# Identifying Undercompensated Groups Defined by Multiple Attributes in Risk
Adjustment

Anna Zink
Harvard University
Cambridge, MA 02138
<EMAIL_ADDRESS>
Sherri Rose
Stanford University
Stanford, CA 94305
<EMAIL_ADDRESS>
###### Abstract

Risk adjustment in health care aims to redistribute payments to insurers based
on costs. However, risk adjustment formulas are known to underestimate costs
for some groups of patients. This undercompensation makes these groups
unprofitable to insurers and creates incentives for insurers to discriminate.
We develop a machine learning method for “group importance” to identify
unprofitable groups defined by multiple attributes, improving on the arbitrary
nature of existing evaluations. This procedure was designed to evaluate the
risk adjustment formulas used in the U.S. health insurance Marketplaces as
well as Medicare. We find that a number of previously unidentified groups with
multiple chronic conditions are undercompensated in the Marketplaces risk
adjustment formula, while groups without chronic conditions tend to be
overcompensated in the Marketplaces. The magnitude of undercompensation when
defining groups with multiple attributes is larger than with single
attributes. No complex groups were found to be consistently under- or
overcompensated in the Medicare risk adjustment formula. Our work provides
policy makers with new information on potential targets of discrimination in
the health care system and a path towards more equitable health coverage.

_Keywords_ Risk adjustment $\cdot$ Subgroups $\cdot$ Random Forests

##  1 Introduction

Risk adjustment is an essential tool in regulated health insurance markets. It
redistributes health plan payments to insurers with higher cost patients,
aiming to decrease the relationship between health and profits to ensure that
sicker individuals are not discriminated against by insurers (McGuire and van
Kleef, 2018). Risk adjustment for health plan payments has been successful in
reducing selection incentives for many individuals, but incentives to
discriminate still exist for groups of individuals whose costs are
underpredicted by the risk adjustment formula, including those with mental
health and substance use disorders (McGuire et al., 2014; Montz et al., 2016).

The ability of insurers to discriminate against some groups of patients has
been curbed since the passage of the Affordable Care Act (ACA) in 2010, which
prevents insurers from refusing enrollment or changing premium prices based on
enrollee health. However, insurers are able to attract more profitable
enrollees or discriminate against less profitable enrollees through
advertising as well as benefit design choices. Changes to benefit design
include which providers comprise their provider networks and which drugs are
placed in higher cost tiers or subject to drug formulary management tools
(Shepard, 2016; Lavetti and Simon, 2018; Rose et al., 2017; Carey, 2017;
Aizawa and Kim, 2018; Geruso et al., 2019). Recently, machine learning methods
have been used to identify individual conditions that are underpaid by the
risk adjustment formula as well as drugs predictive of unprofitability, but
these studies did not focus on groups defined by more than one attribute(Rose
et al., 2017; Rose, 2018). In Germany and the Netherlands, researchers have
incorporated interactions of variables into the risk adjustment formula based
on the partitioning of a single regression tree predicting the residual, but
only considered groups defined within the current risk adjustment formula
(Buchner et al., 2017; van Veen et al., 2018). To date, no systematic method
exists for identifying groups at risk for discrimination in the health care
payment system, or more broadly in the algorithmic fairness literature,
especially groups defined with a more complex set of attributes. Even groups
defined by a single sensitive attribute are typically pre-specified
(Chouldechova and G’Sell, 2017).

In this paper, we present the first data-driven method for identifying
undercompensated groups in health plan payment risk adjustment that are
defined by multiple attributes. Extending the concept of variable importance
for single attributes or predefined groups, we construct a new measure of
“group importance” in the random forests algorithm that relies on multiple
attributes (Grömping, 2009; Louppe et al., 2013; Wehenkel et al., 2018; Fisher
et al., 2019). Groups are defined based on the demographic and clinical
categories susceptible to benefit redesign, namely chronic conditions, age,
and sex. We designed our method for the purposes of studying the risk
adjustment formulas used in the individual health insurance Marketplaces
created by the ACA as well as Medicare. Our newly proposed method for
identifying undercompensated groups overcomes the arbitrary nature of existing
evaluations of risk adjustment, providing policy makers a tool for uncovering
incentives for selection that persist in insurance markets and a path towards
more equitable health coverage.

##  2 Methods

###  2.1 Defining Groups

Our primary objective is to identify complex groups vulnerable to
discrimination in the health insurance market through benefit design.
Therefore, in defining groups, we consider what factors are necessary for this
type of discrimination. To start, the groups must be actionable for insurers,
meaning the insurers must know which groups are profitable or unprofitable and
be able to act upon this information via the design of provider networks or
drug formularies (Ellis and McGuire, 2007). The losses or gains caused by the
group must also be large enough to substantially impact insurers’ revenue.
There are a number of factors that could lead to this type of large impact for
insurers, including the size of the group, the size of the undercompensation
of the group, and the persistence of the undercompensation over time (McGuire
and van Kleef, 2018). For these reasons, we focus on identifying groups
defined by prevalent chronic conditions that require specialist care or drug
treatments. In addition to chronic conditions, we consider whether age and
documented sex affect the size of undercompensation.

###  2.2 Notation

The canonical risk adjustment formula estimates individual annual spending $Y$
using a vector of input variables $X$ that contains $j$ demographic variables,
$D=\\{D_{1},D_{2},\dots,D_{j}\\}$ and $k$ health condition categories,
$H=\\{H_{1},H_{2},\dots,H_{k}\\}$ or $X=\\{D,H\\}$. The risk adjustment
formula is some function $f$ mapping $X$ to $Y$,$f\left(X\right)=Y$. The
residual of the risk adjustment formula, $R=\hat{Y}-Y$, measures the under- or
overprediction for each individual, where $\hat{Y}$ is predicted spending. We
are interested in the set of groups $G$ predictive of the residual $R$. To
define the groups, we select $s$ indicator variables
$I=\\{l_{1},I_{2},\dots,I_{s}\\}$ that will comprise the components of the
groups in $G$: a set of chronic conditions, sex, and age categories. A group
in $G$ is defined as any combination of components in $I$. This allows us to
consider various levels of complex groups. For example, older women with
cancer and mental health disorders would be one group and women with mental
health disorders another. We examine only binary variables; relaxing this
restriction would increase the number of groups substantially.

###  2.3 Estimation

Insurers can adjust their premiums at defined market levels to account for
differences in costs in different geographic regions (CMS, 2020). Because we
are interested in undercompensated groups within markets, we want to adjust
for market-level differences in spending. Therefore, we start by estimating a
slightly modified risk adjustment formula using ordinary least squares
regression:

$Y=\beta X+\textnormal{Market}+\epsilon,$

where, beyond $X$, we additionally control for geographic market, Market, to
account for routine premium adjustments made at the market-level.

To find the groups in $G$ most predictive of the residual $R$, we predict $R$
with the group component variables $I$ using the random forests algorithm. The
random forests algorithm grows many decision trees to predict $R$. Each tree
uses a bootstrapped sample from the original data to decrease the variance.
This is often referred to as bagging, but unlike other forms of bagging, the
random forests algorithm also selects a subset of variables to consider at
each split point in the tree so the same highly correlated variables don’t
appear in every single tree (Breiman, 2001).

Tree-based methods are an intuitive way of forming groups because each
decision tree splits the sample into a set of mutually exclusive groups. These
group are defined based on the intersections of the input variables used for
predicting the outcome. At each split in the tree, a variable is selected that
best partitions the variable space in order to reduce variation within each
new node. By using the component variables $I$ to predict $R$, we naturally
create groups in $G$ that are interactions of the component variables in $I$
used to split the tree. For each tree, we find the set of groups that are
formed in the terminal nodes. We aggregate this information across all trees,
recording the number of times a group appears and its mean estimated residual
$R$ across trees and years. See Figure 1 for an illustration of this process.

Figure 1: Illustration of Random Forests for Group Importance

The hyperparameters of the random forests algorithm are important for defining
group criteria (in addition to algorithmic performance). Through
hyperparameter specification, we set the minimize size of groups by requiring
a minimum size for terminal nodes in the decision tree. We also control the
complexity of the groups by limiting the depth of the tree and the number of
terminal nodes. The deeper the tree grows, the more attributes the groups will
contain. Recall that typically only a subset of randomly selected variables is
considered at each split. By selecting a smaller number of variables at each
split, we can decorrelate not only variables across trees but also groups,
ensuring that the same groups are not favored by every tree. And finally, the
more trees we grow, the more stable our results will be, but this can be
computationally costly and after a certain number of trees only trivially
improves stability (Probst and Boulesteix, 2017).

##  3 Marketplaces Risk Adjustment

In 2014, the Department of Health and Human Services developed a risk
adjustment formula for individual and small group markets to redistribute
insurer payments in the Marketplaces newly established under the ACA (Kautter
et al., 2014). Plans estimated to have lower risk enrollees subsidize plans
with higher risk enrollees through a budget neutral risk adjustment transfer
program regulated by CMS (Semanskee and Claxton, 2016). Premiums are set
within market rating areas defined by counties or Metropolitan Statistical
Areas (MSAs) depending on the state (CMS, 2020). The Marketplaces risk
adjustment formula is built using the IBM MarketScan Commercial Database,
which contains claims on over 23 million individuals receiving employer-
sponsored private health insurance in the U.S. (IBM, 2020).

We estimated individual annual spending using the 2019 Marketplaces risk
adjustment formula inputs adding geographic fixed effects at the MSA level
using the following specification:

$Y=\textnormal{Age}\times\textnormal{Sex}+\textnormal{HCC}+\textnormal{MSA}+\epsilon,$

where Age $\times$ Sex are ten age and sex categories and HCC are a set of
Health Condition Categories mapped from diagnosis codes. We used MSAs as a
proxy for premium rating areas in all states because we did not have county
information for states.

###  3.1 Data

To evaluate the Marketplaces risk adjustment formula, we sampled adults (aged
21-64) insured for a full year in either 2016, 2017, or 2018 from the Truven
data. We calculated individual annual spending as the total spending across
the individual’s outpatient, inpatient, and carrier claims in the given year.
The traditional risk adjustment formula predicts insurer spending, but for
simplicity we used total spending, which included patient cost-sharing. Age,
sex, and MSA information was obtained from the enrollment file. HCC variables
were constructed by mapping ICD-10 diagnosis codes recorded in the inpatient,
outpatient, and carrier claims using the 2019 software. We considered age
categories (21-29, 30-39, 40-49, 50-59, 60-65), documented sex, and twelve
chronic condition indicators (arthritis, asthma and other respiratory
conditions, cancer, diabetes, heart disease, hypertension, kidney disease,
hyperlipidemia, mental health and substance use disorders, nervous system
conditions, osteoporosis, and viral infections) as component variables to
define groups in $G$. These twelve conditions were selected because they are
common chronic conditions monitored by CMS (CMS, 2021). The chronic conditions
indicators were created using individual and multilevel diagnosis-based
categories from the Agency for Healthcare Research and Quality Clinical
Classification Software (see Table A1 for mapping). These categories
incorporate more ICD-10 diagnosis codes than HCCs and therefore allow for
identification of health states not captured by HCCs.

