Research article Special Issues

Altered Levels of Aroma and Volatiles by Metabolic Engineering of Shikimate Pathway Genes in Tomato Fruits

  • The tomato (Solanum lycopersicum) fruit is an excellent source of antioxidants, dietary fibers, minerals and vitamins and therefore has been referred to as a “functional food”. Ripe tomato fruits produce a large number of specialized metabolites including volatile organic compounds. These volatiles serve as key components of the tomato fruit flavor, participate in plant pathogen and herbivore defense, and are used to attract seed dispersers. A major class of specialized metabolites is derived from the shikimate pathway followed by aromatic amino acid biosynthesis of phenylalanine, tyrosine and tryptophan. We attempted to modify tomato fruit flavor by overexpressing key regulatory genes in the shikimate pathway. Bacterial genes encoding feedback-insensitive variants of 3-Deoxy-D-Arabino-Heptulosonate 7-Phosphate Synthase (DAHPS; AroG209-9) and bi-functional Chorismate Mutase/Prephenate Dehydratase (CM/PDT; PheA12) were expressed under the control of a fruit-specific promoter. We crossed these transgenes to generate tomato plants expressing both the AroG209 and PheA12 genes. Overexpression of the AroG209-9 gene had a dramatic effect on the overall metabolic profile of the fruit, including enhanced levels of multiple volatile and non-volatile metabolites. In contrast, the PheA12 overexpression line exhibited minor metabolic effects compared to the wild type fruit. Co-expression of both the AroG209-9 and PheA12 genes in tomato resulted overall in a similar metabolic effect to that of expressing only the AroG209-9 gene. However, the aroma ranking attributes of the tomato fruits from PheA12//AroG209-9 were unique and different from those of the lines expressing a single gene, suggesting a contribution of the PheA12 gene to the overall metabolic profile. We suggest that expression of bacterial genes encoding feedback-insensitive enzymes of the shikimate pathway in tomato fruits provides a useful metabolic engineering tool for the modification of fruits aroma and the generation of new combinations of tomato flavors.

    Citation: Vered Tzin, Ilana Rogachev, Sagit Meir, Michal Moyal Ben Zvi, Tania Masci, Alexander Vainstein, Asaph Aharoni, Gad Galili. Altered Levels of Aroma and Volatiles by Metabolic Engineering of Shikimate Pathway Genes in Tomato Fruits[J]. AIMS Bioengineering, 2015, 2(2): 75-92. doi: 10.3934/bioeng.2015.2.75

    Related Papers:

    [1] Mahmoud El-Morshedy, Zubair Ahmad, Elsayed tag-Eldin, Zahra Almaspoor, Mohamed S. Eliwa, Zahoor Iqbal . A new statistical approach for modeling the bladder cancer and leukemia patients data sets: Case studies in the medical sector. Mathematical Biosciences and Engineering, 2022, 19(10): 10474-10492. doi: 10.3934/mbe.2022490
    [2] Rashad A. R. Bantan, Zubair Ahmad, Faridoon Khan, Mohammed Elgarhy, Zahra Almaspoor, G. G. Hamedani, Mahmoud El-Morshedy, Ahmed M. Gemeay . Predictive modeling of the COVID-19 data using a new version of the flexible Weibull model and machine learning techniques. Mathematical Biosciences and Engineering, 2023, 20(2): 2847-2873. doi: 10.3934/mbe.2023134
    [3] Bochuan Du, Pu Tian . Factorization in molecular modeling and belief propagation algorithms. Mathematical Biosciences and Engineering, 2023, 20(12): 21147-21162. doi: 10.3934/mbe.2023935
    [4] Saleh I. Alzahrani, Wael M. S. Yafooz, Ibrahim A. Aljamaan, Ali Alwaleedi, Mohammed Al-Hariri, Gameel Saleh . AI-driven health analysis for emerging respiratory diseases: A case study of Yemen patients using COVID-19 data. Mathematical Biosciences and Engineering, 2025, 22(3): 554-584. doi: 10.3934/mbe.2025021
    [5] Yufeng Qian . Exploration of machine algorithms based on deep learning model and feature extraction. Mathematical Biosciences and Engineering, 2021, 18(6): 7602-7618. doi: 10.3934/mbe.2021376
    [6] M. E. Bakr, Abdulhakim A. Al-Babtain, Zafar Mahmood, R. A. Aldallal, Saima Khan Khosa, M. M. Abd El-Raouf, Eslam Hussam, Ahmed M. Gemeay . Statistical modelling for a new family of generalized distributions with real data applications. Mathematical Biosciences and Engineering, 2022, 19(9): 8705-8740. doi: 10.3934/mbe.2022404
    [7] Amal S. Hassan, Najwan Alsadat, Christophe Chesneau, Ahmed W. Shawki . A novel weighted family of probability distributions with applications to world natural gas, oil, and gold reserves. Mathematical Biosciences and Engineering, 2023, 20(11): 19871-19911. doi: 10.3934/mbe.2023880
    [8] Gang Chen, Binjie Hou, Tiangang Lei . A new Monte Carlo sampling method based on Gaussian Mixture Model for imbalanced data classification. Mathematical Biosciences and Engineering, 2023, 20(10): 17866-17885. doi: 10.3934/mbe.2023794
    [9] Binjie Hou, Gang Chen . A new imbalanced data oversampling method based on Bootstrap method and Wasserstein Generative Adversarial Network. Mathematical Biosciences and Engineering, 2024, 21(3): 4309-4327. doi: 10.3934/mbe.2024190
    [10] Keyue Yan, Tengyue Li, João Alexandre Lobo Marques, Juntao Gao, Simon James Fong . A review on multimodal machine learning in medical diagnostics. Mathematical Biosciences and Engineering, 2023, 20(5): 8708-8726. doi: 10.3934/mbe.2023382
  • The tomato (Solanum lycopersicum) fruit is an excellent source of antioxidants, dietary fibers, minerals and vitamins and therefore has been referred to as a “functional food”. Ripe tomato fruits produce a large number of specialized metabolites including volatile organic compounds. These volatiles serve as key components of the tomato fruit flavor, participate in plant pathogen and herbivore defense, and are used to attract seed dispersers. A major class of specialized metabolites is derived from the shikimate pathway followed by aromatic amino acid biosynthesis of phenylalanine, tyrosine and tryptophan. We attempted to modify tomato fruit flavor by overexpressing key regulatory genes in the shikimate pathway. Bacterial genes encoding feedback-insensitive variants of 3-Deoxy-D-Arabino-Heptulosonate 7-Phosphate Synthase (DAHPS; AroG209-9) and bi-functional Chorismate Mutase/Prephenate Dehydratase (CM/PDT; PheA12) were expressed under the control of a fruit-specific promoter. We crossed these transgenes to generate tomato plants expressing both the AroG209 and PheA12 genes. Overexpression of the AroG209-9 gene had a dramatic effect on the overall metabolic profile of the fruit, including enhanced levels of multiple volatile and non-volatile metabolites. In contrast, the PheA12 overexpression line exhibited minor metabolic effects compared to the wild type fruit. Co-expression of both the AroG209-9 and PheA12 genes in tomato resulted overall in a similar metabolic effect to that of expressing only the AroG209-9 gene. However, the aroma ranking attributes of the tomato fruits from PheA12//AroG209-9 were unique and different from those of the lines expressing a single gene, suggesting a contribution of the PheA12 gene to the overall metabolic profile. We suggest that expression of bacterial genes encoding feedback-insensitive enzymes of the shikimate pathway in tomato fruits provides a useful metabolic engineering tool for the modification of fruits aroma and the generation of new combinations of tomato flavors.


    The first COVID-19 infected case was identified in late 2019 in China and then spread around the globe with extraordinary speed. By March 11, 2021, the COVID-19 confirmed cases were registered in 213 countries, and the World Health Organization (WHO) declared this disease a global pandemic; see Ngo et al. [1]. Due to this pandemic, every aspect of life has been disturbed and almost every region around the globe has faced unexpected situations. In every region, among other sectors affected by this pandemic, the health sector is the most affected area; see Pfefferbaum and North [2], Kim et al. [3], Campion et al. [4], Gloster et al. [5], Talevi et al. [6], and Wastnedge et al. [7]. As of December 13, 2021, 09:09 GMT (Greenwich Mean Time), totally confirmed cases (TCC) have touched a figure of 270488249, the total number of deaths (TND) has reached 5324113, and 243235043 infected persons have been recovered. For the latest updates and details about the COVID-19 events; see https://www.worldometers.info/coronavirus/.

