IATos: study of an AI-powered pre-screening tool for COVID-19 from cough audio
samples–References

# IATos: study of an AI-powered pre-screening tool for COVID-19 from cough
audio samples

David Trejo Pizzo
Secretaría de Innovavión y Transformación Digital
Ciudad Autonoma de Buenos Aires  Argentina  Santiago Esteban
Subsecretaría de Planificación Sanitaria  Ministerio de Salud
Ciudad Autonoma de Buenos Aires  Argentina

(7 December 2021; 1998)

###### Abstract

OBJECTIVE Our objective is to evaluate the possibility of using cough audio
recordings (spontaneous or simulated) to detect sound patterns in people who
are diagnosed with COVID-19. The research question that led to our work was:
what is the sensitivity and specificity of a machine learning based COVID-19
cough classifier, using RT-PCR tests as the gold standard?

DESIGN Diagnostic test validation study.

SETTING The audio samples that were collected for this study belong to
individuals who were swabbed in the City of Buenos Aires in all public testing
facilities and 1 private facility where RT-PCR studies were carried out on
patients suspected of COVID, and 14 out-of-hospital isolation units for
patients with confirmed COVID mild cases. The audios were collected through
the Buenos Aires city government WhatsApp chatbot that was specifically
designed to address citizen inquiries related to the coronavirus pandemic
(COVID-19).

PARTICIPANTS The first dataset collected corresponds to 2821 individuals who
were swabbed in the City of Buenos Aires, between August 11 and December 2,
2020. Individuals were divided into 1409 that tested positive for COVID-19 and
1412 that tested negative. From this sample group, 52.6% of the individuals
were female and 47.4% were male. 2.5% were between the age of 0 and 20 , 61.1%
between the age of 21 and 40 , 30.3% between the age of 41 and 60 and 6.1%
were over 61 years of age.

UPDATE: A second dataset containing 140.530 audio coughs was collected during
the months of April to October 2021, and has 18.271 audios from individuals
that tested positive for COVID-19 and 122.259 from individuals that tested
negative.

MAIN OUTCOME MEASURES The audio files corresponding to each individual were
categorized as ”positive” and ”negative” based on their PCR swab test results.

RESULTS Using the dataset of 2821 individuals our results showed that the
neural network classifier was able to discriminate between the COVID-19
positive and the healthy coughs with an accuracy of 88%. This accuracy
obtained during the training process was later tested and confirmed with a
second dataset corresponding to 492 individuals.

UPDATE: using a combination of the first and the second dataset, the first
neural network used was re-trained. This new training involved using audio
files from 143.351 individuals (2.821 from the first dataset and 140.530 from
the second dataset). The neural network classifier was able to discriminate
between the COVID-19 positive and the healthy coughs with an accuracy of 86%.
The performance also shows that this new trained NN is better at assessing
negative coughs, and this may be due to the new unbalanced data that has 10%
of positive audio files and 90% of negative audio files.

CONCLUSION Based on the reported results, we consider that the cough audio
analysis could be used as a COVID-19 digital pre-entry screening tool to be
used prior to conventional testing tools such as RT-PCR tests. This could help
reduce wait times, ease stress, and lower exposure risk of healthcare
personnel and other citizens.

FUNDING STATEMENT The project was carried out with the resources of the
Secretariat of Innovation and Digital Transformation and the Ministry of
Health of Buenos Aires City Government and with volunteers from other
government areas for the field duties.

AUTHOR DECLARATIONS

All relevant ethical guidelines have been followed; any necessary IRB and or
ethics committee approvals have been obtained and details of the IRB oversight
body are included in the manuscript. Yes

All necessary patient - participant consent has been obtained and the
appropriate institutional forms have been archived. Yes

The authors have no conflicts of interest to declare. All co-authors have seen
and agree with the contents of the manuscript and there is no financial
interest to report.

##  1 Introduction

As from November 28, 2021, the SARS-CoV-2 Coronavirus has spread to 213
countries around the world, has over 262 million confirmed cases and has
caused over 5.2 million deaths [1] [22].

The SARS-CoV-2 Coronavirus causes a respiratory infection called COVID-19. The
COVID-19 virus spreads primarily through droplets of saliva or discharge from
the nose when an infected person coughs or sneezes [36].

Given that access to vaccines is limited and treatments are still in
development, minimizing the spread through timely testing and isolating
infected people is one of the best effective defenses humanity has at its
disposal against COVID-19. However, the ability to deploy this strategy
depends on each country’s capability to test significant fractions of their
populations, including those that are not yet in contact with the health
system. Agile, scalable and proactive testing have become the key
differentiator for managing this pandemic.

###  1.1 Diagnostic strategies

The main types of tests used for COVID-19 are:

  * •

Rapid serology antibody tests (ICT, LFA)

  * •

Enzyme Linked Immunosorbent Assays (ELISA)

  * •

Real-time polymerase chain reaction (RT-PCR) through swab tests

Rapid serology tests and ELISAs detect the presence of IgG / IgM antibodies
produced by the immune system in response to a SARS-CoV-2 infection, while RT-
PCR tests are molecular tests that detect the presence of viral RNA of SARS-
CoV-2.

Triage remains as the fastest method for pre-screening a person, and this
method is the one that can be scaled faster and at a low cost using
exponential technologies. Triage is the process of sorting people based on
their need for medical treatment as compared to their chance of benefiting
from such care. Triage is usually performed in emergency rooms, disasters and
wars, when limited medical resources must be allocated to maximize the number
of survivors. During infectious disease outbreaks, triage is particularly
important to separate patients likely to be infected with the pathogen of
concern.

The main route of transmission of COVID-19 is through respiratory droplets
generated when an infected person coughs or sneezes. Any person who is in
close contact with someone who has respiratory symptoms (e.g., sneezing,
coughing, etc.) is at risk of being exposed to potentially infective
respiratory droplets [37].

Recently, the US FDA [18] has written about the importance of expanding rapid
COVID-19 serological antibody testing. Current laboratory diagnostic tests for
COVID-19 are based on labor-intensive molecular techniques (RT-PCR swab tests)
and have generally been reserved for patients whose disease severity, age
and/or comorbidities place them in high risk of serious illness, specially in
countries with limited testing resources.

Recent work has also described that SARS-CoV-2 viral shedding often occurs in
early pre-symptomatic stages, which could affect the dynamics and diagnostic
accuracy of SARS-CoV-2 RT-PCR swab tests, especially if patients do not
undergo molecular diagnostic tests until several days after the onset of
symptoms. On the other hand, COVID-19 rapid serology antibody tests detect
IgM/IgG antibodies, which are generated 1-3 weeks after the onset of symptoms.

Together, the SARS-CoV-2 RT-PCR swab test, combined with a rapid COVID-19
serological antibody test provides a more complete picture of COVID-19 disease
progression, in a given individual and across populations.

###  1.2 Importance of early diagnosis

Many authors have highlighted that the key to safely returning back to normal
life [35], especially while we wait for the COVID-19 vaccination process to be
completed, does not lie in a single ”perfect” COVID-19 test - but through
multiple rapid and affordable tests and non-clinic pre-screening tools.

RT-PCR tests have been the gold standard for COVID-19 diagnosis. Nevertheless,
their high cost and relatively long processing times make them a less
appealing option when deploying population wide testing strategies. On the
other hand, antibody and antigen detection tests have proved to be an
excellent complement to RT-PCR tests. Their lower costs, shorter processing
times and higher availability allows for a more wide spread testing strategy
and also shorter testing-isolation times.

###  1.3 Difficulties with early diagnosis

Currently, existing RT-PCR tests take considerably long to deliver results,
are expensive to run frequently, or are often considered uncomfortable, which
discourages people from going to get tested.

There are already some rapid tests for COVID-19 on the market. The Sofia SARS
antigen test performed by the Quidel Corporation is approximately 99%
accurate, takes about 15 minutes to deliver results, and is already in use. A
pharmacy in Arkansas, USA, reported that they had dispensed more than 800 in
about two weeks. The test costs U$D 95, which is a considerable price for most
people’s budget, especially if they have to take the test periodically.

OpenCovidScreen [17], a coalition of scientists and biotech leaders working to
drive innovation in COVID-19 testing and, in turn, accelerate return to work
and school, says that a good COVID-19 testing solution needs four items:
frequency (at least weekly), more test points, an affordable price (around U$D
10-20), and an easy method of collection (e.g. saliva, quick test, cough).
Currently, there are tests available on the market that address one or two of
these considerations, but there is nothing that brings all four together.

###  1.4 Audio signals for pre-screening and disease monitoring

Audio signals generated by the human body (e.g. sighs, breathing, heart,
digestion) have often been used by physicians and clinical researchers in the
diagnosis and monitoring of diseases. However, until recently, such signals
were generally collected by manual auscultation during scheduled visits to a
medical facility. Medical research has begun to use digital technology to
collect body sounds (e.g. digital stethoscopes) and perform automated data
analysis [1], (e.g. for wheezing detection in asthma [2] [3]).

