=Paper=
{{Paper
|id=Vol-3831/paper6
|storemode=property
|title=Prediction of Continuous Targets by Explainable Imbalanced Regression from Omics Data in Childhood Obesity
|pdfUrl=https://ceur-ws.org/Vol-3831/paper6.pdf
|volume=Vol-3831
|authors=María Arteaga,Álvaro Torres-Martos,Augusto Anguita-Ruiz,Jesús Alcalá-Fdez,Rafael Alcalá,María José Gacto
|dblpUrl=https://dblp.org/rec/conf/explimed/ArteagaTAAAG24
}}
==Prediction of Continuous Targets by Explainable Imbalanced Regression from Omics Data in Childhood Obesity==
https://ceur-ws.org/Vol-3831/paper6.pdf
Prediction of Continuous Targets by Explainable
Imbalanced Regression from Omics Data in
Childhood Obesity
María Arteaga1 , Álvaro Torres-Martos2,3 , Augusto Anguita-Ruiz4,2,3 ,
Jesús Alcalá-Fdez1,5 , Rafael Alcalá1,5,* and María José Gacto6,5
1
Dept. of Computer Science and Artificial Intelligence, Univ. of Granada, Granada, Spain
2
Dept. of Biochemistry and Molecular Biology II, “José Mataix Verdú Inst. of Nutrition and Food Technology (INYTA)
and Center of Biomedical Research, Univ. of Granada, Inst. de investigación Biosanitaria ibs.GRANADA, Granada, Spain
3
CIBER de Fisiopatología de la Obesidad y Nutrición (CIBEROBN), Inst. de Salud Carlos III, Madrid, Spain
4
Barcelona Inst. for Global Health (ISGlobal), Barcelona, Spain
5
Research Inst. in Data Science and Computational Intelligence (DaSCI), Univ. of Granada, Granada, Spain
6
Dept. of Software Engineering, Univ. of Granada, Granada, Spain
Abstract
Childhood obesity is a persistent challenge for society since it is highly related to insulin resistance and a
wide range of other chronic diseases, which impair not only the health of the people, but also the health
system itself due to extra costs of treating them. In this context, it is imperative to consider innovative
approaches, like eXplainable Artificial Intelligence, to understand the factors underlying childhood
obesity. Data imbalance has been thoroughly studied in classification, but scarcely studied in regression
since classes do not exist. However, extreme values in continuous domains, which are usually minority
are often the most clinically relevant. Facing the imbalance in regression while obtaining explainable
models has never been studied. In this application work, we adopt a Machine Learning approach to obtain
explainable regression models for imbalanced continuous targets in childhood obesity, as HOMA-IR and
Waist Circumference. We consider 79 variables, including targeted metabolomics, targeted proteomics,
exposomic data (e.g., the physical activity subdomain), hematological parameters and anthropometry
of Spanish children from 3 to 18 years with significant imbalance ratios around 12% in HOMA-IR and
Waist Circumference. Even though we mainly focus on extensions of linguistic fuzzy rule-based systems,
particularly designed for explainability, we also consider highly accurate complementary approaches as
Random Forest that could additionally provide contrast interesting information by post-hoc SHAP. The
models so obtained are better considering the relevant minority information (17% improvements in F1).
Moreover, they seem to properly explain biologically meaningful relations, as in the case of physical
activity data or the one with the follicle stimulating hormone, among others.
Keywords
Childhood Obesity, Insulin Resistance, Accelerometry, eXplainable Regression, Imbalanced Regression
EXPLIMED - First Workshop on Explainable Artificial Intelligence for the medical domain - 19-20 October 2024, Santiago
de Compostela, Spain
*
Corresponding author.
$ m.arteagajover@gmail.com (M. Arteaga); alvarotorresmartos@gmail.com (Á. Torres-Martos);
augusto.anguita@isglobal.org (A. Anguita-Ruiz); jalcala@ugr.es (J. Alcalá-Fdez); alcala@decsai.ugr.es (R. Alcalá);
mjgacto@ugr.es (M. J. Gacto)
0000-0001-6774-6219 (M. Arteaga); 0000-0003-3198-2556 (Á. Torres-Martos); 0000-0001-6888-1041
(A. Anguita-Ruiz); 0000-0002-6190-3575 (J. Alcalá-Fdez); 0000-0003-1140-6156 (R. Alcalá); 0000-0001-9895-9647
(M. J. Gacto)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
1. Introduction
Childhood obesity entails a persistent challenge for society with consequences in the short
and long term. It represents a risk factor for adult obesity, Insulin Resistance (IR) and the
metabolic syndrome (metS), which pose an increased risk of cardiovascular diseases, type 2
diabetes, stroke, etc. Furthermore, it is also related to other chronic diseases, such as respiratory
disorders, musculoskeletal disorders and liver disease. [1, 2, 3] As well as being harmful for
health, childhood obesity is also detrimental for the health system itself due to the additional
cost of treating the other disorder it causes.
Thus, the need for innovative and personalized approaches for the prevention and treatment
of childhood obesity arises. This research involves the analysis of omics data (genomics,
epigenetics, exposomics, metabolomics, etc.), and advanced tools, such as Machine Learning
(ML), which enable the integration of these data and provide a more holistic vision of the issue.
It is evident but crucial that the models generated in health-related issues must not only predict
the variables of interest, but also provide detailed insights on the underlying factors. Following
this idea, eXplainable Artificial Intelligence (XAI) [4] becomes an essential aspect.
In addition, real-world problems frequently involve imbalanced data due to limitations in data
collection, the scarcity of cases, etc. For instance, in a study on childhood obesity, while most
children may present a slightly elevated Body Mass Index (BMI), there may be a lack of children
who are severely obese, the most clinically relevant. A considerable imbalance seriously impacts
and biases the models, so it must be treated. This problem has been extensively studied in
classification, but scarcely studied in regression since classes do not exist.
