In obesity studies, several researchers have been applying machine learning tools to identify factors affecting human body weight. However, a proper review of strength, limitations and evaluation metrics of machine learning algorithms in obesity is lacking. This study reviews the status of application of machine learning algorithms in obesity studies and to identify strength and weaknesses of these methods. A scoping review of paper focusing on obesity was conducted. PubMed and Scopus databases were searched for the application of machine learning in obesity using different keywords. Only English papers in adult obesity between 2014 and 2019 were included. Also, only papers that focused on controllable factors (e.g., nutrition intake, dietary pattern and/or physical activity) were reviewed in depth. Papers on genetic or childhood obesity were excluded. Twenty reviewed papers used machine learning algorithms to identify the relationship between the contributing factors and obesity. Regression algorithms were widely applied. Other algorithms such as neural network, random forest and deep learning were less exploited. Limitations regarding data priori assumptions, overfitting and hyperparameter optimization were discussed. Performance metrics and validation techniques were identified. Machine learning applications are positively impacting obesity research. The nature and objective of a study and available data are key factors to consider in selecting the appropriate algorithms. The future research direction is to further explore and take advantage of the modern methods, i.e., neural network and deep learning, in obesity studies.
Citation: Mohammad Alkhalaf, Ping Yu, Jun Shen, Chao Deng. A review of the application of machine learning in adult obesity studies[J]. Applied Computing and Intelligence, 2022, 2(1): 32-48. doi: 10.3934/aci.2022002
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Abstract
In obesity studies, several researchers have been applying machine learning tools to identify factors affecting human body weight. However, a proper review of strength, limitations and evaluation metrics of machine learning algorithms in obesity is lacking. This study reviews the status of application of machine learning algorithms in obesity studies and to identify strength and weaknesses of these methods. A scoping review of paper focusing on obesity was conducted. PubMed and Scopus databases were searched for the application of machine learning in obesity using different keywords. Only English papers in adult obesity between 2014 and 2019 were included. Also, only papers that focused on controllable factors (e.g., nutrition intake, dietary pattern and/or physical activity) were reviewed in depth. Papers on genetic or childhood obesity were excluded. Twenty reviewed papers used machine learning algorithms to identify the relationship between the contributing factors and obesity. Regression algorithms were widely applied. Other algorithms such as neural network, random forest and deep learning were less exploited. Limitations regarding data priori assumptions, overfitting and hyperparameter optimization were discussed. Performance metrics and validation techniques were identified. Machine learning applications are positively impacting obesity research. The nature and objective of a study and available data are key factors to consider in selecting the appropriate algorithms. The future research direction is to further explore and take advantage of the modern methods, i.e., neural network and deep learning, in obesity studies.
1.
Introduction
Malaria is one of the world's most significant infectious diseases [1]. Malaria is a life-threatening disease caused by parasites that is usually transmitted to persons through the bites of female Anopheles mosquitoes [2]. Malaria gives rise to great pressure for the global prevention and control of infectious diseases [3]. World Health Organization reported [2] that there were an estimated 247 million malaria cases, including 619,000 deaths worldwide in 2021, and the majority of cases and deaths occurred in sub-Saharan Africa. The African region accounted for a disproportionate share of the global malaria burdens [4,5]. In 2021, the African region was home to 95% of global malaria cases and 96% of global malaria deaths, and children under 5 years old accounted for about 80% of all malaria deaths there [2]. There are 5 kinds of parasite species that cause malaria in humans, and two of these species P. falciparum and P. vivax pose the greatest threat [6]. The first malaria symptoms such as headache, fever and chills usually appear 10–15 days after the bite of a malaria mosquito and may be mild and difficult to be recognized as malaria, which implies that malaria exists the incubation period [2]. It was reported that asymptomatic infections were more prevalent in sub-Saharan Africa, where an estimated 24 million people had asymptomatic malaria infections [7]. Thus asymptomatic infections can occur during malaria transmission.
Asymptomatic infected people have no clinical symptoms, but they are contagious and the impact of asymptomatic infections on malaria transmission is enormous [8,9]. Bousema et al. [10] pointed out that asymptomatic carriers contributed to sustained transmission of malaria in local populations, and there was substantial evidence that an increase in the number of asymptomatic carriers at specific time intervals affected the dynamics of malaria transmission. Laishram et al. [4] concluded that asymptomatic malaria infections was a challenge for malaria control programs.
Since the emergence of malaria, scholars at home and abroad have been studying the pathogenesis and transmission dynamics of malaria. In all research methods, mathematical modeling is undoubtedly one of the most intuitive and effective methods. Many researchers have studied the dynamic evolution of malaria transmission by applying some mathematical models of malaria. In 1911, Ross [11] put forward a basic ordinary differential equations (ODEs) malaria model. Afterwards, MacDonald [12] extended Ross's model, and gave first the definition of the basic reproduction number. The extended Ross's model was said to be the Ross-Macdonald model. Subsequently, the Ross-Macdonald model has been extended to higher dimensions and more factors affecting malaria transmission have been taken into account (see, e.g., [1,6,13]). For example, Kingsolver [14] extended the Ross-Macdonald model and explained the greater attraction of infectious humans to mosquitoes in 1987. Safan and Ghazi [1] developed a 4D ODEs malaria transmission model with standard incidence rates, and analyzed the dynamic properties of equilibria of the malaria model. In 2020, Aguilar and Gutierrez [6] established a high-dimensional ODEs malaria model with asymptomatic carriers and standard incidence rate, and dealt with local dynamics of the disease-free equilibrium of the malaria model.
