# Emerging Trends Detection: A Study About Microsoft at the Beginning of
COVID-19 Pandemic

First Author1111Contact Author Second Author2 Third Author2,3&Fourth Author4
1First Affiliation
2Second Affiliation
3Third Affiliation
4Fourth Affiliation
{first<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Anonymous

###### Abstract

Identifying and exploring emerging trends in news is an important task that
helps to understand opportunities through available information. The task is
becoming more essential than ever with many changes occurring around the world
due to the global health crisis. In this paper, we focus our study on emerging
trends detection in news articles about Microsoft, collected before and during
the start of the COVID-19 pandemic (July 2019 to July 2020). We, first,
propose a method for extracting potential keywords that can signify emerging
trends based on topic modelling and term frequency. Next, we detect trends
using an approach based on contextualized and temporal-aware bi-gram
representations to track and rank the similarity relationships between pairs
of keywords. Finally, we evaluate against a gold standard (Google Trends) and
explore our methods in particular scenarios regarding the influence of the
pandemic toward Microsoft.

##  1 Introduction

Digital news, through many means of diffusion (online publishing platforms,
social media, blogs, etc.) is considerably influential, as it not only shapes
and forms public opinion but can also be a factor in the decision-making
process of many industries that uses technology to improve activities and
performance. Therefore, discovering hidden themes and trends residing in news
data is essential to improve analyzing and managing development directions for
many companies. The importance of identifying new trends before they emerge is
further emphasized with the world-changing surrounding the health crisis
triggered by the COVID-19 pandemic.

Emerging trend detection is the task of automatically extracting topics that
are gaining attention and on the verge of being trending QuiDang2016. Emerging
topics usually indicate contents that are more popular in a short period,
while growing in interest and utility over time. Moreover, topics that become
a trend can either be short-lived or last for a long time depending on the
nature of the event (e.g., traffic accidents, natural disasters, election
campaigns, regulation enforcement, etc.).

Based on the platform of publication, the data used for detecting emerging
trends can be classified into two classes: social media text and formal text.
Corpora crawled from social media usually contain text that is short and
concise. usually include social features. Moreover, the main advantage of
social media text in emerging trend detection is that they can benefit from
the social features (e.g., likes and retweets in Twitter) in the metadata,
which can be used to measure the engagement and diffusion rate of content.
Because of this fact, data from social media has been extensively studied in
various research Zarrinkalam2017EventII; Zubiaga2015RealtimeCO;
Saha2012LearningEA; Peng2018EmergingTD; Chen2013EmergingTD;
Peng2018EmergingPT.

Formal text data, unlike the short sub-300 characters post usually seen in
social media, comes with longer, less restricted writing format. An example
includes, but is not limited to news articles, official documents, reports,
which normally do not come with any social features. However, because of these
differences, emerging trend detection on such data is rather under-researched.
While there exist studies on financial data borsje2010semi; malik2011accurate
and news articles Manal2020; BangLiu2020 business model patents
WonSangLee2017, most of them are based on common techniques such as latent
Dirichlet allocation (LDA) topic modelling method Behpour2021;
Bissoyi2020DiscoveringTT, term frequency-inverse document frequency (TF-IDF)
weighting technique Zhu2019; Enrico2020, or different clustering types
linger2020batch; Cao2018; li2020topic for the identification of either “hot”
topics or emerging themes.

Therefore, in this paper, not only that we build a formal dataset that we
extracted from Bloomberg’s Event Driven Feeds (EDF) news collection geared
towards an audience that is interested in finance, but we also propose an
adaptation of an existing work in emerging trends detection in scientific
papers, called Leap2Trend Dridi2019. Because Leap2Trend only considered the
writing style of scientific papers, we modified the keyword extraction phase.
Moreover, we propose the usage of recent contextual embeddings to adapt the
system performing on a news corpus by taking into consideration the
temporality of trending topics.

Finally, our main contributions of our study are: (1) We build a dataset from
the EDF data, containing news about Microsoft in the time interval from July
2019 to July 2020, which is the time period from six-month before and six-
month after the COVID-19 outbreak; (2) We utilize the latest contextual
embeddings to represent the real temporality and variation of the semantics
during different periods; (3) We combine term TF-IDF and LDA for a more
precise generation of keywords (bi-grams); (4) We apply a reverse ranking
method based on an existing work in emerging trends detection in scientific
papers, called Leap2Trend Dridi2019, to observe the temporal evaluation of
pairs of keywords that will provide us with emerging trends.

The article is organized as follows. Section 2 discusses related work on
emerging theme detection. We then describe in Section 3 our EDF dataset.
Section 4 explains our approach for detecting emerging trends in financial-
based data and the experimental setup is established in Section 5 alongside
with the evaluation metrics and the detailed analysis regarding the impact of
the pandemic on Microsoft. Lastly, Section 6 concludes the article with
remarks and points to future work.

##  2 Related Work

#### Trend Detection in Formal Datasets

The general direction for trend detection in formal text data is to use
statistical methods hughes-etal-2020-detecting; Schubert2014SigniTrendSD;
Daud2021FindingRS, topic modeling blei2003latent; Behpour2021, clustering
Goorha2010DiscoveryOS; Chen2007HotTE; BangLiu2020; linger2020batch or
embeddings to represent text Yao2018DynamicWE; Dridi2019. The following
studies describe how these methods were used for the task of emerging themes
detection in recent years. Using financial business patents, the research by
WonSangLee2017 aimed to identify emerging technology trends by applying latent
Dirichlet allocation (LDA) blei2003latent with an exponentially weighted
moving average of LDA probability, which affects whether a topic is “hot” or
“cold”. Survival analysis is utilized within the gap of time when a topic
emerges from “cold” to “hot” status. Generally, most of these approaches are
based on features extracted by using the TF-IDF weighting technique. However,
because TF-IDF has a limitation where the algorithm is less favorable to “hot”
terms with a high frequency of appearances that can signify trends, Zhu2019
proposed a refined version of TF-IDF to discover these terms based on time
distribution information and user attention. word segmentation algorithm.
Thus, to detect “hot” topics according to “hot” terms, the authors used
K-means clustering, with final experimental results that indicated that the
refined TF-IDF algorithm is able to discover “hot” topics efficiently. Unlike
previous work that tackled trend detection in official documents and
newspapers, others focused on proposing new approaches using research and
scientific papers, documents that generally contain citations and
bibliographies that could be considered as additional features Nie2017;
Xu2019; He2009DetectingTE; Israel2020; Tseng2009ACO. However, Dridi2019
pointed out that exploiting bibliographies have disadvantages in timeliness
and content analysis. The authors further proposed an approach using temporal
word embedding, called Leap2Trend. These embeddings were generated by being
trained on the data in an initial period of time, and then fine-tuned in the
upcoming time frames, hence the temporality. This approach also tracked the
similarities between pairs of keywords over time, which yielded results
suggesting the robustness and timeliness characteristics of the Leap2Trend.
Moreover, this work introduced the usage of embedding-based text
representation for trend detection, which proved to be rather promising.

#### Detecting Trends in COVID-19 pandemic

Recent works have also been conducted within the period of the COVID-19
pandemic to study emerging trends within certain communities and gauge the
impact of the outbreak through social media text Kassab2020OnNM;
Chandrasekaran2020TopicsTA. Enrico2020 employed term-frequency analysis,
calculate nutrition and energy metrics, while also using social features in
order to extract hot terms and build a topic graph through co-occurrence
analysis using Twitter data in Italy. Sentiment analysis is another method
applied to Twitter data with the purpose of tracking themes/topics through
evaluating fluctuation of sentiment Li2020AnalyzingCO;
Chandrasekaran2020TopicsTA. Bissoyi2020DiscoveringTT did not only use social
media data, but also news articles to feed into their LDA system to track
emerging trends due to COVID-19 in India. Another research targeted peer-
review papers regarding the COVID-19 virus and apply word embeddings and
machine learning models to track novel insight surrounding the spreading of
the virus Pal2021PredictingET.