Some of the current proposals are pre-processing techniques, such as the over-sampling
method SMOTE for regression [5], which is based on nearest neighbors, and its extension,
SMOGN [6], which incorporates Gaussian noise. Other proposals are modified algorithms that
address the imbalance problem within the learning process itself, such as the modifications
of the linguistic algorithm FSMOGFS𝑒 +TUN𝑒 [7] (here referred to as LING for simplicity) and
the purely approximate method METSK-HD𝑒 [8], detailed in [9]. However, even in the case of
the linguistic algorithm, interpretability aspects were not considered, so that it presents severe
semantic problems (linguistic terms extreme overlapping, significant rule inconsistency, etc.).
In this application work, we explore a pioneering approach that, for the first time, concurrently
addresses both explainability and imbalanced regression. Our aim is to develop interpretable
regression models for two imbalanced continuous targets in childhood obesity: The HOMA-
IR and Waist Circumference z-scores (zHOMA-IR and zWC, respectively). We mainly focus
on the extension of a linguistic fuzzy rule-based system, originally called LING𝐶𝐹 𝐿𝑇 𝑆 [10]
(Composed Fuzzy Linguistic Term Set based approach), which would be able to provide more
interpretable information since it is particularly designed for explainability. For simplicity, we
will call the original algorithm LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 , and our extended proposal LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -
ImbR. Alternatively, in this contribution, we also explore the use of Random Forest [11] combined
with SMOGN pre-processing as complementary approach to contrast the results obtained in
the proposed algorithm. While Random Forest generally outperforms the fuzzy rule-based
models, it provides less directly interpretable information, relying solely on post-hoc SHapley
Additive exPlanations (SHAP) analysis [12]. But contrasting the information obtained from
both approaches could be still interesting.
We consider 79 variables, including targeted metabolomics (e.g., cardiometabolic and hor-
monal biomarkers), targeted proteomics (e.g., inflammatory and hepatic biomarkers), some
subdomains of the exposome (e.g., the physical activity subdomain, obtained by accelerometry,
family history, and socioeconomic and demographic factors), as well as hematological parame-
ters and anthropometry, of Spanish children from 3 to 18 years. Although all children are obese
or overweight, zHOMA-IR and zWC present significant imbalance ratios around 12% due to a
lack of representation of exceptionally high values, which are in fact the most clinically relevant
cases. The said modification of the linguistic algorithm FSMOGFS𝑒 +TUN𝑒 proposed in [9] for
actively addressing the imbalance will be also used for comparison in terms of Mean Square
Errors (MSE) and imbalance consideration.
The models generated through the proposed approaches exhibit notable improvements in
considering the relevant minority information, resulting in F1 1 increases over 17%. These
models effectively enlighten biologically significant relationships among the variables and the
predicted targets, as in the case of physical activity data or the one with the follicle stimulating
hormone (FSH), among others.
This work is organized as follows. In Section 2, we explore the concept of imbalanced
regression and its implications in real-world problems. In Section 3, we focus on childhood
obesity prediction, as we describe the case study (Section 3.1), the challenges of omics and data
pre-processing (Section 3.2) and the importance of considering imbalance in this particular
regression problem (Section 3.3). In Section 4, we present our linguistic rule-based system for
explainable imbalanced regression. Section 5 describes the experimental design, while Section
6 presents the performance results obtained in both problems (Section 6.1) and explores the
relevant variables and explains a model example (Section 6.2). Finally, Section 7 discusses the
findings, limitations and future perspectives.
2. Imbalanced Regression in Real-World Problems
In classification, imbalance refers to an uneven representation of classes. As minority groups are
easily identifiable, this issue has been thoroughly studied. However, in regression, imbalance
refers to a skewed distribution of the data in the target variable, this is, an underrepresentation
of certain specific subdomains, often those clinically relevant. Since classes do not exist, there is
scarce emphasis on the need for a complete representation of the continuous variable domain.
Identifying subdomains seems challenging, as they are usually not uniformly relevant and there
is typically poor representation of the relevant subdomains, so it has been hardly studied.
Real-world problems frequently involve imbalanced data due to various factors, such as
limitations in data collection or the scarcity of cases. For instance, in a study on childhood
obesity, most children fmay present a slightly elevated Body Mass Index (BMI); in a study on
type 2 diabetes, the majority of the patients may present moderately elevated levels of glucose;
in a study in Chronic Obstructive Pulmonary Disease (COPD), most patients present a moderate
reduction in pulmonary capacity. In all these examples, there is a lack of representation of the
most clinically relevant cases (extreme values): children who are severely obese, patients with
significantly elevated glucose levels, and patients with severe reductions in pulmonary capacity,
1
Adaptation of the well-known F1 for classification.
respectively. A considerable imbalance seriously impacts the models, biasing towards the most
frequent values and limiting generalization to underrepresented cases, so it must be treated.