Researchers have also been testing the use of the human voice to aid in the
early diagnosis of a variety of diseases. For example, in Parkinson’s disease
the patient’s speech is analyzed for variations in smoothness to determine if
there exists a lack of coordination in the vocal muscles [4] [5]. Also vocal
tone, rhythm, and volume have also been explored to diagnose illnesses such as
post-traumatic stress disorder [7], traumatic brain injury, and some
psychiatric conditions [8].

The use of human-generated audio as a biomarker for various diseases offers
enormous potential for early diagnosis, as well as for affordable solutions
that could be widely applied if integrated into devices such as cell phones.
This is even more powerful if such solutions could monitor people throughout
their daily lives in a discreet way.

Clinical work has focused on the use of voice analysis for specific diseases:
for example, in Parkinson’s disease, microphone and laryngoscope equipment
have been used to detect the softness of speech resulting from lack of
coordination on the vocal muscles [6] [12]. Voice characteristics have also
been used to diagnose bipolar disorder; and to correlate pitch, rhythm, and
volume with signs of invisible conditions such as post-traumatic stress
disorder [7], traumatic brain injury, and depression. Voice frequency has been
linked to coronary artery disease (as a result of hardening of the arteries
that can affect voice production) [6]. Companies such as Israel-based Beyond
Verbal and Mayo Clinic have indicated in press releases that they are testing
these approaches.

More recently, the microphone in basic devices such as smartphones and
wearables has been used for sound analysis. In [12], the microphone is used to
understand the context of the user and this information is added to generate a
view of the surrounding environment of the recorded sound.. In [13] the
authors analyze the sounds emitted while the user is sleeping, to identify
episodes of sleep apnea. Similar work has also used sound to detect asthma and
wheezing [2] [3].

###  1.5 IATos: proposal and background

Machine learning methods have been designed to recognize and diagnose
respiratory diseases from sounds [1] and more specifically cough: Bales et
al.[14] used convolutional neural networks (CNN) to detect cough within
ambient audio and diagnose three potential diseases (bronchitis, bronchiolitis
and whooping cough) based on its unique audio characteristics.

With the advent of COVID-19, researchers have begun to explore whether cough
sounds could be diagnostic [15]. A COVID-19-related cough detection study is
presented in [15] using a cohort of 48 patients tested with COVID-19 versus
other pathological coughs, in which a set of models is trained. In [11] the
voice recordings of COVID-19 patients are analyzed to automatically classify
the health status of the patients in four aspects, namely, the severity of the
disease, the quality of sleep, fatigue and anxiety. Quatieri et al. [26]
showed that changes in vocal patterns could be a potential biomarker for
COVID-19.

This research project aims to explore the use of human cough sounds as pre-
diagnostic markers for COVID-19. Within the framework of this research study,
we worked on the data acquisition of audio samples from people who quarantined
in out-of-hospital isolation units for COVID-19 mild cases and close contacts
or who were admitted to hospitals.

This research project differs from other similar initiatives in two main
aspects. In the first place, the acquisition of audio samples was done through
WhatsApp, and not through web pages. This allowed for a more streamlined data
collection from the study participants and simplified the signing of informed
consents. On the other hand, only people who underwent a PCR or antigen swab
test participated in the study. Also, their results had to be available in the
Buenos Aires City Health Ministry COVID-19 results database. This helped us
classify the audio samples of each individual for the first 5k audio dataset,
according to the result of their RT-PCR results, considered as the current
”gold standard” for COVID-19 diagnosis. The second dataset, which we used to
re-train the neural network, has 140k audio files. This second dataset is
unbalanced (10% corresponds to COVID-19 positive individuals and 90%
corresponds to COVID-19 negative individuals) and includes both audio files
from people that had undergone an RT-PCR and antigen tests.

The expected outcome is to be able to analyze the patient’s cough as an
additional marker that can be incorporated into the medical triage strategy.
We seek to improve the pre-screening process that is carried out today prior
to undergoing diagnosis testing methods such as PCR or rapid antigen tests.
Having a digital Triage, which can be performed from anywhere using WhatsApp,
allows us to exponentially scale the detection strategy for symptomatic and
asymptomatic cases.

##  2 Methodology

Based on the objective of studying cough as a way to pre screen individuals
for COVID-19, the research question that guided our work was: what is the
sensitivity and specificity of a machine learning based COVID-19 cough
classifier, compared to the standard diagnosis test for COVID-19?

In order to answer this question and prior to the development of the neural
network, we consider the following steps as our methodology of work. The steps
of this process are:

  * •

Generate a data acquisition process

  * •

Generate a dataset

    * –

Define the selection process for study participants

    * –

Contact people who agree to participate in the study

    * –

Start the audio acquisition flow

  * •

Establish a preprocessing strategy for the collected audios

###  2.1 Data description

Observational or experimental studies require a priori sample size calculation
in order to achieve the necessary statistical power to reject the null
hypothesis [25]. In the field of pattern recognition though, there is no
standardized priori determination method to evaluate the generated models
performance based on sample size of the datasets used for training.

Review of previous literature from past research uses an empirical approach
through learning curves and the use of surrogate models to provide an answer
to the problem of sample size [26]. However, these ex-post-facto solutions do
not take into account the nature of the classificatory problem to be solved,
the complexity of the model used, the use of feature augmentation techniques,
or the use of pre-trained models with the transfer learning method.

As an approach to determining the necessary sample size, we revised the
literature that refers to the use of pattern recognition models applied to the
audio signal produced by coughing. We reviewed pre-existing literature and
studied the differences of different samples sizes and their related metrics
(see Table 1).

Table 1: Bibliographic search of works that refer to the use of pattern recognition models applied to the audio signal produced by the act of coughing. Author (Year) | Samples | COVID + | Others111Another non-COVID-19 positive respiratory disease. | Control | S/E | AUC
---|---|---|---|---|---|---
Imran (2020) | 1838 | 70 | 226 | 247 | .96/.95 | NR
Brown (2020) | 491 | 141 | 52 | 298 | NR | 0.875
Bales (2020) | 268 | - | 268 | - | .89/.948 | NR

In light of the studied literature we established that a sample size of 5000
cough audio sounds could enable the correct performance of the generated
model.

The data acquisition process was approved by the Ethics Committee of the
Elizalde Hospital (Comité de Etica - Hospital Elizalde). In August 2020 we
started the process of collecting cough audios through the Buenos Aires City
Government chatbot with the aim of creating an Open Voice dataset for COVID-19
cough discrimination.

We only used samples with two conditions:

  * •

The person providing the cough had undergone a PCR test.

  * •

The audio files were recorded within 3 days of that PCR test.

At the end of the data acquisition process, we had collected 6,000 audio
files, from 2,821 individuals. This collection of audio files was split into
two groups:

  * •

Training and validation: 2,412 audio files from individuals that tested
negative for COVID-19, and 2,477 audio files for individuals that tested
positive These audio files belong to 2,329 individuals.

  * •

Test: 995 audio files that belong to 492 individuals.

Each person included in the study had to generate a variety of representative
audios of the act of coughing. We collected on average 3 coughs per subject
accompanied by general subject information: age, sex, date of the swab/cough,
location, outcome of medical diagnosis, and finally information regarding
timing of onset of COVID-19 signs and symptoms (fever, tiredness, sore throat,
difficulty breathing, persistent pain or pressure in the chest, diarrhoea and
coughing).

Figure 1: Dataset general information.

Data was anonymized before being collected on our secure server and samples
were saved without compression in WAV format. Samples that had no coughs, too
much noise or were silent were removed. No segmentation was performed on the
cough recordings used to train and test the neural network.

###  2.2 Data acquisition

Individuals who participated in the study were asked to send one audio per
day, for 3 consecutive days following the PCR test. The primary goal was to
obtain multiple samples per person. Secondarily, we seek to study the
evolution of audio patterns to detect if there are changes that show evidence
of the evolution of the virus.

When someone went to a testing center, health system personnel invited them to
participate in this study. If the person agreed, they were guided to start a
conversational flow on the Buenos Aires city government WhatsApp chatbot
specifically designed to address citizen inquiries related to the coronavirus
pandemic (COVID-19). From the beginning of the study until September 30, 2020
people signed a printed informed consent to participate in this study. From
September 30, 2020 the informed consent was digitized into the WhatsApp
chatbot conversation.

Figure 2: Data acquisition timeline.

The audio files that were collected for this study belong to individuals who
were swabbed in the City of Buenos Aires in public and private facilities,
out-of-hospital isolation units for patients with confirmed COVID mild cases
and testing points of the strategic testing program ”DetectAR”. The health
personnel who performed the swab briefly explained the study to the
individuals and invited them to participate.

Figure 3: Data acquisition sites withing the Buenos Aires health system.

Data collection sites: 19 Febrile Emergency Units (FEU), 12 testing points of
the strategic testing program ”DetectAR”, 14 out-of-hospital isolation units
for COVID-19 mild cases and close contacts and 1 private medical center as
well as government personnel.

