# Exploring the Scope of Using News Articles to Understand Development
Patterns of Districts in India

Mehak Gupta<EMAIL_ADDRESS>Indian Institute of
TechnologyDelhiIndia ,  Shayan Saifi<EMAIL_ADDRESS>Indian Institute
of TechnologyDelhiIndia ,  Konark Verma<EMAIL_ADDRESS>Indian Institute
of TechnologyDelhiIndia ,  Kumari Rekha<EMAIL_ADDRESS>Indian Institute
of TechnologyDelhiIndia and  Aaditeshwar Seth<EMAIL_ADDRESS>Indian
Institute of TechnologyDelhiIndia

(2021)

###### Abstract.

Understanding what factors bring about socio-economic development may often
suffer from the _streetlight effect_ , of analyzing the effect of only those
variables that have been measured and are therefore available for analysis.
How do we check whether all worthwhile variables have been instrumented and
considered when building an econometric development model? We attempt to
address this question by building unsupervised learning methods to identify
and rank news articles about diverse events occurring in different districts
of India, that can provide insights about what may have transpired in the
districts. This can help determine whether variables related to these events
are indeed available or not to model the development of these districts. We
also describe several other applications that emerge from this approach, such
as to use news articles to understand why pairs of districts that may have had
similar socio-economic indicators approximately ten years back ended up at
different levels of development currently, and another application that
generates a newsfeed of unusual news articles that do not conform to news
articles about typical districts with a similar socio-economic profile. These
applications outline the need for qualitative data to augment models based on
quantitative data, and are meant to open up research on new ways to mine
information from unstructured qualitative data to understand development.

Mass media, socio-economic development, document vectors, document ranking

††copyright: acmcopyright††journalyear: 2021††doi:
xx.xxxx/xxxxxxx.xxxxxxx††conference: The 3rd KDD Workshop on Data Science for
Social Good; The 3rd KDD Workshop on Data Science for Social Good††booktitle:
;††ccs: Computing methodologies Machine learning††ccs: Applied computing
Computing in government

##  1\. Introduction

Indian districts demonstrate a wide diversity in patterns of socio-economic
development and provide a natural experimental setting to understand the
relationships between different development indicators (Goswami et al., 2019).
The discipline of development economics has traditionally used survey and
census methods to measure such indicators, which are then fed into econometric
models to find causal and correlated relationships between them. Doing manual
surveys at regular intervals however is not only expensive, the survey methods
and indicators can also change over time making it hard to obtain comparable
data (Subramanian, 2019). Several modern approaches using big-data sources
such as satellite data, social media, and mass media, are now used to build
proxy indicators for development to counter some of these problems (Lee, 2018;
Bansal et al., 2020; Baker et al., 2016; Antenucci et al., 2014). All these
methods however require a prior identification of development variables that
are collected or obtained through proxies, and then analyzed for their inter-
relationships with one another. Development is a complex process though and
may be affected by variables that were not originally measured or considered
worthwhile as influencing development or getting affected by it (Rodrik,
2021). For example, the effect that communal violence in a location may have
on development, say by affecting the inclination of migrant workers to
relocate there for employment, or the effect that industrialization may have
on environmental factors such as air or water pollution, may go unobserved if
a prior hypothesis is not framed to measure these variables and test for their
influence.

The research question we attempt to investigate is whether qualitative data
such as through news articles or social media that report on various events in
different locations, can provide helpful hints of how these events may affect
different socio-economic development patterns. This can in turn provide useful
inputs to researchers for new hypotheses to frame or additional quantitative
data to collect, to measure indicators based on these events and find out
their relationship with development. Or at the very least, if not identify new
information then such mass media and social media sources can provide
qualitative data to portray the lived experiences of people experiencing
different kinds of development patterns. For example, are districts that
experience rapid industrialization also marked by news about worker protests,
or are less developed districts marked by news about corruption in the
delivery of welfare benefits? The wider motivation for our work arises from
the need to look beyond well recognized quantitative indicators of
development, to understand what these indicators mean in practice for the
people (Wang, 2016), and what other factors may have been overlooked in the
conceptualization of development (Sen, 2001).

We use a unique dataset to study this research question. In prior work, we
built a machine-learning based classifier that uses satellite data from the
Landsat-8 system as a proxy to predict census-based development indicators
such as asset ownership of households, construction material used in
residential housing, access to drinking water, etc (Bansal et al., 2020). The
classifier was trained using data from the Indian census for 2001 and 2011,
and used to obtain development indicators as of 2019, ahead of the next
planned Indian census in 2021 111Once a census is conducted, it takes several
more years for the government to collate and release the data.. These
indicators were linearly summed to build a development index we termed as ADI
(Aggregate Development Index), similar to the HDI (Human Development Index)
methodology that combines economic, literacy, and health based indicators into
a common measure for broad-based development (Programme, 2019). In addition,
the Indian census of 2011 served to categorize the districts on their primary
source of employment, as non-agricultural or agricultural or with high
unemployment. We use these two variables of the type of employment, and change
in ADI between 2011 and 2019, to obtain different categories of Indian
districts that underwent different development patterns, such as industrial
districts that experienced a large change in ADI over the last decade, or
agricultural districts that did not, and so on. Our methodology is generic
though and can be applied to other categorizations as well, for example, to
categorize districts based on health or income or education variables, and
then understand the pace of growth in these categories. In parallel, we have
also built a large corpus of news articles from six primary English language
newspapers in India, since 2011 until now, consisting of all articles
published on a daily basis (Sen et al., 2019a, b). The district location of
these articles is obtained through an entity extraction method, and the media
corpus thus serves as the source of qualitative data to understand the
different development patterns.

Investigating our research question essentially amounts to solving a document
ranking problem, to rank news articles about districts that are most
representative of the development pattern of that district, and most distinct
from articles about districts that underwent other development patterns. We
evaluate different ranking methods based on TFIDF (Term Frequency Inverse
Document Frequency), LDA (Latent Dirichlet Allocation, a topic-modeling
method), and document vectors (an auto-encoder based deep learning approach).
We use these methods to demonstrate several applications, such as to identify
news articles that best explain the development of districts which are
outliers from the dominant development pattern to which they belong, contrast
pairs of districts that were at a similar development status in 2011 but have
diverged significantly since then, and a newsfeed application to identify
interesting news articles that may be counter-intuitive to the dominant
pattern of development of their district. Our analysis shows that news
articles are a promising source of qualitative data to understand the
development of districts in India, and highlights the scope that bottom-up
qualitative data has to complement patterns uncovered by top-down big data
analysis.

##  2\. Related work

A highly public debate has raged since several years about what constitutes
socio-economic development (Jha and Udas-Mankikar, 2019). On the one hand,
economic growth is seen as the “tide that lifts all boats” and needs to be a
precursor before equitable benefits of health, nutrition, education, dignified
livelihood, etc. can reach people (Batabyal, 2013). On the other hand,
instrumental freedoms such as good health, education, and food security are
seen as essential elements that need to be serviced through public
expenditure, to achieve constitutive freedoms for equitable economic
development (Sen, 2001). The diversity of social and economic development
across different Indian districts can potentially serve as a natural
experiment to inform this debate, but comprehensive datasets to build
different socio-economic indicators over time do not exist (Goswami et al.,
2019). Attempts have been made to use proxy indicators such as nightlights
(Lee, 2018), but ultimately any econometric model will be as good as how
comprehensive are the factors that it models. Our goal in this paper is to
inform what factors relevant to development may be missing in development
models, or can at least provide a qualitative understanding of what these
indicators mean in terms of the lived experiences of people.

Our enthusiasm about the importance of qualitative data to supplement
quantitative data is shared by people like Tricia Wang who highlight the
importance of ethnography in generating “thick data” to bridge the social
context-loss between data points obtained through big-data analysis (Wang,
2016). Qualitative datasets such as through social media, mass media, or
Wikipedia pages, can play a role in revealing the underlying dynamics that
cannot be directly observed in large quantitative datasets. Lu et al.
similarly identify correlations between green buildings and construction waste
management using a triangulation of quantitative big data obtained from
government agencies with qualitative “thick data” obtained from case studies
and interviews (Lu et al., 2018). They highlight the importance of
understanding the underlying context before blindly promoting any system such
as the Green Building Rating System (GBRS). Similarly, a report from IBM
writes about the strategy of “mixed analytics” which combines ethnographic
practices with big data analysis in the domain of public services (Ang, 2019).
In recent years, social media platforms have also been considered as a key
source to record social interactions rather than the traditional ways of
ethnography in the form of surveys, group discussions, and interviews. People
tend to be vocal about their experiences and opinions on social media and can
facilitate new approaches of digital ethnography (Dalsgaard, 2016; Bonilla et
al., 2015).

These research directions of using qualitative data to explain or augment
quantitative data is different from some other recent work in improving
quantitative predictions using qualitative data. Researchers have used
Wikipedia pages, for example, to improve satellite-data based predictions of
development (Sheehan et al., 2019). Similarly, news articles about weather
events have been used to improve the predictions of agricultural commodity
prices (Chakraborty et al., 2016b), and methods have been developed for event
identification from news articles (Chakraborty et al., 2016a). News articles
have also been used to build indicators for economic uncertainty (Baker et
al., 2016), and similarly social media data has been used to build indicators
of unemployment (A et al., 2015), and Google search trends have been used to
predict flu outbreaks (Kandula and Shaman, 2019). Our work is distinct and
complementary to this research direction. We use similar techniques for text
analytics, such as topic modeling using LDA (Blei et al., 2001), or document
vectors using deep learning approaches (Le and Mikolov, 2014; Chen et al.,
2017), but we do not aim to solve a prediction problem. Our work is closer to
a ranking problem to identify news articles that are representative of
specific development patterns, and at the same time also distinct from news
articles about other development patterns. These news articles can then give
hints about more extensive surveys or data collection exercises that should be
commissioned to enrich development models with new relevant variables.

##  3\. Dataset

We next describe our dataset that comprises of three parts. First, to obtain a
categorical variable as of 2011 of the primary type of employment in a
district: agricultural, non-agricultural, or high unemployment. Second, to
obtain a categorical variable of the pace of socio-economic development of a
district from 2011 to 2019: fast, medium, slow, based on the relative change
in ADI (Aggregate Development Index). Third, to build a district-wise corpus
of news articles published between 2011 to 2019.