Over the years, considering the incubation period of malaria, lots of researchers established some time-delayed malaria models (see, e.g., [3,5,15,16,17,18]). For instance, in 2008, Ruan et al. [5] first established a class of Ross-Macdonald model with two time delays, and investigated the stability of equilibria of the model and the impact of time delays on the basic reproduction number. In 2019, Ding et al. [3] proposed a malaria model with time delay, and investigated the global stability of the uninfected equilibrium of the model as well as its uniform persistence. For the moment, there are few theoretical analysis of the model of malaria with standard incidence rate. Recently, Guo et al. [13] established a malaria transmission model with time delay and standard incidence rate, and they studied the global dynamic properties of equilibria of the model. Based on this, we extend and improve the model in [13], namely, we establish a malaria transmission model with asymptomatic infections, standard incidence rate and time delay, and then study the global dynamic properties of equilibria of the malaria model.
The remainder of this paper is organized as follows. In Section 2, we put forward a time-delayed dynamic model of malaria with asymptomatic infections and standard incidence rate, and prove the well-posedness as well as dissipativeness of the system. In Section 3, we obtain the existence conditions of malaria-free and malaria-infected equilibria of the system, and verify the local dynamic properties of equilibria in terms of the basic reproduction number R0. In Section 4, to obtain the global dynamic property of the malaria-infected equilibrium for R0>1, we acquire the weak persistence of the system through some analysis techniques. In Section 5, by utilizing the Lyapunov functional method and the limiting system of the model combining stability of partial variables, we obtain the global stability results of malaria-free and malaria-infected equilibria in terms of R0, respectively.
2.
Model formulation
In order to delve into the details of malaria transmission, we develop a time-delayed model with asymptomatic infections and standard incidence rate. The population is classified into four compartments, which are denoted by Sh: susceptible individuals, Ah: asymptomatic infected individuals, Ih: symptomatic infected individuals, Rh: recovered individuals, respectively. The mosquitoes are classified into two compartments, which are denoted by Sm: susceptible mosquitoes and Im: infected mosquitoes, respectively. Then the model of malaria transmission is proposed as follows:
where Nm(t)=Sm(t)+Im(t). Here, time delay τ≥0, and all other parameters of system (2.1) are assumed to be positive and p∈(0,1). The description of parameters are listed in Table 1.
Table 1.
Descriptions of parameters in the model.
Parameter
Description
μm
The natural death rate of mosquitoes
μh
The natural death rate of humans
λm
The natural birth rate of mosquitoes
λh
The natural birth rate of humans
β1
The infection rate of susceptible mosquitoes biting asymptomatic individuals
β2
The infection rate of susceptible mosquitoes biting symptomatic individuals
βh
The infection rate of infected mosquitoes biting susceptible individuals
τ
The incubation period of malaria
γa
The recovery rate of asymptomatic infected individuals
γi
The recovery rate of symptomatic infected individuals
p
The transition probability of asymptomatic infected individuals
where C is the Banach space of continuous functions mapping from [−τ,0] to R6+ with R+=[0,∞) and the supremum norm. In the following, the well-posedness as well as dissipativeness of system (2.1) will be investigated in C+.
Theorem 2.1.The solution u(t)=(Sm(t),Im(t),Sh(t),Ah(t),Ih(t),Rh(t))T of system (2.1) with any ϕ∈C+ exists uniquely, and is non-negative and ultimately bounded on R+. In particular, (Sm(t),Sh(t))T≫0 on (0,∞), and C+ is positively invariant for system (2.1).
Proof. In view of the basic theory of delay differential equations (DDEs) [19,20], the solution u(t) of system (2.1) with any ϕ∈C+ is unique on its maximum interval [0,Tϕ) of existence. Firstly, we will prove that the solution u(t) is non-negative on [0,Tϕ). According to the continuous dependence of solutions of DDEs on parameters [19,20], then for any b∈(0,Tϕ) and a sufficiently small ε>0, the solution u(t,ε)=(u1(t,ε),u2(t,ε),u3(t,ε),u4(t,ε),u5(t,ε),u6(t,ε))T through ϕ of the following model:
uniformly exists on [0,b]. Consequently, we claim u(t,ε)≫0 on [0,b). It is clear that ˙ui(0,ε)>0,i∈I6={1,2,3,4,5,6} whenever ui(0,ε)=0. Next, we prove the claim by contradiction. Suppose that there exists ˉt∈(0,b) such that ui(ˉt,ε)=0 for some i∈I6 and u(t,ε)≫0 for t∈(0,ˉt), where
it follows from (2.2) that ˙ui(ˉt,ε)>0, which yields a contradiction to (2.3). Thus, we have u(t,ε)≫0 for t∈(0,b).
Letting ε→0+ gives that u(t,0)=u(t)≥0 for any t∈[0,b). Note that b∈(0,Tϕ) is chosen arbitrarily, so that u(t)≥0 on [0,Tϕ). It is obvious that Tϕ>τ. Therefore, from system (2.1), we have that for any t≥τ,
As a consequence, by the comparison principle, we can obtain that u(t) is bounded. Accordingly, from the continuation theorem of solutions of DDEs [19], it follows Tϕ=∞. Consequently, we have
Therefore, the solution u(t) with any ϕ∈C+ uniquely exists, and is non-negative and ultimately bounded on R+. Moreover, it is not difficult to get that (Sm(t),Sh(t))T≫0 on R+∖{0}, and C+ is a positive invariant set for system (2.1).
