# CORAE: A Tool for Intuitive and Continuous Retrospective Evaluation of
Interactions

Michael J. Sack1, Maria Teresa Parreira1, Jenny Fu1, Asher Lipman1, Hifza
Javed2, Nawid Jamali2, and Malte Jung1 1Cornell University, 2Honda Research
Institute USA, Inc.

###### Abstract

This paper introduces CORAE, a novel web-based open-source tool for COntinuous
Retrospective Affect Evaluation, designed to capture continuous affect data
about interpersonal perceptions in dyadic interactions. Grounded in behavioral
ecology perspectives of emotion, this approach replaces valence as the
relevant rating dimension with approach and withdrawal, reflecting the degree
to which behavior is perceived as increasing or decreasing social distance. We
conducted a study to experimentally validate the efficacy of our platform with
24 participants. The tool’s effectiveness was tested in the context of dyadic
negotiation, revealing insights about how interpersonal dynamics evolve over
time. We find that the continuous affect rating method is consistent with
individuals’ perception of the overall interaction. This paper contributes to
the growing body of research on affective computing and offers a valuable tool
for researchers interested in investigating the temporal dynamics of affect
and emotion in social interactions.

###### Index Terms:

Affective computing, interpersonal perception, annotation tool,continuous
affect, human-computer interaction

##  I Introduction

Affect is a dynamic phenomenon. Observable behavior, subjective experience,
and physiology all dynamically evolve across time [24]. In interactions,
affective dynamics co-evolve with those of other interactants [2], correlating
in ways that are yet to be fully explored. Understanding these temporal
dynamics requires continuous data streams.

While a growing body of work has addressed strategies and tools to capture
affect data over time across physiology [20], subjective experience [33, 8],
and observable behavior [22, 4], we lack continuous data about how affective
perceptions of others develop dynamically over time.

Retrospective analysis is an established method to collect continuous data
regarding affect [5, 6, 29, 27]. Following this approach, participants are
typically video-recorded during an emotion-eliciting event and are later asked
to continuously rate how they felt while watching the recording [33]. This
method relies on the phenomenon that individuals often re-experience emotions
when reliving a situation [16]. Traditionally, retrospective analysis has
focused on collecting continuous data about subjective experience. For
example, it was used prominently in couples research by Gottman and Levenson
[18]. In this paper, we expand on these works by introducing an approach and a
tool that enables researchers to collect continuous affect data about
interpersonal perceptions.

Our approach is grounded in a behavioral ecology perspective of emotion, which
posits that emotional expressions are primarily social tools used to influence
and learn about others [13, 7, 38]. In line with this perspective, rather than
capturing valence when rating interactions, we focus on a dimension of
approach and withdrawal, i.e., the degree to which behavior is seen as
increasing or decreasing social distance [23, 3, 1].

Figure 1: Annotation dashboard for CORAE version 0.15a. After interacting with
another individual, participants are asked to retrospectively evaluate how the
other person came across by reviewing the recording of the interaction with
only the video from the other participant and audio of both.

We introduce CORAE, a novel browser-based tool for COntinuous Retrospective
Affect Evaluation. This intuitive tool allows participants to retrospectively
rank how another interactant came across immediately following an interaction,
thus allowing us to capture interpersonal affective perceptions rather than
feelings or affective state inferences. In other words, our system allows us
to capture data about how people perceive each other emotionally continuously
over time. We make this tool publicly available and test it in a dyadic
interaction context, drawing insights about how interpersonal dynamics evolve
over time.

Our contributions include: first, an approach that extends existing affect
rating approaches [33] to capture continuous self-report data about
interpersonal affective perceptions. Second, a novel web-based tool (CORAE)
that allows for such data to be captured easily and reliably. Third, an
experimental study where dyads of participants interact and retrospectively
rate each other in terms of social distance, which provides evidence that the
self-report data captured with our system aligns with interpersonal
perceptions people form of others.

##  II Related Work

###  II-A Measuring affect in interactions

A growing body of research in affective computing is investigating how to
develop and validate reliable measures of affect. Traditionally, affect is
represented in two main ways: categorical or dimensional. The first
categorizes affective states into discrete emotions (e.g., happiness, sadness)
[9, 10, 25], while the second distinguishes affective states along dimensions
such as valence or arousal [28, 34]. Dimensional approaches have been shown to
better capture the complexity and nuance of emotional experiences, and can be
used to study affective states that do not fit neatly into discrete categories
[34]. To capture these affective states, continuous representations have been
popularized [19, 30], as they allow for an understanding of how humans
aggregate affect information across time, unveiling regions of “emotional
saliency” which may be pivotal to assessing the emotional experience [30].

There is also growing interest in the social functions of emotions (e.g.,
[38]). Emotional expressions are increasingly recognized as social tools that
people use to influence others and learn about them [13, 7]. From such a
behavioral ecology perspective [13, 7], the expressive meaning of emotions is
not pre-determined by specific behavior patterns or morphologies such as
facial muscle movements or voice tone patterns, but rather constructed in
interaction through a process of ”affective grounding” [23]. Affective
grounding posits that participants of an interaction continuously coordinate
on affect to build shared understanding about how behavior should be
interpreted affectively and how interaction participants are socially
positioned towards each other. This perspective therefore uses “social
distance” as the relevant rating dimension, evaluating the degree to which
social behaviors result in approach and withdrawal [23, 3, 1, 26]. Truly
understanding how affective grounding is built, however, requires the
collection of interpersonal social distance measures [23, 3, 1], continuously.
To the best of our knowledge, this is the first work which addresses
retrospective interpersonal affect annotation.

###  II-B Tools for coding affect

Despite significant efforts to develop automated methods for affect
recognition [12, 39, 41], many studies still rely on human annotators to label
affective states in various modalities, such as speech, text, and images.
Human annotators provide a level of accuracy and nuance that is difficult to
achieve with automated methods, particularly when it comes to complex
emotional experiences that may not fit neatly into discrete categories.
Furthermore, human annotators can provide valuable insights into the
subjective experiences of individuals, enabling researchers to better
understand the cognitive and social factors that shape emotional responses.

Nonetheless, manual annotation brings about its own limitations, as several
factors hinder validity and reliability. For example, annotators’ experience
and personal perceptions, the representations chosen, and even the design of
the annotation tool [30] can affect the final labels. Consequently, a wide
variety of tools have been developed to address these challenges, including
different approaches for continuous affect annotation [15, 32, 40, 5, 6, 29,
27, 14].

Each of these existing tools affords unique research applications and differs
primarily in terms of complexity and accessibility. For example, solutions
such as FeelTrace [5], its successor GTrace [6], and AffectRank [40] each aim
to capture multiple dimensions of affect at once. However, these methods can
be cognitively demanding [29], which is why other approaches, including
RankTrace [27], PAGAN [29], and CARMA [14] focus more narrowly on a single
affective dimension. Further, the mode of annotation for these platforms
differ in their use of bounded [5, 6] and unbounded [27, 29] rating scales.
Additionally, these solutions differ in terms of their affordance of in-person
and remote data collection; of those mentioned above, all but PAGAN require
the collocation of researchers and participants. Conversely, web-based
solutions such as PAGAN may theoretically be deployed for remote data
collection in addition to in-person contexts.

##  III System Design

CORAE, related to the Latin word for “heart” and associated with the
figurative image of affect, was developed through an iterative design process
informed by numerous rounds of pilot testing and user feedback. Broadly, the
CORAE platform enables individuals to intuitively evaluate how a person’s
behavior is interpreted emotionally during interactions. In its current
iteration, the platform consists of two components, namely a frontend web
interface for participants and a command-line-driven backend to facilitate
project management for researchers. Section III-A details our motivations for
various design decisions and situates CORAE in the broader design space.
Section III-B details platform functionality implemented as of the
experimental validation of CORAE described in Section IV. Finally, Section
III-C discusses ongoing improvements made to the platform taking insights from
researcher and participant feedback. We make CORAE publicly available
111https://corae.org as a tool for other researchers.

###  III-A Design

Building upon existing tools, we sought to address features and applications
under-served by the broader design space and incorporate those most aligned
with an intuitive annotation experience. Due to their relative complexity,
many existing annotation tools impose barriers to their effective use. Robust
as such tools may be, they remain inaccessible to the greater population who
lack the necessary training or resources to use them.

fnum@@desciitemLayout:

We designed CORAE to be intuitive and visually minimal (see Figure 1) in its
presentation. CORAE is deployed in a web browser with the central focus being
a video of one’s co-interactant, staged for annotation. This is to ensure
participants are not distracted by their own image on the screen nor by other
visual elements on the platform. Brief instructions above the video player
describe keybindings to control the annotation dashboard (Spacebar to toggle
playback and Left and Right Arrows to control the slider), as well as a brief
description of the terms used for measuring interpersonal perception (Figure
2). In the released version of CORAE, the terms used and respective
descriptions will be easily customizable, enlarging the use potential for this
tool.

Figure 2: Instruction panel for CORAE version 0.15a. The panel includes a
description of available keybindings to control the annotation dashboard
(left), as well as the terms used for measuring interpersonal perception
(right).

An unobtrusive progress bar is displayed below the video player to inform
participants what proportion remains of their evaluation. Finally, below the
video player is displayed the annotation slider. A color gradient (from red to
green) enables participants to more intuitively understand the meaning of each
of the rating terms. The annotation bar is bounded and discretized (a total of
15 points, from $-7$ — Disagreeable — to $+7$ — Agreeable). As mentioned
above, this too will be customizable, as other scales may be interesting to
explore, namely Comfortability [26]. Participants may only change their rating
during video playback and are constrained by the platform to do it
‘continuously” (i.e., they cannot instantaneously change the rating from
Neutral ($0$) to Agreeable ($7$), but rather adjust to each value in
sequence).

fnum@@desciitemIntuitiveness:

By mitigating the need to train annotators on CORAE’s use, our platform
facilitates continuous retrospective annotation immediately following an
interaction. The immediacy of this evaluation allows for stronger salience of
affect compared to a delayed approach. Additionally, valuable time that might
have otherwise been required to train novice annotators may instead be
utilized for more productive ends such as data collection.

Melhart et al. [29] suggest that a lack of intuitiveness can constitute a
barrier for researchers such as in the case of tools like FeelTrace [5],
GTrace [6], CARMA [14], and RankTrace [27]. According to the Melhart, the
complexity of platform deployment may dissuade researchers from using that
platform. Melhart’s solution, PAGAN [29], is a centralized platform accessible
in theory to any interested researcher. In practice, however, the centralized
nature of such platforms renders them dependent upon the continued upkeep of a
central web server and database. Closed-source solutions further compound this
issue by offering little-to-no recourse for researchers who might otherwise
desire to deploy their own implementation locally or even to their own web
server. Our platform addresses both of these concerns through a well-
documented process for third-party deployment and the planned open-source
release of our code repository.

fnum@@desciitemAffective Dimension:

The complexity of annotating two or more concurrent affective dimensions
imposes a nontrivial cognitive demand upon the annotator [6]. This demand, in
turn, may diminish the salience of user annotation along both dimensions.
Similar to PAGAN [29] and others, our design aims to capture affect along a
single dimension.

fnum@@desciitemDistributed Participation:

Whereas existing solutions tend to rely heavily upon the collocation of
researchers and participants for in-person data collection (Section II-B), we
found this approach to be unnecessarily restrictive and rather limiting to the
potential recruitment of more diverse populations. Conversely, remote studies
are unable to guarantee consistency for factors such as participant system
specifications and environmental distractors. We acknowledge the value
afforded by both in-person and remote study formats and sought to develop a
tool capable of facilitating either. To this end, CORAE may be deployed
locally for in-person sessions as well as remotely for distributed
participation.

