# A Rule Search Framework for the Early Identification of Chronic Emergency
Homeless Shelter Clients

Caleb John
University of Calgary
2500 University Dr. NW, Calgary, AB, Canada, T2N 1N4
<EMAIL_ADDRESS>
Geoffrey Messier
University of Calgary
2500 University Dr. NW, Calgary, AB, Canada, T2N 1N4
<EMAIL_ADDRESS>
###### Abstract

This paper uses rule search techniques for the early identification of
emergency homeless shelter clients who are at risk of becoming long term or
chronic shelter users. Using a data set from a major North American shelter
containing 12 years of service interactions with over 40,000 individuals, the
optimized pruning for unordered search (OPUS) algorithm is used to develop
rules that are both intuitive and effective. The rules are evaluated within a
framework compatible with the real-time delivery of a housing program meant to
transition high risk clients to supportive housing. Results demonstrate that
the median time to identification of clients at risk of chronic shelter use
drops from 297 days to 162 days when the methods in this paper are applied.

##  1 Introduction

Homelessness is one of the great challenges facing society today and the
approach to meeting this challenge is changing. In Canada, a country with a
population of approximately 37 million people, at least 235,000 individuals
experience homelessness in a year [1]. Traditionally, emergency shelters have
been a major part of supporting these individuals with 450,000 Canadians using
a shelter at least once between 2010 and 2014 [1]. However, this is changing
as the homelessness serving sector transitions to the Housing First model.
Many emergency shelter clients experience a variety of mental health, physical
health and addictions related challenges. Rather than expecting individuals to
resolve these challenges while living in shelter, Housing First programs
quickly connect people experiencing homelessness to permanent housing without
pre-conditions [2, 3]. People living with long term or chronic homelessness
are a high priority group for referral to housing programs [4, 5, 6].

Identifying members of the chronic homeless population in a timely fashion is
a challenge. There are a variety of chronic homelessness definitions that can
be used to identify which shelter clients fall into this group [7, 8, 9, 10].
However, these definitions require a minimum of 6-12 months of homelessness
before an individual is classified as chronic. This delay places individuals
at risk since conditions in most emergency shelters are difficult and have the
potential to adversely affect the mental and physical health of an already
very vulnerable group.

A second complicating factor is that only a minority of shelter users will
become chronic. Typically, a large majority (over 80%) of first time emergency
shelter clients will exit shelter after a short time with very little help
[11, 12, 13, 14]. Since supportive housing resources in most municipalities
are limited, it would be inefficient to allocate these supports to a group of
individuals who would exit shelter successfully on their own.

The primary contribution of this paper is to present a machine learning
framework for the early identification of individuals at risk of becoming
chronic shelter clients. Our approach will identify the minority of shelter
users who will become chronic as soon as possible so they can be referred to
supportive housing programs. This goal will be achieved using methods with the
additional advantage of being intuitive for non-technical shelter staff and
compatible with the rudimentary information technology (IT) infrastructure
commonly found in most not-for-profit shelter settings.

Machine learning has been applied to the problem of homelessness in a variety
of ways. A commonly studied objective is the identification of individuals
likely to enter or re-enter homelessness after exiting shelter [15, 16, 17,
18, 19, 20, 21, 22]. A smaller body of work addresses the early identification
of people who will experience chronic homelessness [23, 6]. Logistic
regression and decision trees are the most popular algorithms utilized by
these authors with one study [23] exploring the use of neural networks.

Most researchers recognize that their work is more likely to be adopted if it
is interpretable [16, 17, 23]. Any application of machine learning in shelter
will be a collaboration between the machine learning experts who develop and
deploy the machine learning system and front-line shelter staff who use the
system in their interactions with clients. Shelter staff must have an
intuitive understanding of the system in order to trust its decisions. While
interpretability is sometimes cited as the reason to utilize logistic
regression [16, 17], we argue that even logistic regression is largely opaque
to anyone without an advanced mathematical background. In contrast, the
threshold style rules generated by decision trees [6, 16, 21] are much more
intuitive for non-technical staff. For example, if a decision node states that
the number of violent incidents experienced by a client should be less than a
threshold, a non-technical shelter staff member can both understand that rule
and likely validate it with their own experience.

Machine learning will also only make an impact in the fight to end
homelessness if it is compatible with emergency shelter IT infrastructure. Our
experience with 37 different housing and homelessness agencies in Calgary,
Canada reveals an IT landscape consisting primarily of database style systems
that support basic queries and often have closed, proprietary user interfaces.
We feel this is representative of most cities in North America. This places
most machine learning algorithms out of reach for many agencies without a
significant investment in upgrading both their technical infrastructure and
staff expertise. Even the ordered nature of the rules produced by decision
trees are difficult to implement on many systems.

We will meet the dual requirements of interpretability and IT compatibility by
utilizing rule set learning techniques [24] for the early identification of
clients at risk of becoming chronic shelter users. While rule set learning and
decision trees produce similar threshold style rules, rule sets have the
important distinction of being unordered. This makes them much easier to
implement using a database interface that supports basic thresholding and
logical operators. While work has previously been done exploring threshold
testing for chronic shelter use identification [25, 26], this work lacks the
rigor of a formalized rule search framework that optimizes rule selection
based on a performance metric.

Our second contribution is to present a system that requires only data that is
consistent with an inclusive and trauma informed philosophy to emergency
shelter access. Most previous studies utilize data features collected via
intake surveys [15, 16, 17, 18, 19, 20, 21, 22, 23] that include answers to
questions related to employment, health, social stability, past traumatic
experiences, etc. While these data features are powerful predictors,
collecting survey data can be re-traumatizing for some clients. For this
reason, some of the most vulnerable individuals using shelter services refuse
to participate in surveys. The Calgary Drop-In and Rehab Centre (the DI) is
our partner for this research and is an organization that imposes very few
barriers to access their services. New clients at the DI are encouraged but
not required to complete an intake survey. To ensure all clients are included
in our analysis, not just those associated with survey data, our study will
utilize only administrative data from the DI containing events linked to all
clients after they are accepted to shelter. Examples of these events include
accessing sleep services, electing to speak with a counselor and experiencing
a violent incident.

Our final contribution is to evaluate our chronic client identification
performance using an approach that is consistent with how machine learning
would be incorporated into the real-time delivery of an emergency shelter
housing program. Much of the existing literature utilize data collected over
very long observation windows, consisting of several months to several years
[15, 16, 17, 18, 19, 20, 21, 22, 23, 6]. We assume that housing triage
meetings are held on a monthly basis and that clients can only be evaluated
using the data collected between their shelter entry and the meeting date. As
a result, we evaluate our methods using a framework where clients are
evaluated using only the data available at successive monthly meetings. While
this degrades performance relative to very long observation windows, it is
consistent with the philosophy of identifying clients for assistance as soon
as possible. We will also demonstrate that our approach performs well, even
with this constraint.

In the following, Section 2 introduces the dataset used for this study and the
algorithm chosen to identify chronic shelter clients. Section 3 evaluates the
performance of our approach. Discussion and concluding remarks are provided in
Sections 4 and 5, respectively. While implementing the ideas in this paper
will only be possible through a multi-disciplinary collaboration between
machine learning staff and front-line emergency shelter social work staff, the
language and subject matter in this paper is intended for a machine learning
audience.

##  2 Methods

###  2.1 Data

This is a secondary data analysis performed on anonymized shelter records
collected at the Calgary Drop-In Centre (DI) between July 1, 2007 and January
20, 2020. The data anonymization and client privacy protocol used during this
study was approved by the University of Calgary Conjoint Faculties Research
Ethics Board. The dataset consists of 5,060,302 entries for 41,935 unique DI
client profiles. Each entry records a single timestamped interaction with a
client that occurs within the shelter. This interaction could be accessing
sleep services (EntrySleep), accessing counseling services (EntryConsl), a
security incident resulting in a ban or bar from shelter (EntryBar) or a log
note (EntryLog). To mitigate left censoring of the data, we exclude all
clients who have their first sleep date appear prior to July 1, 2008. This
removes a total of 8,589 clients. As a result, 32,346 individuals are retained
in the dataset (77.1% of the original dataset population).