3.
Local stability
To begin with, it follows easily the malaria-free equilibrium E0=(S0m,0,S0h,0,0,0)T, where S0m=λm/μm and S0h=λh/μh. By using the similar method in [21,22], we can calculate the basic reproduction number
R0=√pβhλhβ1μhλm(μh+γa)+(1−p)βhλhβ2μhλm(μh+γi).
To get a malaria-infected equilibrium (i.e., positive equilibrium) E∗=(S∗m,I∗m,S∗h,A∗h,I∗h,R∗h)T, we have the following lemma.
Lemma 3.1.System (2.1) exists a unique E∗≫0 when and only when R0>1.
Proof. First of all, the malaria-infected equilibrium equations can be obtained as follows:
Next, by adopting similar techniques in [23,24,25], we will discuss the local dynamic properties of the malaria-free equilibrium E0 and the the malaria-infected equilibrium E∗ with respect to R0. First of all, for the local stability of the equilibrium E0, we have the theorem as follows.
Theorem 3.1.For any τ≥0, the malaria-free equilibrium E0 is locally asymptotically stable (LAS) when R0<1, and unstable when R0>1.
Proof. With some calculations, the characteristic equation of the linear system of system (2.1) at E0 can be obtained as follows:
Clearly, Eq (3.5) possesses three negative real roots: −μh (double) and −μm. The other roots of Eq (3.5) satisty H(λ)=0. Next, we will prove that any root λ of H(λ)=0 has negative real part. Suppose, by contradiction, λ has the nonegative real part. Then it follows from H(λ)=0 that
for R0<1 and τ≥0, which leads to a contradiction. Therefore, the real part of each root of the Eq (3.5) is negative. Accordingly, E0 is LAS for R0<1 and τ≥0.
Now, we prove that the E0 is unstable for R0>1 and τ≥0 by the zero theorem. Clearly, for R0>1 and τ≥0, we can get
H(0)=μm(μh+γa)(μh+γi)(1−R20)<0,limλ→∞H(λ)=∞.
According to the zero theorem, there must exsit a positive real root in Eq (3.6). Thus, E0 is unstable for R0>1 and τ≥0.
For the local stability of the equilibrium E∗, we can obtain the theorem as follows.
Theorem 3.2.For any τ≥0, the malaria-infected equilibrium E∗ is LAS if and only if R0>1.
Proof. By Lemma 3.1, we just require to demonstrate the sufficiency. Let
Clearly, Eq (3.8) has two negative roots: −μh and −μm. The other roots of Eq (3.8) satisfy g(λ)=0. Then, we will prove that any root λ of g(λ)=0 has negative real part by contradiction. Assume that λ has the non-negative real part. By g(λ)=0, we can get
Obviously, this is a contradiction. Hence, the real part of each root of the Eq (3.8) is negative for R0>1 and τ≥0, which ensures the local stability of the equilibrium E∗.
4.
Weak persistence
Generally, to obtain the global stability of the equilibrium E∗, we need to prove the strong persistence or uniform persistence of system (2.1). However, we study the weak persistence of system (2.1), which can ensure the global stability of the equilibrium E∗. Of course, the weak persistence of system (2.1) is more accessible than its strong or uniform persistence. Now, we define
ϝ={ϕ∈C+:ϕ2(0)>0},
and let
u(t)=(Sm(t),Im(t),Sh(t),Ah(t),Ih(t),Rh(t))T
be the solution of system (2.1) with any ϕ∈ϝ. It follows easily that ϝ is a positive invariant set of system (2.1), and u(t)≫0 for t>0. Hence, we discuss the weak persistence of system (2.1) in ϝ.
According to [26], system (2.1) is said to be weakly persistent if
lim supt→∞ϱ(t)>0,ϱ=Sm,Im,Sh,Ah,Ih,Rh.
We define ut=(Smt,Imt,Sht,Aht,Iht,Rht)T∈C+ to be ut(θ)=u(t+θ), θ∈[−τ,0] for t≥0, and ut is the solution of system (2.1) with ϕ. Inspired by the work in [13], we study the weak persistence of system (2.1). First, we have the following lemma.
Lemma 4.1.Assume that R0>1, θ∈(0,1) and lim supt→∞Im(t)≤θI∗m. Then
Proof. We will use the proof by contradiction to verify this result. Provided that lim supt→∞Im(t)<I∗m. Whereupon, one can find a θ∈(0,1) such that lim supt→∞Im(t)≤θI∗m. Using Lemma 4.1, we can get that there is an ϵ0>0 such that for any ϵ∈(0,ϵ0),
ˉShS0m+ϵ>S∗hS0m,ˉSmS0m+ϵ>S∗mS0m.
(4.1)
Thanks to Lemma 4.1, it follows that for any ϵ∈(0,ϵ0), there can be found T≡T(ϵ,ϕ)>0 such that
Next, we will prove that Im(t)≥c for t≥T. If not, there exists a T0≥0 such that Im(t)≥c for t∈[T,T+τ+T0], Im(T+τ+T0)=c and ˙Im(T+τ+T0)≤0. Then it follows that for t∈[T,T+τ+T0],
Clearly, this contradicts ˙Im(T+τ+T0)≤0. As a result, Im(T)≥c for t≥T. Hence, for t≥τ,
˙L(ut)≥(λmR20ˉSmˉShS0h(S0m+ϵ)−λm)c>0,
which hints L(ut)→∞ as t→∞. Accordingly, this contradicts the boundedness of L(ut).