###  III-B Functionality

We focused our early development efforts on creating an intuitive and seamless
annotation experience for the user. To this end, we withheld several planned
features with the intention of their inclusion in a subsequent release of the
platform (Section III-C). When we refer to CORAE and its functionality in this
section, it is in reference to build 0.15a unless otherwise noted. The
features in this build were those we deemed critical for CORAE’s experimental
validation and ongoing user testing.

fnum@@desciiitemAnnotation Dashboard:

A unique URL is generated for each participant to access their instance of
CORAE’s annotation dashboard (Fig. 1). Upon accessing their instance,
participants are, by default, prompted to enter an identifier which is then
logged by the platform and associated with their session. This prompt may be
disabled by changing a parameter in the project template. CORAE’s annotation
dashboard contains three key components: the instruction panel, the video
player, and the annotation slider. Platform instructions may be altered in the
project template to suit the needs of a study. Dashboard elements responsively
scale to account for differences in viewport size and the relative aspect
ratio of uploaded media. The annotation slider is labeled at each extreme with
the dimension of affect being evaluated.

In terms of interaction, the dashboard affords two primary actions: slider
adjustment and playback control. Annotators may indicate their affect rating
by adjusting the slider using Left and Right Arrow during playback. This
affect rating by default is indicated using a continuous 15-point scale but
may be changed in the project template to suit any granularity. Although some
authors argue in favor of unbounded annotation [27, 29], we opted to bound
ratings as a form of affective grounding across sections. Further, CORAE
eliminates the need to hold input controls when adjusting ratings, which we
found to cause fatigue among participants. Playback may be paused and resumed
using Spacebar at any point. Our motivation for constraining interactability
to these two actions was to minimize cognitive demand and eliminate any
unnecessary sources of distraction during annotation.

fnum@@desciiitemData Logging:

Data is logged for a session in two ways: (1) by default, the mode for data
logging is set to predetermined intervals of one second, which may be adjusted
to any granularity; and (2) to ensure accuracy in the annotation method, CORAE
also logs data whenever a change in the rating occurs. Associated data points
are the slider position (rating), time code, and video frame (in the format
“SliderNumericalPosition”: “Hours:Minutes:Seconds:VideoFrame”), which are
logged in a JSON file. Given that video is recorded at a rate of 30 frames-
per-second, this allows for a resolution of up to $1/30$ s in the annotated
data streams.

###  III-C Release Version

We intend to continue the development of novel features that improve CORAE’s
overall utility. One of the most significant additions moving into the next
release of our platform is the introduction of a more robust project
management system. In the latest build of CORAE, project parameters may be
modified, staged, and published using the administration panel both prior to
and after deployment. Yet another improvement over our 0.15a release is a
server-side pipeline for logging participant data. This data management system
aggregates files generated by the platform within a predetermined directory on
the web server.

Finally, we note that allowing for the open-source modification of our
platform enables researchers and developers to tailor its features to their
specific needs beyond those we have anticipated. Further, the potential impact
of deprecation is lessened by a public release of CORAE’s source code. In the
event that we were to discontinue development, CORAE’s repository would remain
available to access and modify indefinitely.

##  IV Experimental validation of CORAE

We tested a use case for CORAE in the form of an experimental study on
interpersonal dynamics during dyadic interactions. We briefly describe the
study design below.

Our study has two aims: a) understanding how participants interact with the
platform and b) evaluating the tool’s effectiveness in accurately capturing
interpersonal affective evaluations over time. This study took place remotely,
with two participants interacting digitally while completing a task. The study
received IRB approval from Cornell University (IRB0143729).

###  IV-A Experimental Procedure

Participants were recruited through Prolific222https://www.prolific.co/. The
study took place fully online. Before scheduling their slot, each participant
read and signed a consent form. At the scheduled time for the experiment, both
participants received a link to a call on Zencastr333https://zencastr.com/, a
video call platform that allows for high-quality recording of each video and
audio stream separately. Participants read task instructions, including a
description of the discussion topic (Reasons for Poverty task [36], detailed
in Section IV-B). After this, participants were recorded while interacting to
solve the task. When they reached an agreement, or after 10 minutes of
discussion, participants were asked to stop discussing and fill out a survey.
This survey collected demographic data, as well as measures of interpersonal
affect. In the meantime, the researcher downloaded the data streams, merged
the video stream with audio streams from both participants and uploaded them
to CORAE. Each participant was then distributed a unique URL which opened an
instance of CORAE’s annotation platform in their browser. Participants were
each presented with a video of their discussion partner were asked to
continuously rate how their partner came across moment-to-moment. Once
finished, participants were instructed to download the annotation file and
upload it onto an encrypted database. Finally, after completing an exit
survey, participants were compensated for their participation with US$14,
through Prolific.

###  IV-B Reasons for Poverty task

To evaluate our tool, we needed a task that could elicit a broad range of
emotions. We used a modified version of the Reasons for Poverty task [36]. The
task requires participants to rank order a list of “reasons for poverty”
according to their “accuracy”. Half of the items follow a reasoning that sees
the source of poverty in peoples’ situation, i.e., their circumstances,
whereas the other half follows a reasoning that sees the source of poverty in
peoples’ disposition, i.e., their personality. By strategically recruiting
participants with opposing beliefs about poverty, we aimed to elicit an
emotionally engaging interaction. We used the following instructions:

You and the other participant must come to an agreement as to a rank of the 5
most relevant causes of poverty in order of the accuracy of each statement.
The cause of poverty that is evaluated as being most accurate will be ranked
as 1st, and the one that is evaluated as least accurate will be ranked as 5th.

  * •

Poor people lack the ability to manage money.

  * •

Poor people waste their money on inappropriate items.

  * •

Poor people do not actively seek to improve their lives.

  * •

Poor people lack talents and abilities.

  * •

Poor people are exploited by the rich.

  * •

The society lacks justice.

  * •

Distribution of wealth in society is uneven.

  * •

Poor people lack opportunities because they live in poor families.

  * •

Poor people live in places where there are not many opportunities.

  * •

Poor people have encountered personal misfortunes, which limit their
opportunities.

  * •

Poor people are discriminated against in society.

  * •

Poor people have bad fate.

  * •

Poor people lack luck.

Participants were given a maximum of 10 minutes to discuss, to prevent
individuals from getting disengaged when reviewing their discussion on CORAE.

###  IV-C Measures

####  IV-C1 User Interaction and Experience

To evaluate how participants are interacting with the platform, we measured
the Click Rate (interval of time between rating data points) and Rating Range
(range of values for one retrospective annotation session). To measure user
experience, we asked participants for feedback about their interactions with
our tool in the exit survey.

####  IV-C2 Evaluation Accuracy

In order to evaluate the effectiveness of our tool in accurately capturing an
individual’s assessment of the interaction, we compared measures of
Interpersonal Agreeableness and Interpersonal Perception (IP). The
Interpersonal Agreeableness measure was operationalized by asking participants
“How did the other participant come across?” on a 7-point Likert scale (from
disagreeable to agreeable) in the post-interaction survey. The Interpersonal
Perception measure was operationalized from the continuous interpersonal
rating data. First, we increased the resolution of the rating data to a $0.1$s
period ($10$Hz), to ease manipulation of the data within and across sessions.
Then, to calculate the IP and in line with prior work[22, 17], for each
participant, we took the cumulative sum of the ratings during the interaction
and fitted a linear regression to that data. The Interpersonal Perception (IP)
measure is given by the slope of that regression, providing an understanding
of how the perception of the other interactant evolved over the interaction.

####  IV-C3 Demographics

In the post-interaction survey, we collected demographic information (age,
gender, nationality, race/ethnicity) and personality traits through the short-
version of the Big Five Inventory [31]. Participants were also asked to rate
their religiousness (not-at-all religious to very religious) and political
leaning (very liberal to very conservative) with 7-point Likert scales.

###  IV-D Participants

Participants were recruited through Prolific. To elicit disagreement during
the interactions, participants were selected according to their political
leaning (one conservative- and one liberal-leaning), which they disclose a
priori on Prolific. Other recruitment criteria were proficiency in English and
a computer device with a functioning camera and microphone.

##  V Results

###  V-A Participants

A total of 12 interaction sessions (24 participants) were used for this study,
with an average interaction duration of $560.01\pm 111.39$ s (total length of
interaction data of $13440.22$ s, around 224 minutes). Participants’ age
ranged from $20-61$ years ($M\pm SD:39.75\pm 13.49$). Out of the 24
participants, 13 identified as female, 11 as male. Race/ethnicity was mostly
Caucasian/White (17), followed by Asian/Asian American (4), Hispanic/Latino
(4) and African/African American/Black (2) (participants could select
multiple). Most participants were native speakers of English (21), with 3
proficient users.

###  V-B User Interaction and Experience

We evaluated how participants interact with the platform during the
retrospective annotation sessions. Participants’ average Click Rate is
$0.42\pm 0.02s$. Each interaction had an average of $1116\pm 222$ rating data
points. The rating scale numerically ranges from $-7$ (disagreeable) to $+7$
(agreeable). For the sessions analyzed, the average Rating Range was $8.08$.
The distribution of ratings can be seen in Figure 3.

Figure 3: Distribution of rating values in the dataset collected for this
study.

fnum@@desciiiitemFeedback from participants:

Voluntary open-response questions in our exit survey revealed an overall
positive annotation experience as reported by users. Given that these
questions were open-response and inquired broadly about experiences with the
study, respondents, in general, detailed both positive and negative aspects of
their interaction during the discussion task rather than with the platform
itself. Three (3) participants spoke explicitly to CORAE’s “ease of use”, and
seven (7) reported the platform design to be “clear”, “intuitive”, or some
variation thereof. Two (2) participants reported technical difficulties using
the playback feature, where the video would not load upon visiting their
instance of the annotation dashboard. In both cases, the issue was found to be
the result of an unreliable network connection between users and our web
server, and the data was still deemed usable for our study.

###  V-C Evaluation Accuracy

Figure 4: Examples of the dynamics of interpersonal ratings (IR) in two
sessions (top) and respective cumulative sum of rated values (bottom). IP is
calculated by adjusting a linear regression to the latter curves and
extracting the slope of these lines.

In Figure 4, we show examples of the retrospective annotation sessions data.
Linear regression, used to calculate the Interpersonal Perception, was well-
adjusted to the data for all sessions ($R^{2}:0.88\pm 0.34$ across the 24
cumulative rating curves). To understand if the retrospective annotation was
accurately capturing the perception of the interactants, we calculated the
Pearson correlation between the Interpersonal Perception and Interpersonal
Agreeableness (assessed by asking the participant to rate in a single scale
how the other participant came across). We found a strong correlation between
the two measures ($r=0.72,p<0.001$), indicating good agreement between the
continuous rating and the overall assessment of the other participant.

##  VI Discussion

This work introduces CORAE, a novel approach to the collection of
interpersonal data. CORAE is an intuitive tool for continuous retrospective
evaluation of one dimension of the affective content of interactions. Through
an evaluation study, we further provide evidence that the tool is both
accurate and intuitive/easy to use. Our results provide insights into
individuals’ ability to continuously recall affective perceptions of their
interaction participant. We also unveil the intricate dynamics of
interpersonal perception, with complex data streams that are captured with the
use of CORAE as an annotation tool.

CORAE was developed to be intuitive, cutting the need for training sessions
and thus increasing the potential for capturing affective states continuously
over time. The platform allows for the collection of high-resolution data,
with participants changing ratings often and across a broad range of values of
interpersonal distance. CORAE thus constitutes a valuable tool that is also
easy to customize to other interactions or experimental contexts. We make this
tool publicly available and will continue to expand on its functionality.