    The top fifteen countries with the higher TND, include (ⅰ) America with 817956 deaths, (ⅱ) Brazil with 616941 deaths, (ⅲ) India with 475636 deaths, (ⅳ) Mexico with 296672 deaths, (ⅴ) Russia with 290604 deaths, (ⅵ) Peru with 201,650 deaths, (ⅶ) the United Kingdom with 146439 deaths, (ⅷ) Indonesia with 143936 deaths, (ⅸ) Italy with 134831 deaths, (ⅹ) Iran with 130722 deaths, (${\rm{xi}}$) Colombia with 129107 deaths, (${\rm{xii}}$) France with 120431 deaths, (${\rm{xiii}}$) Argentina with 116771 deaths, (${\rm{ivx}}$) Germany with 106331 deaths, and (${\rm{xv}}$) Ukraine with 91215 deaths. For a brief overview of country-wise statistics related to the COVID-19 pandemic, we refer to Bo et al. [8].

    Due to the unprecedented situation of the COVID-19 pandemic, it is necessary to have the best description and efficient modeling of the COVID-19 events. Statistical methodologies are very useful in modeling and predicting lifetime events. Several statistical studies on this pandemic have appeared. For example, Moreau [9] predicted the COVID-19 phenomena in Brazil. Tuli et al. [10] predicted the growth trend of the COVID-19 pandemic. Rahman et al. [11] implemented the Weibull model for the COVID-19 data analysis. Almetwally [12] introduced a new inverted Topp-Leone (NITL) distribution and used it for modeling the COVID-19 mortality rate data.

    These statistical studies are carried out either by implementing the existing models or by proposing new methodologies to update/modify the existing distributions. In the recent era of DT (distribution theory), the development of new methods to introduce new distributions is an important research topic. In this regard, numerous methods to update the distributional flexibility of the existing model have been introduced; see Alizadeh et al. [13] Chipepa et al. [14], Handique et al. [15], Tahir et al. [16], Zaidi et al. [17], Riad et al. [18], and Bakr et al. [19]. For more information about the applicability of statistical models in applied sectors, we refer to Xu et al. [20], and Luo et al. [21].

    In this paper, we further contribute to the literature on DT by proposing a new approach, namely, a new extended exponential (NEExp) family. It can be used to obtain the updated versions of the classical/traditional (such as Weibull, beta, gamma, Gumble, Rayleigh, etc.) or other existing models. The NEExp family is proposed by incorporating the T-X distributions approach of Alzaatreh et al. [22] with the exponential distribution with probability density function (PDF) $ e^{-t}, $ taken as a parent model.

    The novelty and key motivations of the proposed method are the followings:

    ● The method introduced in this paper is new and has not been studied in the literature.

    ● The proposed method is a very simple and convenient approach of adding an extra parameter to obtain the updated versions of the existing models.

    ● The proposed approach helps to improve the flexibility and characteristics of the existing models.

    ● The proposed method provides a close fit to healthcare and other related data sets.

    The reaming work carried out in this paper is organized as follows. In Section 2, we define the proposed family and discuss its special case. Certain mathematical properties of the NEExp family are provided in Section 3. The estimation of the parameters and a simulation study are provided in Section 4. To illustrate the NEExp family, two practical data sets are analyzed in Section 5. To forecast the COVID-19 data sets, the machine learning methods are discussed in Section 6. Finally, some concluding remarks, limitations of the proposed method, and future study plans are discussed in Section 7.

