# Joint clustering with alignment for temporal data in a one-point-per-
trajectory setting

Polina Arsenteva1 Mohamed Amine Benadjaoud2 and Hervé Cardot1
1) Institut de Mathématiques de Bourgogne  UMR CNRS 5584
Université de Bourgogne  Dijon  France
2) IRSN PSE-SANTE/SERAMED  Fontenay aux roses  France

###### Abstract

Temporal data, obtained in the setting where it is only possible to observe
one time point per trajectory, is widely used in different research fields,
yet remains insufficiently addressed from the statistical point of view. Such
data often contain observations of a large number of entities, in which case
it is of interest to identify a small number of representative behavior types.
In this paper, we propose a new method performing clustering simultaneously
with alignment of temporal objects inferred from these data, providing insight
into the relationships between the entities. A series of simulations confirm
the ability of the proposed approach to leverage multiple properties of the
complex data we target such as accessible uncertainties, correlations and a
small number of time points. We illustrate it on real data encoding cellular
response to a radiation treatment with high energy, supported with the results
of an enrichment analysis.
Key words: Complex temporal data, k-medoids clustering, multivariate
statistics.

##  1 Introduction

Temporal data are omnipresent in various domains of research in biology,
physics, econometrics, and many others. Among them we encounter cases where
longitudinal data are observed on the same individuals. Statistical inference
in these cases can be addressed by such frameworks as functional data analysis
(Ramsay and Silverman, 2005), time series and Gaussian process inference
(Rasmussen and Williams, 2005). However, in many studies measurements for
different time points cannot be performed on the same individuals. For
example, the aforementioned context is common in biology, including in vitro
studies, requiring such destructive measurements as real-time qPCR or
microarray (e.g. Gomez-Cabrero et al. (2019); Luo et al. (2023); Sifakis et
al. (2011)), and in vivo, where the sacrifice of the animal is often necessary
(e.g. Bertho et al. (2020); Hu et al. (2023); Kuballa et al. (2011)).

Figure 1.1: A schematic representation of the experimental design used to
obtain the data motivating this paper. Transcriptional profiles of human
endothelial cells were measured with real-time qPCR under control and under a
single irradiation dose of 20 Gy at 0 h using a linear particle accelerator
(LINAC) at 4 MV, 2.5 Gy/min, at 2, 4, 7, 14 and 21 days.

This paper is motivated by an in vitro experiment studying the response of
human endothelial cells to irradiation with a high energy level. Figure 1.1
schematically represents the experimental setting. Since the interest lies in
studying the dynamic of the response to the considered treatment, multiple
time points are measured. Due to the destructive nature of the measurements,
compromising the cells and making them unsuitable for repeated measurements,
every time point is observed on a separate culture flask. This implies that we
do not have access to temporal correlations in the dataset, and cannot observe
an actual temporal signal and treat the data as time series. In practice,
cellular populations used to take measurements for each time point are
separated from one population just prior to the experiments. Therefore, the
flasks used for different time points contain different cells, which implies
that we can reasonably assume the independence with respect to the temporal
dimension.

Datasets considered in this paper share a number of other characteristics that
contribute to their complexity and lead to a non-trivial statistical problem.
One important characteristic is the presence of two conditions, in which case
the focus is put on studying the differences in response between the two
conditions. In case of experimental studies, two conditions, referred to as
case (treated) and control (non-treated), are necessary to study the effect of
the treatment of interest. In the example presented on Figure 1.1, these
treatment conditions are irradiated and non-irradiated. The information on
both conditions can be encoded in a fold change, representing the difference
between the corresponding responses. Fold changes are widely used in biology,
but are generic by definition and can be applied in different fields, for
example in econometrics, where a similar concept is referred to as "treatment
effect" (Wooldridge, 2010).

Furthermore, multiple replicates are necessary in order to account for
measurement uncertainties, which is especially crucial given that different
time points are observed independently. In this case, the mean values over
replicates represent an actual signal, but should not be considered alone for
the inference since this does not include the information on uncertainties.
Consequently, the inference approach has to take into account the available
information on uncertainties and cross section correlations between entities.
This implies the insufficiency of simple techniques such as treating only mean
signals, which is what we observe in literature in the context of clustering
performed on such data (e.g. Dang et al. (2017); Kuballa et al. (2011); Ren et
al. (2022); Sifakis et al. (2011); Shahid et al. (2019)). The uncertainties
are generally only taken into account purely for visualization purposes of a
small number of entities (e.g. Dang et al. (2017); Luo et al. (2023); Shahid
et al. (2019); Ren et al. (2022)).

Finally, the goal in such studies is to evaluate the response of multiple
entities, whose number is often significantly larger than that of time points
that are the same for all entities, making functional data analysis and
stochastic process-based approaches unsuitable for inference. The analysis in
such cases usually involves reducing the datasets to a small number of groups
characterized by typical behavior templates, which methodologically translates
into the task of clustering. Additional information on the joint distribution
of entities is often available through the data and has to be accounted for.
For instance, the data used as an example in this work is transcriptomic,
which means that the expression of up to multiple hundreds of genes is
measured. For a given experimental condition and time point, the measurements
for all genes are taken simultaneously from the cells of one culture flask,
providing access to correlations between the genes. Additionally, in many
cases there exists prior knowledge on underlying causal relationships between
the considered entities. In this respect, the temporal aspect of the data can
be leveraged by integrating the alignment into clustering, thus on the one
hand aiding the clustering itself, and on the other hand gaining information
on the predictive nature of the entities. A number of approaches performing
joint clustering with alignment exist in the framework of functional data
analysis (e.g. Sangalli et al. (2010); Cremona and Chiaromonte (2023)) and
(e.g. Kazlauskaite et al. (2019)), which, while not adapted for dealing with
the data targeted in this paper, will serve as inspiration for the proposed
approach.

In this work, we propose a statistical framework that addresses these
challenges, while leveraging data characteristics to render the computations
time efficient. In Section 2, we introduce estimators of fold changes as well
as a new distance, accounting for the information on uncertainties and cross
section correlations that is available in the considered datasets. We present
a multivariate computationally efficient procedure performing simultaneous
alignment and clustering of the fold changes based on the constructed
distance. We choose the framework of clustering with k-medoids (Kaufmann and
Rousseeuw, 1987), since it is based on comparing elements to a medoid, a
representative element of the given cluster that is an actual member of the
population. The latter distinguishes k-medoids from k-means and many other
clustering methods, and ensures a number of important properties, such as the
ability to keep track of correlations throughout clustering, and the
possibility of rendering alignment clustering-dependent. The proposed approach
is evaluated in the framework of multiple simulation studies in Section 3 in
comparison with a number of alternative choices. Finally, in Section 4 we
present an application to the data acquired in the radiobiological
experimental setting presented earlier. The obtained information is validated
through the bioinformatic enrichment analysis of the clusters.

##  2 Materials and methods

###  2.1 Fold change modeling and estimation

In the multivariate setting, in order to avoid introducing non-existent
information by smoothing the temporal response, we consider time as discrete.
For a given dataset, we define the response variable as $Y_{ikj}^{t}$ for an
entity (gene) $i\in\\{1,2,\dotso,n_{e}\\}$, under the experimental condition
$k=0$ if control (non-irradiated) and $k=1$ if case (irradiated), at a time
point $t\in\\{t_{1},t_{2},\dotso,t_{p}\\}$ and for a replicate
$j\in\\{1,2,\dotso,n_{r}\\}$. Here we assume there are $n_{r}$ observations
for both experimental condition and every time point without loss of
generality, given that $n_{r}\geq 2$.

Two constraints with respect to the covariances between the responses follow
from the specificity of the experimental design. On the one hand, the measures
of expressions for all genes for a given experimental condition and time point
are collected from the same plate, which allows to estimate cross section
covariances between genes, i.e.
$\mathrm{Cov}\left(Y_{ikj}^{t},Y_{i^{\prime}kj}^{t}\right)$ for $i\neq
i^{\prime}$. On the other hand, we do not have access to the temporal
covariance structure due to the destructive technique used in collecting
measures from a plate for a given time point. Thus, measures for different
time points are produced individually on different cells and are not
correlated, i.e. given distinct time points $t\neq t^{\prime}$, for any
replicate pair $(j,j^{\prime})\in\\{1,2,\dotso,n_{r}\\}^{2}$ and entity pair
$(i,i^{\prime})\in\\{1,2,\dotso,n_{e}\\}^{2}$ we have
$\mathrm{Cov}\left(Y_{ikj}^{t},Y_{i^{\prime}kj^{\prime}}^{t^{\prime}}\right)=0$.

Classical estimators of the fold changes in the multivariate setting are the
pointwise estimators: a set of empirical individual fold changes is denoted by
$\Gamma=\left(\Gamma_{1},\dotso,\Gamma_{n_{e}}\right)$ where
$\Gamma_{i}=\left(\Gamma_{i}^{t_{1}},\dotso,\Gamma_{i}^{t_{p}}\right)$ such
that
$\Gamma_{i}^{t}=\frac{\sum_{j=1}^{n_{r}}Y_{i1j}^{t}-\sum_{j=1}^{n_{r}}Y_{i0j}^{t}}{n_{r}}=\overline{Y_{i1}^{t}}-\overline{Y_{i0}^{t}}$,
representing the difference between the means of the control and the case
response. However, these estimators do not take into account the information
of uncertainties and correlations present in the data. We propose a new
definition of fold changes estimators in order to fully take into account all
the information about their estimated distributions:

######  Definition 2.1.

The estimator of the fold change of entity $i$ is denoted by
$\widehat{\Gamma}_{i}$, assumed to be a random Gaussian vector and is defined
as
$\widehat{\Gamma}_{i}|\Gamma_{i},\Sigma_{\Gamma_{i}}\sim\mathcal{N}\left(\Gamma_{i},\Sigma_{\Gamma_{i}}\right)$
where $\Gamma_{i}^{t}=\overline{Y_{i1}^{t}}-\overline{Y_{i0}^{t}}$ (the
pointwise estimator), and $\Sigma_{\Gamma_{i}}$ is a diagonal matrix with the
diagonal
$(\sigma^{2}_{\Gamma_{i}^{t_{1}}},\dots,\sigma^{2}_{\Gamma_{i}^{t_{p}}})$ such
that
$\sigma^{2}_{\Gamma_{i}^{t}}=\frac{\sum_{j=1}^{n_{r}}\left[(Y_{i1j}^{t}-\overline{Y_{i1}^{t}})^{2}+(Y_{i0j}^{t}-\overline{Y_{i0}^{t}})^{2}\right]}{n_{r}-1}$.