Figure 1: Monthly news volume distribution about Microsoft from Bloomberg EDF.

##  3 Trend Detection Dataset

Figure 2: Summary of the proposed methodology.

For the current study, we built a dataset by extracting a portion of the
Event-Driven Feeds (EDF) data, a proprietary dataset provided by
Bloomberg222https://www.bloomberg.com. Bloomberg L.P. provides financial
software tools and enterprise applications such as analytics and equity
trading platform, data services, and news to financial companies and
organizations. The EDF data is massive and contains more than ten years of
news collection that Bloomberg published in multiple languages from 2010 up
until the present.

Considering the scope of the research, specific time frames and a company were
chosen as follows: we studied the period from July 2019 to July 2020 - six
months before the COVID-19 outbreak up until its peak, split into monthly
collections of news. This is a very particular time span since the COVID-19
triggered a massive global health crisis and drastically changed the way
people live, which created new trends.

For the specific company, we chose Microsoft which, at the time of writing
this article, was known to play a major role in remote working trends during
the pandemic by providing a stable and convenient platform for online
workplace communication. Moreover, Microsoft is widely known as one of the
leading companies in the field of cloud computing. Hence, it is interesting to
investigate how trends in Microsoft shifted during the pandemic.

With these criteria, we extracted a total of 11,923 news articles about
Microsoft within the said time span. The monthly news volume distribution is
in Figure 1. The figure shows an enormous surge in the number of articles
during the month of October 2019, which consisted of around 7,500 documents
and accounted for more than 60% of the total volume. In contrast, the number
of documents during the beginning phase of the COVID-19 pandemic, in the span
of 7 months from January 2020 to July 2020, is much lower in comparison.
Averaging at about 300 articles per month, the period, the 7-month period only
consisted just under 17% of the total amount of articles in the dataset. With
these figures, it is expected that October 2019 could be the month that
Microsoft started a trend due to the huge spike in news articles volume, while
the COVID-19 period may seem uneventful to the company.

##  4 Experimental Methodology

Our approach adapts the Leap2Trend method proposed by Dridi2019 on detecting
emerging themes in scientific papers, to financial-based documents in our
case. The authors generated keywords representations were in different periods
of time using static embeddings (i.e., Word2Vec mikolov2013distributed).
Afterward, they assessed the similarity evolution of keyword pairs over time
to depict which keywords are trending, thus forming topics based on the
closeness between keywords. Leap2Trend also tackled the matter of lacking a
gold standard to evaluate the result of trend detection by using Google
Trends333https://trends.google.com/trends/?geo=US. Google Trends collects
search data of keywords and present them as interest rate over time, starting
from 2004 to the present, which can be used to project emerging trends
prediction results to gauge the performance of the system.

Nonetheless, Leap2Trend has some disadvantages that needed to be addressed.
First and foremost, Leap2Trend used a straightforward approach to extract the
main keywords by inspecting titles of scientific papers for the most frequent
bi-grams. The solution is justified by the fact that titles from scientific
publications are written with the purpose of being self-explanatory and
conveying the methods/problems clearly. The writing style leads to titles
often containing a substantial amount of keywords. News articles, on the other
hand, have condensed headlines that will only be expanded further in the main
content of the documents, where most keywords reside. Thus, using raw
frequency to extract keywords is inefficient due to noisy text overshadowing
important phrases. Secondly, unlike in the scientific corpus that Leap2Trend
used, where the context (which is mostly about the computer science field) is
rather consistent, news. However, it can contain numerous subjects ranging
from technology, finance, economy to media. Having global static word
embeddings, that have a diversified number of abstract topics, might not yield
the best results.

Figure 3: Selected potential keywords’ TF-IDF value.

Thus, we propose to address the aforementioned disadvantages and to adapt them
to our dataset an approach that is detailed in Figure 2. We first extract
keywords using TF-IDF scores and LDA topic modelling, while using contextual
embeddings to replace static word embeddings in order to model the temporality
of the different contexts in different periods of time. In order to transform
a set of incidents into intervals trend detection analysis, we propose a
method with six steps. First, we pre-process our dataset to remove unwanted
text in order to focus on the main content of the news article. Moreover, the
dataset is divided into monthly sub-corpora (Section 4.1). The next step is to
identify potential keywords from the corpus. We calculate the TF-IDF value as
well as apply LDA to generate a list of top-rated bi-grams (Section 4.2) we
generate contextualized representations for these trending keywords for each
month (Section 4.3). We then rank the pair of keywords based on their
representation similarity (Section 4.4). Lastly, we assess the contextual
trend evolution in Section 4.5 by analyzing the change in ranking in each pair
of keywords over each time span in regards to Microsoft.

###  4.1 Data Pre-processing

From our analysis of the data, there are text patterns that appear relatively
frequent in the corpus and mainly talk about Bloomberg’s own publisher’s
detail, more exactly, the Bloomberg’s standardized text as this provides no
valuable knowledge to our task. Contrary, this standardized text could
actually hinder the performance of a system as they could cause potential
keywords to be more subtle, thus harder to be recognized. Moreover, since
Bloomberg News is geared towards an audience that is interested in finance,
articles usually contain an abundance of words that belong to the financial
glossary (e.g. “dividend”, “bond”, “equity”). The existence of this vocabulary
could cause the same problem as with Bloomberg’s standardized text, thus is it
also removed from the text. Afterward, we perform a lemmatization to return to
the canonical form of the token in the text, for reducing the size of the
vocabulary to process in later steps. Lastly, any article that has less than
ten tokens is considered uninformative and is removed444For this step, we
manually checked different values of this threshold and the minimum quality of
the remaining per-processed article..

###  4.2 Keyword Identification

The research on keyword identification discovery originates from the topic
detection and tracking (TDT) technology that was first studied by scholars in
1996 and its goal was making new detection and tracking within streams of
broadcast news stories Zhu2019. Emerging trends are usually signified by terms
and phrases that are later considered as defining keywords for such themes.
Hence, correctly identifying potential keywords will lead to the right
direction in discovering promising dormant trends. While keywords can be
n-grams, in the scope of this research, only bi-grams were considered due to
the fact that compared to uni-grams, they are less ambiguous, while appearing
more frequently than other n-grams.

Next, we present our methods of extracting potent keywords, using TF-IDF
values to generate the list of bi-grams, then contextualizing these bi-grams
with the results from LDA. Afterward, based on the generated list of bi-grams,
only those that are services, products or technology terms were considered,
while names, locations were discarded.

#### TF-IDF Bi-gram Generation

TF-IDF captures how important a word is in the corpus by considering its
frequency and penalizing it for appearing in too many entries in the corpus.
Hence, words that are too common have considerably lower TF-IDF values than
those that are less frequent. We exploited this method to extract important
bi-grams from the corpus. Per month, TF-IDF values are generated for every bi-
gram in the news collection and, afterward, we evaluated and produced lists of
bi-grams that have either the highest average TF-IDF value in the collection
or the highest single TF-IDF value across all documents. A high average TF-IDF
value may indicate that a keyword associates closely with the company
throughout the month, while a single high TF-IDF value signifies a sudden
change in the context surrounding the company.