###  3.2 Implementation

We sampled 1 million individuals for each sample year and ran separate
ordinary least squares regressions to predict annual spending. We then
deployed the random forests algorithm to predict the residual for each sample
year. We identified groups under different size and complexity hyperparameters
holding the number of trees (1,000) and the number of component variables
selected at each branch (10) constant. This resulted in 4 hyperparameter
settings: (1) minimum size = 100 and maximum nodes = 8, (2) minimum size = 100
and maximum nodes = 64, (3) minimum size = 10,000 and maximum nodes = 8, and
(4) minimum size = 10,000 and maximum nodes = 64. The random forests algorithm
was run in R using the randomForest package (Liaw and Weiner, 2002). To ensure
that identified groups persisted in all three sample years with some frequency
across trees, we limited the results to groups that appeared in at least 1% of
the trees in each sample year. Predicted versus observed residuals for each
identified group were compared to assess the accuracy of our predictions.

##  4 Results

Figure 2: Top Under- and Overcompensated Groups in the Marketplaces Risk
Adjustment (minimum node size: 10,000, maximum nodes: 8)

Note: Unfilled circles indicate the lack of a condition. Only 7
overcompensated groups were identified in this setting, after removing groups
that appeared in less than 1% of trees across the three years.

Table 1 summarizes the Marketplaces sample characteristics for each sample
year. About 5% of the sample was documented as female. The sample was fairly
evenly distributed across the five age categories with more enrollees in their
50s than any other age group (26.6 to 27.2%). Of the 12 chronic conditions,
hypertension was the most common, documented in about 14% of the sample.
Osteoporosis, chronic kidney disease, and chronic viral infections were the
least common, appearing in <1% of the sample. For each sample year, we
observed an average undercompensation of around -$1,500 to -$5,000 for most of
the selected chronic conditions. Diabetes was the least undercompensated group
with an average undercompensation of a few hundred dollars (see Table 2).
Average annual spending ranged from $6,500 to $7,000 in the study period.

In general, all four hyperparameter settings yielded similar results.
Individuals with multiple chronic conditions, in particular enrollees with
some combination of asthma, heart disease, arthritis, and mental health and
substance use disorders, tended to be undercompensated whereas individuals
with no chronic conditions were overcompensated. In Figure 2, we present the
top under- and overcompensated groups (measured by the average residual across
the three sample years) limiting the group size to 10,000 and maximum nodes to
8. Predicted undercompensation was substantially larger than overcompensation.
The groups we identified with multiple chronic conditions were
undercompensated, on average, by at least -$10,000 and up to -$29,600, whereas
overcompensated groups were all overcompensated by less than $1,000. Also, the
predicted residuals for groups defined by multiple attributes were much larger
than the average residuals for each condition individually. For example,
individuals documented as having a combination of asthma, heart disease, and
mental health and substance use disorders were estimated to be
undercompensated by up to $12,000 on average, which is more than the sum of
their observed average residuals.

Table 1: Marketplaces Sample Characteristics (in Percents) Variables | 2016 | 2017 | 2018
---|---|---|---
Documented Sex, Female | 52.3 | 52.2 | 51.6
Age |  |  |
21-29 | 17.9 | 17.9 | 18.0
30-39 | 20.4 | 20.6 | 21.0
40-49 | 23.7 | 23.7 | 23.7
50-59 | 27.2 | 26.9 | 26.6
60-65 | 10.8 | 10.8 | 10.7
Arthritis | 4.5 | 4.5 | 4.5
Asthma | 10.6 | 10.7 | 10.8
Cancer | 7.1 | 7.0 | 6.8
Diabetes | 8.9 | 9.0 | 8.9
Heart Disease | 9.1 | 9.1 | 9.3
Hypertension | 14.1 | 13.9 | 13.7
Kidney Disease | 0.6 | 0.6 | 0.6
Lipid Disorders | 10.2 | 9.7 | 9.5
Mental Health | 11.1 | 11.7 | 12.6
Nervous | 0.7 | 0.7 | 0.7
Osteoporosis | 0.6 | 0.6 | 0.6
Viral Infections | 0.4 | 0.4 | 0.4
N | 1,000,000 | 1,000,000 | 1,000,000
Table 2: Marketplaces Average Residual ($) by Chronic Condition | 2016 | 2017 | 2018
---|---|---|---
Chronic Condition | No | Yes | No | Yes | No | Yes
Arthritis | 263 | -5611 | 293 | -6213 | 283 | -5957
Asthma | 325 | -2751 | 335 | -2783 | 458 | -3785
Cancer | 163 | -2120 | 165 | -2200 | 189 | -2583
Diabetes | 23 | -236 | 19 | -188 | 47 | -481
Heart Disease | 393 | -3924 | 383 | -3813 | 502 | -4903
Hypertension | 263 | -1596 | 255 | -1583 | 315 | -1985
Kidney Disease | 18 | -2838 | 20 | -3126 | 28 | -4596
Lipid Disorders | 121 | -1065 | 113 | -1044 | 98 | -930
Mental Health | 312 | -2499 | 330 | -2492 | 398 | -2770
Nervous | 33 | -4775 | 30 | -4382 | 36 | -5229
Osteoporosis | 12 | -1963 | 15 | -2593 | 10 | -1760
Viral Infections | 2 | -419 | 6 | -1508 | 5 | -1276

When we allowed for smaller (minimum group size = 100) and more complex groups
(maximum nodes = 64) we saw groups with more conditions appear in the top ten
(Figure 3). Namely, we identified more groups with asthma, cancer,
hypertension, chronic kidney disease, and mental health and substance use
disorders. Interestingly, we also found that most top overcompensated groups
were older, but relatively healthy individuals. In general, tuning the maximum
node size, which limits the depth of the trees, was more important for
determining group complexity than limiting the size of the groups. Some
chronic conditions, such as chronic viral infections and disorders of lipid
metabolism as well as documented sex, rarely if ever appeared as components in
our identified groups.

When we disaggregated the results by year, we observed the same pattern of
undercompensation for enrollees with multiple chronic conditions and
overcompensations for those without, but there was some temporal variation in
the types of conditions that were identified (Figure A1). For example, in
2016, five out of the ten top undercompensated groups were identified as
having a hereditary and generative nervous system in addition to other chronic
conditions, but this attribute didn’t appear in later years.

In the Appendix, we present results from the other setting we considered:
identifying undercompensated groups in the Medicare risk adjustment formula
used by CMS to set capitation rates for Medicare Advantage plans. We found no
groups that were consistently under- or overcompensated across the three
sample years.

Figure 3: Top Under- and Overcompensated Groups in the Marketplaces Risk
Adjustment (minimum node size: 100, maximum nodes: 64)

Note: Unfilled circles indicate the lack of a condition.

##  5 Discussion

In this research, we presented a new approach for identifying undercompensated
groups defined by multiple attributes, extending the random forests algorithm
to determine group importance without requiring groups to be pre-specified. We
implemented our group importance algorithm on the Marketplaces and Medicare
risk adjustment formulas. The Marketplaces formula favored healthier
individuals compared to those with multiple chronic conditions. In particular,
groups that had some combination of asthma, heart disease, arthritis, and
mental health and substance use disorder were largely undercompensated. These
results expand on and reenforce earlier work exploring issues of
undercompensation in risk adjustment for people with multiple chronic
conditions Pope et al. (2011). Undercompensation in the Medicare risk
adjustment formula was smaller in magnitude compared to the Marketplaces
formula and no groups were persistently under- or overcompensated across the
three sample years. This suggests that the Medicare risk adjustment formula
may induce fewer incentives to discriminate with respect to the twelve
selected conditions.

Our results raise the question of how to address incentives in markets,
especially for people with multiple health conditions. There are numerous
approaches for improving risk adjustment performance aimed at removing
selection incentives in regulated insurance markets. The Marketplaces aim to
partly address undercompensation for enrollees with two chronic health
conditions, where one condition is severe, by including interactions in the
risk adjustment formula (Kautter et al., 2014). Given our results, additional
interactions for those with multiple chronic conditions could be beneficial.
However, in Germany and the Netherlands, researchers found mixed results when
they included the interactions found in a single regression tree predicting
the residual, sometimes leading to negative consequences for other groups
(Buchner et al., 2017; van Veen et al., 2018). While not yet used in practice,
machine learning methods have also been proposed as an alternative to predict
spending and may be better able to capture non-linear spending trends for
enrollees with multiple chronic conditions (Rose, 2016). Another plausible
option could be to enforce improved fit for selected groups through
constrained or penalized regression, incorporating group fairness directly
into the loss function and where negative consequences for other groups were
not observed (Zink and Rose, 2020; McGuire et al., 2021). Although, solutions
to discrimination via insurance benefit design may lie outside of the risk
adjustment formula itself and be better addressed by, for example, additional
legal remedies, such as requiring adequate coverage of treatments and services
by insurers for the set of conditions we identified.

This work has several limitations. We developed a methodology that focused on
groups with the largest magnitude average residual that appeared in all three
sample years. The groups were composed of conditions that we considered
actionable for insurers, but we didn’t consider a priori whether certain
combinations of conditions (or lack of conditions) would be more actionable
for insurers compared to others. In our algorithm, we set the size and number
of terminal nodes in order to restrict the size and complexity of a group, but
future work could develop more sophisticated algorithms that aim to
empirically examine whether specific combinations of chronic conditions are
actionable for insurers within the decision rules of the tree rather than post
hoc.

The techniques we presented are relevant to other insurance markets relying on
risk adjustment. They can be used by developers and evaluators of risk
adjustment formulas to identify and assess performance across many different
groups defined by multiple attributes. In this paper, we focused on twelve
chronic conditions, but our method allows for the inclusion of additional
chronic conditions and other variables appropriate for identifying groups,
subject to computational constraints and ethical factors. Variable choices
will be limited by data availability: while countries like the U.S. and the
Netherlands have rich data sources to construct and assess their risk
adjustment formulas, not all countries have access to such detailed data
(McGuire and van Kleef, 2018).

Our group importance method may be relevant in a wide range of applications
beyond risk adjustment, although we caution researchers to carefully consider
the context before attempting to identify potentially marginalized groups. In
some settings, identifying groups could actively cause harm, e.g., may involve
collecting or amplifying stigmatizing information. In other cases, our tool
may be useful in mitigating ongoing harms. Machine learning predictions for
clinical outcomes have been found to be less accurate for groups defined by
age, race, or other attributes, contributing to health disparities Chen et al.
(2021). Our new method could help ensure that algorithms deployed in such
settings remedy inequities for currently unidentified marginalized groups. We
recommend researchers create a social impact statement and follow an ethical
pipeline for building algorithms when considering adapting our tool to any
setting (Chen et al., 2021; Diakopoulos et al., 2020).