####  2.2.1 Update on data acquisition

A second dataset containing 140.530 audio coughs was collected during the
months of April to October 2021, and has 18.271 audios from individuals that
tested positive for COVID-19 and 122.259 from individuals that tested
negative.

This second data acquisition process was carried on from April 2021 until
October 2021, and the same method for audio collection was applied. In this
case, we incorporated information related to the clinical triage.

Each individual answered if they had symptoms of Covid-19 at the time of
sending the audio, if they were in close contact with someone with COVID-19,
if they had breathing problems or if they were part of a risk group.
Additionally, the data corresponding to the type of test that was carried out
was stored as a variable for future studies. On this occasion, the samples
were not analyzed looking for the percentage of men and women who
participated, nor the age groups, assuming that the sample is a representation
of the population of the City of Buenos Aires that uses the public testing
system.

###  2.3 Data preprocessing

The preprocessing pipeline consists of sampling, denoising and cough
discrimination from noise and silence audio chunks.

Although these could be extended to more stages, our current proposal (Fig. X)
consists of:

  * •

Detects the presence of a cough in an audio, and cuts an audio segment of 1
second maximum duration.

  * •

Normalize the audios in amplitude so that they all have a similar amplitude
domain.

  * •

Denoise audio files that have a low signal-to-noise ratio (SNR).

Figure 4: Pre-processing pipeline

As the first stage of preprocessing the audios, they are converted from the
.ogg format to the .wav format, and all the audios are sampled equally at 16
kHz.

Subsequently, the audio files are cut into 1 second chunks. Then, using YAMNet
and a network trained from the collected audio files, audio chunks that
contain coughs are separated from those that are noise or silence. Finally,
the resulting fragments are filtered to remove any noise that the signals may
have.

The audios are then preprocessed in order to extract their main
characteristics. This was achieved through a spectral analysis and selection
of features of the signal. Finally the audio samples were normalized in
amplitude.

From the information extracted, images were generated and introduced into a
deep learning model, in order to be able to classify new cough audio signals.

For data processing, we mainly used a transformation that has broad consensus
in the scientific community. In the first place, from the audio signal, the
Mel [27] spectrogram was obtained, extracting the most relevant
characteristics of the sound using human auditory perception as a criterion.
The transformation consists of splitting the signal into short frames,
applying the discrete Fourier transform (DFT) to generate the spectrogram,
then applying the Mel Filterbank -to calculate the energy in each Mel filter-
and finally calculating the logarithm of the energy for each filterbank [28].

We used regularization techniques in our data since, given the complexity of
the model to be applied, this will allow a better generalization of them.
Specifically, to feed the model we used data augmentation, that is, we
performed small random transformations to create new images, for example
scaling and/or rotating the data. This technique allowed us to generalize the
information and also obtain an improvement in the performance over audio
signal data.

##  3 Neural network model

A deep learning-based COVID-19-associated cough detector was created. That is,
a three-class classifier based on Convolutional Neural Networks that uses
Mel’s spectrograms as input information. Due to the subtle differences that
exist between images of coughs corresponding to healthy patients, with various
respiratory diseases, or with COVID-19, a neural network with more layers was
required. Therefore, we used a CNN architecture.

Figure 5: CNN Architecture.

Regarding the architecture, CNNs have a set of convolutional layers, to which
pooling layers are then added and the dimensions of the matrix are adjusted.
This is repeated successively and in the end, a fully connected layer is added
to perform the prediction through a softmax.

For this project, three widely accepted scientific metrics will be used:
Specificity, Recall/Sensitivity and Accuracy. In the training process of the
model, a constant monitoring of the precision metric was carried out in order
to generate the least amount of false positives. Due to the way COVID-19
spreads, it will always be a priority to maintain a high criterion of
reliability and certainty before recommending a patient to seek assistance at
a health facility. In parallel, the AUC measurement was monitored, which
represents the area under the ROC curve that allows visualizing the
sensitivity versus specificity.

We started by using a classification threshold of 0.5. Our proposed
architecture, drawn in Fig. 5, takes a recording with one or more coughs,
performs pre-processing steps and inputs it into a CNN based model to output a
pre-screening result. As pre-processing, each input audio file is split into 1
second audio chunks, padded as needed, processed with the MFCC package and
subsequently passed through the neural network classifier. The output of these
steps becomes the input to a CNN.

In order to train the classifier model, the set of spectrograms obtained were
divided into train (80%), validation (10%) and internal test (10%). Finally,
with the model obtained, inferences were made to a second set of audio files
that belong to a test set of 492 individuals. In this way we were able to
evaluate the performance of the model on new data that it had never analyzed
before.

##  4 Results

As we explained in the previous section, in order to evaluate the model we use
the performance metrics of accuracy, specificity and sensitivity/recall on the
test set. The accuracy here refers to the overall accuracy of the model. These
performance metrics are based on mean confusion matrices from cross-
validation. Tuning of the various hyper-parameters (number of hidden layers,
learning rate, activation functions, dropout rate) of deep neural network-
based models has also been performed, based on the cross-validation accuracy.
Furthermore, the decay of model loss versus the number of epochs has been
investigated to rule out the possibility of overfitting.

###  4.1 Cough discrimination accuracy

When we receive an audio file, the first step is to discriminate whether it
has a cough or not. For this step we combine the use of YAMNet and a neural
network that we specially trained to distinguish coughs. The analyzed audio
file is cut into 1-second chunks and these fragments are evaluated by this
detection stage, to discard the fragments that do not have the presence of
cough. We achieved an accuracy of 97% determining which fragments have a cough
and which have noise compared to a random selected, manually classified sample
of 100 1 second audio chunks. This allowed us to move on to the next stage,
which is to determine if the audio fragments tagged as “cough” by the YAMNet
algorithm, correspond to that of an individual who is potentially COVID-19
positive or not.

###  4.2 COVID-19 detection

At the moment and based on the collected data, the overall accuracy of the
deep learning based classifier is 86.00%, Recall is 0.89 and F1-score is 0.87.
The mean normalized confusion matrix resulting from this approach is shown in
Fig. 6. Future work will continue to improve this model as more training data
becomes available. Fig. XX shows the mean loss versus epochs of the neural
network classifier, for both training and validation data sets. Both the
curves start to saturate after around 70 epochs, indicating a reasonable
learning time, without overfitting.

Figure 6: Confusion matrix.

##  5 Discussion and future work

We have shown that COVID-19 can be discriminated with 86% accuracy using only
a forced-cough audio recording. We find most remarkable that our model
performs better with coughs from positive patients than those from negative
ones.

This first stage of developing the model focused on training it on a large
dataset to learn good features for discriminating COVID-19 forced-coughs. The
results when evaluating the testing set (made with coughs from individuals
diagnosed with a validated PCR and antigen tests) serve as an indicator that
the model would have similar accuracy when deployed. The next stage of this
project is turning on the NN classifier together with the triage that is
currently being used in the Buenos Aires City WhatsApp chatbot. We will also
gather more quality data that can further train, fine-tune, and validate the
model in a real world environment.

Since there might be cultural and age differences in coughs, future work could
also focus on tailoring the model to different age groups and regions of the
world using the metadata captured, and possibly including other sounds or
input modalities such as vision or natural language symptom descriptions.

###  5.1 Potential use cases of IATos

Motivated by an urgent need, we developed a research protocol that was
intended to evaluate the usefulness of an AI-based preliminary diagnostic tool
for COVID-19, ubiquitously scalable through WhatsApp audio files containing a
recording of an individual’s forced cough.

The central idea of such a tool is inspired by previous studies [10] [16]
showing that coughing can be used as a pre-screening tool for the diagnosis of
a variety of respiratory diseases using AI techniques. The main objective is
that the cough analysis is a complement and never a replacement for other
diagnostic methods.

Figure 7: Triage flow with IATOS.

This non-invasive, free to access and real-time pre-screening tool may prove
to have a great potential to complement current efforts to contain the disease
spread in many different contexts.

Possible use cases could include:

  * •

Population daily screening tool: As workers go back to work, students go back
to school, and commuters use public transport, to name a few, methods are
required to screen infected COVID-19 carriers, especially asymptomatics. The
only screening method currently available is using thermometers, however this
study [36] showed only 45% of mild-moderate COVID-19 cases have fever (this
represents 9% of COVID-19 positives when asymptomatics are included).
Meanwhile our AI tool has shown to discriminate against a forced cough of
COVID-19 positives with 86% accuracy and could act as a complement to the
medical triage questions.

  * •

Pre-selection of candidates for test pooling: The test pooling strategy is
expected to be employed in many countries, especially in low-incidence areas
to rapidly identify a subgroup of individuals likely to be infected, however,
“preliminary results show there is no dilution and no decrease on test
sensitivity when mini pools of five samples each are used” [37]. Group testing
with our tool as shown in Fig. 7, could pre-screen school classrooms,
factories or even countries on a daily basis signalling probable infected
candidate groups for smaller test pooling batches.