###  3.1. Employment type

The Government of India conducts a household-level population census every 10
years. We used data from the 2011 census, available from its official website
(Chandramouli and General, 2011), to build a categorical variable of the
primary type of employment in each district. The census reports for each
district the population employed in agriculture, construction, mining,
technical services, public administration, and other industry sectors. We
first grouped these mutually exclusive variables into three broad parameter
types of unemployment (unemp), agricultural (agri), and non-agricultural (non-
agri) employment. For example, cultivators and agricultural workers were
considered as employed in agriculture, other types of services and trades were
grouped together as non-agricultural employment, and the rest of the adult
non-working population was considered as not employed. We then did a k-means
clustering of the districts based on the vector of the percentage of
population in these three categories: agricultural, non-agricultural, and
unemployed. Figure 2 shows a box-plot for the distribution of districts across
these categories (k = 3). The unemp category contains districts where there is
heavy unemployment and significantly less agricultural or non-agricultural
employment. The rest of the districts are split into agri and non-agri
categories depending upon which one of them is more dominant. A detailed
explanation of the discretization method and robustness for different values
of k can be found in (Goswami et al., 2019).

Figure 1. Distribution of the % of population under different types of
employment (unemployed, agricultural, non-agricultural) across three district
clusters

###  3.2. Pace of growth

Similar to the district-wise population distribution of employment in
different industry sectors, the census reports seven other indicators:
literacy, access to drinking water, access to sanitation facilities, fuel used
for cooking, main source of lighting, asset ownership, and construction
condition of households. Each of these indicators is reported in terms of
constituent variables, for example, the fuel used for cooking is reported with
the number of households that use firewood, kerosene, LPG (Liquid Petroleum
Gas), PNG (Piped Natural Gas), and biogas. We group these variables into three
broad parameter types of rudimentary (RUD), intermediate (INT), and advanced
(ADV) forms of fuel used for cooking. Firewood is considered as a rudimentary
type of fuel, kerosene and cow dung are grouped together as an intermediate
type, and PNG, LPG and bio-gas are grouped as advanced types of fuels for
cooking. Following a method similar to the one used by Fabini et al. (Fabini
et al., 2014) to discretize a collection of constituent variables, a k-means
clustering is done on this 3-tuple vector (% RUD, % INT, % ADV) to obtain an
ordinal label for each districts as level-1/2/3 in terms of the primary type
of fuel for cooking used in the district. Figure 2 shows a box-plot for the
districts mapped to these three levels (k = 3). Level-1 districts
predominantly use rudimentary types of fuel for cooking, level-2 districts use
intermediate types of fuel, and level-3 districts use advanced types of fuel
for cooking. A detailed explanation of the discretization method and its
robustness for different values of k is given in prior work (Goswami et al.,
2019). In a similar manner, each of the seven indicators are used to label
districts in terms of three levels.

Figure 2. An example of discretizing the level of development of districts in
terms of the fuel used for cooking in the district households. Shown is the
distribution of the % of households using different types of fuel rudimentary
(RUD), intermediate(INT), or advanced(ADV) across three district clusters.

In another prior work, we used these district labels obtained from the Indian
census of 2001 and 2011 to build a machine learning classifier using satellite
data (Bansal et al., 2020). This classifier uses daytime satellite imagery
obtained from the Landsat-8 system (open dataset available through the Google
Earth Engine), to obtain the corresponding socio-economic development level
for the district under the seven different indicators. We used this classifier
to take the satellite imagery as of 2019 as an input, and obtain a prediction
of the current levels of socio-economic development of all the Indian
districts since a census has not been conducted in recent years as yet. An
evaluation of the model on the ten-yearly data from 2001-2011 gave a steady
performance of between 70-80% accuracy for each of the seven indicators.
Interestingly, we found that a simple summation of the levels of all these
indicators to create an Aggregate Development Index (ADI), on the same lines
as how the HDI (Human Development Index) is computed as a summation of three
indicators on life expectancy, literacy rate, and per capita GDP, gave a low
normalized RMSE of 0.04, likely due to a cancellation of errors made in the
prediction of different indicators. We were thus able to deploy the classifier
to obtain the ADI as of 2019, as a value between 7 (all seven indicators at
level-1) to 21 (all seven indicators at level-3).

We then use the difference in ADI between the predicted values for 2019 and
the known values for 2011 to calculate the pace of socio-economic development
for each district. The difference in ADI values was in the range -3 to 9, as
shown in the CDF plot in Figure 3. We consider the bottom 36% of the districts
as slow developing districts (with a change in ADI between -3 to 0), the next
37% as medium developing (with a change in ADI between 1 to 3), and last 27%
as fast developing districts (with a change in ADI between 4 to 9). In
addition, those districts that had a very high ADI value of 20 or 21 for both
2011 and 2019, were considered as fast growing districts as well. Thus, by
categorizing districts on the two axes of primary type of employment (agri,
non-agri, unemp) and pace of growth (fast, medium, slow) we are able to obtain
9 sub-classes of districts that underwent different development patterns, as
shown in Table 1.

| Slow | Average | Fast
---|---|---|---
Unemp | 97 (92) | 100 (97) | 31 (31)
Agri | 91 (90) | 102 (101) | 34 (34)
N-Agri | 23 (23) | 30 (30) | 81 (81)
Table 1. Total number of districts in each subclass (number of districts with
enough news articles) Figure 3. CDF Plot for change in ADI values

###  3.3. Mass media dataset

We next build a corpus of news articles for each of the 9 sub-classes. Through
web-crawling of archives from January 1st 2011 and ongoing daily data
collection, we have built a large corpus of news articles containing all the
news published by six prominent national English newspapers in India:
Hindustan Times, Times of India, The Hindu, The Indian Express, Deccan Herald,
and the New Indian Express. This corpus contains more than 5 million articles.
In prior work, we used this corpus to study media bias in the coverage of
several important economic and technology policies in India (Sharma et al.,
2020; Sen et al., 2019b, a).

For this paper, we use the corpus to isolate news articles along the following
five broad themes: agriculture, industrialization, social welfare,
environment, and politics and social activities. We chose these themes because
most analysis of development revolves around them (Drèze and Sen, 2013).
Agriculture is considered as the primary industrial sector of the Indian
economy. Manufacturing and services are the secondary and tertiary sectors,
which we clubbed together under the industrialization theme. Social welfare is
a strong determinant of development, comprising almost 10% of the Indian GDP.
Environment is a cross-cutting theme which has become an increasingly
important category and mediates many development strategies. The final theme
of social activities and politics was selected as reflective of what aspects
of development are expressed and considered important by society. This
includes topics like crime, traffic, communal violence, sports, and elections.

We developed a simple keyword based method to map news articles to one or more
of these themes. An initial seed set of keywords was put together by the
authors based on their knowledge of the themes, and a query expansion process
was then followed to identify new keywords from among frequently occurring
keywords in the current document selections. New documents corresponding to
the expanded queries were then obtained, with up to three iterations after
which saturation was noticed for all the themes. The final document
collections were made for the five themes by selecting news articles that
contained at least three keywords from the keyword set for that theme. The
keywords used are given in Table 7 in the Appendix. Articles with overlapping
themes were mapped to multiple themes.

Once the theme-wise collections were obtained, entity extraction of the
district location was used to identify the districts referred to in a news
article. We first used the OpenCalais service (Reuters, [n.d.]) to identify
locations mentioned in the article, and then followed several steps to resolve
these location entities against the list of districts as per the Indian
census. We only considered entities mentioned in the opening paragraph of an
article, but excluded the news bureau location which is typically prefixed at
the start of an article. We then ran an edit distance match between the
extracted location entities and the census list of districts. Any entities
with a poor match were manually examined to check for spelling variations. We
also took care of cases where districts could be referred to by multiple names
(such as _Poorvi_ /_Pashim_ Champaran for _East_ /_West_ Champaran
respectively), or the names of districts were changed (such as _Prayagraj_ for
_Allahabad_), or a district was split into smaller districts (such as _Thane_
was split into _Thane_ and _Palghar_). In the case of split districts, we only
considered the merged entities as of the 2011 census. Through an iterative
process, we thus built a comprehensive district resolution list, and upon
evaluating 100 randomly chosen entities we obtained precision and recall
values of 93.18% and 95.35% respectively (Sen et al., 2018).

The district mapping thus enabled us to categorize the theme-wise document
collections into the 9 sub-classes based on the type of employment and pace of
growth seen in the districts. Table 2 shows the count of news articles for the
five themes against the 9 sub-classes. Finally, some pre-processing steps were
undertaken on the documents for stop-word removal, stemming, and entity
blinding so that the subsequent models we build using the documents are
agnostic to names of people, organizations, political parties, etc. We also
did not consider those districts for which the number of news articles was
less than 100.

Collections |  Total articles | Unemp districts | Agri districts | Non Agri districts
---|---|---|---|---
Slow | Medium | Fast | Slow | Medium | Fast | Slow | Medium | Fast
Agriculture | 128487 | 11097 | 5968 | 3968 | 9639 | 12204 | 4183 | 6762 | 5085 | 69581
Social welfare | 25709 | 2160 | 1085 | 647 | 1373 | 1565 | 570 | 1633 | 1199 | 15477
Environment | 155205 | 10248 | 6662 | 5020 | 7734 | 8665 | 3518 | 9776 | 5890 | 97692
Industrialization | 172430 | 8379 | 4568 | 3648 | 5798 | 6269 | 2084 | 8337 | 4964 | 128383
Politics and social activities | 370509 | 29770 | 13957 | 10458 | 14513 | 12628 | 5238 | 26743 | 12748 | 244454
Table 2. Total number of articles in each subclass

##  4\. Methodology

Based on district-wise categorical variables of the primary type of employment
and the pace of growth of a district, we define three types of development
patterns that we attempt to investigate:

  * •

_Pattern 1:_ To understand what factors relate to a different pace of growth
among high-unemployment districts

  * •

_Pattern 2:_ Similarly, among agricultural districts

  * •

_Pattern 3:_ And among non-agricultural districts

Note that the framework we have outlined is flexible to define other
development patterns as well based on which variables are to be contrasted
with one another. Our objective is simply to evaluate through a demonstration
of whether this approach of using mass media can help explain or provide
qualitative insights about the dynamics of different development trajectories
followed by the districts. For each pattern, we therefore solve a document
ranking problem to identify news articles that are most representative of the
sub-class to which they belong, and most different from other sub-classes in
the pattern under investigation. We compare three methods for this purpose.

###  4.1. TFIDF based selection

TFIDF is a common information retrieval technique to obtain a weighting factor
for each word in a document in the corpus. Term frequency TF($w$, $d$) is the
frequency of word $w$ in document $d$. IDF($w$, $D$) is the logarithmically
scaled inverse of the number of documents in the corpus $D$ that contain word
$w$. TFIDF($t$, $d$, $D$) is then computed as TF($w$, $d$) $*$ IDF($w$, $D$),
and essentially gives a higher weightage to words that occur more frequently
in a document, but also less frequently in the overall corpus so as to
identify words that are unique to the given document. In our case, we
concatenated all the articles of a sub-class to form one document for that
sub-class, and then computed TFIDF scores for all the words in each sub-class
document. From each sub-class, we then obtained top-100 news articles by
summing the TFIDF scores of all the words in the news article. We then
compared the quality of this selection of the top-100 articles with selections
obtained using other competing methods, as described next.