#### LDA Bi-gram Generation

LDA blei2003latent derives from textual data the probabilities of words
belonging to a predetermined number of topics. As such, the method excels at
providing easily interpretable insights into what consists of a text corpus.
Taking advantage of this fact, we used the results of applying LDA on the
collection of texts in each month of our dataset to contextualize bi-grams by
topics, which we obtained in the previous step. This is done by searching for
topics that contain bi-grams in their list of words with the highest
probability that represent topics. To find the optimal number of topics, we
built numerous LDA models with different values of the number of topics and
measured their topic coherence score Rder2015ExploringTS. We chose the topic
number that gives the highest coherence value. Coherence is a measure to
evaluate to which degree the induced topics of an LDA model are correlated to
one another, thus choosing the optimal number of topics that marks the end of
the rapid growth of topic coherence usually offers meaningful and
interpretable topics. We experimented with different number of topics ($k$)
that ranged in $[10,\ldots,30]$.

Table 1: Bi-grams extracted using TF-IDF and LDA (without being pre-processed for readability). Method |  Bi-grams | Total
---|---|---
LDA |  video game, mobile game, SQL server, Star Wars, intelligent cloud, Surface Book, InfiniBand Internet, mobile device, Windows Phone, hybrid cloud, etc. | 14
TF-IDF |  Windows Desktop, global pandemic, physical store, cloud solution, mobile security, security solution, public cloud, storage management, Big Data, communication service, etc. | 46
LDA & TF-IDF |  Microsoft Teams, Artificial Intelligence, digital transformation, Microsoft Azure, cloud computing, cloud solution, health care, storage server, data center, remote work, etc. | 22

###  4.3 Bi-gram Contextual Representation Generation

Unlike static word embeddings that capture the global representations of words
in a vocabulary, contextual embeddings aim at representing each word or sub-
word in the corpus depending on the words surrounding it. Therefore, each
appearance of the word will have a unique vector assigned to it, which
differentiates the same token but appears in context. For example, the token
“teams” in “Microsoft Teams” and “team management” should be represented by
distinct vectors, as their context and usage are entirely different. Thus, a
contextual embedding is generally better than static word embedding,
especially in a scenario where the corpus consists of news articles and not
documents containing specialized knowledge.

Because contextual embeddings usually derive vectors for one single token or a
character n-gram separately, we employ the following strategy to generate bi-
gram embedding to suit our need: for each occurrence of two tokens belonging
to a bi-gram appears next to each other and in the right order, the embeddings
of both tokens are taken into account, and the final vector of the bi-gram is
calculated by averaging all detected embeddings555For this step, we used spaCy
library https://spacy.io/ that provides freely available trained pipelines for
several languages. In our case, we chose the en_core_web_sm English pre-
trained model.

###  4.4 Keyword Pairs Contextual Ranking

With the contextual embeddings for the set of chosen keywords, we proceed to
compute the similarity between each pair of keywords and rank them based on
the value calculated. The idea is that when a number of terms appear
frequently together, they usually share the same set of surrounding
vocabulary, thus having similar context. For this, we computed the cosine
similarity. Regarding how to establish the ranking, we first utilized the
algorithm employed by Leap2Trend Dridi2019, sorting the similarity of pairs of
keywords in descending order, where the higher the similarity is, the lower
the rank the pair of keywords has.

###  4.5 Assessment of Contextual Trend Evolution

Following the ranking calculation for each month, we attempted to assess the
contextual evolution of each pair of keywords to identify potential emerging
trends relating to the selected keywords. To achieve this, we analyze how much
the rank increase/decrease between each month and set a threshold to decide
whether changes in ranking signify emerging keywords that can lead to emerging
trends. If the differences in ranking between the current month and the
previous month of a pair of keywords are greater than the chosen threshold, we
identify that this set of keywords will become the emerging terms. On the
other hand, if the shift in ranking does not meet the threshold, the pair of
keywords is regarded as having no potential in its emergence, either is
falling off or is at standstill in terms of growth for being the next trend.

##  5 Evaluation and Experimental Setup

In this section, we present the experiments on previously presented dataset
from Bloomberg News about Microsoft from July 2019 to July 2020. This includes
the description of the evaluation process, the results of our proposed system
for emerging trend detection by comparing the shift in similarity ranking of
pairs of keywords, and the elaboration of the story behind some keyword pairs
that were in trend, and we deemed as interesting.

###  5.1 Evaluation

To our knowledge at the current time of writing, there is no annotated dataset
that is publicly available to experiment our method on. Instead, we follow a
procedure from existing work Dridi2019, to produce a gold standard. The
process involves examining Google Trends data, a platform for tracking
terms/phrases popularity based on Google’s search history, of the chosen
keywords to identify where their emergence is. This was done by calculating
the regression of interest rate evolution from a selected timestamp to $N$
months forward, with a positive value indicating an increase in attention
toward the keywords, thus signifying the possibility of the keywords belonging
to emerging topics. Formula 1 describes the regression of interest rate
evolution in the next $N$ months, denoted as $m_{hits}$:

$m_{hits}=\frac{\sum_{i=1}^{N}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i=1}^{N}(x_{i}-\bar{x})^{2}}$
(1)

where $x_{i}$ and $y_{i}$ represent the month number and the interest rate of
that month, respectively. $\bar{x}$ corresponds to the mean of the month
number, while $\bar{y}$ is the mean of interest rate.

With the way Google Trends treats a string in a search query, requesting data
on just the literal phrase, we experienced difficulties in extracting interest
rate data for combining a pair of keywords. Thus, we devised a solution by
looking for data on each keyword separately and averaging the two results to
get a final interest rate of the pair of keywords. The hypothesis behind this
is the assumption that both keywords are part of the same theme/topic, thus,
their evolution should have a similar tendency to rise/fall, albeit having
different magnitude, making the combined signal stays relatively close to the
two originals in terms of signal progression. Vice versa, phrases that do not
fall into the same category cannot produce a good signal, which automatically
makes them irrelevant to each other.

After obtaining the gold standard, we proceed to treat the task as a
classification problem, where the system will classify whether changes in
ranking context can lead to the same type of movement in the gold standard in
the next $N$ months. Accordingly, the main evaluation metrics are precision,
recall, and F1-measure. Not only do we consider the macro metrics, but we also
take into account the aforementioned metrics on the true class detection
specifically, since our focus is leaning toward correctly identifying emerging
trends, which is signified by the true class. Additionally, we report the
receiver operating characteristic (ROC) curves according to the selected time
span onward with the purpose of assessing the detection capability on how many
months forward can our system detect trends emergent effectively.

###  5.2 Experiments

First, after the keyword generation step (bi-gram generation with TF-IDF and
LDA), we noticed that while extracting bi-grams using raw TF-IDF value yielded
many potential keywords, the amount of bi-gram that also existed in the LDA
results was significantly lower. This could mean that there is an abundant
number of terms selected based on TF-IDF value that could not be categorized
into topics by LDA methods, to which we discard them.

(a) Original Leap2Trend

(b) Contextual Leap2Trend

Figure 4: Compared performance (time span onward = 3).

Moreover, some bi-grams acquired from LDA did not appear in the top TF-IDF
list, yet they were quite interesting, even if they only existed in a single
time period. For example, Stranger Things surfaced only in July, yet they were
the topic of that month due to Microsoft co-operation with Netflix for a
promotion campaign for the third season of the
series666https://www.bloomberg.com/news/articles/2019-07-08/-stranger-things-
is-netflix-s-hunt-for-billion-dollar-franchise. Star Wars also appear in this
conversation as a subtopic due to comparison between the two franchises. Table
1 details further on the list of extracted bi-grams and a total number of bi-
grams yielded by each method.

Finally, we selected a set of interesting keywords that we deemed as what
Microsoft is mostly about to conduct our study: Microsoft Teams, artificial
intelligence, digital transformation, Microsoft Azure, cloud computing, cloud
solution, health care, storage server, data server, remote work. The semantics
of these selected keywords have been chosen with the aim of offering a wide
coverage of the main trending topics, not focusing only on the COVID-19
pandemic, but also on a more general direction and development of Microsoft.