######  Remark 2.1.

The fact that the covariance matrix in Definition 2.1 is diagonal is a direct
consequence of the second covariance constraint mentioned above.

###  2.2 Introducing a new distance between fold change estimators

Since the task at hand is clustering of the estimators of fold changes, and
thus distribution clustering, there is a need to choose an appropriate
distance. First, we expand the Definition 2.1 to a pair of fold changes by
specifying their joint distribution:

######  Definition 2.2.

The estimator of a pair of fold changes of entities $i$ and $i^{\prime}$ is
denoted as
$\left[\widehat{\Gamma}_{i}^{\intercal}\widehat{\Gamma}_{i^{\prime}}^{\intercal}\right]^{\intercal}$,
assumed to be a random Gaussian vector and is defined as follows:

$\begin{bmatrix}\widehat{\Gamma}_{i}\\\
\widehat{\Gamma}_{i^{\prime}}\end{bmatrix}\sim\mathcal{N}\left(\begin{bmatrix}\Gamma_{i}\\\
\Gamma_{i^{\prime}}\end{bmatrix},\begin{bmatrix}\Sigma_{\Gamma_{i}}&\mathrm{P}_{\Gamma_{i}\Gamma_{i^{\prime}}}\\\
\mathrm{P}_{\Gamma_{i}\Gamma_{i^{\prime}}}^{\intercal}&\Sigma_{\Gamma_{i^{\prime}}}\end{bmatrix}\right),$

where the quantities $\Gamma_{i}$, $\Gamma_{i^{\prime}}$,
$\Sigma_{\Gamma_{i}}$ and $\Sigma_{\Gamma_{i^{\prime}}}$ describing marginal
distributions of $\widehat{\Gamma}_{i}$ and $\widehat{\Gamma}_{i^{\prime}}$
are defined according to Definition 2.1, and the diagonal cross-covariance
matrix $\mathrm{P}_{\Gamma_{i}\Gamma_{i^{\prime}}}$ with the diagonal
$(\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{1}}},\dots,\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{p}}})$
such that
$\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t}}=\frac{\sum_{j=1}^{n_{r}}\left[(Y_{i1j}^{t}-\overline{Y_{i1}^{t}})(Y_{i^{\prime}1j}^{t}-\overline{Y_{i^{\prime}1}^{t}})+(Y_{i0j}^{t}-\overline{Y_{i0}^{t}})(Y_{i^{\prime}0j}^{t}-\overline{Y_{i^{\prime}0}^{t}})\right]}{n_{r}-1}$.

The chosen distance is constructed based on $L^{2}$-distance between normally
distributed fold changes estimators $\widehat{\Gamma}_{i}$ and
$\widehat{\Gamma}_{i^{\prime}}$, the latter is constructed as
$d_{2}^{2}\left(\widehat{\Gamma}_{i},\widehat{\Gamma}_{i^{\prime}}\right)=\mathbb{E}\|\widehat{\Gamma}_{i}-\widehat{\Gamma}_{i^{\prime}}\|^{2}_{2}=\|\Gamma_{i}-\Gamma_{i^{\prime}}\|^{2}_{2}+\mathrm{Tr}(\Sigma_{\Gamma_{i}})+\mathrm{Tr}(\Sigma_{\Gamma_{i^{\prime}}})-2\mathrm{Tr}(\mathrm{P}_{\Gamma_{i}\Gamma_{i^{\prime}}})$
where $\|\cdot\|_{2}$ is the the Euclidean norm (Givens and Shortt, 1984).

######  Definition 2.3.

The squared $L^{2}$-distance between fold changes estimators
$\widehat{\Gamma}_{i}$ and $\widehat{\Gamma}_{i^{\prime}}$, with the joint
distribution given in Definition 2.2, will be denoted as
$\mathbf{\widehat{d_{2}^{2}}}$ and defined as follows:

$\mathbf{\widehat{d_{2}^{2}}}\left(\widehat{\Gamma}_{i},\widehat{\Gamma}_{i^{\prime}}\right)=\sum_{l=1}^{p}\left(\Gamma_{i}^{t_{l}}-\Gamma_{i^{\prime}}^{t_{l}}\right)^{2}+\sum_{l=1}^{p}\sigma^{2}_{\Gamma_{i}^{t_{l}}}+\sum_{l=1}^{p}\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{l}}}-2\sum_{l=1}^{p}\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{l}}}.$

###  2.3 Fold change alignment

In this section, we introduce all the mathematical quantities necessary to
perform the temporal alignment of the fold changes, which will be further
applied jointly with clustering. The idea behind alignment is illustrated in
Figure 2.1. In this example, fold changes are very similar up to a time shift,
which means that alignment should significantly reduce the distance between
them and thus force them to belong to the same cluster. First, we define a
transformation of a pair of time vectors that will be referred to as a time
warp, in analogy with a similar concept in functional data analysis.

Figure 2.1: The effect of time warping illustrated on a figure, where means
with standard deviation of a pair of normalized transcriptomic fold changes
are plotted. It can be observed that after warping the fold change of the gene
CSF2 backwards (continuous line represents the original mean and the dashed
one represents the warped mean), its mean practically coincides with that of
the fold change of the gene FAS.

######  Definition 2.4.

Let $\mathcal{T}$ and $\mathcal{T}_{s}$ be sets of time vectors for considered
omic datasets. A time warp $\mathcal{W}_{s}$ of step $s\in\mathbbm{Z}$ is a
transformation of two time vectors, defined as follows:

$\displaystyle\mathcal{W}_{s}\colon\mathcal{T}^{2}$
$\displaystyle\to\mathcal{T}_{s}^{2}$
$\displaystyle\begin{pmatrix}\mathbf{t^{0}}\\\\[-8.0pt]
\mathbf{t^{0}}\end{pmatrix}$
$\displaystyle\mapsto\begin{pmatrix}\mathbf{t^{11}}\\\\[-8.0pt]
\mathbf{t^{12}}\end{pmatrix}\text{ where: }$
$\mathbf{t^{0}}=\\{t_{l}\\}_{l=1}^{p},\hskip
15.0pt\mathbf{t^{11}}=\begin{cases}\\{t_{l}\\}_{l=1}^{p-s}\text{ if
}s>0\\\\[-10.0pt] \\{t_{l}\\}_{l=1-s}^{p}\text{ if }s<0\\\\[-10.0pt]
\mathbf{t^{0}}\text{ if }s=0\end{cases}\ \text{ and }\hskip
15.0pt\mathbf{t^{12}}=\begin{cases}\\{t_{l}\\}_{l=1+s}^{p}\text{ if
}s>0\\\\[-10.0pt] \\{t_{l}\\}_{l=1}^{p+s}\text{ if }s<0\\\\[-10.0pt]
\mathbf{t^{0}}\text{ if }s=0\end{cases}.\ $

In this definition, we distinguish three major warping types: backward warp
($s<0$), forward warp ($s>0$) and identity warp ($s=0$). Next, we define a
warped fold changes pair in terms of the original fold changes:

######  Definition 2.5.

Let $\mathbf{t^{0}}\in\mathcal{T}$ be a $p$-dimensional time vector, and
$s\in\mathbbm{Z}$ a warp step. We denote as
$\left[\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}^{\intercal}\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}^{\intercal}\right]^{\intercal}$
an $s$-warped fold changes pair
$\left[\widehat{\Gamma}_{i}^{\intercal}\widehat{\Gamma}_{i^{\prime}}^{\intercal}\right]^{\intercal}$
such that:

$\begin{bmatrix}\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}\\\
\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\end{bmatrix}=\begin{bmatrix}\left(\widehat{\Gamma}_{i}^{\mathbf{t^{1}_{1}}},\dotso,\widehat{\Gamma}_{i}^{\mathbf{t^{1}_{p-|s|}}}\right)^{\intercal}\\\
\left(\widehat{\Gamma}_{i^{\prime}}^{\mathbf{t^{1}_{p-|s|+1}}},\dotso,\widehat{\Gamma}_{i^{\prime}}^{\mathbf{t^{1}_{2(p-|s|)}}}\right)^{\intercal}\end{bmatrix}\text{
where $\mathbf{t^{1}}=\mathcal{W}_{s}(\mathbf{t^{0}})$.}$

######  Remark 2.2.

In order to be able to refer directly to an individual warped fold change, we
denote it as $\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}$ with a slight abuse of
notation, since the warping transformation is applied to a pair of fold
changes.

For every fold changes pair only the first fold change is being moved since it
allows for a more convenient manipulation of warping results while being able
to examine all warping possibilities if considering both forward and backward
type warping. According to the definitions presented above, the first fold
change in the pair is being warped with a subsequent cutoff of extraneous
parts. The second fold change in the pair does not move, however its parts
that do not correspond to remaining post-warping points of the first one are
also being cut off. The calculations are detailed in the proof of Proposition
2.1.

We introduce a new dissimilarity measure between the random fold changes
estimators that is a generalization of the distance
$\mathbf{\widehat{d^{2}_{2}}}$ in order to take all the covariances into
account in the case where time warping is applied:

######  Definition 2.6.

Let $s\in\mathbbm{Z}$, and $X$ and $Y$ be $p$-dimensional Gaussian random
variables with a joint distribution

$\begin{bmatrix}X\\\
Y\end{bmatrix}\sim\mathcal{N}\left(\begin{bmatrix}\mu_{X}\\\
\mu_{Y}\end{bmatrix},\begin{bmatrix}\Sigma_{X}&\mathrm{P}_{XY}\\\
\mathrm{P}_{XY}^{\intercal}&\Sigma_{Y}\end{bmatrix}\right)\text{ such that
}\left[\mathrm{P}_{XY}\right]_{ij}=0\text{ if }j\neq i-s.$

We define a dissimilarity measure $\mathbf{\widehat{diss}_{s}}$ between $X$
and $Y$ as follows:

$\mathbf{\widehat{diss}_{s}}\left(X,Y\right)=\|\mu_{X}-\mu_{Y}\|^{2}+\mathrm{Tr}(\Sigma_{X})+\mathrm{Tr}(\Sigma_{Y})-2\sum_{l=1}^{p-|s|}\left[\mathrm{P}_{XY}\right]_{\left(l+s\mathbbm{1}_{\mathbbm{Z}_{+}^{*}}(s),l+s\mathbbm{1}_{\mathbbm{Z}_{-}^{*}}(s)\right)}.$

Applying the dissimilarity measure to the warped fold changes, we get the
expression resembling the value of $\mathbf{\widehat{d^{2}_{2}}}$ in the case
of non-warped fold changes, with extraneous part getting cut off as a result
of warping:

######  Proposition 2.1.

Let $s\in\mathbbm{Z}$ be a warp step, and
$\left[\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}^{\intercal}\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}^{\intercal}\right]^{\intercal}$
an $s$-warped fold changes pair. The value of dissimilarity
$\mathbf{\widehat{diss}_{s}}$ between the fold changes
$\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}$ and
$\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}$ can be expressed in the
following form:

$\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)=\sum_{l=l^{*}}^{p^{*}}\left(\Gamma_{i}^{t_{l}}-\Gamma_{i^{\prime}}^{t_{l+s}}\right)^{2}+\sum_{l=l^{*}}^{p^{*}}\sigma^{2}_{\Gamma_{i}^{t_{l}}}+\sum_{l=l^{*}}^{p^{*}}\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{l+s}}}-2\sum_{l=1+|s|}^{p-|s|}\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{l}}},$

where $l^{*}=1-s\mathbbm{1}_{\mathbbm{Z}_{-}^{*}}(s)$ and
$p^{*}=p-s\mathbbm{1}_{\mathbbm{Z}_{+}^{*}}(s)$.