###  4.2. Topic modeling based selection

LDA (Latent Dirichlet Allocation) is a common topic modeling technique that
identifies hidden topics as having been generated through a mixtures of
keywords, and a mixture of the topics as further generating the documents in
the corpus. We ran LDA over the news articles for each sub-class to identify
the constituent topics for each sub-class. Based on the probability for an
article to belong to a topic as given by the LDA routine, we then picked up
$100/k$ articles from each topic, where $k$ was the optimal number of topics
identified through LDA in the sub-class. If a topic has less than $100/k$
articles, additional articles were then picked up equally from other topics to
finally select the top-100 articles for a sub-class. We used topic coherency
scores to determine a good value of $k$ (Röder et al., 2015).

Our intuition behind sampling articles from across the different topics
identified by LDA was to choose a diverse set of articles from the sub-class,
so that diverse descriptions of common development processes for that sub-
class could be obtained. As compared to the previously described TFIDF method,
the LDA method is therefore likely to give a more complete representation of
articles for a sub-class. However, this method does not select articles that
are distinct from other sub-classes, which the IDF component of TFIDF is able
to achieve. In the next method using document vectors, we try to achieve both
goals of choosing diverse articles in a sub-class, and also picking up
articles that are distinctive to the sub-class as compared to other sub-
classes.

###  4.3. Clustering over vector embeddings

This method works through a three-step process as follows.

  1. (1)

Vector embeddings using DocTag2Vec. Doc2Vec is a deep-learning based
unsupervised approach to obtain a vector representation for a document (Le and
Mikolov, 2014). It is known to work better for information retrieval tasks
than TFIDF based vector representations since it also takes the order of words
into account, thus making it more sensitive to the context in which different
words are used. An extension to Doc2Vec is the DocTag2Vec architecture which
additionally allows modeling of document tags: The vector representation of
documents thus becomes sensitive to the tags attached to the documents, and
vectors are also obtained for the tags as well (Chen et al., 2017). In our
case, we consider the categorical variables of the type of employment and the
pace of growth as tags. Districts are also uniquely identified through tags.

Figure 4. Doc-Tag-2-Vec Model (Chen et al., 2017)

Figure 4 shows the DocTag2Vec architecture. Documents (news articles) and tags
(type of employment, pace of growth, district) are taken as inputs, and
corresponding embeddings are generated for the documents and tags. We trained
this model separately for the article collections for each theme, thus
obtaining five different models. The training parameters are described in more
detail in the Appendix Appendices. After empirical evaluation, 50-dimensional
vectors were found to work well.

  2. (2)

Hierarchical clustering of document embeddings. Within each sub-class, we use
agglomerative hierarchical clustering on the DocTag2Vec document embeddings to
obtain diverse topic clusters (Cohen-addad et al., 2019). We remove outlier
clusters that contain very few articles, and finally obtain $k$ clusters for
each sub-class where the value of $k$ is decided based upon a cophenetic
distance statistic that determines which clusters can be merged with one
another. Different sub-classes may have a different number of topic clusters.

  3. (3)

Document selection. Finally, we then compute a global centroid for each sub-
class based on the document vectors of the news articles in the sub-class.
Similarly, we compute a cluster centroid for the $k$ topic clusters in each
sub-class. Based upon the appropriate ranking criteria for different
applications, as we explain next, we are then able to use the document
vectors, cluster centroid vectors, and sub-class centroid vectors, to identify
the highest ranked news articles for the application. We calculate a score
$s_{dt}$ as the cosine similarity between a document vector $d$ and the
centroid of the topic cluster $t$ to which the document belongs, $s_{ds}$ as
the cosine similarity between a document vector $d$ and the centroid of the
sub-class $s$ to which the document belongs, and $s_{ds^{\prime}}$ as the
cosine similarity between a document vector $d$ and centroid of a contrasting
sub-class $s^{\prime}$. In applications where our goal is to select news
articles that are most representative of their sub-class but most distinct
from a contrasting sub-class, we rank the news articles based on the metric
$s_{ds}*s_{dt}-s_{ds^{\prime}}$ and select $100/k$ articles from each topic
cluster in the sub-class $s$. Similarly, in applications where news articles
need to be selected based on how much a district conforms to typical districts
in its sub-class, we use the metric $s_{ds}*s_{dt}$ to rank news articles
about that district. A document vector representation can thus be flexibly
used for different applications.

##  5\. Evaluation through contrasting patterns

We next compare the three methods of TFIDF, LDA, and DocTag2Vec, to identify
the most suitable method for article selection and ranking for our primary
use-case to identify articles most representative of their sub-class as well
as most distinctive from other sub-classes. We evaluate using both
quantitative metrics as well a qualitative user evaluation.

For a quantitative evaluation, we use the intuition that the top-100 articles
identified for a sub-class should have the least overlap of keywords with the
top-100 articles from other sub-classes. For each pair of sub-classes, we
therefore compute a Jaccard similarity coefficient on keywords in the
articles. A low Jaccard similarity would imply that the article selection
method was able to identify articles containing keywords quite distinct
between pairs of sub-classes. Another metric we develop based on the same
intuition is an entropy based measure that assigns a probability value to a
keyword based on the most likely sub-class to which it belongs. A low entropy
value for a sub-class would imply that the set of keywords of dominant use
within it, are distinct from dominant keywords in other classes. We build two
additional metrics as well based on TFIDF vectors for the top-100 articles in
each sub-class: A mean cosine based and a Euclidean distance based similarity
score between the articles in pairs of sub-classes. Low cosine similarities or
high Euclidean distances would denote greater separation between the classes.
The detailed values of the metrics are given in the Appendix, and Table 3
shows the best performing method based of each of these four metrics. Clearly,
the DocTag2Vec method seems to work the best for most cases.

We also conducted qualitative evaluations for each of the three patterns,
based on a self-assessment of the identified articles, and another assessment
by volunteers from among graduate students in our research group. Table 13,
14, 15 shows the primary topics for each (sub-class, theme) pair identified by
the authors through a manual inspection of the top identified articles.
Several expected topics emerge for various development patterns, such as
articles about irregularities in job schemes funded by the government, and
crimes including thefts and violence, are common among slow-growing and high-
unemployment districts (belonging to pattern-1). Fast-growing and high-
unemployment districts had more news articles about industrial projects,
development research, and festivals. Pattern-2 related to agricultural
districts had topics such as farmer suicides and food adulteration surfacing
among slow-growing districts, while fast-growing districts had more articles
related to government schemes and good agricultural harvests. Topics like
illegal mining and electricity power crises came up among slow-growing non-
agricultural districts, while economic revival and revenue generation by the
state were reported among fast-growing non-agricultural districts.

We also conducted a user evaluation by recruiting 11 volunteers from among
non-authors of this paper. A training session was held to communicate the
purpose of the study and each rater was asked to assess based on 10 randomly
chosen articles (from among the top-100 articles identified by the three
ranking methods described earlier) for each pattern about whether the articles
were representing commonly known development pathways within the patterns.
Three questions were put forth to the raters for each theme, to assess based
on a scale of 0..5 whether the articles were (a) relevant to the sub-classes,
(b) they depicted events or topics that could be associated with positive or
negative development, (c) the article selections for different sub-classes
within each pattern was diverse and denoted different aspects. A 2-sample
t-test was then conducted for each question between the ratings given for the
different methods. In most cases, ratings given to the DocTag2Vec method came
out as better than the other methods. The detailed t-test results are shown in
Table 12 in the appendix.

Pattern | Metric | Agriculture | Social welfare | Environment | Industrialization |  Politics and social activities
---|---|---|---|---|---|---
Slow vs fast unemp |  TFIDF based jaccard similarity | DocTag2Vec | LDA | DocTag2Vec | DocTag2Vec |  DocTag2Vec
Global similarity | DocTag2Vec | TFIDF | DocTag2Vec | DocTag2Vec |  DocTag2Vec
Euclidean distance | DocTag2Vec | TFIDF | DocTag2Vec | DocTag2Vec |  DocTag2Vec
|  Entropy | TFIDF | DocTag2Vec | TFIDF | TFIDF |  LDA
|  Qualitative evaluation | DocTag2Vec | LDA | TFIDF | DocTag2Vec |  DocTag2Vec
Slow vs fast agri |  TFIDF based jaccard similarity | DocTag2Vec | TFIDF | DocTag2Vec | DocTag2Vec |  DocTag2Vec
Global similarity | LDA | DocTag2Vec | TFIDF | DocTag2Vec |  DocTag2Vec
Euclidean distance | TFIDF | DocTag2Vec | DocTag2Vec | DocTag2Vec |  LDA
|  Entropy | DocTag2Vec |  | TFIDF LDA | DocTag2Vec |  DocTag2Vec
|  Qualitative evaluation | DocTag2Vec | LDA | DocTag2Vec | DocTag2Vec |  TFIDF
Slow vs fast non-agri |  TFIDF based jaccard similarity | LDA | DocTag2Vec | TFIDF | LDA |  DocTag2Vec
Global similarity | TFIDF | TFIDF | DocTag2Vec | LDA |  LDA
Euclidean distance | TFIDF | TFIDF | DocTag2Vec | LDA |  TFIDF
|  Entropy | DocTag2Vec | LDA | LDA | LDA |  DocTag2Vec
|  Qualitative evaluation | DocTag2Vec | LDA | LDA | DocTag2Vec |  DocTag2Vec
Table 3. Quantitative evaluation results

##  6\. Applications

Given the superior performance of the DocTag2Vec method in most cases, we next
use this method to demonstrate a number of applications we can build from the
models.

###  6.1. Contrasting pairs of districts

The development indicators for 2011 and 2019 can identify districts that were
at the same development status (in terms of ADI) in 2011 and with the same
type of primary employment, but achieved a markedly different development
status by 2019. An interesting application can be to find pairs of such
districts that followed divergent development trajectories, and understand
through the news articles about the events that transpired in the districts
during the intervening years of 2011 to 2019 that may help explain this
difference.