The monthly frequency of the ten top-ranking bi-grams is shown in the Figure
LABEL:fig:frequency. We observe that Health care, data center, and cloud
computing had a surge in appearances in October 2019 while mostly having a low
presence throughout other months. On the other hand, bi-grams such as
artificial intelligence and Microsoft Azure consistently appeared every month,
though each to a different amount. Figure 3 illustrates the importance of each
keyword through the TF-IDF score. It can be observed that health care, data
center, and cloud computing had a poor TF-IDF value in their peak months, as
well as getting an average score at best. In contrast, Microsoft Azure were
usually among keywords with the highest TF-IDF value, if not the highest,
which suggested the strong significance and relevance of the bi-gram every
time it appeared.

###  5.3 Results

(a) Original Leap2Trend

(b) Contextual Leap2Trend

Figure 5: Compared ROC curves based on timespan adjustment (threshold = 0).

We compare the performance of two ranking systems: original Leap2Trend and our
Contextual Leap2Trend. Figure 5(a) and Figure 5(b) demonstrate the predictive
capability of what is the optimal number of months forward origin Leap2Trend
and Contextual Leap2Trend ranking respectively can perform the task
efficiently. For this experiment, the threshold was set at $-1$ as it can be
seen from both Figure 4(a) and 4(b) that threshold=$-1$ yielded the most
balance performance across all evaluation metrics. Both ranking methods showed
their best performance in predicting growth and decline of pairs of keywords
with $3$ months onward. However, with Area Under the Curve (AUC) of 3-month
forward time span prediction of both ranking systems are only at 0.57,
emerging trends detection proved to be a difficult task. On other selected
values, Contextual Leap2Trend ranking performed slightly better than original
Leap2Trend on time span onward = $\\{4,\ldots,6\\}$, while the opposite
happened on time span onward = $2,5$.

On our performance experiment, we chose a timespan onward value of $3$ due to
the fact that the gold standard generated using this value is fairly balanced,
in which the True class consists of 57% of across all pairs of keywords. Per
selected keyword pairs, 31 out of 45 pairs have balanced class representation,
with neither “True” nor “False” class occupying more than 60% of said pairs.

Based on Figure 5(b), our proposed system is more efficient at predicting
trends three months in advance. However, with the current dataset, the task is
considerably challenging. In order to gauge the performance of our system, we
compared the results of a system with threshold of $-1,0$ to a zero rule
baseline where every possible bi-gram was considered as trendy (True class),
thus True class metrics were not taken into account for this experiment. This
threshold value was specifically chosen as the performance was quite balanced.
Table 2 shows that our system out-performed the zero rule baseline in all
three metrics, but are only marginally better in terms of recall value. This
observation further supports the difficulty of the task present using the
current data.

Another observation is when comparing the change in ranking to Google Trends
data, we notice that delays sometimes exists, usually of one month due to data
split in the pre-processing phase, in how soon the ranking change with respect
to Google Trends, meaning the timeliness aspect of detecting emerging trends
might be affected. One possible explanation is that Bloomberg News could be
behind in terms of timing compare to public response as Bloomberg mostly
covers big events that were already happened, and does not cover innovations
process that can lead to such event.

In following sections, we discuss several example pairs of keywords that
signified ongoing trends and emerging ones, mainly about what was happening
surround the keywords while comparing the graph between Google Trends and
contextual ranking evolution of Contextual Leap2Trend system.

Table 2: Proposed approach vs. zero rule baseline, timespan onward =3. Method | Thresh | Precision | Recall | Macro F1
---|---|---|---|---
Zero-rule baseline |  | 0.3000 | 0.5000 | 0.3700
Contextual Leap2Trend | 0 | 0.5600 | 0.5476 | 0.5388
Original Leap2Trend | 0 | 0.5384 | 0.5330 | 0.5276

(a) A.I./Digital Transformation

(b) Digital Transformation/Cloud Computing

(c) Microsoft Teams/Remote Work

Figure 6: Trends for different pairs of keywords.

###  5.4 A. I. - Digital Transformation

Compared to Google Trends data, for the artificial intelligence (A.I.) -
digital transformation pair of keywords, our proposed contextual ranking
reflects their trends accordingly, as shown in Figure 6(a). Digital
transformation and artificial intelligence have been an ongoing conversation
in technology and business in recent years as the movement seeks an effort to
incorporate A.I. into digitizing business processes, customer experiences,
etc. This is illustrated through the European Union’s strategy to apply A.I.
to digital
transformation777https://ec.europa.eu/commission/presscorner/detail/en/qanda_20_264.
As for Microsoft, the company supports this trend with multiple projects, one
of them being a major collaboration was mentioned in the EDF
data888https://www.bloomberg.com/press-releases/2020-03-26/c3-ai-microsoft-
and-leading-universities-launch-c3-ai-digital-transformation-institute.

###  5.5 Digital Transformation - Cloud Computing

Within the context of the corpus, the digital transformation - cloud computing
pair of keywords describes how Microsoft’s cloud computing service contribute
to their involvement in advancing Digital Transformation with their business
partners by providing Azure, Microsoft’s leading cloud platform, to enhance
the digital capability of their business
partner999https://www.bloomberg.com/press-releases/2020-04-07/blackrock-and-
microsoft-form-strategic-partnership-to-host-aladdin-on-azure-as-blackrock-
readies-aladdin-for-next-chapter-of. According to figure 6(b), the contextual
ranking changes of the pair, in general, are in-line with Google Trends data,
albeit displayed some level of differences.

One thing to note is that it is the consensus, mentioned throughout the
COVID-19 period in our corpus, regarding Digital Transformation that because
of the pandemic pushing society to stay distant and work remotely, the Digital
Transformation process will need to be developed faster to adapt to the
current situation. In Figures 6(b) and 6(a), it can be seen that this opinion
were reflected through an increase in rankings starting March 2020. This
period is also where the interest rate on Google Trends on this matter started
to increase again after a dip.

###  5.6 Microsoft Teams - Remote Work

While the magnitude displayed in Figure 6(c) was definitely lower than Google
Trends’s signal, Leap2Trend’s signal visually still showed signs that the
trends of the Microsoft Teams - remote work pair of keywords have potential.
Remote work, while being lesser known in 2019, has been a staple since the
COVID-19 pandemic began. Zoom, a platform for online meetings, was booming at
the start of the pandemic, yet fell in popularity due to security reasons. The
slump of Zoom paved the way for the rise of Microsoft Teams as the reliable
platform for workplace
communication101010https://www.computerweekly.com/news/252485100/Microsoft-
Teams-usage-growth-surpasses-Zoom. In our EDF dataset, the development of
Microsoft Teams with new and better features also aid in its popularity, which
was
mentioned111111https://www.bloomberg.com/news/articles/2020-03-19/microsoft-
teams-boosts-work-at-home-effort.

##  6 Conclusions and Future Work

In this work, we proposed modifications of Leap2Trend method Dridi2019 to
adapt to our own dataset of news articles from Bloomberg with the intent of
detecting trends associated with the Microsoft company since the beginning of
the COVID-19 crisis.

This research addressed the drawback of existing methods in keyword
extraction, bi-gram representation due to the differences in writing style
between scientific papers and news articles. We, instead, introduced a
combination of TF-IDF and LDA for generating potential keywords, utilized
contextual embeddings to replace static word embeddings. Our modified version
of the original system shows considerable improvements in results as compared
to the original method in some scenarios. Yet, the task remains difficult
since the best AUC we can get is 0.57, and we need to imagine how to combine
the original method and our alternative to get more reliable results.

Our research is not without limitations. Our current data might be affecting
timeliness in our results due to the nature of reporting events results.
Moreover, the system still relied quite heavily on human judgment to perform
efficiently. Another drawback to adding was that our gold standard was based
on Google Trends, which returns trend data via rate and not a definite search
volume. We intend to find a better solution, as a dataset with ground truth
for emerging trends detection is rare to find.