Figure 4 demonstrates how different sessions (and different sets of
participants) can entail disparate interpersonal ratings. The plots on the
left reflect an interaction where individuals were perceived as mostly
agreeable. Interesting synchronization in the ratings can be observed (e.g.,
the rating drops at around $250s$ and $300s$), even though the annotation
sessions took place individually. This indicates that a shared understanding
of the interaction’s affective content may have been established, in line with
theories about affective grounding [23]. The plots on the right show a session
where one participant rated the other much more positively. This becomes
particularly apparent when looking at the cumulative IR curves, which
reinforces the applicability of the Interpersonal Perception measure as a good
single-value indicator of the quality of an interaction as perceived by the
individual. In line with this, we found that the IP was strongly correlated
with the overall, “static” measure of Interpersonal Agreeableness. This shows
that CORAE can accurately capture the affective content of an interaction as
it relates to social distance.

Our findings demonstrate the value of using continuous affect rating methods
to capture the temporal dynamics of affect and emotion in social interactions.
By allowing participants to rate their affective experience both continuously
and retrospectively, our tool provides a more fine-grained understanding of
the emotional experiences of individuals throughout an interaction. This can
help researchers to better understand how affective experiences influence
behavior and how behavior, in turn, shapes affective experiences [30].

Our approach of replacing valence with approach and withdrawal as the relevant
rating dimension reflects a shift towards a more ecologically valid
perspective on emotion [13, 2, 38]. This approach acknowledges that emotional
experiences are not simply positive or negative but are shaped by the context
in which they occur. By focusing on approach and withdrawal, our tool provides
a more nuanced understanding of the social dynamics underlying emotional
experiences and highlights the importance of considering the social context in
which affective experiences occur.

Finally, we believe that CORAE and its subsequent developments have great
potential for studying the dynamics of affect and emotion in a range of social
contexts beyond dyadic negotiation, such as team collaboration [21], or
romantic relationships [18, 22]. Expanding the tool beyond dyadic use could
imply a longer annotation process, as each interactant would have to be
annotated individually, or pairwise-annotation could occur, where each
participant only evaluates one of the interactants; nonetheless, this could
provide interesting insights into the differences between individual
retrospective evaluation and group interpersonal perception, as the latter
requires attention to be split among all the participants. The ability to
capture continuous affect data in real-time and retrospectively provides a
powerful tool for investigating the complex interplay between affect,
behavior, and social context and has the potential to inform the development
of more effective interventions and technologies for improving social
interactions.

###  VI-A Limitations and Future Work

Despite yielding promising results through our experimental validation of
CORAE, we recognize potential limitations to both the platform and broader
generalizability of the dataset collected from these sessions.

####  VI-A1 CORAE

Like any annotation tool, CORAE data might suffer from threats to validity
that are common when using human annotators, such as anchoring [35], framing
[37] and recency [11] effects. The use of a retrospective, rather than
simultaneous, annotation method can be questioned in terms of validity.
Continuous simultaneous rating has been used before [17], but the cognitive
load and meta-affective analysis that needs to take place can hinder the
authenticity of the social behaviors demonstrated. The retrospective
annotation method addresses these challenges, although the ratings might be
affected by future events in an interaction, as individuals may find it hard
to abstract from their recalling of how the interaction developed.
Nonetheless, our experimental results prove that CORAE ratings effectively
reflect interpersonal perceptions of the interactions. Future work may compare
simultaneous with retrospective use of the platform. Further, analyses of the
affective data can take into account different horizons of the interaction, as
insights about how we cognitively model social distance may emerge.

####  VI-A2 Experimental Validation

On the matter of generalizability, constraints relating to participant
recruitment skewed our sample heavily in favor of native English speakers from
the United States. It would be interesting to see if different cultural
backgrounds, which may be attached to different social signaling behaviors
[7], also correlate with more disagreement in the ratings of interpersonal
perceptions. While we kept the analysis of the data focused mostly on the
validation of the platform, we see potential in this experiment to shed light
on affective grounding mechanisms and implicit and explicit behaviors that
signal social distance. For example, disagreement in the ratings may be
measured with mean squared error or even inter-rater reliability measures.
Future work may also look into personality traits similarities or differences
and evaluate if these predict agreement between how participants rated the
interaction. Finally, we see potential for the use of machine learning tools
to predict interpersonal perception through social behaviors that can be
captured through multimodal systems (e.g., audiovisual data).

##  VII Conclusion

In this work, we introduced CORAE, a novel approach to capturing continuous
affect data along an approach-withdrawal dimension, reflecting the degree to
which behavior is perceived as increasing or decreasing social distance. Our
work contributes to the growing body of research on affective computing and
provides a valuable tool for investigating the temporal dynamics of affect and
emotion in social interactions. By making our tool publicly available, we hope
to promote the adoption of intuitive continuous affect rating methods and
facilitate further research into the role of affect in social interactions,
namely for machine learning applications. We believe that our tool has the
potential to shed light on the complex and nuanced nature of human emotional
experiences and inform the development of more effective interventions and
technologies for improving social interactions.

## Ethical Impact Statement

Our study involves human participants, and we have taken several steps to
ensure that their privacy and well-being are protected. All participants
provided informed consent before participating in the study, and we obtained
ethical approval from our institution’s ethics board. Participants were duly
informed of privacy concerns and our steps to address them. According to Open
Science practices, all of the anonymized data used in the current study is
made available. In addition, we recognize that our tool has the potential to
be used in a variety of contexts, and that its use could have ethical
implications. Models developed from this tool could be used to monitor
emotional experiences in workplace interactions, which could potentially lead
to negative consequences for employees. We make the tool publicly available
with the understanding that researchers who use it will abide by ethical
guidelines and take steps to protect the privacy and well-being of their
participants. We acknowledge concerns about the use of affective computing
technologies and their potential impact. We hope that our research will
contribute to a broader conversation about these issues and inform the
development of ethical guidelines for the use of affective computing
technologies in social interactions.

## Acknowledgment

This work was supported by Honda Research Institute USA, Inc..

## References

  * Andersen and Guerrero [1996] P. A. Andersen and L. K. Guerrero.  Principles of communication and emotion in social interaction.  In _Handbook of communication and emotion_ , pages 49–96. Elsevier, 1996.
  * Butler and Randall [2013] E. A. Butler and A. K. Randall.  Emotional coregulation in close relationships.  _Emotion Review_ , 5(2):202–210, 2013.
  * Caffi and Janney [1994] C. Caffi and R. W. Janney.  Toward a pragmatics of emotive communication.  _Journal of pragmatics_ , 22(3-4):325–373, 1994\.
  * Coan and Gottman [2007] J. A. Coan and J. M. Gottman.  The specific affect coding system (spaff).  _Handbook of emotion elicitation and assessment_ , 267:285, 2007.
  * Cowie et al. [2000] R. Cowie, E. Douglas-Cowie, S. Savvidou, E. McMahon, M. Sawey, and M. Schröder.  ’feeltrace’: An instrument for recording perceived emotion in real time.  01 2000.
  * Cowie et al. [2013] R. Cowie, M. Sawey, C. Doherty, J. Jaimovich, C. Fyans, and P. Stapleton.  Gtrace: General trace program compatible with emotionml.  In _2013 Humaine Association Conference on Affective Computing and Intelligent Interaction_ , pages 709–710, 2013.  doi: 10.1109/ACII.2013.126.
  * Crivelli and Fridlund [2019] C. Crivelli and A. J. Fridlund.  Inside-out: From basic emotions theory to the behavioral ecology view.  _Journal of Nonverbal Behavior_ , 43(2):161–194, 2019.
  * Csikszentmihalyi et al. [2014] M. Csikszentmihalyi, M. Csikszentmihalyi, and R. Larson.  Validity and reliability of the experience-sampling method.  _Flow and the foundations of positive psychology: The collected works of Mihaly Csikszentmihalyi_ , pages 35–54, 2014.
  * Ekman [1992] P. Ekman.  An argument for basic emotions.  _Cognition and Emotion_ , 6(3-4):169–200, 1992\.  doi: 10.1080/02699939208411068.  URL https://doi.org/10.1080/02699939208411068.
  * Ekman and Friesen [2003] P. Ekman and W. Friesen.  _Unmasking the Face: A Guide to Recognizing Emotions from Facial Clues_.  Number v. 10 in Spectrum book. Malor Books, 2003.  ISBN 9781883536367.  URL https://books.google.com/books?id=TukNoJDgMTUC.
  * Erk et al. [2003] S. Erk, M. Kiefer, J. Grothe, A. Wunderlich, M. Spitzer, and H. Walter.  Emotional context modulates subsequent memory effect.  _NeuroImage_ , 18:439–47, 03 2003.  doi: 10.1016/S1053-8119(02)00015-0.
  * Filippini et al. [2022] C. Filippini, A. Di Crosta, R. Palumbo, D. Perpetuini, D. Cardone, I. Ceccato, A. Di Domenico, and A. Merla.  Automated affective computing based on bio-signals analysis and deep learning approach.  _Sensors_ , 22(5), 2022.  ISSN 1424-8220.  doi: 10.3390/s22051789.  URL https://www.mdpi.com/1424-8220/22/5/1789.
  * Fridlund [2014] A. J. Fridlund.  _Human facial expression: An evolutionary view_.  Academic press, 2014.
  * Girard [2014] J. M. Girard.  Carma: Software for continuous affect rating and media annotation.  _Journal of Open Research Software_ , Jul 2014.  doi: 10.5334/jors.ar.
  * Gottman and Levenson [1985a] J. M. Gottman and R. W. Levenson.  A valid procedure for obtaining self-report of affect in marital interaction.  _Journal of Consulting and Clinical Psychology_ , 53:151–160, 1985a.
  * Gottman and Levenson [1985b] J. M. Gottman and R. W. Levenson.  A valid procedure for obtaining self-report of affect in marital interaction.  _Journal of consulting and clinical psychology_ , 53(2):151, 1985b.
  * Gottman and Levenson [1992] J. M. Gottman and R. W. Levenson.  Marital processes predictive of later dissolution: behavior, physiology, and health.  _Journal of personality and social psychology_ , 63(2):221, 1992.
  * Gottman and Levenson [2000] J. M. Gottman and R. W. Levenson.  The timing of divorce: Predicting when a couple will divorce over a 14-year period.  _Journal of Marriage and Family_ , 62(3):737–745, 2000.
  * Gunes and Schuller [2013] H. Gunes and B. Schuller.  Categorical and dimensional affect analysis in continuous input: Current trends and future directions.  _Image and Vision Computing_ , 31(2):120–136, 2013.  ISSN 0262-8856.  doi: https://doi.org/10.1016/j.imavis.2012.06.016.  URL https://www.sciencedirect.com/science/article/pii/S0262885612001084.  Affect Analysis In Continuous Input.
  * Healey and Picard [2005] J. A. Healey and R. W. Picard.  Detecting stress during real-world driving tasks using physiological sensors.  _IEEE Transactions on intelligent transportation systems_ , 6(2):156–166, 2005.
  * Jung et al. [2012] M. Jung, J. Chong, and L. Leifer.  Group hedonic balance and pair programming performance: Affective interaction dynamics as indicators of performance.  In _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_ , CHI ’12, page 829–838, New York, NY, USA, 2012. Association for Computing Machinery.  ISBN 9781450310154.  doi: 10.1145/2207676.2208523.  URL https://doi.org/10.1145/2207676.2208523.
  * Jung [2016] M. F. Jung.  Coupling interactions and performance: Predicting team performance from thin slices of conflict.  _ACM Trans. Comput.-Hum. Interact._ , 23(3), jun 2016.  ISSN 1073-0516.  doi: 10.1145/2753767.  URL https://doi.org/10.1145/2753767.
  * Jung [2017] M. F. Jung.  Affective grounding in human-robot interaction.  In _Proceedings of the 2017 ACM/IEEE International Conference on Human-Robot Interaction_ , pages 263–273, 2017.
  * Kuppens and Verduyn [2017] P. Kuppens and P. Verduyn.  Emotion dynamics.  _Current Opinion in Psychology_ , 17:22–26, 2017.
  * Lazarus and Lazarus [1994] R. S. Lazarus and B. N. Lazarus.  _Passion and Reason: Making Sense of Our Emotions_.  Oxford University Press USA, 1994.
  * Lechuga Redondo et al. [2022] M. E. Lechuga Redondo, R. Niewiadomski, R. Francesco, and A. Sciutti.  Comfortability recognition from visual non-verbal cues.  In _Proceedings of the 2022 International Conference on Multimodal Interaction_ , ICMI ’22, page 207–216, New York, NY, USA, 2022. Association for Computing Machinery.  ISBN 9781450393904.  doi: 10.1145/3536221.3556631.  URL https://doi.org/10.1145/3536221.3556631.
  * Lopes et al. [2017] P. Lopes, G. N. Yannakakis, and A. Liapis.  Ranktrace: Relative and unbounded affect annotation.  In _2017 Seventh International Conference on Affective Computing and Intelligent Interaction (ACII)_ , pages 158–163, 2017.  doi: 10.1109/ACII.2017.8273594.
  * Mehrabian [1996] A. Mehrabian.  Pleasure-arousal-dominance: A general framework for describing and measuring individual differences in temperament.  _Current Psychology_ , 14:261–292, 1996.
  * Melhart et al. [2019] D. Melhart, A. Liapis, and G. N. Yannakakis.  Pagan: Video affect annotation made easy.  pages 130–136. IEEE, 9 2019.  ISBN 978-1-7281-3888-6.  doi: 10.1109/ACII.2019.8925434.  URL https://ieeexplore.ieee.org/document/8925434/.
  * Metallinou and Narayanan [2013] A. Metallinou and S. Narayanan.  Annotation and processing of continuous emotional attributes: Challenges and opportunities.  In _2013 10th IEEE International Conference and Workshops on Automatic Face and Gesture Recognition (FG)_ , pages 1–8, 2013.  doi: 10.1109/FG.2013.6553804.
  * Rammstedt and John [2007] B. Rammstedt and O. P. John.  Measuring personality in one minute or less: A 10-item short version of the big five inventory in english and german.  _Journal of Research in Personality_ , 41(1):203–212, 2007.  ISSN 0092-6566.  doi: https://doi.org/10.1016/j.jrp.2006.02.001.  URL https://www.sciencedirect.com/science/article/pii/S0092656606000195.
  * Ruef and Levenson [2007a] A. Ruef and R. Levenson.  Continuous measurement of emotion: The affect rating dial.  In J. Coan and J. Allen, editors, _Handbook of emotion elicitation and assessment_. Oxford University Press, New York, Ny, 2007a.
  * Ruef and Levenson [2007b] A. M. Ruef and R. W. Levenson.  Continuous measurement of emotion.  _Handbook of emotion elicitation and assessment_ , pages 286–297, 2007b.
  * Russell [1980] J. Russell.  A circumplex model of affect.  _Journal of Personality and Social Psychology_ , 39:1161–1178, 12 1980.  doi: 10.1037/h0077714.
  * Seymour and McClure [2008] B. Seymour and S. M. McClure.  Anchors, scales and the relative coding of value in the brain.  _Current Opinion in Neurobiology_ , 18(2):173–178, 2008.  ISSN 0959-4388.  doi: https://doi.org/10.1016/j.conb.2008.07.010.  URL https://www.sciencedirect.com/science/article/pii/S0959438808000676.  Cognitive neuroscience.
  * Shek [2002] D. T. L. Shek.  Chinese adolescent’ explanations of poverty: the perceived causes of poverty scale.  _Adolescence_ , 37:789–804, 12 2002.  ISSN 00018449.
  * Tversky and Kahneman [1981] A. Tversky and D. Kahneman.  The framing of decisions and the psychology of choice.  _Science_ , 211(4481):453–458, 1981.  doi: 10.1126/science.7455683.  URL https://www.science.org/doi/abs/10.1126/science.7455683.
  * Van Kleef [2010] G. A. Van Kleef.  The emerging view of emotion as social information.  _Social and Personality Psychology Compass_ , 4(5):331–343, 2010.
  * Wang et al. [2019] S. Wang, S. Chen, and Q. Ji.  Content-based video emotion tagging augmented by users’ multiple physiological responses.  _IEEE Transactions on Affective Computing_ , 10(2):155–166, 2019.  doi: 10.1109/TAFFC.2017.2702749.
  * Yannakakis and Martínez [2015] G. N. Yannakakis and H. P. Martínez.  Grounding truth via ordinal annotation.  In _2015 International Conference on Affective Computing and Intelligent Interaction (ACII)_ , pages 574–580, 2015.  doi: 10.1109/ACIwI.2015.7344627.
  * Zhao et al. [2020] S. Zhao, Y. Ma, Y. Gu, J. Yang, T. Xing, P. Xu, R. Hu, H. Chai, and K. Keutzer.  An end-to-end visual-audio attention network for emotion recognition in user-generated videos.  _Proceedings of the AAAI Conference on Artificial Intelligence_ , 34(01):303–311, Apr. 2020.  doi: 10.1609/aaai.v34i01.5364.  URL https://ojs.aaai.org/index.php/AAAI/article/view/5364.