According to Theorem 4.1, we have the following result.
Corollary 4.1.If R0>1, then for any τ≥0, system (2.1) is weakly persistent.
5.
Global stability
We will study the global asymptotic stability of the equilibria E0 and E∗ with respect to R0. For this purpose, we get from (2.4) the following limiting system of system (2.1):
Adopting a similar argument as in the proof of Theorem 2.1, it follows that the solution
z(t)=(Sm(t),Im(t),Sh(t),Ah(t),Ih(t),Rh(t))T
of system (5.1) through any φ=(φ1,φ2,φ3,φ4,φ5,φ6)T∈C+ uniquely exists, and is non-negative and ultimately bounded on [0,∞). Setting
zt(θ)=z(t+θ),θ∈[−τ,0]
gives that zt=(Smt,Imt,Sht,Aht,Iht,Rht)T∈C+ is also the solution of system (5.1) through φ for t≥0. We can find easily that E∗ and E0 are also the equilibria of system (5.1), and C+ is a positive invariant set of system (5.1). By the way, (Sm(t),Sh(t))T≫0 for t>0. Define H(v)=v−1−lnv,v>0. Thereupon, for the global dynamic property of the equilibrium E0 of system (2.1), we have the theorem as follows.
Theorem 5.1.For any τ≥0, the malaria-free equilibrium E0 is GAS when R0<1 and GA when R0=1 in C+.
Proof. By Theorem 3.1, it follows that for R0<1, E0 is LAS. Thus, we only need to prove that for R0≤1, E0 is GA. Let ut be the solution of system (2.1) with any ϕ∈C+ and zt be the solution of system (5.1) though any φ∈C+. Let ω(ϕ) be the ω-limit set of ϕ with respect to system (2.1). In order to prove the global attractivity of E0, we just need to show that ω(ϕ)={E0}. By Theorem 2.1, we know that ut is bounded on C+. Hence, it follows from (2.4) that ω(ϕ) is a compact set, and is also a subset of C+.
Let us define the following functional V on L1={φ∈C+:φ1(0)>0,φ3(0)>0}⊆C+
where h=βhμm/λm. Considering (5.2) and (5.3), we can conclude that both Sh(t) and Sm(t) are persistent. In other words, there exists a σ=σ(φ)>0 such that lim inft→∞Sh(t)>σ and lim inft→∞Sm(t)>σ. As a result, ω(φ)⊆L1, where ω(φ) is the ω-limit set of φ with respect to system (5.1). It is evident that V is a Lyapunov functional on {zt:t≥1}⊆L1. Then it follows from [27,Corollry 2.1] that ˙V(ψ)=0, ∀ψ∈ω(φ).
Assume that zt is the solution of system (5.1) through any ψ∈ω(φ). Then the invariance of ω(φ) gives that zt∈ω(φ) for t∈R. According to (5.3), we have Sm(t)=S0m and Sh(t)=S0h for t∈R. From system (5.1) and the invariance of ω(φ), it follows that Im(t)=Ah(t)=Ih(t)=Rh(t)=0 for t∈R. Thus for R0≤1, it holds that ω(φ)={E0}, which implies that Ws(E0)=C+, where Ws(E0) is the stable set of E0 with respect to system (5.1).
Now, we prove that the equilibrium E0 of system (5.1) is uniformly stable for R0≤1 by using the similar approach in [28,29]. Observe that the first five equations of system (5.1) can constitute an independent subsystem
is a positive invariant set with respect to system (5.4). Clearly, system (5.4) has a malaria-free equilibrium X0=(S0m,0,S0h,0,0)T. According to (5.2), (5.3), [27,Corollary 3.3] hints that X0 is uniformly stable. Define
k:=min{μhγa,μhγi,1}.
By the definition of uniform stability of X0, it follows that for any ϵ>0, there is δ≤2ϵ/3 such that for any ξ∈C+ and ‖ξ−X0‖<δ, there holds
‖Xt−X0‖<ϵk3,∀t≥0,
where Xt is the solution of system (5.4) with ξ. Then considering the sixth equation of system (5.1), we can get
Thus, the equilibrium E0 is uniformly stable for system (5.1).
Next, we claim that ω(ϕ)={E0} for R0≤1. We first have E0∈ω(ϕ) since ω(ϕ)⊆C+=Ws(E0). Assume that there exists ψ∈ω(ϕ) such that ψ≠E0. Let α(ψ) be the α-limit set of ψ for system (5.1). Then it follows from the invariance and the compactness of ω(ϕ) that α(ψ)⊆ω(ϕ). The invariance of α(ψ) and the stable set C+ of E0 yield that E0∈α(ψ). Obviously, this contradicts to the stability of E0 for system (5.1). Therefore, ω(ϕ)={E0}.
Remark 5.1.In fact, the stability of the malaria-free equilibrium E0 of system (5.1) can be acquired for R0=1 in the proof of Theorem 5.1. But using the proof of Theorem 3.1, we can not obtain the stability of the equilibrium E0 for R0=1.
For the global dynamic property of the equilibrium E∗ of system (2.1), we can draw the following theorem.
Theorem 5.2.For any τ≥0, the malaria-infected equilibrium E∗ is globally asymptotically stable if and only if R0>1 in ϝ.
