# Envisioning MedCLIP: A Deep Dive into Explainability for Medical Vision-
Language Models

###### Abstract

Explaining Deep Learning models is becoming increasingly important in the face
of daily emerging multimodal models, particularly in safety-critical domains
like medical imaging. However, the lack of detailed investigations into the
performance of explainability methods on these models is widening the gap
between their development and safe deployment. In this work, we analyze the
performance of various explainable AI methods on a vision-language model,
MedCLIP, to demystify its inner workings. We also provide a simple methodology
to overcome the shortcomings of these methods. Our work offers a different new
perspective on the explainability of a recent well-known VLM in the medical
domain and our assessment method is generalizable to other current and
possible future VLMs.

Index Terms—  Explainable AI, Vision-Language Models, Multimodal Models

##  1 Introduction

Deep Learning (DL) models provide a remarkable performance on many tasks,
however, they often work as a black box and their internal working mechanism
is hidden from the end-user [1]. This induces skepticism and a lack of trust
in these models, particularly for cases like clinical diagnosis. Thereby
hindering the real-life deployment and adaptation of DL models in critical
domains like healthcare.

Explainable AI (XAI) [2] offers a remedy to this problem by providing insights
into the inner workings of DL models. It enables the end-users to see and
understand the rationale behind models’ prediction, enhancing fairness and
confidence in DL models. In recent years, a number of XAI methods have been
introduced to demystify these black-box behavior in DL. These methods differ
from each other mainly based on the information they offer, their
applicability (model specific or agnostic), and the scope of their explanation
[2]. Furthermore, XAI methods can be categorized as intrinsic or post-hoc
depending on the provided interpretation. Intrinsic or model-based methods
rely on the structure of the model itself and hence are limited to models like
linear regression. Whereas post-hoc methods are applied to more complex models
that are often difficult to explain. Examples of post-hoc methods include
gradient backpropagation [3], layer-wise relevance propagation [4], and class
activation mapping (CAM) [5].

Multimodal learning has shown great success in combining the information from
various modalities to increase the performance of deep learning models [6, 7].
Particularly, vision-language models (VLM) utilize both visual and textual
information to learn a meaningful representation [8, 9] and to answer complex
questions related to the associated data [10, 11]. These models bring AI a
step closer to the human-like working paradigm where AI models can process
multiple types of inputs simultaneously to perform a task. It also enables the
utilization of large-scale pre-trained vision and language models for
downstream tasks, which reduces the cost and hustle of training models from
scratch.

VLMs have also demonstrated significant utility in medical applications, as
evidenced by studies such as [12, 13, 14]. Moreover, the medical domain
contains the paired scans-reports that can be utilized to train VLMs for
enhanced performance in downstream tasks like classification, segmentation,
and image generation [15, 16, 14]. A recent VLM, MedCLIP [14] efficiently
trained the CLIP-like [17] model using a modified objective for X-Ray image-
report dataset. However, despite being very useful, the complexity of VLMs
makes them less plausible causing obstruction to their wide applications.
Explaining these VLMs is of high importance but also challenging as typical
XAI methods work with one modality i.e. text or image alone [18], which can be
misleading as VLMs heavily rely on the interaction between text and image
features. Furthermore, it is unclear whether single-modality-based
explainability methods can explain VLMs.

To address the problem of explainability in VLMs, we analyze the effectiveness
of different existing XAI methods to explain a recently introduced VLM,
MedCLIP [14]. Furthermore, we propose a simple and effective method to
overcome the shortcomings of these methods by combining the XAI methods with
the text and image interaction in VLM. This increases the plausibility as well
as provides a framework to understand what the VLM actually looks for when
making a certain prediction.

##  2 Methodology

We selected a diverse set of XAI methods for our analysis. A brief overview of
these methods is as follows:

  * •

Gradient backpropagation (GB) is a saliency-map-based method that
backpropagates the gradients to find the features of high importance [19]. It
assigns each input feature an importance score which indicates that feature’s
contribution to the model prediction.

  * •

Occlusion method is used for analyzing the importance of each region in the
input. It partly occludes an image and observes the change in the network
activation and the model’s prediction [20]. The input regions that bring the
most change are considered to be the most important.

  * •

Integrated-Gradients (IG) interpolates a baseline image to the actual input
image to quantify the importance of each pixel. The baseline image can be
random noise or pure white pixels. This is an efficient method that integrates
the gradients at different levels [21].

  * •

Grad-Shapley (GS) combines the gradients of the model predictions with respect
to the input pixels and combines them with the Shapley value [22].

We selected these four XAI methods for several reasons. First, they represent
a diverse range of techniques that offer distinct insights into the inner
workings of the DL model. GB helps us understand feature importance through
the gradient flow in the model. Occlusion provides insights into the model’s
sensitivity to different image regions by systematically occluding parts of
the input. IG method offers a way to attribute predictions to specific
features by tracking their influence across input variations. GS method, on
the other hand, enables us to quantify the contribution of each feature’s
interaction to the final prediction. Moreover, these methods are well-
established and widely recognized in the field of XAI. They have been applied
successfully across various domains, making them suitable choices for our
specific use case.

For our analysis, we first applied these methods to MedCLIP in order to
analyze their performance through explainability maps for a given image. This
is followed by using our proposed method to acquire the explainability maps
and analyze the difference in interpretability.

Fig. 1: Class-specific feature maps generated for the prompt-classifier. These
maps do not provide any significant class-specific information and therefore
are not suitable for explaining VLM like MedCLIP. Fig. 2:  Feature activation
maps were generated using the proposed methods. These activation maps are
focused and clearly explain MedCLIP. The top part shows show the VLM is able
to focus on specific areas of the input image based on the input prompts.
Whereas, the lower part shows the activation maps for the used models when
provided with the class labels as text inputs. These maps comprehensively
explain the difference in the performance of MedCLIP based on the type of text
input provided.