######  Remark 2.3.

It can be noted that
$\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)=\mathbf{\widehat{d_{2}^{2}}}\left(\widehat{\Gamma}_{i},\widehat{\Gamma}_{i^{\prime}}\right)$
in the case of the identity warp $s=0$.

Algorithm 1 Joint clustering and alignment algorithm: classical version

1:Fold changes
$\widehat{\Gamma}=\left(\widehat{\Gamma}_{1},\dots,\widehat{\Gamma}_{n_{e}}\right)$,
$K\in\mathbbm{N}$, $it_{max}\in\mathbbm{N}$, $n_{init}\in\mathbbm{N}$,
$\epsilon>0$.

2:$TC\leftarrow\infty$

3:for $init\in\\{1,\dots,n_{init}\\}$ do

4: Initialize centroids
$C=\left(C_{1},\dots,C_{K}\right)\subset\\{1,\dots,n_{e}\\}$ with kmeans++

5: $TC_{it}\leftarrow\infty$

6: $\Delta TC\leftarrow\infty$

7: $it\leftarrow 1$

8: while $\Delta TC>\epsilon$ and $it<it_{max}$ do

9: $TC_{it}^{new}\leftarrow 0$

10: 1\. Assign step:

11: for $i\in\\{1,\dots,n_{e}\\}$ do

12: $d_{min}\leftarrow\infty$

13: for $k\in\\{1,\dots,K\\}$ do

14:
$s_{k}\leftarrow\operatorname*{arg\,min}_{s\in\mathcal{S}}\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{C_{k}}\circ\mathcal{W}_{s}}\right)$
$\triangleright$ Align FCs with the centroid

15:
$d_{k}\leftarrow\mathbf{\widehat{diss}_{s_{k}}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s_{k}}},\widehat{\Gamma_{C_{k}}\circ\mathcal{W}_{s_{k}}}\right)$

16: if $d_{k}<d_{min}$ then $\triangleright$ Assign FCs to centroids

17: $d_{min}\leftarrow d_{k}$

18: $Cl_{i}\leftarrow k$

19: end if

20: end for

21: end for

22: 2\. Update step:

23: for $k\in\\{1,\dots,K\\}$ do

24: $d_{min}\leftarrow\infty$

25: for $i\in cluster_{k}=\\{i\in\\{1,\dotso,n_{e}\\}|Cl_{i}=k\\}$ do
$\triangleright$ Candidate for a centroid

26: $d_{cluster_{k}}\leftarrow 0$

27: for $i^{\prime}\in cluster_{k}$ do

28:
$s_{i^{\prime}i}\leftarrow\operatorname*{arg\,min}_{s\in\mathcal{S}}\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}\right)$$\triangleright$
Align FCs with the candidate

29: $d_{cluster_{k}}\leftarrow
d_{cluster_{k}}+\mathbf{\widehat{diss}_{s_{i^{\prime}i}}}\left(\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s_{i^{\prime}i}}},\widehat{\Gamma_{i}\circ\mathcal{W}_{s_{i^{\prime}i}}}\right)$

30: end for

31: if $d_{cluster_{k}}<d_{min}$ then $\triangleright$ Choose new centroid

32: $d_{min}\leftarrow d_{cluster_{k}}$

33: $C_{k}^{new}\leftarrow i$

34: end if

35: end for

36: $TC_{it}^{new}\leftarrow TC_{it}^{new}+d_{min}$

37: end for

38: 3\. Calculate the change in total cost:

39: $\Delta TC\leftarrow TC_{it}-TC_{it}^{new}$

40: if $\Delta TC>\epsilon$ then

41: $C\leftarrow\left(C_{1}^{new},\dots,C_{K}^{new}\right)$

42: $TC_{it}\leftarrow TC_{it}^{new}$

43: end if

44: $it\leftarrow it+1$

45: end while

46: if $TC_{it}<TC$ then

47: $C\leftarrow\left(C_{1},\dots,C_{K}\right)$$\triangleright$ centroids
labels

48: $Cl\leftarrow\left(Cl_{1},\dots,Cl_{n_{e}}\right)$$\triangleright$ cluster
labels

49:
$\mathcal{W}\leftarrow\left(s_{1Cl_{1}},\dots,s_{n_{e}Cl_{n_{e}}}\right)$$\triangleright$
warps

50: $TC\leftarrow TC_{it}$

51: end if

52:end for

53:return $C$, $Cl$, $\mathcal{W}$

######  Remark 2.4.

We chose to construct time warping in a way that some parts of the fold
changes that move outside of the temporal domain considered, get cut off. This
choice was deemed as preferable to alternatives based on extending the fold
changes instead of cutting them, since they imply adding unobserved
information. However, it potentially introduces a bias in comparison between
warped and unwarped sequences, hence it is important to normalize the
dissimilarities with respect to the number of post-warping time points in
order to render them comparable (implemented in our package ScanOFC).

###  2.4 Joint clustering with alignment

The main idea behind our approach to key features selection for a given
dataset is reducing the fold changes to a small number of behavior types up to
a time shift, which translates into clustering of aligned fold changes. In
order to combine dissimilarities between fold changes with optimal alignments,
we introduce the following matrix:

######  Definition 2.7.

Let $\mathcal{S}=\\{-s_{max},\dots,s_{max}\\}\subset\mathbbm{Z}$ be a finite
set of allowed warp steps, given a maximal warping step
$s_{max}\in\mathbbm{N}$. The Optimal Warping Dissimilarity matrix, denoted
$\mathcal{OWD}$, is a matrix containing the values of the dissimilarity
measure $\mathbf{\widehat{diss}_{s}}$ for all pairs of fold changes in case of
their optimal pairwise alignment over the set of all possible warps with steps
in $\mathcal{S}$, or formally:

$\mathcal{OWD}=\left[\min_{s\in\mathcal{S}}\left[\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)\right]\right]_{1\leq
i,i^{\prime}\leq n_{e}}.$

Algorithm 2 Joint clustering and alignment algorithm based on $\mathcal{OWD}$
and $\mathcal{OW}$ matrices

1:Fold changes
$\widehat{\Gamma}=\left(\widehat{\Gamma}_{1},\dots,\widehat{\Gamma}_{n_{e}}\right)$,
$K\in\mathbbm{N}$, $it_{max}\in\mathbbm{N}$, $n_{init}\in\mathbbm{N}$,
$\epsilon>0$.

2:Compute $\mathcal{OWD}$ and $\mathcal{OW}$

3:$TC\leftarrow\infty$

4:for $init\in\\{1,\dots,n_{init}\\}$ do

5: Initialize centroids
$C=\left(C_{1},\dots,C_{K}\right)\subset\\{1,\dots,n_{e}\\}$ with kmeans++

6: $TC_{it}\leftarrow\infty$

7: $\Delta TC\leftarrow\infty$

8: $it\leftarrow 1$

9: while $\Delta TC>\epsilon$ and $it<it_{max}$ do

10: $TC_{it}^{new}\leftarrow 0$

11: 1\. Assign step:

12: for $i\in\\{1,\dots,n_{e}\\}$ do

13:
$Cl_{i}\leftarrow\operatorname*{arg\,min}_{k\in\\{1,\dotso,K\\}}\mathcal{OWD}_{iC_{k}}$
$\triangleright$ Assign aligned FCs to centroids

14: end for

15: 2\. Update step:

16: for $k\in\\{1,\dots,K\\}$ do

17: $d_{min}\leftarrow\infty$

18: for $i\in cluster_{k}=\\{i\in\\{1,\dotso,n_{e}\\}|Cl_{i}=k\\}$ do
$\triangleright$ Candidate for a centroid

19: $d_{cluster_{k}}\leftarrow\sum_{i^{\prime}\in
cluster_{k}}\mathcal{OWD}_{ii^{\prime}}$

20: if $d_{cluster_{k}}<d_{min}$ then $\triangleright$ Choose new centroid

21: $d_{min}\leftarrow d_{cluster_{k}}$

22: $C_{k}^{new}\leftarrow i$

23: end if

24: end for

25: $TC_{it}^{new}\leftarrow TC_{it}^{new}+d_{min}$

26: end for

27: 3\. Calculate the change in total cost:

28: $\Delta TC\leftarrow TC_{it}-TC_{it}^{new}$

29: if $\Delta TC>\epsilon$ then

30: $C\leftarrow\left(C_{1}^{new},\dots,C_{K}^{new}\right)$

31: $TC_{it}\leftarrow TC_{it}^{new}$

32: end if

33: $it\leftarrow it+1$

34: end while

35: if $TC_{it}<TC$ then

36: $C\leftarrow\left(C_{1},\dots,C_{K}\right)$$\triangleright$ centroids
labels

37: $Cl\leftarrow\left(Cl_{1},\dots,Cl_{n_{e}}\right)$$\triangleright$ cluster
labels

38:
$\mathcal{W}=\left(\mathcal{OW}_{1Cl_{1}},\dotso,\mathcal{OW}_{n_{e}Cl_{n_{e}}}\right)$$\triangleright$
warps

39: $TC\leftarrow TC_{it}$

40: end if

41:end for

42:return $C$, $Cl$, $\mathcal{W}$

According to Definition 2.4, alignments are uniquely defined not for a given
fold change, but for a given pair of fold changes. In other words, a fold
change may have different optimal warps when paired with different fold
changes, and thus the elements of the $\mathcal{OWD}$ matrix are not directly
comparable. Hence, the task can only be approached by iterating between
clustering and alignment until convergence. Using k-medoids allows to make
alignment clustering-dependent: while comparing elements to medoids for
clustering, their warps can also be chosen in a unique way with respect to
medoids. In comparison, many other methods such as k-means or its extensions
are unsuitable for the task. For example, k-means applied directly to the
$\mathcal{OWD}$ matrix is potentially applicable, but it is not suitable for
alignment since it does not allow to choose fold change warps uniquely.
Another option could be performing k-means through constructing a fold change
barycenter, which is unsuitable since the barycenters are not observed from
the data, and after the first iteration the information on joint distributions
is lost.