    This section is divided into two subsections. In the very first subsection, we define the proposed NEExp family of distributions. Whereas, the second subsection is devoted to studying a special case of the NEExp family of distributions

    Let V has a NEExp family, if its distribution function (DF) $ F\left(v; \delta, \pmb{\vartheta}\right) $ is given by

    $ F(v;δ,ϑϑ)=1δ[1M(v;ϑϑ)]δ+M(v;ϑϑ),vR,
    $
    (2.1)

    where $ \delta > 0 $ is an additional parameter and $ \pmb{\vartheta} $ is a vector of parameters associated with the baseline DF $ M\left(v; \pmb{\vartheta} \right). $

    Furthermore, in link to $ F\left(v; \delta, \pmb{\vartheta}\right) $, the PDF $ f\left(v; \delta, \pmb{\vartheta}\right) $ of the NEExp family is

    $ f(v;δ,ϑϑ)=δ(δ+1)m(v;ϑϑ)[δ+M(v;ϑϑ)]2,vR.
    $
    (2.2)

    For $ v \in \mathbb{R}, $ the survival function (SF) $ S\left(v; \delta, \pmb{\vartheta}\right) = 1-F\left(v; \delta, \pmb{\vartheta}\right) $, and hazard function (HF) $ h\left(v; \delta, \pmb{\vartheta}\right) = \frac{f\left(v; \delta, \pmb{\vartheta}\right)}{1-F\left(v; \delta, \pmb{\vartheta}\right)} $ of the NEExp family are given by

    $ S\left(v; \delta, \pmb{\vartheta}\right) = \frac{\delta \left[1-M\left(v; \pmb{\vartheta} \right) \right]}{\delta+ M\left(v; \pmb{\vartheta}\right) }, $

    and

    $ h\left(v; \delta, \pmb{\vartheta}\right) = \frac{\left(\delta+1 \right) m\left(v; \pmb{\vartheta}\right) }{\left[1-M\left(v; \pmb{\vartheta} \right) \right] \left[\delta+ M\left(v; \pmb{\vartheta}\right) \right]}, $

    respectively.

    A special member of the NEExp family called, a new extended exponential Weibull (NEExp-Weibull) model is discussed in the next subsection. The NEExp-Weibull model is introduced by using the DF of the Weibull model in Eq (2.1). The DF $ M\left(v; \pmb{\vartheta} \right) $ of the Weibull model is given by

    $ M(v;ϑϑ)=1eφ2vφ1,v0,φ1>0,φ2>0,
    $
    (2.3)

    with PDF $ m\left(v; \pmb{\vartheta} \right) $ given by

    $ m\left(v; \pmb{\vartheta} \right) = \varphi_{1} \varphi_{2} v^{\varphi_{1-1}} e^{-\varphi_{2} v^{\varphi_{1}}}, {\;\;\;\;\;\;\;} v > 0, \varphi_{1} > 0, \varphi_{2} > 0, $

    where $ \pmb{\vartheta} = \left(\varphi_{1}, \varphi_{2}\right). $ By incorporating Eq (2.3) in Eq (2.1), we reach at the DF of the NEExp-Weibull model; see Subsection 2.2.

    A random variable V has the NEExp-Weibull model with parameters $ \varphi_{1} > 0, \varphi_{2} > 0 $, and $ \delta > 0, $ if its DF $ F\left(v; \delta, \pmb{\vartheta}\right) $ and PDF $ F\left(v; \delta, \pmb{\vartheta}\right) $, are given by

    $ F(v;δ,ϑϑ)=1δeφ2vφ1δ+1eφ2vφ1,v0,
    $
    (2.4)

    and

    $ f(v;δ,ϑϑ)=δ(δ+1)φ1φ2vφ11eφ2vφ1[δ+1eφ2vφ1]2,v>0,
    $
    (2.5)

    respectively.