In future work, we plan to reduce the influence and bias of human knowledge in
the keyword selection phase. Moreover, having a better variety of data is also
in our upcoming work, with news articles covering more companies and sectors,
plus having wider time intervals to assess trends longevity.


In $x$ direction, the entropy-scaled right eigenvectors corresponding to the
above eigenvalues in the term of primitive variables are the following,

$\displaystyle\tilde{R}_{\lambda_{1}}=\begin{pmatrix}\frac{\sqrt{\rho}(2p_{\parallel}+\rho)}{2\sqrt{5p_{\parallel}^{2}+4p_{\parallel}\rho+\rho^{2}}}\\\
0\\\ 0\\\ 0\\\
\frac{p_{\parallel}\sqrt{\rho}}{2\sqrt{5p_{\parallel}^{2}+4p_{\parallel}\rho+\rho^{2}}}\\\
0\\\ 0\\\ 0\\\ 0\end{pmatrix},\tilde{R}_{\lambda_{2}}=\begin{pmatrix}0\\\ 0\\\
0\\\ 0\\\ 0\\\ 0\\\ \sqrt{\frac{p_{\perp}}{2\rho}}\\\ 0\\\
0\end{pmatrix},\tilde{R}_{\lambda_{3}}=\begin{pmatrix}\frac{p_{\parallel}\sqrt{\rho}}{2\sqrt{5p_{\parallel}^{2}+4p_{\parallel}\rho+\rho^{2}}}\\\
0\\\ 0\\\ 0\\\
\frac{p_{\parallel}(5p_{\parallel}+2\rho)}{2\sqrt{\rho(5p_{\parallel}^{2}+4p_{\parallel}\rho+\rho^{2})}}\\\
0\\\ 0\\\ 0\\\ 0\end{pmatrix},$
$\displaystyle\tilde{R}_{\lambda_{4,5}}=\frac{1}{2}\begin{pmatrix}0\\\ 0\\\
\pm\frac{\sqrt{p_{\perp}}}{\rho}{\beta_{z}}\\\
\mp\frac{\sqrt{p_{\perp}}}{\rho}{\beta_{y}}\\\ 0\\\ 0\\\ 0\\\
-\sqrt{\frac{p_{\perp}}{\rho}}{\beta_{z}}\\\
\sqrt{\frac{p_{\perp}}{\rho}}{\beta_{y}}\\\
\end{pmatrix},\tilde{R}_{\lambda_{6,7}}=\frac{1}{2\sqrt{2}}\begin{pmatrix}\alpha_{f}\sqrt{\rho}\\\
\pm\frac{\alpha_{f}c_{f}}{\sqrt{\rho}}\\\
\mp\frac{\alpha_{s}c_{s}\beta_{y}}{\sqrt{\rho}}\\\
\mp\frac{\alpha_{s}c_{s}\beta_{z}}{\sqrt{\rho}}\\\
\alpha_{f}\frac{p_{\parallel}}{\sqrt{\rho}}\\\ \alpha_{f}\sqrt{\rho}a^{2}\\\
0\\\ \alpha_{s}a\beta_{y}\\\
\alpha_{s}a\beta_{z}\end{pmatrix},\tilde{R}_{\lambda_{8,9}}=\frac{1}{2\sqrt{2}}\begin{pmatrix}\alpha_{s}\sqrt{\rho}\\\
\pm\frac{\alpha_{s}c_{s}}{\sqrt{\rho}}\\\
\pm\frac{\alpha_{f}c_{f}\beta_{y}}{\sqrt{\rho}}\\\
\pm\frac{\alpha_{f}c_{f}\beta_{z}}{\sqrt{\rho}}\\\
\alpha_{s}\frac{p_{\parallel}}{\sqrt{\rho}}\\\ \alpha_{s}\sqrt{\rho}a^{2}\\\
0\\\ -\alpha_{f}a\beta_{y}\\\ -\alpha_{f}a\beta_{z}\end{pmatrix},$

In $y$-direction, the entropy-scaled right eigenvectors corresponding to the
above eigenvalues in the term of primitive variables are the following,

$\displaystyle\tilde{R}_{\lambda_{1}}=\begin{pmatrix}\frac{\sqrt{\rho}(2p_{\parallel}+\rho)}{2\sqrt{5p_{\parallel}^{2}+4p_{\parallel}\rho+\rho^{2}}}\\\
0\\\ 0\\\ 0\\\
\frac{p_{\parallel}\sqrt{\rho}}{2\sqrt{5p_{\parallel}^{2}+4p_{\parallel}\rho+\rho^{2}}}\\\
0\\\ 0\\\ 0\\\ 0\end{pmatrix},\tilde{R}_{\lambda_{2}}=\begin{pmatrix}0\\\ 0\\\
0\\\ 0\\\ 0\\\ 0\\\ 0\\\ \sqrt{\frac{p_{\perp}}{2\rho}}\\\
0\end{pmatrix},\tilde{R}_{\lambda_{3}}=\begin{pmatrix}\frac{p_{\parallel}\sqrt{\rho}}{2\sqrt{5p_{\parallel}^{2}+4p_{\parallel}\rho+\rho^{2}}}\\\
0\\\ 0\\\ 0\\\
\frac{p_{\parallel}(5p_{\parallel}+2\rho)}{2\sqrt{\rho(5p_{\parallel}^{2}+4p_{\parallel}\rho+\rho^{2})}}\\\
0\\\ 0\\\ 0\\\ 0\end{pmatrix},$
$\displaystyle\tilde{R}_{\lambda_{4,5}}=\frac{1}{2}\begin{pmatrix}0\\\
\pm\frac{\sqrt{p_{\perp}}}{\rho}{\beta_{z}}\\\ 0\\\
\mp\frac{\sqrt{p_{\perp}}}{\rho}{\beta_{x}}\\\ 0\\\ 0\\\
-\sqrt{\frac{p_{\perp}}{\rho}}{\beta_{z}}\\\ 0\\\
\sqrt{\frac{p_{\perp}}{\rho}}{\beta_{x}}\\\
\end{pmatrix},\tilde{R}_{\lambda_{6,7}}=\frac{1}{2\sqrt{2}}\begin{pmatrix}\alpha_{f}\sqrt{\rho}\\\
\mp\frac{\alpha_{s}c_{s}\beta_{x}}{\sqrt{\rho}}\\\
\pm\frac{\alpha_{f}c_{f}}{\sqrt{\rho}}\\\
\mp\frac{\alpha_{s}c_{s}\beta_{z}}{\sqrt{\rho}}\\\
\alpha_{f}\frac{p_{\parallel}}{\sqrt{\rho}}\\\ \alpha_{f}\sqrt{\rho}a^{2}\\\
\alpha_{s}a\beta_{x}\\\ 0\\\
\alpha_{s}a\beta_{z}\end{pmatrix},\tilde{R}_{\lambda_{8,9}}=\frac{1}{2\sqrt{2}}\begin{pmatrix}\alpha_{s}\sqrt{\rho}\\\
\pm\frac{\alpha_{f}c_{f}\beta_{x}}{\sqrt{\rho}}\\\
\pm\frac{\alpha_{s}c_{s}}{\sqrt{\rho}}\\\
\pm\frac{\alpha_{f}c_{f}\beta_{z}}{\sqrt{\rho}}\\\
\alpha_{s}\frac{p_{\parallel}}{\sqrt{\rho}}\\\ \alpha_{s}\sqrt{\rho}a^{2}\\\
-\alpha_{f}a\beta_{x}\\\ 0\\\ -\alpha_{f}a\beta_{z}\end{pmatrix},$

##  Appendix B Proof of
$\textbf{V}^{\top}\cdot\textbf{C}_{x}(\textbf{U})=\textbf{V}^{\top}\cdot\textbf{C}_{y}(\textbf{U})=0$