In order to formally introduce the concept of imbalanced regression [6, 13], let us contemplate
an unknown function 𝑓 (𝑋1 , 𝑋2 , ..., 𝑋𝑝 ), where 𝑝 denotes the number of predictor variables,
aiming to approximate the output variable’s defining function with maximal accuracy and
proximity. Next, we possess a training dataset 𝐷 = {⟨𝑥𝑖 , 𝑦𝑖 ⟩}𝑛𝑖=1 with 𝑛 instances from which
we will derive our approximate function 𝑓 , where 𝑥 and 𝑦 are the inputs and the continuous
output values, respectively. To take into account a possible imbalance in 𝑌 , we need to define
a relevance distribution function Φ(𝑌 ), so we can create a “minority” subset 𝐷𝑟 of relevant
data (i.e. those instances with a relevance value above 𝑡𝑟 ), as well as a “majority” subset 𝐷𝑛 of
non-relevant data (those with relevance values less than or equal to 𝑡𝑟 ). Φ(𝑌 ) is based on the
concept of extreme 𝑌 values and established in the following sigmoid function by Torgo and
Branco et al. as follows:
1
Φ(𝑌 ) = (1)
1 + 𝑒−𝑠*(𝑌 −𝑐)
where 𝑐 is the center of the sigmoid, that is, the value where Φ(𝑌 ) = 0.5, and 𝑠 is the shape
of the sigmoid (see [13] for more details). Since both, low extreme values and high extreme
values could exist, Φ(𝑌 ) is defined with two different sigmoid functions. Moreover, they also
adapted some of the metrics used in imbalanced domains, as the well-known F-Measure, F1
when 𝛽 = 1 (harmonic mean of Recall and Precision):
(𝛽 2 + 1) * 𝑃 𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 * 𝑅𝑒𝑐𝑎𝑙𝑙
𝐹 = (2)
𝛽 2 * 𝑃 𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + 𝑅𝑒𝑐𝑎𝑙𝑙
This adaptation draws upon well-established classification concepts: Recall ( 𝑇 𝑃𝑇+𝐹𝑃
𝑁 where
TP represents True Positives and FN represents False Negatives) and Precision ( 𝑇 𝑃 +𝐹 𝑃 where
𝑇𝑃
FP represents False Positives). To achieve this, they establish a non-trivial connection between
instances exhibiting an acceptable margin of error (effectively classified in classification prob-
lems) and their relevance, so that they can further quantify those relevant bad predictions (see
[13] for further insights into this adaptation).
As stated previously, imbalanced regression has been hardly studied. The few existing
techniques can be categorized as [9]:
• Pre-processing techniques: These re-sampling techniques balance data distribution via
pre-processing so that algorithms focus on the most relevant instances. Their main
advantage is that they can be applied to any existing algorithm, whereas their main
disadvantages are that they are dependent on the data and sensitive to the data quality
and parameters adjustment. These could be considered “passive” techniques since they
do not affect the learning process.
• Algorithmic approaches: These are modified algorithms that address the imbalance
problem within the learning process itself. These could be considered “active” techniques
since they affect the learning process.
Within “passive” techniques, SMOTE for regression is outlined [5]. It is based on SMOTE
(Synthetic Minority Over-sampling Technique) [14], an over-sampling method for classification
that generates synthetic instances by interpolating between minority class samples based on
their nearest neighbors and is usually combined with under-sampling. In SMOTE for regression,
synthetic instances are generated by considering the regression line between neighboring
minority class samples. SMOGN (Synthetic Minority Over-sampling Technique for Regression
with Gaussian Noise) [6] is an extension of SMOTE for regression that introduces Gaussian
noise when interpolating between samples, so it preserves the diversity of the data. Regarding
"active" approaches, the state-of-the-art is limited. We must highlight two algorithms based on
fuzzy rules: a linguistic algorithm called FSMOGFS𝑒 +TUN𝑒 [7] (in this work referred to as LING
for simplicity) and an approximate algorithm called METSK-HD𝑒 [8], which were modified to
actively address imbalance [9].
As said, even in the case of the linguistic algorithm, it was not particularly designed to
provide fully transparent and therefore explainable regression models. In this work, we extend
LING𝐶𝐹 𝐿𝑇 𝑆 [10] (Composed Fuzzy Linguistic Term Set based approach, in this work referred to
as LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 ), which does not natively handle imbalanced regression, to actively address
the imbalance, since this algorithm was particularly designed to obtain interpretable/transparent
linguistic models.
3. Childhood Obesity Prediction Problem
In this section, we will delve into the challenge of predicting continuous variables related to
pediatric obesity in overweight and obese children. We will begin with a description of the case
study and predictive targets, followed by a discussion on the challenges in omic data analysis
and pre-processing. Finally, we will explore the relevance of considering imbalanced regression.
3.1. Description of the Case Study and Continuous Prediction Targets
Childhood obesity presents an enduring challenge for nowadays society given its strong relation
with insulin resistance, metabolic syndrome, and numerous other chronic diseases. These not
only affect the health of the population, but also the health system itself due to the additional
costs of treating them. In this case study, we consider a database from national project of
the Carlos III Health Institute [15]. It corresponds to the IBEROMICS project [16], which
focuses on overweight or obese children between 3 and 18 years old in Santiago de Compostela
and Zaragoza. Since we aimed to provide a comprehensive analysis that captures potential
interactions and influences on the target variables, we consider 79 variables of interest, including
targeted metabolomics (e.g., cardiometabolic and hormonal biomarkers), targeted proteomics
(e.g., inflammatory and hepatic biomarkers), some subdomains of the exposome (e.g., the
physical activity subdomain, obtained by accelerometry, family history, and socioeconomic
and demographic factors), as well as hematological parameters and anthropometry. This
approach let us explore the relationships between a wide range of variables. Fuzzy rule-based
algorithms ensure that only the most relevant variables contribute to each rule, maintaining
model interpretability and relevance, while Random Forest are known for their ability to handle
high-dimensional data and their robustness to irrelevant features.