    For the NEExp-Weibull model with DF in Eq (2.4) and PDF in Eq (2.5), the SF $ \bar{F}\left(v; \delta, \pmb{\vartheta}\right) = 1-F\left(v; \delta, \pmb{\vartheta}\right) $, HF $ h\left(v; \delta, \pmb{\vartheta}\right) = \frac{f\left(v; \delta, \pmb{\vartheta}\right) }{\bar{F}\left(v; \delta, \pmb{\vartheta}\right)} $, and CHF $ H\left(v; \delta, \pmb{\vartheta}\right) = -\log \left(\bar{F}\left(v; \delta, \pmb{\vartheta}\right)\right) $ are given by

    $ \bar{F}\left(v; \delta, \pmb{\vartheta}\right) = \frac{\delta e^{-\varphi_{2} v^{\varphi_{1}}}}{\delta+1-e^{-\varphi_{2} v^{\varphi_{1}}}}, {\;\;\;\;\;\;\;\;\;} v > 0, $
    $ h\left(v; \delta, \pmb{\vartheta}\right) = \frac{ \left(\delta+1 \right) \varphi_{1} \varphi_{2} v^{\varphi_{1-1}}}{\left[\delta+1-e^{-\varphi_{2} v^{\varphi_{1}}} \right]}, {\;\;\;\;\;\;\;\;\;} v > 0, $

    and

    $ H\left(v; \delta, \pmb{\vartheta}\right) = -\log \left(\frac{\delta e^{-\varphi_{2} v^{\varphi_{1}}}}{\delta+1-e^{-\varphi_{2} v^{\varphi_{1}}}} \right), {\;\;\;\;\;\;\;\;\;} v > 0, $

    respectively.

    A visual behavior of $ f\left(v; \delta, \pmb{\vartheta}\right) $ for (i) $ \varphi_{1} = 5.5, \varphi_{2} = 0.2, \delta = 9.5 $ (red curve) (ii) $ \varphi_{1} = 4.5, \varphi_{2} = 0.6, \delta = 2.5, $ (green curve) (iii) $ \varphi_{1} = 0.5, \varphi_{2} = 1.6, \delta = 1.5, $ (blue curve) and (iv) $ \varphi_{1} = 1.9, \varphi_{2} = 2.6, \delta = 2.5, $ (gold curve) is provided in Figure 1.

    Figure 1.  A visual behavior of $ f\left(v; \delta, \pmb{\vartheta}\right) $ for different values of $ \varphi_{1}, \varphi_{2}, $ and $ \delta $.

    From the visual illustration in Figure 1, we can see that $ f\left(v; \delta, \pmb{\vartheta}\right) $ possess different behaviors. For example, it takes (i) the left-skewed form (red curve), (ii) the symmetrical shape (green curve), (iii) the reverse-J shape (blue curve), and (iv) the right-skewed (gold curve).

    Here, we obtain different mathematical properties of the NEExp family with PDF $ f\left(v; \delta, \pmb{\vartheta}\right) $. These properties include QF (quantile function) expressed by $ Q\left(u\right) $, $ r^{th} $ moment denoted by $ \mu_{r}^{/} $, MGF (moment generating function) represented by $ M_{V}\left(t\right) $, RL (residual life), and RRL (reverse residual life) functions.

    The QF of the NEExp family can be obtained by inverting Eq (2.1). Let V have the NEExp family with DF $ F\left(v; \delta, \pmb{\vartheta}\right) $, then, its QF is given by

    $ v_{q} = Q\left(u\right) = F^{-1} \left(\frac{\delta u}{\delta + 1 - u}\right) , $

    where $ u \in \left(0, 1\right) $.

    Suppose $ V $ follows the NEExp family of distributions, then the $ r^{th} $ moment of the NEExp distributions is derived as

    $ μ/r=Ωvrδ(δ+1)m(v;ϑϑ)[δ+M(v;ϑϑ)]2dv,μ/r=(δ+1)δΩvrm(v;ϑϑ)[1+M(v;ϑϑ)δ]2dv.
    $
    (3.1)

    Consider the series

    $ 1(1+k)2=a=1(1)a1aka1.
    $
    (3.2)

    By implementing Eq (3.2) with $ k = \frac{M\left(v; \pmb{\vartheta}\right)}{\delta}, $ we have

    $ 1(1+[M(v;ϑϑ)δ])2=a=1(1)a1a(M(v;ϑϑ)δ)a1.
    $
    (3.3)