###  2.1 MedCLIP

MedCLIP is a recently introduced powerful VLM for chest X-ray (CXR)
classification. It combines a
BioClinicalBERT111https://huggingface.co/emilyalsentzer/Bio_ClinicalBERT-based
text-encoder backbone and a SwinTransformer [23] vision-encoder, pre-trained
on the ImageNet dataset [24]. This BERT model has been pre-trained on the
MIMIC-III dataset [25] containing the electronic health records from ICU
patients. These transformer-based large models possess the ability to
efficiently learn complex features from the input data.

First, a 224x224 input image (Ximg) is passed through the vision encoder
(Venc) to produce image embeddings ($I\in\mathbb{R}^{D}$). These embeddings
are then further projected into a lower-dimensional vector
$I_{p}\in\mathbb{R}^{M}$ via a projection head denoted as $P_{v}$.

$\displaystyle I$ $\displaystyle=V_{enc}(X_{img})$ (1) $\displaystyle I_{p}$
$\displaystyle=P_{v}(I)$ (2)

Similarly, the input text ($X_{txt}$) is tokenized and then encoded using the
text encoder $T_{enc}$. The resultant vector ($T\in\mathbb{R}^{E}$) is
subsequently projected to $T_{p}\in\mathbb{R}^{M}$ using the text projection
head $P_{t}$, to match the vision and text embeddings dimensions denoted as M
(M=512 in MedCLIP).

$\displaystyle T$ $\displaystyle=T_{enc}(X_{txt})$ (3) $\displaystyle T_{p}$
$\displaystyle=P_{t}(T)$ (4)

These text and image embeddings ($I_{p}$ and $T_{p}$) are normalized before
calculating the dot product $M_{dot}$ in a contrastive manner.

$\displaystyle M_{dot}$ $\displaystyle=I_{p}\cdot T_{p}$ (5) $\displaystyle L$
$\displaystyle=M_{dot}*\tau$ (6)

Here $L$ represents the final output logit, indicating the similarity between
the input image and the text and $\tau$ is the learnable temperature
parameter.

###  2.2 Proposed Approach

Our proposed approach is based on the idea of applying the selected XAI
methods to the embedding space of the VLM, instead of the final output of the
model. This is motivated by the fact that VLMs process each input separately
before fusing them to generate the final output (see MedCLIP section) and each
of the encoders can be treated as separate models. Hence applying the existing
XAI methods for each of the input modalities and then combining it with the
other modality can lead to better explainability. The proposed method has
three main steps:

  1. 1.

Firstly, an XAI method of choice $M_{xai}$ is applied to the image embeddings
generated by the vision-encoder of $V_{enc}$ of the MedCLIP to generate an
explainability map $F_{map}^{i}\in\mathbb{R}^{AxA}$ for each embedding
dimension $i$. This yields M distinct maps, each highlighting the input image
pixels important for one specific image embedding dimension.

$F_{map}^{i}=M_{xai}(model=V_{enc},target=i)$

where $i$ is the index of the image embedding $Ip\in\mathbb{R}^{1x512}$.

  2. 2.

In the second step, a text input $X_{t}xt$ is selected and encoded through the
text encoder $T_{end}$ to generate the embeddings $T_{p}\in\mathbb{R}^{1xM}$.
These generated embeddings are scaled by the learned temperature parameter of
the VLM.

  3. 3.

In the final step, a dot product between the image explainability maps
generated in step 1 and text embeddings is calculated to get a
$\mathbb{R}^{AxA}$ weighted average of these maps $F_{map}^{out}$. This step
calculates the similarity of each text embedding to the corresponding image
embedding as well as quantifies their contribution to the final output,
thereby providing a measure of how important each explainability map is for
the models’ prediction.

$F_{map}^{out}=T_{p}\cdot F_{map}^{all}$

where $F_{map}^{all}\in\mathbb{R}^{MxAxA}$ is a list containing $M$ generated
feature maps.

This results in a single explainability map that highlights the specific image
pixels influencing the model’s confidence with regard to the given input. It
is important to note that this method is very efficient as the second and
third steps can be repeated for different prompt embeddings once the complete
set of feature maps for an image is obtained, obviating the need for repeating
step 1 for each input text. We experimented with both class labels as well as
text prompts (sentences) as text input to the model. A set of 10 prompts was
developed for each class label. Each generated text prompt encompasses
information about the specific pathology, its location, intensity, and
variations in sub-types (e.g. ”mild linear atelectasis at the mid lung zone”).
This comparison between text prompts and class labels will help us visualize
the effect of different input types on the VLM.

####  2.2.1 Dataset

We used the MIMIC-CXR [26] dataset which is a large CXR dataset with free-text
radiology reports collected from the Beth Israel Deaconess Medical Center in
Boston, MA. This dataset contains approximately 377,110 CXR-report pairs with
14 classes (13 pathologies and 1 healthy class). We incorporated a subset of
2000 randomly selected samples along with the class labels for our analysis.

####  2.2.2 Implementation Details

We performed all experiments on a single Nvidia Quadro RTX 6000 GPU with 24GB
of memory. The MedCLIP model was implemented using the PyTorch library [27],
while we used the Captum library [28] for off-the-shelf XAI methods.