This technology could be assembled with low levels of effort and risk within
the virtual triage that is carried out as a follow-up protocol for close
contacts of positive patients.

In the future, subject to the learning curve and validation of the model,
additional use cases could include [10]:

  * •

Complementing temperature scanners at airports, borders or other key places
where the virus circulates.

  * •

Allowing remote pre-monitoring for anyone, anywhere, at any time, regardless
of the existing infrastructure/testing facilities.

  * •

Providing a centralized record of tests with location and time stamps. The
data collected from the app could serve as input for long-term health care
planning and health policy formulation.

Despite the good performance that is preliminarily observed in these tools,
they are not intended as a replacement for clinical tests but as a complement.
The goal is to investigate the potential for a single functional tool to
monitor, track and control the rampant spread of the global pandemic in a
timely, cost-effective and most importantly safe manner, by allowing a pre-
test for anyone with access to a smartphone and an internet connection.

##  6 Conclusion

We presented an AI pre-screening tool that uses a neural network classifier
that is able to discriminate COVID-19 positives with 86% accuracy from a
forced-cough recording, at essentially no cost and with an exponential scaling
capacity.

Despite its performance, IATOS is not meant to compete with clinical testing.
Instead, it offers a unique functional tool for timely, cost-effective and
most importantly safe monitoring, tracing, tracking and thus, controlling the
rampant spread of the global pandemic by virtually enabling testing for
everyone.

The proposed solution could be seen as a group outbreak detection tool for
pre-screening whole-populations on a daily basis, taking the pre-screening to
each house.. At the same time, this tool will allow us to send more people,
potentially asymptomatic, to be tested. In this way we may be able to detect
many positive asymptomatic cases that would normally go undetected.

As part of our ongoing pilots, data pipelines with FEUs and detection points
have been set up to continue to improve the neural network. We plan on
leveraging this data to further train and validate our models with the aim of
improving pandemic management practices.

As part of our efforts to make data public and contribute to open innovation,
we are opening our research methods and our dataset to collaborate and inspire
others to develop similar tools to fight the COVID-19 pandemic, and hopefully
be better prepared for the next pandemic.

##  7 Ethical aspects

The study will be guided by the rules of good clinical practice, the
Declaration of Helsinki and the regulations of the Government of the City of
Buenos Aires in force. Authorization was requested from the Research Ethics
Committee, the DGDIYDP (Dirección General de Docencia, Investigación y
Desarrollo Profesional) and the SSPLSAN (Subsecretaría de Planificación
Sanitaria. Ministerio de Salud). Informed consent of those responsible,
subjects and / or consent of minors will be requested as appropriate. Due to
the context of the study and the specific SOPs of the Ethics Committee of the
Elizalde Hospital (CEI-HGNPE, Hospital General de Niños Pedro de Elizalde),
the documentation of the process will be protected in images of the forms duly
signed by the subjects or their managers, who will keep the originals. The
audio data of the patients and their COVID-19 test results will be stored
without direct or indirect identifiers. Because it is an observational study
and is framed in the context of the COVID-19 pandemic, the CEI-HGNPE requested
its expeditious evaluation and approval.

Data Availability All the data referred to in the manuscript are available in
https://data.buenosaires.gob.ar/.

Declaration of competing interest The authors declare that they have no known
competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.

Acknowledgement This work is dedicated to those affected by the COVID-19
pandemic and those who are helping to fight this battle in any way they can.

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# Solvable and non-solvable finite groups
of the same order type

Paweł Piwek  Mathematical Institute, Andrew Wiles Building, Observatory
Quarter, University of Oxford, Oxford OX2 6GG, UK<EMAIL_ADDRESS>
(Date: February 2024)

###### Abstract.

We construct two finite groups of size $2^{365}\cdot 3^{105}\cdot 7^{104}$: a
solvable group $G$ and a non-solvable group $H$, such that for every integer
$n$ the groups have the same number of elements of order $n$. This answers a
question posed in 1987 by John G. Thompson.

##  1\. Introduction

The following meta-question has been of considerable interest in the field of
finite group theory.

######  Meta-question.

Which properties of a group $G$ can we discern from knowing only the orders of
its elements?

While some addressed this question by looking at just the _set_ of orders of
elements of $G$ – see [11], we are interested in also using the _multiplicity_
of these orders, which prompts the following definition.

######  Definition 1.

For a group $G$ we define its _order type_ to be the function

$o_{G}:n\mapsto|\\{g\in G\ |\ g\text{ has order }n\\}|.$

It isn’t difficult to give examples of an abelian group $G$ and a non-abelian
group $H$ such that $o_{G}=o_{H}$. Indeed, take $G=C_{3}\times C_{3}\times
C_{3}$ and $H$ to be the Heisenberg group $(C_{3}\times C_{3})\rtimes C_{3}$,
which is non-abelian of exponent $3$.

On the other hand, the order type allows us to count the number of elements of
order $p^{\alpha}$ for $p$ a prime, so it can also tell us if a $p$-Sylow
subgroup is normal. This implies that the order type can distinguish nilpotent
groups from non-nilpotent ones. A similar argument can be given for super-
solvable groups.

A problem posed in 1987 by John G. Thompson in a private communication with
Wujie Shi (see [9, 10]; also Question 2.13 in [3], and Problem 12.37 in [5])
is: which is the case for solvability?

######  Question.

Let $G$ be a finite solvable group and $H$ be any finite group such that
$o_{G}=o_{H}$. Is $H$ necessarily solvable?

It was shown that the answer is positive if we assume that the prime graph of
$G$ is disconnected – see [9], or in the case where there are at most three
different orders of elements – see [8]. Some works also showed results about
solvability of groups (and super-solvability, nilpotence and cyclicity too)
based on the average element order – see [4, 6].

In this paper, we give a negative answer to Thompson’s question. No claim is
made that the presented examples are of minimal size.

######  Theorem A.

Let $G_{i}$ and $H_{i}$ be the collections of finite groups described in
Tables 1(a) and 1(b), and let $m_{i}$ and $n_{i}$ be the associated natural
numbers from the tables. Let $G$ and $H$ be the direct products

$G=\prod G_{i}^{m_{i}},\quad H=\prod H_{i}^{n_{i}}.$

Then $o_{G}=o_{H}$, $G$ is solvable, and $H$ is not solvable.

$i$ | $G_{i}$ | Id | $m_{i}$
---|---|---|---
1 | $C_{4}$ | (4, 1) | 9
2 | $D_{3}$ | (6, 1) | 6
3 | $C_{7}$ | (7, 1) | 1
4 | $D_{4}$ | (8, 3) | 9
5 | $D_{7}$ | (14, 1) | 18
6 | $\mathrm{SL}(2,3)$ | (24, 3) | 21
7 | $C_{24}\rtimes C_{2}$ | (48, 6) | 3
8 | $C_{7}\rtimes D_{4}$ | (56, 7) | 3
9 | $C_{7}\rtimes C_{12}$ | (84, 1) | 6
10 | $\operatorname{\mathrm{Dic}}_{21}$ | (84, 5) | 6
11 | $C_{7}\rtimes A_{4}$ | (84, 11) | 21
12 | $C_{7}\rtimes D_{7}$ | (98, 4) | 2
13 | $C_{4}\rtimes F_{7}$ | (168, 9) | 21
14 | $C_{21}\rtimes D_{4}$ | (168, 15) | 9
15 | $C_{7}\rtimes D_{12}$ | (168, 17) | 6
16 | $F_{8}\rtimes C_{3}$ | (168, 43) | 3
17 | $D_{8}\rtimes D_{7}$ | (224, 106) | 3
18 | $C_{7}\rtimes D_{24}$ | (336, 31) | 3
(a) Groups $G_{i}$ and numbers $m_{i}$.

$i$ | $H_{i}$ | Id | $n_{i}$
---|---|---|---
1 | $C_{2}$ | (2, 1) | 21
2 | $C_{3}$ | (3, 1) | 3
3 | $\operatorname{\mathrm{Dic}}_{3}$ | (12, 1) | 6
4 | $A_{4}$ | (12, 3) | 21
5 | $\operatorname{\mathrm{SD}}_{16}$ | (16, 8) | 3
6 | $C_{7}\rtimes C_{3}$ | (21, 1) | 4
7 | $D_{12}$ | (24, 6) | 6
8 | $C_{3}\rtimes D_{4}$ | (24, 8) | 6
9 | $\operatorname{\mathrm{Dic}}_{7}$ | (28, 1) | 15
10 | $F_{7}$ | (42, 1) | 18
11 | $D_{21}$ | (42, 5) | 6
12 | $D_{24}$ | (48, 7) | 3
13 | $D_{28}$ | (56, 5) | 27
14 | $\operatorname{\mathrm{Dic}}_{7}\rtimes C_{6}$ | (168, 11) | 3
15 | $C_{14}.A_{4}$ | (168, 23) | 21
16 | $\mathrm{GL}(3,2)$ | (168, 42) | 3
17 | $C_{7}\rtimes F_{7}$ | (294, 10) | 2
18 | $D_{12}.D_{7}$ | (336, 36) | 3
(b) Groups $H_{i}$ and numbers $n_{i}$.