    Using Eq (3.3) in Eq (3.1), we get

    $ μ/r=(δ+1)δaa=1(1)a1aΩvrm(v;ϑϑ)M(v;ϑϑ)a1dv,μ/r=(δ+1)δaa=1(1)a1Ωvrka(v;ϑϑ)dv,
    $
    (3.4)

    where $ k_{a} \left(v; \pmb{\vartheta}\right) = a \; m\left(v; \pmb{\vartheta}\right) M\left(v; \pmb{\vartheta}\right)^{a-1} $ is the exponentiated PDF, and $ a $ is a power parameter. We can also write Eq (3.4), as follows

    $ μ/r=(δ+1)δaa=1(1)a1λa,r,
    $
    (3.5)

    where

    $ λa,r=Ωvrka(v;ϑϑ)dv.
    $
    (3.6)

    Using the DF and PDF of the Weibull model in Eq (3.6), we get

    $ λa,r=0vrφ1φ2vφ11eφ2vφ1(1eφ2vφ1)a1dv.
    $
    (3.7)

    On solving Eq (3.7), we get

    $ λa,r=a1k=0(1)k(a1k)0vrφ1φ2vφ11eφ2(k+1)vφ1dv,λa,r=a1k=0(1)k(a1k)Γ(rφ1+1)(φ2)rφ1(k+1)rφ1+1.
    $
    (3.8)

    Using Eq (3.8) in Eq (3.5), we get the $ r^{th} $ moment of the NEExp-Weibull model, given by

    $ μ/r=(δ+1)δaa=1a1k=0(1)a+k1(a1k)Γ(rφ1+1)(φ2)rφ1(k+1)rφ1+1.
    $
    (3.9)

    Using $ r = 1, $ in Eq (3.9), we get the first $ r^{th} $ moment of the NEExp-Weibull model, given by

    $ \mu_{1}^{/} = \frac{\left(\delta+1 \right)}{\delta^{a}} \sum\limits_{a = 1}^{\infty} \sum\limits_{k = 0}^{a-1} \left(-1\right)^{a+k-1} {a-1 \choose k} \frac{\Gamma\left(\frac{1}{\varphi_{1}}+1\right)}{\left(\varphi_{2}\right)^{\frac{1}{\varphi_{1}}} \left(k+1\right)^{\frac{1}{\varphi_{1}}+1}}. $

    Using $ r = 2, $ in Eq (3.9), we obtain the second $ r^{th} $ moment of the NEExp-Weibull model, given by

    $ \mu_{2}^{/} = \frac{\left(\delta+1 \right)}{\delta^{a}} \sum\limits_{a = 1}^{\infty} \sum\limits_{k = 0}^{a-1} \left(-1\right)^{a+k-1} {a-1 \choose k} \frac{\Gamma\left(\frac{2}{\varphi_{1}}+1\right)}{\left(\varphi_{2}\right)^{\frac{2}{\varphi_{1}}} \left(k+1\right)^{\frac{2}{\varphi_{1}}+1}} . $

    Using $ r = 3, $ in Eq (3.9), we obtain the third $ r^{th} $ moment of the NEExp-Weibull model, given by

    $ \mu_{3}^{/} = \frac{\left(\delta+1 \right)}{\delta^{a}} \sum\limits_{a = 1}^{\infty} \sum\limits_{k = 0}^{a-1} \left(-1\right)^{a+k-1} {a-1 \choose k} \frac{\Gamma\left(\frac{3}{\varphi_{1}}+1\right)}{\left(\varphi_{2}\right)^{\frac{3}{\varphi_{1}}} \left(k+1\right)^{\frac{3}{\varphi_{1}}+1}} . $

    Using $ r = 4, $ in Eq (3.9), we obtain the second $ r^{th} $ moment of the NEExp-Weibull model, given by

    $ \mu_{4}^{/} = \frac{\left(\delta+1 \right)}{\delta^{a}} \sum\limits_{a = 1}^{\infty} \sum\limits_{k = 0}^{a-1} \left(-1\right)^{a+k-1} {a-1 \choose k} \frac{\Gamma\left(\frac{4}{\varphi_{1}}+1\right)}{\left(\varphi_{2}\right)^{\frac{4}{\varphi_{1}}} \left(k+1\right)^{\frac{4}{\varphi_{1}}+1}} . $