##  3 Results

Figure 1 shows the class-wise explainability maps generated using the four
selected XAI methods. It’s worth noting that these explainability maps exhibit
a remarkable degree of similarity despite having significantly different
inputs. What’s more, they collectively assign a substantial portion of the
image as important, which results in a high rate of false positives. This
consistent high false-positive behavior is observed across all four methods.
In fact, these methods often highlight pixels outside the human body in chest
X-rays as equally crucial for the final prediction. Furthermore, it is evident
that the class labels do not exert a discernible influence on the final
output. These results fail to align with the established medical diagnostic
methodologies, which are typically lesion-specific, as different regions of a
chest X-ray are critical for different diagnoses. This discrepancy further
underscores the limitations of conventional XAI methods in effectively
elucidating the mechanisms underlying the MedCLIP model. Our proposed approach
produces the feature maps depicted in Figure 2. We generated explainability
maps using both text prompts (sentences) and class labels to investigate the
influence of different text inputs. To begin, we applied our method to
MedCLIP, utilizing images and text prompts describing various lesions. As
depicted in the top section of Figure 2, our approach stands in contrast to
conventional XAI methods by avoiding false positives and accurately
highlighting the most important images. The highlighted pixel locations
closely align with established clinical diagnostic procedures for the
specified pathology. Additionally, our method effectively illustrates how
MedCLIP’s focus shifts based on the input text prompt, providing strong
evidence of this VLMs’ capacity to comprehend text and identify relevant image
pixels. It is also evident this method is capable of delivering nuanced,
comprehensive, and meaningful insights into the model’s operation. Secondly,
we employed class labels in conjunction with chest X-ray (CXR) images to
evaluate our method, as shown in the bottom section of Figure 2. The
highlighted image pixels are localized and exhibit variations across different
class labels.

##  4 Discussion and Conclusion

In this paper, we analyzed the usefulness of existing XAI methods for VLMs and
provided a simple and highly effective method to overcome their shortcomings
for the multimodal models. Through our work, we demonstrate the efficacy of
our proposed approach in comprehensively explaining the functioning of the
MedCLIP VLM, a feat that the conventional approach fails to achieve. Moreover,
our approach shows the combined effect of both input modalities on the VLM
prediction which can be of high importance in VLMs. Furthermore, the
explainability maps can help us understand the discrepancy in the performance
of DL models. We visualize the difference in activation maps for two different
text input forms in order to understand the effect of different types of text
inputs. One major benefit of this work is the flexibility to use any off-the-
shelf XAI method for VLMs, making the method versatile. Furthermore, it can
also be adapted to other VLMs by following the image and text embedding fusion
approach used in that specific model.

In conclusion, pretrained VLMs like MedCLIP have enormous potential to be used
for downstream tasks without fine-tuning. However, it is very important to
make AI more trustworthy by making these models explainable to the end user.
Further research is required to design new VLM-specific XAI methods or
frameworks for practical use of already existing methods.

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# Assessing time series irreversibility through micro-scale trends

Massimiliano Zanin  Instituto de Física Interdisciplinar y Sistemas Complejos
IFISC (CSIC-UIB), Campus UIB, 07122 Palma de Mallorca, Spain

###### Abstract

Time irreversibility, defined as the lack of invariance of the statistical
properties of a system or time series under the operation of time reversal,
has received an increasing attention during the last decades, thanks to the
information it provides about the mechanisms underlying the observed dynamics.
Following the need of analysing real-world time series, many irreversibility
metrics and tests have been proposed, each one associated with different
requirements in terms of e.g. minimum time series length or computational
cost. We here build upon previously proposed tests based on the concept of
permutation patterns, but deviating from them through the inclusion of
information about the amplitude of the signal and how this evolves over time.
We show, by means of synthetic time series, that the results yielded by this
method are complementary to the ones obtained by using permutation patterns
alone, thus suggesting that “one irreversibility metric does not fit all”. We
further apply the proposed metric to the analysis of two real-world data sets.

> Given a system, or more generally a time series representing the observable
> dynamics of a system, the first step is usually to try to characterise it
> through one or more metrics. Among these, tests assessing the irreversible
> nature of a time series, i.e. whether a time series can or cannot be
> recognised from its time-reversed version, are gaining attention.
> Irreversibility stems both from non-linearities and memory in the dynamics,
> and represents the entropy production of a system out of equilibrium; in
> short, it can be used to infer information about the physical processes
> generating a time series, even when these are not directly accessible. We
> here leverage on a previously proposed metric to estimate the
> irreversibility of a time series through the concept of permutation
> patterns, introducing information about the amplitude of the signal and how
> this changes over time. Most notably, we show that the proposed metric and
> the original one are complementary, i.e. their relative performance depends
> on the characteristics of the system under study.

##  I Introduction

The time reversibility of a time series, or more generally of a process,
refers to the fact that its statistical properties are invariant under the
operation of time reversal; in turns, a time series is said to be irreversible
when the result of applying a general function over it changes according to
the direction of the arrow of time. Time irreversibility is a fundamental
property of non-equilibrium systems, and stems from two properties observed in
many real-world systems: the presence of non-conservative forces, i.e. of
memory Zwanzig (1961); Puglisi and Villamaina (2009), and of non-linear
dynamics Lawrance (1991). Most importantly, the irreversibility of a time
series provides information about the physical mechanism generating it, even
when their details are unknown Roldán and Parrondo (2010).