We established two variables as continuous prediction targets in childhood obesity: zHOMA-
IR (n = 190) and zWC (n = 204). HOMA-IR (Homeostasis Model Assessment for Insulin Resistance)
is an indicator of insulin resistance, which is strongly related to obesity since fat cells release
biochemical substances that can interfere with the normal insulin response. Waist Circumference
is an indicator to evaluate central obesity and visceral fat distribution. It is related to insulin
resistance since visceral abdominal fat is metabolically active and produces several biochemicals,
including adipokines and cytokines, that may contribute to it. A z-score is a statistical measure
that represents the number of standard deviations a data point is from the mean of a reference
population dataset: negative z-scores indicate below-average values and positive z-scores
indicate above-average values, with the absolute value representing the deviation from the
mean. This standardization ensures that comparisons are fair and meaningful, especially when
analyzing variables where values can vary significantly based on age and height, as in this case.
The HOMA-IR z-score was calculated using Stavnsbo as reference (see [17] for more details),
whereas the Waist Circumference z-score was calculated using Ferrández as reference (see [18]
for further insights).
This case study mirrors a real-world health-related scenario, necessitating interpretability
(models must offer not just variables but also meaningful explanations) and a consideration of
imbalanced data. This dual focus represents a novel approach.
3.2. Challenges in Omics ML and Data Pre-processing
The application of ML to Omics data implies unique challenges in pre-processing and analysis.
Additionally, to ensure robust and reliable models, it was crucial to effectively handle missing
values, noise, outliers and batch effects, among others.
In this case study, we conducted data cleaning to ensure the reliability of the dataset. Raw
accelerometry data were transformed using the Actilife v.6.13.3 software into variables of interest,
such as the number of steps or measures of sedentary lifestyle and exercise of different levels
corresponding to weekdays, weekends, or in total. Also, several variables (e.g., HOMA-IR, BMI,
triglycerides, blood pressure, etc.) were transformed to z-scores. Specifically, these variables are
WC ([18]), HOMA-IR, HDL and TG ([17]), BMI ([19]), DBP and SBP ([20]). Additionally, to ensure
robust and reliable models, it was crucial to effectively handle missing values, noise, outliers
and batch effects, among others. Categorical variables were transformed by label encoding.
Variables with more than 15% missing values were removed, followed by a non-parametric
missing value imputation using the MissForest algorithm, based on Random Forest [21]. An
exploratory analysis ensured that this imputation did not significantly affect data distribution.
3.3. Regression Imbalance Consideration
Real-world problems frequently involve imbalanced data due to various factors, such as limi-
tations in data collection or the scarcity of cases. A considerable imbalance seriously impacts
the models, biasing towards the most frequent values and limiting generalization to under-
represented cases, so it must be treated. For example, if the majority of the children of a
hypothetical dataset had an average body mass index (BMI), the model would be biased towards
normal-weight children.
According to Section 2, the imbalance ratio is calculated as the percentage of instances with
a phi(Y) value over 0.8, that is,
number of instances with Φ(𝑌 ) > 0.8
· 100
total number of instances
In this case study, although all the children are overweight obese, there is still a lack of
representation for those with exceptionally high values in zHOMA-IR and zWC, which present
significant imbalance ratios (12.105% and 12.745%, respectively). This means that children
with severe insulin resistance and children with a severe accumulation of abdominal fat, which
are in fact the most clinically relevant cases, are underrepresented. Since it may appreciably
affect and bias the models, the imbalance existent in these datasets must be considered for
obtaining not only transparent/interpretable regression models, but also more reliable ones.
4. Linguistic Rule-based System for explainable imbalanced
regression
In [10], the authors propose a novel method for evolutionary regression, which aims to predict
numerical outcomes based on input variables while maintaining simplicity and transparen-
cy/interpretability, measured by two semantic interpretability indexes, Geometric Mean of
3 complementary Metrics (GM3M) and Rule Meaning Index (RMI). As stated before, in this
work this method is referred to as LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 . GM3M and RMI are two well-known in-
terpretability measures from the specialized literature that account for the preservation of the
initial linguistic terms definitions and/or the rule meaning/consistency, respectively.
1
0
Figure 1: Illustration of linguistic partition with a GM3M value of 0.8 (blue) in comparison to the
corresponding strong fuzzy partition (gray).
GM3M measures (ranging from 0.0 to 1.0) the semantic interpretability at the level of linguistic
partitions, where values close to 1.0 represent a really high proximity to the membership
functions associated with the initial linguistic terms (in our case, equally distributed strong
linguistic partitions). See an example of linguistic partition with GM3M equal to 0.8 in Figure 1.
RMI measures (also raging from 0.0 to 1.0) the semantic interpretability at the rule base level,
with values close to 1.0 representing the absence of inconsistent rules (i.e., the output of the
rule is consistent with the whole model output in the corresponding rule core domain). GM3M
or RMI associated to a whole model are computed as the worst case from all the linguistic terms
or rules, respectively. Thus, having GM3M=0.7 and RMI=0.8 for one regression model means
that all membership functions have GM3M equal or higher than 0.7 and that all the rules have
RMI values equal or higher than 0.8.
The algorithm combines two key elements: A new linguistic fuzzy grammar and an in-
terpretable linear extension. This approach helps to effectively reduce the number of rules,
maximize the semantic interpretability (both GM3M and RMI), minimize errors, and maintain
the rule length (number of conditions) within reasonable bounds. It operates on a two-stage
tree-based hybrid evolutionary multiobjective algorithm:
• First stage: This involves learning the initial linguistic partitions and associated rules.
Here, an embedded multiobjective evolutionary learning of the linguistic partitions
(number of linguistic terms and their definitions) is employed, aiming to minimize both
number of rules and errors measured as the 𝑀 𝑆𝐸/2 (MSE divided by 2). It utilizes a
multiobjective evolutionary approach to learn the linguistic partitions and applies a
rapid method to derive rule sets for each definition of the linguistic partitions. It builds
upon a hybridization with an innovative linguistic tree-based rule learning technique,
extending the renowned M5-prime method to derive rule sets for each evolved linguistic
partitioning.