## References

  * Aizawa and Kim (2018) Aizawa, N. and Kim, Y. S. (2018), “Advertising and Risk Selection in Health Insurance Markets,” American Economic Review, 108, 828–867.
  * Breiman (2001) Breiman, L. (2001), “Random Forests,” Machine Learning, 45, 5–32.
  * Buchner et al. (2017) Buchner, F., Wasem, J., and Schillo, S. (2017), “Regression Trees Identify Relevant Interactions: Can This Improve the Predictive Performance of Risk Adjustment?” Health Economics, 26, 74–85.
  * Carey (2017) Carey, C. (2017), “Technological Change and Risk Adjustment: Benefit Design Incentives in Medicare Part D,” American Economic Journal: Economic Policy, 9, 38–73.
  * Chen et al. (2021) Chen, I. Y., Pierson, E., Rose, S., Joshi, S., Ferryman, K., and Ghassemi, M. (2021), “Ethical Machine Learning in Healthcare,” Annual Review of Biomedical Data Science, 4, https://doi.org/10.1146/annurev-biodatasci-092820-114757.
  * Chouldechova and G’Sell (2017) Chouldechova, A. and G’Sell, M. (2017), “Fairer and more accurate, but for whom?” arXiv:1707.00046 [cs, stat], arXiv: 1707.00046.
  * CMS (2020) CMS (2020), “Market Rating Reforms | CMS,” Accessed December 14, 2020. https://www.cms.gov/cciio/programs-and-initiatives/health-insurance-market-reforms/state-gra.
  * CMS (2021) — (2021), “Chronic Conditions | CMS,” Accessed June 25, 2021\. https://www.cms.gov/Research-Statistics-Data-and-Systems/Statistics-Trends-and-Reports/Chronic-Conditions/CC_Main.
  * Diakopoulos et al. (2020) Diakopoulos, N., Friedler, S., Arenas, M., Barocas, S., Hay, M., Howe, B., Jagadish, H., Unsworth, K., Sahuguet, A., Venkatasubramanian, S., Wilson, C., Yu, C., and Zevenbergen, B. (2020), “Principles for Accountable Algorithms and a Social Impact Statement for Algorithms :: FAT ML,” Accessed December 30, 2020. https://www.fatml.org/resources/principles-for-accountable-algorithms.
  * Ellis and McGuire (2007) Ellis, R. P. and McGuire, T. G. (2007), “Predictability and predictiveness in health care spending,” Journal of Health Economics, 26, 25–48.
  * Fisher et al. (2019) Fisher, A., Rudin, C., and Dominici, F. (2019), “All Models are Wrong, but Many are Useful: Learning a Variable’s Importance by Studying an Entire Class of Prediction Models Simultaneously,” Journal of Machine Learning Research, 20, 1–81.
  * Geruso et al. (2019) Geruso, M., Layton, T., and Prinz, D. (2019), “Screening in Contract Design: Evidence from the ACA Health Insurance Exchanges,” American Economic Journal: Economic Policy, 11, 64–107.
  * Grömping (2009) Grömping, U. (2009), “Variable Importance Assessment in Regression: Linear Regression versus Random Forest,” The American Statistician, 63, 308–319.
  * IBM (2020) IBM (2020), “MarketScan Research Databases - databases,” Accessed September 16, 2020. https://www.ibm.com/products/marketscan-research-databases/databases.
  * Kautter et al. (2014) Kautter, J., Pope, G. C., Ingber, M. J., Freeman, S., Patterson, Cohen, and Keenan (2014), “The HHS-HCC Risk Adjustment Model for Individual and Small Group Markets under the Affordable Care Act,” .
  * Lavetti and Simon (2018) Lavetti, K. and Simon, K. (2018), “Strategic Formulary Design in Medicare Part D Plans,” American Economic Journal. Economic Policy, 10, 154–192.
  * Liaw and Weiner (2002) Liaw, A. and Weiner, M. (2002), “Classification and Regression by randomForest,” R News, 2, 18–22.
  * Louppe et al. (2013) Louppe, G., Wehenkel, L., Sutera, A., and Geurts, P. (2013), “Understanding variable importances in forests of randomized trees,” in Advances in Neural Information Processing Systems 26, eds. Burges, C. J. C., Bottou, L., Welling, M., Ghahramani, Z., and Weinberger, K. Q., Curran Associates, Inc., pp. 431–439.
  * McGuire et al. (2014) McGuire, T. G., Newhouse, J. P., Normand, S.-L., Shi, J., and Zuvekas, S. (2014), “Assessing Incentives for Service-Level Selection In Private Health Insurance Exchanges,” Journal of health economics, 0, 47–63.
  * McGuire and van Kleef (2018) McGuire, T. G. and van Kleef, R. (eds.) (2018), Risk Adjustment, Risk Sharing and Premium Regulation in Health Insurance Markets., Elsevier.
  * McGuire et al. (2021) McGuire, T. G., Zink, A., and Rose, S. (2021), “Improving the performance of risk adjustment systems: Constrained regressions, reinsurance, and variable selection,” American Journal of Health Economics, in press.
  * Montz et al. (2016) Montz, E., Layton, T., Busch, A. B., Ellis, R. P., Rose, S., and McGuire, T. G. (2016), “Risk-Adjustment Simulation: Plans May Have Incentives To Distort Mental Health And Substance Use Coverage,” Health Affairs, 35, 1022–1028, publisher: Health Affairs.
  * Pope et al. (2011) Pope, G. C., Kautter, J., Ingber, M. J., Freeman, S., Sekar, R., and Newhart, C. (2011), “Evaluation of the CMS-HCC Risk Adjustment Model, Final Report,” , 127.
  * Probst and Boulesteix (2017) Probst, P. and Boulesteix, A.-L. (2017), “To tune or not to tune the number of trees in random forest?” Journal of Machine Learning Research, 18.
  * Rose (2016) Rose, S. (2016), “A Machine Learning Framework for Plan Payment Risk Adjustment,” Health Services Research, 51, 2358–2374.
  * Rose (2018) — (2018), “Robust Machine Learning Variable Importance Analyses of Medical Conditions for Health Care Spending,” Health Services Research, 53, 3836–3854.
  * Rose et al. (2017) Rose, S., Bergquist, S. L., and Layton, T. J. (2017), “Computational health economics for identification of unprofitable health care enrollees,” Biostatistics (Oxford, England), 18, 682–694.
  * Semanskee and Claxton (2016) Semanskee, A. and Claxton, G. (2016), “Explaining Health Care Reform: Risk Adjustment, Reinsurance, and Risk Corridors,” Accessed September 16, 2020. https://www.kff.org/health-reform/issue-brief/explaining-health-care-reform-risk-adjustment-reinsurance-and-risk-corridors/.
  * Shepard (2016) Shepard, M. (2016), “Hospital Network Competition and Adverse Selection: Evidence from the Massachusetts Health Insurance Exchange,” Working Paper 22600, National Bureau of Economic Research, series: Working Paper Series.
  * van Veen et al. (2018) van Veen, S. H. C. M., van Kleef, R., van de Ven, W. P. M. M., and van Vliet, R. C. J. A. (2018), “Exploring the predictive power of interaction terms in a sophisticated risk equalization model using regression trees,” Health Economics, 27, e1–e12.
  * Wehenkel et al. (2018) Wehenkel, M., Sutera, A., Bastin, C., Geurts, P., and Phillips, C. (2018), “Random Forests Based Group Importance Scores and Their Statistical Interpretation: Application for Alzheimer’s Disease,” Frontiers in Neuroscience, 12, publisher: Frontiers.
  * Zink and Rose (2020) Zink, A. and Rose, S. (2020), “Fair regression for health care spending,” Biometrics, 76, 973–982.

##  Appendix A Appendix

###  A.1 Medicare Risk Adjustment

The Medicare Risk Adjustment formula was first implemented in 2004 by Centers
for Medicare & Medicaid Services (CMS) to set capitation rates for Medicare
Advantage plans and ensure that plans with higher risk beneficiaries (on
average) are compensated more than plans with lower risks beneficiaries (on
average). The formula is built and calibrated using Traditional Medicare
beneficiaries. Our risk adjustment estimates were based on the 2017 Medicare
risk adjustment formula for beneficiaries who qualified for Medicare based on
age and were not enrolled in Medicaid. We additionally included a control for
county, the level at which premiums are set:

$Y=\textnormal{Age}\times\textnormal{Sex}+\textnormal{HCC}+\textnormal{County}+\epsilon,$

where Age $\times$ Sex are eight age and sex categories and HCC are a set of
Health Condition Categories. Unlike the Marketplaces formulas, the Medicare
risk adjustment formula is prospective, meaning it uses diagnosis-based
condition categories from the previous year to predict current year spending.

####  A.1.1 Data

To evaluate the Medicare risk adjustment formula, we sampled individuals age
65+ enrolled in Traditional Medicare for at least two years starting in either
2015, 2016, or 2017. Because the risk adjustment formula is prospective, we
required these two years of enrollment in order to have a complete year of
data for measuring HCCs and a subsequent complete year of data for annual
spending. Individual annual spending, measured in 2016, 2017, and 2018, was
calculated as the total spending across the individual’s outpatient,
inpatient, and carrier claims in the given year. Diagnosis codes recorded in
inpatient, outpatient, and carrier claims were mapped to HCC categories using
the CMS-HCC software. The component variables in $I$ to define groups in $G$
were age categories (65-69, 70-79, 80-89, 90+), sex, and the same four chronic
conditions (mental health disorders, cancer, heart disease, and diabetes).
Similar to the Marketplaces formula, chronic conditions were defined using the
individual diagnosis-based categories from the Agency for Healthcare Research
and Quality Clinical Classification Software but measured in the prior year.
Age and sex information was acquired from the Master Beneficiary Summary File.
We excluded individuals with any months of Medicaid eligibility in their
starting year.

####  A.1.2 Results

The Medicare sample had more individuals with female documented sex than the
Marketplaces sample (55% compared to 52%), and nearly half the sample was in
their 70s (45.1% to 47.7%). As expected for an older sample, chronic
conditions were much more prevalent than in the younger commercial sample: a
majority of the conditions were found in at least 25% of the sample
(hypertension and lipid disorders were the most common conditions with a
prevalence of approximately 65%). Chronic viral infections and nervous
conditions were the rarest conditions and were only recorded in 0.4% and 6.8%
of the sample, respectively. Recall that chronic conditions were recorded in
the year prior, and the slight differences in prevalence for the 2016 sample
(with 2015 diagnosis) may be due to the switch from ICD-9 to ICD-10 on October
1, 2015. Table A2 presents the sample statistics for the Medicare sample. The
observed average residual for every chronic condition except chronic viral
infections was trivial (all less than $100, or less than 1% of the average
annual spending of $7,500). Chronic viral infections had an average
undercompensation of -$358 in 2016 and average overcompensation in both 2017
and 2018 ($138 and $184, respectively). In general, under- versus
overcompensation did not appear uniformly across years (Table A3). We found no
groups that were consistently under- and overcompensated over the three-year
observation period. In Figure A6, we display the top under- and
overcompensated groups for each sample year when limiting the group size to
10,000 and maximum nodes to 8. The top identified groups are different in each
year, with certain conditions appearing in some years and not others. For
example, in 2017 groups with chronic kidney disease appeared in six out of the
ten top undercompensated groups, but not consistently in other years. Some
conditions appeared in both under- and overcompensated groups within and
across years. For example, nerve disorders was a common condition in groups
that were under- and overcompensated in 2017 and 2018. Six out of the ten top
undercompensated groups in 2016 included beneficiaries with viral infections,
but groups with this condition tended to be overcompensated in 2017 and 2018.
Recall that chronic viral infections had the largest observed negative average
residual in 2016 and largest positive residual in 2017 and 2018, thus, this
result aligns with those findings. Compared to the Marketplaces risk
adjustment, the magnitude of the predicted average residual for
undercompensation was smaller (about $500 rather than thousands) and more
evenly balanced between the under- and overcompensated groups. When we allowed
the groups to be more complex and smaller in size, we identified very few
groups that appeared in more than 1% of trees.