An example is the pair of districts of Thiruvarur and Thanjavur in the state
of Tamil Nadu, both of which were high unemployment districts in 2011 with an
ADI of 16, but while Thiruvarur’s ADI in 2019 only improved marginally to 17,
the ADI of Thanjavur improved to 20. Table 4 shows the primary topics in the
top ranked news articles for the two districts, based on the ranking metric of
$s_{ds}*s_{dt}-s_{ds^{\prime}}$ as described in Section 4, for each of the
five themes. The articles from Thanjavur are progressive in nature and include
news about job schemes, tourism, agricultural waivers, and scholarships, while
Thiruvarur has articles on crop damage, protests by political parties
(_bandhs_), and cyclones. Such an analysis method can therefore help provide a
description of different development pathways followed by the districts.

|  Agriculture |  Social welfare |  Environment |  Industrialization |  Politics and social activities
---|---|---|---|---|---
Thiruvarur |  • Crop damage • Paddy cultivation • Farmer suicide due to indebtedness |  • Poultry Farms • Road blockade • Job reservation |  • Crop training • Large scale strike • Viral disease |  • Methane issue • Cyclone • Data theft |  • Diseases • Crimes • Adulteration
Thanjavur |  • Good monsoon • Loan waiver • Water conservation |  • Check dams • Entrepreneur-ship • Job scheme |  • Tourism promotion • Agri Technologies • Scholarship |  • Handloom expo • Entrepre-neurs • Festivals |  • Cultural programme • Tourist activities • Dengue control
Table 4. Subjective assessment results for contrasting pairs of districts:
Thiruvarur and Thanjavur

Another similar example is the pair of districts of Bargarh and Sambalpur in
the state of Odisha. Both are agricultural districts and had an ADI of 10 in
2011. Bargarh’s ADI decreased to 9 by 2019, while Sambalpur’s ADI grew rapidly
to 15. Table 5 shows subjective assessment results for the news articles from
these districts. Articles about Sambalpur are on progressive topics such as
good rainfall, job schemes, women hostels, and healthcare, while articles from
Bargarh are about farmer suicides, pest attacks, and floods, among others.

|  Agriculture |  Social welfare |  Environment |  Industrialization |  Politics and social activities
---|---|---|---|---|---
Bargarh |  • Crop loss • Pest attack • Farmer suicide due to indebtedness |  • Skill institute • Cancer survivor |  • Murders • Animal deaths • Flood damages |  • Builders fraud • Mining auction • Asset Raids |  • Political attacks • Dance fest • Builders fraud
Sambalpur |  • Good rainfall • Festivals • Farming activities |  • Arrests of scam accused • Rural job scheme |  • Tourism activities • Thermal plant • Save River campaign |  • Industri-alization effects • Women hostel • Develop-ment projects |  • Folk festivals • Tourist activities • Ambula-nce service
Table 5. Subjective assessment results for contrasting pairs of districts:
Bargarh and Sambalpur

###  6.2. Outlier analysis

As another interesting application, we visualize using TSNE (van der Maaten
and Hinton, 2008) the district centroids for each sub-class and theme, and
manually spot any outlier districts. The articles from these outlier districts
may reveal some counter-intuitive development pathways for the sub-classes.
For example, Figure 5 shows the TSNE plots for four sub-classes: based on the
politics and social activities theme for slow-growing high unemployment
districts, agriculture theme for fast-growing high unemployment districts,
industrialization theme for fast-growing agricultural districts, and
agriculture theme for medium-growing non-agricultural districts. There are
some clear outliers and Table 6 describes the prominent topics covered in the
news about these districts:

  1. (1)

Rampur: Upon analysing the topics covered in the news articles, this district
seems to have a large number of articles about violence and murder, while the
other districts in the collection prominently feature articles on politics and
the development of essential amenities.

  2. (2)

Champawat: News articles about this district discuss plans of pharmaceutical
firms and government aided industrialization, while the rest of the districts
see more emphasis on farming activities. Champawat therefore seems to have
come up as an outlier because it is probably deviating towards an
industrialization led development.

  3. (3)

Ganganagar: Topics featured in Ganganagar are about the Information Technology
sector and financial investment in agricultural industrialization, while the
rest of the districts see more emphasis on core agricultural improvement such
as irrigation. The outlier district therefore seems to have taken a path of
attracting private sector investment for growth.

  4. (4)

Lakshadweep: The Lakshadweep Islands are a group of islands off the Indian
mainland and therefore see articles about the monsoons and cyclones, distinct
from the rest of the districts.

This method can thus help identify some unusual districts and explain possible
reasons behind their distinctive properties.

Figure 5. Outlier visualization using TSNE embeddings Sub-class |  Collection |  Outlier district |  Analysis on Outlier |  Rest
---|---|---|---|---
Slow-growing unemployment districts |  Politics and social activities |  Rampur |  • Mob Violence • Molestation • Murder |  • Assembly polls • Tribute to martyr • Medical College
Fast-growing unemployment districts |  Agriculture |  Champawat |  • Pharma Firms • Govt production plans • Poll promises |  • Flood crisis • Farmer protest • Organic farming
Fast-growing agricultural districts |  Industrialization |  Ganganagar |  • IT sector • Upbeat investment • Agro industry |  • Cleanliness activities • Water lease • Tourist flow
Average-growing non-agricultural districts |  Agriculture |  Lakshadweep |  • Cyclonic storm • Pre-monsoon showers • Monsoon activities |  • Railway routes • Improve vegetable production • Festivals
Table 6. Subjective assessment results for outlier analysis: Rampur,
Champawat, Ganganagar, Lakshadweep

###  6.3. Unusual development pattern descriptions

A third application we develop on similar lines as the previous application,
is to identify any topics that seem unusual for a particular sub-class of
development patterns. This can provide hints towards possibly new development
variables that should be measured and tracked to understand their impact. We
use the metric $s_{ds}-\tilde{s_{ds^{\prime}}}$ to identify news articles that
are representative of their sub-class but different from other sub-classes,
and several surprising topics do emerge. For example, arrests of Maoist rebels
and news about communal clashes is seen among slow-growing high-unemployment
districts. Most districts under Maoist influence are indeed under-developed,
but it is rare to find quantitative econometric research that captures the
degree of Maoist activity, or issues like communal violence in many other
districts, and relates them to socio-economic development (Gomes, 2015).
Similarly, we find that news about falling sex ratio was common among slow-
growing agricultural districts, possibly indicative of growing female
foeticide with poor economic development, something which again needs further
investigation (Bhattacharyya and Haldar, 2019). Another interesting dominant
topic we identified was about cybercrime in fast-growing non-agricultural
districts. This may indicate that an increasing use of digital financial
services but among a newly prosperous population that is yet to become very
familiar with these services, can leave the people vulnerable to financial
fraud (Kshetri, 2016). Such unusual topics that emerge as dominant in a sub-
class can provide hints about new variables or relationships that should be
tracked to understand socio-economic development.

###  6.4. Newsfeed for novel information

A fourth application we demonstrate is to generate a newsfeed from among
unseen (future) news articles. We consider an article about a district as
providing some novel information if it deviates from typical news articles
based on the development pattern of the district. Such news articles could
indicate if the district is witnessing some interesting events. We do this as
follows. Given a new article about a district and belonging to a particular
theme, we first find its document vector using the trained DocTag2Vec model
for the theme. Since the district is known to have a particular employment
type (agricultural, non-agricultural, high-unemployment) and pace of growth
(fast, medium, slow), we then compute the cosine similarity between the
document vector of this article and the global centroid of the document
vectors for the articles about that employment type. If the cosine similarity
is higher between this article and an employment type different from the one
known for the district, it may indicate that the district could be
transitioning to a different dominant employment type. In the same manner, a
similarity between this article and the global centroid for sub-classes based
on the different pace of growth categories, can indicate if the district is
witnessing expected events or those that might indicate a departure from the
known development pattern of the district. Note that it is also possible that
the media coverage about the district could be biased to cause a deviation
from the socio-economic development pattern predicted through satellite based
data, but we feel that the political and ideological diversity of newspapers
we have included may guard against such a possibility.

We used the models trained on data before 2018, to generate document vectors
for news articles publishing during 2018-19. Figure 6 shows CDFs of the
fraction of articles in a district that appeared as outliers, i.e. the cosine
similarity between the document vector of these articles was different from
the document vector centroids of the employment type and the pace of growth
categories to which the district belonged. Many districts indeed have a high
fraction of recent news articles that diverge from the expected development
pattern of the district, hinting at the need for further investigation. For
example, we find that recent articles about the district of Vishakapatnam
(slow-growing high unemployment district) are outliers with both the
employment type outlier fraction at 0.92 and pace of growth outlier fraction
at 0.87. Many of these articles are about agricultural diversification into
fruits like mangoes and vegetables like bitter gourd, along with political
manifestos aimed at increasing rural economic growth, possibly indicating that
the district may have recently taken a turn towards improved development. In
contrast, we find that articles about Jammu (fast-growing non-agricultural
district) also have a high outlier fraction for the pace of growth (0.92) and
are about the retraction of several aspects of Article 370 of the Indian
constitution about the special autonomous status given to the state of Jammu
and Kashmir. This retraction was followed with the imposition of an Internet
shutdown throughout the state, and several other restrictions, likely to
negatively impact the economy of the state. We thus see that such a newsfeed
application can keep alerting development researchers about deviations from
normal patterns, which can then be tracked and monitored more closely.

Figure 6. CDF plot for outlier ratios for Pace of Growth and Employment Type

##  7\. Conclusions

We showed in this paper that qualitative data from sources such as news
articles published in the mass media can provide valuable information to
explain observations made using quantitative data. We built methods using
unsupervised machine learning models applied on news articles to explain
different socio-economic development patterns seen by Indian districts, during
an approximate ten-year span from 2011 to 2019. We demonstrated several
interesting applications, such as being able to understand divergent socio-
economic development patterns followed by pairs of districts that were
originally similar to each other, unusual topics observed through news
articles that do not conform to typical topics witnessed in a given
development pattern, and a newsfeed application to aggregate and rank news
articles that may be indicative of deviations of a district from development
patterns it has seen so far. Our work is an effort to draw attention to the
relevance of qualitative data to augment quantitative data, especially from
big-data sources that build indicators for socio-economic development, and
encourage similar work using other bottom-up information sources such as
social media and rapid local surveys.

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## Appendices

  1. (1)

The final set of keywords to identify articles belonging to the five themes,
are given in Table 7. An article was selected for a theme only if contained at
least three of these keywords.