• Second stage: This stage entails post-processing to further refine the learned solutions. It
employs an advanced multiobjective evolutionary algorithm to fine-tune membership
functions and for rule selection. This refinement process aids in minimizing number
of rules, maximizing linguistic interpretability (GM3M and RMI), and reducing errors
(𝑀 𝑆𝐸/2 ) in the initial global structure obtained in the first stage, which was initially
based on strong fuzzy partitions.
Unlike other methods, the adaptability of evolutionary algorithms allows them to adjust to
specific problems in a versatile manner. Leveraging this characteristic, in this contribution, we
introduce a solution called LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR, that guides the algorithm towards the said
outcomes (number of rules, GM3M, RMI and 𝑀 𝑆𝐸/2 ) but by also considering the imbalance
through a refined evaluation function enriched with additional information. The idea is to
replace 𝑀 𝑆𝐸/2 in the second stage by an objective function maximizing the performance
obtained in the set of relevant data 𝐷𝑟 (guided by the relevance function), without unbalancing
the performance that we obtain in the rest of the data 𝐷𝑛 (guided by 𝑀 𝑆𝐸/2 ). In order to do
this, we define a Weighted MSE (𝑀 𝑆𝐸/2 𝑊 ) attending to the following equations:
(︁(︀ |𝐷
∑︁𝑟| )︁
(3)
)︀
𝑑𝑤 = 𝑎 + Φ(𝑦𝑖 ) + |𝐷𝑛 | .
𝑖=1
𝑑𝑤 represents a weight factor that combines the contributions from the relevant and non-
∑︀|𝐷 |
relevant data, since 𝑖=1𝑟 𝑎 + Φ(𝑦𝑖 ) is the sum of a constant value (𝑎) and the relevance
)︀
function (Φ(𝑦𝑖 )), and 𝐷𝑛 is the number of instances in the non-relevant set. Based on another
work [9], constant 𝑎 has been set at 5, since it was the value that obtained the best results.
|𝐷𝑟 |
∑︁
𝑆𝐸𝐷𝑟 = (𝐹 (𝑥𝑖 ) − 𝑦 𝑖 )2 * (𝑎 + Φ(𝑦𝑖 )), (4)
𝑖=1
where 𝑆𝐸𝐷𝑟 is the sum of the squared errors for the relevant data, weighted by 𝑎 + Φ(𝑦𝑖 ).
|𝐷𝑛 |
∑︁
𝑆𝐸𝐷𝑛 = (𝐹 (𝑥𝑖 ) − 𝑦 𝑖 )2 , (5)
𝑖=1
and 𝑆𝐸𝐷𝑛 is the sum of the squared errors for the non-relevant data, without additional
weighting.
𝑆𝐸𝐷𝑟 + 𝑆𝐸𝐷𝑛
𝑊
𝑀 𝑆𝐸/2 = (6)
𝑑𝑤 * 2
where 𝑀 𝑆𝐸/2
𝑊 is the weighted MSE, normalized by 𝑑 .
𝑤
5. Experimental design
In this work, we mainly focus on the proposed extension of the linguistic fuzzy rule-based system
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 [10], which could provide more interpretable information since it is particularly
designed for explainability. To evaluate the performance of this algorithm, we apply various
algorithms (Random Forest [11], LING [7], and LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 [10]) to the stratified partitions
from the original dataset, so that imbalance is not considered, and again after applying SMOGN
to the training sets, so that imbalance is considered via pre-processing the data. Moreover, we
employ the modified version of LING [9] (LING-ImbR) and our proposal for LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 ,
that actively address data imbalance (see Section 4), to the stratified partitions from the original
dataset. Finally, the methods considered in the experiments are summarized in Table 1. Except
for the number of trees in Random Forest, which was established to 500, the default parameters
suggested by authors were used for all the algorithms.
Table 1
Methods considered in the experiments. MOEA: Multi-Objective Evolutionary Algorithm, TS: Tuning
and Selection, ImbR: Imbalanced Regression, *: Proposed here.
Method Ref. Description
SMOGN+? [6] Preprocessing plus other method (?): SMOTE for
regression with the introduction of Gaussian noise
RF [11] Random Forest Regressor
LING [7] MOEA for embedded DB Learning, RB wrapper
(FSmogfs𝑒 +Tun𝑒 ) generation and TS MOEA
LING-ImbR [9] MOEA for embedded linguistic partitions learning,
rule base wrapper generation and TS MOEA &
ImbR
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 [10] eXplainable-based MOEA for embedded learning of
transparent linguistic partitions with wrapper
linguistic tree-based rule base generation and TS
MOEA
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 - * LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 & ImbR
ImbR
Since error estimation depends on the training and test data, we used a repeated 10-fold
stratified cross-validation to reduce the loss of diversity when partitioning the data and preserve
the minority set representation. For each dataset (the dataset with zHOMA-IR as target and the
one with zWC), we use a 10-fold cross-validation strategy, that is, the original data is divided
into 10 subsets or folds. This is done two times. Thus, we created 20 partitions per dataset.
The stratification used in these regression problems consisted in the discretization of the
continuous target variables into in 𝑐 cuts, that is, a specified number of bins which result from
dividing the number of examples of our dataset by the number of desired partitions. However,
it must be applied separately to relevant and non-relevant data. In order to achieve this, based
on another work [9], the threshold 𝑡𝑟 for the relevance function was established at 0.8, so
that higher values were considered relevant, and lower values, normal. Once relevant data
were identified, stratification was applied to the relevant and non-relevant data separately. For
each set (relevant and non-relevant), we divided the data into 𝑐 cuts, ensuring that each bin
had approximately the same number of examples (equal-frequency bins). After creating the
bins, we randomly and proportionally distributed the relevant and non-relevant instances from
each cut among the relevant and non-relevant data partitions, respectively, to ensure balanced
representation. Finally, both relevant and non-relevant data partitions were joined.