We perform clustering using "k-means like" version of k-medoids (Park and Jun,
2009) based on a series of random initializations of type k-means++ (Arthur
and Vassilvitskii, 2007). Pseudocode for the state-of-the-art version of joint
clustering and alignment, applied in the context discussed in this work, is
presented in Algorithm 1. We propose a modification of the algorithm, that
reduces the computation time by leveraging the low temporal dimensionality of
the data in the multivariate setting. Since the number of time points is
typically small, the number of possible warps has to be even smaller and known
in advance, and since the distributions of the fold change pairs under
different warps are known, it is possible to calculate all alignment options
before performing clustering, while reducing computation time and in a non-
memory-intensive way. Consequently, we introduce the following quantity, that
will be used in the modified version of the algorithm:

######  Definition 2.8.

Let $\mathcal{S}=\\{-s_{max},\dots,s_{max}\\}\subset\mathbbm{Z}$ be a finite
set of allowed warp steps, given a maximal warping step
$s_{max}\in\mathbbm{N}$. The Optimal Warp matrix, denoted $\mathcal{OW}$, is a
matrix containing, for all pairs of fold changes, the values in $\mathcal{S}$
corresponding to the warp steps allowing to achieve their optimal pairwise
alignment with respect to the dissimilarity measure
$\mathbf{\widehat{diss}_{s}}$, or formally:

$\mathcal{OW}=\left[\operatorname*{arg\,min}_{s\in\mathcal{S}}\left[\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)\right]\right]_{1\leq
i,i^{\prime}\leq n_{e}}.$

######  Proposition 2.2.

The following statements are true for matrices $\mathcal{OWD}$ and
$\mathcal{OW}$:

  1. 1.

$\mathcal{OWD}$ is symmetric.

  2. 2.

$\mathcal{OW}$ is anti-symmetric.

######  Remark 2.5.

$\mathcal{OW}$ allows to interpret the main warping types. For a given fold
changes pair, if the optimal warp is the identity warp, they are referred to
as simultaneous. If not, then one fold change in the pair is warped forward
with respect to the other, whereas the other fold change is being warped
backwards with respect to the first. In this case, the fold change that is
warped forward is referred to as ’predictive’ of other one, whereas the latter
is labeled as ’predicted’, or ’regulated’.

The modified version of the previous algorithm, presented in Algorithm 2, is
based on integrating time warping in the clustering process through pre-
calculated matrices $\mathcal{OWD}$ and $\mathcal{OW}$. The former replaces a
standard dissimilarity matrix, the latter is used to extract final warps. The
comparison between the two algorithms leads to the following result:

######  Theorem 2.1.

The following is true about the joint clustering and alignment algorithms:

  1. 1.

Algorithm 2 converges in a finite number of iterations.

  2. 2.

Algorithms 1 and 2 are equivalent, in the sense that for the same input they
produce the same output.

  3. 3.

Algorithms 1 and 2 have polynomial time complexities, that are given in the
proof. Moreover, the degree of the largest polynomials of the time complexity
of Algorithm 1 is greater than that of Algorithm 2, meaning that the latter is
less complex.

######  Remark 2.6.

In practice, the improvement of Algorithm 2 in terms of the runtime can be
very important because a large value often has to be chosen for $n_{init}$.
Since such clustering algorithms tend to be rather initialization sensitive,
it is beneficial to perform such a number of random initializations that
covers a sufficiently big range of initial combinations, which becomes
important with higher values of $n_{e}$.

######  Remark 2.7.

In line 38 of Algorithm 2, the indexing order of $\mathcal{OW}$ is important,
since this matrix is anti-symmetric, as shown in Proposition 2.1. This
specific indexing implies that the fold changes are being warped with respect
to their centroids, which remains static.

###  2.5 Sign penalty

The following feature is motivated by the fact that, in terms of
interpretation, there is an important difference between positively and
negatively expressed fold changes. As a means to reinforce this distinction in
the obtained clusters, we introduce a penalty term that increases the
dissimilarity for those pairs of entities with different signs for one or more
corresponding instances. For a warp step $s$ and entity index pair
$(i,i^{\prime})\in\\{1,\dots,n_{e}\\}^{2}$, the penalty term represents the
proportion of time points where the means of the two considered fold changes
have different signs:

$Pen\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)=\frac{1}{p-|s|}\sum_{l=1}^{p-|s|}\mathbbm{1}_{\mathbbm{R}_{-}}\left((\Gamma_{i}\circ\mathcal{W}_{s})_{t_{l}}\times(\Gamma_{i^{\prime}}\circ\mathcal{W}_{s})_{t_{l}}\right).$

By analogy with the distance matrix, a penalty matrix can be formulated:

$\left[Pen_{ii^{\prime}}\right]_{1\leq i,i^{\prime}\leq n_{e}}\text{ such that
}Pen_{ii^{\prime}}=Pen\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right).$

Finally, the penalized dissimilarity is defined with a penalization
hyperparameter $\lambda\geq 0$:

$Pen\left(d\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)\right)=\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)+\lambda\times
Pen_{ii^{\prime}}.$

This penalized dissimilarity is integrated into the aligned clustering
procedure by replacing
$\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)$
in Definitions 2.7 and 2.8.

##  3 Simulation study

A series of simulation studies was performed in order to evaluate the proposed
methodology in the framework similar to that of temporal fold changes with
respect to the existing state-of-the-art alternatives. In every scenario,
$n_{e}=300$ fold changes were simulated over $p=8$ time points ranging from
0.5 to 21. Simulated fold changes are defined by their means and their
covariance matrix. Using the same notation as previously in the context of
real datasets, let the means be represented by
$\Gamma=\left(\Gamma_{1},\dotso,\Gamma_{n_{e}}\right)$, where
$\Gamma_{i}=\left(\Gamma_{i}^{t_{1}},\dotso,\Gamma_{i}^{t_{p}}\right)$ for
$i\in\\{1,\dotso,n_{e}\\}$. The covariance matrices will be denoted by
$\Psi=(\Psi_{ii^{\prime}})_{(i,i^{\prime})\in\\{1,\dotso,n_{e}\\}^{2}}$, where
$\Psi_{ii^{\prime}}$ is a diagonal matrix, with the diagonal
$(\psi_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{1}}},\dots,\psi_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{p}}})$,
such that, for $t\in\\{t_{1},\dotso,t_{p}\\}$,
$\psi_{\Gamma_{i}\Gamma_{i^{\prime}}^{t}}=\sigma^{2}_{\Gamma_{i}^{t}}$ if
$i=i^{\prime}$, and
$\psi_{\Gamma_{i}\Gamma_{i^{\prime}}^{t}}=\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t}}$
otherwise. The differences in the design of different scenarios are presented
in Table LABEL:tab:sim_scenarios.

Table 3.1: Details on the simulation design in all scenarios, denoted by codes, such that M1 and M2 indicate two different approaches to the simulation of means, and C1-C6 indicate different ways to simulate covariances. The covariances are defined through a matrix and a scaling transformation: $\Psi_{ii^{\prime}}=f_{\Psi}(\Psi^{\prime}_{ii^{\prime}})$. The scaling constants are set as follows: $\\{cst_{1},cst_{2},cst_{3},cst_{4}\\}=\\{\max\\{\psi^{\prime}_{\Gamma_{i}\Gamma_{i^{\prime}}^{t}}|i\neq i^{\prime},t\in\\{t_{1},\dots,t_{p}\\}\\},20,100,50\\}$. |  | Nb. of
---
clusters
$\Gamma_{i}$ distribution |  $\Psi^{\prime}_{ii^{\prime}}$ distribution | $f_{\Psi}$
M1-C1 | 4 | $\mathcal{U}(\\{f_{1},f_{2},f_{3},f_{4}\\})$ | $\\!\begin{aligned} \mathcal{N}(0,2^{2})\text{ if }i=i^{\prime},0\text{ otherwise}\end{aligned}$ | $f_{\Psi}=id$
M1-C2 | 2 | $\mathcal{U}(\\{f_{1},f_{2}\\})$ | $\\!\begin{aligned} \mathcal{N}(0,2^{2})\text{ if }i=i^{\prime},0\text{ otherwise}\end{aligned}$ | $f_{\Psi}=id$
M1-C3 | 2 | $\mathcal{U}(\\{f_{1},f_{2}\\})$ | $\\!\begin{aligned} \lvert\mathcal{N}(0,2^{2})\rvert&\text{ if $\mathbbm{1}_{cl_{1}}(i)=\mathbbm{1}_{cl_{1}}(i^{\prime})$},\\\\[-3.0pt] &0\text{ otherwise}\end{aligned}$ | $f_{\Psi}(x)=\tfrac{x^{2}}{cst_{1}}$
M1-C4 | 2 | $\mathcal{U}(\\{f_{1},f_{2}\\})$ | $\\!\begin{aligned} \lvert\mathcal{N}(0,2^{2})\rvert&\text{ if $\mathbbm{1}_{cl_{1}}(i)=\mathbbm{1}_{cl_{1}}(i^{\prime})$},\\\\[-3.0pt] &0\text{ otherwise}\end{aligned}$ | $f_{\Psi}(x)=\tfrac{x^{2}}{cst_{2}}$
M1-C5 | 2 | $\mathcal{U}(\\{f_{1},f_{2}\\})$ | $\\!\begin{aligned} \mathcal{U}([0,1])\text{ if }\mathbbm{1}&{}_{cl_{1}}(i)=\mathbbm{1}_{cl_{1}}(i^{\prime})=1,\mathcal{U}([-1,0])\\\\[-7.0pt] \text{ if }\mathbbm{1}_{cl_{2}}(i)&=\mathbbm{1}_{cl_{2}}(i^{\prime})=1,0\text{ otherwise}\end{aligned}$ | $f_{\Psi}(x)=\tfrac{x^{2}}{cst_{3}}$
M1-C6 | 2 | $\mathcal{U}(\\{f_{1},f_{2}\\})$ | $\\!\begin{aligned} \mathcal{U}([0,1])\text{ if }\mathbbm{1}&{}_{cl_{1}}(i)=\mathbbm{1}_{cl_{1}}(i^{\prime})=1,\mathcal{U}([-1,0])\\\\[-7.0pt] \text{ if }\mathbbm{1}_{cl_{2}}(i)&=\mathbbm{1}_{cl_{2}}(i^{\prime})=1,0\text{ otherwise}\end{aligned}$ | $f_{\Psi}(x)=\tfrac{x^{2}}{cst_{4}}$
M2 | 4 | $\mathcal{U}(\\{f_{1},f_{2},f_{3},f_{4}\\})$ | $\\!\begin{aligned} \mathcal{N}(0,2^{2})\text{ if }i=i^{\prime},0\text{ otherwise}\end{aligned}$ | $f_{\Psi}=id$

The design of the fold change means is detailed in Table LABEL:tab:sim_means.
It can be noted that the means were simulated based on 4 behavior types that
were meant to reproduce the characteristics of the real fold changes similar
to those observed on real data. The simulations are characterized by a
relatively low level of model-imposed features. In particular, we only assume
that the fold changes have estimated probability distributions described by
means and covariances, thus we simulate directly the fold changes estimators.
The temporal complexity of the data is independent of the framework we
propose, and is only inspired by the functional patterns we observe in real
data.