Proof. From Lemma 3.1 and Theorem 3.2, we just require to prove that E∗ is GA for R0>1. Let ut be the solution of system (2.1) with any ϕ∈ϝ and zt be the solution of system (5.1) through any φ∈ϝ. We can obtain that ϝ is positively invariant for system (5.1), and z(t)≫0 for t≥0. In order to show that E∗ is GA, we only need to show that ω(ϕ)={E∗}. It follows from Theorem 2.1 that ut is bounded on ϝ. Thus, it holds that ω(ϕ) is compact.
Let us define a functional V on L2={φ∈C+:φi(0)>0,i=1,2,3,4,5}⊆ϝ as follows
By (5.6) and (5.7), it follows that ω(φ)⊆L2. It is clear that V is a Lyapunov functional on {zt:t≥τ+1}⊆L2. As a consequence, [27,Corollary2.1] implies that ˙V(ψ)=0 for any ψ∈ω(φ).
Let zt be the solution of system (5.1) for any ψ∈ω(φ). Then the invariance of ω(φ) indicates that zt∈ω(φ) for any t∈R. Thus, from (5.7), it follows that for any t∈R,
By (5.8) and the third equation of system (5.1), we get that for any t∈R,Im(t)=I∗m,Ah(t)=A∗h and Ih(t)=I∗h. Consequently, by the invariance of ω(φ) and system (5.1), it holds that Rh(t)=R∗h for any t∈R. Therefore, it follows that z0=ψ=E∗, and then ω(φ)={E∗}, which implies Ws(E∗)=ϝ, where Ws(E∗) is the stable set of E∗ with respect to system (5.1).
Now, we prove that the equilibrium E∗ of system (5.1) is uniformly stable by using the similar argument in [28,29]. Note that system (5.4) has a unique malaria-infected equilibrium X∗=(S∗m,I∗m,S∗h,A∗h,I∗h). It follows from (5.6), (5.7) and [27,Corollary 3.3] that X∗ is uniformly stable. By the definition of uniform stability of X∗, it follows that for any ϵ>0, there is δ≤2ϵ/3 such that for any ξ∈C+ and ‖ξ−X∗‖<δ, there holds
‖Xt−X∗‖<ϵk3,∀t≥0,
where Xt is the solution of system (5.4) through ξ. Hence, for any ϕ∈L2 and ‖ϕ−E∗‖<δ, it follow from (5.5) that
which gives that the equilibrium E∗ is uniformly stable with respect to system (5.1).
Next, we claim that ω(ϕ)={E∗}. From Theorem 4.1, it follows that ω(ϕ)∩ϝ≠∅, which gives that E∗∈ω(ϕ). Assume that there is ψ∈ω(ϕ) such that ψ≠E∗. Then the invariance and the compactness of ω(ϕ) implies that α(ψ)⊆ω(ϕ). By Theorem 3.2, we have that E∗∈α(ψ). Obviously, this contradicts to the stability of E∗ with respect to system (5.1). Therefore, ω(ϕ)={E∗}.
Remark 5.2.Indeed, the proof of Theorem 5.2 can be simplified, i.e., the stability of system (5.1) is not a must, because E∗∈ω(ϕ) and Theorem 3.2 can imply that ω(ϕ)={E∗}. But if we use [30,Theorem 4.1] to prove that ω(ϕ)={E∗}, then the stability of system (5.1) is required.
Acknowledgements
This work is partially supported by the National NSF of China (No. 11901027), the Major Program of the National NSF of China (No. 12090014), the State Key Program of the National NSF of China (No. 12031020), the China Postdoctoral Science Foundation (No. 2021M703426), the Pyramid Talent Training Project of BUCEA (No. JDYC20200327), and the BUCEA Post Graduate Innovation Project (No. PG2022143).
Conflict of interest
The authors declare there is no conflict of interest.
A. Hruby, J. E. Manson, L. Qi, V. S. Malik, E. B. Rimm, Q. Sun, W. C. Willett, F. B. Hu, Determinants and consequences of obesity, Am. J. Public Health, 106 (2016), 1656-1662. https://doi.org/https://doi.org/10.2105/AJPH.2016.303326 doi: 10.2105/AJPH.2016.303326
J. Cawley, C. Meyerhoefer, The medical care costs of obesity: An instrumental variables approach, J. Health Econ., 31 (2012), 219-230. https://doi.org/10.1016/j.jhealeco.2011.10.003 doi: 10.1016/j.jhealeco.2011.10.003
[6]
L. Angrisani, A. Santonicola, P. Iovino, G. Formisani, H. Buchwald, N. Scopinaro, Bariatric Surgery Worldwide 2013, Obes. Surg., 25 (2015), 1822-1832. https://doi.org/10.1007/s11695-015-1657-z doi: 10.1007/s11695-015-1657-z
[7]
T. Bhurosy, R. Jeewon, Overweight and obesity epidemic in developing countries: A problem with diet, physical activity, or socioeconomic status? Scientific World Journal, 2014 (2014). https://doi.org/10.1155/2014/964236
[8]