    Furthermore, the MGF of the NEExp-Weibull model is given by

    $ M_{V} \left(t\right) = \frac{\left(\delta+1 \right)}{\delta^{a}} \sum\limits_{a = 1}^{\infty} \sum\limits_{k = 0}^{a-1} \sum\limits_{r = 0}^{\infty} \frac{\left(-1\right)^{a+k-1} t^{r}}{r!} {a-1 \choose k} \frac{\Gamma\left(\frac{r}{\varphi_{1}}+1\right)}{\left(\varphi_{2}\right)^{\frac{r}{\varphi_{1}}} \left(k+1\right)^{\frac{r}{\varphi_{1}}+1}} . $

    The RL of the NEExp-Weibull model represented by $ R_{t} \left(v\right), $ is given by

    $ R_{t} \left(v\right) = \frac{\delta e^{-\varphi_{2} (v+t)^{\varphi_{1}}}}{\delta+1-e^{-\varphi_{2} (v+t)^{\varphi_{1}}}} \times \frac{\delta+1-e^{-\varphi_{2} v^{\varphi_{1}}}}{\delta e^{-\varphi_{2} v^{\varphi_{1}}}} . $

    Furthermore, the RRL of the NEExp-Weibull model represented by $ \bar{R}_{t} \left(v\right), $ is given by

    $ \bar{R}_{t} \left(v\right) = \frac{\delta e^{-\varphi_{2} (v-t)^{\varphi_{1}}}}{\delta+1-e^{-\varphi_{2} (v-t)^{\varphi_{1}}}} \times \frac{\delta+1-e^{-\varphi_{2} v^{\varphi_{1}}}}{\delta e^{-\varphi_{2} v^{\varphi_{1}}}} . $

    In this section, we obtain the estimators $ \left(\hat{\varphi_{1}}, \hat{\varphi_{2}}, \hat{\delta}\right) $ of the parameters $ \left(\varphi_{1}, \varphi_{2}, \delta\right) $ by implementing the maximum likelihood estimation approach. Furthermore, for the evaluation of $ \hat{\varphi_{1}}, \hat{\varphi_{2}}, $ and $ \hat{\delta} $, a simulation is also provided.

    Let $ V_{1}, V_{2}, ..., V_{p} $ be a sample of size p observed from the PDF $ f\left(v; \delta, \pmb{\vartheta}\right) $. In link to $ f\left(v; \delta, \pmb{\vartheta}\right) $, the likelihood function (LH) $ \lambda\left(\delta, \pmb{\vartheta}| v_{1}, v_{2}, ..., v_{p} \right) $ is given by

    $ λ(δ,ϑϑ|v1,v2,...,vp)=pa=1f(va;δ,ϑϑ).
    $
    (4.1)

    Using Eq (2.5) in Eq (4.1), we get

    $ λ(δ,ϑϑ|v1,v2,...,vp)=pa=1δ(δ+1)φ1φ2vφ11aeφ2vφ1a[δ+1eφ2vφ1a]2.
    $
    (4.2)

    Corresponding to $ \lambda\left(\varphi_{1}, \varphi_{2}, \delta| v_{1}, v_{2}, ..., v_{p} \right) $, the log LH $ \pi \left(\delta, \pmb{\vartheta}| v_{1}, v_{2}, ..., v_{p} \right) $ is given by

    $ π(δ,ϑϑ|v1,v2,...,vp)=plogδ+plog(δ+1)+plogφ1+plogφ2+(φ11)pa=1logvapa=1φ2vφ1a2pa=1log(δ+1eφ2vφ1a).
    $

    The partial derivatives of $ \pi \left(\delta, \pmb{\vartheta}| v_{1}, v_{2}, ..., v_{p} \right) $ are given by

    $ φ1π(δ,ϑϑ|v1,v2,...,vp)=pφ1+pa=1logvaφ2pa=1(logva)vφ1a2φ2pa=1(logva)vφ1aeφ2vφ1a(δ+1eφ2vφ1a),
    $
    $ φ2π(δ,ϑϑ|v1,v2,...,vp)=pφ2pa=1vφ1a2pa=1vφ1aeφ2vφ1a(δ+1eφ2vφ1a),
    $

    and

    $ δπ(δ,ϑϑ|v1,v2,...,vp)=pδ+p(δ+1)2pa=11(δ+1eφ2vφ1a),
    $

    respectively.