# Probing the Structure and Evolution of BASS AGN through Eddington Ratios

Tonima Tasnim Ananna Department of Physics and Astronomy, Dartmouth College,
6127 Wilder Laboratory, Hanover, NH 03755, USA  C. Megan Urry Department of
Physics and Yale Center for Astronomy & Astrophysics, Yale University, P.O.
Box 201820, New Haven, CT 06520, USA  Claudio Ricci Núcleo de Astronomía de la
Facultad de Ingeniería, Universidad Diego Portales, Av. Ejército Libertador
441, Santiago 22, Chile  Kavli Institute for Astronomy and Astrophysics,
Peking University, Beijing 100871, People’s Republic of China  Priyamvada
Natarajan Department of Astronomy, Yale University, 52 Hillhouse Avenue, New
Haven, CT 06520, USA  Department of Physics, Yale University, P.O. Box 208121,
New Haven, CT 06520, USA  Black Hole Initiative, Harvard University, 20 Garden
Street, Cambridge MA 02138, USA  Ryan C. Hickox Department of Physics and
Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA
Benny Trakhtenbrot School of Physics and Astronomy, Tel Aviv University, Tel
Aviv 69978, Israel  Ezequiel Treister Instituto de Astrofísica, Facultad de
Física, Pontificia Universidad Católica de Chile, Campus San Joaquín, Av.
Vicuña Mackenna 4860, Macul Santiago, Chile, 7820436  Anna K. Weigel Modulos
AG, Technoparkstrasse 1, CH-8005 Zurich, Switzerland  Yoshihiro Ueda
Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku,
Kyoto 606-8502, Japan  Michael J. Koss Eureka Scientific, 2452 Delmer Street
Suite 100, Oakland, CA 94602-3017, USA  Space Science Institute, 4750 Walnut
Street, Suite 205, Boulder, Colorado 80301, USA  F. E. Bauer Instituto de
Astrofísica, Facultad de Física, Pontificia Universidad Católica de Chile,
Campus San Joaquín, Av. Vicuña Mackenna 4860, Macul Santiago, Chile, 7820436
Centro de Astroingeniería, Facultad de Física, Pontificia Universidad Católica
de Chile, Campus San Joaquín, Av. Vicuña Mackenna 4860, Macul Santiago, Chile,
7820436  Millennium Institute of Astrophysics, Nuncio Monseñor Sótero Sanz
100, Of 104, Providencia, Santiago, Chile  Matthew J. Temple Núcleo de
Astronomía de la Facultad de Ingeniería, Universidad Diego Portales, Av.
Ejército Libertador 441, Santiago 22, Chile  Mislav Baloković Department of
Physics and Yale Center for Astronomy & Astrophysics, Yale University, P.O.
Box 201820, New Haven, CT 06520, USA  Department of Astronomy, Yale
University, 52 Hillhouse Avenue, New Haven, CT 06520, USA  Richard Mushotzky
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
Connor Auge Institute for Astronomy, University of Hawai`i, 2680 Woodlawn
Drive, Honolulu, HI 96822, USA  David B. Sanders Institute for Astronomy,
University of Hawai`i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Darshan
Kakkad Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
21218 MD, USA  Lia F. Sartori ETH Zurich, Institute for Particle Physics and
Astrophysics, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland  Stefano
Marchesi INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna,
Via Piero Gobetti, 93/3, 40129, Bologna, Italy  Department of Physics and
Astronomy, Clemson University, Kinard Lab of Physics, Clemson, SC 29634, USA
Fiona Harrison  Cahill Center for Astronomy and Astrophysics, California
Institute of Technology, Pasadena, CA 91125, USA  Daniel Stern Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Drive, MS
169-224, Pasadena, CA 91109, USA  Kyuseok Oh Korea Astronomy & Space Science
institute, 776, Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea
Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku,
Kyoto 606-8502, Japan  Turgay Caglar Leiden Observatory, PO Box 9513, 2300 RA,
Leiden, the Netherlands  Meredith C. Powell Kavli Institute of Particle
Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA
94305, USA  Stephanie A. Podjed Department of Physics and Astronomy, Dartmouth
College, 6127 Wilder Laboratory, Hanover, NH 03755, USA  Julian E. Mejía-
Restrepo European Southern Observatory, Casilla 19001, Santiago 19, Chile

###### Abstract

We constrain the intrinsic Eddington ratio ($\lambda_{\rm Edd}$) distribution
function for local AGN in bins of low and high obscuration [$\log(N_{\rm
H}/{\rm cm}^{-2})\leq 22$ and $22<\log(N_{\rm H}/{\rm cm}^{-2})<25$], using
the Swift-BAT 70-month/BASS DR2 survey. We interpret the fraction of obscured
AGN in terms of circum-nuclear geometry and temporal evolution. Specifically,
at low Eddington ratios ($\log~{}\lambda_{\rm Edd}<-2$), obscured AGN
outnumber unobscured ones by a factor of $\sim 4$, reflecting the covering
factor of the circum-nuclear material (0.8, or a torus opening angle of $\sim
34^{\circ}$). At high Eddington ratios ($\log~{}\lambda_{\rm Edd}>-1$), the
trend is reversed, with $<30$% of AGN having $\log(N_{\rm H}/{\rm
cm}^{-2})>22$, which we suggest is mainly due to the small fraction of time
spent in a highly obscured state. Considering the Eddington ratio distribution
function of narrow-line and broad-line AGN from our prior work, we see a
qualitatively similar picture. To disentangle temporal and geometric effects
at high $\lambda_{\rm Edd}$, we explore plausible clearing scenarios such that
the time-weighted covering factors agree with the observed population ratio.
We find that the low fraction of obscured AGN at high $\lambda_{\rm Edd}$ is
primarily due to the fact that the covering factor drops very rapidly, with
more than half the time spent with $<10\%$ covering factor. We also find that
nearly all obscured AGN at high-$\lambda_{\rm Edd}$ exhibit some broad-lines.
We suggest that this is because the height of the depleted torus falls below
the height of the broad-line region, making the latter visible from all lines
of sight.