Table 1. The groups $G_{i}$ and $H_{i}$ involved in A and their multiplicities
$m_{i}$ and $n_{i}$. The column labelled ‘Id’ contains the Small Groups
isomorphism type identifier (see [1]), while columns $G_{i}$ and $H_{i}$ give
some idea of the group structure, but they don’t necessarily identify the
groups uniquely in the case of extensions.

### Acknowledgements

The author is grateful to his friend and colleague Adam Klukowski for a
fruitful conversation about the problem, and to his PhD supervisor Martin
Bridson for the general guidance.

##  2\. Proof of A

Before presenting the proof, let’s first rephrase the question slightly.

######  Definition 2.

For a group $G$ we define its _exponent type_ to be the function

$e_{G}:n\mapsto|\\{g\in G\ |\ g^{n}=1\\}|.$

######  Lemma 1.

For groups $G$ and $H$ we have: $o_{G}=o_{H}\iff e_{G}=e_{H}$.

###### Proof.

$g^{n}=1$ if and only if $\text{ord}(g)$ divides $n$. Thus
$e_{G}(n)=\sum_{d|n}o_{G}(d)$, and $o_{G}(n)=\sum_{d|n}e_{G}(n/d)\cdot\mu(d)$,
where $\mu$ is the Möbius function. ∎

######  Lemma 2.

Let $G$ and $H$ be any groups. Then $e_{G\times H}=e_{G}\cdot e_{H}$.

###### Proof.

For $(g,h)\in G\times H$ we have $(g,h)^{n}=(g^{n},h^{n})$, so $(g,h)^{n}$ is
trivial if and only if $g^{n}$ and $h^{n}$ are trivial. ∎

###### Proof of A.

With the help of MAGMA computational package [2] we can check that—with the
notable exception of $H_{16}$—all of the groups $G_{i}$ and $H_{i}$ are
solvable. This means that $G$ is solvable as a product of solvable groups, and
$H$ isn’t as it contains a non-solvable subgroup, namely
$H_{16}\cong\mathrm{GL}(3,2)$.

To prove $e_{G}=e_{H}$ we need to compute the exponent types $e_{G_{i}}$ and
$e_{H_{i}}$, and check that $\prod e_{G_{i}}^{m_{i}}=\prod e_{H_{i}}^{n_{i}}$.

The exponent types of groups $G_{i}$ and $H_{i}$ were computed with the help
of MAGMA too (see the ancillary file exponents_computation.m) and are listed
in Tables 2 and 4. Only the columns labelled with divisors of $168$ were
included as $e_{G}(n)=e_{G}\left(\gcd(n,E_{G})\right)$ for $E_{G}$ being the
exponent of the group $G$. Since the products $\prod e_{G_{i}}(n)^{m_{i}}$ and
$\prod e_{H_{i}}(n)^{n_{i}}$ become unmanageably large, we factorise them into
their prime factors to compute the exponent types of $G$ and $H$ listed in
Tables 3 and 5. This computation is done in the ancillary file
exponents_verification.ipynb. ∎

##  3\. How these groups were found

Finding groups $G$ and $H$ presented in A manually would be very difficult.
Instead, we employed a computer-based search method outlined below.

Step 1. We used MAGMA to access the Small Groups database and to we go through
all groups of size at most 2000 excluding groups of size divisible by 128, and
to compute for each group

  1. (1)

whether it is solvable—using IsSolvable function,

  2. (2)

whether it is a non-trivial direct product—using a custom function
IsDirectProduct (see the ancillary file functions.m), which searches for two
normal subgroups with trivial intersection and whose sizes multiply to the
size of the investigated group,

  3. (3)

its order type—using a simple custom function OrderType.

Step 2. We parsed the outputs into a .csv table and handled the remaining
tasks using various Python libraries. Given our objective of constructing an
example through the direct product of groups, we focused solely on those
groups that were not direct products themselves.

Step 3. We rephrased the question as follows.

  1. (1)

First we converted the order types $o_{G}$ to exponent types $e_{G}$ as
described in Section 2.

  2. (2)

Then we used multiplicative Möbius inversion on $e_{G}$ defining the _revolved
exponent type_ $r_{G}:\mathbb{N}\to\mathbb{Q}$ as

$r_{G}(n):=\prod_{d|n}e_{G}(n/d)^{\mu(d)}.$

The advantage of doing so was that $r_{G}(n)=1$ if $n\nmid E_{G}$. Indeed, for
$k\mid E_{G}$ and $\gcd(m,E_{G})=1$ we get the following.

$\displaystyle r_{G}(mk)$
$\displaystyle=\prod_{d|mk}e_{G}(mk/d)^{\mu(d)}=\prod_{e|m}\prod_{f|k}e_{G}(m/e\cdot
k/f)^{\mu(ef)}$
$\displaystyle=\prod_{e|m}\prod_{f|k}e_{G}(k/f)^{\mu(e)\mu(f)}=\prod_{e|m}\Big{(}\prod_{f|k}e_{G}(k/f)^{\mu(f)}\Big{)}^{\mu(e)}$
$\displaystyle=r_{G}(k)^{\sum_{e|m}\mu(e)}=\begin{cases}r_{G}(k)&\text{ if
}m=1,\\\ 1&\text{ otherwise}.\end{cases}$

Additionally, we maintained the multiplicativity with respect to direct
products: $r_{G_{1}\times G_{2}}=r_{G_{1}}\cdot r_{G_{2}}$. In order to
operate with rational numbers we used a Python datatype Fraction.

  3. (3)

Finally, we factorised the revolved exponent types into their prime factors by
defining $v_{G}(n,p)$ to be the exponent of a prime $p$ in the factorisation
of $r_{G}(n)$ into prime factors. Thus, for $G=\prod G_{i}^{m_{i}}$ we have

$v_{G}(n,p)=\sum m_{i}\cdot v_{G_{i}}(n,p).$

Additionally, for a fixed $G$, the values of $v_{G}(n,p)$ are zero for all but
finitely many combinations $(n,p)$.

At this point the question became: is there a positive rational linear
combination $\sum k_{i}\cdot v_{N_{i}}$ for non-solvable groups $N_{i}$ in the
rational span of $v_{S_{i}}$ for $S_{i}$ solvable groups?

Step 4. An even simpler question, although not equivalent, arises: is any
$v_{N}$ for a non-solvable group $N$ in the span of $v_{S_{i}}$ for solvable
groups $S_{i}$? This question is equivalent to determining whether the
equation $Vx=v_{N}$ has a solution, where $V$ is a matrix of $v_{S_{i}}(n,p)$
with rows indexed by pairs $(n,p)$ and columns indexed by solvable groups
$S_{i}$. A significant challenge lies in that the matrix $V$ were in our case
of size $9945\times 100972$, albeit very sparse.

Step 5. We first attempted solving this numerically using Least Squares
methods of the SciPy library [12], which gave two conclusions.

  1. (1)

For many of the non-solvable groups considered (e.g. $N=A_{5}$) the
(numerical) projection of their $v_{N}$ onto the column space of $V$ was at
distance about $1$ from $v_{N}$, meaning that $v_{N}$ was almost certainly not
in the span.

  2. (2)

For some of the groups (e.g. the $\mathrm{GL}(3,2)$) the algorithm converged
to a solution up to a small error of $10^{-6}$. This lead us to search for an
exact solution to $Vx=v_{N}$ for these groups.

Step 6. For $N=\mathrm{GL}(3,2)$ we solved $Vx=v_{N}$ exactly using the
symbolic computation library SymPy [7]. A family of solutions was found; we
chose one of them.

Step 7. We converted the solution back into the language of groups and checked
that it holds using more direct methods.