Many database entries also have comments fields where shelter staff can
describe an incident or consultation. To maintain client privacy, these
comment fields are stripped by DI IT staff as part of the anonymization
process. Each comment field is replaced by a series of boolean flags
indicating whether certain keywords appear in that field. Based on
consultation with DI shelter staff, the keywords are divided into the
following categories: Violence, Overdose, Police/Justice, Mental Health,
Physical Health, Emergency Medical Services (EMS), Addiction, Conflict, Bar
and Supports. If one or more keywords in a category appear in a comment field,
the boolean flag for that category is set to true. For example, a common
Police word was “CPS” (Calgary Police Services) and a common Violence word was
“fight”. A comment field with one or more occurrences of “CPS” and one or more
occurrences of “fight” would have its Police/Justice and Violence flags set to
true. All other category flags for that comment would be false.

Since this purpose of this study is to identify clients at risk of chronic
shelter use, it is important to assign a flag to each individual in the
database indicating whether they did in fact become a chronic shelter user
over the time interval considered for this study. This flag is determined by
applying the Canadian federal government definition of chronic homelessness to
each client’s entire record of shelter use. This definition states that the
chronically homeless group includes “individuals who are currently
experiencing homelessness and who meet at least 1 of the following criteria:
they have a total of at least 6 months (180 days) of homelessness over the
past year or they have recurrent experiences of homelessness over the past 3
years, with a cumulative duration of at least 18 months (546 days)” [10]. A
total of 3,191 of the 32,346 clients (9.9%) satisfy this definition and are
designated as chronic in the dataset.

To better understand the people being considered in this study, it is also
useful to compare the shelter access characteristics of the clients that do
and do not meet the Canadian definition for chronic homelessness (ie. the
chronic and non-chronic groups, respectively). In the following, a stay is a
24 hour period where sleep services are accessed at least once. An episode of
shelter use is a series of stays where the separation between consecutive stay
dates is less than 30 days [7]. Statistical results are presented for the
total shelter stays per client, total episodes per client, shelter tenure per
client (number of days between first and last stay), shelter use percentage
(stays divided by tenure) and the average gap length between episodes of
shelter use. Since shelter client characteristics are often exponentially
distributed, we show median, 10th percentile and 90th percentile in addition
to average. The results for the chronic and non-chronic groups are provided in
Tables 1 and 2, respectively. The statistics shown in Tables 1 and 2 are
broadly consistent with other studies of chronic and non-chronic shelter users
in North America [12, 27, 13, 14].

| Average | Median | 10th Pct. | 90th Pct.
---|---|---|---|---
Total Stays | 963.6 | 671.0 | 277.0 | 2140.0
Total Episodes | 6.9 | 5.0 | 1.0 | 15.0
Tenure (days) | 2254.4 | 2237.0 | 612.0 | 3960.0
Usage Percentage | 48.1 | 40.9 | 15.3 | 93.4
Avg. Gap Length (days) | 3.2 | 2.4 | 1.0 | 6.5
Table 1: Chronic user shelter access characteristics. | Average | Median | 10th Pct. | 90th Pct.
---|---|---|---|---
Total Stays | 34.6 | 5.0 | 1.0 | 108.0
Total Episodes | 3.3 | 2.0 | 1.0 | 8.0
Tenure (days) | 766.1 | 162.0 | 1.0 | 2590.0
Usage Percentage | 46.6 | 13.7 | 0.4 | 100.0
Avg. Gap Length (days) | 164.9 | 20.0 | 1.0 | 430.3
Table 2: Non-chronic user shelter access characteristics.

###  2.2 Feature Generation

We assume that shelter case workers convene housing meetings on the first of
each month to determine which clients should be referred to housing programs.
Case workers will base their housing decisions on all data collected for a
client. This would include the responses to survey questions, if available.
However, in order to be as inclusive as possible and avoid potential bias
towards clients who elect to do intake surveys, we will restrict ourselves to
using only the shelter event data described in Section 2.1 since it is
available for all clients.

We assume that housing recommendations are made only for clients who have
recently interacted with shelter services and staff. An active client is
defined as one who has accessed shelter services at least once in the 30 days
prior to the housing meeting. Attribute summary tables are prepared that
summarize the shelter administration data available for each active client as
of the date of the housing meeting. These tables are a sum of the number of
data entry types described in Section 2.1 (ie. EntrySleep, EntryLog, etc.) and
the number of data entries with particular keyword flags set (ie. number of
entries with the Violence keyword set, number of entries with Addiction set,
etc.).

The appropriate length of time a client should be in shelter before being
assessed for housing is an important question. If this time window is too
short, there may be insufficient data to accurately predict if a client will
become chronic. On the other hand, longer time windows will improve accuracy
at the cost of increased time in shelter before receiving help. To explore
this tradeoff, attribute summary tables are generated for window sizes of
$w\in\\{30,60,90,120\\}$ days. Section 2.1 describes $K$ = 14 data attributes
consisting of 3 one hot entry type indicators and 11 keyword boolean flags.
The attribute table for a window size of $w$ contains sums of each client’s
$K$ data attributes calculated at the first monthly meeting where they have
accumulated at least $w$ days in shelter. Attribute tables are only generated
for clients who meet the active client criterion on the meeting date. Each
entry in an attribute table also contains a boolean flag indicating whether
the client satisfies the definition for chronic homelessness described in
Section 2.1. Therefore, the dimensions of an attribute summary table for
window size $w$ are equal to $N_{w}\times(K+1)$, where $N_{w}$ is the number
of active clients in the table. Each row in an attribute summary table is
referred to as an example.

Table 3 shows the number of clients contained in each attribute summary table
(ie. the number of examples in each table) relative to the total study
population. As expected, the number of clients included in each attribute
summary table drops for the longer window sizes since most clients remain only
a short time in shelter. Some clients may appear in attribute summary tables
for more than one window size.

Win. Size | Number of Examples, $N_{w}$ | Num. Features, $L$
---|---|---
30 | 12175/32346 (37.64%) | 491
60 | 8671/32346 (26.81%) | 655
90 | 7440/32346 (23.00%) | 779
120 | 6650/32346 (20.56%) | 903
Table 3: Example and feature counts.

A feature is the binary result of a threshold test applied to one of the $K$
attribute columns in an attribute summary table [24]. The two threshold
comparison operators are $\\{\geq,<\\}$. Each distinct combination of a
comparison operator and threshold value corresponds to a different feature. To
prevent the feature space from getting too large, the feature generation
algorithm presented in [28] is used where only threshold examples that
discriminate at least one pair of positive and negative examples are included.
The resulting features are contained in $N_{w}\times(L+1)$ coverage tables
where $L>K$ is the number of features and an additional column containing the
chronic shelter use label is retained. The rows of each coverage table are
determined by applying the $L$ feature tests to the corresponding row in an
attribute summary table. The number of features in each coverage table are
presented in Table 3. The number of features grows with window size since the
attribute summary tables contain a larger range of values for longer windows.

A rule set consists of the conjunctive and disjunctive combination of a small
subset of the $L$ features contained in a coverage table that yield the best
performance according to a chosen metric like accuracy, precision, etc.
Searching for this subset among the very large number of features indicated in
Table 3 has the potential to require considerable computation. The search
space can be reduced by discarding attributes that are either highly
correlated with another attribute and/or poorly correlated with the chronic
shelter use outcome label.

This attribute pruning is implemented by first determining the correlation
coefficients between each of the $K$ attributes in each attribute summary
table and the correlation coefficient between those attributes and the chronic
homelessness outcome label. Attributes are placed in pairs in decreasing order
of cross correlation between the pair members. For example, the two attributes
with the highest cross-correlation are placed in the first pair, the remaining
two with the next highest cross correlation are placed in the next pair and so
on. Then, the member of each pair that is the least correlated with the output
label is discarded. This process is repeated until all attributes have been
discarded. The reverse of the discard order defines the order that attributes
should be retained.