The entropy variable
$\textbf{V}:=\frac{\partial\mathcal{E}}{\partial\textbf{U}}$ is given by

V
$\displaystyle=\Bigg{\\{}5-s-\beta_{\perp}\textbf{u}^{2},~{}2\beta_{\perp}\textbf{u},~{}-\beta_{\parallel}+\beta_{\perp},~{}-2\beta_{\perp},~{}2\beta_{\perp}\boldsymbol{B}\Bigg{\\}}^{\top}$

where $\beta_{\perp}=\frac{\rho}{p_{\perp}},$ and
$\beta_{\parallel}=\frac{\rho}{p_{\parallel}}$. Now the dot product of
$\textbf{V}^{\top}$ and first coloum of $\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot$
$\displaystyle\left\\{0,-\frac{b_{x}^{2}\textbf{u}^{2}}{2},-\frac{b_{x}b_{y}\textbf{u}^{2}}{2},-\frac{b_{x}b_{z}\textbf{u}^{2}}{2},-\frac{2p_{\parallel}b_{x}}{\rho}(\boldsymbol{b}\cdot\textbf{u}),\Upsilon_{1}^{x},0,0,0\right\\}=$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)\left(-\frac{b_{x}^{2}\textbf{u}^{2}}{2}\right)+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)\left(-\frac{b_{x}b_{y}\textbf{u}^{2}}{2}\right)+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)\left(-\frac{b_{x}b_{z}\textbf{u}^{2}}{2}\right)$
$\displaystyle\hskip
99.58464pt+\left(\frac{\rho(p_{\parallel}-p_{\perp})}{p_{\parallel}p_{\perp}}\right)\left(-\frac{2p_{\parallel}b_{x}}{\rho}(\boldsymbol{b}\cdot\textbf{u})\right)+\left(-2\frac{\rho}{p_{\perp}}\right)(\Upsilon_{1}^{x})$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)\left(-\frac{b_{x}^{2}\textbf{u}^{2}}{2}\right)+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)\left(-\frac{b_{x}b_{y}\textbf{u}^{2}}{2}\right)+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)\left(-\frac{b_{x}b_{z}\textbf{u}^{2}}{2}\right)$
$\displaystyle\hskip
99.58464pt+\left(\frac{\rho(p_{\parallel}-p_{\perp})}{p_{\parallel}p_{\perp}}\right)\left(-\frac{2p_{\parallel}b_{x}}{\rho}(\boldsymbol{b}\cdot\textbf{u})\right)$
$\displaystyle\hskip
99.58464pt+\left(-2\frac{\rho}{p_{\perp}}\right)\left(-b_{x}(b\cdot\textbf{u})\frac{\textbf{u}^{2}}{2}-\frac{(p_{\parallel}-p_{\perp})b_{x}(\boldsymbol{b}\cdot\textbf{u})}{\rho}\right)$
$\displaystyle=0$

Now the dot product of $\textbf{V}^{\top}$ and second coloum of
$\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot$
$\displaystyle\left\\{0,b_{x}^{2}u_{x},b_{x}b_{y}u_{x},b_{x}b_{z}u_{x},\frac{2p_{\parallel}b_{x}}{\rho}b_{x},\Upsilon_{2}^{x},0,0,0\right\\}=$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)(b_{x}^{2}u_{x})+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)(b_{x}b_{y}u_{x})+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)(b_{x}b_{z}u_{x})$
$\displaystyle+\left(\frac{\rho(p_{\parallel}-p_{\perp})}{p_{\parallel}p_{\perp}}\right)\left(\frac{2p_{\parallel}b_{x}}{\rho}b_{x}\right)+\left(-2\frac{\rho}{p_{\perp}}\right)\left(b_{x}(\boldsymbol{b}\cdot\textbf{u})u_{x}+\frac{(p_{\parallel}-p_{\perp})b^{2}_{x}}{\rho}\right)$
$\displaystyle=0$

Now the dot product of $\textbf{V}^{\top}$ and third coloum of
$\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot$
$\displaystyle\left\\{0,b_{x}^{2}u_{y},b_{x}b_{y}u_{y},b_{x}b_{z}u_{y},\frac{2p_{\parallel}b_{x}}{\rho}b_{y},\Upsilon_{3}^{x},0,0,0\right\\}=$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)(b_{x}^{2}u_{y})+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)(b_{x}b_{y}u_{y})+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)(b_{x}b_{z}u_{y})$
$\displaystyle+\left(\frac{\rho(p_{\parallel}-p_{\perp})}{p_{\parallel}p_{\perp}}\right)\left(\frac{2p_{\parallel}b_{x}}{\rho}b_{y}\right)+\left(-2\frac{\rho}{p_{\perp}}\right)\left(b_{x}(\boldsymbol{b}\cdot\textbf{u})u_{y}+\frac{(p_{\parallel}-p_{\perp})b_{x}b_{y}}{\rho}\right)$
$\displaystyle=0$

Now the dot product of $\textbf{V}^{\top}$ and fourth coloum of
$\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot$
$\displaystyle\left\\{0,b_{x}^{2}u_{z},b_{x}b_{y}u_{z},b_{x}b_{z}u_{z},\frac{2p_{\parallel}b_{x}}{\rho}b_{z},\Upsilon_{4}^{x},0,0,0\right\\}=$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)(b_{x}^{2}u_{z})+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)(b_{x}b_{y}u_{z})+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)(b_{x}b_{z}u_{z})$
$\displaystyle+\left(\frac{\rho(p_{\parallel}-p_{\perp})}{p_{\parallel}p_{\perp}}\right)\left(\frac{2p_{\parallel}b_{x}}{\rho}b_{z}\right)+\left(-2\frac{\rho}{p_{\perp}}\right)\left(b_{x}(\boldsymbol{b}\cdot\textbf{u})u_{z}+\frac{(p_{\parallel}-p_{\perp})b_{x}b_{z}}{\rho}\right)$
$\displaystyle=0$

Now the dot product of $\textbf{V}^{\top}$ and fifth coloum of
$\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot$
$\displaystyle\left\\{0,\frac{3}{2}b_{x}^{2},\frac{3}{2}b_{x}b_{y},\frac{3}{2}b_{x}b_{z},0,\frac{3}{2}b_{x}(b\cdot\textbf{u}),0,0,0\right\\}=$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)(\frac{3}{2}b_{x}^{2})+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)(\frac{3}{2}b_{x}b_{y})+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)(\frac{3}{2}b_{x}b_{z})$
$\displaystyle+\left(-2\frac{\rho}{p_{\perp}}\right)\left(\frac{3}{2}b_{x}(\boldsymbol{b}\cdot\textbf{u})\right)$
$\displaystyle=0$

Now the dot product of $\textbf{V}^{\top}$ and sixth coloum of
$\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot$
$\displaystyle\left\\{0,-b_{x}^{2},-b_{x}b_{y},-b_{x}b_{z},0,-b_{x}(b\cdot\textbf{u}),0,0,0\right\\}=$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)(-b_{x}^{2})+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)(-b_{x}b_{y})+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)(-b_{x}b_{z})$
$\displaystyle+\left(-2\frac{\rho}{p_{\perp}}\right)\left(-b_{x}(\boldsymbol{b}\cdot\textbf{u})\right)$
$\displaystyle=0$

The seventh coloum, 8th coloum and ninth coloum of
$\textbf{C}_{x}(\textbf{U})$ denoted by $C_{7},~{}C_{8}$ and $C_{9}$ is given
below,