Collection |  Set of keywords
---|---
Agriculture |  agri, pesticide, insecticide, kharif crop, kharif-crop, rabi crop, rabi-crop, crops, monsoon, irrigate, farmer (includes farmer protest, farmers rally farmers distress), loan waiver, bhartiya kisan sangathan, bks, pradhan mantri fasal bima, pm fasal bima,national agriculture market, enam, pmksy, pkvy, pradhan mantri krishi sinchayee, pm krishi sinchayee, paramparagat krishi vikas, pm kisan yojna, pradhan mantri kisan yojna
Social welfare |  development scheme, development program, pradhan mantri gram sadak, national rural employment guarantee, mgnrega, nrega, pmgsy, make in india, jan dhan yojna, beti bachao beti padhao, digital india, stand up india, prime minister ujjwala plan,pm ujjwala plan gramoday se bharat uday, shramew jayate , ujjwala scheme, udan, regional connectivity scheme, smart cities mission, skill india mission, national career service, egovernance, egov, aadhaar, pds, ration, nutrition, malnutrition, sanitation, hygiene, immunization, vaccines, ayusman bharat, rsby
Environment |  forest, eco, environment, deforestation, wildlife, pollution, swachh bharat mission, swachchh bharat mission, swachhgram, clean india mission, pmfby, fra, integrated conservation and development, icdp, jfm, poaching, ntfp, tiger, leopard, zool
Industrialization |  coal, lignite,steel product, industry, leather product,crude petroleum, metal product, textile, fertilizer, pesticide, enterprise,prime minister employment generation programme,estate, pmegp,credit guarantee trust fund for micro & small enterprises,cgt sme,mine, mining, stock market, equity market, share market, factory
Politics and social activities |  lifestyle, life-style, fashion, art, art and culture, health tips, tourism, culture, travel, tech, spirituality, astrology, celebrity, riot, movie, crime, violence, communal, hatred, fake news, misinformation, migration, suicide
Table 7. Final set of keywords used for theme-wise collections

  2. (2)

The parameters used for training the DocTag2Vec models were:

     * •

Learning rate: 0.1

     * •

Minimum learning rate: 0.001

     * •

Vector size: 50

     * •

Minimum occurrences of a word: 3

     * •

Word representation: Distributed bag of words

     * •

Window size for words: 3

     * •

Learning rate decays linearly

  3. (3)

Data from quantitative evaluation metrics for document ranking are given in
Tables 8, 9, 10, and 11. Evaluations based on Jaccard similarity among the
top-100 articles for each sub-class are shown in Table 8, a lower value
denotes more separation, and we can see that the DocTag2Vec method gives the
lowest values in most cases. Distances between articles of a sub-class and
global centroids of other sub-classes, are calculated using cosine similarity
in Table 9 and Euclidean distances in Table 10. Lower values of cosine
similarity and higher values of Euclidean distances denote greater separation.
Table 11 shows results from an entropy-based measure, where the entropy of a
sub-class is calculated based on the probability of keywords belonging more to
that sub-class than other competing sub-classes. Methods producing lower
values of entropy are better.

|  | Agriculture | Social welfare | Environment | Industrialization |  Politics and social activities
---|---|---|---|---|---|---
Slow vs fast unemp | DocTag2Vec Method | 0.12 | 0.13 | 0.13 | 0.12 |  0.11
TFIDF based selection | 0.12 | 0.13 | 0.14 | 0.14 |  0.12
LDA | 0.14 | 0.12 | 0.13 | 0.14 |  0.12
Slow vs fast agri | DocTag2Vec Method | 0.14 | 0.15 | 0.12 | 0.11 |  0.14
TFIDF based selection | 0.15 | 0.14 | 0.12 | 0.13 |  0.12
LDA | 0.15 | 0.15 | 0.13 | 0.11 |  0.12
Slow vs fast non-agri | DocTag2Vec Method | 0.13 | 0.13 | 0.13 | 0.13 |  0.11
TFIDF based selection | 0.12 | 0.13 | 0.12 | 0.12 |  0.13
LDA | 0.12 | 0.13 | 0.14 | 0.09 |  0.12
Table 8. Jaccard similarity metric |  | Agriculture | Social welfare | Environment | Industrialization |  Politics and social activities
---|---|---|---|---|---|---
Slow vs fast unemp | DocTag2Vec Method | 0.66 | 0.71 | 0.67 | 0.76 |  0.61
TFIDF based selection | 0.77 | 0.61 | 0.76 | 0.8 |  0.7
LDA | 0.78 | 0.66 | 0.78 | 0.76 |  0.71
Slow vs fast agri | DocTag2Vec Method | 0.64 | 0.58 | 0.66 | 0.73 |  0.65
TFIDF based selection | 0.67 | 0.72 | 0.65 | 0.77 |  0.74
LDA | 0.63 | 0.71 | 0.70 | 0.80 |  0.72
Slow vs fast non-agri | DocTag2Vec Method | 0.44 | 0.43 | 0.29 | 0.51 |  0.44
TFIDF based selection | 0.37 | 0.37 | 0.44 | 0.56 |  0.47
LDA | 0.51 | 0.40 | 0.56 | 0.45 |  0.41
Table 9. Global centroid similarity metric |  | Agriculture | Social welfare | Environment | Industrialization |  Politics and social activities
---|---|---|---|---|---|---
Slow vs fast unemp | DocTag2Vec Method | 4.33 | 4.24 | 4.80 | 4.86 |  5.05
TFIDF based selection | 3.9 | 5.43 | 4.67 | 3.98 |  4.85
LDA | 3.49 | 4.29 | 3.85 | 4.30 |  4.38
Slow vs fast agri | DocTag2Vec Method | 4.88 | 4.55 | 4.41 | 5.16 |  5.46
TFIDF based selection | 5.4 | 4.09 | 4.91 | 4.8 |  4.2
LDA | 4.74 | 4.23 | 4.72 | 4.99 |  5.49
Slow vs fast non-agri | DocTag2Vec Method | 5.04 | 5.33 | 6.53 | 5.20 |  4.34
TFIDF based selection | 5.66 | 5.63 | 4.71 | 5.3 |  5.47
LDA | 4.94 | 4.95 | 3.92 | 4.96 |  4.85
Table 10. Euclidean distance metric |  | Agriculture | Social welfare | Environment | Industrialization |  Politics and social activities
---|---|---|---|---|---|---
Slow vs fast unemp | DocTag2Vec Method | 5.96 | 5.96 | 5.84 | 5.73 |  6.0
TFIDF based selection | 5.57 | 6.08 | 5.77 | 5.94 |  5.69
LDA | 6.02 | 6.03 | 5.89 | 5.92 |  5.95
Slow vs fast agri | DocTag2Vec Method | 5.77 | 6.06 | 5.94 | 5.97 |  5.73
TFIDF based selection | 5.80 | 5.78 | 5.84 | 4.73 |  5.96
LDA | 5.78 | 5.93 | 5.84 | 6.01 |  5.88
Slow vs fast non-agri | DocTag2Vec Method | 5.86 | 6.01 | 5.95 | 5.98 |  5.95
TFIDF based selection | 5.91 | 5.98 | 5.99 | 5.87 |  5.92
LDA | 5.96 | 5.96 | 5.74 | 5.82 |  6.0
Table 11. Entropy metric

  4. (4)

Qualitative evaluation: Standard two sample t-tests were performed on the
review ratings given by eleven raters to the document selections obtained from
the three different methods. A t-test was done to compare these methods in
pairs with each other, and the tests were done separately for each question
out of the three questions put up to the raters. The null hypothesis that
pairs of ratings were similar to each other, was rejected in most cases with
p-values less than 0.05. Table 12 shows the results. The questions put up to
the raters are shown in Table 16.

|  | Agriculture | Development | Environment | Industrialization | Politics and social activities
---|---|---|---|---|---|---
|  | p1=DocTag2Vec | p1=DocTag2Vec | p1=DocTag2Vec | p1=DocTag2Vec | p1=DocTag2Vec | p1=DocTag2Vec | p1=DocTag2Vec | p1=DocTag2Vec | p1=DocTag2Vec | p1=DocTag2Vec
|  | p2=TFIDF | p2=LDA | p2=TFIDF | p2=LDA | p2=TFIDF | p2=LDA | p2=TFIDF | p2=LDA | p2=TFIDF | p2=LDA
|  |  t-value |  p-value |  t-value |  p-value |  t-value |  p-value |  t-value |  p-value |  t-value |  p-value |  t-value |  p-value |  t-value |  p-value |  t-value |  p-value |  t-value |  p-value |  t-value |  p-value
Slow vs Fast unemp |  Q1 |  1.112 |  0.279 |  0.933 |  0.362 |  2.196 |  0.04 |  1.077 |  0.294 |  1.627 |  0.119 |  1.978 |  0.062 |  4.041 |  0.001 |  6.264 |  0.0004 |  3.149 |  0.005 |  2.864 |  0.01
|  Q2 |  2.149 |  0.044 |  2.344 |  0.03 |  0.806 |  0.429 |  0.0 |  1.0 |  1.102 |  0.284 |  0.464 |  0.648 |  2.229 |  0.037 |  2.752 |  0.012 |  2.569 |  0.018 |  2.372 |  0.028
|  Q3 |  2.401 |  0.026 |  3.305 |  0.004 |  -0.501 |  0.622 |  0.264 |  0.794 |  0.761 |  0.455 |  1.204 |  0.243 |  1.668 |  0.111 |  1.0 |  0.329 |  1.479 |  0.155 |  1.352 |  0.191
Slow vs Fast agri |  Q1 |  3.796 |  0.001 |  3.551 |  0.002 |  -1.373 |  0.185 |  0.753 |  0.46 |  4.008 |  0.001 |  2.911 |  0.009 |  2.142 |  0.045 |  2.397 |  0.026 |  0.267 |  0.792 |  -1.432 |  0.168
|  Q2 |  2.548 |  0.019 |  4.561 |  0.0 |  -0.801 |  0.433 |  0.598 |  0.557 |  3.227 |  0.004 |  2.813 |  0.011 |  1.593 |  0.127 |  0.767 |  0.452 |  0.755 |  0.459 |  -0.503 |  0.621
|  Q3 |  3.212 |  0.004 |  1.0 |  0.329 |  1.708 |  0.103 |  1.006 |  0.326 |  1.48 |  0.154 |  1.725 |  0.1 |  2.677 |  0.014 |  2.236 |  0.037 |  1.204 |  0.243 |  0.977 |  0.34
Slow vs Fast non-agri |  Q1 |  3.493 |  0.002 |  4.472 |  0.0002 |  2.41 |  0.026 |  1.021 |  0.32 |  3.117 |  0.005 |  0.75 |  0.462 |  2.85 |  0.01 |  3.07 |  0.006 |  6.708 |  0.0001 |  3.833 |  0.001
|  Q2 |  2.57 |  0.018 |  3.774 |  0.001 |  2.381 |  0.027 |  0.169 |  0.867 |  2.193 |  0.04 |  1.472 |  0.156 |  3.608 |  0.002 |  4.146 |  0.0001 |  2.008 |  0.058 |  1.48 |  0.154
|  Q3 |  0.342 |  0.736 |  0.775 |  0.447 |  -0.439 |  0.665 |  0.706 |  0.489 |  -0.644 |  0.527 |  1.337 |  0.196 |  0.0 |  1.0 |  -0.338 |  0.739 |  1.513 |  0.146 |  2.665 |  0.015
Table 12. T-test results from qualitative evaluation

  5. (5)

A subjective assessment of the patterns is given in Tables 13, 14, and 15, for
the three patterns, done by manually examining the dominant topics covered in
the top-100 articles for each sub-class.