The feature retention order results are provided in Table 4 where features at
the top of the table are more valuable than the features below. The table
shows that the set of most valuable features converge for window sizes greater
than 30. A particularly important note is that the top three best attributes
are entry types (EntrySleep, EntryBar and EntryConsl). Database entry types
are easy to access in most IT systems but keyword counts may be beyond the
technical capabilities of many shelters. A rule set that operates only by
counting the number of database category entry types will be much easier to
implement.

Based on Table 4, we define two attribute sets. The core attribute set is
{EntrySleep, EntryBar, EntryConsl}. The extended attribute set is {EntrySleep,
EntryBar, EntryConsl, Violence, Overdose, Addiction}. These sets are used to
generate core coverage tables and extended coverage tables, respectively.

$w$ = 30 days |  $w$ = 60 days |  $w$ = 90 days |  $w$ = 120 days
---|---|---|---
EntrySleep | EntrySleep | EntrySleep | EntrySleep
Violence | EntryBar | EntryBar | EntryBar
Overdose | EntryConsl | EntryConsl | EntryConsl
EntryConsl | Violence | Violence | Violence
EntryBar | Overdose | Overdose | Overdose
Addiction | Addiction | MentalHealth | MentalHealth
PhysicalHealth | PhysicalHealth | PhysicalHealth | PhysicalHealth
PoliceJustice | PoliceJustice | PoliceJustice | PoliceJustice
EMS | EMS | EMS | EMS
MentalHealth | MentalHealth | Addiction | Addiction
Table 4: Attribute retention order.

###  2.3 Rule Search Algorithm

Rule search operates by creating a disjunctive set of rules that combine to
identify or cover as many positive examples in the data set as possible. Each
rule in these sets is a conjunctive combination of certain data features
extracted from the coverage tables. The “search” part of rule search involves
finding the best possible conjunctive rule based on a metric like accuracy or
precision. These search algorithms are generally divided into two categories:
heuristic and exhaustive search. Heuristic searches are efficient at the
potential cost of settling on a locally optimum rule. Exhaustive searches are
guaranteed to find the global optimum with the potential penalty of being
inefficient.

The OPUS algorithm [29] is chosen for this study since it uses exhaustive
search to reliably find the globally optimum rule while improving efficiency
by pruning regions of search space that represent either redundant rules or
rules that will never exceed the current best rule candidate. OPUS will
optimize according to a variety of quality criteria but F-score was chosen for
this study [24]. F-score finds a compromise between precision and recall that
can be adjusted according to a factor $\beta^{2}$ where $\beta^{2}$ = 0
results in optimizing solely on precision and $\beta^{2}=\infty$ will optimize
solely on recall. Recall is true positive rate defined as $|\hat{P}|/|P|$,
where $|\hat{P}|$ is the number of true positives identified by the rule and
$|P|$ is the total number of positives. Precision is
$|\hat{P}|/(|\hat{P}|+|\hat{N}|)$, where $|\hat{P}|+|\hat{N}|$ is the total
number of examples identified by the rule (true positives plus false
positives). In the context of referring shelter clients to housing programs,
precision is important to be confident that scarce housing resources are being
efficiently allocated to shelter users that will become chronic. On the other
hand, recall is also important to ensure that no one with the potential to
become a chronic shelter user is missed. In addition to the standard optimal
search OPUS termination criterion, the algorithm is modified to terminate
after searching all rules of maximum length, $M_{R}$.

The OPUS algorithm is wrapped in a simple disjunctive rule set learning
algorithm. OPUS is used to find a single conjunctive rule. Then, all examples
covered by that rule are removed and OPUS is re-run on the remaining examples
to find a second rule that is combined disjunctively with the first. This
continues until all positive examples are covered or the rule set reaches a
maximum length, $M_{S}$.

###  2.4 Real Time Program Delivery

Our objective is to integrate the rule set search techniques in this paper
into the real time delivery of housing programs within an emergency shelter.
This process begins by having machine learning experts run the algorithms in
Section 2.3 on a historical shelter client dataset. Ideally, each shelter
using this system would retain a machine learning consulting firm to perform
this on data from their own clients. However, this is not strictly necessary
since the rules presented in Section 3 could be used as a reasonable starting
point for most North American shelters.

Once the rules are determined, they are provided to shelter staff to be used
in preparing a list of clients for discussion during monthly housing referral
meetings. Due to the unordered, threshold style nature of the rules, the
shelter staff can implement the rules using any database or spreadsheet system
that allows basic query functionality. For each meeting, rules for a window
size of $w$ days are run on the most recent $w$ days of data for all active
clients. Any clients detected by the rules are placed on a triage list for
discussion during the meeting. It is important to note that clients have
multiple chances of being detected. For example, a client who perhaps fails
the active client criterion or does not meet the rule set thresholds for one
meeting will remain in the shelter system and will be re-evaluated for triage
lists at subsequent meetings.

##  3 Results

###  3.1 Rule Search Parameter Selection

The rule search algorithm described in Section 2.3 has a number of parameters
that need to be selected. This exploration begins with an initial maximum rule
length and rule set size of $M_{R}$ = 3 and $M_{S}$ = 2, respectively. While
longer rules and rule sets are certainly computationally feasible, one of our
main objectives is to generate rule sets that are both intuitive for shelter
workers to understand and practical to enter by hand into database query
tools. Selecting modest values for $M_{R}$ and $M_{S}$ achieves this
objective. The following explores the rule search parameter settings that find
a compromise between performance, interpretability and search efficiency. The
results in this section are determined using stratified K-fold cross
validation [11] with 10 folds.

#### F-Score $\beta^{2}$ Setpoint

Values of $\beta^{2}\in\\{0.01,0.25,0.5,1\\}$ are evaluated with the extended
attribute set described in Section 2.2 and a $w$=90 day window size. The
$\beta^{2}$ values yield (precision, recall) values of (0.8644, 0.1978),
(0.6254, 0.6793), (0.5236, 0.7978) and (0.5273, 0.8569), respectively. As
expected, increasing $\beta^{2}$ improves recall at the cost of precision with
the value of $\beta^{2}$ = 0.25 yielding a good compromise between the two
parameters.

#### Extended vs. Core Attribute Set

Values $\beta^{2}$ = 0.25 and a $w$ = 90 day window size are then used to
explore how simplifying the number of input attributes affects performance.
The core and extended rule sets are compared for $M_{R}$ = 3 and $M_{S}$ = 2
which yields (precision, recall) values of (0.6260, 0.6832) and (0.6254,
0.6793), respectively. Since these results are very comparable, the core
attribute set is used for the remainder of this paper. This has the advantage
of reducing search times and, as noted in Section 2.2, uses features easily
extracted from most shelter databases.

#### Effect of Rule Length

Next, reducing rule set size and rule length is explored for the core
attribute set, $\beta^{2}$=0.25 and a 90 day window size. Values of $M_{R}$=3
and $M_{S}$=1 result in (precision, recall) values of (0.765, 0.5034). As
expected, $M_{S}$=1 reduces recall since the disjunctive combination of
multiple rules works to include more positive individuals. Values of $M_{R}$=2
and $M_{S}$=1 result in comparable (precision, recall) values of (0.7427,
0.5336). The values of $M_{R}$=2 and $M_{S}$=1 are used for the remainder of
this paper since shorter rules have the practical benefit of both reducing
search time and being very simple for shelter staff to work with. As we will
demonstrate in the next section, using the rules in the monthly triage
framework described in Section 2.4 will recover the loss in recall that
results from choosing $M_{S}$=1.

#### Effect of Window Size

Finally, the overall performance for the different window sizes can be
evaluated using $M_{R}$ = 2, $M_{S}$ = 1, $\beta^{2}$ = 0.25 and the core
attribute set. Table 5 shows the performance estimated using cross-validation
along with the rules that result from training on the entire coverage table
for each window size. Rule performance does improve with longer window size
but not dramatically so. A window size of 90 days appears to offer the best
compromise between rapid identification and performance.