While the concept of time irreversibility is an old one, going back to the
philosophy of Aristotle Akih-Kumgeh (2017), only recently it has been applied
to the study of real-world systems, with an increasing attention being devoted
to biological ones. Examples include Parkinson’s disease and time series
representing the tremors it generates Timmer  _et al._ (1993); brain dynamics,
as e.g. electroencephalographic (EEG) recordings of epileptic patients Van der
Heyden  _et al._ (1996); Schindler  _et al._ (2016); Yao  _et al._ (2020) and
in other pathologies Zanin  _et al._ (2020); human gait Orellana  _et al._
(2018); Martín-Gonzalo  _et al._ (2019); and cardiac dynamics in different
conditions Costa, Goldberger, and Peng (2005); Costa, Peng, and Goldberger
(2008); Casali  _et al._ (2008); Yao, Yao, and Wang (2019). Besides biology,
time irreversibility has also been studied in, e.g., ecological and
epidemiological time series Grenfell  _et al._ (1994); Stone, Landan, and May
(1996), and finance Ramsey and Rothman (1996); Zumbach (2009); Yamashita Rios
de Sousa, Takayasu, and Takayasu (2017).

Many irreversibility metrics have been developed to support these analyses. To
illustrate, the most notable ones include the analysis of consecutive values
Ramsey and Rothman (1996), the use of symbolic methods Daw, Finney, and Kennel
(2000), data compression dictionaries Kennel (2004), visibility graphs Lacasa
_et al._ (2012); Donges, Donner, and Kurths (2013), and permutation patterns
Zanin  _et al._ (2018); Martínez, Herrera-Diestra, and Chavez (2018).

The existence of multiple metrics to detect irreversibility does not only
respond to the development of better and more efficient ways of estimating it,
but also to the lack of concretion in its definition. As previously stated,
irreversibility is defined as any (statistically significant) change in the
result of applying a general function on a time series when the arrow of time
is reversed; yet, no restriction is imposed on that function. In other words,
time irreversibility can appear in any statistical property of the time
series, and different metrics have naturally focused on different properties.
This may lead to situations in which one time series may be assessed as
irreversible by one approach, and as reversible by another one.

In order to understand why an irreversibility metric may fail at correctly
classifying a time series, let us introduce the concept of permutation
patterns, initially proposed to assess the degree of complexity (or
determinism vs. stochasticity) of time series Bandt and Pompe (2002); Zanin
_et al._ (2012). Given a (usually short) window, the corresponding permutation
pattern is defined as the order that has to be applied to its elements to sort
them - such that, for instance, the pattern associated to values $(8,4,5)$
would be $(2,3,1)$, as the second element is the smallest one, followed by the
third and the first. It has recently been proposed that such patterns can be
used to assess irreversibility. In short, the probability of finding a pattern
in the original time series should be similar (in a statistic sense) to the
probability of finding the same pattern in the time reversed version; if this
does not hold true, then that pattern can be used to fix the direction of the
time arrow Zanin  _et al._ (2018); Martínez, Herrera-Diestra, and Chavez
(2018).

Figure 1: Irreversibility in time series. (Left) Example of an irreversible
time series (black line) and of its time reversed version (red line). (Centre)
Probability distributions of the six permutation patterns extracted from the
original time series (black bars) and from the time-reversed version (red
bars); note that the two distributions are not different in a statistical
significant way. (Right) Distribution of the slopes for the original (black
line) and time reversed (red line) time series - see Section II.2 for
definitions. The two distributions are different in a statistical significant
way ($p$-value of $0.0019$, Kolmogorov-Smirnov test).

In spite of some important advantages, like their simplicity and reduced
computational cost, permutation patterns also present the drawback of
disregarding the amplitude of the signal - to illustrate, the two time series
$(8,4,5)$ and $(16,8,10)$ will always result in the same pattern. This well-
known fact has important consequences in the estimation of the associated
entropy Fadlallah  _et al._ (2013); Rostaghi and Azami (2016); but further
leads to situations in which an irreversible time series may be classified as
not irreversible, only because such irreversibility manifests itself in
amplitude. As an example, consider the time series represented in Fig. 1 (left
panel, black line) and its time reversed version (red line). It can be
appreciated by naked eye that it is irreversible, as in fact it is not
stationary and displays a clear upward trend. Nevertheless, when the
probabilities of the permutation patterns (here considering window sizes of
$3$, hence $6$ distinct patterns can appear) are calculated and compared to
those of the time reversed version, the difference is not statistically
significant - see Fig. 1, central panel. In other words, triplets of ascending
and descending values (respectively corresponding to patterns $(1,2,3)$ and
$(3,2,1)$) appear with approximately the same probability. In fact,
irreversibility here does not stem from different pattern probabilities, but
from their steepness - i.e. from the amplitude variation within them.

In this contribution we propose an alternative irreversibility metric, based
on calculating micro-scale trends, i.e. the slope of a polynomial fit
performed over small overlapping windows of the original time series; and on
comparing their probability distribution with the one observed in the time-
reversed time series. While conceptually similar to the permutation patterns
approach, we will show that the inclusion of information about amplitude
yields significant advantages when analysing synthetic and real time series,
while retaining benefits like being almost parameter-free and of low
computational complexity.

##  II Assessing irreversibility in time series

###  II.1 The permutation patterns method

For the sake of completeness, we here synthesise how irreversibility can be
detected by means of permutation patterns. Note that several methods have
independently been proposed in the last years Zanin  _et al._ (2018);
Martínez, Herrera-Diestra, and Chavez (2018); Li, Shang, and Zhang (2019); Yao
_et al._ (2019); while the underlying concept is the same, namely extracting
permutation patterns and comparing their frequency, details about how the
statistical significance of results is assessed vary. While we here use the
method proposed in Zanin  _et al._ (2018), the reader should be aware of the
available alternatives Martínez, Herrera-Diestra, and Chavez (2018); Li,
Shang, and Zhang (2019); Yao  _et al._ (2019).