|  |  Agriculture |  Social welfare |  Environment |  Industrialization |  Politics and social activities
---|---|---|---|---|---|---
DocTag2Vec method |  Slow growing Unemployment districts |  • Food godowns • Policy execution • Heavy rainfall |  • Irregularities in job scheme • Compensation to killed • Policy implementation reviews |  • Pitch for quality education • Springs under threat • Review power pyre |  • Gang wars, Thefts • Free wifi • Growth problems |  • Trade Fair • Healthcare promise • Jail Raid
|  Fast growing Unemployment districts |  • Production improvement • Farmer Suicide • Rain & Floods |  • Health Fair • Development projects • Industrial progress |  • Industrial fuel • Wildlife protection • Air purifiers |  • Biomining project • Fine for burning • Air pollution |  • Curb crime • Textile celebrations • Farming experience
TFIDF based selection |  Slow growing Unemployment districts |  • Political events • Power plants • Dalit issues |  • Crimes against women • Digital India • Financial assistance |  • Free wifi • Toilets • Environment Day |  • Holy Festival • Illegal mining • Ailing industry |  • Riots • Deaths • Monument protection
|  Fast growing Unemployment districts |  • Floods • Health risks • Pesticides |  • MGNREGA • Investment activities • Education problems |  • Air Pollution • Hunter activities • Clean Ganga |  • Technology • Cyclone recovery • Industrial growth |  • Bad air quality • Murders • Democracy crisis
LDA |  Slow growing Unemployment districts |  • Agri production • Development packages • Political march |  • Air connectivity • Plantations • Rural development |  • Highway alignment • Development project • Tiger Reserve |  • Agitation • Farming • Chit fund scam |  • Health camp • Unlawful activities • Riots
|  Fast growing Unemployment districts |  • Farmer death • Seed subsidy • Awards |  • Agri business • MGNREGA • Mass marriage |  • NGO activities • Suicide • Zoo animals |  • Cleanliness drive • Property stealing • Deaths |  • Riots • Infra development plan • Plastic ban
Table 13. Subjective assessment results for Pattern 1 |  |  Agriculture |  Social welfare |  Environment |  Industrialization |  Politics and social activities
---|---|---|---|---|---|---
DocTag2Vec method |  Slow growing Agricultural districts |  • Rain & Floods • Farmer deaths • Arts expo |  • Farmer politics • Pending wage payments • Raise in air fare |  • Fuel adulteration • Rapes • Pollution |  • Power staff strike • Bus service suspended • Sanitation drive |  • Protests • Violence & Crimes • Art & Design
|  Fast growing Agricultural districts |  • Crop loss compensation • Hike in power tariff • Best practices for farmers |  • Conservation projects • Plantation scheme • Intellectual property meet |  • Protest against poaching • River protection petition • Wild-life habitat |  • Mining activities • Milk price hike • Transportation activites |  • Tech Summit • Celebrations • Elections
TFIDF based selection |  Slow growing Agricultural districts |  • Food prices • Water conservation • Better harvest |  • Job scheme • Boost agriculture • MGNREGA dilution |  • Forest degradation • Tiger reserve • Waste water |  • Violence • Smart apps • Coal mine |  • Train services • Cow violence • Liquor
|  Fast growing Agricultural districts |  • Water industry • Fodder substitute • Drought |  • Aviation policy • Business expo • Air connectivity |  • Infra project • Awareness on afforestation • Fight against pollution |  • Income Tax raids • Water crisis • Mining activities |  • Narcotics • Protests • Bullet train
LDA |  Slow growing Agricultural districts |  • Monsoon • Farm crisis • Price rise |  • Village development • Skill programme • Water project |  • Tourism • Killings • Reservoirs overflow |  • Free wifi • Illegal mining • Protest march |  • Murder • Suspension • Entrance exam
|  Fast growing Agricultural districts |  • Hill highway • MGNREGA • Drought |  • Rail budget • Development MoUs • Taxation |  • Tourist spot • Road project • Textile activities |  • Waste segregation • Telecom firms • Blast suspect |  • Road accident • Health check-up • Arrests
Table 14. Subjective assessment results for Pattern 2 |  |  Agriculture |  Social welfare |  Environment |  Industrialization |  Politics and social activities
---|---|---|---|---|---|---
DocTag2Vec method |  Slow growing Non-Agricultural districts |  • Rain & Floods • Farmer deaths • Arts expo |  • Folk programmes • Awareness training • Pension scheme |  • Plantation drive • Poor air quality • Clean Ganga cursade |  • Demonetization • Power crisis • Illegal mining |  • Elections • Violence • Politics
|  Fast growing Non-Agricultural districts |  • Farmers immolation • Capital city project • Strike warning |  • Air link scheme • Anti-tank missiles • Firm activities |  • Production plant • Pollution spike • Wildlife tourists |  • Economy revival • Infra growth • Revenue generation |  • Modern economy threat • Hacking • Politics
TFIDF based selection |  Slow growing Non-Agricultural districts |  • Air quality • Demonetisation • Elections |  • Connectivity scheme • Debt waiver scheme • Transportation |  • Child rights • Air pollution • Urban planning |  • GST impact • CEAT company • Co-living spaces |  • Accidential deaths • Protest activities • Politics
|  Fast growing Non-Agricultural districts |  • Dam Safety Bill • Pollution • Rainfalls |  • Food prices • Policy change • National meet |  • Protest site • Influence of rivers • Politics |  • PM speeches • Criminal activities • Banking sector |  • Cricket • Election results • Underworld crimes
LDA |  Slow growing Non-Agricultural districts |  • River link • Medical University • Drains |  • E-toilets • Consulting firms • Housing scheme |  • Pollution • Green plan • LPG scheme |  • Illegal mining • Telecom providers • Insurance firms |  • Disputes • Rain • Arrests
|  Fast growing Non-Agricultural districts |  • Suicide • Drought • Food bill |  • Thermal unit • Fund release • Tourism |  • Mining • Youth Fest • Debt markets |  • Telcos • Drug prices • Bank norms |  • Crime • Health programme • Sensex
Table 15. Subjective assessment results for Pattern 3 Collection | Questions
---|---
Agriculture |  • Are the articles depicting pace of growth i.e. slow vs fast in agriculture sector like slow pace is describing about floods, monsoon, droughts, suicides while fast is describing about compensations, crop prices, relief schemes etc? • Are the articles relevant to the sub-classes i.e. are they discussing about farming activities, crops, compensations to farmers etc? • Are the topics discussed in sub-classes different from each other?
Social welfare |  • Are the articles depicting pace of growth i.e. slow vs fast in terms of development as slow ones are talking about failure in development schemes, unemployment and fast is talking about success of schemes and new schemes? • Are the articles relevant to the sub-classes i.e. they are talking about job schemes, investments, development activities? • Are the topics discussed in sub-classes different from each other?
Environment |  • Are the articles depicting pace of growth i.e. slow vs fast in terms of environment as slow ones are talking about environmental problems as pollution, animal killing while fast is talking about activities for wildlife protection, prevention of poaching activities? • Are the articles relevant to the sub-classes i.e. slow ones are talking about environmental and wildlife problems while fast ones are talking about steps towards their protection? • Are the topics discussed in sub-classes different from each other?
Industrialization |  • Are the articles depicting pace of growth i.e. slow vs fast in terms of industrialization as slow ones are talking about problems in industry, failed investments while fast ones are focusing on development activities in industry as revenue generation, trades etc? • Are the articles relevant to the sub-classes i.e. slow ones are focusing on problems related to different industrial sectors while fast ones are inclined towards industrial development, investment, schemes? • Are the topics discussed in sub-classes different from each other?
Politics and social activities |  • Are the articles depicting pace of growth i.e. slow vs fast in terms of lifestyle like slow ones are focusing on women problems, violence, crimes while fast ones are focusing on celebrations, fight against crimes, special cells ? • Are the articles relevant to the sub-classes i.e. slow is talking about crimes, rapes, arrests while fast is talking about police activities against crime, law and order, celebrations etc? • Are the topics discussed in sub-classes different from each other?
Table 16. Questionnaire for qualitative review


# On distribution of the depth index on perfect matchings

Yonah Cherniavsky  and  Yuval Khachatryan-Raziel  Ariel University, Israel
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study the restriction of depth index statistic on the set of perfect
matchings. In particular, we provide additional combinatorial description of
the statistic for perfect matchings and calculate the generating polynomial.
The main result of the present short paper is that the depth index on perfect
matchings is equidistributed with the rank function of the Bruhat order.

##  1\. Introduction

This paper studies the distributions of the depth index statistics on the set
of perfect matchings in $PM_{2n}$. The depth-index of a set partition, was
introduced and studied in [2] by the first author together with Can. In [3]
together with Rubey they established connection to another statistic of set
partitions and provided a combinatorial interpretation of these two
statistics. As was shown in [2], the depth index statistic is equal to the
rank function of a certain graded EL-shellable order on $\Pi_{n}$, the set
partions $[n]$. That order is isomorphic to the Bruhat-Chevalley-Renner order
on the upper-triangular matrices, and thus, the depth index appears to be
equal to the dimension of a certain matrix variety. The present paper studies
the restriction of the depth index to the set of partitions with blocks of
size 2, which can be interpreted as perfect matchings on the set $[2n]$.

The set of perfect matchings is a classical combinatorial object that was
studied in various contexts. Perfect matchings, when viewed as fixed-point
free involutions on $[2n]$ have a natural poset structure – namely, the
restriction of the strong Bruhat order on $\mathfrak{S}_{2n}$. This poset was
studied in detail in [1]. It was shown that the length function (let us denote
it $\ell$) of Bruhat order on $PM_{2n}$ is $[2n-1]_{q}!!$, and it equals to
the length function of the poset that was studied by Deodhar and Srinivasan in
[4].

It can be easily checked (for example, for the case $n=3$), that the classical
strong Bruhat order on $PM_{2n}$ and the order on $PM_{2n}$ induced from the
order on set partitions studied in [2], are different. However, we prove in
this paper that their generating polynomials satisfy the following identity.

$\sum_{\pi\in PM_{2n}}q^{\mathrm{t}(\pi)}=q^{\binom{n+1}{2}}[2n-1]_{q}!!$

.

##  2\. Background

$1$$2$$3$$4$$5$$6$$7$$8$ Figure 2.1. The arc diagram of the set partition
$\pi=1378|26|45$.

######  Definition 2.1.

A collection of subsets $\left\\{S_{i}\right\\}_{i}\in I$ of a set $S$ is said
to be a set partition of $S$ if the sets $S_{i}$ are mutually disjoint and we
have $\bigcup\limits_{i\in I}S_{i}=S$. The sets $S_{i}$ are called the blocks
of the partition. For $n>0$, the set of all set partitions of $[n]$ is denoted
by $\Pi_{n}$. We will often drop set parentheses and commas and just put
vertical bars between blocks. If $B_{1},\dots,B_{k}$ are the blocks of the set
partition $\pi\in\Pi_{n}$, then the standard form of $\pi$ is defined as
$B_{1}|\dots|B_{k}$ where we assume that $\min B_{i}<\min B_{i+1}$ for every
$1\leq i\leq k$, and the elements of every block are listed in increasing
order. For example, $\pi=1378|26|45$ is a set partition in $\Pi_{8}$.