Window Size (days) | Precision | Recall | Rule
---|---|---|---
30 | 0.4611 | 0.4514 | EntrySleep $\geq$ 28 $\cap$ EntryBar $<$ 0.5
60 | 0.6092 | 0.4744 | EntrySleep $\geq$ 54 $\cap$ EntryConsl $<$ 10.5
90 | 0.7427 | 0.5336 | EntrySleep $\geq$ 78 $\cap$ EntryBar $<$ 3.5
120 | 0.7979 | 0.5362 | EntrySleep $\geq$ 99 $\cap$ EntryBar $<$ 4.5
Table 5: Window size performance.

###  3.2 Real Time Program Delivery

This section evaluates the performance of the rules in Table 5 when integrated
into a monthly housing program triage process as described in Section 2.4.
Unlike the results in Section 3.1 which are generated using coverage tables
containing only the clients indicated in Table 3, the results in this section
are generated using the entire population in the DI shelter data set. Table 6
shows the resulting precision, recall and confusion matrix values. The table
also shows median time to identification (TTI) for all chronic clients. TTI is
the number of days a chronic client spends in shelter before being identified
by the rules in Table 5 for referral to a monthly housing meeting. TTI for
those clients who are never detected by a rule is calculated using the first
day they satisfy the Canadian definition of chronic homelessness described in
Section 2.1.

Win |  |  | True | False | False | True | Median
---|---|---|---|---|---|---|---
Size | Recall | Precision | Pos. | Neg. | Pos. | Neg | TTI (days)
30 | 0.85 | 0.60 | 2726 | 465 | 1792 | 27363 | 162.00
60 | 0.77 | 0.71 | 2452 | 739 | 991 | 28164 | 225.00
90 | 0.75 | 0.78 | 2409 | 782 | 680 | 28475 | 254.00
120 | 0.73 | 0.83 | 2338 | 853 | 465 | 28690 | 272.00
Table 6: Program delivery performance.

##  4 Discussion

Table 6 demonstrates that rule search methods for identifying individuals at
risk of chronic shelter use provide very strong performance when integrated
into a monthly housing triage program. The authors in [23] achieve similar
results using neural networks but rule sets are much easier to adopt in
shelter, as noted in Section 1. The results in Table 6 are also much better
than Table 5. Table 5 is produced using the subset of the client population
used to generate the coverage tables. This subset consists only of those
clients who meet the active client criterion after their first $w$ days of
being in the client database. This is a more challenging dataset since this
sub-population contains a higher proportion of clients with heavy initial
shelter use but who still do not become chronic. Table 6 is produced using the
entire population in the data set which contains more easy to distinguish
individuals and results in a performance improvement.

The best choice of window size is $w$ = 30 days which achieves the best recall
and median TTI. Ideally, TTI would be even lower but 162 days is still a
considerable improvement over using the Canadian federal definition of chronic
homelessness to identify clients which would result in a median TTI of 297
days for this data set. The high recall of using $w$ = 30 days ensures that as
many vulnerable clients as possible are referred to housing. However, this
comes at the cost of also having the lowest precision in Table 6.

A low precision raises two potential concerns: the false positives may
overwhelm housing program capacity and/or the false positives are not good
candidates for limited housing resources. Regarding program capacity, the 30
day rule set identifies a total of 4,518 clients in a data set that stretches
over 101 months. This corresponds to 44.7 clients per month which the DI has
already demonstrated is within the capacity of their housing programs [8].
Regarding whether the false positives are good candidates for housing, Table 7
shows the shelter access statistics for the 1,792 false positive clients
identified by the 30 day rule. While these individuals are in shelter less
time than the chronic clients in Table 1, they are still very heavy shelter
users. With median shelter stays of over 150 days and tenures of interaction
over 1000 days, we feel that this false positive group are still excellent
candidates for housing. Referring these individuals for support would not be a
misallocation of resources.

| Average | Median | 10th Pct. | 90th Pct.
---|---|---|---|---
Total Stays | 186.3 | 153.0 | 72.0 | 350.0
Total Episodes | 6.1 | 4.0 | 1.0 | 14.0
Tenure (days) | 1493.0 | 1210.0 | 122.0 | 3410.0
Usage Percentage | 30.0 | 16.1 | 5.0 | 86.0
Avg. Gap Length (days) | 8.9 | 6.2 | 1.1 | 20.1
Table 7: False positive shelter access characteristics.

Interestingly, all of the rules in Table 5 have the same structure: they are
looking for clients who have a lot of sleep stays in shelter and very few bar
or counseling events. It is not surprising that accumulating a lot of shelter
stays is a strong indicator of future chronic shelter use. However, imposing
an upper limit on bars and counseling is more interesting. Anecdotally, DI
shelter staff indicate that chronic shelter users often “fly under the radar”.
They accumulate a lot of shelter stays but otherwise avoid attracting
attention. A bar event tends to occur for clients who are barred from
accessing shelter for a period of time due to being violent or otherwise
breaking shelter rules. Counseling services are associated with clients
actively trying to exit shelter. Clients that maintain a low profile would
naturally not accumulate many of either type of event. Low profile clients are
also exactly the clients that benefit from a machine learning identification
system, since they will not necessarily be top of mind due to their limited
interactions with staff and may otherwise “fall through the cracks”.

It is important to note the limitations of performing this type of study on
data from a single emergency shelter. First, our results are necessarily
biased towards individuals who make use of shelter services. Many people
living with homelessness sleep outside and would not be captured in our data.
This shortcoming can be addressed by applying the techniques in this paper to
a data set that has been augmented to include individuals sleeping outside,
possibly using input from street outreach teams. Cluster analysis has
previously demonstrated that the client shelter access patterns in the DI data
set are broadly representative of most North American emergency shelters [25].
While the rules presented in this paper should be a reasonable starting place
for most shelters, it is still best for each shelter seeking to apply our
techniques to first conduct a new rule search on their own client data.
Finally, rather than keeping the rule set static, we recommend that the rules
are re-examined and potentially refreshed on a regular basis. This could be
done by having the rule search algorithm re-run on an updated data set. The
thresholds could also be adjusted by the shelter staff based on their
experience with the suitability of the clients on past lists.

##  5 Conclusions

Homelessness is one of the most significant challenges facing society today.
The Housing First philosophy recognizes that people experiencing homelessness
are often struggling with many challenges and that placing them in a stable
housing situation is the first step towards addressing them. The data science
community has an obligation to do its part to assist with this effort.

This paper has presented a rule search framework for the early identification
of emergency shelter clients who have the potential to become long term
shelter users. Our approach achieves performance similar to neural networks
and has a median time to identification of 162 days compared to 297 days when
standard chronic homelessness definitions are used to identify high priority
housing candidates. We are the first to discuss in detail how machine learning
can be integrated into real-time housing program delivery. Finally, our
approach is far simpler than any of the machine learning work in the
homelessness sector to date, will make intuitive sense to non-technical staff
and will be compatible with the basic IT infrastructure found in most not-for-
profit shelters. This increases the likelihood of our approach making the
transition from the research domain to practice.

Finally, we conclude by expressing our strong opinion that it would be a
mistake to rely exclusively on any machine learning method for supportive
housing referrals. Chronic shelter users are extremely complex and benefit
from a range of assessment and triaging methods [5]. We feel the techniques
presented in this paper are best used to identify low profile clients who may
otherwise escape notice by staff, particularly in very busy shelters. However,
the final decision regarding which individuals are selected for housing and
where they are placed should be made by experienced case managers working
collaboratively with the clients themselves.

##  6 Acknowledgments

The authors would like to acknowledge the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC), the Calgary Drop-In Centre and
the Government of Alberta. This study is based in part on data provided by
Alberta Community and Social Services. The interpretation and conclusions
contained herein are those of the researchers and do not necessarily represent
the views of the Government of Alberta. Neither the Government of Alberta nor
Alberta Community and Social Services express any opinion related to this
study.

