [Lønfeldt et al., 2023]. Our model will be tested on the Wrist Angel data both as pre-trained with and without fine-tuning and as non-pre-trained, to examine if pre-training and fine-tuning improves performance. The model will be evaluated with and without removing the observations classified as physical activity to investigate if a simple physical activity classifier helps in stress prediction.
The signals will be shifted with different gaps between the end of the signal and the OCD event tag, such that we can explore the performance of predicting OCD events at different times ahead.
The Wrist Angel data is being split in windows of both 1min and 5min, to examine if the length of the time series influences the performance.

By publishing a statistical analysis plan before handling the Wrist Angel data, we ensure that the model and data parameters are set, such that, no data interpretation bias influences our results and decisions.

The Wrist Angel dataset introduces some limitations. First, the limited number of participants, and thus the generalizability of the model can not be properly evaluated.
Second, given the in-the-wild nature of the data, a significant portion will most likely be influenced by noise, for instance, because the wristband is not properly worn.

[Lønfeldt et al., 2023]
Nicole Nadine Lønfeldt, Kristoffer Vinther Olesen, Sneha Das, Cecilie Mora-Jensen, Anne Katrine Pagsberg, and Line Katrine Harder.
Predicting obsessive-compulsive disorder episodes in adolescents using a wearable biosensor - a wrist angel feasibility study.

[Schmidt et al., 2018]
Philip Schmidt, Attila Reiss, Robert Duerichen, Claus Marberger, and Kristof Van Laerhoven.
Introducing wesad, a multimodal dataset for wearable stress and affect detection.
In Proceedings of the 20th ACM International Conference on Multimodal Interaction, ICMI '18, page 400–408, New York, NY, USA, 2018. Association for Computing Machinery.
ISBN 9781450356923.
URL <https://doi.org/10.1145/3242969.3242985>.

[Kirschbaum et al., 1993]
Clemens Kirschbaum, Karl-Martin Pirke, and Dirk Hellhammer.
The ‘trier social stress test’ – a tool for investigating psychobiological stress responses in a laboratory setting.
Neuropsychobiology, 28:0 76–81, 02 1993.

[Haouij et al., 2018]
Neska El Haouij, Jean-Michel Poggi, Sylvie Sevestre-Ghalila, Raja Ghozi, and Mériem Jaïdane.
Affectiveroad system and database to assess driver's attention.
In Proceedings of the 33rd Annual ACM Symposium on Applied Computing, SAC '18, page 800–803, New York, NY, USA, 2018. Association for Computing Machinery.
ISBN 9781450351911.
URL <https://doi.org/10.1145/3167132.3167395>.

[Sah et al., 2022]
Ramesh Kumar Sah, Michael McDonell, Patricia Pendry, Sara Parent, Hassan Ghasemzadeh, and Michael J. Cleveland.
Adarp: A multi modal dataset for stress and alcohol relapse quantification in real life setting.
0 (arXiv:2206.14568), June 2022.
URL <http://arxiv.org/abs/2206.14568>.
arXiv:2206.14568 [cs, eess].

[6]
Jorge Mota, Paula Santos, Sandra Guerra, José C. Ribeiro, and José A. Duarte.
Patterns of daily physical activity during school days in children and adolescents.
150 (4):0 547–553.
ISSN 1520-6300.
URL <https://onlinelibrary.wiley.com/doi/abs/10.1002/ajhb.10163>.
_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/ajhb.10163.

[Mitchell et al., 2018]
Margaret Mitchell, Simone Wu, Andrew Zaldivar, Parker Barnes, Lucy Vasserman, Ben Hutchinson, Elena Spitzer, Inioluwa Deborah Raji, and Timnit Gebru.
Model cards for model reporting.
CoRR, abs/1810.03993, 2018.
URL <http://arxiv.org/abs/1810.03993>.

[Deznabi and Fiterau, 2023]
Iman Deznabi and Madalina Fiterau.
Multiwave: Multiresolution deep architectures through wavelet decomposition for multivariate time series prediction, 2023.

[Olesen et al., 2023]
Kristoffer Vinther Olesen, Nicole Nadine Lønfeldt, Sneha Das, Anne Katrine Pagsberg, and Line Katrine Harder Clemmensen.
Predicting obsessive-compulsive disorder events in children and adolescents in the wild using a wearable biosensor (wrist angel): Protocol for the analysis plan of a nonrandomized pilot study.
JMIR Res Protoc, 12:0 e48571, Nov 2023.
ISSN 1929-0748.
URL <https://www.researchprotocols.org/2023/1/e48571>.

§ APPENDIX

§.§ Model Card [Mitchell et al., 2018]

[title=Model Card - CNN Stress Classification,
breakable, sharp corners, boxrule=0.7pt]

Model Details

* Developed by Harald, Mia, and Simon at DTU, 2023 - Inspired by [Deznabi and Fiterau, 2023].