######  Definition 2.2.

Let $\pi\in\Pi_{n}$ be a set partition. The minimal elements of the blocks are
the openers, and the maximal elements of are the closers of $\pi$.

######  Example 2.3.

The partition $\pi=1378|26|45$ in $\Pi_{8}$ has openers $1,4,7$ and closers
$5,6,8$.

######  Definition 2.4.

Let $\pi\in\Pi_{n}$ be a set partition with the standard form
$B_{1}|\dots|B_{k}$. The arc diagram of $\pi$ is obtained by placing labels
$1,\dots,n$ on horizontal line and connecting consecutive elements of each
block by arcs, such the arcs do not cross each other if it is possible to draw
them without a crossing as in Figure 2.1.

######  Definition 2.5.

Let $\pi\in\Pi_{n}$ be a set partition of $[n]$. The extended arc diagram of
$\pi$ is obtained from the arc diagram of $\pi$ by adding a half-arc
$(-\infty,i)$ from far left to each opener $i$ and a half-arc $(j,\infty)$
from each closer $j$ to far right. These arcs are drawn in such a way that the
half-arcs to the left do not cross, and half-arcs to the right do not cross
either. See Figure 2.2 for an example.

$1$$2$$3$$4$$5$$6$$7$$8$ Figure 2.2. The extended arc diagram of the set
partition $\pi=1378|26|45$.

######  Remark 2.6.

The correspondence between set partitions of $\pi$ and their arc diagrams is a
bijection, and in the sequel we will identify the partition $\pi$ of $[n]$
with its arc diagram $A(\pi)$.

The intertwining number was introduced in [5]. Using the extended diagram
representations of set partitions, it can be defined equivalently as in [3].

######  Definition 2.7.

Let $A\in\Pi_{n}$ be a set partition of $[n]$. Two (generalized) arcs $(i,j)$
and $(k,l)$ in the extended arc diagram of $A$ are said to cross in $\pi$ if
$i<k<j<l$. The number of crossings in the extended arc diagram of $A$ is
called the intertwining number of $A$ and denoted
$\operatorname{\mathtt{i}}(A)$. (See [3] for further details and references).

A closely related statistic to intertwining number is the depth-index, defined
as follows.

######  Definition 2.8.

Let $A\in\Pi_{n}$ be a set partition of $[n]$. Denote by
$\operatorname{\text{A}rcs}(A)$ the set of arcs of $A$. For every $1\leq v\leq
n$, the depth of vertex $v$, denoted by $\operatorname{depth}(v)$ is the
number of arcs $(i,j)\in Arcs(A)$ with $1\leq i<v<j\leq n$. For every
$\alpha=(u,v)\in\operatorname{\text{A}rcs}(A)$, the depth of the arc $\alpha$
is the number of arcs $(i,j)\in\operatorname{\text{A}rcs}(A)$ with $1\leq
i<u<v<j\leq n$.

The depth index $\operatorname{\mathtt{t}}(A)$ of $A$ is

$\operatorname{\mathtt{t}}(A)=\sum_{i=1}^{|\operatorname{\text{A}rcs}(A)|}(n-i)-\sum_{v=1}^{n}\operatorname{depth}(v)+\sum_{\alpha\in\operatorname{\text{A}rcs}(A)}\operatorname{depth}(\alpha).$

######  Remark 2.9.

Intuitively, the depth of a vertex or an arc is the number of arcs obove it in
the arc diagram.

Intertwining number and depth-index satisfy the following identity.

######  Theorem 2.10.

[3, Theorem 1] For any set partition $A\in\Pi_{n}$, we have

$\operatorname{\mathtt{t}}(A)+\operatorname{\mathtt{i}}(A)=\binom{n}{2}.$

We are interested in the distribution of the depth-index on perfect matchings
on $[2n]$.

######  Example 2.11.

For partition $\pi=1378|26|45$ as in Figure 2.2,
$\operatorname{\mathtt{i}}(\pi)$ is the number of crossings in the extended
arc diagram, thus we have $\operatorname{\mathtt{i}}(\pi)=8$. The number of
arcs in $\pi$ is $5$, the only arc of nonzero depth is $(4,5)$ with
$\operatorname{depth}((4,5))=2$, and we have

$\displaystyle\operatorname{depth}(2)=\operatorname{depth}(3)=\operatorname{depth}(6)=1,$
$\displaystyle\operatorname{depth}(4)=\operatorname{depth}(5)=2,$
$\displaystyle\operatorname{depth}(1)=\operatorname{depth}(8)=0.$

Hence, by definition we have

$\displaystyle\operatorname{\mathtt{t}}(\pi)$
$\displaystyle=\sum_{i=1}^{8}(8-i)-\sum_{v=1}^{8}\operatorname{depth}(v)+\sum_{\alpha\in\operatorname{\text{A}rcs}(\pi)}\operatorname{depth}(\alpha)$
$\displaystyle=(7+6+5+4+3)-(1+2+2+1+1)+2=20.$

It is easily verified that
$\operatorname{\mathtt{i}}(\pi)+\operatorname{\mathtt{t}}(\pi)=28=\binom{8}{2}$.

######  Definition 2.12.

A perfect matching on $[2n]$ is a set partition of $[2n]$ in which every block
has size $2$, or equivalently an arc diagram in which every vertex touches
exactly one arc. The set of perfect matchings on $[2n]$ will be denoted by
$PM_{2n}$.

######  Remark 2.13.

By interpreting the arcs in the arc diagram of a perfect matching $\pi\in
PM_{2n}$ as transpositions in $\mathfrak{S}_{2n}$, we obtain a fixed point
free involution in $\mathfrak{S}_{2n}.$ In fact, this is a bijection from
perfect matchings to fixed point free involutions in $\mathfrak{S}_{2n}$. In
the rest of the text, every perfect matching $\pi\in PM_{2n}$ will be
identified with its corresponding fixed point free involution.

In particular, it was proved that the length functions for Bruhat-Chevalley
order and the order described in [4] are equal (see [1, Corollary 4]). We
denote this function by $\ell$.

We proceed to describe this function in terms of [4].

######  Definition 2.14.

Let $\pi\in PM_{2n}$ be a perfect matching on $[2n]$. The crossing number
$\operatorname{\mathtt{c}}(\pi)$ of $\pi$ is the number of arcs
$(i,j),(k,l)\in\operatorname{\text{A}rcs}(\pi)$ that cross. The span of an arc
$\alpha=(i,j)\in\operatorname{\text{A}rcs}(\pi)$ is
$\operatorname{\mathtt{s}pan}(\alpha)=j-i-1$, which is the number of dots
below the arc.

######  Theorem 2.15.

[4, Theorem 1.3]

  1. 1.

For every $\pi\in PM_{2n}$,

$\ell(\pi)=\sum_{\alpha\in\operatorname{\text{A}rcs}(\pi)}\operatorname{\mathtt{s}pan}(\alpha)-\operatorname{\mathtt{c}}(\pi)$

  2. 2.

The generating polynomial of $\ell$ in $PM_{2n}$ is

$L_{PM_{2n}}(q)=\sum_{\pi\in PM_{2n}}q^{\ell(\pi)}=[2n-1]_{q}!!.$

Recall, that $[2n-1]_{q}!!$ is the $q$-analogue of the double factorial, that
is defined as follows

$[2n-1]_{q}!!=\prod_{i=1}^{n}\frac{1-q^{2i-1}}{1-q}.$

##  3\. Main result

We prove the following relation between the depth-index and the length
function of Bruhat-Chevalley order on fixed point free involutions.

######  Theorem 3.1.

The depth index on $PM_{2n}$ is distributed as follows:

$T_{PM_{2n}}(q)=\sum_{\pi\in
PM_{2n}}q^{\operatorname{\mathtt{t}}(\pi)}=q^{\binom{n+1}{2}}[2n-1]_{q}!!.$

We examine the length function and depth index restricted to perfect matchings
first. The definitions that follow are valid for arcs in the regular arc
diagrams and not extended ones.

######  Definition 3.2.

[6] Let $\pi\in PM_{2n}$. We say that two arcs
$e=(i,j),f=(k,l)\in\operatorname{\text{A}rcs}(\pi)$ form:

  * (i)

a crossing with $e$ as initial edge if $i<k<j<l$;

  * (ii)

a nesting with $e$ as initial edge if $i<k<l<j$;

  * (iii)

a alignment with $e$ as initial edge if $i<j<k<l$.

The number of crossings, nestings and alignments will be denoted by
$\operatorname{\mathtt{c}r},\operatorname{\mathtt{n}e}$ and
$\operatorname{\mathtt{a}l}$ respectively.

$i$$k$$j$$l$$i$$k$$l$$j$$i$$j$$k$$l$$(i)$$(ii)$${(iii)}$ Figure 3.1. Crossing,
nesting and alignment of two edges

######  Observation 3.3.

For every perfect matching $\pi$ in $PM_{2n}$ there are exactly $\binom{n}{2}$
pairs
$(e,f)\in\operatorname{\text{A}rcs}(\pi)\times\operatorname{\text{A}rcs}(\pi)$
with $e=(i,j)$, $f=(k,l)$ and $i<k$. Therefore, we have

$\operatorname{\mathtt{c}r}(\pi)+\operatorname{\mathtt{n}e}(\pi)+\operatorname{\mathtt{a}l}(\pi)=\binom{n}{2}$

for every $\pi\in PM_{2n}$.

By changing the summation order, we obtain the following identities:

######  Observation 3.4.

For every $\pi\in PM_{2n}$, we have:

  1. 1.

$\sum_{v=1}^{2n}depth(v)=\sum_{\alpha\in\operatorname{\text{A}rcs}(\pi)}\operatorname{\mathtt{s}pan}(\alpha)$;

  2. 2.

$ne(\pi)=\sum_{\alpha\in\operatorname{\text{A}rcs}(\pi)}depth(\alpha)$.

######  Definition 3.5.

Let $\pi\in PM_{2n}$. The sum
$\sum_{v=1}^{2n}depth(v)=\sum_{\alpha\in\operatorname{\text{A}rcs}(\pi)}\operatorname{\mathtt{s}pan}(\alpha)$
is called the total vertex depth of $\pi$ and denoted by
$\operatorname{\mathtt{t}vd}(\pi)$.

By combining Observations 3.3 and 3.4 we obtain the following equalities for
the depth index.