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# Regular Black Holes in $D=2+1$ with $f(R)$ Gravity

A. C. L. Santos<EMAIL_ADDRESS>Universidade Federal do Ceará
(UFC), Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  R. V. Maluf
<EMAIL_ADDRESS>Universidade Federal do Ceará (UFC), Departamento de
Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  Departamento
de Física Teórica and IFIC, Centro Mixto Universidad de Valencia - CSIC.
Universidad de Valencia, Burjassot-46100, Valencia, Spain.  C. R. Muniz
<EMAIL_ADDRESS>Universidade Estadual do Ceará (UECE), Faculdade de
Educação, Ciências e Letras de Iguatu, Av. Dário Rabelo s/n, Iguatu-CE,
63.500-00 - Brazil.

(August 28, 2024)

###### Abstract

Despite the experimental success of general relativity in the scales it has
been tested, there are still some inconsistencies, such as explaining the
acceleration of the universe and singularities. $f(R)$ gravity has emerged to
provide corrections to the Ricci scalar, leading to new equations of motion
that could explain this acceleration. On the other hand, Regular Black Holes,
supported by non-linear electrodynamics, have well-defined curvature scalars
but often violate energy conditions. In this work, we investigate how to
obtain regular solutions in $(2+1)$ dimensions by considering $f(R)$ gravity
and non-linear electrodynamics. Subsequently, we find that non-linear
electrodynamics models that guarantee regular solutions in General Relativity
do not produce regular solutions in $f(R)$ theory. Finally, we observe that
energy conditions are still violated in $f(R)$ gravity.

Regular Black Holes. Non-linear Electrodynamics. Gravity Modified Theories.

##  I Introduction

The experimental success of General Relativity at the scales it has been
tested is undeniable. From classical tests (such as the accurate prediction of
light deflection and the explanation for the precession of Mercury, for
instance Will:2005va ), to the prediction of Black Holes and Gravitational
Waves, which were recently confirmed EventHorizonTelescope:2019dse ;
EventHorizonTelescope:2022wkp ; LIGOScientific:2016aoc , the theory presents
considerable robustness. However, some aspects remain misunderstood: the
confirmed accelerated expansion of the universe SupernovaSearchTeam:1998fmf
and singularities Penrose:1964wq . Over the years, various theoretical
approaches have emerged in an attempt to shed light on these questions.

In the search for corrections to the equations of motion to explain the
accelerated expansion of the universe, extensions or modifications of General
Relativity have emerged – since the beginning of the theory – such as the
inclusion of Torsion and non-metricity (a great compilation can be found in
CANTATA:2021ktz ). In this way, $f(R)$ gravity Olmo:2011uz  predicts
corrections given by nonlinear terms of the curvature scalar in the Lagrangian
that would lead to cosmological self-accelerating solutions without resorting
to the existence of dark energy. With this framework, this extension has been
extensively studied in astrophysical and cosmological scenarios delaCruz-
Dombriz:2006kob ; Li:2007xn ; Bamba:2013fha ; Capozziello:2008ch ;
Muniz:2022eex . Also, an interesting feature of this model is the mapping into
a scalar-tensor theory, which has also undergone deep investigation in
cosmological scenarios Olmo:2005hc .

On the other hand, since Bardeen’s pioneering solution BARDEEN , Regular Black
Holes have emerged in a classical context: Black Holes that possess well-
defined curvature scalars at every point in space Lan:2023cvz . These
solutions are obtained when we consider nonlinear electrodynamics as source
Ayon-Beato:2000mjt  or fluids with certain specific energy densities
Maluf:2022jjc . The cost of this regularity is the violation of energy
conditions, most of the time. In this sense, many regular solutions have been
investigated in the literature Rodrigues:2020pem ; Olmo:2023lil ;
Guerrero:2022msp , including quantum corrections Maluf:2018ksj .

Despite some problems, like the absence of dynamical degrees of freedom, since
the first Black Hole solution in ($2+1$) dimensions found by Bañados,
Teitelboim and Zanelli Banados:1992wn , the interest in solutions in low-
dimensional contexts has been expanding Karakasis:2021lnq ; Karakasis:2021ttn
. Among the reasons, we can emphasize the various predicted and detected
effects in lower dimensions in other branches, such as Statistical Mechanics
and Quantum Mechanics. It is worth mentioning that the unexpected and
intriguing Unruh effect was initially predicted in $(1+1)$ dimensions
Unruh:1976db  and the most promising experimental investigation of the EP=EPR
conjecture was carried out in a two-dimensional gravity dual system
Jafferis:2022crx . Also, the possibility of spontaneous dimensional reduction
is predicted in some models to occur at extremely high energy scales
Carlip:2016qrb .

In this context, to address a solution that deals with both the accelerated
expansion of the universe and singularities, we dare to ask: Just as in
($3+1$) dimensions Rodrigues:2015ayd ; Rodrigues:2016fym , do we find regular
solutions in ($2+1$) dimensions in $f(R)$ gravity? If so, what is the most
general method to ensure that we will have a regular solution? Do nonlinear
electrodynamics and energy densities that ensured regular solutions in General
Relativity still ensure regular solutions in $f(R)$ gravity? Are the energy
conditions still violated?

To answer these questions, we have organized the paper as follows: in section
II, without specifying a $f(R)$ model, we find the necessary conditions for
having a regular solution supported by non-linear electrodynamics.
Subsequently, we consider models of non-linear electrodynamics that guarantee
regular solutions in General Relativity to investigate if these models still
support regular solutions in a $f(R)$ gravity and we also study the energy
conditions. In section III, considering a regular geometry, we investigate
whether the energy conditions are violated. Finally, in section V, we outline
our perspectives and conclusions.

##  II General Regular Black Holes Solutions With NED

In this section, we are interested in creating the most general way to find a
regular solution in a $f(R)$ theory in ($2+1$) dimensions considering
nonlinear electrodynamics. In this sense, we start with the following action

$S=\int d^{3}x\sqrt{-g}\left[\frac{f(R)}{16\pi}+L(F)\right],$ (1)

where $g$ stands for the determinant of the metric $g_{\mu\nu}$, $f(R)$ is an
arbitrary function of the Ricci scalar and $L(F)$ is the Lagrangian of the
nonlinear electrodynamics with $F=F^{\mu\nu}F_{\mu\nu}$. Varying (1) with
respect to the metric, we obtain the equations of motion

$f_{R}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f(R)+g_{\mu\nu}\Box
f_{R}-\nabla_{\mu}\nabla_{\nu}f_{R}=8\pi T_{\mu\nu},$ (2)

where $f_{R}$ represents the derivative of $f(R)$ with respect to $R$. The
energy-momentum tensor is given by

$T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}=g_{\mu\nu}L(F)-4L_{,F}F_{\mu\alpha}F_{\nu}^{\leavevmode\nobreak\
\alpha},$ (3)

and $L_{,F}$ represents the derivative of $L(F)$ with respect to $F$. The
electromagnetic field equations are determined by varying with respect to the
potential $A_{\mu}$, remembering that the field-strength tensor $F_{\mu\nu}$
is defined as $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, we
obtain

$\nabla_{\mu}(L_{,F}F^{\mu\nu})=0.$ (4)

We will assume a circularly symmetric spacetime

$ds^{2}=-b(r)dt^{2}+b(r)^{-1}dr^{2}+r^{2}d\phi^{2},$ (5)

where $b(r)$ is an arbitrary function of the radial coordinate $r$. It is
worth mentioning that the choice $\displaystyle g_{tt}g_{rr}=-1$, it is
naturally obtained in General Relativity, but that is not the case in $f(R)$
gravity theories, where such a choice represents a further constraint on the
system Rodrigues:2015ayd . Considering radial electric and scalar magnetic
fields Cataldo:2000ns , then