Given a time series $X=\\{x_{1},x_{2},\ldots,x_{N}\\}$ composed of $N$ values,
this is divided into $N-\delta+1$ overlapping windows of length $\delta$, such
that the $i$-th window is defined as
$w_{i}=\\{x_{i},\ldots,x_{i+\delta-1}\\}$. Values composing each window are
then sorted from smaller to larger, and the permutation $\pi_{i}$ needed to
perform this sorting is extracted. To illustrate, consider a time series
$X=\\{4,7,5,6,\ldots\\}$; for $\delta=3$, $w_{1}$ is composed of values
$\\{4,7,5\\}$, and the corresponding permutation pattern is $\pi_{1}=(1,3,2)$.
$\pi_{i}$ thus represents how values should be reordered to sort them, and
hence the structure by them created Bandt and Pompe (2002); Zanin  _et al._
(2012).

Irreversibility can be assessed by noting that the distribution of the
permutation patterns probability $p(\pi)$ should be the same in the original
and time reversed time series. Such similarity can be assessed through the
Jensen-Shannon divergence, a symmetric version of the Kullback-Leibler
divergence that measures the similarity between two probability distributions
Grosse  _et al._ (2002). Finally, a test based on surrogate time series can be
performed to check the statistical significance of the difference between the
two distributions, and eventually obtain a $p$-value.

###  II.2 Introducing amplitude: the micro-scale trends method

As in the previous case, let us suppose a time series
$X=\\{x_{1},x_{2},\ldots,x_{N}\\}$ composed of $N$ values, also divided into
$N-\delta+1$ overlapping windows of length $\delta$, such that the $i$-th
window is defined as $w_{i}=\\{x_{i},\ldots,x_{i+\delta-1}\\}$. Afterwards, a
least squares polynomial fit of degree $d<\delta$ is applied to each window,
and the highest power coefficient $a$ is extracted. Finally, two probability
distributions of $a$, for the original and time-reversed time series, are
extracted and compared through a Kolmogorov-Smirnov test.

The behaviour of this test can be clarified considering the simplest situation
of $\delta=2$ and $d=1$. In this case, any window $i$ will be composed of two
values $\\{x_{i},x_{i+1}\\}$, the fit will be a linear one, and the
coefficient $a_{i}$ will correspond to the slope, i.e. $a_{i}=x_{i+1}-x_{i}$.
The two probability distributions $P_{a}$ and $P_{a^{t}}$ then represent the
distribution of the discrete derivative of the original and time reversed time
series, or, in other words, of their slopes. If these are different in a
statistically significant way, as evaluated by the Kolmogorov-Smirnov (K-S)
test, then the irreversibility hypothesis is accepted. This process is
illustrated in Fig. 1 right panel, which depicts the two distributions $P_{a}$
(black line) and $P_{a^{t}}$ (red line) for the time series represented in the
left panel. The instances of sharp increases in the original time series
(slope $>0.5$) are not seen in the time reversed version, as these become
sharp decreases (slope $<-0.5$). This asymmetry in the distribution of slopes
then highlights the presence of an irreversible process.

The similarities and differences with the permutation patterns approach are
easy to visualise. In the case of $\delta=2$ and $d=1$, this test is
equivalent to performing an analysis based on permutation patterns of length
$2$; yet, we here consider the slope, or excursion between consecutive values,
instead of its sign only. Similarly, the case of $\delta=3$ and $d=1$ is a
linear fit between three consecutive values; the middle one is then
disregarded, and this is equivalent to considering the permutation pattern
created by values $x_{i}$ and $x_{i+2}$.

For the sake of simplicity, in this contribution we only consider the cases
for $\delta=2\ldots 10$ and $d=1$, i.e. linear fits on windows of small
length. Nevertheless, the method here proposed can easily be extended to more
complex situations, which can yield richer views of the dynamics. First of
all, polynomials of any order can be extracted; the distributions $P_{a}$ and
$P_{a^{t}}$ will then represent the dominant trend in the windows. Secondly,
as customary in the evaluation of permutation patterns, one can add lags in
the reconstruction of the windows, such that
$w_{i}=\\{x_{i},x_{i+\gamma},x_{i+2\gamma},\ldots\\}$. Finally, any
transformation of the original time series can be used; for instance, one
could create a new time series with the standard deviation of each window,
thus representing the evolution of the dispersion of values, for then
evaluating the irreversibility of this new time series.

##  III Evaluation on synthetic time series

In order to evaluate the capacity of this method to detect irreversible
dynamics, we firstly test it with a set of time series created by the
following standard processes:

  * •

A Gaussian noise of zero mean and unitary standard deviation.

  * •

The Arnold Cat map, defined as: $x_{n+1}=(x_{n}+y_{n})mod(1)$,
$y_{n+1}=(x_{n}+2y_{n})mod(1)$. The analysed time series corresponds to the
evolution of the $x$ variable.

  * •

The Ornstein-Uhlenbeck process, i.e. a mean-reverting linear Gaussian process
$T$ Weiss (1975).

  * •

The Generalised Autoregressive Conditional Heteroskedasticity (GARCH) model
Bollerslev (1986), defined as: $x_{t}=\sigma_{t}z_{t}$, with
$\sigma^{2}_{t}=\alpha^{*}(1+\sum_{i=1}^{3}2^{-i})x^{2}_{t-i}+\sum_{i=1}^{3}2^{-j}\sigma^{2}_{t-j}$,
and $z_{t}$ being independent random numbers drawn from an uniform
distribution $\mathcal{U}(0,1)$. $\alpha^{*}$ (here set to $1.5$) is a
parameter controlling the strength of the time dependence between present and
past values of $x$, and hence the irreversibility of the time series.