Also, we need to consider metrics to properly evaluate the algorithms performance. In this
study, we mainly consider two metrics: the said adaptation of F1 for regression (see Section 2) and
the Mean Square Error (MSE). F1 is a metric that allows us to evaluate how well are the models
addressing the imbalance problem. MSE is a classical metric to evaluate accuracy. It measures
the average squared difference between the predicted and the actual values. Considering both
will help us assess the models performance not only on the relevant data but also on the overall
set.
6. Main Results and Insights: zHOMA-IR and zWC
In this section, we will delve into the performance results obtained in both problems (zHOMA-IR
and zWC). Also, we will explore the relevant variables and explain a model example.
6.1. Performance results obtained in both problems
Firstly, we will compare the results for all the algorithms, which are shown in Table 2. Each row
corresponds to an algorithm, with those actively addressing imbalanced marked in grey. The
table is divided in two sections, one per problem (zHOMA-IR and zWC). Within each section,
for each algorithm, the first column shows the number of variables (when applicable) and the
second column, the number of rules (when applicable). The third and forth columns show the
average MSE and F1 values in the test sets, respectively.
The results show that when SMOGN is applied, the F1 for all the algorithms values are better
compared to those obtained with the original data. This means that addressing imbalance
through the previously mentioned passive imbalance pre-processing technique improves the
performance in the relevant set. However, the results show that applying SMOGN implies a
cost in MSE, that is, an increased error in predicting our continuous outcomes. The results
obtained in F1 by the algorithms that actively address imbalance within the learning process
Table 2
Results obtained by the studied algorithms. MSE and F1 values are the average errors calculated on the
corresponding test sets (generalization). Those actively considering imbalanced regression are marked
in grey.
zHOMA-IR zWC
Methods #Vars. #Rules MSE𝑡𝑠𝑡 F1𝑡𝑠𝑡 #Vars. #Rules MSE𝑡𝑠𝑡 F1𝑡𝑠𝑡
RF - - 1.018 0.00 - - 0.944 0.55
SMOGN+RF - - 1.241 0.45 - - 1.210 0.71
LING 6.7 23.5 1.662 0.09 3.9 8.7 1.200 0.48
SMOGN+LING 5.8 27.7 2.905 0.15 4.8 20.5 1.931 0.49
LING-ImbR 6.7 21.7 2.070 0.12 3.9 8.0 1.402 0.52
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 3.4 6.7 1.804 0.36 2.2 4.5 1.329 0.62
SMOGN+LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 2.8 7.0 3.647 0.50 2.4 5.7 2.217 0.73
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR 3.4 6.4 2.305 0.40 2.2 4.3 1.329 0.75
(those marked in gray) are slightly worse or better (depending on the problem) than those
obtained when using SMOGN, whereas MSE values are notoriously better, similar to the results
obtained without considering the imbalance, specially for zWC. LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR presents
the best values for F1 in both problems. While Random Forest outperforms the fuzzy rule-based
models in MSE in both problems, it provides less detailed information, relying solely on post-hoc
SHapley Additive exPlanations (SHAP) analysis. For both linguistic algorithms, the number
of variables and rules are similar in the original algorithm, the SMOGN approach and the
extended version. Nevertheless, the number of variables and rules is lower in LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒
compared to LING, which contributes to the interpretability of the model, since LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒
is particularly designed for explainability. Furthermore, the errors are in 10 to 15% of the output
domains, so the models explain from 85 to 90% of the domain.
Table 3
GM3M and RMI: Interpretability metrics obtained by the linguistic fuzzy algorithms
zHOMA-IR zWC
Methods GM3M RMI GM3M RMI
LING 0.2 0.3 0.2 0.5
SMOGN+LING 0.2 0.3 0.2 0.4
LING-ImbR 0.2 0.4 0.2 0.5
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 0.7 1.0 0.8 1.0
SMOGN+LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 0.7 0.9 0.7 1.0
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR 0.7 1.0 0.7 1.0
Secondly, we will compare the linguistic algorithms. Table 3 shows the interpretability
metrics obtained by the linguistic fuzzy algorithms. Each row corresponds to an algorithm.
The columns are also divided in two sections: the results for the zHOMA-IR and the zWC
dataset. In each section, the first columns represents the GM3M value and the second column,
the RMI value. As we stated previously, GM3M measures the semantic interpretability at the
level of linguistic partitions, whereas RMI measures the semantic interpretability at the rule
base level. Thus, having GM3M=0.7 and RMI=1.0 for one algorithm in the table means that,
in average the 20 runs, all membership functions have GM3M equal or higher than 0.7 and
that all the rules have RMI values equal to 1.0 (all the rules in the 20 runs are fully consistent
rules). For both zHOMA-IR and zWC, LING, LING using SMOGN and the modified version
of LING have consistently low GM3M and RMI values, indicating the existence of significant
interpretability problems. In contrast, LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 , LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 using SMOGN and
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR show quite high GM3M and RMI values, indicating a very high level of
transparency. GM3M values for this algorthim, which are close to 1.0, represent a high proximity
to the membership functions associated with the initial linguistic terms. RMI values, which
are even 1.0 in the case of LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 and LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR, represent the absence of
inconsistent rules.
Table 4
Variables that appear in the models shown by importance: More to less times used #T and best to worst
SHAP ranking/position respectively. Those related to physical activity are marked in blue.
zHOMA-IR zWC
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR SMOGN+RF LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR SMOGN+RF
#T Vars. Vars. #T Vars. Vars.