$F_{\mu\nu}=\left(\begin{array}[]{cccc}0&E(r)&0\\\ -E(r)&0&-B(r)\\\
0&B(r)&0\end{array}\right).$ (6)

Using the ansatz to the metric and the energy-momentum tensor, we find

$\displaystyle rf(R)+16\pi
rL(F)+b^{\prime}(r)f_{R}(r)-2b(r)f_{R}^{\prime}(r)-rb^{\prime}(r)f_{R}^{\prime}(r)$
$\displaystyle+64\pi
rE(r)^{2}L_{,F}+rf_{R}(r)b^{\prime\prime}(r)-2rb(r)f_{R}^{\prime\prime}(r)=0.$
(7) $\displaystyle 16\pi
r^{2}L(F)+rf_{R}(r)b^{\prime}(r)-2rb(r)f_{R}^{\prime}(r)-r^{2}b^{\prime}(r)f_{R}^{\prime}(r)+r^{2}f(R)$
$\displaystyle+64\pi r^{2}E(r)^{2}L_{,F}-64\pi
b(r)B(r)^{2}L_{,F}+r^{2}f_{R}(r)b^{\prime\prime}(r)=0.$ (8)
$\displaystyle-r^{2}f(R)+2(-8\pi
r^{2}L(F)-rf_{R}(r)b^{\prime}(r)+r^{2}b^{\prime}(r)f_{R}^{\prime}(r)$
$\displaystyle+32\pi b(r)B(r)^{2}L_{,F}+r^{2}b(r)f_{R}^{\prime\prime}(r))=0.$
(9) $32\pi b(r)E(r)B(r)L_{,F}=0.$ (10)

Where the prime $(^{\prime})$ stands for the total derivative with respect to
the radial coordinate $r$. From equation (10), we obtain that, curiously, the
radial magnetic field or the radial electric field should be zero. We will
consider the radial electric field just like in General Relativity
Cataldo:2000ns . Therefore, the equation of motion from the matter sector (4)
is

$E(r)L_{,F}=-\frac{q}{r}.$ (11)

Where $q$ is an integration constant related to the electric charge. Since
$B(r)=0$, subtracting (II) from (II), we have:

$rb(r)f^{\prime\prime}_{R}(r)=0.$ (12)

The solution of equation (12) is given by

$f_{R}(r)=\alpha r+c_{1},$ (13)

where $c_{1}$ and $\alpha$ are constants of integration. The integration of
(13) yields

$f(R)=R-2\Lambda+\alpha\int r(R)dR,$ (14)

where we have set an integration constant to $-2\Lambda$ to match with General
Relativity in the particular case with $\alpha=0$ Maluf:2022jjc . And,
therefore, we have corrections to Ricci scalar and consequently to General
Relativity, as expected. At this point, it is interesting to note that the
cosmological constant emerges naturally from the equations of motion as an
integration constant of the $f(R)$ theory. On the other hand, with the metric
ansatz (5), the invariant scalars, Ricci and Kretschmann, are given by,
respectively

$R=-b^{\prime\prime}(r)-\frac{2b^{\prime}(r)}{r},$ (15)
$K=b^{\prime\prime}(r)^{2}+\frac{2b^{\prime}(r)^{2}}{r^{2}}.$ (16)

To find regular solutions, we will adopt the He and Ma procedure He:2017ujy .
We begin by imposing a regular geometry, which means these scalars are well-
defined in all domains, and so we should have the limit satisfied

$\lim_{r\to 0}\frac{b^{\prime}(r)}{r}=constant.$ (17)

Since in a vacuum solution, we need to recover the Bañados, Teitelboim and
Zanelli case Banados:1992wn , we take the form to the metric

$b(r)=-m+\frac{r^{2}}{l^{2}}+k(r),$ (18)

where $k(r)$ is a function responsible for guaranteeing the regularity of the
solution and should be expanded in the following way He:2017ujy

$k(r)=k_{0}+k_{2}r^{2}+O(r^{3}),$ (19)

avoiding the existence of poles. Deriving (II) with respect to $r$ and
replacing (13), (15) and (18), we obtain

$\displaystyle-k^{(3)}(r)(1+\alpha r)-2\alpha
k^{\prime\prime}(r)+\frac{2\alpha k^{\prime}(r)}{r}+16\pi\frac{\partial
L(F)}{\partial F}\frac{\partial F}{\partial r}=0.$ (20)

At this point, we want to obtain more physical results imposing that the
nonlinear electrodynamics should reduce to the Maxwell theory in the weak
field limit, namely $L(F)\to-F$, for large $r$ Maluf:2022jjc . So, in the
asymptotic limit

$-k^{(3)}(r)(1+\alpha r)-2\alpha k^{\prime\prime}(r)+\frac{2\alpha
k^{\prime}(r)}{r}-\frac{64\pi q^{2}}{r^{3}}=0.$ (21)

And when $\displaystyle\alpha\to 0$, $\displaystyle k^{(3)}(r)=\frac{64\pi
q^{2}}{r^{3}}$, which it is in agreement with General Relativity, where
$\displaystyle k^{\prime}(r)\propto\frac{1}{r}$ and we have a cosmological
constant in the action (1) He:2017ujy . In this way, we can construct regular
black holes with functions $k(r)$ satisfying the equations (19), (20) and
(21). These equations were obtained without specifying the $f(R)$ gravity.
However, we naturally obtained that the constraint of the equation (13) should
be respected. To find such a theory, we must find the $r(R)$ based on the
Ricci scalar and, finally, solve the integral (14).

##  III New Solutions Considering NED’s Models

First of all, let us analyze the BTZ-like charged solution, which means the
case when $L(F)=-F$, where $\displaystyle F(r)=-\frac{2q^{2}}{r^{2}}$, so we
obtain:

$\displaystyle k(r)=32\pi q^{2}\left(\ln(r)\left(\alpha^{2}r^{2}\ln(r)+2\alpha
r(\alpha r+1)-1\right)+4\alpha r\right.$
$\displaystyle\left.+2\alpha^{2}r^{2}\text{Li}_{2}(r\alpha+1)-2\alpha^{2}r^{2}(-\ln(\alpha(-r))+\ln(r)+1)\ln(\alpha
r+1)\right).$ (22)

And when $\alpha\to 0$, we obtain

$k(r)=-32\pi q^{2}\ln(r).$ (23)

Is it a regular solution? The Ricci scalar is given by:

$\displaystyle R$ $\displaystyle=$ $\displaystyle
192\pi\alpha^{2}q^{2}(2(-\ln(-\alpha r+\ln(r+1)\ln(\alpha
r+1)-\ln(r(\ln(r+2))-\frac{6}{l^{2}}$ (24) $\displaystyle+$
$\displaystyle\frac{1}{r^{2}(\alpha r+1)^{2}}(32\pi
q^{2}(-12\alpha^{2}r^{2}((\alpha r+1)^{2}\text{Li}_{2}(\alpha r+1)+\alpha
r(\ln(r+2))$ $\displaystyle-$ $\displaystyle 3\alpha r(13\alpha r+4)-2\alpha
r(9\alpha r+2)\ln(r+1)).$

In the asymptotic limit, $r\to 0$

$R\approx\frac{32\pi q^{2}}{r^{2}},$ (25)

which diverges independently of $\alpha$ and, consequently, $f(R)$ theory.
Therefore, the $f(R)$ gravity does not "cure" the singularity of the solution.
On the other hand, He and Ma He:2017ujy  construct in the context of General
Relativity some $(2+1)$-dimensional regular black holes with specifics $k(r)$
functions and the associated nonlinear electrodynamics which guarantees it. As
examples of the generality of the equation (20), we are interested in
investigating if these nonlinear electrodynamics continue to ensure regular
solutions in a $f(R)$ theory.