Table 3.2: Distributions of the simulated fold change means for each of the 4 behavior types (clusters) in 2 different scenarios. Every simulated fold change is an independent realization of one of these templates, with all the coefficients being random (Gaussian or uniform). In particular, $s$ is the time shift parameter appearing only in the second simulation study M2, with the distribution $s\sim\mathcal{U}([-7,7])$. A continuous distribution is chosen since in real life time shifts can be reasonably assumed to be continuous, which cannot be captured since we only have access to measurements at discrete time points. | M1 | M2
---|---|---
Cluster 1 | $f_{1}(x)=\tfrac{a}{2}x^{2}+bx+c$ | $f_{1}(x)=\tfrac{a}{2}(x-s)^{2}+b(x-s)+c$
Cluster 2 | $\\!\begin{aligned} f_{2}(x)=&\tfrac{a}{3}x^{3}-\tfrac{a(r_{1}+r_{2})}{2}x^{2}\\\\[-10.0pt] &+(ar_{1}r_{2}+c)x+d\end{aligned}$ | $\\!\begin{aligned} f_{2}(x)=&\tfrac{a}{3}(x-s)^{3}-\tfrac{a(r_{1}+r_{2})}{2}(x-s)^{2}\\\\[-10.0pt] &+(ar_{1}r_{2}+c)(x-s)+d\end{aligned}$
Cluster 3 | $\\!\begin{aligned} f_{3}(x)=&\tfrac{a}{3}x^{3}-\tfrac{a(r_{1}+r_{2})}{2}x^{2}\\\\[-10.0pt] &+(ar_{1}r_{2}+c)x+d\end{aligned}$ | $\\!\begin{aligned} f_{3}(x)=&\tfrac{a}{3}(x-s)^{3}-\tfrac{a(r_{1}+r_{2})}{2}(x-s)^{2}\\\\[-10.0pt] &+(ar_{1}r_{2}+c)(x-s)+d\end{aligned}$
Cluster 4 | $\\!\begin{aligned} f_{4}(x)=&\tfrac{a}{4}x^{4}-\tfrac{a(r_{1}+r_{2}+r_{3})}{3}x^{3}\\\\[-7.0pt] &+\tfrac{a\left(r_{1}r_{2}+r_{3}(r_{1}+r_{2})\right)}{2}x^{2}\\\\[-7.0pt] &-(ar_{1}r_{2}r_{3}+b)x+c\end{aligned}$ | $f_{4}(x)=a\sin(b(x-s))+c$

The first simulation study (scenarios M1-C1 to M1-C6) focuses on the
evaluating different distances and clustering algorithms. Attempting to mimic
the real data as closely as possible, the time points were spaced in the same
way as in of of the real omic datasets:
$(t_{1},\dotso,t_{p})=(0.5,1,2,3,4,7,14,21)$. The study confirmed the
superiority of performing $\mathbf{\widehat{d_{2}^{2}}}$-based k-medoids on
fold changes-like data111See Supplementary Figure C.1 for details.. In
particular, the success of $\mathbf{\widehat{d_{2}^{2}}}$ k-medoids compared
to Wasserstein k-means and hierarchical clustering within the independent
scenarios M1-C1 and M1-C2 demonstrates that k-medoids algorithm is more
adapted to perform clustering of fold changes. The remaining scenarios
incorporating different levels of intra-cluster covariances show that
$\mathbf{\widehat{d_{2}^{2}}}$ manages to leverage the correlations for
clustering more successfully than Wasserstein or Hellinger distances.

The second simulation study (scenario M2) is aimed at evaluating the effect of
integrating time warping into clustering. Here the time points were set almost
equally spaced, with the goal of correcting the important changes in scale
arising from the horizontal shifts of the fold changes applied in order to
study the effect of alignment: $(t_{1},\dotso,t_{p})=(0.5,3,6,9,12,15,18,21)$.
Since the approach remains non-functional, it does not affect the
interpretability of the findings.

Table 3.3:  Results of the simulation study of the effect of incorporating time warping into clustering. Standard deviations of the ARI (Chacón and Rastrojo, 2023) and V-measure (Rosenberg and Hirschberg, 2007) scores are 0.04 for the $\mathcal{D}$-based k-medoids, and of order $10^{-3}$ and less for the rest of the methods. | Without alignment | With alignment
---|---|---
|  |  $\mathcal{D}$-based
---
k-medoids
|  $\mathcal{D}$-based spectral
---
clustering
|  $\mathcal{OWD}$-based
---
k-medoids
|  $\mathcal{OWD}$-based spectral
---
clustering
Mean ARI | 0.22 | 0.26 | 0.61 | 0.26
Mean V-measure | 0.39 | 0.45 | 0.67 | 0.47

The results of this simulation study are presented in Table LABEL:tab:sim2. We
compared the proposed algorithms performing k-medoids clustering jointly with
alignment to k-medoids clustering without alignment, more specifically applied
to the distance matrix $\mathcal{D}$ containing observed distances instead of
post-warping ones. We also tested spectral clustering with and without
aligning, applied to the dissimilarity matrices transformed into similarity
matrices with a transformation
$f(\mathcal{M})=\tfrac{\max\mathcal{M}-\mathcal{M}}{\max\mathcal{M}}$ for a
given matrix $\mathcal{M}$. The spectral clustering algorithm (von Luxburg,
2007) was performed using the framework implemented in Python package scikit-
learn, which consists in performing k-means clustering on relevant
eigenvectors of a normalized Laplacian matrix of the similarity matrix. On the
one hand, the study demonstrated the interest of integrating time warping in
order to take temporal shifts into account. On the other hand, using the
example of spectral clustering, often being preferred to k-medoids, the study
showed that the specific k-medoids based clustering framework proposed in this
paper leverages the information on the alignments for clustering much better
than other state-of-the-art clustering approaches applied on the same
dissimilarity matrix $\mathcal{OWD}$.

##  4 Application to real data

We demonstrate the proposed approach on real data from an in vitro study
taking place in the radiobiological context, with the goal of assessing the
effect of irradiation with higher energy level on human endothelial cells
(HUVEC) through their transcriptomic expression. This study is a part of a
research project initiated at the French Radioprotection and Nuclear Safety
Institute (IRSN) that seeks to improve our knowledge on the adverse effects
induced by radiotherapy on healthy tissues, with potential applications in
clinical setting. The transcriptomic fold changes of 157 genes, measured at 2,
4, 7, 14 and 21 days after irradiation, were estimated based on 3 replicates
from the log-transformed data according to the procedure presented in Section
2.1. We also preprocessed the fold changes in order to amplify more
significant values and reduce the difference in scale effect, the details are
presented in Appendix B. We performed clustering coupled with alignment based
on the sign-penalized optimal warping dissimilarity matrix.

(a)

(b)

Figure 4.1: Clustering of the LINAC dataset with $\mathcal{OWD}$-based
k-medoids in 5 clusters. a) Means of original normalized fold changes
(unaligned). b) Means of warped normalized fold changes (aligned).

The five clusters222These plots only contain the means of preprocessed fold
changes, giving a rough idea of the genes’ behavior but can be at times
misleading since clustering is performed on full fold changes, containing not
only means but all the information on correlations and uncertainties that can
be inferred from the replicates. that were obtained are presented in Figure
4.1. The number of clusters was set at five since this is the smallest one
allowing for a sufficient separation of features from the radiobiological
point of view. The colorcode and the legend on the plots allow to identify
which warp group (warped backward with respect to the centroid, simultaneous
with the centroid and warped forward with respect to the centroid) each gene
belongs to. The aligned version is the one used for clustering, allowing to
identify global behavior types up to a time shift, whereas the unaligned
version allows to identify temporal cascades inside every cluster, i.e. the
forward-warped predict simultaneous that predict the backward-warped. As a
result, five distinct behavior types have been identified (top to bottom): up-
regulated and tending towards zero, up-regulated initially and down-regulated
later on, steady growth, down-regulated initially and up-regulated later on,
down-regulated and tending towards zero. The clusters will be hereafter
referred to by numbers according to this order. In comparison, spectral
clustering on the similarity constructed from the $\mathcal{OWD}$ matrix, was
also performed on this data in the same manner as on simulated data in the
previous section. The obtained behavior types are much less
distinguishable333See Supplementary Figure C.3 for a visualization of the
clusters., which is also confirmed by the silhouette score (Rousseeuw, 1987):
0.36 for our k-medoids-based approach against 0.24 for spectral clustering.

In order to gain insight into the biological interpretation of the quantities
and patterns identified using our methodology, we performed the enrichment
analysis of clusters with cellular processes using Pathway Studio. As a result
of this analysis, for a list of genes contained in the given cluster, we
obtained the cellular processes matching with an important proportion of these
genes according to literature, and for each cellular process a subset of
implicated genes, as well as some statistics such as the associated p-value.
The considered processes were filtered out with respect to the p-value at the
level of 0.01, and with respect to the overlap at the level that was chosen in
order to get a sufficient amount of information for larger clusters and avoid
getting excessive information for smaller clusters. The barplot with all of
the detected cellular processes and their distributions across clusters is
presented in Appendix, Figure C.4.