Table A1: Mapping Diagnosis Codes and Clinical Classification Software (CCS) to Chronic Conditions |  | Single Level | Multiple 2nd Level | ICD-10
---|---|---|---|---
|  | CCS Diagnosis | CCS Diagnosis | Diagnosis
Chronic Condition | Short Name | Categories | Categories | Codes
Arthritis (osteoarthritis & rheumatoid) | Arthritis | 202,203 |  |
Asthma, COPD, and Other Chronic Lung Diseases | Asthma |  | 8.2-8.8 |
Cancer | Cancer | 11-45 |  |
Diabetes | Diabetes | 49, 50 |  |
Diseases of the Heart | Heart | 103 |  |
Hypertension | Hypertension | 98,99 |  |
Chronic Kidney Disease | Kidney | 158 |  |
Disorders of Lipid Metabolism | Lipid | 53 |  |
Mental Health and Substance Use Disorders | Mental | 650-663, 670 |  |
Hereditary and Degenerative Nervous System | Nervous |  | 6.2 |
Osteoporosis | Osteoporosis | 206 |  |
Chronic Viral Infections | Viral | 5 |  | B18
Table A2: Medicare Sample Characteristics (in Percents) Variables | 2016 | 2017 | 2018
---|---|---|---
Documented Sex, Female | 55.2 | 55.1 | 55.0
Age |  |  |
65-69 | 27.9 | 27.6 | 26.6
70-79 | 45.1 | 46.2 | 47.6
80-89 | 22.1 | 21.3 | 21.0
90+ | 4.9 | 4.8 | 4.7
Arthritis | 28.9 | 30.1 | 30.7
Asthma | 37.4 | 37.4 | 38.8
Cancer | 33.0 | 33.3 | 33.9
Diabetes | 35.7 | 37.2 | 38.3
Heart Disease | 45.6 | 45.3 | 45.8
Hypertension | 68.6 | 68.5 | 68.3
Kidney Disease | 11.6 | 12.2 | 12.8
Lipid Disorders | 65.7 | 65.4 | 66.0
Mental Health | 34.2 | 35.6 | 37.7
Nervous | 6.8 | 7.3 | 7.5
Osteoporosis | 11.1 | 10.8 | 11.0
Viral Infections | 0.4 | 0.4 | 0.4
N | 1,000,000 | 1,000,000 | 1,000,000
Table A3: Medicare Average Residual ($) by Chronic Condition | 2016 | 2017 | 2018
---|---|---|---
Chronic Condition | No | Yes | No | Yes | No | Yes
Arthritis | -19 | 47 | 6 | -14 | 4 | -8
Asthma | 6 | -10 | -6 | 10 | 8 | -13
Cancer | 17 | -34 | -11 | 23 | -1 | 3
Diabetes | -11 | 20 | -8 | 13 | 9 | -14
Heart Disease | 2 | -2 | 12 | -14 | 10 | -11
Hypertension | 12 | -5 | 5 | -2 | 22 | -10
Kidney Disease | -5 | 40 | 11 | -80 | 3 | -21
Lipid Disorders | -16 | 8 | -5 | 3 | -8 | 4
Mental Health | 10 | -20 | -8 | 15 | 7 | -12
Nervous | -1 | 20 | 2 | -26 | -4 | 46
Osteoporosis | 0 | 3 | -12 | 96 | -1 | 10
Viral Infections | 1 | -358 | 0 | 138 | 0 | 184

Figure A1: Top Under- and Overcompensated Groups in the Marketplaces Risk
Adjustment by Year (minimum node size: 10,000, maximum nodes: 8)

Note: Unfilled circles indicate the lack of a condition.

Figure A2: Top Under- and Overcompensated Groups in the Marketplaces Risk
Adjustment, Observed vs Predicted Residuals (minimum node size: 10,000,
maximum nodes: 8)

Note: Unfilled circles indicate the lack of a condition.

Figure A3: Top Under- and Overcompensated Groups in the Marketplaces Risk
Adjustment, Observed vs Predicted Residuals (minimum node size: 100, maximum
nodes: 64)

Note: Unfilled circles indicate the lack of a condition.

Figure A4: Top Under- and Overcompensated Groups in the Marketplaces Risk
Adjustment, Observed vs Predicted Residuals (minimum node size: 10,000,
maximum nodes: 64)

Note: Unfilled circles indicate the lack of a condition.

Figure A5: Top Under- and Overcompensated Groups in the Marketplaces Risk
Adjustment, Observed vs Predicted Residuals (minimum node size: 100, maximum
nodes: 8)

Note: Unfilled circles indicate the lack of a condition.

Figure A6: Top Under- and Overcompensated Groups in the Medicare Risk
Adjustment by Year (minimum node size: 10,000, maximum nodes: 8)

Note: Unfilled circles indicate the lack of a condition.


# The classification of constant weighted curvature curves in the plane with a
log-linear density

Doan The Hieu
Departement of Mathematics
College of Education, Hue University, Hue, Vietnam
<EMAIL_ADDRESS>
Tran Le Nam
Departement of Mathematics
Dong Thap University, Dong Thap, Vietnam
<EMAIL_ADDRESS>
###### Abstract

In this paper, we classify the class of constant weighted curvature curves in
the plane with a log-linear density, or in other words, classify all traveling
curved fronts with a constant forcing term in $\mathbb{R}^{2}.$ The
classification gives some interesting phenomena and consequences including:
the family of curves converge to a round point when the weighted curvature of
curves (or equivalently the forcing term of traveling curved fronts) goes to
infinity, a simple proof for a main result in [13] as well as some well-known
facts concerning to the isoperimetric problem in the plane with density
$e^{y}.$

AMS Subject Classification (2000): Primary 53C44, 53A04; Secondary 53C21,
35J60
Keywords: Curve flow with a forcing term, traveling curved front, self similar
translator, planes with a log-linear density, weighted curvature.

##  1 Introduction

Consider the following equation

$\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla
u|^{2}}}\right)=\frac{a}{\sqrt{1+|\nabla u|^{2}}}+b,$ (1)

where $u:U\subset\mathbb{R}^{n}\longrightarrow\mathbb{R},$ $U$ is an open
domain and $b$ is a constant. This is the equation for self-similar
translators with constant velocity $a$ of a mean curvature flow (MCF) with
constant forcing term ${\cal H}=b$

$\frac{\partial u}{\partial t}=\sqrt{1+|\nabla
u|^{2}}\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+\nabla u|^{2}}}-{\cal
H}\right).$ (2)

MCF (2) can be thought of as a generalization of the classical MCF when ${\cal
H}=0$ and $a=1$ or as a modified version of MCF with constant forcing term in
Minkowski space, a problem studied by Ecker and Huisken [7], followed by
Aarons [1], H. Jian et al. [12], [14] and others.

Equation (1) appears in the study of traveling curved fronts, originated by
the research of vortex motion of Ginzburg-Landau superconduction equation or
by the research of interfacial phenomena, and in several other fields in
Physics (see [20], [13]).

It also appears in the study of constant weighted mean curvature hypersurfaces
in $\mathbb{R}^{n+1}$ with a log-linear density.

The study of the geometry of manifolds with density has increased in the last
seven years after the appearance of Morgan’s paper “Manifolds with density”
published in the Notices Amer. Math. Soc. journal in 2005. A manifold with
density is a Riemannian manifold with a positive function, called the density,
used to weight both volume and hypersurface area. On a manifold with density
$f=e^{\varphi},$ the weighted mean curvature of a hypersurface with unit
normal $N$ is defined as follows

$H_{f}:=H-\frac{1}{n}\frac{d\varphi}{d{N}},$ (3)

where $H$ is the (Riemannian) mean curvature of the hypersurface.

For more details about manifolds with density and some relative topics we
refer the reader to [2]-[6], [8], [9], [15]-[19], [21].

A typical example of manifolds with density is Gauss space $G^{n},$ the
Euclidean space $\mathbb{R}^{n}$ with Gaussian probability density
$f=(2\pi)^{-\frac{n}{2}}e^{-\frac{r^{2}}{2}},$ that is very interesting to
probabilists. Weighted minimal hypersurfaces, i. e. $H_{f}=0,$ in Gauss spaces
are known as self-similar shrinkers. This is one of three types of self-
similar solutions of MCF (self-similar shrinkers, self-similar translators and
self-similar expanders) that are most studied recently. Self-similar
translators are just weighted minimal hypersurfaces in $\mathbb{R}^{n+1}$ with
a log-linear density $f,$ that is $f=e^{\varphi},$ where
$\varphi(x_{1},x_{2},\ldots,x_{n+1})=\sum_{i=1}^{n+1}a_{i}x_{i}+a_{0}$ (see
[8]).

Indeed, let $\Gamma_{u}\subset\mathbb{R}^{n+1}$ be the graph of the function
$u.$ Because $\operatorname{div}\left({\nabla u}/{\sqrt{1+|\nabla
u|^{2}}}\right)=nH,$ and $a/{\sqrt{1+|\nabla u|^{2}}}=a\langle
N,e_{n}\rangle,$ where $H$ is the mean curvature of $\Gamma_{u}$ and $N$ is
the unit normal of the graph; Equation (1) can be rewritten as

$H_{f}=c,$ (4)

where $c=b/n$ and $H_{f}:=H-(a/n)\langle N,e_{n+1}\rangle$ is the weighted
mean curvature of $\Gamma_{u}$ with density $e^{ax_{n+1}}.$

Therefore, solutions of Equation (1) are constant weighted mean curvature
hypersurfaces in $\mathbb{R}^{n+1}$ with a log-linear density
$f=e^{ax_{n+1}}.$ Weighted minimal hypersurfaces in $\mathbb{R}^{n+1}$ with
the density $f=e^{x_{n+1}}$ (self-similar translators of MCF) are solutions of
(4) for $a=1$ and $c=0.$

The classification of all constant weighted curvature hypersurfaces as well as
weighted minimal hypersurfaces in $\mathbb{R}^{n+1}$ with a general log-linear
density is equivalent to the one with the density $f=e^{x_{n+1}}$ (see Section
2). In other words, we can assume that the velocity $a=1$ in the
classification of all self-similar translators of the MCF (2).

Self-similar translators of MCF play a crucial role in the study type II
singularities of the flow (see [10], [11]). Beside lines, the first example of
self-similar translators in the plane, the Grim Reaper, was discovered by
Calabi. For higher dimensional cases, only a few specific examples of self-
similar translators are known. In terms of weighted minimal surfaces in
$\mathbb{R}^{3}$ with density $e^{z},$ some cylindrical type examples have
been showed in [8].