15 zBMI zTriglycerides 20 zBMI zBMI
13 Adiponectin-leptin ratio MCH 18 Age LightWEd
11 TSH SedentaryWd 11 FSH LH
9 Urea, Calcium zBMI 7 %Fat_mass cpmWd
8 Age, Origin, StepsWd, LDL 3 Creatinine, Erythrocytes, Iron
zHDL, Light, Sex
zTryglicerides zWC 2 %ACT, ALT, GGT, FSH
Haemoglobin,
7 Sedentary, zWC MCV LightWEd, Moderate, Ori- Age
gin,
6 Tanner_stage Calcium Tanner_stage, Testos- Cortisol
terone, zSBP,
5 Cortisol, SedentaryWEd zHDL zTriglycerides zDBP
4 CpmWd, Haemoglobin, ALP Monocytes
ModerateWd,
#days, SedentaryWd Total_Bilirrubin Erythrocytes
3 LDLc, LightWEd, mvpaWd, Creatinine Calcium
MCV
6.2. Relevant Variables and Explained Model Example
Table 4 shows the variables that appeared in the models shown by importance. For
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 , variables are shown from more to less times used (#T), whereas for Ran-
dom Forest, variables are shown from more to less importance for the model output based on
SHAP rankings.
In the zHOMA-IR dataset, it is noteworthy that all but one of the rule bases include a variable
associated with accelerometry (see those in blue in Table 4). However, in the zWC dataset,
accelerometry variables are present in only half of the generated rule bases, suggesting that
these variables may not always be significant. This suggests complex factors beyond physical
activity influence in waist circumference during childhood obesity. Further analysis and context
exploration are necessary for deeper insights into this fact.
Moreover, the rules for zWC consistently incorporate the variable zBMI in each line. However,
in the zHOMA-IR dataset, the variable zBMI appears 15 times, while the variable Adiponectin-
leptin ratio appears 13 times, indicating the presence of specific rule bases containing both zBMI
and Adiponectin-leptin ratio. The Adiponectin-leptin ratio has been associated with insulin
resistance in overweight and obese children [22]. The thyroid-stimulating hormone (TSH)
appears 11 times, as abnormal levels of TSH seem positively correlated to insulin resistance
[23] independently of the body status [24].
SHAP rankings let us contrast to the most variables used in Random Forest models. For the
zHOMA-IR problem, the z-score of triglycerides is the most important variable in the Random
Forest models, followed by the mean cell hemoglobin (MCH), an accelerometry measure, zBMI,
which is the most repeated variable for the linguistic models, etc. For the zWC problem, the
most important variables for the output are zBMI, some accelerometry measures, the follicle
stimulating hormone (FSH), which is the third most repeated variable for the linguistic models,
etc.
Figure 2: Top 20 variable’s contribution in the SMOGN+RF model in the zHOMA-IR problem (SHAP)
Figure 2 represents a summary graph for a model generated via Random Forest in the
zHOMA-IR dataset. It shows the contribution of the 20 most important variables, ordered
from more important (top) to less important (bottom). Each point represents an instance,
where its horizontal position indicates the impact that the feature has in the model output
(positive or negative), and its color indicates the feature value (blue for low values, red for high
values). Triglycerides, MCH, and SedentaryWd are the most influential features for predicting
zHOMA-IR. Thus, both clinical measures (e.g., lipids, blood counts) and lifestyle factors (e.g.,
sedentary physical activity) are important, demonstrating the multifactorial nature of insulin
resistance. Nevertheless, other characteristics with lower SHAP values contribute cumulatively,
emphasizing the need for a broader view in the prediction and management of insulin resistance.
Figure 3: Top 20 variable’s contribution in the SMOGN+RF model in the zWC problem (SHAP)
Figure 3 represents a summary graph for a model generated via Random Forest in the zWC
dataset. This graph provides an overview of how features affect the model predictions, with
zBMI being notoriously the variable with the highest impact on the model output. However,
this does not imply that the other variables are irrelevant, but that their individual impacts are
smaller compared to zBMI. This means that these features contribute to the model through
their cumulative effects, even if their individual SHAP values are smaller. Furthermore, features
can interact with each other in complex ways, so a small individual SHAP value might still be
crucial when considered in combination with other features.
Figure 4 represents an example of a linguistic model obtained for zWC. The arrangement
of variables in the figure corresponds to the sequence of splits generated in the decision tree
during the rule-learning process. Each split is a division of the data, from more general to more
specific in each split. Colors are used to aid in distinguishing the various cases depicted by
the rules, with each split and variable sharing the same color. Gray text elements are not part
of the rules, but provide extra information, as well as percentages of covered instances and
the GM3M and RMI values. These values provide insights into the semantic quality of each
partition and rule, respectively. Notably, all rules exhibit an RMI value of 1.0, indicating that any
R1RMI:1.0 (19% of the instances):
IF FSH up to Low AND Age up to Low AND LightWd up to High
THEN zWC betw. Low and High (centered on 4.2)
+/-1.26 per +/- unit in zBMI from 3.76(moving in [Low,High])
R2RMI:1.0 (65.2% of the instances)
IF FSH up to Low AND Age from Med AND LightWd up to High
THEN zWC is Low (centered on 1.78 )
+/- 1.08 per +/- unit in zBMI from 4.05(moving in [Low,High])
R3RMI:1.0 (15.8% of the instances)
IF FSH from High
THEN zWC is Low (centered on 1.78 )
+/- 0.27 per +/- unit in zBMI from 1.71(moving in [Low,High])
Variable labels and definition points of the
membership functions:
Follicle-stimulating hormone (FSH) Age - GM3M:0.9
GM3M:0.7 1.7, 4.8, 7.6 VeryLow
-1.1, 1.7, 5.1 VeryLow 4.4, 6.8, 10.3 Low
0.6, 6.2, 7.7 Low 7.0, 10.6, 12.9 Med
4.7, 8.1, 9.8 High 10.1, 13.0, 15.8 High
8.3, 10.2, 14.6 VeryHigh 13.9, 15.8, 18.7 VeryHigh
Light exercise on weekdays (LightWd) Waist Circumference z-score
GM3M:0.9 (zWC) Ferrández - GM3M:0.9
-42.2, 35.9, 112.7 VeryLow -3.0, 2.3, 7.5 Low
44.2, 128.7, 213.3 Low 2.8, 8.0, 12.9 High
128.7, 213.3, 297.8 MedLow
213.3, 297.8, 382.4 MedHigh Body Mass Index z-score
307.1, 362.5, 470.0 High (zBMI) Orbegozo
382.4, 467.0, 551.5 VeryHigh Range: [0.7, 17.3]
Figure 4: Linguistic model obtained by LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR in the zWC problem. MSE𝑡𝑠𝑡 = 0.572,
F1𝑡𝑟𝑎 = 0.85 and F1𝑡𝑠𝑡 = 0.76.