$i$ | $G_{i}$ | Id | $m_{i}$ | $E_{G_{i}}$ | 1 | 2 | 3 | 4 | 6 | 7 | 8 | 12 | 14 | 21 | 24 | 28 | 42 | 56 | 84 | 168
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | $C_{4}$ | (4, 1) | 9 | 4 | 1 | 2 | 1 | 4 | 2 | 1 | 4 | 4 | 2 | 1 | 4 | 4 | 2 | 4 | 4 | 4
2 | $D_{3}$ | (6, 1) | 6 | 6 | 1 | 4 | 3 | 4 | 6 | 1 | 4 | 6 | 4 | 3 | 6 | 4 | 6 | 4 | 6 | 6
3 | $C_{7}$ | (7, 1) | 1 | 7 | 1 | 1 | 1 | 1 | 1 | 7 | 1 | 1 | 7 | 7 | 1 | 7 | 7 | 7 | 7 | 7
4 | $D_{4}$ | (8, 3) | 9 | 4 | 1 | 6 | 1 | 8 | 6 | 1 | 8 | 8 | 6 | 1 | 8 | 8 | 6 | 8 | 8 | 8
5 | $D_{7}$ | (14, 1) | 18 | 14 | 1 | 8 | 1 | 8 | 8 | 7 | 8 | 8 | 14 | 7 | 8 | 14 | 14 | 14 | 14 | 14
6 | $\mathrm{SL}(2,3)$ | (24, 3) | 21 | 12 | 1 | 2 | 9 | 8 | 18 | 1 | 8 | 24 | 2 | 9 | 24 | 8 | 18 | 8 | 24 | 24
7 | $C_{24}\rtimes C_{2}$ | (48, 6) | 3 | 24 | 1 | 14 | 3 | 28 | 18 | 1 | 32 | 36 | 14 | 3 | 48 | 28 | 18 | 32 | 36 | 48
8 | $C_{7}\rtimes D_{4}$ | (56, 7) | 3 | 28 | 1 | 18 | 1 | 32 | 18 | 7 | 32 | 32 | 42 | 7 | 32 | 56 | 42 | 56 | 56 | 56
9 | $C_{7}\rtimes C_{12}$ | (84, 1) | 6 | 84 | 1 | 2 | 15 | 16 | 30 | 7 | 16 | 72 | 14 | 21 | 72 | 28 | 42 | 28 | 84 | 84
10 | $\operatorname{\mathrm{Dic}}_{21}$ | (84, 5) | 6 | 84 | 1 | 2 | 3 | 44 | 6 | 7 | 44 | 48 | 14 | 21 | 48 | 56 | 42 | 56 | 84 | 84
11 | $C_{7}\rtimes A_{4}$ | (84, 11) | 21 | 42 | 1 | 4 | 57 | 4 | 60 | 7 | 4 | 60 | 28 | 63 | 60 | 28 | 84 | 28 | 84 | 84
12 | $C_{7}\rtimes D_{7}$ | (98, 4) | 2 | 14 | 1 | 50 | 1 | 50 | 50 | 49 | 50 | 50 | 98 | 49 | 50 | 98 | 98 | 98 | 98 | 98
13 | $C_{4}\rtimes F_{7}$ | (168, 9) | 21 | 84 | 1 | 30 | 15 | 32 | 114 | 7 | 32 | 144 | 42 | 21 | 144 | 56 | 126 | 56 | 168 | 168
14 | $C_{21}\rtimes D_{4}$ | (168, 15) | 9 | 84 | 1 | 22 | 3 | 64 | 54 | 7 | 64 | 96 | 70 | 21 | 96 | 112 | 126 | 112 | 168 | 168
15 | $C_{7}\rtimes D_{12}$ | (168, 17) | 6 | 84 | 1 | 50 | 3 | 64 | 54 | 7 | 64 | 96 | 98 | 21 | 96 | 112 | 126 | 112 | 168 | 168
16 | $F_{8}\rtimes C_{3}$ | (168, 43) | 3 | 42 | 1 | 8 | 57 | 8 | 120 | 49 | 8 | 120 | 56 | 105 | 120 | 56 | 168 | 56 | 168 | 168
17 | $D_{8}\rtimes D_{7}$ | (224, 106) | 3 | 56 | 1 | 52 | 1 | 96 | 52 | 7 | 128 | 96 | 112 | 7 | 128 | 168 | 112 | 224 | 168 | 224
18 | $C_{7}\rtimes D_{24}$ | (336, 31) | 3 | 168 | 1 | 98 | 3 | 100 | 102 | 7 | 128 | 108 | 182 | 21 | 192 | 196 | 210 | 224 | 252 | 336
Table 2. The groups $G_{i}$, their multiplicities $m_{i}$, their exponents
$E_{G_{i}}$, and their exponent types.

1 | 1
---|---
2 | $2^{221}\cdot 3^{36}\cdot 5^{37}\cdot 7^{9}\cdot 11^{9}\cdot 13^{3}$
3 | $3^{126}\cdot 5^{27}\cdot 19^{24}$
4 | $2^{500}\cdot 3^{3}\cdot 5^{10}\cdot 7^{3}\cdot 11^{6}$

6 | $2^{215}\cdot 3^{174}\cdot 5^{34}\cdot 13^{3}\cdot 17^{3}\cdot 19^{21}$
---|---
7 | $7^{107}$
8 | $2^{530}\cdot 5^{4}\cdot 11^{6}$
12 | $2^{464}\cdot 3^{144}\cdot 5^{28}$

14 | $2^{191}\cdot 3^{33}\cdot 5^{9}\cdot 7^{113}\cdot 13^{3}$
---|---
21 | $3^{147}\cdot 5^{3}\cdot 7^{104}$
24 | $2^{488}\cdot 3^{132}\cdot 5^{28}$
28 | $2^{374}\cdot 3^{3}\cdot 7^{110}$

42 | $2^{185}\cdot 3^{177}\cdot 5^{3}\cdot 7^{104}$
---|---
56 | $2^{398}\cdot 7^{104}$
84 | $2^{347}\cdot 3^{114}\cdot 7^{104}$
168 | $2^{365}\cdot 3^{105}\cdot 7^{104}$

Table 3. The exponent type of $G$. $i$ | $H_{i}$ | Id | $n_{i}$ | $E_{H_{i}}$ | 1 | 2 | 3 | 4 | 6 | 7 | 8 | 12 | 14 | 21 | 24 | 28 | 42 | 56 | 84 | 168
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | $C_{2}$ | (2, 1) | 21 | 2 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2
2 | $C_{3}$ | (3, 1) | 3 | 3 | 1 | 1 | 3 | 1 | 3 | 1 | 1 | 3 | 1 | 3 | 3 | 1 | 3 | 1 | 3 | 3
3 | $\operatorname{\mathrm{Dic}}_{3}$ | (12, 1) | 6 | 12 | 1 | 2 | 3 | 8 | 6 | 1 | 8 | 12 | 2 | 3 | 12 | 8 | 6 | 8 | 12 | 12
4 | $A_{4}$ | (12, 3) | 21 | 6 | 1 | 4 | 9 | 4 | 12 | 1 | 4 | 12 | 4 | 9 | 12 | 4 | 12 | 4 | 12 | 12
5 | $\operatorname{\mathrm{SD}}_{16}$ | (16, 8) | 3 | 8 | 1 | 6 | 1 | 12 | 6 | 1 | 16 | 12 | 6 | 1 | 16 | 12 | 6 | 16 | 12 | 16
6 | $C_{7}\rtimes C_{3}$ | (21, 1) | 4 | 21 | 1 | 1 | 15 | 1 | 15 | 7 | 1 | 15 | 7 | 21 | 15 | 7 | 21 | 7 | 21 | 21
7 | $D_{12}$ | (24, 6) | 6 | 12 | 1 | 14 | 3 | 16 | 18 | 1 | 16 | 24 | 14 | 3 | 24 | 16 | 18 | 16 | 24 | 24
8 | $C_{3}\rtimes D_{4}$ | (24, 8) | 6 | 12 | 1 | 10 | 3 | 16 | 18 | 1 | 16 | 24 | 10 | 3 | 24 | 16 | 18 | 16 | 24 | 24
9 | $\operatorname{\mathrm{Dic}}_{7}$ | (28, 1) | 15 | 28 | 1 | 2 | 1 | 16 | 2 | 7 | 16 | 16 | 14 | 7 | 16 | 28 | 14 | 28 | 28 | 28
10 | $F_{7}$ | (42, 1) | 18 | 42 | 1 | 8 | 15 | 8 | 36 | 7 | 8 | 36 | 14 | 21 | 36 | 14 | 42 | 14 | 42 | 42
11 | $D_{21}$ | (42, 5) | 6 | 42 | 1 | 22 | 3 | 22 | 24 | 7 | 22 | 24 | 28 | 21 | 24 | 28 | 42 | 28 | 42 | 42
12 | $D_{24}$ | (48, 7) | 3 | 24 | 1 | 26 | 3 | 28 | 30 | 1 | 32 | 36 | 26 | 3 | 48 | 28 | 30 | 32 | 36 | 48
13 | $D_{28}$ | (56, 5) | 27 | 28 | 1 | 30 | 1 | 32 | 30 | 7 | 32 | 32 | 42 | 7 | 32 | 56 | 42 | 56 | 56 | 56
14 | $\operatorname{\mathrm{Dic}}_{7}\rtimes C_{6}$ | (168, 11) | 3 | 84 | 1 | 18 | 15 | 32 | 102 | 7 | 32 | 144 | 42 | 21 | 144 | 56 | 126 | 56 | 168 | 168
15 | $C_{14}.A_{4}$ | (168, 23) | 21 | 84 | 1 | 2 | 57 | 8 | 114 | 7 | 8 | 120 | 14 | 63 | 120 | 56 | 126 | 56 | 168 | 168
16 | $\mathrm{GL}(3,2)$ | (168, 42) | 3 | 84 | 1 | 22 | 57 | 64 | 78 | 49 | 64 | 120 | 70 | 105 | 120 | 112 | 126 | 112 | 168 | 168
17 | $C_{7}\rtimes F_{7}$ | (294, 10) | 2 | 42 | 1 | 50 | 15 | 50 | 162 | 49 | 50 | 162 | 98 | 147 | 162 | 98 | 294 | 98 | 294 | 294
18 | $D_{12}.D_{7}$ | (336, 36) | 3 | 168 | 1 | 14 | 3 | 100 | 18 | 7 | 128 | 108 | 98 | 21 | 192 | 196 | 126 | 224 | 252 | 336
Table 4. The groups $H_{i}$, their multiplicities $n_{i}$, their exponents
$E_{H_{i}}$, and their exponent types.