In this paper, we solve Equation (4) for parametric curves in the simplest
case $n=1$ and therefore give the classification of constant weighted
curvature curves in the plane with a log-linear density, or in other words,
the classification of all traveling curved fronts with a constant forcing term
in $\mathbb{R}^{2}.$ Although the classification is elementary, it gives some
interesting phenomena and consequences including: the family of curves
converge to a round point when the weighted curvature of curves goes to
infinity (see Section 5); a simple proof for a main result in [13] as well as
some well-known facts concerning to the isoperimetric problem in the plane
with density $e^{y}.$

Acknowledgement. We would like to thank Professor Frank Morgan for his
encouragement and for reading the first draft of the paper. The authors are
supported in part by the National Foundation for Science and Technology
Development, Vietnam (Grant No. 101.01-2011.26).

##  2 Spaces with log-linear densities

Consider $\mathbb{R}^{n}$ with a log-linear density $f=e^{\varphi},$ where
$\varphi(x_{1},\ldots,x_{n+1})=\sum_{i=1}^{n+1}a_{i}x_{i}+a_{0}.$ In this
space, the weighted mean curvature of a hypersurface with the unit normal $N$
is

$H_{f}:=H-\frac{1}{n}\langle N,e_{n}\rangle,$

The quantity $\langle N,e_{n+1}\rangle$ is just the projection of $N$ on the
$x_{n+1}$-axis. This quantity and the mean curvature are invariant under rigid
motions. Therefore, when study any problem concerning only to weighted mean
curvature, without loss of generality we can assume that $\varphi=ax_{n+1}.$
Moreover, we can assume $a=1.$

Now consider the plane with density $e^{ay}.$ This is a plane of zero
generalized Gauss curvature, $G_{\varphi}=G-\triangle\varphi,$ where $G$ is
the classical Gauss curvature. The notion of generalized Gauss curvature was
introduced in Corwin et al. [4] to study the generalization of the Gauss-
Bonnet formula for surfaces with densities, which is finally found in the most
general form for the case of unrelated densities in [5].

Let $\alpha(s)=(x(s),y(s))$ be a parametric curve with arc length parameter
$s,$ and let $\beta(s)=(1/a)(x(as),y(as)).$ Denote by $k_{f}(\alpha)$ the
weighted curvature of $\alpha$ corresponding to the density $f=e^{y}$ and
$k_{\widetilde{f}}(\beta)$ the weighted curvature of $\beta$ corresponding to
the density $\widetilde{f}=e^{ay}.$ By a simple computation, it is showed that
the weighted curvature of $\alpha$ in $\mathbb{R}^{2}$ with density $f=e^{y}$
is

$k_{f}(\alpha)=x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime}-x^{\prime},$
(5)

while the weighted curvature of $\beta$ in $\mathbb{R}^{2}$ with density
$\widetilde{f}=e^{ay}$ is

$k_{\widetilde{f}}(\beta)=ax^{\prime}y^{\prime\prime}-ax^{\prime\prime}y^{\prime}-ax^{\prime}.$
(6)

Therefore, we get

######  Lemma 1.

$ak_{f}(\alpha)=k_{\widetilde{f}}(\beta).$

##  3 Constant weighted curvature curves in the plane with density $e^{y}$

For solving Equation (4) to classify all constant weighted curvature curves in
the plane with density $e^{ay},$ by Lemma 1, we can assume that $a=1.$
Moreover, because the weighted curvature of a curve is invariant under
translations, some constant of integration will be omitted for simplicity.

Equation (4) for a parametric curve $\alpha(s)=(x(s),y(s))$ with arc length
parameter is

$x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime}-x^{\prime}=c.$ (7)

Set

$\displaystyle\begin{cases}x^{\prime}=-\cos(2\xi)\\\
y^{\prime}=\sin(2\xi)\end{cases}.$ (8)

Then, the equation (7) reads as follows:

$\displaystyle-2\xi^{\prime}+\cos(2\xi)=c.$ (9)

If $\cos(2\xi)\neq c$ for all $s,$ then the equation (9) is rewritten as
follows

$\displaystyle-2\xi^{\prime}=\dfrac{c-1+(c+1)\tan^{2}\xi}{1+\tan^{2}\xi}.$
(10)

###  3.1 The case of $\boldsymbol{c<-1}$ (see Figure 6.1)

Equation (10) can be rewritten as

$\displaystyle-\dfrac{2d\tan\xi}{\tan^{2}\xi+a^{2}}=(c+1)ds,\;\mbox{where
}\,a=\sqrt{\dfrac{c-1}{c+1}}.$

Thus,

$-\dfrac{2}{a}\arctan\dfrac{\tan\xi}{a}=(c+1)s$

or

$\tan\xi=a\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)},$ (11)

because $c+1<0.$ From (11), we have

$\begin{cases}x^{\prime}(s)=\dfrac{a^{2}\tan^{2}\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}-1}{1+a^{2}\tan^{2}\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}},\\\
y^{\prime}(s)=\dfrac{2a\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{1+a^{2}\tan^{2}\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}},\\\
\end{cases}$

A direct computation yields

$\displaystyle\begin{cases}x^{\prime}(s)=\dfrac{-2}{\sqrt{c^{2}-1}}\dfrac{\Big{(}1-a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}d\Big{(}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{\Big{(}1+a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}\Big{(}1+\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}},\\\\[22.76228pt]
y^{\prime}(s)=\dfrac{-4}{(c+1)}\dfrac{\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}d\Big{(}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{\Big{(}1+a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}\Big{(}1+\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}.\end{cases}$
(12)

Integrating both sides of (12), we obtain

$\displaystyle\begin{cases}x(s)=\dfrac{2(a^{2}+1)}{\sqrt{c^{2}-1}(a^{2}-1)}\dfrac{\sqrt{c^{2}-1}}{2}s-\dfrac{4a}{\sqrt{c^{2}-1}(a^{2}-1)}\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right),\\\\[11.38092pt]
y(s)=\dfrac{2}{(c+1)(a^{2}-1)}\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$
(13)

Substitute $a=\sqrt{\dfrac{c-1}{c+1}}$ into (13), we get

$\displaystyle\begin{cases}x(s)=-2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt]
y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$

###  3.2 The case of $\boldsymbol{c=-1}$ (see Figure 6.2)

If there exists $s_{0}$ so that $\cos[2\xi(s_{0})]=-1,$ then
$\xi(s)=\xi(s_{0})$ is the unique solution of the equation (9) and the
corresponding curves are straight lines parallel to the $x$-axis.

If $\cos 2\xi(s)\neq-1$ for all $s,$ then the equation (10) becomes

$\xi^{\prime}=\dfrac{1}{1+\tan^{2}\xi}.$

Thus,

$d(\tan\xi)=ds,$

and therefore,

$\tan\xi=s.$

Then, by Equation (8)

$\displaystyle\begin{cases}x^{\prime}(s)=-\dfrac{1-s^{2}}{1+s^{2}},\\\\[5.69046pt]
y^{\prime}(s)=\dfrac{2s}{1+s^{2}}.\end{cases}$ (14)

Integrating both sides (14), we obtain

$\displaystyle\begin{cases}x(s)=-2\arctan s+s,\\\
y(s)=\ln(1+s^{2}).\end{cases}$ (15)

###  3.3 The case of $\boldsymbol{-1<c<1}$ (see Figures 6.3, 6.4, 6.5)

If there exists $s_{0}$ so that $\cos[2\xi(s_{0})]=c,$ then
$\xi(s)=\xi(s_{0})$ is the unique solution of the equation (9) and the
corresponding curves are straight lines with the slope
$-\frac{\sqrt{1-c^{2}}}{c}.$ In the case of $c=0,$ the slope of the lines is
$\infty,$ i.e. the lines are vertical.

If this is not to be the case, Equation (10) is equivalent to

$\displaystyle-\dfrac{2d(\tan\xi)}{\tan^{2}\xi-b^{2}}=(c+1)ds,\;\mbox{where
}\,b=\sqrt{\dfrac{1-c}{c+1}}\neq 0.$ (16)

Solving (16) yields

$\dfrac{1}{b}\ln\Big{|}\dfrac{\tan\xi+b}{\tan\xi-b}\Big{|}=(c+1)s,$

or

$\Big{|}\dfrac{\tan\xi+b}{\tan\xi-b}\Big{|}={e}^{\sqrt{1-c^{2}}s}.$

We have two cases to consider:

Case 1: $\dfrac{\tan\xi+b}{\tan\xi-b}={e}^{\sqrt{1-c^{2}}s},$ or equivalently,
$\tan\xi=b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}.$

In this case, we have

$\begin{cases}x^{\prime}(s)=-\dfrac{1-\Big{(}b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}\Big{)}^{2}}{1+\Big{(}b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}\Big{)}^{2}},\\\
y^{\prime}(s)=\dfrac{2\Big{(}b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}\Big{)}}{1+\Big{(}b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}\Big{)}^{2}};\end{cases}$

or

$\begin{cases}x^{\prime}(s)=-\dfrac{(1-b^{2}){e}^{2\sqrt{1-c^{2}}s}-2(b^{2}+1){e}^{\sqrt{1-c^{2}}s}+1-b^{2}}{(1+b^{2}){e}^{2\sqrt{1-c^{2}}s}+2(b^{2}-1){e}^{\sqrt{1-c^{2}}s}+b^{2}+1},\\\\[5.69046pt]
y^{\prime}(s)=\dfrac{2b\big{(}{e}^{2\sqrt{1-c^{2}}s}-1\big{)}}{(1+b^{2}){e}^{2\sqrt{1-c^{2}}s}+2(b^{2}-1){e}^{\sqrt{1-c^{2}}s}+b^{2}+1}.\end{cases}$
(17)

Substitute $b=\sqrt{\dfrac{1-c}{1+c}}$ into (17), we get

$\begin{cases}x^{\prime}(s)=-\dfrac{\frac{2c}{c+1}{e}^{2\sqrt{1-c^{2}}s}-2\frac{2}{c+1}{e}^{\sqrt{1-c^{2}}s}+\frac{2c}{c+1}}{\frac{2}{c+1}{e}^{2\sqrt{1-c^{2}}s}+2\frac{-2c}{c+1}{e}^{\sqrt{1-c^{2}}s}+\frac{2}{c+1}},\\\\[5.69046pt]
y^{\prime}(s)=\dfrac{2\sqrt{\frac{1-c}{1+c}}\big{(}{e}^{2\sqrt{1-c^{2}}s}-1\big{)}}{\frac{2}{c+1}{e}^{2\sqrt{1-c^{2}}s}+2\frac{-2c}{c+1}{e}^{\sqrt{1-c^{2}}s}+\frac{2}{c+1}};\end{cases}$

or

$\displaystyle\begin{cases}x^{\prime}(s)=-\dfrac{c{e}^{2\sqrt{1-c^{2}}s}-2{e}^{\sqrt{1-c^{2}}s}+c}{{e}^{2\sqrt{1-c^{2}}s}-2c{e}^{\sqrt{1-c^{2}}s}+1},\\\\[11.38092pt]
y^{\prime}(s)=\dfrac{\sqrt{1-c^{2}}({e}^{\sqrt{1-c^{2}}s}-{e}^{-\sqrt{1-c^{2}}s})}{{e}^{\sqrt{1-c^{2}}s}-2c+{e}^{-\sqrt{1-c^{2}}s}}.\end{cases}$
(18)

Integrating both sides of (18), we obtain

$\begin{cases}x(t)=2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[11.38092pt]
y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right).\end{cases}$
(19)

Case 2: $\dfrac{\tan\xi+b}{\tan\xi-b}={e}^{\sqrt{1-c^{2}}s},$ or equivalently,
$\tan\xi=b\dfrac{{e}^{\sqrt{1-c^{2}}s}-1}{{e}^{\sqrt{1-c^{2}}s}+1}.$ After a
quite similar computation as in Case 1, we obtain

$\begin{cases}x(s)=-2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}+c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[8.5359pt]
y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}+2c\right).\end{cases}$
(20)

######  Remark 2.

  1. 1.