E. Alpaydin, Introduction to Machine Learning, Cambridge: MIT press, 2014.
[9]
N. S. Rajliwall, R. Davey, G. Chetty, Machine learning based models for cardiovascular risk prediction, International Conference on Machine Learning and Data Engineering 2018, (iCMLDE), (2018), 142-148. https://doi.org/10.1109/iCMLDE.2018.00034
[10]
J. B. Heaton, N. G. Polson, J. H. Witte, Deep learning for finance: deep portfolios, Appl. Stoch. Model. Bus., 33 (2017), 3-12. https://doi.org/10.1002/asmb.2209 doi: 10.1002/asmb.2209
[11]
J. Kim, J. Canny, Interpretable Learning for Self-Driving Cars by Visualizing Causal Attention, Proceedings of the IEEE International Conference on Computer Vision, (2017), 2942-2950. https://doi.org/10.1109/ICCV.2017.320
[12]
D. Gruson, T. Helleputte, P. Rousseau, D. Gruson, Data science, artificial intelligence, and machine learning: Opportunities for laboratory medicine and the value of positive regulation, Clin. Biochem., 69 (2019), 1-7. https://doi.org/10.1016/j.clinbiochem.2019.04.013 doi: 10.1016/j.clinbiochem.2019.04.013
[13]
D. Panaretos, E. Koloverou, A. C. Dimopoulos, G. M. Kouli, M. Vamvakari, G. Tzavelas, C. Pitsavos, D. B. Panagiotakos, A comparison of statistical and machine-learning techniques in evaluating the association between dietary patterns and 10-year cardiometabolic risk (2002-2012): The ATTICA study, Brit. J. Nutr., 120 (2018), 326-334. https://doi.org/10.1017/S0007114518001150 doi: 10.1017/S0007114518001150
[14]
H. C. Koh, G. Tan, Data Mining Applications in Healthcare, Journal of Healthcare Information Management, 19 (2011), 64-72.
[15]
K. Kourou, T. P. Exarchos, K. P. Exarchos, M. V. Karamouzis, D. I. Fotiadis, Machine learning applications in cancer prognosis and prediction, Comput. Struct. Biotec., 13 (2015), 8-17. https://doi.org/10.1016/j.csbj.2014.11.005 doi: 10.1016/j.csbj.2014.11.005
[16]
V. Gulshan, L. Peng, M. Coram, M. C. Stumpe, D. Wu, A. Narayanaswamy, S. Venugopalan, K. Widner, et al., Development and validation of a deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs, JAMA - Journal of the American Medical Association, 316 (2016), 2402-2410. https://doi.org/10.1001/jama.2016.17216 doi: 10.1001/jama.2016.17216
[17]
Y. Xing, J. Wang, Z. Zhao, Combination data mining methods with new medical data to predicting outcome of Coronary Heart Disease, International Conference on Convergence Information Technology, (ICCIT) 2007, (2007), 868-872. https://doi.org/10.1109/ICCIT.2007.4420369
[18]
P. Fränti, S. Sieranoja, K. Wikströ m, T. Laatikainen, Clustering diagnoses from 58M patient visits in Finland during 2015-2018, JMIR Medical Informatics, (2022). https://doi.org/10.2196/35422
[19]
Z. Obermeyer, E. J. Emanuel, Predicting the Future: Big Data, Machine Learning, and Clinical Medicine, The New England journal of medicine, 375 (2016), 1216-1219. https://doi.org/doi:10.1056/NEJMp1606181 doi: 10.1056/NEJMp1606181
[20]
M. A. Morris, E. Wilkins, K. A. Timmins, M. Bryant, M. Birkin, C. Griffiths, Can big data solve a big problem? Reporting the obesity data landscape in line with the Foresight obesity system map, Int. J. Obesity, 42 (2018), 1963-1976. https://doi.org/10.1038/s41366-018-0184-0 doi: 10.1038/s41366-018-0184-0
[21]
C. Y. J. Peng, K. L. Lee, G. M. Ingersoll, An introduction to logistic regression analysis and reporting, J. Educ. Res., 96 (2002), 3-14. https://doi.org/10.1080/00220670209598786 doi: 10.1080/00220670209598786
[22]
D. Dietrich, B. Heller, Y. Beibei, Data Science and Big Data Analytics: Discovering, Analyzing, Visualizing and Presenting Data, Indianapolis: Wiley, 2015.
[23]
H. O. Alanazi, A. H. Abdullah, K. N. Qureshi, A Critical Review for Developing Accurate and Dynamic Predictive Models Using Machine Learning Methods in Medicine and Health Care, J. Med. Syst., 41 (2017), 1-10. https://doi.org/10.1007/s10916-017-0715-6 doi: 10.1007/s10916-017-0715-6
[24]
Y. Y. Song, L. U. Ying, Decision tree methods: applications for classification and prediction, Shanghai Archives of Psychiatry, 27 (2015), 130-135. https://doi.org/10.11919/j.issn.1002-0829.215044 doi: 10.11919/j.issn.1002-0829.215044
[25]
M. Pal, Random forest classifier for remote sensing classification, Int. J. Remote Sens., 26 (2005), 217-222. https://doi.org/10.1080/01431160412331269698 doi: 10.1080/01431160412331269698
[26]
S. V. Vishwanathan, M. N. Murty, SSVM: A simple SVM algorithm, International Joint Conference on Neural Networks (IJCNN) 2002, 3 (2002), 2393-2398. https://doi.org/10.1109/IJCNN.2002.1007516
[27]