Active galactic nuclei (16), Supermassive black holes (1663), X-ray surveys
(1824), X-ray active galactic nuclei (2035), Luminosity function (942),
Accretion (14)

##  1 Introduction

To study active galactic nuclei (AGN) comprehensively, across multiple
wavelengths, we must disentangle confounding observational selection effects,
to measure the underlying physical quantities like AGN mass, luminosity, and
accretion rate. For example, although unobscured AGN dominate optically-
selected samples of highly luminous sources (e.g., Richards et al., 2002) they
end up being a minority in the census of the total population (e.g., Treister
et al., 2004; Gilli et al., 2007; Treister et al., 2009; Ananna et al., 2019).
Similarly, radio-selected samples are dominated by radio-loud AGN, which also
constitutes a very small fraction of the overall AGN population (e.g., Best et
al., 2005; Stawarz, 2010). To get at the underlying demographics, it is
important to account for both selection biases and measurement uncertainties.
In the last fifteen years, high-energy X-ray telescopes such as Swift-BAT,
INTEGRAL and NuSTAR have provided nearly unbiased X-ray-detected samples of
AGN, where biases only become significant for the most heavily obscured
sources (see Ricci et al., 2015; Koss et al., 2016; Ananna et al., 2022). The
Swift-BAT 70–month survey provided the most sensitive map above 10 keV
(Baumgartner et al., 2013). Using optical and infrared spectroscopy, the BAT
AGN spectroscopic survey (BASS) provided morphology, masses, redshifts,
luminosity and obscuring column density for 752 non-blazar AGN, including 292
Type 2 AGN (Ricci et al., 2017a; Koss et al., 2022a, b). The 98% completeness
in black hole mass estimates for unbeamed AGN outside of the Galactic plane
comes from broad emission lines and from velocity dispersion of stars within
the host galaxy bulges (Mejía-Restrepo et al., 2022; Koss et al., 2022b, c).

Figure 1:  The distribution of Type 1 (green crosses) and Type 2 (orange
crosses) AGN in column density-Eddington ratio space. There are 276 obscured
AGN ($\log(N_{\rm H}/{\rm cm}^{-2})\geq 22$), compared to 220 Type 2 AGN
(narrow lines). Similarly, there are 301 AGN with $\log(N_{\rm H}/{\rm
cm}^{-2})<22$, and 366 Type 1 AGN, some of which have high column densities.

In Ananna et al. (2022; henceforth A22) we used a Bayesian inference
methodology (described in detail in §3 of A22, summarized here in §2) to
calculate the bias-corrected intrinsic black hole mass function (BHMF) and
Eddington ratio distribution function (ERDF) of local AGN (i.e., $0.01\leq
z\leq 0.3$), divided into optical broad-line/Type 1 and narrow-line/Type 2 AGN
categories. The bias correction accounted for both Eddington bias and the
effect of obscuration on apparent source brightness (and thus, obscuration-
dependent survey depth). A22 found the shape of the BHMFs of Type 1 and Type 2
AGN to be in agreement. However, the distributions of Eddington ratios of Type
1 and Type 2 AGN were significantly different, and these differences prompt an
interesting interpretation with far-reaching implications for AGN unification.

The original AGN unification scheme was purely geometric: viewing angle
explained the major distinctions amongst different types of AGN. However, AGN
are variable sources, sometimes changing from one type to another (e.g. Urry &
Padovani, 1995; Marchese et al., 2012; Braito et al., 2014; Trakhtenbrot et
al., 2019; Green et al., 2022). A luminosity-dependent interpretation referred
to as the receding torus model (e.g., Lawrence, 1991; Simpson, 2005; Oh et
al., 2015) suggested that as AGN luminosity increases, since the dust
sublimation radius increases, this causes the inner edge of the obscuring
torus to recede. This idea was supported by the results of Ueda et al. (2003),
La Franca et al. (2005), Barger et al. (2005), Simpson (2005), Treister et al.
(2008), who found a decreasing fraction of obscured AGN with luminosity.

Analysis of the BASS sample supports an alternate interpretation, where
Eddington ratio rather than luminosity is the parameter that regulates how
obscuring matter is distributed around AGN. Ricci et al. (2017b) used the
observed BAT sample to study fractions of obscured AGN in two luminosity bins
as a function of Eddington ratio. The obscured fractions in the two bins were
very similar, and decreased with increasing Eddington ratio, implying that the
torus structure is more fundamentally dependent on Eddington ratio than on
luminosity. This interpretation is known as the radiation-regulated
unification model. The intrinsic Eddington ratio distribution function
analysis for broad-line/narrow-line AGN in A22 is also in agreement with these
results. Ricci et al. (2022), which is a companion paper to our analysis in
this work, uses the updated BASS DR2 data (Koss et al., 2022a, b) to reaffirm
this version of the unification scheme, and discusses evolution along the
obscuration${-}$Eddington ratio plane.

Here, we report the intrinsic space density of AGN as a function of Eddington
ratio in bins of obscuration, quantified by equivalent hydrogen column
density, $\log(N_{\rm H}/{\rm cm}^{-2})$. Specifically, we divide the AGN into
sub-samples according to the $\log(N_{\rm H}/{\rm cm}^{-2})$: $\log(N_{\rm
H}/{\rm cm}^{-2})<22$ and $22\leq\log(N_{\rm H}/{\rm cm}^{-2})<25$. Intrinsic
ERDFs constrained in the X-ray-based column density measurements allow us to
study this AGN sample from a different perspective and compare with the ERDFs
derived earlier by A22 using Type 1/Type 2 (optical broad-line/narrow-line)
categorization. Moreover, X-ray-based column densities provide direct insight
into line-of-sight circum-nuclear obscuration, and facilitate comparison with
theoretical predictions. Figure 1 shows the distribution of observed Type 1
and Type 2 AGN in $\lambda_{\rm Edd}$-$\log~{}{\rm N}_{\rm H}$ space. The
number of objects in each obscuration bin is given in Table 1.

Our paper is structured as follows: in §2, we discuss the data and the
analysis methodology. In §3, we present our results, and in § 4 we offer a
physically motivated interpretation of our results, divided into high- and
low-$\lambda_{\rm Edd}$ regimes, in the context of geometric unification and
transition. In § 5, we present our conclusions. We adopt a $\Lambda$CDM
cosmology with $h_{0}=0.7$, $\Omega_{m}=0.3$, and $\Omega_{\Lambda}=0.7$
throughout this paper.

Table 1: Intrinsic Eddington ratio distribution function in two obscuration
binsa

$\log(N_{\rm H}/{\rm cm}^{-2})$ bin |  ${\rm N}_{\rm obs}$b | $\log\xi$ | $\delta_{1}$ | $\epsilon_{\lambda}$ | $\log(\lambda_{*})$
---|---|---|---|---|---
$\log(N_{\rm H}/{\rm cm}^{-2})<22$ | 301 (6) |  |  |  |
$\sigma=0.3$ |  | -4.29 | $0.10^{+0.16}_{-0.00}$ | $2.73^{+0.22}_{-0.16}$ | $-1.064^{+0.055}_{-0.052}$
$\sigma=0.5$ |  | -4.26 | $0.10^{+0.55}_{-0.05}$ | $2.51^{+0.44}_{-0.46}$ | $-1.165^{+0.112}_{-0.076}$
$22\leq\log(N_{\rm H}/{\rm cm}^{-2})\leq 25$ | 285 (4) |  |  |  |
$\sigma=0.3$ |  | -3.42 | $-0.01^{+0.27}_{-0.34}$ | $2.51^{+0.32}_{-0.19}$ | $-1.750\pm 0.093$
$\sigma=0.5$ |  | -3.42 | $0.03^{+0.30}_{-0.65}$ | $2.91^{+0.38}_{-0.35}$ | $-1.67^{+0.13}_{-0.14}$

aafootnotetext: Parameters defined in Equation 3.

bbfootnotetext:  Number of AGN with $\log\lambda_{\rm Edd}>0$ in parentheses.

##  2 Analysis

Our main sample follows the same selection criteria as A22, and includes all
unbeamed AGN over an absolute galactic latitude of $5^{\circ}$, and fall
within $0.01\leq z\leq 0.3$, $6.5\leq\log~{}({\rm M}_{\rm BH}/{\rm
M}_{\odot})\leq 10.5$ and $-3\leq\log~{}\lambda_{\rm Edd}\leq 1$ (i.e., a
total of 586 sources from the mass, redshift and $\lambda_{\rm Edd}$ range
where the BASS sample is most complete; see Figure 1 in A22). We use the
masses, redshifts (Koss et al., 2022a), X-ray luminosities and obscuration
measurements (Ricci et al., 2017a) for BASS DR2 sample to calculate Eddington
ratios for these AGN.

###  2.1 Intrinsic Eddington Ratio Distribution Functions

Figure 2:  Top panel: Eddington ratio distribution functions for AGN in
$\log(N_{\rm H}/{\rm cm}^{-2})=20{-}22$ and $\log(N_{\rm H}/{\rm
cm}^{-2})=22{-}25$ bins, as well as for Type 1 and Type 2 AGN (from A22).
Shaded regions show 1$\sigma$ uncertainties. Bottom panel: Ratio of obscured
to overall space densities as a function of $\lambda_{\rm Edd}$. The red line
shows ratio of $\log(N_{\rm H}/{\rm cm}^{-2})$= 22–25 AGN over all AGN, and
the orange line shows ratio of Type 2 AGN/all AGN. Black vertical lines show
effective Eddington limits for $\log(N_{\rm H}/{\rm cm}^{-2})\geq 22$ gas
according to Fabian et al. (2008). At low Eddington ratios ($\lambda_{\rm
Edd}$$<-1.7)$, this ratio is high because gas is retained, while for higher
Eddington ratios, gas is blown away. We suggest that at low-$\lambda_{\rm
Edd}$ this ratio is equal to the pure geometric covering factor of the torus,
while at high $\lambda_{\rm Edd}$, it is a time-averaged covering factor. We
calculate the torus opening angle at low-$\lambda_{\rm Edd}$ and the covering
factor decay rate at high $\lambda_{\rm Edd}$ using these ratios.

While previous studies have constrained the Eddington ratio distribution
function for Type 1/broad-line AGN (e.g., Kelly & Shen, 2013; Schulze et al.,
2015), A22 was the first to constrain the intrinsic Eddington ratio
distribution function for obscured Type 2/narrow-line AGN. We briefly describe
the method here (more details in §3.4 of A22). Using a Bayesian ensemble
sampler with 50 walkers (Foreman-Mackey et al., 2013), we maximize the
likelihood for the following function:

$\ln\mathcal{L}=\sum_{i}^{N_{\rm obs}}\ln p_{i}(\log M_{\mathrm{BH},i},\log
N_{\mathrm{H},i},\log\lambda_{\mathrm{E},i},z_{i}),$ (1)

where $p_{i}$ is a convolution of the intrinsic AGN mass function and
$\log~{}\lambda_{\rm Edd}$ distribution (constrained together), co-moving
volume element, the area-flux curve (i.e., selection function) of the
survey(s), redshift evolution function, and absorption function. As $p_{i}$ is
a probability distribution function, it is normalized by integrating this
product over all observables:

$\displaystyle p_{\rm i,conv}$ $\displaystyle(\log
M_{\mathrm{BH},i},\log\lambda_{\mathrm{E},i},\log
N_{\mathrm{H},i},z_{i})=\frac{N_{\rm i,conv}}{N_{\rm tot}}$ (2)
$\displaystyle=\frac{1}{N_{\rm tot}}\iiiint\Psi(\log
M_{\mathrm{BH}},\log\lambda_{\mathrm{E}},\log N_{\mathrm{H}},z)$
$\displaystyle\ \times\Omega_{\rm sel}(\log
M_{\mathrm{BH}},\log\lambda_{\mathrm{E}},\log N_{\rm H},z)$ $\displaystyle\
\times p(\log N_{\rm H})~{}p(z)~{}\frac{dV_{C}(z)}{dz}$ $\displaystyle\
\times\omega(\log M_{\mathrm{BH},i},\log\lambda_{\mathrm{E},i},\log
N_{\mathrm{H},i},z_{i}|$ $\displaystyle\log M_{\rm BH},\log\lambda_{\rm
Edd},\log N_{\mathrm{H}},z)$ $\displaystyle~{}~{}d\log M_{\rm
BH}~{}~{}d\log\lambda_{\rm Edd}~{}~{}d{\log N_{\rm H}}~{}~{}dz\,,$

where $\Psi$ is the convolution of mass and Eddington ratio distribution
functions. The parametric form of the intrinsic Eddington ratio distribution
function is a double-power law:

$\xi(\log\lambda_{\rm Edd})=\frac{dN}{d\log\lambda_{\rm
Edd}}\propto\xi^{*}\times\left[\left(\frac{\lambda_{\rm Edd}}{\lambda_{\rm
Edd}^{*}}\right)^{\delta_{1}}+\left(\frac{\lambda_{\rm Edd}}{\lambda_{\rm
Edd}^{*}}\right)^{\delta_{2}}\right]^{-1}\,.$ (3)

$\Omega_{\rm sel}$ is the selection function (area-flux curve for BAT 70-month
survey; Baumgartner et al., 2013), $p(\log N_{\mathrm{H},i})$ is the intrinsic
absorption function for BAT AGN (from Ricci et al., 2015), $p(z_{i})=1$ as we
assume negligible redshift evolution over the $0.01\leq z\leq 0.3$ range, and
$\frac{dV_{C}(z_{i})}{dz}$ is the co-moving volume element. The $\omega$ term
allows us to convolve uncertainty in mass measurement. We assume Gaussian
scatter in mass and luminosity measurement with different dispersions in A22.
We report results for both $\sigma_{\rm M}=\sigma_{\lambda_{\rm Edd}}=0.3$ and
$0.5$ cases in § 3, and find that the functions agree within 1$\sigma$ over
the range of $\log~{}\lambda_{\rm Edd}$ considered here, so only the first
case is shown in the figures for clarity. The 1$\sigma$ random errors on the
functions are calculated using the covariance matrix, which we derive from the
MCMC chain. The formula for this error estimation is given in Appendix C of
A22. Note that assuming a functional form such as double-power law leads to
smaller errors than more flexible approaches for constraining space densities
(e.g., Buchner et al., 2015; Ananna et al., 2020). We assume that the shape of
the ERDF is independent of mass, because in § 4.4 of A22, we showed that for
the mass range considered for this sample [$6.5\leq\log~{}({\rm M}_{\rm
BH}/{\rm M}_{\odot})\leq 10.5$], the shape of the ERDF does not change when
constrained in two mass bins independently: [$\log~{}({\rm M}_{\rm BH}/{\rm
M}_{\odot})\leq 7.8$ and $\log~{}({\rm M}_{\rm BH}/{\rm M}_{\odot})\geq 8.2$].

###  2.2 Evolution of AGN Covering factor at high Eddington Ratio

When the torus is stable, the ratio of obscured to overall AGN space density
is equal to the population averaged covering factor of the torus. When the
radiation pressure is very high (e.g., at high $\lambda_{\rm Edd}$, the torus
is unstable, and its geometry is time-dependent). To disentangle the geometric
and temporal aspects at high $\lambda_{\rm Edd}$ (theoretically at
$\log~{}\lambda_{\rm Edd}>-1.7$; see § 4.1), we consider two simple parametric
models of how the covering factor varies over time at high $\lambda_{\rm
Edd}$. We assume that after AGN are triggered from a low-$\lambda_{\rm Edd}$
phase to high $\lambda_{\rm Edd}$, they typically start with a high covering
factor ($\sim 83$% for $\log(N_{\rm H}/{\rm cm}^{-2})=22-25$ torus and $\sim
60\%$ for narrow-line only AGN, as shown in the lower panel of Figure 2). As
obscuring matter is removed because of the increased radiation pressure around
an AGN, its covering factor should decrease. We consider two scenarios for the
temporal dependence: (i) the rate at which the covering factor decreases is
highest when there is more obscuring matter, or (ii) the covering factor
initially decreases slowly due to shielding from obscuring matter, and then
decreases more rapidly as more and more matter is removed. Both scenarios end
with the covering factor decreasing asymptotically such that it may approach
zero, depending on the obscured/overall ratio. We do not force the final
covering factor to equal zero at the end of the high $\lambda_{\rm Edd}$ phase
as IR studies of luminous quasars show residual dust even for luminous optical
quasars (e.g., Hall et al., 2004; Hopkins et al., 2004), essentially allowing
the data to decide the final covering factor. We use exponential and sigmoid
functions to model the behavior for scenarios (i) and (ii), respectively:

$\displaystyle(i)~{}C_{\rm exp}(t)=C_{0}e^{-kt}$ (4)
$\displaystyle(ii)~{}C_{\rm
sigmoid}(t)=C_{0}\frac{1+e^{-kt_{0}}}{1+e^{k(t-t_{0})}}$

Figure 3 shows the simulated evolution derived for a sample of 1000 AGN with
the average observed covering factors [drawn from a normal distribution of
$\mu=0.83,\sigma=0.08$ for $\log(N_{\rm H}/{\rm cm}^{-2})=22-25$ torus, and
$\mu=0.6,\sigma=0.07$ for the narrow-line AGN]. Given a set of parameters, we
calculate the average covering factor for these 1000 AGN over 100 time steps.
To calculate the time-averaged covering factor for these 1000 objects, we draw
random times from the lifetime of each AGN, and average over the whole
population. We repeat this 100 times for each set of parameters to account for
the stochasticity in selecting different time steps, and choose the median. We
use a Bayesian ensemble sampler to optimize these functions to reproduce the
obscured/overall ratio (shown in Figure 2) using a Gaussian likelihood
function:

$\displaystyle L(k,t_{0};{\rm obs~{}ratio})=$
$\displaystyle\frac{1}{\sqrt{2\pi\sigma_{\rm obs~{}ratio}^{2}}}\times$ (5)
$\displaystyle\exp{\left(-\frac{(\langle C(t)\rangle-\mu_{\rm
obs~{}ratio})^{2}}{2\sigma_{\rm obs~{}ratio}^{2}}\right)}$

where C(t) is defined in Eqn. 4, and only the parameter k is constrained for
scenario (i).

##  3 Results

Figure 3:  Decline of the covering factor after an AGN reaches high enough
$\lambda_{\rm Edd}$ that radiation pressure exceeds the gravitational pull on
obscuring matter (at $\log~{}\lambda_{\rm Edd}\geq-1.7$; Fabian et al., 2008),
shown for two different assumed temporal dependences (dashed and solid black
lines represent sigmoid and exponential functions, respectively, as in Eq. 4),
evaluated for two different definitions of obscured versus unobscured classes
(sorted by $\log~{}N_{\rm H}$ or spectral line width). The colored lines show
the decay for a 1000 AGN starting from different covering factors (see § 2.2).
Assuming that the ratio of obscured to overall AGN is a time-averaged covering
factor (see bottom panel of Figure 2), we use an ensemble sampler to constrain
the model parameters (shown in legend). All black lines and associated shaded
regions show covering factor decay for $-1\leq\log~{}\lambda_{\rm Edd}\leq
0.5$. Exponential decay functions at two lower Eddington ratios are shown
using blue lines: blue dotted lines show results for $\log~{}\lambda_{\rm
Edd}=-1.5$ and blue dashed lines show results for $\log~{}\lambda_{\rm
Edd}=-1.2$. The residual covering factor is higher at lower $\lambda_{\rm
Edd}$.  Figure 4:  Schematic diagram of AGN circum-nuclear structure at low
and high Eddington ratios. Top panel: For $\lambda_{\rm Edd}<-2$, radiation
pressure is too low to remove dust and gas around the AGN, so the covering
factor remains high and differences between obscured and unobscured AGN are
likely due to viewing angle. That is, because 80% of AGN have $\log(N_{\rm
H}/{\rm cm}^{-2})>22$, the torus rises to $56^{\circ}$ above the equatorial
plane; since optical broad lines emission are seen in 60% of AGN, the densest
gas lies below 37∘. Bottom panel: At high $\lambda_{\rm Edd}$, radiation
pressure removes the obscuring matter, leaving at most a small persistent
covering factor ($<5\%$). We find that BLR is visible from nearly all angles
after 30% of the time at this $\lambda_{\rm Edd}$, which may indicate that the
height of the BLR region rises above that of the depleted torus.

The intrinsic Eddington ratio distribution function in bins of $\log N_{\rm
H}$ is presented in Table 1 and Figure 2. The top panel of Figure 2 shows the
selection bias and measurement uncertainty-corrected Eddington ratio
distributions of unabsorbed [$\log(N_{\rm H}/{\rm cm}^{-2})$: $20{-}22$] and
absorbed AGN bins [$\log(N_{\rm H}/{\rm cm}^{-2})$: $22{-}25$], along with the
Type 1/Type 2 ERDFs from A22. The bottom panel shows the fraction of Type 2
AGN and $\log(N_{\rm H}/{\rm cm}^{-2})$: $22{-}25$ AGN as a function of
$\lambda_{\rm Edd}$, calculated by dividing the ERDFs of these populations by
the ERDF of all AGN.

Figure 3, shows the evolution of the geometric covering factor at high
$\lambda_{\rm Edd}$ (see Eq. 4). These model parameters ($k$ and $t_{0}$) are
constrained using the ratio of obscured to the total population shown in Fig.
2, at $\log~{}\lambda_{\rm Edd}=-1.5$, -1.2, and averaged over
$-1\leq\log~{}\lambda_{\rm Edd}\leq 0.5$. The average ratios for the last bin
are $0.23\pm 0.08$ and $0.06\pm{+0.07}$, for X-ray and optical measure of
obscuration, respectively.

##  4 Discussion

In this work, we calculate the Eddington ratio distribution functions for
obscured and unobscured AGN using X-ray measures of $N_{\rm H}$. We compare
these functions to the ERDF for Type 1 and Type 2 AGN calculated in A22 using
the same sample (all ERDFs shown in Fig. 2).

At low Eddington ratio, when the circum-nuclear torus is presumably stable,
the ratio of the space density of obscured AGN to all AGN reflects the average
covering factor of the torus. In § 4.1, we discuss the torus at
low-$\lambda_{\rm Edd}$, below the theoretical threshold where radiation
pressure exceeds the gravitational pull. At high $\lambda_{\rm Edd}$, in
contrast, the gas and dust are blown away, so this ratio reflects the time-
averaged covering factor. In § 4.2, we consider the ERDF and obscured/overall
ratio at higher $\lambda_{\rm Edd}$, and explore the time dependence of the
covering factor using simple physically motivated models described in § 2.2.

###  4.1 Low-$\lambda_{\rm Edd}$ Region: Geometric Unification

Fabian et al. (2006, 2008) pointed out that the standard Eddington ratio is
calculated for fully ionized gas, whereas for partially ionized cold gas, the
scattering cross-section is higher, and therefore the “effective Eddington
limit” ($\lambda_{\rm eff}$) for such gas is lower. That is, colder, less
ionized gas can be removed from near the AGN at lower levels of radiation
pressure and thus lower Eddington limit.

In order to compare our results quantitatively with this theoretical picture,
we check whether $\lambda^{*}$, the break in the power-law form of the
Eddington ratio distribution function (see Eqn. 3), corresponds to the
effective Eddington ratio for dusty gas at high densities. The value of $A$
from Fig. 1 in Fabian et al. (2008) represents the ratio of scattering cross
sections for dusty gas compared to ionized gas (see Eqn. 2 from Fabian et al.,
2008), or equivalently, to $\lambda_{\rm Edd}$/$\lambda_{\rm eff}$. That is,
the break, $\lambda^{*}$, should occur at $\lambda_{\rm eff}/A$ (for
$\lambda_{\rm Edd}=1$). For $\log(N_{\rm H}/{\rm cm}^{-2})=22{-}25$,
$\log\lambda^{*}=$ $-1.67^{+0.13}_{-0.14}$ for
$\sigma_{\log~{}M}=\sigma_{\log~{}\lambda}=0.5$ , which is in excellent
agreement with the Fabian et al. (2008) predicted value, $\log A\sim 1.67$.