It can be observed that a number of processes are highly represented in
multiple clusters, it is the case for example of cell proliferation and
adhesion. We are particularly interested in those that are highly represented
in only one cluster and not the others, thus allowing to conclude that the
cellular process in question potentially characterize this cluster. The
identified cluster-specific cellular processes are consistent with the
biological expectations. The up-regulated clusters 1 and 3 are shown to be
associated respectively with cellular aging and oxidative stress. Indeed, both
functions are related to senescence/apoptosis (Pole et al., 2016), and the
implicated genes are expected to be significantly up-regulated under high
energy irradiation. Moreover, 80% of genes from cluster 1 that were shown to
be associated with cellular aging are either forward-warped or simultaneous,
corresponding to an early expression pattern. This indicates the additional
predictive potential of warp groups identified within given clusters. On the
other hand, the genes from cluster 3 that have been linked to oxidative stress
demonstrate stable growth in expression, peaking in the late phase, that may
have been influenced by the aforementioned genes from cluster 1.

The genes from cluster 5, which has a down-regulated pattern symmetrically to
cluster 1, are associated with cell survival. This function is of opposing
nature with respect to apoptosis, and the corresponding genes are expected to
show strong down-regulation in this context. The remaining clusters are of
mixed nature and are more complex to analyze. Cluster 2 is the smallest in
size and appears to contain mainly genes with endothelium-specific functions.
Lastly, cluster 4 has been matched with DNA replication. The early down-
regulation with a subsequent up-regulation of these genes is consistent with
the idea of the DNA replication process initially slowing down as a reaction
to stress, and being reinforced further on in order to restore the damaged
tissues.

##  5 Discussion

In this article we propose a novel procedure designed for complex temporal
data in the context when only one time point can be observed per trajectory.
We designed a computationally efficient algorithm performing clustering
jointly with temporal alignment of the fold changes, chosen as the object
encoding the response in the context of two different treatment groups. Our
approach to modeling the fold changes as well as the proposed distance, with a
further generalization to a dissimilarity measure, as well as the main
algorithm have been shown to be better adapted for the treated setting than a
number of alternative options. Moreover, having applied our method to real
transcriptomic data, we managed to uncover biologically meaningful groups of
genes with interpretable patterns. The proposed approach could potentially be
extended to a more sophisticated form of clustering that may produce even
better results on complex data.

## Software availability

Python package and the LINAC data are freely available at
https://github.com/parsenteva/scanofc.

## Acknowledgments

We thank Olivier Guipaud, Fabien Milliat and Vincent Paget for the
experimental data and useful discussions. This work is supported by the
European Union through the PO FEDER-FSE Bourgogne 2014/2020 programs as part
of the project ModBioCan2020, and by Institut de Radioprotection et de Sûreté
nucléaire as part of the project ROSIRIS.

Conflict of Interest: none declared.

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##  Appendix A Proofs

Proof of Proposition 2.1
We will denote the joint distribution of the fold changes pair
$\left[\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}^{\intercal}\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}^{\intercal}\right]^{\intercal}$
in the following way:

$\begin{bmatrix}\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}\\\
\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\end{bmatrix}\sim\mathcal{N}\left(\begin{bmatrix}\Gamma_{i}\circ\mathcal{W}_{s}\\\
\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}\end{bmatrix},\begin{bmatrix}\Sigma_{\Gamma_{i}}\circ\mathcal{W}_{s}&\mathrm{P}_{\Gamma_{i}\Gamma_{i^{\prime}}}\circ\mathcal{W}_{s}\\\
(\mathrm{P}_{\Gamma_{i}\Gamma_{i^{\prime}}}\circ\mathcal{W}_{s})^{\intercal}&\Sigma_{\Gamma_{i^{\prime}}}\circ\mathcal{W}_{s}\end{bmatrix}\right).$

Using Definitions 2.5 and 2.2, the means can be expressed depending on the
warp type:

$\begin{bmatrix}\Gamma_{i}\circ\mathcal{W}_{s}\\\
\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}\end{bmatrix}=\begin{cases}\left[\Gamma_{i}^{t_{1}}\ldots\Gamma_{i}^{t_{p-s}}\Gamma_{i^{\prime}}^{t_{1+s}}\ldots\Gamma_{i^{\prime}}^{t_{p}}\right]^{\intercal}\text{
if }s>0\\\
\left[\Gamma_{i}^{t_{1-s}}\ldots\Gamma_{i}^{t_{p}}\Gamma_{i^{\prime}}^{t_{1}}\ldots\Gamma_{i^{\prime}}^{t_{p+s}}\right]^{\intercal}\text{
if }s<0\\\
\left[\Gamma_{i}^{\intercal}\Gamma_{i^{\prime}}^{\intercal}\right]^{\intercal}\text{
if }s=0\end{cases},$

Similarly, we can express the elements of the covariance matrix:

$\Sigma_{\Gamma_{i}}\circ\mathcal{W}_{s}=\begin{cases}\begin{bmatrix}\sigma^{2}_{\Gamma_{i}^{t_{1}}}&&0\\\
&\ddots\\\ 0&&\sigma^{2}_{\Gamma_{i}^{t_{p-s}}}\end{bmatrix}\text{ if
}s>0\\\\[40.0pt] \begin{bmatrix}\sigma^{2}_{\Gamma_{i}^{t_{1-s}}}&&0\\\
&\ddots\\\ 0&&\sigma^{2}_{\Gamma_{i}^{t_{p}}}\end{bmatrix}\text{ if
}s<0\\\\[40.0pt] \Sigma_{\Gamma_{i}}\text{ if }s=0\end{cases},\hskip
15.0pt\Sigma_{\Gamma_{i^{\prime}}}\circ\mathcal{W}_{s}=\begin{cases}\begin{bmatrix}\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{1+s}}}&&0\\\
&\ddots\\\ 0&&\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{p}}}\end{bmatrix}\text{ if
}s>0\\\\[40.0pt] \begin{bmatrix}\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{1}}}&&0\\\
&\ddots\\\ 0&&\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{p+s}}}\end{bmatrix}\text{ if
}s<0\\\\[30.0pt] \Sigma_{\Gamma_{i^{\prime}}}\text{ if }s=0\end{cases}.$
$\text{ and }\hskip 15.0ptK\circ\mathcal{W}_{s}=\begin{cases}\leavevmode\hbox
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}s<0\\\\[35.0pt] \Sigma_{\Gamma_{i^{\prime}}}\text{ if }s=0\end{cases}$

It can be noted that since the non-zero elements of the matrix
$K\circ\mathcal{W}_{s}$ have been moved from the diagonal to either sub-
diagonal or super-diagonal of order $s$, its trace is now equal to zero. Since
the condition on the joint distribution given in Definition 2.6 is satisfied,
we can calculate the value of $\mathbf{\widehat{diss}_{s}}$ element by
element, starting with the square norm of the difference between means:

$\|\Gamma_{i}\circ\mathcal{W}_{s}-\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}\|^{2}=\begin{cases}\sum_{l=1}^{p-s}\left(\Gamma_{i}^{t_{l}}-\Gamma_{i^{\prime}}^{t_{l+s}}\right)^{2}\text{
if }s>0\\\
\sum_{l=1-s}^{p}\left(\Gamma_{i}^{t_{l}}-\Gamma_{i^{\prime}}^{t_{l+s}}\right)^{2}\text{
if }s<0\\\
\sum_{l=1}^{p}\left(\Gamma_{i}^{t_{l}}-\Gamma_{i^{\prime}}^{t_{l+s}}\right)^{2}\text{
if }s=0\end{cases}.\ $

Next, the trace of the covariance matrix of the first warped fold change in
the pair:

$\mathrm{Tr}(\Sigma_{\Gamma_{i}}\circ\mathcal{W}_{s})=\begin{cases}\sum_{l=1}^{p-s}\sigma^{2}_{\Gamma_{i}^{t_{l}}}\text{
if }s>0\\\ \sum_{l=1-s}^{p}\sigma^{2}_{\Gamma_{i}^{t_{l}}}\text{ if }s<0\\\
\sum_{l=1}^{p}\sigma^{2}_{\Gamma_{i}^{t_{l}}}\text{ if }s=0\end{cases}.\ $

Similarly for the second fold change:

$\mathrm{Tr}(\Sigma_{\Gamma_{i^{\prime}}}\circ\mathcal{W}_{s})=\begin{cases}\sum_{l=1+s}^{p}\sigma^{2}_{\Gamma_{i}^{t_{l}}}=\sum_{l=1}^{p-s}\sigma^{2}_{\Gamma_{i}^{t_{l+s}}}\text{
if }s>0\\\
\sum_{l=1}^{p+s}\sigma^{2}_{\Gamma_{i}^{t_{l}}}=\sum_{l=1-s}^{p}\sigma^{2}_{\Gamma_{i}^{t_{l+s}}}\text{
if }s<0\\\
\sum_{l=1}^{p}\sigma^{2}_{\Gamma_{i}^{t_{l}}}=\sum_{l=1}^{p}\sigma^{2}_{\Gamma_{i}^{t_{l+s}}}\text{
if }s=0\end{cases}.\ $

Finally, the last term containing the cross-covariances is calculated:

$\sum_{l=1}^{p-|s|}\left[\mathrm{P}_{\Gamma_{i}\Gamma_{i^{\prime}}}\circ\mathcal{W}_{s}\right]_{\left(l+s\mathbbm{1}_{\mathbbm{Z}_{+}^{*}}(s),l+s\mathbbm{1}_{\mathbbm{Z}_{-}^{*}}(s)\right)}=\begin{cases}\sum_{l=1+s}^{p-s}\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{l}}}\text{
if }s>0\\\
\sum_{l=1-s}^{p+s}\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{l}}}\text{ if
}s<0\\\ \sum_{l=1}^{p}\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{l}}}\text{ if
}s=0\end{cases}.\ $

Hence, we obtain the value of the dissimilarity by writing the expression for
any step $s\in\mathbbm{Z}$. $\Box$

Proof of Proposition 2.2
Let $(i,i^{\prime})\in\\{1,\dots,n_{e}\\}^{2}$ be an entity pair. The
statements of the proposition are equivalent to saying that, for any warp step
$s\in\mathcal{S}$, we have:

$\begin{cases}\min_{s\in\mathcal{S}}\left[\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)\right]=\min_{s\in\mathcal{S}}\left[\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}\right)\right]\\\
\arg\min_{s\in\mathcal{S}}\left[\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)\right]=-\arg\min_{s\in\mathcal{S}}\left[\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}\right)\right]\end{cases}.\
$ (A.1)

Let us denote
$s_{*}=\arg\min_{s\in\mathcal{S}}\left[\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}}\right)\right]$.
To prove both parts of the proposition, it suffices to show that the following
is true:

$\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s_{*}}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s_{*}}}\right)=\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{-s_{*}}},\widehat{\Gamma_{i}\circ\mathcal{W}_{-s_{*}}}\right)$
(A.2)

Using the expression of the dissimilarity given in Proposition 2.1, we can
develop the left-hand side of (A.2):

$\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s_{*}}},\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s_{*}}}\right)=\sum_{l=l^{*}}^{p^{*}}\left(\Gamma_{i}^{t_{l}}-\Gamma_{i^{\prime}}^{t_{l+s_{*}}}\right)^{2}+\sum_{l=l^{*}}^{p^{*}}\sigma^{2}_{\Gamma_{i}^{t_{l}}}+\sum_{l=l^{*}}^{p^{*}}\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{l+s_{*}}}}-2\sum_{l=1+|s_{*}|}^{p-|s_{*}|}\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{l}}}.$
(A.3)

where $l^{*}=1-s_{*}\mathbbm{1}_{\mathbbm{Z}_{-}^{*}}(s_{*})$ and
$p^{*}=p-s_{*}\mathbbm{1}_{\mathbbm{Z}_{+}^{*}}(s_{*})$.