  * •

The Henon map (defined as $x_{n+1}=1+y_{n}-ax_{n}^{2}$, $y_{n+1}=bx_{n}$, with
$a=1.4$ and $b=0.3$) and the logistic map ($x_{n+1}=ax_{n}(1-x_{n})$, with
$a=4.0$). Both maps are dissipative systems, and are by definition
irreversible Mori and Kuramoto (2013).

  * •

The Lorenz chaotic system, a continuos system defined as
$x^{\prime}=\sigma(y-x)$, $y^{\prime}=x(\rho-z)-y$, and $z^{\prime}=xy-\beta
z$, with $\rho=28$, $\sigma=10$ and $\beta=8/3$. The system has been solved
with integration steps of $dt=0.01$. The three variables $x$, $y$ and $z$ are
here analysed independently.

Both the Gaussian noise, the Arnold Cat map (an example of a conservative
chaotic map) and the Ornstein-Uhlenbeck process generate reversible time
series, while all others are irreversible.

Figure 2: Evolution of the probability of finding a statistically significant
time series as a function of its length, for the processes described in Sec.
III. Lines with color between cyan and purple correspond to the proposed
method, for $\delta$ between $2$ and $5$; black lines correspond to the
permutation patterns method.

Fig. 2 reports the evolution of the probability of finding an irreversible
time series as a function of its length, for the nine systems considered, and
for $\delta=2\ldots 5$ and $d=1$. As a reference, the black lines of the same
figure report the results obtained with the permutation patterns approach
described in Sec. II.1. It can be appreciated that the proposed metric is able
to correctly detect the presence or absence of irreversibility, provided
enough data (i.e. long enough time series) are available. Also, results are
generally similar to the ones yielded by the permutation patterns approach.
Still, several interesting differences can also be observed. Specifically, the
proposed method requires smaller time series in the case of the GARCH model,
and is slightly better in detecting the irreversibility of the Lorenz system.
On the other hand, it requires longer time series (i.e. it is less sensitive)
in the cases of the Henon and logistic maps.

Fig. 2 also indicates that the role of $\delta$, i.e. of the length of the
windows on which the fit is calculated, is a complex one. On one hand, in the
case of the Lorenz chaotic system, different $\delta$s yield the same result.
On the other hand, results differ substantially for the GARCH, Henon and
logistic time series; the test is more sensitive for large $\delta$s in the
former case, while the opposite can be observed for the two chaotic maps. Such
heterogeneity can easily be explained by considering the nature of these time
series. Data generated by GARCH are not stationary; larger $\delta$s thus
allow to filter out the local noise and extract the main trend. Conversely,
time series generated by the Henon and logistic maps are bounded and
stationary; calculating the slope of a linear fit over long windows is
effectively smoothing out the dynamics, and the slope actually becomes zero in
the limit of infinitely long windows, thus erasing differences between the
original and time reversed time series. Exceptions are nevertheless present:
for instance, the test is more sensitive in the case of the logistic map for
$\delta=4$ than for $\delta=3$. In order to understand this behaviour, Fig. 3
reports several graphs associated to the map, for $\delta$ varying between $2$
(top row) and $5$ (bottom row). When one considers the probability
distributions of the slopes for $\delta=3$ and $4$, normalised according to
the stationary distributions of $X$, both present differences between the
original (black lines) and time reserved (red lines) data; these differences
are nevertheless centred around zero in the case of $\delta=4$, thus yielding
a larger K-S statistics and a smaller $p$-value. In synthesis, these results
highlight that, whenever the characteristics of the underlying dynamics are
not known a priori, one should compare the results corresponding to different
$\delta$s to achieve an optimal detection.

Figure 3: Analysis of the irreversibility of the logistic map. From top to
bottom, rows correspond to $\delta=2$, $3$, $4$ and $5$. From left to right,
columns correspond to: i) Poincaré plot of the map, depicting the evolution of
$f^{\delta-1}(x)$ as a function of $x$. ii) Slope of the linear fit of values
$(x_{t},\ldots,x_{t+\delta})$ as a function of $x$. iii) Histogram of the
probability distribution of the slopes. iv) Histogram of the probability
distribution of the slopes, when values are normalised according to the
stationary distribution of $x$. v) Cumulative probability distributions of the
normalised slopes; the text inside each panel reports the statistics of the
K-S test assessing the equality of both distributions. In the third, fourth
and fifth columns, black and red lines respectively correspond to the original
and time reversed time series.

As shortly introduced in Sec. II.2, the proposed method can be applied to any
modification of the original time series. To illustrate, a new time series
$y_{t}$ can be created by calculating the second (standard deviation) and
third (skewness) central moments of the sub-time series
$\\{x_{t},\ldots,x_{t+\Delta}\\}$ (note that $\Delta\neq\delta$, and that
usually $\Delta\gg\delta$); the method can then be applied to the new time
series $y$. This allows detecting situations in which, for instance, the time
series is alternating between upward and downward movements with an increasing
(or decreasing) frequency; while the same number of positive and negative
slopes will appear, and hence no irreversibility will be detected in the raw
time series, the change in frequency will reduce (or increase) the deviation
from the mean, yielding an irreversibility in the time series of the standard
deviation. This idea is applied in Fig. 4, depicting the evolution of the
irreversibility in the Lorenz system for the raw, standard deviation and
skewness time series (for $\Delta=20$). It can be appreciated that, while in
most cases the derived time series underperform, calculating the
irreversibility over the time series of the standard deviation for the Y
channel completely reverses the result. The reason is readily identifiable by
looking at an example of the corresponding time series (see inset in the
central panel): the raw time series (cyan line) increases in amplitude while
maintaining a constant average, which translates to a clear non-stationarity
(and hence, irreversibility) of the time series of the standard deviation
(blue line). When a priori information about the time series is not available,
a solution may entail performing the test on the three time series, for then
accepting the original time series as irreversible if any of the three tests
yielded a statistically significant result after correcting for multiple
comparisons. This is illustrated in Fig. 4 by the grey lines, depicting the
evolution of the fraction of irreversible time series when the three tests are
combined ($\alpha=0.01$ with a Bonferroni correction).