single rule assert actually matches the real output of the model. Moreover, all variables display
GM3M values exceeding 0.7, and even reaching 0.9, suggesting equidistant strong linguistic
partitions. The MSE in test (0.572) is almost a half of the one obtained by RF in average without
imbalance consideration (0.944) while F1 in test is over 0.75. Consequently, it seems that the
method effectively characterizes the overall behavior of the dataset and actually improves the
consideration of the relevant data, which could help to appropriately discern the principal
relationships among predictor variables.
The model was able to accurately retrieve the zWC of children with obesity using just 4
variables (which included the Follicle stimulating hormone (FSH) blood concentrations, the child
age (decimal age), the amount of light physical activity, and the zBMI). Interestingly, despite
the presence of clear zWC indicators among predictors (such is the case of zBMI), the model
chose a non-typical metabolic risk marker to be placed in the first level of division (at the rule
tree-based generation splitting) for each rule; the FSH hormone. FSH showed an intriguing and
non-obvious relationship with the zWC in this population. As can be seen in rule R3, high and
very high ("from High") values of FSH were directly linked to low zWC values, with a zBMI-
mediated effect modification of +/- 0.27 per unit +/-. Individuals with higher concentrations of
this hormone will therefore tend to present less central obesity, with a small contribution of
increases in their zBMI.
In rules R1 and R2, on the other hand, the opposite zWC-FSH relationship was evidenced.
While in R1, "up to Low" values of FSH were linked to medium ("between Low and High") zWC
(as expected from R1), in R2, these same values (up to low values of FSH) were linked to "Low"
zWC, with child’s age jumping on the scene as an effect modifier in this case (i.e., younger
children with up to low FSH tend to present "between Low and High" zWC, while medium aged
and older children with up to low FSH present low zWC). Interestingly, the cutoffs in the age
variable proposed by the method for generating the fuzzy intervals that differentiate these two
rules expand exactly in the estimated ages for normal puberty onset, suggesting a plausible
involvement of the sexual maturation procedure in the FSH-zWC relationship.
Scientific literature has extensively pointed FSH as a significant player in human metabolic
disorders. Epidemiological studies have established a strong correlation between FSH levels and
metabolic diseases, while experimental research has delved into the underlying mechanisms
both in vivo and in vitro [25]. From in vitro and in vivo studies, now we know that FSH is
a risk marker for metabolic dysfunction given its direct role in promoting adipogenesis and
insulin resistance. Nevertheless, epidemiological studies have shown an inverse relationship;
individuals with obesity tend to present lower FSH values (which goes in line with our findings
from R3 and R1). To account for this apparent contradiction, a hypothesis has been proposed:
FSH stimulates obesity, leading to elevated estrogen levels, which subsequently diminish FSH
levels through negative feedback mechanisms. Moreover, since FSH is a sexual hormone, its
production gets increased after puberty initiation, and its effects could be influenced by even
more complex feedback mechanisms, highlighting the relevance of this period for the study of
its effects in obesity and cardiometabolic health (as evidenced by our model in R2) [26].
Our model therefore adds some evidence to the existing findings relating high FSH with
lower obesity (R1 and R3), reporting in this case an important and novel contribution of sexual
development in this relationship (R2). Further epidemiological studies and in vitro studies would
be needed to deepen into presented hypotheses.
7. Conclusions
In this application work, we have adopted a Machine Learning approach to obtain explainable
regression models for two imbalanced continuous targets in childhood obesity, the z-score of
HOMA-IR and the Waist Circumference, with significant imbalance ratios around 12%. The
models obtained by the proposed approaches are better considering the relevant minority infor-
mation, with F1 improvements over 17%. Furthermore, the linguistic fuzzy models generated by
LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 and the proposal, LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR exhibit superior interpretability (ele-
vated GM3M and RMI and low number of variables and rules), since they have been specifically
designed for explainability. LING𝑒𝑋𝑝𝑙𝑎𝑖𝑛𝑎𝑏𝑙𝑒 -ImbR presents the best values for F1, that is, the
metric that allows us to evaluate the models in the imbalanced set, while relatively preserving
the MSE value. Moreover, they seem to properly explain biologically meaningful relations
among the involved factors and the predicted targets. The identification of accelerometry
variables, specially in zHOMA-IR, and novel relationships, such as FSH levels in predicting zWC,
could inform clinical practice by highlighting new biomarkers and pathways for intervention
in childhood obesity.
8. Acknowledgements
This research was supported by the Instituto de Salud Carlos III co-funded by ERDF - A way of
making Europe - and the European Union (grant numbers PI20/00711, PI20/00563, PI23/00129,
PI23/00165 and PI23/00028).
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