Under $\lambda_{\rm eff}$ for obscured gas, both theoretically and as
indicated by observations ($\log~{}\lambda_{\rm Edd}<-1.67$ according to
Fabian et al., 2008 and our results, and $\log~{}\lambda_{\rm Edd}<-2$ more
conservatively), the radiation pressure is too low to blow away the obscuring
matter. The simple scenario that emerges from this low-$\lambda_{\rm Edd}$
region is that the pure geometric unification model (Antonucci, 1993; Urry &
Padovani, 1995; Netzer, 2015) applies in this $\lambda_{\rm Edd}$ regime,
where the covering factor of the torus (equal to the ratio of obscured to all
AGN in this regime $\lambda_{\rm Edd}$) could be as high as 83%. However, the
BLR is completely blocked for only 60% of the total solid angle, which could
be indicative of clumpiness in the torus structure (e.g., Ramos Almeida et
al., 2009; Stalevski et al., 2012; Elitzur, 2012) allowing some broad-line
visibility even with obscuring matter along our line of sight.

Semi-analytical models (e.g., Venanzi et al., 2020) find that the densest gas
sinks closer to the equatorial plane of the AGN, due to asymmetry of the
radiation field ($\propto\cos{\theta_{\rm axis}}$). If the pure geometric
unification model applies, this would mean that our view of the BLR is blocked
when looking through angles closer to the equatorial region, whereas at angles
higher above the equatorial plane, obscuring matter is distributed more
sparsely, and therefore the BLR is more likely to be visible. According to our
calculations with the BASS sample, at $-3<\log~{}\lambda_{\rm Edd}<-2$, the
torus rises as high as $56^{\circ}$ [calculated using $\theta=90^{\circ}-{\rm
arccos(covering~{}factor)}$; similarly, the torus opening angle $={\rm
arccos(covering~{}factor)}=34^{\circ}$], while the BLR is completely blocked
by dense matter up to $37^{\circ}$ above the equatorial plane. A schematic of
the geometric unification model, along with these angles and the clumpy
structure of the torus, is shown in the top panel of Figure 4.

###  4.2 High-$\lambda_{\rm Edd}$ Region: Transitional Timescales

The high-$\lambda_{\rm Edd}$ region gives rise to some seemingly contradictory
observational signatures. While overall population studies such as ours
indicate that the covering factor at high $\lambda_{\rm Edd}$ should be very
low if geometric unification applies (e.g., lower panel of Figure 2, Figure 3
of Ricci et al., 2017b), observed obscured AGN which are found at these
$\lambda_{\rm Edd}$ have very high covering factors. Ricci et al. (2017c)
finds that among local ultra-luminous infrared galaxies (ULIRGs), the torus
covering factor is as high as 95%, and most of these AGN are in late stages of
mergers. A recent study of 57 ULIRGs (Yamada et al., 2021) also finds a high
covering factor of 66% at $\lambda_{\rm Edd}\simeq 1$. This is quantitatively
at odds with a population-averaged covering factor of $<30$% at these
$\lambda_{\rm Edd}$ (from Fig. 2). While ULIRGs represent only a small
fraction of the overall AGN population, they dominate infrared samples because
they have a lot of dust and they have high covering factors and high
$\lambda_{\rm Edd}$. Other obscured high-$\lambda_{\rm Edd}$ populations
include red quasars (e.g., Glikman et al., 2004, 2012; Banerji et al., 2015;
Glikman et al., 2018), and Hot Dust-Obscured Galaxies (HotDOGs, e.g., Assef et
al., 2016; Vito et al., 2018). Some X-ray selected studies also find that
heavily dust-reddened quasars are in a radiatively driven blow-out phase
(Lansbury et al., 2020).

In other words, infrared-selected quasars have high covering factors and high
$\lambda_{\rm Edd}$. Yet, according to Figure 2, we should find a much smaller
obscured/overall ratio, implying a much smaller torus covering factor if
simple geometric unification were the cause. As most of these studies conclude
that these high-$\lambda_{\rm Edd}$, obscured AGN are in a transitional state,
we suggest that the low obscured/overall ratio at high $\lambda_{\rm Edd}$ is
indicative of the duration of the obscured phase. That is, the time-averaged
covering factor is low ($\sim 30$%), while infrared studies select
luminous/high-$\lambda_{\rm Edd}$ sources that are still in a dust-obscured
phase.

To disentangle the geometric and temporal aspects of the evolution of the
covering factor, we consider two simple time-dependent parametric models of
evolution, described in § 2.2. In Figure 3, we show these functions
constrained using the obscured/overall ratio (from Figure 2) at
$\log~{}\lambda_{\rm Edd}=-1.5$ and $-1.2$, and over the
$-1\leq\log~{}\lambda_{\rm Edd}\leq 0.5$ range. We find that for the first
model (the sigmoid function), in which the decay of the covering factor might
start slowly due to shielding from dense obscuring matter, then accelerate as
more matter is removed, the covering factor for the torus approaches zero
within 40% of the time spent at $\log~{}\lambda_{\rm Edd}>-1$. The sigmoid
function therefore somewhat contradicts infrared studies that find evidence of
residual dust around luminous SDSS quasars. The second model (exponential
function) accommodates slower decay of the X-ray detected covering factor,
even though the part of the torus that completely blocks the BLR is depleted
inwithin 30% of the AGN lifetime. The persistent covering factors, calculated
using X-ray obscuration measurements, at $\log~{}\lambda_{\rm Edd}=-1.5$,
$-1.2$ and $\sim 0$ are $\sim 40\%$, 5% and 2%, respectively, indicate that at
lower $\lambda_{\rm Edd}$, more obscuring matter remains around the AGN at the
end of the active phase. The bottom panels of Figure 4 shows a schematic of
this transition, with the residual dust ($<5\%$ by the end of the phase) in
the right panel.

While the exponential function allows a gradual decrease in covering factor,
the sigmoid function behaves almost like a step function. If we interpret the
sigmoid function as a limiting case where the high-$\lambda_{\rm Edd}$
obscured/total ratio is decided purely by timescale (i.e., an obscured/total
ratio of $x$% means $x$% of the time is spent fully obscured, and ($100-x$)%
of the time is spent in the almost fully unobscured state), then the
percentage of time spent fully obscured by $\log(N_{\rm H}/{\rm
cm}^{-2})=22{-}25$ torus is $24.7^{+4.1}_{-3.4}$%. The narrow-line only
covering factor will be depleted within $10.3^{+5.4}_{-6.7}$% of the time, so
that for about 15% of the time the AGN will appear broad-line and obscured.
This estimate agrees well with the Glikman et al. (2012) and Glikman et al.
(2018) results, which suggested that the duration of the red-quasar phase is
15–20% of the total quasar lifetime.

As both Figures 2 and 3 show, the X-ray measure of obscured/overall ratio at
high-$\lambda_{\rm Edd}$ is higher than the Type 2/overall ratio. A likely
interpretation for this is that the former indicates a time-averaged covering
factor, while the latter indicates an increase in the size of the broad-line
region relative to the torus. When the scale height of the BLR region is
higher than that of the torus, the BLR should be visible even from equatorial
lines of sight, as shown in the bottom right panel of Figure 4. This depletion
of torus/rise of the BLR happens within 30% of the time spent at
$\log~{}\lambda_{\rm Edd}>-1$ (see Fig. 3). We do an order-of-magnitude
calculation using well-established constraints on the duration of unobscured
quasar accretion, which is roughly the Salpeter time of $10^{7}{-}10^{8}$
years (Martini, 2003; Hopkins et al., 2006; Worseck et al., 2007; Gonçalves et
al., 2008). Using the functions shown in Figure 3, we estimate that after
being triggered to $\log~{}\lambda_{\rm Edd}>-1$, it takes 20–30% of the time
(3–30 Myr) to transition into a completely broad-line phase.

##  5 Conclusions

This work presents the first intrinsic Eddington ratio distribution functions
for X-ray-obscured and -unobscured AGN, constrained using the local BASS
sample. These ERDFs show that there are $\sim 4$ times as many obscured as
unobscured AGN at low $\lambda_{\rm Edd}$, while the reverse is true at high
$\lambda_{\rm Edd}$, with $\sim 3$ times as many unobscured as obscured AGN.

Reasoning that the circum-nuclear obscuration is relatively stable at
$\log~{}\lambda_{\rm Edd}<-2$, we interpret that population ratio in purely
geometric terms. A ratio of 4:1 corresponds to an $\sim 80$% covering factor,
meaning that a simple obscuring torus would rise 56∘ above the equatorial
plane. The Type 2 to overall AGN space density ratio, determined by A22, is
somewhat smaller, $\sim 60$%, with broad optical lines seen in roughly one
quarter of high-$N_{\rm H}$ AGN. This suggests that the obscuring torus is
clumpy. Recent simulations suggest that obscuration is densest near the
equatorial plane (e.g., Venanzi et al., 2020). In that case, all broad-lines
are blocked within 37∘ of the plane by tightly packed obscuring matter, with
some open lines of sight between 37∘ and 56∘. The top panel of Figure 4 shows
a schematic representation of this clumpy torus, with geometry derived from
the ERDF ratios.

At high Eddington ratios, the fraction of obscured AGN is much smaller.
Additionally, the obscured fraction is much lower for narrow-line AGN ($\sim
5$%) than for AGN with $\log(N_{\rm H}/{\rm cm}^{-2})>22$ ($\sim 30$%, see
bottom panel of Fig. 2). Infrared-selected luminous, high-$\lambda_{\rm Edd}$,
obscured AGN have very high covering factors (66-95%; see § 4.2), —much higher
than the observed population average. This tension can be resolved if the
highly covered phase is short-lived and infrared-luminous, so that infrared
selection preferentially finds ULIRGs, red quasars, and HotDOGs.

To disentangle the geometric and temporal aspects at high $\lambda_{\rm Edd}$,
we considered two simple physically motivated models of the decline in
covering factor with time (Fig. 3), constrained by the obscured/overall ratios
at high $\lambda_{\rm Edd}$. Using these models, we find that it takes
approximately 50% of the lifetime of the high-$\lambda_{\rm Edd}$ phase (i.e.,
at $\log~{}\lambda_{\rm Edd}>-1$) to reduce the covering factor from 80% to
$\sim$ 10%. The covering factor is $<5\%$ at the end of the high-$\lambda_{\rm
Edd}$ phase. Additionally, the broad-line region becomes visible along all
lines of sight within 20–30% of the time (orange and yellow lines in Fig. 3),
possibly indicating that the height of the broad-line region is higher than
that of the obscuring torus, making it visible along all lines of sight, as
shown in the bottom right panel of Figure 4.