Similarly, we develop the right-hand side:

$\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{-s_{*}}},\widehat{\Gamma_{i}\circ\mathcal{W}_{-s_{*}}}\right)=\sum_{l=l_{*}}^{p_{*}}\left(\Gamma_{i^{\prime}}^{t_{l}}-\Gamma_{i}^{t_{l-s_{*}}}\right)^{2}+\sum_{l=l_{*}}^{p_{*}}\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{l}}}+\sum_{l=l_{*}}^{p_{*}}\sigma^{2}_{\Gamma_{i}^{t_{l-s_{*}}}}-2\sum_{l=1+|-s_{*}|}^{p-|-s_{*}|}\rho_{\Gamma_{i^{\prime}}\Gamma_{i}^{t_{l}}}.$
(A.4)

where $l_{*}=1-(-s_{*})\mathbbm{1}_{\mathbbm{Z}_{-}^{*}}(-s_{*})$ and
$p_{*}=p-(-s_{*})\mathbbm{1}_{\mathbbm{Z}_{+}^{*}}(-s_{*})$. These quantities
can be rewritten as
$l_{*}=1+s_{*}\mathbbm{1}_{\mathbbm{Z}_{+}^{*}}(s_{*})=l^{*}+s_{*}$ and
$p_{*}=p+s_{*}\mathbbm{1}_{\mathbbm{Z}_{-}^{*}}(s_{*})=p^{*}+s_{*}$. It can
also be noticed that $|-s_{*}|=|s_{*}|$, and
$\rho_{\Gamma_{i^{\prime}}\Gamma_{i}^{t_{l}}}=\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{l}}}$
by the symmetry of the covariance. Hence, we can rewrite (A.4) as follows:

$\displaystyle\begin{split}\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{-s_{*}}},\widehat{\Gamma_{i}\circ\mathcal{W}_{-s_{*}}}\right)=&\sum_{l=l^{*}+s_{*}}^{p^{*}+s_{*}}\left(\Gamma_{i}^{t_{l-s_{*}}}-\Gamma_{i^{\prime}}^{t_{l}}\right)^{2}+\sum_{l=l^{*}+s_{*}}^{p^{*}+s_{*}}\sigma^{2}_{\Gamma_{i}^{t_{l-s_{*}}}}\\\
&+\sum_{l=l^{*}+s_{*}}^{p^{*}+s_{*}}\sigma^{2}_{\Gamma_{i^{\prime}}^{t_{l}}}-2\sum_{l=1+|s_{*}|}^{p-|s_{*}|}\rho_{\Gamma_{i}\Gamma_{i^{\prime}}^{t_{l}}}.\end{split}$
(A.5)

It can be noticed that the expression in (A.5) is identical to (A.3), which
concludes the proof. $\Box$
Proof of Theorem 2.1
To prove the first statement, it suffices to show that the total cost always
decreases, that is, for every iteration $it$, $TC_{it}\geq TC_{it+1}$.

We denote $\left(Cl_{1},\dots,Cl_{n_{e}}\right)$ and
$\left(Cl_{1}^{*},\dots,Cl_{n_{e}}^{*}\right)$ cluster labels at iterations
$it$ and $it+1$ respectively. For a given initialization, the cost at
iteration $it$ can be written as follows:

$TC_{it}=\sum_{k=1}^{K}\sum_{i\in cluster_{k}}\mathcal{OWD}_{iC_{k}},$ (A.6)

given, for every cluster label $k\in\\{1,\dots,K\\}$,
$cluster_{k}=\\{i\in\\{1,\dotso,n_{e}\\}|Cl_{i}=k\\}$ the current composition
of the cluster, and $C_{k}$ the corresponding centroid. Similarly, the cost at
iteration $it+1$ can be expressed:

$TC_{it+1}=\sum_{k=1}^{K}\sum_{i\in
cluster_{k}^{*}}\mathcal{OWD}_{iC_{k}^{*}},$ (A.7)

given, for every cluster label $k\in\\{1,\dots,K\\}$,
$cluster_{k}^{*}=\\{i\in\\{1,\dotso,n_{e}\\}|Cl_{i}^{*}=k\\}$ the current
composition of the cluster, and $C_{k}^{*}$ the corresponding centroid.
Additionally, for a given cluster label $k$, we denote the migrating sub-
clusters:

  * –

the sub-cluster of elements that left cluster $k$ at $it+1$:

$cluster_{k}^{*C}=\\{i\in\\{1,\dotso,n_{e}\\}|Cl_{i}=k\text{ and
}Cl_{i}^{*}\neq k\\},$

  * –

the sub-cluster of elements that joined cluster $k$ at $it+1$:

$cluster_{k}^{C}=\\{i\in\\{1,\dotso,n_{e}\\}|Cl_{i}\neq k\text{ and
}Cl_{i}^{*}=k\\}.$

Noticing that $cluster_{k}=\left(cluster_{k}^{*}\cup
cluster_{k}^{*C}\right)\setminus cluster_{k}^{C}$, the quantity $TC_{it}$ can
be decomposed as follows:

$TC_{it}=\underbrace{\sum_{k=1}^{K}\sum_{i\in
cluster_{k}^{*}}\mathcal{OWD}_{iC_{k}}}_{\text{A}}+\underbrace{\sum_{k=1}^{K}\sum_{i\in
cluster_{k}^{*C}}\mathcal{OWD}_{iC_{k}}}_{\text{B}}-\underbrace{\sum_{k=1}^{K}\sum_{i\in
cluster_{k}^{C}}\mathcal{OWD}_{iC_{k}}}_{\text{C}}.$ (A.8)

First, it follows from the "Update" step by construction that

$A=\sum_{k=1}^{K}\sum_{i\in
cluster_{k}^{*}}\mathcal{OWD}_{iC_{k}}\geq\sum_{k=1}^{K}\sum_{i\in
cluster_{k}^{*}}\mathcal{OWD}_{iC_{k}^{*}}=TC_{it+1}.$

Next, we consider the quantities B and C. It can be noticed, by construction
of the "Assign" step, that for every $i\in cluster_{k}^{*C}$ there exists a
unique $k*\in\\{1,\dots,K\\}\setminus{k}$ such that $i\in cluster_{k*}^{C}$
and $\mathcal{OWD}_{iC_{k^{*}}}\leq\mathcal{OWD}_{iC_{k}}$. In other words,
there is a bijection between the indices in B and C, such that the
corresponding elements of the sum in B are larger than those in C. Therefore,
$\text{B}-\text{C}\geq 0$, and $TC_{it}=\text{A}+\text{B}-\text{C}\geq
TC_{it+1}$.

Thus, the total cost sequence is decreasing, and, noticing that total cost is
positive, it can be concluded that the sequence has a limit. Finally, there is
a finite number of cluster configurations possible, therefore the sequence of
total costs contains a finite number of values. Hence, the algorithm converges
in a finite number of iterations.

The second statement follows directly from Definitions 2.7 and 2.8. In
particular, we have:

  * –

Line 13 of Algorithm 2 is equivalent to lines 12-20 of Algorithm 1, since:

$\operatorname*{arg\,min}_{k\in\\{1,\dotso,K\\}}\mathcal{OWD}_{iC_{k}}=\operatorname*{arg\,min}_{k\in\\{1,\dotso,K\\}}\left(\min_{s\in\mathcal{S}}\left[\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i}\circ\mathcal{W}_{s}},\widehat{\Gamma_{C_{k}}\circ\mathcal{W}_{s}}\right)\right]\right)=\operatorname*{arg\,min}_{k\in\\{1,\dotso,K\\}}(d_{k}),$

where $d_{k}$ is the quantity from Algorithm 1 of the final value after the
for loop terminating at line 20.

  * –

Lines 19-23 of Algorithm 2 are equivalent to lines 26-34 of Algorithm 1,
since:

$\sum_{i^{\prime}\in
cluster_{k}}\mathcal{OWD}_{ii^{\prime}}=\sum_{i^{\prime}\in
cluster_{k}}\mathcal{OWD}_{i^{\prime}i}=\sum_{i^{\prime}\in
cluster_{k}}\mathbf{\widehat{diss}_{s}}\left(\widehat{\Gamma_{i^{\prime}}\circ\mathcal{W}_{s}},\widehat{\Gamma_{i}\circ\mathcal{W}_{s}}\right),$

due to the symmetry of $\mathcal{OWD}$, the final quantity being equivalent to
the value of $d_{cluster_{k}}$ at line 34 of Algorithm 1.

  * –

Line 38 of Algorithm 2 is equivalent to line 49 of Algorithm 1.

We obtain the following complexities for different parts of the algorithms:

  1. 1.

Assign step:  $\mathcal{O}(n_{e}K|\mathcal{S}|)$ for Algorithm 1, and
$\mathcal{O}(n_{e}K)$ for Algorithm 2.

  2. 2.

Update step:  $\mathcal{O}(n_{e}^{2}K|\mathcal{S}|)$ for Algorithm 1, and
$\mathcal{O}(n_{e}^{2}K)$ for Algorithm 2.