Case I:

$L(r)=-\frac{a^{2}q^{2}}{16\pi(a^{2}+r^{2})^{\frac{3}{2}}}.$ (26)

So, the equation (20) becomes

$\displaystyle-k^{(3)}(r)(1+\alpha r)-2\alpha
k^{\prime\prime}(r)+\frac{2\alpha
k^{\prime}(r)}{r}+\frac{3a^{2}q^{2}r}{\left(a^{2}+r^{2}\right)^{5/2}}=0.$ (27)

Wich solution it is

$\displaystyle k(r)=q^{2}\left(\sqrt{a^{2}+r^{2}}(2\alpha
r-1)+r^{2}\alpha\left(2a\alpha\ln(r)-2a\alpha\ln\left(a\left(\sqrt{a^{2}+r^{2}}+a\right)\right)\right.\right.$
$\displaystyle\left.\left.+\frac{\left(2a^{2}\alpha^{2}+1\right)\left(\ln\left(\alpha
a^{2}+\sqrt{\left(a^{2}\alpha^{2}+1\right)\left(a^{2}+r^{2}\right)}-r\right)-\ln(\alpha
r+1)\right)}{\sqrt{a^{2}\alpha^{2}+1}}\right)\right),$ (28)

and when $\alpha\to 0$, we recover the General Relativity result, where:

$k(r)=-q^{2}\sqrt{a^{2}+r^{2}}.$ (29)

The Ricci scalar is a long equation, which in the asymptotic limit $r\to 0$,
we obtain the dominant term

$R\approx-\frac{4\sqrt{a^{2}}\alpha q^{2}}{r},$ (30)

that diverges. Evidently, when $\alpha\to 0$ we recover the regularity
obtained in the General Relativity result.

Case II:

$L(r)=\frac{a^{2}q^{2}(ar-q)}{8\pi(ar+q)^{3}}.$ (31)

So, the equation (20) becomes

$\displaystyle-k^{(3)}(r)(c_{1}+\alpha r)-2\alpha
k^{\prime\prime}(r)+\frac{2\alpha
k^{\prime}(r)}{r}-\frac{4a^{3}q^{2}(ar-2q)}{(ar+q)^{4}}=0.$ (32)

Wich solution is given by

$\displaystyle
k(r)=\frac{2}{3}q^{2}\left(\frac{3a\left(a^{2}r^{2}+aqr+q^{2}\right)}{q(\alpha
q-a)(ar+q)}-\frac{12\alpha
q}{a}+\frac{3ar}{q}+3\alpha^{2}r^{2}(2\ln(q)+5)\ln(r)+15\alpha
r+\pi^{2}\right.$ $\displaystyle-\frac{3\alpha^{2}r^{2}\left(2a^{2}-8\alpha
aq+5\alpha^{2}q^{2}\right)\ln(\alpha r+1)}{(a-\alpha
q)^{2}}-\frac{6\ln\left(\frac{\alpha(ar+q)}{\alpha
q-a}\right)\left(\left(a-a\alpha^{2}r^{2}\right)\ln(\alpha r+1)+\alpha
q\right)}{a}$ $\displaystyle+3\ln(ar+q)\left(-\frac{a^{4}r^{2}}{q^{2}(a-\alpha
q)^{2}}+\frac{a^{2}r^{2}}{q^{2}}+2\alpha\left(\frac{ar^{2}}{q}+\frac{q}{a}+r\right)-2\left(\alpha^{2}r^{2}-1\right)\ln(\alpha
r+1)-1\right.$ $\displaystyle\left.\left.-2\ln\left(\frac{\alpha
ar+a}{a-\alpha
q}\right)\right)-6\alpha^{2}r^{2}\text{Li}_{2}\left(-\frac{ar}{q}\right)-6\text{Li}_{2}\left(\frac{(q+ar)\alpha}{q\alpha-a}\right)+6\left(\alpha^{2}r^{2}-1\right)\text{Li}_{2}\left(\frac{r\alpha
a+a}{a-q\alpha}\right)\right).$ (33)

And when $\alpha\to 0$, we recover the General Relativity result, where

$k(r)=-\frac{2q^{3}}{ar+q}-2q^{2}\ln(ar+q).$ (34)

In the asymptotic limit, $r\to 0$, we have that the Ricci scalar is given by

$R\approx-\frac{4\alpha q^{2}(2\ln(q)+5)}{r},$ (35)

that diverges and also when $\alpha\to 0$ we recover the regularity obtained
in the General Relativity case. In fact, the result that these solutions are
not regular is not surprising, since the $k(r)$ functions obtained from the
nonlinear electrodynamics do not satisfy the condition (19). Therefore, the
modifications induced by a $f(R)$ theory remove the regularity of solutions
conferred by this nonlinear electrodynamics in $(2+1)$ dimensions.

###  III.1 Energy Conditions

To describe the energy conditions with the appropriate physical
interpretation, we follow the approach of Rodrigues:2015ayd  and rewrite the
equations of motion (2) as

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=f_{R}^{-1}\left[8\pi
T_{\mu\nu}+\frac{1}{2}g_{\mu\nu}(f-Rf_{R})+(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box)f_{R}\right]\equiv
8\pi\mathcal{T}_{\mu\nu}^{(eff)},$ (36)

where,

${\mathcal{T}}^{0(eff)}_{0}=-\rho^{(eff)},\;{\mathcal{T}}^{1(eff)}_{1}=p_{r}^{(eff)},\;{\mathcal{T}}^{2(eff)}_{2}=p_{\theta}^{(eff)}.$
(37)

This redefinition could be interpreted as the possibility of transferring
nonlinearity from the geometric sector to the matter sector Afonso:2018hyj  to
study the energy conditions in the background of the geometry of General
Relativity where the theorems of singularities were described. Using the
ansatz to the metric and the energy-momentum tensor, we find

$\displaystyle\rho^{(eff)}$ $\displaystyle=$ $\displaystyle\frac{-1}{16\pi
rf_{R}(r)}(rf(R)+16\pi rL(F)+2f_{R}(r)b^{\prime}(r)-2b(r)f_{R}^{\prime}(r)$
(38) $\displaystyle-$ $\displaystyle rb^{\prime}(r)f_{R}^{\prime}(r)+64\pi
rE^{2}(r)L_{F}+rf_{R}(r)b^{\prime\prime}(r)),$ $\displaystyle p_{r}^{(eff)}$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi rf_{R}(r)}(rf(R)+16\pi
rL(F)+2f_{R}(r)b^{\prime}(r)-2b(r)f_{R}^{\prime}(r)$ (39) $\displaystyle-$
$\displaystyle rb^{\prime}(r)f_{R}^{\prime}(r)+64\pi
rE^{2}(r)L_{F}+rf_{R}(r)b^{\prime\prime}(r)),$ $\displaystyle
p_{\theta}^{(eff)}=-\frac{(rf(R)+16\pi
rL(F)+2f_{R}(r)b^{\prime}(r)-2rb^{\prime}(r)f_{R}^{\prime}(r)+rf_{R}(r)b^{\prime\prime}(r)}{16\pi
rf_{R}(r)}.$ (40)

Considering (II), (II) and (II), we can reduce to:

$\rho^{(eff)}=-p_{r}^{(eff)}=-\frac{b^{\prime}(r)}{16\pi r},$ (41)
$p_{\theta}^{(eff)}=\frac{b^{\prime\prime}(r)}{16\pi}.$ (42)

Compatible with the relationships imposed between
$\mathcal{T}_{\mu\nu}^{(eff)}$ and the Einstein tensor, $G_{\mu\nu}$. Now we
can draw a parallel with General Relativity to study the energy conditions
that impose: the energy density plus any of the pressures cannot be negative
(Null Energy Condition), the energy density must be greater than or equal to
the absolute value of any of the pressures (Dominant Energy Condition), the
sum of energy density and pressure cannot be negative and NEC condition
(Strong Energy Condition). Finally, the energy density cannot be negative and
NEC condition (Weak Energy Condition). In Figure 1, we analyze these
conditions graphically for the solutions previously found.