## Acknowledgments

We thank the referee for thoughtful comments that helped us greatly improve
this paper. This paper is part of a series presented by the BASS
Collaboration. Specifically, this is BASS XXXVIII. T.T.A. support from NASA
through ADAP award NNH22ZDA001N. T.T.A. and R.C.H. acknowledge support from
NASA through ADAP award 80NSSC19K0580, and the National Science Foundation
through CAREER award 1554584. C.M.U. acknowledges support from the National
Science Foundation under Grant No. AST-1715512, and from NASA through ADAP
award 80NSSC18K0418. P.N. acknowledges the Black Hole Initiative (BHI) at
Harvard University, which is supported by grants from the Gordon and Betty
Moore Foundation and the John Templeton Foundation. B.T. acknowledges support
from the Israel Science Foundation (grant number 1849/19) and from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (grant agreement number 950533). M.K.
acknowledges support from NASA through ADAP award NNH16CT03C. M.B.
acknowledges support from the YCAA Prize Postdoctoral Fellowship. K.O.
acknowledges the support of the Korea Astronomy and Space Science Institute
under the R&D program (Project No. 2022-1-830-06) supervised by the Ministry
of Science and ICT and from the National Research Foundation of Korea
(NRF-2020R1C1C1005462). We acknowledge funding support through ANID programs:
Millennium Science Initiative NCN19_058 (E.T.), and ICN12_009 (FEB); CATA-
BASAL - ACE210002 (E.T., F.E.B.) and FB210003 (E.T., F.E.B., C.R.); FONDECYT
Regular - 1190818 (E.T., F.E.B.) and 1200495 (F.E.B., E.T.); and FONDECYT
Iniciacion 11190831 (C.R.). This work was performed in part at the Aspen
Center for Physics, which is supported by National Science Foundation grant
PHY-1607611.

## References

  * Ananna et al. (2020) Ananna, T. T., Treister, E., Urry, C. M., et al. 2020, ApJ, 889, 17, doi: 10.3847/1538-4357/ab5aef
  * Ananna et al. (2019) —. 2019, ApJ, 871, 240, doi: 10.3847/1538-4357/aafb77
  * Ananna et al. (2022) Ananna, T. T., Weigel, A. K., Trakhtenbrot, B., et al. 2022, ApJS, 261, 9, doi: 10.3847/1538-4365/ac5b64
  * Antonucci (1993) Antonucci, R. 1993, ARA&A, 31, 473, doi: 10.1146/annurev.aa.31.090193.002353
  * Assef et al. (2016) Assef, R. J., Walton, D. J., Brightman, M., et al. 2016, The Astrophysical Journal, 819, 111, doi: 10.3847/0004-637X/819/2/111
  * Banerji et al. (2015) Banerji, M., Alaghband-Zadeh, S., Hewett, P. C., & McMahon, R. G. 2015, Monthly Notices of the Royal Astronomical Society, 447, 3368, doi: 10.1093/mnras/stu2649
  * Barger et al. (2005) Barger, A. J., Cowie, L. L., Mushotzky, R. F., et al. 2005, AJ, 129, 578, doi: 10.1086/426915
  * Baumgartner et al. (2013) Baumgartner, W. H., Tueller, J., Markwardt, C. B., et al. 2013, ApJS, 207, 19, doi: 10.1088/0067-0049/207/2/19
  * Best et al. (2005) Best, P. N., Kauffmann, G., Heckman, T. M., et al. 2005, MNRAS, 362, 25, doi: 10.1111/j.1365-2966.2005.09192.x
  * Braito et al. (2014) Braito, V., Reeves, J. N., Gofford, J., et al. 2014, The Astrophysical Journal, 795, 87, doi: 10.1088/0004-637X/795/1/87
  * Buchner et al. (2015) Buchner, J., Georgakakis, A., Nandra, K., et al. 2015, ApJ, 802, 89, doi: 10.1088/0004-637X/802/2/89
  * Elitzur (2012) Elitzur, M. 2012, The Astrophysical Journal, 747, L33, doi: 10.1088/2041-8205/747/2/L33
  * Fabian et al. (2006) Fabian, A. C., Celotti, A., & Erlund, M. C. 2006, Monthly Notices of the Royal Astronomical Society, 373, L16, doi: 10.1111/j.1745-3933.2006.00234.x
  * Fabian et al. (2008) Fabian, A. C., Vasudevan, R. V., & Gandhi, P. 2008, Monthly Notices of the Royal Astronomical Society, 385, L43, doi: 10.1111/j.1745-3933.2008.00430.x
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Gilli et al. (2007) Gilli, R., Comastri, A., & Hasinger, G. 2007, A&A, 463, 79, doi: 10.1051/0004-6361:20066334
  * Glikman et al. (2004) Glikman, E., Gregg, M. D., Lacy, M., et al. 2004, ApJ, 607, 60, doi: 10.1086/383305
  * Glikman et al. (2012) Glikman, E., Urrutia, T., Lacy, M., et al. 2012, ApJ, 757, 51, doi: 10.1088/0004-637X/757/1/51
  * Glikman et al. (2018) Glikman, E., Lacy, M., LaMassa, S., et al. 2018, ApJ, 861, 37, doi: 10.3847/1538-4357/aac5d8
  * Gonçalves et al. (2008) Gonçalves, T. S., Steidel, C. C., & Pettini, M. 2008, The Astrophysical Journal, 676, 816, doi: 10.1086/527313
  * Green et al. (2022) Green, P. J., Pulgarin-Duque, L., Anderson, S. F., et al. 2022, The Astrophysical Journal, 933, 180, doi: 10.3847/1538-4357/ac743f
  * Hall et al. (2004) Hall, P., Hopkins, P., Strauss, M., Richards, G., & Brinkmann, J. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 311, AGN Physics with the Sloan Digital Sky Survey, ed. G. T. Richards & P. B. Hall, 65.  https://arxiv.org/abs/astro-ph/0312281
  * Hopkins et al. (2006) Hopkins, P. F., Somerville, R. S., Hernquist, L., et al. 2006, ApJ, 652, 864, doi: 10.1086/508503
  * Hopkins et al. (2004) Hopkins, P. F., Strauss, M. A., Hall, P. B., et al. 2004, AJ, 128, 1112, doi: 10.1086/423291
  * Kelly & Shen (2013) Kelly, B. C., & Shen, Y. 2013, ApJ, 764, 45, doi: 10.1088/0004-637X/764/1/45
  * Koss et al. (2016) Koss, M. J., Assef, R., Baloković, M., et al. 2016, ApJ, 825, 85, doi: 10.3847/0004-637X/825/2/85
  * Koss et al. (2022a) Koss, M. J., Ricci, C., Trakhtenbrot, B., et al. 2022a, ApJS, 261, 2, doi: 10.3847/1538-4365/ac6c05
  * Koss et al. (2022b) Koss, M. J., Trakhtenbrot, B., Ricci, C., et al. 2022b, ApJS, 261, 1, doi: 10.3847/1538-4365/ac6c8f
  * Koss et al. (2022c) —. 2022c, ApJS, 261, 6, doi: 10.3847/1538-4365/ac650b
  * La Franca et al. (2005) La Franca, F., Fiore, F., Comastri, A., et al. 2005, ApJ, 635, 864, doi: 10.1086/497586
  * Lansbury et al. (2020) Lansbury, G. B., Banerji, M., Fabian, A. C., & Temple, M. J. 2020, Monthly Notices of the Royal Astronomical Society, 495, 2652, doi: 10.1093/mnras/staa1220
  * Lawrence (1991) Lawrence, A. 1991, MNRAS, 252, 586, doi: 10.1093/mnras/252.4.586
  * Marchese et al. (2012) Marchese, E., Braito, V., Della Ceca, R., Caccianiga, A., & Severgnini, P. 2012, Monthly Notices of the Royal Astronomical Society, 421, 1803, doi: 10.1111/j.1365-2966.2012.20445.x
  * Martini (2003) Martini, P. 2003, QSO Lifetimes, arXiv.  http://arxiv.org/abs/astro-ph/0304009
  * Mejía-Restrepo et al. (2022) Mejía-Restrepo, J. E., Trakhtenbrot, B., Koss, M. J., et al. 2022, The Astrophysical Journal Supplement Series, 261, 5, doi: 10.3847/1538-4365/ac6602
  * Netzer (2015) Netzer, H. 2015, ARA&A, 53, 365, doi: 10.1146/annurev-astro-082214-122302
  * Oh et al. (2015) Oh, K., Yi, S. K., Schawinski, K., et al. 2015, ApJS, 219, 1, doi: 10.1088/0067-0049/219/1/1
  * Ramos Almeida et al. (2009) Ramos Almeida, C., Levenson, N. A., Rodríguez Espinosa, J. M., et al. 2009, ApJ, 702, 1127, doi: 10.1088/0004-637X/702/2/1127
  * Ricci et al. (2015) Ricci, C., Ueda, Y., Koss, M. J., et al. 2015, ApJ, 815, L13, doi: 10.1088/2041-8205/815/1/L13
  * Ricci et al. (2017a) Ricci, C., Trakhtenbrot, B., Koss, M. J., et al. 2017a, ApJS, 233, 17, doi: 10.3847/1538-4365/aa96ad
  * Ricci et al. (2017b) —. 2017b, Nature, 549, 488, doi: 10.1038/nature23906
  * Ricci et al. (2017c) Ricci, C., Bauer, F. E., Treister, E., et al. 2017c, MNRAS, 468, 1273, doi: 10.1093/mnras/stx173
  * Ricci et al. (2022) Ricci, C., Ananna, T. T., Temple, M. J., et al. 2022, arXiv e-prints, arXiv:2209.00014.  https://arxiv.org/abs/2209.00014
  * Richards et al. (2002) Richards, G. T., Fan, X., Newberg, H. J., et al. 2002, AJ, 123, 2945, doi: 10.1086/340187
  * Schulze et al. (2015) Schulze, A., Bongiorno, A., Gavignaud, I., et al. 2015, MNRAS, 447, 2085, doi: 10.1093/mnras/stu2549
  * Simpson (2005) Simpson, C. 2005, MNRAS, 360, 565, doi: 10.1111/j.1365-2966.2005.09043.x
  * Stalevski et al. (2012) Stalevski, M., Fritz, J., Baes, M., Nakos, T., & Popović, L. Č. 2012, MNRAS, 420, 2756, doi: 10.1111/j.1365-2966.2011.19775.x
  * Stawarz (2010) Stawarz, L. 2010, in Astronomical Society of the Pacific Conference Series, Vol. 427, Accretion and Ejection in AGN: a Global View, ed. L. Maraschi, G. Ghisellini, R. Della Ceca, & F. Tavecchio, 357
  * Trakhtenbrot et al. (2019) Trakhtenbrot, B., Arcavi, I., MacLeod, C. L., et al. 2019, The Astrophysical Journal, 883, 94, doi: 10.3847/1538-4357/ab39e4
  * Treister et al. (2008) Treister, E., Krolik, J. H., & Dullemond, C. 2008, ApJ, 679, 140, doi: 10.1086/586698
  * Treister et al. (2009) Treister, E., Urry, C. M., & Virani, S. 2009, ApJ, 696, 110, doi: 10.1088/0004-637X/696/1/110
  * Treister et al. (2004) Treister, E., Urry, C. M., Chatzichristou, E., et al. 2004, ApJ, 616, 123, doi: 10.1086/424891
  * Ueda et al. (2003) Ueda, Y., Akiyama, M., Ohta, K., & Miyaji, T. 2003, ApJ, 598, 886, doi: 10.1086/378940
  * Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, PASP, 107, 803, doi: 10.1086/133630
  * Venanzi et al. (2020) Venanzi, M., Hönig, S., & Williamson, D. 2020, The Astrophysical Journal, 900, 174, doi: 10.3847/1538-4357/aba89f
  * Vito et al. (2018) Vito, F., Brandt, W. N., Stern, D., et al. 2018, Monthly Notices of the Royal Astronomical Society, 474, 4528, doi: 10.1093/mnras/stx3120
  * Worseck et al. (2007) Worseck, G., Fechner, C., Wisotzki, L., & Dall’Aglio, A. 2007, Astronomy and Astrophysics, Volume 473, Issue 3, October III 2007, pp.805-818, 473, 805, doi: 10.1051/0004-6361:20077585
  * Yamada et al. (2021) Yamada, S., Ueda, Y., Tanimoto, A., et al. 2021, The Astrophysical Journal Supplement Series, 257, 61, doi: 10.3847/1538-4365/ac17f5