* Convolutional Neural Network.

* Pre-trained for stress classification. Will be fine-tuned for obsessive-compulsive disorder OCD event classification in children and adolescents.

Intended Use

* Intended use is to predict OCD events.

* Intended users are children with OCD, wearing an Empatica E4 wristband, such that the model can predict and warn, before an OCD event.

* Not specifically pre-trained for either children or OCD events, hence could be used for several age groups and various acute stress episodes.

* Relevant factors that might affect the model performance are personal factors such as a person's age, health, shape, and environmental factors such as temperature, and humidity.

* Evaluation factors are 1: with or without physical activity classification, 2: using respectively 1min and 5min intervals, 3: varying the gap between the signal and the event tag (0 to 5min), and 4: the five different ways of applying the model as described in <ref>.

* Accuracy and F1-score of stress classification are calculated.

* Metrics are reported as an average with a standard deviation of 10 runs.

Training Data

* The DTU dataset

Physiological data from Empatica E4 from 28 subjects. The stress categorization comes from a timed puzzle task and the non-stress categorization from a resting phase.

* WESAD [Schmidt et al., 2018]

Physiological data from Empatica E4 from 15 subjects. The stress categorization comes from a Trier Social Stress Test (TSST) [Kirschbaum et al., 1993] and the non-stress categorization from a resting phase and watching funny videos.

* AffectiveROAD dataset [Haouij et al., 2018]

Physiological data from Empatica E4 from 10 subjects (14 drives). The categorization comes from a subjective continuous stress evaluation of the driver during various driving scenarios.

* ADARP dataset [Sah et al., 2022]

Physiological data from Empatica E4 from 11 subjects diagnosed with alcohol use disorder (AUD). Data recorded in the wild. The stress categorization is from the subject tagging when they felt stressed.

Evaluation Data

* Running evaluations of the model have been done by leaving one training data set out as test data.

* The final evaluation will be done on a test set of the WristAngel data.

* The WristAngel dataset [Olesen et al., 2023]

Physiological data from Empatica E4 from 8 children with OCD. Data recorded in the wild. The stress categorization is from the child tagging when they felt stressed due to OCD symptoms.

Ethical Considerations

* Physiological data as used for the model, is very personal and should be handled in accordance with GDPR.

Caveats and Recommendations

* The final evaluation of the model has not been done yet, hence usefulness and considerations are subject to change.

Quantitative Analyses

To be written after testing on the WristAngel data.

§.§ Initial daily plan for WristAngel data analysis
(Week 50, 2023: Monday to Friday)

Day 1 (Dec 11.): Understand the setup and how the data is formatted

Day 2 (Dec 12.): Data preprocessing (this includes generating the datasets according to our models)

Day 3 (Dec 13.): The initial in-clinic data for movement

Day 4 (Dec 14.) – Day 5 (Dec 15.): Training and evaluating using WristAngel data

Additionally, a 3-day buffer where we could be there sporadically.

§.§ Flow chart

§.§ Example of how to evaluate model performance

1c     4cPrediction time

1c-2*Condition 1     0     1     $\cdots$     5

Model 1     [HTML]FFFC9E

Model 2     [HTML]FFFC9E

⋮    [HTML]FFFC9E
1c     4cPrediction time

1c-2*Condition 2     0     1     $\cdots$     5

Model 1     [HTML]FFFC9E

Model 2     [HTML]FFFC9E

⋮    [HTML]FFFC9E

Initial step: Each model in each condition with prediction lead time 0.

1c     4cPrediction time

1c-2*Condition 1     0     1     $\cdots$     5

Model 1     [HTML]EFEFEF0.2

Model 2     [HTML]9AFF990.7     [HTML]FFFC9E      [HTML]FFFC9E      [HTML]FFFC9E

$\vdots$     [HTML]EFEFEF$\vdots$
1c     4cPrediction time

1c-2*Condition 2     0     1     $\cdots$     5

Model 1     [HTML]EFEFEF0.3

Model 2     [HTML]EFEFEF0.3

$\vdots$     [HTML]EFEFEF$\vdots$

Second step: Pick the row/model and condition with the best performance. Run this model for each prediction lead time.

1c     4cPrediction time

1c-2*Condition 1     0     1     $\cdots$     5

Model 1     [HTML]EFEFEF0.2                [HTML]FFFC9E

Model 2     [HTML]EFEFEF0.7     [HTML]EFEFEF0.4
[HTML]EFEFEF$\cdots$     [HTML]9AFF990.75

⋮    [HTML]EFEFEF$\vdots$               [HTML]FFFC9E
1c     4cPrediction time

1c-2*Condition 2     0     1     $\cdots$     5

Model 1     [HTML]EFEFEF0.3                [HTML]FFFC9E

Model 2     [HTML]EFEFEF0.3             [HTML]FFFC9E

$\vdots$     [HTML]EFEFEF$\vdots$               [HTML]FFFC9E

Thirds step: Pick column/prediction lead time with best performance. Run each model and condition with this prediction lead time.