######  Observation 3.6.

For every $\pi\in PM_{2n}$ we have the following equality:

$\displaystyle\operatorname{\mathtt{t}}(\pi)$
$\displaystyle=\sum_{i=1}^{|\operatorname{\text{A}rcs}(\pi)|}(n-i)-\operatorname{\mathtt{t}vd}(\pi)+\operatorname{\mathtt{n}e}(\pi)$
$\displaystyle=n^{2}+\binom{n}{2}-\operatorname{\mathtt{t}vd}(\pi)+\binom{n}{2}-\operatorname{\mathtt{c}r}(\pi)-\operatorname{\mathtt{a}l}(\pi)$
$\displaystyle=n^{2}+2\binom{n}{2}-\operatorname{\mathtt{t}vd}(\pi)-\operatorname{\mathtt{c}r}(\pi)-\operatorname{\mathtt{a}l}(\pi).$

We are interested in expressing the term $\operatorname{\mathtt{t}vd}(\pi)$
using $\operatorname{\mathtt{c}r}(\pi),\operatorname{\mathtt{n}e}(\pi)$ and
$\operatorname{\mathtt{a}l}(\pi$).

Recall the definition of the intertwining number (Definition 2.7)- the number
of crossings in the extended arc diagram.

######  Lemma 3.7.

Let $\pi\in PM_{2n}$. Then
$\operatorname{\mathtt{i}}(\pi)=3\operatorname{\mathtt{c}r}(\pi)+2\operatorname{\mathtt{n}e}(\pi)+\operatorname{\mathtt{a}l}(\pi)$.

###### Proof.

Let $e=(i,j)$ and $f=(k,l)$ with $i<k$. If the pair $(e,f)$ is a nesting - it
adds exactly two crossings to the the extended arc diagram: $(-\infty,k)$ with
$(i,j)$ and $(i,j)$ with $(l,\infty)$. If the pair is a crossing, it adds 3
crossings in the extended arc diagram: $e$ with $f$, $(-\infty,k)$ with $e$,
and $f$ with $(j,\infty)$. Finally, if $(e,f)$ is an alignment, we get a
single crossing $(j,\infty)$ with $(k,-\infty)$. By summing everything, we get
the desired result. ∎

By using Theorem 2.10 one can express $\operatorname{\mathtt{t}vd}(\pi)$ in
terms of $\operatorname{\mathtt{n}e}(\pi),\operatorname{\mathtt{a}l}(\pi)$ and
$\operatorname{\mathtt{c}r}(\pi)$.

######  Lemma 3.8.

For every $\pi\in PM_{2n}$ we have:

$\operatorname{\mathtt{t}vd}(\pi)=2\binom{n}{2}-2\operatorname{\mathtt{a}l}(\pi)=2(\operatorname{\mathtt{c}r}(\pi)+\operatorname{\mathtt{n}e}(\pi)).$

###### Proof.

By Theorem 2.10, for $\pi\in PM_{2n}$, and using Observation 3.6 and Lemma 3.7
we obtain

$\displaystyle\binom{2n}{2}$
$\displaystyle=\operatorname{\mathtt{t}}(\pi)+\operatorname{\mathtt{i}}(\pi)$
$\displaystyle=n^{2}+2\binom{n}{2}-\operatorname{\mathtt{t}vd}(\pi)-\operatorname{\mathtt{c}r}(\pi)-\operatorname{\mathtt{a}l}(\pi)+3\operatorname{\mathtt{c}r}(\pi)+2\operatorname{\mathtt{n}e}(\pi)+\operatorname{\mathtt{a}l}(\pi)$
$\displaystyle=n^{2}+2\binom{n}{2}-\operatorname{\mathtt{t}vd}(\pi)+2\operatorname{\mathtt{c}r}(\pi)+2\operatorname{\mathtt{n}e}(\pi).$

By rearranging the equation we obtain:

$\displaystyle\operatorname{\mathtt{t}vd}(\pi)$
$\displaystyle=n^{2}+2\binom{n}{2}-\binom{2n}{2}+2\operatorname{\mathtt{c}r}(\pi)+2\operatorname{\mathtt{n}e}(\pi)$
$\displaystyle=2\operatorname{\mathtt{c}r}(\pi)+2\operatorname{\mathtt{n}e}(\pi)$
$\displaystyle=2\binom{n}{2}-2\operatorname{\mathtt{a}l}(\pi),$

as desired. ∎

We can now express $\ell(\pi)$ and $\operatorname{\mathtt{t}}(\pi)$ using
$\operatorname{\mathtt{n}e}(\pi)$ and $\operatorname{\mathtt{c}r}(\pi)$.

######  Corollary 3.9.

For every $\pi\in PM_{2n}$ we have:

$\ell(\pi)=\operatorname{\mathtt{c}r}(\pi)+2\operatorname{\mathtt{n}e}(\pi)$

and

$\operatorname{\mathtt{t}}(\pi)=n^{2}+\binom{n}{2}-2\operatorname{\mathtt{c}r}(\pi)-\operatorname{\mathtt{n}e}(\pi)$

.

###### Proof.

Substitute $\operatorname{\mathtt{t}vd}(\pi)$ with
$2\operatorname{\mathtt{n}e}(\pi)+\operatorname{\mathtt{c}r}(\pi)$ in the
expressions
$\ell(\pi)=\operatorname{\mathtt{t}vd}(\pi)-\operatorname{\mathtt{c}r}(\pi)$
and
$\operatorname{\mathtt{t}}(\pi)=n^{2}+\binom{n}{2}-\operatorname{\mathtt{t}vd}(\pi)+\operatorname{\mathtt{n}e}(\pi)$.
∎

We are ready to prove the main theorem. In order to do it, we rely on the
following facts.

######  Theorem 3.10 (Theorem 1.2).

[6] There exists an involution $\phi:PM_{2n}\to PM_{2n}$ that preserves the
number of alignments and exchanges the number of crossings and nestings. In
other words, for each $\pi\in PM_{2n}$ we have

$\operatorname{\mathtt{a}l}(\phi(\pi))=\operatorname{\mathtt{a}l}(\pi),\operatorname{\mathtt{c}r}(\phi(\pi))=\operatorname{\mathtt{n}e}(\pi),\operatorname{\mathtt{n}e}(\phi(\pi))=\operatorname{\mathtt{c}r}(\pi).$

######  Fact 3.11.

The polynomial $[2n-1]_{q}!!$ has degree $n^{2}-n$ and is palindromic. Namely,
the coefficients of $q^{r}$ and $q^{n^{2}-n-r}$ in $[2n-1]_{q}!!$ are equal
for each $0\leq r\leq n^{2}-n$.

######  Corollary 3.12.

There exists a bijection $\psi:PM_{2n}\to PM_{2n}$ such that

$\ell(\psi(\pi))=n^{2}-n-\ell(\pi)$

for every $\pi\in PM_{2n}$.

###### Proof.

By Theorem 2.15, the generating polynomial on $PM_{2n}$ of the length function
is $[2n-1]_{q}!!$. Together with Fact 3.11, it implies that for every $0\leq
r\leq n^{2}-n$, the number of elements of $PM_{2n}$ with length $r$ is equal
to the number of elements with length $n^{2}-n-r$, as desired. ∎

We proceed to prove the main theorem.

###### Proof of Theorem 3.1.

By Corollary 3.9 and Theorem 3.10 we have:

$\ell(\phi(\pi))=2\operatorname{\mathtt{n}e}(\phi(\pi))+\operatorname{\mathtt{c}r}(\phi(\pi))=2\operatorname{\mathtt{c}r}(\pi)+\operatorname{\mathtt{n}e}(\pi).$

By applying Corollary 3.9 again we obtain:

$\operatorname{\mathtt{t}}(\pi)+\ell(\phi(\pi)=n^{2}+\binom{n}{2}.$

Using $\psi$ from Corollary 3.12 and the relation
$\ell(\phi(\pi))=n^{2}-n-\ell(\pi)$ we obtain:

$\operatorname{\mathtt{t}}(\pi)+n^{2}-n-\ell(\psi(\phi(\pi)))=n^{2}+\binom{n}{2}.$

Finally, by rearranging the equation, we obtain
$t(\pi)=\binom{n+1}{2}+\ell(\psi(\phi(\pi)))$. Since, $\psi\circ\phi$ is a
bijection, the statistics $\operatorname{\mathtt{t}}$ and
$\binom{n+1}{2}+\ell$ are equi-distributed, and the main theorem follows. ∎

By using the identity
$\operatorname{\mathtt{i}}(\pi)+\operatorname{\mathtt{t}}(\pi)=\binom{2n}{2}$
for every $\pi\in\Pi_{2n}$ we obtain a similar formula for the generating
polynomial of the intertwining number.

We rely on the following observation.

######  Fact 3.13.

If $p(q)$ is a palindromic polynomial in $q$ of degree $m$, then

$q^{m}p\left(\frac{1}{q}\right)=p(q)$

######  Corollary 3.14.

The generating polynomial of the intertwining number $I_{PM_{2n}}(q)$ is

$I_{PM_{2n}}(q)=\sum_{\pi\in
PM_{2n}}q^{\operatorname{\mathtt{i}}(\pi)}=q^{\binom{n}{2}}[2n-1]_{q}!!.$

###### Proof.

By Theorem 2.10 we have

$\sum_{\pi\in PM_{2n}}q^{\operatorname{\mathtt{i}}(\pi)}=\sum_{\pi\in
PM_{2n}}q^{\binom{2n}{2}-\operatorname{\mathtt{t}}(\pi)}=q^{\binom{2n}{2}}\sum_{\pi\in
PM_{2n}}\frac{1}{q^{-\operatorname{\mathtt{t}}(\pi)}},$

By applying Theorem 3.1, we obtain

$q^{\binom{2n}{2}}\sum_{\pi\in
PM_{2n}}\frac{1}{q^{-\operatorname{\mathtt{t}}(\pi)}}=q^{\binom{2n}{2}}\frac{1}{q^{\binom{n+1}{2}}}[2n-1]_{\frac{1}{q}}!!.$

By Facts 3.13 and Fact 3.11, $q^{n^{2}-n}[2n-1]_{\frac{1}{q}}!!=[2n-1]_{q}!!$,
and therefore

$\displaystyle
q^{\binom{2n}{2}}\frac{1}{q^{\binom{n+1}{2}}}[2n-1]_{\frac{1}{q}}!!$
$\displaystyle=q^{\binom{2n}{2}}\frac{1}{q^{\binom{n+1}{2}}q^{n^{2}-n}}[2n-1]_{q}!!$
$\displaystyle=q^{\binom{2n}{2}-\binom{n+1}{2}-n^{2}-n}[2n-1]_{q}!!=q^{\binom{n}{2}}[2n-1]_{q}!!,$

as desired. ∎

Acknowledgements. We are grateful to Ron Adin, Yuval Roichman and Mahir Bilen
Can for helpful discussions.

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