Set
$f_{1}(c,s)=2\arctan\Big{(}{\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\Big{)}-cs$,
$f_{2}(c,s)=-2\,\arctan\Big{(}{\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}+c}{\sqrt{1-{c}^{2}}}}\Big{)}-cs$,
$g_{1}(c,s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right)$
and
$g_{2}(c,s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}+2c\right),$
we can verify that

$\displaystyle\begin{cases}f_{1}(c,s)=-f_{2}(-c,s),\\\
g_{1}(c,s)=g(-c,s).\end{cases}$

Therefore the traces of the curves defined by equation (19) and (20) are
symmetric to the $x$-axis.

  2. 2.

When $c=0,$ Equation (19) and (20) are the equations of the Grim Reapers (see
Figure 6.4)

$\begin{cases}x(s)=\pm 2\arctan(e^{s}),\\\ y(s)=\ln(e^{s}+e^{-s});\end{cases}\
\ \ \ \ s\in\mathbb{R}.$ (21)

  3. 3.

From Equation (21) we can deduce some simple facts (see also Figure 6.4).

    1. (a)

Two zero weighted curvature curves intersect in at most one point. Therefore,
the geodesic connecting two points if exists is unique.

    2. (b)

If the difference of the $x$-coordinates of two given points is not less than
$\pi,$ then there exists no geodesic connecting these points.

    3. (c)

If the difference of the $x$-coordinates of two given points is less than
$\pi,$ then there exists a unique zero weighted curvature curve connecting
these points and this curve is the shortest path which can be proved directly.

###  3.4 The case of $\boldsymbol{c=1}$ (see Figure 6.2)

If there exists $s_{0}$ so that $\cos[2\xi(s_{0})]=1,$ then
$\xi(s)=\xi(s_{0})$ is the unique solution of the equation (9) and the
corresponding curves are straight lines parallel to the $x$-axis.

If $\cos 2\xi(s)\neq 1$ for all $s,$ then the equation (10) becomes

$-\xi^{\prime}=\dfrac{\tan^{2}\xi}{1+\tan^{2}\xi}=\dfrac{1}{1+\cot^{2}s}.$

Thus,

$d(\cot\xi)=ds,$

and therefore,

$\cot\xi=s.$

Then, by Equation (8)

$\displaystyle\begin{cases}x^{\prime}(s)=\dfrac{1-s^{2}}{1+s^{2}},\\\\[5.69046pt]
y^{\prime}(s)=\dfrac{2s}{1+s^{2}}.\end{cases}$ (22)

Integrating both sides (22) yields

$\displaystyle\begin{cases}x(s)=2\arctan s-s,\\\
y(s)=\ln(1+s^{2}).\end{cases}$ (23)

We can see that the curves determined by (15) and (23) have the same traces
but have opposite directions.

###  3.5 The case of $\boldsymbol{c>1}$ (see Figure 6.6)

With similar arguments as in the case of $c<-1$, we have

$\tan\xi=-a\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}.$ (24)

Therefore,

$\displaystyle\begin{cases}x^{\prime}(s)=\dfrac{-2}{\sqrt{c^{2}-1}}\dfrac{\Big{(}1-a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}d\Big{(}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{\Big{(}1+a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}\Big{(}1+\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}},\\\\[22.76228pt]
y^{\prime}(s)=\dfrac{-4}{(c+1)}\dfrac{\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}d\Big{(}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{\Big{(}1+a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}\Big{(}1+\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}.\end{cases}$
(25)

Integrating both sides of (25) and substituting $a=\sqrt{\dfrac{c-1}{c+1}}$
into the result, we get

$\displaystyle\begin{cases}x(s)=2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt]
y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$

##  4 Classification of constant weighted curvature curves

Combining the above results, up to translations, the classification of
constant weighted curvature curves in the plane with density $e^{y}$ is stated
as follows.

######  Theorem 3.

  1. 1.

A curve with weighted curvature zero is either a straight line (parallel to
the $y$-axis) or the Grim Reaper defined by (see Figure 6.4)

$\begin{cases}x(s)=2\arctan(e^{s}),\\\
y(s)=\ln(e^{s}+e^{-s}).\end{cases}s\in\mathbb{R}.$ (26)

  2. 2.

A curve with constant weighted curvature $|k_{\varphi}|<1$ is either a
straight line or the one defined by (see Figure 6.3, Figure 6.5)

$\begin{cases}x(s)=2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[11.38092pt]
y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right).\end{cases}s\in\mathbb{R}.$

  3. 3.

A curve with constant weighted curvature $\pm 1$ is either a straight line
(parallel to the $x$-axis) or the one defined by (see Figure 6.2)

$\begin{cases}x(s)=2\arctan s-s,\\\
y(s)=\ln(1+s^{2}).\end{cases}s\in\mathbb{R}.$

  4. 4.

A curve with constant weighted curvature $|k_{\varphi}|>1$ is defined by (see
Figure 6.1, Figure 6.6)

$\displaystyle\begin{cases}x(s)=\pm
2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt]
y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}s\in\Big{(}-\dfrac{\pi}{\sqrt{c^{2}-1}},\dfrac{\pi}{\sqrt{c^{2}-1}}\Big{)}.$

##  5 Some consequenses

  1. 1.

The figures in the next section seem to show that if the weighted curvature
$c$ goes to $\pm\infty$ then the limit of the curves is a point. For looking
the behaviour of curves near the limit point we can use the standard technique
of rescaling the curves by a scale factor $\sqrt{c^{2}-1}.$ We give a proof
for the case of $c\rightarrow\infty.$ The proof for $c\rightarrow-\infty$ is
quite similar. Suppose that $\alpha(c)$ is a curve with arc length parameter
and of constant weighted curvature $c>1.$ Let
$\beta(c)=\sqrt{c^{2}-1}\alpha(c).$ The curvature of $\beta$ is
$(1/\sqrt{c^{2}-1})k(\alpha(c)),$ where $k(\alpha(c))$ is the curvature of the
curve $\alpha(c).$ By Equation (7), $k(\alpha(c))=x^{\prime}+c.$ Therefore

$\frac{1}{\sqrt{c^{2}-1}}k(\alpha(c))=\frac{1}{\sqrt{c^{2}-1}}\left(\dfrac{-c\cos(\sqrt{c^{2}-1}s)-1}{c+\cos(\sqrt{c^{2}-1}s)}+c\right).$

It is not hard to check that

$\lim_{c\rightarrow\infty}\Big{|}\frac{1}{\sqrt{c^{2}-1}}k(\alpha(c))\Big{|}=1.$

Thus, the family of curves converges to a round point when $c$ goes to
infinity.

  2. 2.

The study of traveling fronts of curve flow with external force field for the
simple case $\nabla w=(c_{1},c_{2})$ leads to the following equation (see
[13]):

$c=\frac{\varphi^{\prime\prime}(x)}{1+\varphi(x)^{2}}+c_{2}-c_{1}\varphi^{\prime}(x).$
(27)

One of the main results in [13] stated that, the solution of (27) with an
initial conditions is either a line or the Grim Reaper. In terms of weighted
curvature, solutions of (27) are just zero weighted curvature curves in the
plane with density $f=e^{-c_{1}x+(c_{2}-c)y},$ which are also zero weighted
curvature curves in the plane with density $f=e^{y}$ under a suitable change
of coordinates. In other words, the solutions are self-similar translators in
the plane which are known to be either lines or Grim Reapers.

  3. 3.

Proposition 4.8 in [2], which states “The plane with density $e^{x}$ contains
no isoperimetric region”, can be deduced from the classification. It is clear
from the classification (and the figures) that isoperimetric curves, i.e.
curves bounding isoperimetric regions (in case of existence), must have
infinite weighted length or singularities.

##  6 Figures of constant weighted curvature curves

Figure 6.1. Curves of $k_{\varphi}<-1$. |  Figure 6.2. Curves of $k_{\varphi}=\pm 1$.
---|---
$\begin{cases}x(s)=-2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt] y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$ | $\begin{cases}x(s)=s-2\arctan s,\\\ y(s)=\ln(1+s^{2}).\end{cases}$

Figure 6.3. Curves of $k_{\varphi}\in(-1,0)$. |  Figure 6.4. Curve of $k_{\varphi}=0$.
---|---
$\begin{cases}x(s)=2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[11.38092pt] y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right).\end{cases}$ |  (the Grim Reaper curve) $\begin{cases}x(s)=2\arctan(e^{s}),\\\ y(s)=\ln(e^{s}+e^{-s}).\end{cases}$
Figure 6.5. Curves of $k_{\varphi}\in(0,1)$. |  Figure 6.6. Curves of $k_{\varphi}>1$.
---|---
$\begin{cases}x(s)=2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[11.38092pt] y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right).\end{cases}$ | $\begin{cases}x(s)=2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt] y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$

## References

  * [1] M. A. S. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var. Partial Differential Equations 25 (2006), no. 2, 205-246.
  * [2] C. Carroll, A. Jacob, C. Quinn, R. Walters, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc. 78 (2008), no. 2, 177–197.
  * [3] A. Cañete, M. Miranda and D. Vittone, Some isoperimetric problems in planes with density, J. Geo. Anal.,Vol. 20, No. 2, 243-290.
  * [4] I. Corwin, N. Hoffman, S. Hurder, V. Sesum, and Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J., 7 (1) (2006).
  * [5] I. Corwin, F. Morgan, The Gauss-Bonnet formula on surfaces with densities, Involve 4 (2011), no. 2, 199–202.
  * [6] J. Dahlberg, A. Dubbs, E. Newkirk, H. Tran, Isoperimetric regions in the plane with density $r^{p}$, New York J. Math. 16 (2010) 31-51.
  * [7] K. Ecker, G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys. 135 (1991), no. 3, 595-613.
  * [8] D. T. Hieu, N. M. Hoang, Ruled minimal surfaces in $\mathbb{R}^{3}$ with density $e^{z}$, Pacific J. Math. 243 (2009), no. 2, 277-285.
  * [9] D. T. Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys. 61 (2011), no. 8, 1625-1629.
  * [10] G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. PDE 8 (1999), no. 1, 1-14.
  * [11] G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), no. 1, 45-70.
  * [12] H. Jian, H. Ju, Y. Liu, W. Sun, Symmetry of translating solutions to mean curvature flows, Acta Math. Sci. Ser. B Engl. Ed. 30 (2010), no. 6, 2006-2016.
  * [13] H. Jian, H. Ju, Y. Liu, W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal. 9 (2010), no. 4, 975-986.
  * [14] H. Ju, J. Lu, H. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space Commun. Pure Appl. Anal. 9 (2010), no. 4, 963-973.
  * [15] Q. Maurmann, F. Morgan, Isoperimetric comparison theorems for manifolds with density, Calc. Var. PDE 36 (2009), No. 1, 1-5.
  * [16] F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858.
  * [17] F. Morgan, Myers Theorem with density, Kodai Math. J. 29 (2006), 454-460.
  * [18] F. Morgan, “Geometric Measure Theory: a Beginner s Guide”, Academic Press, fourth edition, 2009.
  * [19] F. Morgan, Manifolds with density and Perelman’s proof of the Poincaré Conjecture, Amer. Math. Monthly 116 (Feb., 2009), 134-142.
  * [20] H. Ninomiya, M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force Free boundary problems: theory and applications, I (Chiba, 1999), 206-221, GAKUTO Internat. Ser. Math. Sci. Appl., 13, Gakkotosho, Tokyo, 2000.
  * [21] C. Rosales, A. Cañete, V. Bayle and F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. PDE 31 (2008), no. 1, 27-46.