1 | 1
---|---
2 | $2^{221}\cdot 3^{36}\cdot 5^{37}\cdot 7^{9}\cdot 11^{9}\cdot 13^{3}$
3 | $3^{126}\cdot 5^{27}\cdot 19^{24}$
4 | $2^{500}\cdot 3^{3}\cdot 5^{10}\cdot 7^{3}\cdot 11^{6}$

6 | $2^{215}\cdot 3^{174}\cdot 5^{34}\cdot 13^{3}\cdot 17^{3}\cdot 19^{21}$
---|---
7 | $7^{107}$
8 | $2^{530}\cdot 5^{4}\cdot 11^{6}$
12 | $2^{464}\cdot 3^{144}\cdot 5^{28}$

14 | $2^{191}\cdot 3^{33}\cdot 5^{9}\cdot 7^{113}\cdot 13^{3}$
---|---
21 | $3^{147}\cdot 5^{3}\cdot 7^{104}$
24 | $2^{488}\cdot 3^{132}\cdot 5^{28}$
28 | $2^{374}\cdot 3^{3}\cdot 7^{110}$

42 | $2^{185}\cdot 3^{177}\cdot 5^{3}\cdot 7^{104}$
---|---
56 | $2^{398}\cdot 7^{104}$
84 | $2^{347}\cdot 3^{114}\cdot 7^{104}$
168 | $2^{365}\cdot 3^{105}\cdot 7^{104}$

Table 5. The exponent type of $H$, equal to the exponent type of $G$.

## References

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  * [2] Wieb Bosma, John Cannon, and Catherine Playoust.  The Magma algebra system. I. The user language.  J. Symbolic Comput., 24(3-4):235–265, 1997.  Computational algebra and number theory (London, 1993).
  * [3] Robert M Guralnick and Al Weiss.  Same genus and embeddings of groups.  In Linear Algebraic Groups and Their Representations: Conference on Linear Algebraic Groups and Their Representations, March 25-28, 1992, Los Angeles, California, volume 153, page 21. American Mathematical Soc., 1993.
  * [4] Marcel Herzog, Patrizia Longobardi, and Mercede Maj.  Another criterion for solvability of finite groups.  Journal of Algebra, 597:1–23, 2022.
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$\displaystyle=b\left(w_{0}+\delta_{K}(\lambda^{\ast}(T-\tau,B(\tau)))\right)+(\kappa\odot\theta)^{\prime}B(\tau)+\frac{1}{2}\|\sigma^{\prime}B(\tau)\|^{2}$
$\displaystyle\qquad+\frac{1}{2}\frac{b}{1-b}\Bigg{(}\left\|\eta+\big{(}b(T-\tau;\hat{T})^{\prime}\sigma\big{)}^{-1}\lambda^{\ast}(T-\tau,B(\tau))\right\|^{2}$
$\displaystyle\hskip
71.13188pt+2\left(\sigma\eta+\left(b(t;\hat{T})^{\prime}\right)^{-1}\lambda^{\ast}(t,B(\tau))\right)^{\prime}B(\tau)+\left\|\sigma^{\prime}B(\tau)\right\|^{2}\Bigg{)}$
$\displaystyle=bw_{0}+(\kappa\odot\theta)^{\prime}B(\tau)+\frac{1}{2}\|\sigma^{\prime}B(\tau)\|^{2}$
$\displaystyle\qquad+\frac{1}{2}\frac{b}{1-b}\Bigg{(}2(1-b)\delta_{K}(\lambda^{\ast}(T-\tau,B(\tau)))+\left\|\eta+\sigma^{\prime}B(\tau)+\left(b(T-\tau;\hat{T})^{\prime}\right)^{-1}\lambda^{\ast}(T-\tau,B(\tau))\right\|^{2}\Bigg{)}$
$\displaystyle=bw_{0}+(\kappa\odot\theta)^{\prime}B(\tau)+\frac{1}{2}\|\sigma^{\prime}B(\tau)\|^{2}+\frac{1}{2}\frac{b}{1-b}\inf_{\lambda\in\mathbbm{R}^{d}}\left(2(1-b)\delta_{K}(\lambda)+\left\|\eta+\sigma^{\prime}B(\tau)+\left(b(T-\tau;\hat{T})^{\prime}\right)^{-1}\lambda\right\|^{2}\right)$

and

$\displaystyle B_{\tau}(\tau)$ $\displaystyle=bw_{1}-\kappa\odot B(\tau).$

Hence, according to Theorem 3.10,

$G(t,v,z)=\frac{1}{b}v^{b}\exp(A(T-t)+B(T-t)^{\prime}z)$

is a solution to the dual HJBI PDE (3.6). The corresponding candidate optimal
portfolio (18) is given through

$\displaystyle\underaccent{\bar}{\pi}^{\ast}(t,v,z)$
$\displaystyle=\frac{1}{1-b}\left(\Sigma(t,z)\Sigma(t,z)^{\prime}\right)^{-1}\Big{[}\mu(t,z)+\underaccent{\bar}{\lambda}^{\ast}(t,v,z)-r(t,z)\mathbbm{1}+\left(\Sigma^{z}(t,z)\rho(t,z)\Sigma(t,z)^{\prime}\right)^{\prime}B(T-t)\Big{]}$
$\displaystyle=\frac{1}{1-b}\left(\sigma^{\prime}b(t;\hat{T})\right)^{-1}\Big{(}\eta+\left(b(t;\hat{T})^{\prime}\sigma\right)^{-1}\lambda^{\ast}(t,B(T-t))+\sigma^{\prime}B(T-t)\Big{)}.$

∎

###### Proof of Corollary 4.9.

We again verify Condition ($\text{UI}_{\lambda}$) by showing the $L^{q}$
boundedness of
$G\left(\tau^{0}_{n,t},V^{v_{0},\underaccent{\bar}{\pi}^{\ast}}_{0}(\tau^{0}_{n,t}),z(\tau^{0}_{n,t})\right)$
in $n\in\mathbbm{N}$ for arbitrary $q>1.$
As per Remark 4.8, there exists a closed-form expression for $B$ which is
continuously differentiable. Moreover, the matrix $b(t;\hat{T})$ is
continuously differentiable in $t$ and non-singular for all $t\in[0,T]$ and
therefore uniformly bounded in $t\in[0,T].$ Following the same arguments as in
the proof of Corollary 4.3, this has the consequence that the minimizer
$\lambda^{\ast}(t,B(T-t)),$ $A(T-t)$ and the candidate optimal portfolio
$\pi^{\ast}(t):=\underaccent{\bar}{\pi}^{\ast}(t,v,z)$ are uniformly bounded
in $t\in[0,T].$

For arbitrary $q>1,$ we can thus find a constant $C_{q}>0$ such that for all
$t\in[0,T]$

$\displaystyle\Big{|}G(t,V^{v_{0},\underaccent{\bar}{\pi}^{\ast}}(t),z(t))\Big{|}^{q}$
$\displaystyle=\frac{1}{|b|}\exp\Bigg{(}bq\int_{0}^{t}w_{0}+w_{1}^{\prime}z(s)+\eta^{\prime}\sigma^{\prime}b(s;\hat{T})\pi^{\ast}(s)-\frac{1}{2}\|\sigma^{\prime}b(s;\hat{T})\pi^{\ast}(s)\|^{2}ds$
$\displaystyle\hskip
56.9055pt+bq\int_{0}^{t}\pi^{\ast}(s)^{\prime}b(s;\hat{T})^{\prime}\sigma
dW(s)+qA(T-t)+qB(T-t)^{\prime}z(t)\Bigg{)}$ $\displaystyle\leq
C_{q}\exp\left(\underbrace{bq\int_{0}^{t}w_{1}^{\prime}z(s)ds+bq\int_{0}^{t}\pi^{\ast}(s)^{\prime}b(s;\hat{T})^{\prime}\sigma
dW(s)+qB(T-t)^{\prime}z(t)}_{=:X_{t}}\right)$
$\displaystyle=C_{q}\exp\left(X_{t}\right).$ (34)

Since, $B$ is continuously differentiable, we can use Itô’s product rule to
rewrite