Y. Qu, B. Fang, W. Zhang, R. Tang, M. Niu, H. Guo, Y. Yu, X. He, Product-Based Neural Networks for User Response Prediction over Multi-Field Categorical Data, ACM T. Inform. Syst., 37 (2019), 1-35. https://doi.org/10.1145/3233770 doi: 10.1145/3233770
[28]
T. Chen, C. Guestrin, XGBoost: A scalable tree boosting system, Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (2016), 785-794. https://doi.org/10.1145/2939672.2939785
[29]
A. T. C. Goh, Back-propagation neural networks for modeling complex systems, Artificial Intelligence in Engineering, 9 (1995), 143-151. https://doi.org/10.1016/0954-1810(94)00011-S doi: 10.1016/0954-1810(94)00011-S
[30]
Y. Lecun, Y. Bengio, G. Hinton, Deep learning, Nature, 521 (2015), 436-444. https://doi.org/10.1038/nature14539 doi: 10.1038/nature14539
[31]
A. K. Jain, M. N. Murty, P. J. Flynn, Data clustering: A review, ACM Comput. Surv., 31 (1999), 264-323. https://doi.org/10.1145/331499.331504 doi: 10.1145/331499.331504
[32]
H. Arksey, L. O'Malley, Scoping studies: towards a methodological framework, Int. J. Soc. Res. Method., 8 (2005), 19-32. https://doi.org/10.1080/1364557032000119616 doi: 10.1080/1364557032000119616
[33]
H. So, L. McLaren, G. C. Currie, The relationship between health eating and overweight/obesity in Canada: cross-sectional study using the CCHS, Obesity Science and Practice, 3 (2017), 399-406. https://doi.org/10.1002/osp4.123 doi: 10.1002/osp4.123
[34]
N. Daud, N. L. Mohd Noor, S. A. Aljunid, N. Noordin, N. I. M. F. Teng, Predictive Analytics: The Application of J48 Algorithm on Grocery Data to Predict Obesity, 2018 IEEE Conference on Big Data and Analytics, ICBDA, (2018), 1-6. https://doi.org/10.1109/ICBDAA.2018.8629623
[35]
J. F. Easton, H. Román Sicilia, C. R. Stephens, Classification of diagnostic subcategories for obesity and diabetes based on eating patterns, Nutr. Diet., 76 (2019), 104-109. https://doi.org/10.1111/1747-0080.12495 doi: 10.1111/1747-0080.12495
[36]
J. Dunstan, M. Aguirre, M. Bastías, C. Nau, T. A. Glass, F. Tobar, Predicting nationwide obesity from food sales using machine learning, Health Inform. J., 26 (2019), 652-663. https://doi.org/10.1177/1460458219845959 doi: 10.1177/1460458219845959
[37]
N. Kanerva, J. Kontto, M. Erkkola, J. Nevalainen, S. Mannisto, Suitability of random forest analysis for epidemiological research: Exploring sociodemographic and lifestyle-related risk factors of overweight in a cross-sectional design, Scand. J. Public Health, 46 (2018), 557-564. https://doi.org/10.1177/1403494817736944 doi: 10.1177/1403494817736944
[38]
K. W. DeGregory, P. Kuiper, T. DeSilvio, J. D. Pleuss, R. Miller, J. W. Roginski, C. B. Fisher, D. Harness, et al., A review of machine learning in obesity, Obes. Rev., 19 (2018), 668-685. https://doi.org/10.1111/obr.12667 doi: 10.1111/obr.12667
[39]
D. Kim, W. Hou, F. Wang, C. Arcan, Factors Affecting Obesity and Waist Circumference Among US Adults, Prev. Chronic Dis., 16 (2019). https://doi.org/10.5888/pcd16.180220
[40]
R. L. Figueroa, C. A. Flores, Extracting Information from Electronic Medical Records to Identify the Obesity Status of a Patient Based on Comorbidities and Bodyweight Measures, J. Med. Syst., 40 (2016). https://doi.org/10.1007/s10916-016-0548-8 doi: 10.1007/s10916-016-0548-8
[41]
M. A. Green, M. Strong, F. Razak, S. V. Subramanian, C. Relton, P. Bissell, Who are the obese? A cluster analysis exploring subgroups of the obese, J. Public Health (UK), 38 (2016), 258-264. https://doi.org/10.1093/pubmed/fdv040 doi: 10.1093/pubmed/fdv040
[42]
P. P. Brzan, Z. Obradovic, G. Stiglic, Contribution of temporal data to predictive performance in 30-day readmission of morbidly obese patients, PeerJ, 5 (2017), e3230. https://doi.org/10.7717/peerj.3230 doi: 10.7717/peerj.3230
[43]
A. Kupusinac, E. Stokić, R. Doroslovački, Predicting body fat percentage based on gender, age and BMI by using artificial neural networks, Comput. Meth. Prog. Bio., 113 (2014), 610-619. https://doi.org/10.1016/j.cmpb.2013.10.013 doi: 10.1016/j.cmpb.2013.10.013
[44]
M. Batterham, L. Tapsell, K. Charlton, J. O'shea, R. Thorne, Using data mining to predict success in a weight loss trial, J. Hum. Nutr. Diet., 30 (2017), 471-478. https://doi.org/10.1111/jhn.12448 doi: 10.1111/jhn.12448
[45]
Z. Feng, L. Mo, M. Li, A Random Forest-based ensemble method for activity recognition, 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2015 EMBS, (2015), 5074-5077. https://doi.org/10.1109/EMBC.2015.7319532
[46]
M. Batterham, E. Neale, A. Martin, L. Tapsell, Data mining: Potential applications in research on nutrition and health, Nutr. Diet., 74 (2017), 3-10. https://doi.org/10.1111/1747-0080.12337 doi: 10.1111/1747-0080.12337