$\displaystyle C_{7}=\begin{pmatrix}0\\\
b_{x}^{2}B_{x}+\frac{2\Gamma_{x}b_{x}}{|\boldsymbol{B}|}\\\
b_{x}b_{y}B_{x}+\frac{\Gamma_{x}b_{y}-\Delta Pb_{yxx}}{|\boldsymbol{B}|}\\\
b_{x}b_{z}B_{x}+\frac{\Gamma_{x}b_{z}-\Delta Pb{zxx}}{|\boldsymbol{B}|}\\\
0\\\ b_{x}(\boldsymbol{b}\cdot\textbf{u})B_{x}+\Theta^{x}_{1}\\\ 0\\\ 0\\\
0\end{pmatrix},C_{8}=\begin{pmatrix}0\\\ b_{x}^{2}B_{y}-\frac{2\Delta
Pb_{yxx}}{|\boldsymbol{B}|}\\\ b_{x}b_{y}B_{y}+\frac{\Gamma_{y}b_{x}-\Delta
Pb_{xyy}}{|\boldsymbol{B}|}\\\ b_{x}b_{z}B_{y}-\frac{2\Delta
Pb_{xyz}}{|\boldsymbol{B}|}\\\ 0\\\
b_{x}(\boldsymbol{b}\cdot\textbf{u})B_{y}+\Theta^{x}_{2}\\\ 0\\\ 0\\\
0\end{pmatrix},$ $\displaystyle C_{9}=\begin{pmatrix}0\\\
b_{x}^{2}B_{z}-\frac{2\Delta Pb_{zxx}}{|\boldsymbol{B}|}\\\
b_{x}b_{y}B_{z}-\frac{2\Delta Pb_{xyz}}{|\boldsymbol{B}|}\\\
b_{x}b_{z}B_{z}+\frac{\Gamma_{z}b_{x}-\Delta Pb_{xzz}}{|\boldsymbol{B}|}\\\
0\\\ b_{x}(\boldsymbol{b}\cdot\textbf{u})B_{z}+\Theta^{x}_{3}\\\ 0\\\ 0\\\
0\end{pmatrix}.$

Now the dot product of $\textbf{V}^{\top}$ and seventh coloum of
$\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot C_{7}$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)\left(b_{x}^{2}B_{x}+\frac{2(p_{\parallel}-p_{\perp})b_{x}(1-b_{x}^{2})}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)\left(b_{x}b_{y}B_{x}+\frac{(p_{\parallel}-p_{\perp})b_{y}(1-b_{x}^{2})}{|\boldsymbol{B}|}-\frac{(p_{\parallel}-p_{\perp})b_{y}b_{x}^{2}}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)\left(b_{x}b_{z}B_{x}+\frac{(p_{\parallel}-p_{\perp})b_{z}(1-b_{x}^{2})}{|\boldsymbol{B}|}-\frac{(p_{\parallel}-p_{\perp})b_{z}b_{x}^{2}}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(-2\frac{\rho}{p_{\perp}}\right)\Bigg{(}b_{x}(\boldsymbol{b}\cdot\textbf{u})B_{x}+\left\\{(p_{\parallel}-p_{\perp})(\boldsymbol{b}\cdot\textbf{u})+(p_{\parallel}-p_{\perp})b_{x}u_{x}\right\\}\Bigg{(}\frac{1-b^{2}_{x}}{|\boldsymbol{B}|}\Bigg{)}$
$\displaystyle\hskip
142.26378pt-(p_{\parallel}-p_{\perp})\frac{b^{2}_{x}b_{y}u_{y}}{|\boldsymbol{B}|}-(p_{\parallel}-p_{\perp})\frac{b^{2}_{x}b_{z}u_{z}}{|\boldsymbol{B}|}\Bigg{)}$
$\displaystyle=0$

Now the dot product of $\textbf{V}^{\top}$ and 8th coloum of
$\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot C_{8}$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)\left(b_{x}^{2}B_{y}-\frac{2(p_{\parallel}-p_{\perp})b_{y}b_{x}^{2}}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)\left(b_{x}b_{y}B_{y}+\frac{(p_{\parallel}-p_{\perp})b_{x}(1-b_{y}^{2})}{|\boldsymbol{B}|}-\frac{(p_{\parallel}-p_{\perp})b_{x}b_{y}^{2}}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)\left(b_{x}b_{z}B_{y}-\frac{2(p_{\parallel}-p_{\perp})b_{x}b_{y}b_{z}}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(-2\frac{\rho}{p_{\perp}}\right)\Bigg{(}b_{x}(\boldsymbol{b}\cdot\textbf{u})B_{y}+(p_{\parallel}-p_{\perp})b_{x}u_{y}\Bigg{(}\frac{1-b^{2}_{y}}{|\boldsymbol{B}|}\Bigg{)}-(p_{\parallel}-p_{\perp})\frac{b_{x}b_{y}b_{z}u_{z}}{|\boldsymbol{B}|}$
$\displaystyle\hskip
142.26378pt-\\{(p_{\parallel}-p_{\perp})(\boldsymbol{b}\cdot\textbf{u})+(p_{\parallel}-p_{\perp})b_{x}u_{x}\\}\frac{b_{x}b_{y}}{|\boldsymbol{B}|}\Bigg{)}$
$\displaystyle=0$

Now the dot product of $\textbf{V}^{\top}$ and ninth coloum of
$\textbf{C}_{x}(\textbf{U})$ is:

$\displaystyle\textbf{V}^{\top}\cdot C_{9}$
$\displaystyle=\left(2\frac{\rho}{p_{\perp}}u_{x}\right)\left(b_{x}^{2}B_{z}-\frac{2(p_{\parallel}-p_{\perp})b_{z}b_{x}^{2}}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(2\frac{\rho}{p_{\perp}}u_{y}\right)\left(b_{x}b_{y}B_{z}-\frac{2(p_{\parallel}-p_{\perp})b_{x}b_{y}b_{z}}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(2\frac{\rho}{p_{\perp}}u_{z}\right)\left(b_{x}b_{z}B_{z}+\frac{(p_{\parallel}-p_{\perp})b_{x}(1-b_{z}^{2})}{|\boldsymbol{B}|}-\frac{(p_{\parallel}-p_{\perp})b_{x}b_{z}^{2}}{|\boldsymbol{B}|}\right)$
$\displaystyle+\left(-2\frac{\rho}{p_{\perp}}\right)\Bigg{(}b_{x}(\boldsymbol{b}\cdot\textbf{u})B_{z}+(p_{\parallel}-p_{\perp})b_{x}u_{z}\Bigg{(}\frac{1-b^{2}_{z}}{|\boldsymbol{B}|}\Bigg{)}-(p_{\parallel}-p_{\perp})\frac{b_{x}b_{y}b_{z}u_{y}}{|\boldsymbol{B}|}$
$\displaystyle\hskip
142.26378pt-\\{(p_{\parallel}-p_{\perp})(\boldsymbol{b}\cdot\textbf{u})+(p_{\parallel}-p_{\perp})b_{x}u_{x}\\}\frac{b_{x}b_{z}}{|\boldsymbol{B}|}\Bigg{)}$
$\displaystyle=0$

Similarly we can proof that
$\textbf{V}^{\top}\cdot\textbf{C}_{y}(\textbf{U})=0$ for $y$-direction.