$\displaystyle B(T-t)^{\prime}z(t)$
$\displaystyle=B(T)^{\prime}z_{0}+\int_{0}^{t}B(T-s)^{\prime}dz(s)+\int_{0}^{t}z(s)^{\prime}d\big{(}B(T-s)\big{)}+\underbrace{\langle
z,B(T-\cdot)\rangle_{t}}_{=0}$
$\displaystyle=B(T)^{\prime}z_{0}+\int_{0}^{t}B(T-s)^{\prime}\kappa\odot\left[\theta-z(s)\right]-z(s)^{\prime}B_{\tau}(T-s)ds+\int_{0}^{t}B(T-s)^{\prime}\sigma
dW^{z}(s)$ (35)

Due to $\rho(t,z)=I_{m},$ we know that $W^{z}(t)=W(t)$ holds
$\mathcal{L}[0,T]\otimes Q$-a.e.. Hence, using (35) and disregarding terms of
finite variation, the quadratic variation of $X$ can be computed as

$\displaystyle\langle X\rangle_{t}$
$\displaystyle=\Bigg{\langle}bq\int_{0}^{t}w_{1}^{\prime}z(s)ds+bq\int_{0}^{\cdot}\pi^{\ast}(s)^{\prime}b(s;\hat{T})^{\prime}\sigma\underbrace{dW(s)}_{=dW^{z}(s)}+qB(T-\cdot)^{\prime}z(\cdot)\Bigg{\rangle}_{t}$
$\displaystyle=\left\langle
bq\int_{0}^{\cdot}\pi^{\ast}(s)^{\prime}b(s;\hat{T})^{\prime}\sigma+\frac{1}{b}B(T-s)^{\prime}\sigma
dW^{z}(s)\right\rangle_{t}$
$\displaystyle=b^{2}q^{2}\int_{0}^{t}\left\|\sigma^{\prime}\left(b(s;\hat{T})\pi^{\ast}(s)+\frac{1}{b}B(T-s)\right)\right\|^{2}ds.$

Since all involved funtions are bounded and deterministic, $\langle
X\rangle_{t}\leq\langle X\rangle_{T}<\infty$ yields a deterministic upper
bound on $\langle X\rangle_{t}$ for all $t\in[0,T].$ Therefore, we can
continue equation (34) to obtain for all $t\in[0,T]$

$\displaystyle\Big{|}G(t,V^{v_{0},\underaccent{\bar}{\pi}^{\ast}}(t),z(t))\Big{|}^{q}$
$\displaystyle\overset{(\ref{eq: first Lq bound value function
OU})}{\leq}C_{q}\exp(X_{t})\leq\underbrace{C_{q}\exp\left(\frac{1}{2}\langle
X\rangle_{T}\right)}_{=:\tilde{C}_{q}}\underbrace{\exp\left(X_{t}-\frac{1}{2}\langle
X\rangle_{t}\right)}_{=:M_{t}}=\tilde{C}_{q}M_{t}$ (36)

The process $M=\left(M_{t}\right)_{t\in[0,T]}$ is a non-negative local
martingale and thus a supermartingale. Doob’s optional sampling theorem (
?O.S.?) implies

$\displaystyle\sup_{n\in\mathbbm{N}}\mathbbm{E}\left[\left|G\left(\tau^{0}_{n,t},V^{v_{0},\underaccent{\bar}{\pi}^{\ast}}_{0}(\tau^{0}_{n,t}),z(\tau^{0}_{n,t})\right)\right|^{q}\right]\overset{(\ref{eq:
second Lq bound value function
OU})}{\leq}\tilde{C}_{q}\sup_{n\in\mathbbm{N}}\mathbbm{E}\left[M_{\tau^{0}_{n,t}}\right]\overset{O.S.}{\leq}\tilde{C}_{q}M_{0}=\tilde{C}_{q}<\infty.$

Hence,

$\left(G\left(\tau^{0}_{n,t},V^{v_{0},\underaccent{\bar}{\pi}^{\ast}}_{0}(\tau^{0}_{n,t}),z(\tau^{0}_{n,t})\right)\right)_{n\in\mathbbm{N}}$

is bounded in $L^{q}$ for any $q>1$ and $t\in[0,T]$ and is thus uniformly
integrable for any $t\in[0,T]$ (see Theorem 4.5.9 in [41] with $G(t)=t^{q}$).
Hence, Condition ($\text{UI}_{\lambda}$) is satisfied and
$\underaccent{\bar}{\pi}^{\ast}$ is optimal for $\mathbf{(P)}$ by virtue of
Theorem 3.12. ∎

###### Notes

  1. 1 We use $\|\cdot\|$ to denote the standard Euclidian norm.
  2. 2 A proper concave function $f:\mathbbm{R}^{d}\rightarrow\bar{\mathbbm{R}}$ is a concave function with $f(x)<\infty$ for all $x\in\mathbbm{R}^{d}$ and there exists at least one $x_{0}\in\mathbbm{R}^{d}$ such that $f(x_{0})>-\infty$.
  3. 3 It is important to emphasize that the ensuing duality relations as well as their applicability are dependent on the specific choice of the pertubation $F(x,u)$ of $f(x)$.
  4. 4 In the convex analysis literature (see e.g. equation (3.25) in [67]), it is customary to define the conjugate of $F$ in the concave sense (w.r.t. $u$) as  $F^{\ast}(x,\lambda)=-\sup_{u\in\mathbbm{R}^{d}}\big{(}F(x,u)-\lambda^{\prime}u\big{)}=\inf_{u\in\mathbbm{R}^{d}}\big{(}\lambda^{\prime}u-F(x,u)\big{)}.$ In contrast to our definition, this has the satisfying consequence that the bi-conjugate of $F(x,\cdot)$ is obtained by taking the conjugate of $F^{\ast}(x,\cdot)$, i.e. $F^{\ast\ast}(x,\cdot)=(F^{\ast}(x,\cdot))^{\ast}(x,\cdot)$. According to our definition, we have the (slightly) less elegant version $F^{\ast\ast}(x,\cdot)=-(-F^{\ast}(x,\cdot))^{\ast}(x,\cdot)$. However, in the mathematical finance literature, in particular in a portfolio optimization context, our definition is more prevalent. Since we are going to apply the above theory in a portfolio optimization context, we decided to adhere to the corresponding convention.
  5. 5 In fact, the result of [62] even includes the more general case of an exponentially quadratic separation. The approach we present below can be extended to include quadratic separation in a natural manner. However, such an extension would complicate the involved notation and thus diminish the presentation of the core concepts involved. Moreover, we were not able to construct realistic working examples that require quadratic separation in an allocation constrained setting. Hence, we restrict our analysis in this work to exponentially affine separation.
  6. 6 The functions $h_{1}[\cdot]$ and $l_{1}[\cdot]$ are three-dimensional tensors, which are a generalization of vectors and matrices to higher dimensions. In our context, we may think of $h_{1}$ and $l_{1}$ as being represented by matrices, whose entries $(h_{1})_{ij}$ and $(l_{1})_{ij}$ are $\mathbbm{R}^{m}$-valued. Upon being evaluated at a $z\in\mathbbm{R}^{m},$ each entry of $h_{1}[z]$ and $l_{1}[z]$ is obtained by computing the scalar product $z^{\prime}(h_{1})_{ij}$ and $z^{\prime}(l_{1})_{ij}$. Hence, for any $x,y\in\mathbbm{R}^{m}$ and applying the rules of ordinary vector-matrix multiplication, the products $x^{\prime}h_{1}[\cdot]y$ and $x^{\prime}l_{1}[\cdot]y$ are vectors in $\mathbbm{R}^{m}$. In particular, $x^{\prime}h_{1}[z]y=z^{\prime}\left(x^{\prime}h_{1}[\cdot]y\right)$ and $x^{\prime}l_{1}[z]y=z^{\prime}\left(x^{\prime}l_{1}[\cdot]y\right)$ for any $x,y,z\in\mathbbm{R}^{m}$.
  7. 7 Here, $A_{\tau}$ and $B_{\tau}$ denote the derivatives of $A$ and $B$ with respect to $\tau\in[0,T]$.
  8. 8 We could equivalently consider $m=0$ and completely disregard the stochastic factor $z$ and its drift, diffusion and correlation coefficients in the definition of the Black-Scholes model $\mathcal{M}_{BS}.$
  9. 9 We will later see in Lemma 4.5 that Condition $(\text{EAS})$ is only satisfied in $\mathcal{M}_{CIR}$ if the structure of the allocation constraints allows for a convenient separabiltiy in (15). In the definition of $\mathcal{M}_{CIR},$ we have intentionally limited the covariance $\Sigma$ of risky assets to be of block-diagonal structure to facilitate the presentation of this fact. However, we can in principle also choose more complex models for $\Sigma,$ such as the general PCSV model, and adjust the allocation constraints accordingly without changing the underlying theory in a significant way.
  10. 10 Compare to the derivation of (33) in the proof of the subsequent Lemma 4.5 for details.
  11. 11 Note that this assumption is valid due to the exponential structure of the wealth process $V^{v_{0},\pi}$ and the strict monotonicity of the utility function $U$.