# AUDIO CODING WITH UNIFIED NOISE SHAPING AND PHASE CONTRAST CONTROL

###### Abstract

Over the past decade, audio coding technology has seen standardization and the
development of many frameworks incorporated with linear predictive coding
(LPC). As LPC reduces information in the frequency domain, LP-based frequency-
domain noise-shaping (FDNS) was previously proposed. To code transient signals
effectively, FDNS with temporal noise shaping (TNS) has emerged. However,
these mainly operated in the modified discrete cosine transform domain, which
essentially accompanies time domain aliasing. In this paper, a unified noise-
shaping (UNS) framework including FDNS and complex LPC-based TNS (CTNS) in the
DFT domain is proposed to overcome the aliasing issues. Additionally, a
modified polar quantizer with phase contrast control is proposed, which saves
phase bits depending on the frequency envelope information. The core coding
feasibility at low bit rates is verified through various objective metrics and
subjective listening evaluations.

Index Terms—  Audio coding, quantization, DFT, noise shaping, TCX

##  1 Introduction

Psychoacoustic-based audio coding has evolved as a method to preserve
perceptual quality while decreasing bit rates [1]. In contrast, speech coding
has been refined using linear prediction (LP) models [2, 3]. Separately
developed audio and speech coding evolved into the unified speech and audio
coding frameworks (MPEG USAC [4]), such as LP-based frequency-domain noise
shaping (FDNS). In addition, LP models are used to reduce temporal information
by temporal noise shaping (TNS), which is a representative tool conforming to
the audio coding standards [5, 6]. The sequential combination of FDNS and TNS
in the modified discrete cosine transform (MDCT) domain [7] is feasibly
realized in enhanced voice services (EVS) [8] and MPEG-H 3D audio [9].

TNS is mainly incorporated in MDCT, which may incur the undesired time-domain
aliasing noises after decoding [10, 11]. Specifically, the high noise level at
the edge of an attack signal, time-domain aliasing noise, and noise spread due
to the order of the TNS filter are three undesired artifacts, which can be
reduced by window-switching or using a small overlapped window [12]. However,
the window-switching increases the complexity and algorithmic delay. In
addition, using a small overlapped window can minimize the aliasing noise but
does not avoid it fundamentally [5]. Thus, this study presents a unified noise
shaping (UNS) model in the time and frequency domain without any aliasing-
related artifacts.

In the UNS framework, FDNS and TNS are sequentially operated using discrete
Fourier transform (DFT). Specifically, a complex linear predictive coding
(LPC)-based TNS (CTNS) model is operated, which predicts a temporal envelope
close to the Hilbert envelope. The model effectively reduces the amount of
temporal information, but the residual signal has a complex value. Hence, an
efficient quantization of the complex value should be developed. Exclusively,
the UNS framework with the DFT can reduce the structural complexity, as it
does not perform any additional inverse DFT or MDCT in the encoder. Since the
quantization target of the UNS framework is a complex-valued signal, several
quantizers in the polar coordinates should be considered, such as a strict
polar quantizer (SPQ) [13], a conventional polar quantizer [14], and an
unrestricted polar quantizer (UPQ) [15]. Among them, the UPQ utilizes
magnitude levels for phase quantization, and its performance outperforms the
other polar quantizers [16]. Recently, an entropy-constrained UPQ (ECUPQ) and
its optimal solution at low rates were reported [16, 17]. Yoon and Malvar
presented an audio coding framework with a UPQ for modulated complex lapped
transform (MCLT) coefficients [18]. The authors’ previous study proposed a
modified UPQ (mUPQ) combined with FDNS at low bit rates [19]. Thus, this study
presents an extended mUPQ with a phase contrast control (PCC) utilizing the
frequency envelope ratio (FER) calculated from the quantized LPC model.

##  2 TNS with MDCT

TNS is used to avoid pre-echo artifacts, which may occur when coding a
transient signal with a long window [5]. Considering the duality between the
spectral and Hilbert envelopes, TNS is operated with an open-loop prediction
in the frequency domain. TNS can be adapted as an independent tool in any
audio coder, and it works effectively for ”pitched” signals where the window-
switching tool does not operate well [5]. However, when TNS is operated in the
MDCT domain, it may cause the undesired time-domain aliasing noises [11]. The
MDCT operation can be decomposed into a time-domain aliasing block and shifted
DFT [20]. Hence, the MDCT coefficient predictions are accompanied by time-
domain aliased terms, such that the estimated temporal envelope is distorted.
The artifacts can be avoided fundamentally by using CTNS in the DFT domain.

Figure 1 shows the reconstructed TNS residual signals in the (b) MDCT and (c)
DFT domains. For analyzing and synthesizing the TNS residual signals in the
two domains, the same window configuration was utilized. The TNS orders were
set to $16$. Since the prediction is incomplete in the MDCT domain, the TNS
residual signal with the MDCT in the transient period has relatively high
power ($1\sim 2.5$ sec). Contrarily, the TNS residual signal with the DFT in
the transient period is whitened in the time domain, which implies that the
Hilbert envelope is well predicted by the CTNS. TNS with DFT reduces the
temporal information, which leads to a decrease in the entropy of the
quantized coefficients.

Fig. 1: Illustration of TNS in two different domains. The (a) downsampled
audio signal (music1), (b) residual signal in the MDCT domain, and (c)
residual signal in the DFT domain

##  3 Proposed Audio Coding System

###  3.1 Overview

Fig. 2: Block diagram of the encoder and decoder. Thin and solid lines
indicate signal flows. Dotted lines indicate control signals. The encoded
signals are described as thick and solid lines.

Figure 2 shows the proposed audio coding system. Unlike the transform coded
excitation (TCX) in USAC [21], the residual signal is computed by the UNS
using DFT. First, the audio signal is windowed and fed to the DFT and LP
analysis blocks simultaneously. For DFT, the window can be chosen without
considering the time domain aliasing cancellation (TDAC) condition. Therefore,
a small overlap window without zero sides is used. Unlike the FDNS of MPEG
USAC [4], the FDNS of the proposed system is done by dividing the magnitude of
the DFT coefficients by the reconstructed frequency envelope (FE) generated
using the quantized LPC model without interpolation. Second, the complex-
valued FDNS residual signal undergoes complex LP analysis, which is performed
using the same LP algorithm _e.g._ Levinson-Durbin recursion. The filtering
process using the complex LPC is conducted above $312$ Hz. The filtered signal
($x_{CT}\in\mathbb{C}$) is utilized for calculating the predicted gain as

$G(n)=10\log_{10}{\frac{\sum_{f}\left|x_{FD}(n,f)-x_{CT}(n,f)\right|^{2}}{\sum_{f}\left|x_{FD}(n,f)\right|^{2}}},$
(1)

where $n$ and $f$ are the frame and frequency indices, respectively, and the
complex value $x_{FD}$ denotes the FDNS residual signal. The CTNS block
operates when the prediction gain is higher than the pre-defined threshold.
The flag indicating the on/off state of the CTNS block for each frame is
allocated a $1$ bit and transmitted to the decoder.

The effects of the proposed UNS are illustrated in Fig. 3. For all
simulations, the same window configuration was utilized. It is observed that
the reconstructed signal with only FDNS contains quantization noises spread
around the attack period, shown in Fig. 3 (b). Conversely, the reconstructed
signal with the proposed UNS contains well-shaped quantization noises as shown
in Fig. 3 (c). Figures 3 (d) and (e) show the prediction gain and CTNS flag
for each frame, respectively. In this study, the threshold for predicting gain
was set to $-4.5$ dB.

Fig. 3: Illustration of CTNS and CTNS determination. The (a) downsampled audio
signal (cast01), (b) reconstructed signal with FDNS, (c) reconstructed signal
with FDNS and CTNS, (d) plot of CTNS gain per frame indices and its threshold,
and (e) plot of CTNS flag per frame indices.

Before quantization, the complex-valued UNS residual signal is divided by a
scale factor for each sub-band to reduce the dynamic range perceptually. The
boundaries of the sub-bands are set to [$40$, $90$, $140$, $200$, $260$,
$330$, $410$, $512$], which is the modified equivalent-rectangular bandwidth
(ERB) scale. The scale factor for each sub-band is determined by the allocated
bits and power of each sub-band SQ_gain in MPEG USAC [4]. It is uniformly
quantized in the decibel scale with $1$ dB step, and then entropy coded. The
input bits for the SQ_gain were set to [$45$, $34$, $30$, $23$, $19$, $16$,
$16$, $16$] for $12$ kbps and [$67$, $50$, $45$, $34$, $29$, $23$, $23$, $23$]
for $16$ kbps. The scaled residual signal is then quantized by mUPQ with PCC.
The detailed quantization procedure is elaborated in section 3.2. The
quantized magnitude and phase indices are entropy coded and transmitted to the
decoder. Since the methodology of memory-efficient entropy coding is not the
main focus of this study, the sample entropies of the adjacent four indices
within the same frame were computed to coordinate the target bits.

The decoder used is shown at the bottom of Fig. 2. Decoding procedures are
opposite to that of an encoder. The decoded magnitude and phase indices are
fed to the inverse mUPQ, which produces a de-quantized signal. The de-
quantized complex signal is rescaled for each sub-band by multiplying it with
the decoded scale factor. The rescaled signal is optionally fed to the inverse
CTNS filter when the decoder CTNS flag is on. The inversely filtered or
bypassed signals are multiplied by the reconstructed FE computed from the
quantized LPC. Finally, the output signal is reconstructed by using inverse
DFT (IDFT).

###  3.2 Polar quantizer with PCC

Fig. 4: Schematic of mUPQ with PCC and its counterpart (de-quantizer). Thin
and solid lines indicate signal flows. Dotted lines indicate control signals.
The encoded signals are described as thick and solid lines.