Figure 4: Evolution of the probability of finding a statistically significant
time series as a function of its length, for the three channels of a Lorenz
system, and considering: the raw time series (cyan lines), the time series of
standard deviation (blue) and skewness (purple), the joint test over the
previous three time series (grey), and by using the permutation pattern test
(black). The inset in the central panel shows an example of such time series,
and specifically a raw one (cyan) and the corresponding standard deviation
time series (blue).

##  IV Evaluation on real-world time series

We then move to the evaluation of the proposed method in real-world time
series, and specifically consider two examples: the analysis of brain
electroencephalographic (EEG) data, and time series representing the evolution
of delays in the air transport system.

###  IV.1 Brain electroencephalographic data

Given that the irreversibility of a dynamical system is related to its entropy
production and to its performance as a thermal machine, it is not surprising
to find numerous applications of this concept to the study of the human brain
Van der Heyden  _et al._ (1996); Schindler  _et al._ (2016); Yao  _et al._
(2020); Zanin  _et al._ (2020). Specifically, if a disease or condition is
impairing the self-organising capabilities of the brain, this should reflect
in an abnormal (either higher or smaller) time irreversibility, and the latter
could therefore be used as a marker of the former. Previous works have
nevertheless shown that irreversibility is not an easy to detect property of
brain signals, and that long time series are usually required (Martínez,
Herrera-Diestra, and Chavez, 2018; Zanin  _et al._ , 2020). This may preclude
the use of this property in the study of brain dynamics developing on short
time scales, as for instance the response to stimuli.

As a first real-world test, we here apply the proposed irreversibility metric
to a data set of electroencephalographic (EEG) recordings, comprising both
control subjects and patients suffering from alcoholism (Zhang  _et al._ ,
1995; Cao  _et al._ , 2014; Zanin  _et al._ , 2021) and available at
https://archive.ics.uci.edu/ml/datasets/EEG+Database. Each recording
corresponds to the execution of a standard object recognition task (Snodgrass
and Vanderwart, 1980), and includes $64$ time series (i.e. one for each of the
$64$ electrodes) of $256$ elements (i.e. one second of brain activity). Note
the reduced length of these time series, which would preclude obtaining
statistically significant results with the method proposed in (Zanin  _et al._
, 2018) \- see also (Zanin  _et al._ , 2020) for a more complete analysis. On
a positive side, a large number of trials are available, specifically $4,024$
for control people and $7,033$ for patients.

Figure 5: Irreversibility of electroencephalographic (EEG) recordings. (Left)
Scatter plot of the fraction of patients’ recordings detected as irreversible,
as a function of the same fraction for control subjects. Each point represents
an electrode; magenta points further indicate the electrodes for which the
difference between the two fractions is statistically significant, according
to a binomial test and for $\alpha=0.01$. (Right) Histogram of the number of
electrodes in the left hemisphere as a function of the corresponding fraction
of irreversible time series, for control subjects (black columns) and patients
(red columns).

Fig. 5 Left reports a scatter plot of the fraction of time series that were
detected as irreversible by the proposed method for patients, as a function of
the fraction of irreversible time series in control subjects, with each point
representing a different EEG channel. In other words, channels laying close to
the main diagonal, represented by the black dashed line, display a similar
degree of irreversibility both in patients and control subjects. It is firstly
worth noting that the fraction of irreversible time series is globally quite
high, i.e. between $89\%$ and $93\%$; obtaining such strong signals using a
permutation patterns-based method would require time series $100$ times longer
(Zanin  _et al._ , 2020). Secondly, interesting differences between the two
groups can be observed, and specifically that time series of P1 and F1
electrodes are more irreversible in patients, and of P2 and PO8 in control
subjects; alcoholic patients seem thus to have a more irreversible dynamics in
the left hemisphere and a more reversible one in the right hemisphere. This is
further confirmed by Fig. 5 Right, depicting an histogram of the number of
electrodes in the left hemisphere as a function of the fraction of
irreversible time series, both for control subjects (black columns) and
patients (red columns); here again time series corresponding to electrodes in
the left hemisphere of patients are associated with higher irreversibility
than patients’ ones. While the relationship between cerebral laterality and
alcoholism has long been studied (London, 1987; Ellis and Oscar-Berman, 1989;
Akshoomoff, Delis, and Kiefner, 1989; McNamara  _et al._ , 1994; Zhu  _et al._
, 2014), to the best of our knowledge this is the first time irreversibility
of brain dynamics is introduced in the picture.

###  IV.2 Air transport delay data

As a second real-world example, we here consider time series representing the
evolution of delays at the $30$ largest European airports. Relatively few
works have studied the dynamics of delays from the point of view of
statistical physics, and this in spite of their importance for the cost-
efficiency Cook and Tanner (2011), safety Duytschaever (1993), and
environmental impact Carlier  _et al._ (2007) of this transportation mode. One
may prima facie expect delays to be random and independent events - as e.g.
when one aircraft experiences some technical problems, and has to delay the
take off; as such, the time series representing the evolution of the average
delay should resemble a random process, and hence not be irreversible. On the
other hand, if delays are not independent (as e.g. when the delay of one
flight is caused by the late arrival of the previous one), the memory present
in the system will result in an irreversible dynamics. Note that
discriminating between these two cases is not just an academic exercise, as
delays of the first type are unpredictable, while on the contrary those of the
second type are the result of inefficiencies in the system and can be avoided.