As observed in graph (a) the conditions are not violated, since $\alpha=0$ and
we have the classical electromagnetism case. In graph (b), when we include
contributions from $f(R)$ gravity - with $\alpha\neq 0$ \- some energy
conditions are violated. In graphs (c) and (d), some energy conditions are
already violated, because we have regular geometries whose sources are
nonlinear electrodynamics. By including $f(R)$ gravity corrections to these
solutions in (d) and (f) - with $\alpha\neq 0$ \- we notice that these energy
conditions are still violated. Therefore, $f(R)$ gravity did not prevent the
violation of energy conditions.

---
(a) (b)
(c) (d)
(e) (f)
Figure 1:  The graphical representation of the radial dependence for
$\rho^{(eff)}$ and combinations with $p_{r}^{(eff)}$ and $p_{\theta}^{(eff)}$
given from solutions (III), (III), (III), respectively, with
$l=0.7,q=10,m=80,a=2,\alpha=0$ (left panels) and
$l=0.7,q=10,m=80,a=2,\alpha=1$ (right panels).

##  IV A Regular Geometry

Now we are interested in knowing whether, as in General Relativity, given a
regular geometry, the energy conditions will be violated. Therefore, let us
consider a hypothetical scenario in which we have a regular geometry within
the context of $f(R)$ gravity. However, it turns out to be singular in the
case of General Relativity:

$b(r)=-m+\frac{r^{2}}{l^{2}}-q^{2}\ln{\left(\alpha^{2}l^{2}+\frac{r^{2}}{l^{2}}\right)},$
(43)

where $\alpha$ is the parameter related to the correction to $R$ in the $f(R)$
model. The Ricci and Kretschmann scalars are given by

$R=\frac{2\left(-3\alpha^{4}l^{8}+3\alpha^{2}l^{6}q^{2}-6\alpha^{2}l^{4}r^{2}+l^{2}q^{2}r^{2}-3r^{4}\right)}{l^{2}\left(\alpha^{2}l^{4}+r^{2}\right)^{2}}.$
(44)
$K=4\left[\frac{3}{l^{4}}+\frac{q^{4}\left(3\alpha^{4}l^{8}+2\alpha^{2}l^{4}r^{2}+3r^{4}\right)}{\left(\alpha^{2}l^{4}+r^{2}\right)^{4}}-\frac{2q^{2}\left(3\alpha^{2}l^{4}+r^{2}\right)}{l^{2}\left(\alpha^{2}l^{4}+r^{2}\right)^{2}}\right].$
(45)

These functions exhibit well-defined properties across their entire domain,
ensuring a regular geometry, notably evident when approaching $r\to 0$.
However, the regularity is compromised in the scenario where $\alpha=0$,
namely, when we return to General Relativity. Following, we can perform an
inversion to obtain $r(R)$ and integrate the equation (14) to determine the
associated $f(R)$ gravity Rodrigues:2015ayd :

$\displaystyle f(R)=R-2\Lambda+\frac{1}{4\alpha
l^{8}q^{2}}[(\sqrt{l^{4}q^{4}+4\alpha^{2}l^{6}q^{2}\left(l^{2}R+6\right)}-l^{2}q^{2})^{2}$
$\displaystyle\sqrt{\frac{(l^{2}q^{2}-l^{4}(6+l^{2}R)\alpha^{2}+\sqrt{l^{4}q^{4}+4l^{6}q^{2}(6+l^{2}R)\alpha^{2})}}{(6+l^{2}R)}}-16\alpha
l^{6}q^{4}\tan^{-1}\beta(R)],$ (46)

where,

$\beta(R)=\frac{1}{\alpha
l^{2}}\sqrt{\frac{l^{2}q^{2}-\left(\alpha^{2}l^{4}\left(l^{2}R+6\right)\right)+\sqrt{l^{4}q^{4}+4\alpha^{2}l^{6}q^{2}\left(l^{2}R+6\right)}}{l^{2}R+6}}$
(47)

and derivating with respect to $R$, we obtain

$f_{R}(R)=1+\alpha\sqrt{\frac{L^{2}q^{2}-\left(\alpha^{2}L^{4}\left(L^{2}R+6\right)\right)+\sqrt{L^{4}q^{4}+4\alpha^{2}L^{6}q^{2}\left(L^{2}R+6\right)}}{L^{2}R+6}},$
(48)

and as expected $\displaystyle f_{R}(R)=1$, when $\alpha\to 0$. With the
equation(20) and the $k(r)$ function of (43), we can find the nonlinear
electrodynamics associated

$L(r)=\frac{q^{2}}{4\pi}\left(\frac{\left(r^{2}-\alpha^{2}l^{4}\right)-2\alpha^{3}l^{4}r}{2\left(\alpha^{2}l^{4}+r^{2}\right)^{2}}+\frac{\cot^{-1}\left(\frac{\alpha
l^{2}}{r}\right)}{l^{2}}\right)+d_{1},$ (49)

where $d_{1}$ is a constant. Unfortunately, the equation (49) does not
actually represent a nonlinear electrodynamics, since

$E(r)=\int\frac{rL^{\prime}(r)}{4q}dr=\frac{q}{32\pi}\left(\frac{2cr^{3}-\alpha^{5}l^{8}-3\alpha^{3}l^{4}r^{2}}{\left(\alpha^{2}l^{4}+r^{2}\right)^{2}}+\alpha\ln\left(\alpha^{2}l^{4}+r^{2}\right)\right),$
(50)

and in the asymptotic limit, $r\to\infty$

$E(r)\approx\frac{\alpha q\ln(r)}{16\pi}.$ (51)

Which is different from classical electromagnetism. This conclusion reinforces
the idea that it seems in a $f(R)$ gravity, we do not have regular solutions
sourced by nonlinear electrodynamics in $(2+1)$ dimensions.

###  IV.1 Energy Conditions

With the functions $f(R)$ and $L(r)$ defined, we graphically examine the
energy conditions from the solution (43) in Figure (2). In graph (a), the
energy conditions are not violated, since for $\alpha\neq 0$ we return to the
charged BTZ solution. However, when we regularize the solution with a term
from $f(R)$ gravity, we observe that the energy conditions are violated. Thus,
in principle, $f(R)$ gravity, even when exhibiting regularity, does not ensure
the fulfillment of energy conditions.

---
(a) (b)
Figure 2:  The graphical representation of the radial dependence for
$\rho^{(eff)}$ and combinations with $p_{r}^{(eff)}$ and $p_{\theta}^{(eff)}$
given from solutions (43) with $l=0.7,q=10,m=80,\alpha=0$ (left panel) and
$l=0.7,q=10,m=80,a=2,\alpha=1$ (right panel).

##  V Conclusion

In summary, we identified the necessary conditions to find a regular solution
in $(2+1)$ dimensions supported by non-linear electrodynamics in any $f(R)$
gravity theory. We obtained a charged BTZ-like solution in $f(R)$ gravity and
investigated two models of non-linear electrodynamics that provided regular
solutions in General Relativity. We found that all solutions are non-regular.
Therefore, the corrections in $f(R)$ gravity do not "cure" the singularity of
the charged BTZ solution and it remove the regularity that non-linear
electrodynamics had conferred in General Relativity. When analyzing the energy
conditions, we observed that the energy conditions continue to be violated.

Attempting to answer the question: given a hypothetical geometry regularized
by $f(R)$ gravity, would the energy conditions still be violated? We
considered a regular geometry for $\alpha\neq 0$ and non-regular for
$\alpha=0$. We defined the $f(R)$ gravity, as well as the associated non-
linear electrodynamics (since the electric field did not recover classical
electromagnetism at infinity, regularity would not be effectively conferred by
non-linear electrodynamics). We found that even in this scenario, the energy
conditions would still be violated.

## Acknowledgments

The authors thank the Fundação Cearense de Apoio ao Desenvolvimento Científico
e Tecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES), and the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Grants no. 88887.822058/2023-00 (ACLS), no. 308268/2021-6
(CRM), and no. 200879/2022-7 (RVM) for financial support. R. V. Maluf
acknowledges the Department de Física Teòrica de la Universitat de València
for the kind hospitality.

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