1c     4cPrediction time

1c-2*Condition 1     0     1     $\cdots$     5

Model 1     [HTML]EFEFEF0.2               [HTML]EFEFEF0.4

Model 2     [HTML]EFEFEF0.7     [HTML]EFEFEF0.4      [HTML]EFEFEF$\cdots$     [HTML]9AFF990.75

$\vdots$     [HTML]EFEFEF$\vdots$              [HTML]EFEFEF$\vdots$
1c     4cPrediction time

1c-2*Condition 2     0     1     $\cdots$     5

Model 1     [HTML]EFEFEF0.3                [HTML]EFEFEF0.3

Model 2     [HTML]EFEFEF0.3                [HTML]EFEFEF0.6

$\vdots$     [HTML]EFEFEF$\vdots$               [HTML]EFEFEF$\vdots$

Final step: Performance has not improved.

The steps of how we will evaluate the models are shown. The yellow cells are steps that will be taken. The green cells show the combination that created the best results and is the one we will continue. After the final step if time allows, we will continue with the best non-filled row or column.

[Lønfeldt et al., 2023]
Nicole Nadine Lønfeldt, Kristoffer Vinther Olesen, Sneha Das, Cecilie Mora-Jensen, Anne Katrine Pagsberg, and Line Katrine Harder.
Predicting obsessive-compulsive disorder episodes in adolescents using a wearable biosensor - a wrist angel feasibility study.

[Schmidt et al., 2018]
Philip Schmidt, Attila Reiss, Robert Duerichen, Claus Marberger, and Kristof Van Laerhoven.
Introducing wesad, a multimodal dataset for wearable stress and affect detection.
In Proceedings of the 20th ACM International Conference on Multimodal Interaction, ICMI '18, page 400–408, New York, NY, USA, 2018. Association for Computing Machinery.
ISBN 9781450356923.
URL <https://doi.org/10.1145/3242969.3242985>.

[Kirschbaum et al., 1993]
Clemens Kirschbaum, Karl-Martin Pirke, and Dirk Hellhammer.
The ‘trier social stress test’ – a tool for investigating psychobiological stress responses in a laboratory setting.
Neuropsychobiology, 28:0 76–81, 02 1993.

[Haouij et al., 2018]
Neska El Haouij, Jean-Michel Poggi, Sylvie Sevestre-Ghalila, Raja Ghozi, and Mériem Jaïdane.
Affectiveroad system and database to assess driver's attention.
In Proceedings of the 33rd Annual ACM Symposium on Applied Computing, SAC '18, page 800–803, New York, NY, USA, 2018. Association for Computing Machinery.
ISBN 9781450351911.
URL <https://doi.org/10.1145/3167132.3167395>.

[Sah et al., 2022]
Ramesh Kumar Sah, Michael McDonell, Patricia Pendry, Sara Parent, Hassan Ghasemzadeh, and Michael J. Cleveland.
Adarp: A multi modal dataset for stress and alcohol relapse quantification in real life setting.
0 (arXiv:2206.14568), June 2022.
URL <http://arxiv.org/abs/2206.14568>.
arXiv:2206.14568 [cs, eess].

[6]
Jorge Mota, Paula Santos, Sandra Guerra, José C. Ribeiro, and José A. Duarte.
Patterns of daily physical activity during school days in children and adolescents.
150 (4):0 547–553.
ISSN 1520-6300.
URL <https://onlinelibrary.wiley.com/doi/abs/10.1002/ajhb.10163>.
_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/ajhb.10163.

[Mitchell et al., 2018]
Margaret Mitchell, Simone Wu, Andrew Zaldivar, Parker Barnes, Lucy Vasserman, Ben Hutchinson, Elena Spitzer, Inioluwa Deborah Raji, and Timnit Gebru.
Model cards for model reporting.
CoRR, abs/1810.03993, 2018.
URL <http://arxiv.org/abs/1810.03993>.

[Deznabi and Fiterau, 2023]
Iman Deznabi and Madalina Fiterau.
Multiwave: Multiresolution deep architectures through wavelet decomposition for multivariate time series prediction, 2023.

[Olesen et al., 2023]
Kristoffer Vinther Olesen, Nicole Nadine Lønfeldt, Sneha Das, Anne Katrine Pagsberg, and Line Katrine Harder Clemmensen.
Predicting obsessive-compulsive disorder events in children and adolescents in the wild using a wearable biosensor (wrist angel): Protocol for the analysis plan of a nonrandomized pilot study.
JMIR Res Protoc, 12:0 e48571, Nov 2023.
ISSN 1929-0748.
URL <https://www.researchprotocols.org/2023/1/e48571>.

line above to use inline references.