The magnitude and phase of the complex-valued UNS residual signal are
sequentially quantized by mUPQ with PCC. The magnitude is mostly quantized by
the ECUPQ solution at a rate of $2.495$ bits/samples reported in [17].
However, when the ECUPQ is exclusively used, the local spectral peaks may not
be precisely reconstructed as presented in [19]. Alternatively, a hybrid
quantization is adopted, as shown in Fig. 4. The magnitude is quantized by the
ECUPQ when its value is less than the highest threshold ($r_{7}=5.056$) of the
ECUPQ. Then, the magnitude index $1$ is obtained. Otherwise, the magnitude is
determined by comparing its value to be greater than or equal to the second
threshold ($\tilde{r}_{7}=8.5^{4/3}$). In the former, the magnitude
quantization index $1$ ($I^{(1)}_{m}$) is computed as a nonlinear equation.

$I^{(1)}_{m}={\left\lfloor{{A^{3/4}}+0.5}\right\rfloor}.$ (2)

While in the latter, the magnitude is uniformly quantized by a rounding
operation, where the magnitude quantization index $2$ is calculated
($I^{(2)}_{m}$). The magnitude indices ($I^{(1)}_{m}$ and $I^{(2)}_{m}$) are
entropy coded and then transmitted to the decoder.

The phase is uniformly quantized because the phases are assumed to be a
uniformly distributed random variable [13]. To determine the number of phase
cells, the conventional UPQ only considers its magnitude. However, the
proposed PCC additionally considers the FER calculated from the quantized LPC
to control the phase contrast. The FER for each sub-band is calculated by the
reconstructed FE, $H_{dB}(n,f)$ in decibels, as

$FER(n,b)=\frac{\bar{H}_{dB}(n,b)}{\sum_{b}{\bar{H}_{dB}(n,b)}},$ (3)

where the maximum FE per sub-band is computed as

$\bar{H}_{dB}(n,b)=\max_{f\in B_{b}}{H_{dB}(n,f)},$ (4)

Here, $b$ denotes the sub-band index and $B_{b}$ indicates the set of
frequency indices within the $b$-th sub-band. For a sub-band with FER greater
than the FER threshold, the number of phase-cells set for the corresponding
sub-band is assigned as [$1$, $8$, $16$, $16$, $32$, $32$, $64$, $64$] for
uniform quantization, else as [$1$, $4$, $8$, $8$, $16$, $16$, $32$, $32$],
which has the half resolution. The FER threshold is set to $0.125$, such that
the segmental signal-to-noise ratio (SNR) is maximum. Depending on the FER,
which contains the coarsely modeled FE information, the phase resolution is
controlled by allocating fewer bits to the sub-band with FE relatively small
within the frame. This reduces the phase bits for perceptually less dominant
spectral components.

The decoder first decodes magnitude index $1$ and then feds it to the hybrid
magnitude de-quantizer. When the magnitude index $1$ equals to $8$ indicating
that the magnitude is within the outlier region, the magnitude is
reconstructed by magnitude index $2$ directly. Else, inverse quantization is
performed according to the ECUPQ solution or inversion procedure of nonlinear
quantization, depending on the magnitude index $1$. For nonlinear de-
quantization, the magnitude is reconstructed to the power of $4/3$. To
determine the number of phase cells for uniform phase de-quantizer, the usage
of the de-quantized magnitudes and calculated FER is similar to that in the
encoder.

##  4 Performance evaluation

###  4.1 Setup

Objective evaluation and subjective listening tests were conducted to assess
the coding performance. For objective evaluation, segmental SNR (segSNR),
noise-to-mask ratio (NMR), and perceptual evaluation of audio quality (PEAQ)
were calculated [22, 23], while the listening test was conducted based on the
MUSHRA methodology [24]. The tests were conducted with $10$ experienced
listeners using reference quality headphones, and $10$ audio signals including
$3$ speeches, $4$ music scores, and $3$ mixed items were extracted from the
MPEG audio sequences. Bit rates at $12$ kbps and $16$ kbps were selected for
evaluation. For comparison, the TCX of MPEG USAC was chosen as the base [4].
To evaluate the coding performance of the core-band primarily, noise-filling
and bandwidth extensions were deactivated. The core-band was coded with a
downsampled audio signal at $12.8$ kHz.

A cosine-tapered window of $1024$ samples with $256$-sized overlap was used
for DFT analysis and synthesis. The LPC orders for both FDNS and CTNS were set
to $16$, while the perceptual weights for LP analysis were used at $0.98$ and
$0.9$, respectively. The LPC for FDNS was transformed to line spectral
frequencies (LSFs) and quantized using a two-stage vector quantizer (VQ) with
codewords of $20$ bits. Conversely, the complex LPC for CTNS was directly
quantized using a three-stage VQ with codewords of $30$ bits for the magnitude
and phase, respectively. The allocated bits are shown in Table 1. The bits of
the complex LPC for CTNS can be efficiently reduced by CTNS switching, similar
to the conventional TNS.

###  4.2 Evaluation results

Table 1: Bit allocation for the proposed audio coding system

Table 2: Objective scores of TCX with USAC and the proposed system for bit
rates of $12$ kbps and $16$ kbps

Fig. 5: MUSHRA scores and their $95$% confidence intervals at (a) $12$ kbps
and (b) $16$ kbps.

The objective scores are summarized in Table 2, where the scores of the
proposed system outperformed the baseline. The score differences of the three
metrics at $16$ kbps were higher than those at $12$ kbps. Figure 5 shows the
MUSHRA scores of the proposed system and the baseline with a $95$ % confidence
interval. At $12$ kbps, the coding performances of the proposed system for
speech and music were statistically better than those of the baseline. For
music, though the confidence intervals of both systems overlap, the proposed
system mean was higher than the baseline. At $16$ kbps, the coding performance
of the proposed system for speech was statistically better than that of the
baseline. Nonetheless, the mean of the proposed system for both music and
mixed audio was higher than that of the baseline. Thus, the proposed system
has statistically better sound quality at $12$ and $16$ kbps for all audio
items. Although different window configurations are considered against the
baseline, the proposed DFT-based audio coding notably produces a perceptually
better sound at low bit rates compared to the conventional MDCT-based coding
system.

##  5 Conclusion

In this paper, an audio coding framework that includes UNS and mUPQ with PCC
in the DFT domain is proposed. The proposed UNS model is composed of FDNS
without interpolation and CTNS model in the DFT domain, which efficiently
reduces the time and frequency-domain information during transient periods
without window-switching. The UNS residual signal is quantized by an mUPQ with
PCC, which selectively economizes the phase bits based on the computed FER for
sub-bands. The objective metric scores of the proposed system outperform those
of the baseline. In addition, the total scores of the listening tests are
statistically better than those of the baseline at lower bit rates.

## References

  * [1] Marina Bosi and Richard E Goldberg,  Introduction to digital audio coding and standards, vol. 721,  Springer Science & Business Media, 2002.
  * [2] Jerry D Gibson,  “Speech coding methods, standards, and applications,”  IEEE Circuits Syst. Mag., vol. 5, no. 4, pp. 30–49, 2005.
  * [3] 3GPP TS 26.290,  “Audio codec processing functions; Extended Adaptive Multi-Rate - Wideband (AMR-WB+) codec; Transcoding functions,”  .
  * [4] ISO/IEC 23003-3,  “Information technology - MPEG audio technologies - Part 3: Unified speech and audio coding,”  2012\.
  * [5] Jürgen Herre and James D Johnston,  “Enhancing the performance of perceptual audio coders by using temporal noise shaping (TNS),”  in Audio Eng. Soc. Conv., 1996.
  * [6] ISO/IEC 14496,  “Information technology - Coding of Audiovisual Objects - Part 3, Audio,”  1999\.
  * [7] Guillaume Fuchs, Christian R Helmrich, Goran Marković, Matthias Neusinger, Emmanuel Ravelli, and Takehiro Moriya,  “Low delay LPC and MDCT-based audio coding in the EVS codec,”  in IEEE Int. Conf. Acoust. Speech Signal Process., 2015, pp. 5723–5727.
  * [8] 3GPP TS 26.441,  “Codec for Enhanced Voice Services (EVS); General overview,”  .
  * [9] ISO/IEC 23008-3,  “Information technology - High efficiency coding and media delivery in heterogeneous environments - Part 3: 3D audio - Amendment 1: MPEG-H, 3D audio profile and levels,”  2015\.
  * [10] Jürgen Herre,  “Temporal noise shaping, quantization and coding methods in perceptual audio coding: A tutorial introduction,”  in Audio Eng. Soc. Conf., 1999.
  * [11] Chi-Min Liu, Han-Wen Hsu, and Wen-Chieh Lee,  “Compression artifacts in perceptual audio coding,”  IEEE/ACM Trans. Audio, Speech, Language Process., vol. 16, no. 4, pp. 681–695, 2008.
  * [12] Eric Allamanche, Ralf Geiger, Jürgen Herre, and Thomas Sporer,  “MPEG-4 low delay audio coding based on the AAC codec,”  in Audio Eng. Soc. Conv., 1999.
  * [13] W Pearlman,  “Polar quantization of a complex Gaussian random variable,”  IEEE Trans. Commun., vol. 27, no. 6, pp. 892–899, 1979.
  * [14] David L Neuhoff,  “Polar quantization revisited,”  in IEEE Int. Symp. Information Theory, 1997, p. 60.
  * [15] S Wilson,  “Magnitude/phase quantization of independent Gaussian variates,”  IEEE Trans. Commun., vol. 28, no. 11, pp. 1924–1929, 1980.
  * [16] Renat Vafin and W Bastiaan Kleijn,  “Entropy-constrained polar quantization and its application to audio coding,”  IEEE Speech Audio Process., vol. 13, no. 2, pp. 220–232, 2005.
  * [17] Huihui Wu and Sorina Dumitrescu,  “Design of optimal entropy-constrained unrestricted polar quantizer for bivariate circularly symmetric sources,”  IEEE Trans. Commun., vol. 66, no. 5, pp. 2169–2180, 2018.
  * [18] Byung-Jun Yoon and Henrique S Malvar,  “Coding overcomplete representations of audio using the MCLT,”  in Data Compress. Conf., 2008, pp. 152–161.
  * [19] Byeongho Jo, Seungkwon Beack, and Taejin Lee,  “Modified unrestricted polar quantization with the psychoacoustic parameter for audio coding,”  in Int. Congress. Acoustics, 2022.
  * [20] Ye Wang, Leonid Yaroslavsky, and Miikka Vilermo,  “On the relationship between mdct, sdft and dft,”  in Int. Conf. Signal Process., 2000, vol. 1, pp. 44–47.
  * [21] Guillaume Fuchs, Markus Multrus, Max Neuendorf, and Ralf Geiger,  “MDCT-based coder for highly adaptive speech and audio coding,”  in Eur. Signal Process. Conf., 2009, pp. 1264–1268.
  * [22] ITU-R BS. 1387,  “Method for objective measurements of perceptual audio quality,”  1998\.
  * [23] Peter Kabal et al.,  “An examination and interpretation of ITU-R BS. 1387: Perceptual evaluation of audio quality,”  TSP Lab Technical Report, McGill University, pp. 1–89, 2002.
  * [24] ITU-R BS.1534,  “Method for the subjective assessment of intermediate sound quality (MUSHRA),”  .