Air traffic data have been extracted from the EUROCONTROL’s R&D data archive
EUROCONTROL (2020), a large and freely available repository of information
about the European airspace and all commercial flights crossing it. The data
set covers four years, from 2015 to 2018, with four months being available for
each year (March, June, September and December). All flights landing at the
$30$ largest European airports (according to their number of passengers in
2015) have been extracted, for then calculating their delay at landing as the
difference between the actual and the scheduled landing times. All flights
that have landed at a given airport and in a given hour have then been
aggregated, to obtain a time series of average hourly delay at each airport.

Each of the $16$ available months have been analysed independently, and the
irreversibility has thus been calculated over time series of length $720-744$
(i.e. $24$ values per day, $30$ or $31$ days depending on the month). When
including $30$ airports, a total of $480$ time series have been analysed. Fig.
6 reports the scatter plot of the $p$-values obtained with the proposed
method, as a function of the $p$-value yielded by the permutation patterns
approach, for $\delta=2$ (left panel) and $\delta=10$ (central panel). It can
be observed that, while for $\delta=2$ the $p$-values obtained by the former
method are usually larger than those corresponding to the latter one, the
opposite happens for $\delta=10$. Specifically, it is possible to observe a
cluster of points for which the $p$-value yielded by the permutation pattern
is not statistically significant ($\approx 10^{-1}$), while the one yielded by
the trends method is around $10^{-5}-10^{-7}$. The right panel of Fig. 6
finally report the evolution of the $p$-value of the irreversibility, as
yielded by both methods, for the Vienna International Airport (i.e. the
airport with the highest average irreversibility). While the permutation
pattern approach yields increasing $p$-values, thus suggesting less systemic
delays, the proposed method points towards the presence of non-random delays
in June and December 2016, December 2017 and June 2018. In short, it can be
observed that both methods are not equivalent, but actually complementary. The
proposed method is able to detect irreversibility for some time series that
are identified as reversible through permutation patterns; and in this case,
it is more sensitive to irreversibility for large values of $\delta$, as
previously seen in the case of the GARCH model.

Figure 6: Irreversibility of airport delay time series. Left and central
panels report the $p$-value of the irreversibility of the time series of each
airport (see main text for definitions), as a function of the $p$-value
obtained by the permutation patterns method Zanin  _et al._ (2018), for
$\delta=2$ (left panel) and $\delta=10$ (central panel). (Right) Monthly
evolution of the $p$-value of the irreversibility for Vienna International
Airport, according to the permutation patterns method (black line) and the
proposed method (from $\delta=2$, cyan lines, to $\delta=10$, magenta lines).

##  V Discussion and conclusions

The recently increasing interest in the concept of irreversibility has been
followed closely by an increase in the number of metrics designed to detect
irreversibility in real-world time series. These approaches do not represent
differential increments, e.g. minor improvements in the computational cost,
but instead yield complementary views on the concept of irreversibility, as
the concept itself is only loosely defined. A number of these metrics leverage
on the concept of permutation patterns and have been independently proposed by
several research groups Zanin  _et al._ (2018); Martínez, Herrera-Diestra, and
Chavez (2018); Li, Shang, and Zhang (2019); Yao  _et al._ (2019), being
increasingly applied to the study of experimental data sets. We nevertheless
here show that permutation patterns alone cannot identify irreversibility in
some pathological cases, due to the fact that they disregard information about
the amplitude (or magnitude) of the values composing the time series. In other
words, from the permutation patterns point of view, the sequences $(0,1,2)$,
$(0,10,20)$ and $(100,101,102)$ are identical. In this contribution we thus
propose an alternative method, based on evaluating the changes in the
amplitude of the time series’ values, through micro-scale regressions of the
time series and of transformations of it.

The evaluation of the proposed method through synthetic time series depicts an
interesting picture: its performance strongly depends on the dynamical system
under analysis. To illustrate, the proposed method is able to detect the
irreversible nature of the Y channel of a Lorenz system, something not
achieved by a permutation pattern approach (see Fig. 4); yet, the latter is
substantially more efficient at detecting irreversibility in Henon and
logistic maps (see Fig. 2). In other words, and consistently with its
definition, the proposed method ought to be used when the amplitude of the
signal is not constant, for instance due to local non-stationarities. This
yields major benefits in the case of the analysis of real-world time series,
which are not necessarily stationary. To illustrate, the proposed method was
able to detect the irreversibility of brain EEG recordings even for time
series composed of only 256 points, while previous attempts required
substantially longer recordings Zanin  _et al._ (2020).

On the other hand, it is also important to highlight some limitations of the
present approach. First of all, while it yields better results in the case of
some dynamical systems, it is not clear when this is the case; in other words,
we cannot provide a decision algorithm that suggests the best test to be used
given one time series - beyond, of course, the brute-force approach of trying
all possible algorithms. This is not only limited to the proposed approach,
but is instead an open research question. Secondly, the proposed approach
includes some parameters that have to be tuned to maximise the sensitivity of
the irreversibility test, including the sub-window length $\delta$ and the use
of transformed time series. Note that their tuning is more complex than e.g.
tuning the embedding dimension of permutation patterns, as, provided enough
data are available, higher embedding dimensions are usually better. On the
contrary, and as shown in Figs. 2 and 3, increasing $\delta$ may lead to a
reduced statistical significance.

###### Acknowledgements.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 851255). M.Z. acknowledges the Spanish State Research
Agency, through the Severo Ochoa and María de Maeztu Program for Centers and
Units of Excellence in R&D (MDM-2017-0711).

## Data availability

The data that supports the findings of this study are available within the
article.

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