[47]
W. J. Heerman, N. Jackson, M. Hargreaves, S. A. Mulvaney, D. Schlundt, K. A. Wallston, R. L. Rothman, Clusters of Healthy and Unhealthy Eating Behaviors Are Associated With Body Mass Index Among Adults, J. Nutr. Educ. Behav., 49 (2017), 415-421. https://doi.org/10.1016/j.jneb.2017.02.001 doi: 10.1016/j.jneb.2017.02.001
[48]
I. Sarasfis, C. Diou, I. Ioakimidis, A. Delopoulos, Assessment of In-Meal Eating Behaviour using Fuzzy SVM, 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), (2019), 6939-6942. https://doi.org/10.1109/EMBC.2019.8857606
[49]
P. Pouladzadeh, S. Shirmohammadi, A. Bakirov, A. Bulut, A. Yassine, Cloud-based SVM for food categorization, Multimed.Tools Appl., 74 (2015), 5243-5260. https://doi.org/10.1007/s11042-014-2116-x doi: 10.1007/s11042-014-2116-x
[50]
E. J. Heravi, H. Habibi Aghdam, D. Puig, A deep convolutional neural network for recognizing foods, Eighth International Conference on Machine Vision (ICMV), 9875 (2015), 98751D. https://doi.org/10.1117/12.2228875 doi: 10.1117/12.2228875
[51]
E. Disse, S. Ledoux, C. Bétry, C. Caussy, C. Maitrepierre, M. Coupaye, M. Laville, C. Simon, An artificial neural network to predict resting energy expenditure in obesity, Clin. Nutr., 37 (2018), 1661-1669. https://doi.org/10.1016/j.clnu.2017.07.017 doi: 10.1016/j.clnu.2017.07.017
[52]
N. Cesare, P. Dwivedi, Q. C. Nguyen, E. O. Nsoesie, Use of social media, search queries, and demographic data to assess obesity prevalence in the United States, Palgrave Communications, 5 (2019), 1-9. https://doi.org/10.1057/s41599-019-0314-x doi: 10.1057/s41599-019-0314-x
[53]
P. Kuhad, A. Yassine, S. Shimohammadi, Using distance estimation and deep learning to simplify calibration in food calorie measurement, IEEE International Conference on Computational Intelligence and Virtual Environments for Measurement Systems and Applications, CIVEMSA, (2015), 1-6. https://doi.org/10.1109/CIVEMSA.2015.7158594
[54]
K. Shameer, K. W. Johnson, B. S. Glicksberg, J. T. Dudley, P. P. Sengupta, Machine learning in cardiovascular medicine: Are we there yet? Heart, 104 (2018), 1156-1164. https://doi.org/10.1136/heartjnl-2017-311198 doi: 10.1136/heartjnl-2017-311198
[55]
B. A. Goldstein, A. M. Navar, R. E. Carter, Moving beyond regression techniques in cardiovascular risk prediction: Applying machine learning to address analytic challenges, Eur. Heart J., 38 (2017), 1805-1814. https://doi.org/10.1093/eurheartj/ehw302 doi: 10.1093/eurheartj/ehw302
[56]
N. Jothi, N. A. A. Rashid, W. Husain, Data Mining in Healthcare - A Review, Procedia Computer Science, 72 (2015), 306-313. https://doi.org/10.1016/j.procs.2015.12.145 doi: 10.1016/j.procs.2015.12.145
[57]
A. L. Beam, I. S. Kohane, Big data and machine learning in health care, JAMA - Journal of the American Medical Association, 319 (2018), 1317-1318. https://doi.org/10.1001/jama.2017.18391 doi: 10.1001/jama.2017.18391
[58]
A. Mozumdar, G. Liguori, Corrective Equations to Self-Reported Height and Weight for Obesity Estimates among U.S. Adults: NHANES 1999-2008, Res. Q. Exercise Sport, 87 (2016), 47-58. https://doi.org/10.1080/02701367.2015.1124971 doi: 10.1080/02701367.2015.1124971
[59]
M. Stommel, C. A. Schoenborn, Accuracy and usefulness of BMI measures based on self-reported weight and height: Findings from the NHANES & NHIS 2001-2006, BMC Public Health, 9 (2009), 1-10. https://doi.org/10.1186/1471-2458-9-421 doi: 10.1186/1471-2458-9-421
[60]
D. Rativa, B. J. T. Fernandes, A. Roque, Height and Weight Estimation from Anthropometric Measurements Using Machine Learning Regressions, IEEE J. Transl. Eng. He., 6 (2018), 1-9. https://doi.org/10.1109/JTEHM.2018.2797983 doi: 10.1109/JTEHM.2018.2797983
[61]
J. A. Sáez, J. Luengo, F. Herrera, Predicting noise filtering efficacy with data complexity measures for nearest neighbor classification, Pattern Recogn., 46 (2013), 355-364. https://doi.org/10.1016/j.patcog.2012.07.009 doi: 10.1016/j.patcog.2012.07.009
[62]
T. Ferenci, L. Kovács, Predicting body fat percentage from anthropometric and laboratory measurements using artificial neural networks, Applied Soft Computing Journal, 67 (2018), 834-839. https://doi.org/10.1016/j.asoc.2017.05.063 doi: 10.1016/j.asoc.2017.05.063
[63]
S. P. Goldstein, F. Zhang, J. G. Thomas, M. L. Butryn, J. D. Herbert, E. M. Forman, Application of Machine Learning to Predict Dietary Lapses During Weight Loss, Journal of Diabetes Science and Technology, 12 (2018), 1045-1052. https://doi.org/10.1177/1932296818775757 doi: 10.1177/1932296818775757
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