##  Appendix C Non-symmetrizability of CGL System

In this section, we have discussed the symmetrizability of the following CGL
equations. Consider CGL equations in one dimension as follows:

$\frac{\partial\textbf{U}}{\partial t}+\frac{\partial\textbf{f}_{x}}{\partial
x}+\textbf{C}_{x}(\textbf{U})\frac{\partial\textbf{U}}{\partial x}=0.$ (60)

where $\textbf{U},~{}\textbf{f}_{x}$ and $\textbf{C}_{x}(\textbf{U})$ are
defined in (3). Follow yadav2023entropy , the CGL equations are said to be
symmetrizable if the change of variable $\textbf{U}\rightarrow\textbf{V}$
applied to (60) and can be written as:

$\frac{\partial\textbf{U}}{\partial\textbf{V}}\frac{\partial\textbf{V}}{\partial{t}}+\left(\frac{\partial\textbf{f}_{x}}{\partial\textbf{U}}+\textbf{C}_{x}(\textbf{U})\right)\frac{\partial\textbf{U}}{\partial\textbf{V}}\frac{\partial\textbf{V}}{\partial{x}}=0.$

the matrix $\frac{\partial\textbf{U}}{\partial\textbf{V}}$ is a symmetric,
positive definite matrix and
$\aleph(\textbf{U})=\left(\frac{\partial\textbf{f}_{x}}{\partial\textbf{U}}+\textbf{C}_{x}(\textbf{U})\right)\frac{\partial\textbf{U}}{\partial\textbf{V}}$
is symmetric matrix. For the CGL equation (60), we calculate matrix
$\aleph(\textbf{U})$ to examine its symmetry and conclude that CGL equations
are non-symmetrizable. The resultant matrix
$\aleph(\textbf{U})-\aleph(\textbf{U})^{\top}$ is given below:

$\displaystyle\begin{pmatrix}0&-\frac{\Delta Pb_{x}^{2}}{2}&-\frac{\Delta
Pb_{x}b_{y}}{2}&-\frac{\Delta Pb_{x}b_{z}}{2}&0&-\frac{\Delta
Pb_{x}(\boldsymbol{b}\cdot\textbf{u})}{2}&0&0&0\\\ \frac{\Delta
Pb_{x}^{2}}{2}&0&\frac{\Delta Pb_{x}(b_{x}u_{y}-b_{y}u_{x})}{2}&\frac{\Delta
Pb_{x}(b_{x}u_{z}-b_{z}u_{x})}{2}&\frac{3\Delta
Pp_{\parallel}b_{x}^{2}}{2\rho}&-\frac{\alpha_{xyz}}{4\rho}&-\frac{p_{\perp}\omega_{1}}{2\rho}&-\frac{\Delta
Pp_{\perp}b_{y}b_{x}^{2}}{\rho|\boldsymbol{B}|}&-\frac{\Delta
Pp_{\perp}b_{z}b_{x}^{2}}{\rho|\boldsymbol{B}|}\\\ \frac{\Delta
Pb_{x}b_{y}}{2}&\frac{\Delta Pb_{x}(b_{y}u_{x}-b_{x}u_{y})}{2}&0&\frac{\Delta
Pb_{x}(b_{y}u_{z}-b_{z}u_{y})}{2}&\frac{3\Delta
Pp_{\parallel}b_{x}b_{y}}{2\rho}&-\frac{\alpha_{yxz}}{4\rho}&-\frac{p_{\perp}\omega_{2}}{2\rho}&\frac{\Delta
Pp_{\perp}b_{x}(1-2b_{y}^{2})}{2\rho|\boldsymbol{B}|}&-\frac{\Delta
Pp_{\perp}b_{x}b_{y}b_{z}}{\rho|\boldsymbol{B}|}\\\ \frac{\Delta
Pb_{x}b_{z}}{2}&\frac{\Delta Pb_{x}(b_{z}u_{x}-b_{x}u_{z})}{2}&\frac{\Delta
Pb_{x}(b_{z}u_{y}-b_{y}u_{z})}{2}&0&\frac{3\Delta
Pp_{\parallel}b_{x}b_{z}}{2\rho}&-\frac{\alpha_{zxy}}{4\rho}&-\frac{p_{\perp}\omega_{3}}{2\rho}&-\frac{\Delta
Pp_{\perp}b_{x}b_{y}b_{z}}{\rho|\boldsymbol{B}|}&\frac{\Delta
Pp_{\perp}b_{x}(1-2b_{z}^{2})}{2\rho|\boldsymbol{B}|}\\\ 0&-\frac{3\Delta
Pp_{\parallel}b_{x}^{2}}{2\rho}&-\frac{3\Delta
Pp_{\parallel}b_{x}b_{y}}{2\rho}&-\frac{3\Delta
Pp_{\parallel}b_{x}b_{z}}{2\rho}&0&-\frac{3\Delta
Pp_{\parallel}b_{x}(\boldsymbol{b}\cdot\textbf{u})}{2\rho}&0&0&0\\\
\frac{\Delta
Pb_{x}(\boldsymbol{b}\cdot\textbf{u})}{2}&\frac{\alpha_{xyz}}{4\rho}&\frac{\alpha_{yxz}}{4\rho}&\frac{\alpha_{zxy}}{4\rho}&\frac{3\Delta
Pp_{\parallel}b_{x}(\boldsymbol{b}\cdot\textbf{u})}{2\rho}&0&-\frac{p_{\perp}\beta_{1}}{2\rho}&\frac{p_{\perp}\beta_{2}}{2\rho}&\frac{p_{\perp}\beta_{3}}{2\rho}\\\
0&\frac{p_{\perp}\omega_{1}}{2\rho}&\frac{p_{\perp}\omega_{2}}{2\rho}&\frac{p_{\perp}\omega_{3}}{2\rho}&0&\frac{p_{\perp}\beta_{1}}{2\rho}&0&\frac{p_{\perp}u_{y}}{2\rho}&\frac{p_{\perp}u_{z}}{2\rho}\\\
0&\frac{\Delta Pp_{\perp}b_{y}b_{x}^{2}}{\rho|\boldsymbol{B}|}&-\frac{\Delta
Pp_{\perp}b_{x}(1-2b_{y}^{2})}{2\rho|\boldsymbol{B}|}&\frac{\Delta
Pp_{\perp}b_{x}b_{y}b_{z}}{\rho|\boldsymbol{B}|}&0&-\frac{p_{\perp}\beta_{2}}{2\rho}&-\frac{p_{\perp}u_{y}}{2\rho}&0&0\\\
0&\frac{\Delta Pp_{\perp}b_{z}b_{x}^{2}}{\rho|\boldsymbol{B}|}&\frac{\Delta
Pp_{\perp}b_{x}b_{y}b_{z}}{\rho|\boldsymbol{B}|}&-\frac{\Delta
Pp_{\perp}b_{x}(1-2b_{z}^{2})}{2\rho|\boldsymbol{B}|}&0&-\frac{p_{\perp}\beta_{3}}{2\rho}&-\frac{p_{\perp}u_{z}}{2\rho}&0&0\end{pmatrix}.$

where, $\Delta P=(p_{\parallel}-p_{\perp})$ and
$\alpha_{lmn}=2p_{\perp}B_{x}B_{l}+\Delta Pb_{x}\left(2\rho
u_{l}(b_{m}u_{m}+b_{n}u_{n})-b_{l}(3p_{\parallel}+2p_{\perp}+\rho(u_{m}^{2}+u_{n}^{2}-u_{l}^{2}))\right);~{}~{}\forall~{}l,m,n\in\\{x,y,z\\}$

$\beta_{1}=B_{y}u_{y}+B_{z}u_{z}+\frac{\Delta
P\left(-b_{y}u_{y}-b_{z}u_{z}+2b_{x}((\boldsymbol{b}\cdot\textbf{u})b_{x}-u_{x})\right)}{|\boldsymbol{B}|}$
and $\beta_{l}=B_{x}u_{l}-\frac{\Delta
Pb_{x}(2b_{l}(\boldsymbol{b}\cdot\textbf{u})-u_{l})}{|\boldsymbol{B}|};~{}~{}\forall~{}l\in\\{y,z\\}$

$\omega_{1}=B_{x}-\frac{2\Delta Pb_{x}(1-b_{x}^{2})}{|\boldsymbol{B}|}$ and
$\omega_{l}=B_{l}-\frac{\Delta
Pb_{l}(1-2b_{x}^{2})}{|\boldsymbol{B}|};~{}~{}\forall~{}l\in\\{y,z\\}$