Thus, adding the complexity of calculating the matrices $\mathcal{OWD}$ and
$\mathcal{OW}$ beforehand, we obtain in total
$\mathcal{O}(n_{init}it_{max}n_{e}K|\mathcal{S}|(1+n_{e}))$ for Algorithm 1,
and $\mathcal{O}(n_{init}it_{max}n_{e}K(1+n_{e})+n_{e}^{2}|\mathcal{S}|)$ for
Algorithm 2. The degree of the largest polynomial of the former is 6, and that
of the latter is 5, hence Algorithm 2 is less complex. $\Box$

##  Appendix B Data preprocessing

Before performing clustering, certain transformations have to be applied to
the raw data in order to amplify those characteristics that are of particular
interest, and reduce those that can be ignored. We perform data scaling with
respect to the following criteria:

  * •

Scaling by standard deviation: performed in order to account for
uncertainties, so that the observations with high uncertainty caused by
individual variability appear with lower weight compared to those with low
uncertainty. Standard deviation estimates are calculated as follows: for
$i\in\\{1,2,\dotso,n_{e}\\}$ we denote
$\sigma_{\Gamma_{i}}=(\sigma_{\Gamma_{i}^{t_{1}}},\dotso,\sigma_{\Gamma_{i}^{t_{p}}})$
where $\sigma_{\Gamma_{i}^{t}}=\sqrt{\sigma^{2}_{\Gamma_{i}^{t}}}$.

  * •

Scaling by the fold change norm: performed with the purpose of diminishing the
effect of scale differences between the fold changes. The norm of
$\widehat{\Gamma}_{i}$ associated with the distance
$\mathbf{\widehat{d_{2}^{2}}}$ or the dissimilarity
$\mathbf{\widehat{diss}_{s}}$ can be expressed as follows:

$Norm(\widehat{\Gamma}_{i})=\sqrt{\|\Gamma_{i}\|_{2}^{2}+\mathrm{Tr}(\Sigma_{\Gamma_{i}})}=\sqrt{\|\Gamma_{i}\|_{2}^{2}+\sum_{l=1}^{p}\sigma^{2}_{\Gamma_{i}^{t_{l}}}}.$
(B.1)

The two scaling transformations described above are applied in a consecutive
manner: the fold change norm scaling is calculated based on the result of the
scaling by standard deviation, which implies that the norm of the final output
is equal to 1. Thus, we obtain a processed dataset, from which new pairs of
random fold changes estimators are constructed, and finally the pairwise
distances are calculated. An illustrative example for the effect of
preprocessing on the fold changes can be found in Figure C.2.

###  B.1 Fold change estimation from preprocessed data

After applying the preprocessing to the response $Y_{ikj}^{t}$ of an entity
$i$ at the time point $t$ for a replicate $j$ under the experimental condition
$k$, the response becomes:

$\widetilde{Y}_{ikj}^{t}=\dfrac{Y_{ikj}^{t}}{\sigma_{\Gamma_{i}^{t}}\times
Norm\left(\Sigma_{\Gamma_{i}}^{-1}\widehat{\Gamma}_{i}\right)}\text{, where
}\sigma_{\Gamma_{i}^{t}}=\sqrt{\sigma^{2}_{\Gamma_{i}^{t}}}.$ (B.2)

We obtain the following expression by applying the norm defined in (B.1) to
the fold change $\widehat{\Gamma}_{i}$ after the scaling by standard
deviation:

$Norm\left(\Sigma_{\Gamma_{i}}^{-1}\widehat{\Gamma}_{i}\right)=\sqrt{\sum_{l=1}^{p}\left[\left(\dfrac{\Gamma_{i}^{t_{l}}}{\sigma^{2}_{\Gamma_{i}^{t_{l}}}}\right)^{2}+1\right]}.$
(B.3)

The joint distribution of a fold change pair obtained from the preprocessed
data can be rewritten in the following way:

$\begin{bmatrix}\widehat{\widetilde{\Gamma}}_{i}\\\
\widehat{\widetilde{\Gamma}}_{i^{\prime}}\end{bmatrix}\sim\mathcal{N}\left(\begin{bmatrix}\widetilde{\Gamma}_{i}\\\
\widetilde{\Gamma}_{i^{\prime}}\end{bmatrix},\begin{bmatrix}\Sigma_{\widetilde{\Gamma}_{i}}&\mathrm{P}_{\widetilde{\Gamma}_{i}\widetilde{\Gamma}_{i^{\prime}}}\\\
\left(\mathrm{P}_{\widetilde{\Gamma}_{i}\widetilde{\Gamma}_{i^{\prime}}}\right)^{\intercal}&\Sigma_{\widetilde{\Gamma}_{i^{\prime}}}\end{bmatrix}\right)\text{
such that:}$

  * •

Means for $x\in\\{i,i^{\prime}\\}$:

$\begin{split}\widetilde{\Gamma}_{x}&=\left(\dfrac{\sum_{j=1}^{n_{r}}(Y_{i1j}^{t_{1}}-Y_{i0j}^{t_{1}})}{n_{r}\sigma_{\Gamma_{x}^{t_{1}}}Norm\left(\Sigma_{\Gamma_{x}}^{-1}\widehat{\Gamma}_{x}\right)},\dotso,\dfrac{\sum_{j=1}^{n_{r}}(Y_{i1j}^{t_{p}}-Y_{i0j}^{t_{p}})}{n_{r}\sigma_{\Gamma_{x}^{t_{p}}}Norm\left(\Sigma_{\Gamma_{x}}^{-1}\widehat{\Gamma}_{x}\right)}\right)\\\
&=\dfrac{1}{Norm\left(\Sigma_{\Gamma_{x}}^{-1}\widehat{\Gamma}_{x}\right)}\left(\dfrac{\Gamma_{x}^{t_{1}}}{\sigma_{\Gamma_{x}^{t_{1}}}},\dotso,\dfrac{\Gamma_{x}^{t_{p}}}{\sigma_{\Gamma_{x}^{t_{p}}}}\right),\end{split}$
(B.4)

  * •

Covariance matrices for $x\in\\{i,i^{\prime}\\}$:
$\Sigma_{\widetilde{\Gamma}_{x}}=\begin{bmatrix}\sigma^{2}_{\widetilde{\Gamma}_{x}^{t_{1}}}&&0\\\
&\ddots&\\\ 0&&\sigma^{2}_{\widetilde{\Gamma}_{x}^{t_{p}}}\end{bmatrix},$

$\begin{split}\text{with
}\sigma^{2}_{\widetilde{\Gamma}_{x}^{t}}=\dfrac{\sum_{j=1}^{n_{r}}\left[(\widetilde{Y}_{i1j}^{t}-\overline{\widetilde{Y}_{i1}^{t}})^{2}+(\widetilde{Y}_{i0j}^{t}-\overline{\widetilde{Y}_{i0}^{t}})^{2}\right]}{n_{r}-1}&=\dfrac{\sigma^{2}_{\Gamma_{x}^{t}}}{\sigma^{2}_{\Gamma_{x}^{t}}\left(Norm\left(\Sigma_{\Gamma_{x}}^{-1}\widehat{\Gamma}_{x}\right)\right)^{2}}\\\
&=\dfrac{1}{\left(Norm\left(\Sigma_{\Gamma_{x}}^{-1}\widehat{\Gamma}_{x}\right)\right)^{2}}\end{split}.$
(B.5)

  * •

Cross-covariance matrix:
$\mathrm{P}_{\widetilde{\Gamma}_{i}\widetilde{\Gamma}_{i^{\prime}}}=\begin{bmatrix}\rho_{\widetilde{\Gamma}_{i^{\prime}}^{t_{1}}\widetilde{\Gamma}_{x}^{t_{1}}}&&0\\\
&\ddots&\\\
0&&\rho_{\widetilde{\Gamma}_{i^{\prime}}^{t_{p}}\widetilde{\Gamma}_{x}^{t_{p}}}\end{bmatrix},$

$\begin{split}\text{with
}\rho_{\widetilde{\Gamma}_{i}^{t}\widetilde{\Gamma}_{i^{\prime}}^{t}}&=\dfrac{\sum_{j=1}^{n_{r}}\left[(\widetilde{Y}_{i1j}^{t}-\overline{\widetilde{Y}_{i1}^{t}})(\widetilde{Y}_{i^{\prime}1j}^{t}-\overline{\widetilde{Y}_{i^{\prime}1}^{t}})+(\widetilde{Y}_{i0j}^{t}-\overline{\widetilde{Y}_{i0}^{t}})(\widetilde{Y}_{i^{\prime}0j}^{t}-\overline{\widetilde{Y}_{i^{\prime}0}^{t}})\right]}{n_{r}-1}\\\
&=\dfrac{\rho_{\Gamma_{i}^{t}\Gamma_{i^{\prime}}^{t}}}{\sigma_{\Gamma_{i^{\prime}}^{t}}\sigma_{\Gamma_{i^{\prime}}^{t}}Norm\left(\Sigma_{\Gamma_{i}}^{-1}\widehat{\Gamma}_{i}\right)Norm\left(\Sigma_{\Gamma_{i^{\prime}}}^{-1}\widehat{\Gamma}_{i^{\prime}}\right)}.\end{split}$
(B.6)

##  Appendix C Supplementary figures

Figure C.1: Results of the simulation study comparing distances and clustering
algorithms. Figure C.2: The effect of raw data scaling illustrated on a
figure, where means with standard deviation of a pair of transcriptomic fold
changes are plotted, inferred from the original data on the left and from
scaled data on the right. As a result of scaling, the fold changes of genes
TBXAS1 and FUT8 are rendered significantly closer than the original. For
instance, it can be observed that the original fold changes are both
characterized by almost monotonous growth during the whole period after
irradiation. On the one hand, the curve of gene TBXAS1 is more concave, which
can be neglected, and the difference is reduced by the scaling with respect to
the norm. On the other hand, the scaling with respect to the standard
deviation reduces the peak in the mean of gene TBXAS1 observed at day 14,
which is also negligible due to very high standard deviation at that point.
Figure C.3: Means of the fold changes in each of the 5 clusters obtained as a
result of the clustering of the LINAC dataset with $\mathcal{OWD}$-based
spectral clustering. The order of clusters is chosen to maximize the pairwise
intersections with the clusters presented in Figure 4.1. Figure C.4: Summary
of the enrichment with cellular processes of the clusters obtained for the
LINAC dataset. Cellular processes, listed on the left, are sorted from the
most represented to the least. The enriched clusters are indicated with
different colors. The numbers on the bars indicate the percentage of overlap
with the given process for the given cluster.