    On solving $ \frac{\partial}{\partial \varphi_{1}}\pi \left(\delta, \pmb{\vartheta}| v_{1}, v_{2}, ..., v_{p} \right) = 0 $, $ \frac{\partial}{\partial \varphi_{2}}\pi \left(\delta, \pmb{\vartheta}| v_{1}, v_{2}, ..., v_{p} \right) = 0 $, and $ \frac{\partial}{\partial \delta}\pi \left(\delta, \pmb{\vartheta}| v_{1}, v_{2}, ..., v_{p} \right), $ we get the estimators $ \hat{\varphi_{1}}, \hat{\varphi_{2}}, $ and $ \hat{\delta} $, respectively.

    Now, we evaluate the performances of $ \hat{\varphi_{1}}, \hat{\varphi_{1}} $, and $ \hat{\varphi} $ by conducting a SiSt (simulation study). The SiSt is performed by adopting the following steps

    ● To carry out the SiSt, the RNs (random numbers) from the NEExp-Weibull model are generated using the inverse DF, given by

    $ v_{q} = Q\left(u\right) = F^{-1} \left(\frac{\delta u}{\delta + 1 - u}\right). $

    ● The SiSt is performed for two different combination sets of $ \varphi_{1}, \varphi_{2} $, and $ \delta, $ such as (a) $ \varphi_{1} = 0.7, \varphi_{2} = 0.9, \delta = 1.4 $, and (b) $ \varphi_{1} = 1.5, \varphi_{2} = 1.4, \delta = 1.25. $

    ● For both two sets of $ \varphi_{1} $, $ \varphi_{2} $, and $ \delta, $ RNs of sizes $ p = 25, 50, 75, ..., 500 $ are generated using the inverse DF method.

    ● The numerical values of the maximum likelihood estimators (MLEs) $ \left(\hat{\varphi_{1}}, \hat{\varphi_{2}}, \hat{\delta}\right) $ of the parameters $ \left(\varphi_{1}, \varphi_{2}, \delta\right) $ are obtained.

    ● Two statistical quantities/measures such as (i) mean square errors (MSEs) and (ii) Bias, are selected for assessing $ \hat{\varphi_{1}} $, $ \hat{\varphi_{2}} $, and $ \hat{\delta} $. The values of these quantities are given by

    $ MSE \left(\pmb{\hat{\phi}}\right) = \frac{1}{500} \sum\limits_{a = 1}^{500} \left(\pmb{\hat{\phi}}_{a} - \pmb{\phi}\right)^{2}, $

    and

    $ Bias \left(\pmb{\hat{\phi}}\right) = \frac{1}{500} \sum\limits_{a = 1}^{500} \left(\pmb{\hat{\phi}}_{a} - \pmb{\phi}\right), $

    respectively, where $ \pmb{\phi} = \left(\varphi_{1}, \varphi_{2}, \delta\right). $

    All the numerical and simulation results are obtained using $ \mathsf{optim()} $ $ \mathsf{R} $-$ \mathsf{function} $ with the argument $ \mathsf{method = "L} $-$ \mathsf{BFGS} $-$ \mathsf{B"} $. The results of the SiSt of the NEExp-Weibull distribution are reported in Tables 1 and 2, and presented visually in Figures 2 and 3.

    Table 1.  The results of the SiSt of the NEExp-Weibull model for $ \varphi_{1} = 0.7, \delta = 1.4, \varphi_{2} = 0.9 $.
    n Parameters MLEs MSEs Biases
    $ \varphi_{1} $ 0.7439555 0.01816230 0.043955486
    20 $ \delta $ 2.5224710 4.62452592 1.122470958
    $ \varphi_{2} $ 0.9819897 0.08933255 0.081989688
    $ \varphi_{1} $ 0.7261523 0.00675241 0.026152290
    40 $ \delta $ 2.2436960 3.50148162 0.843695947
    $ \varphi_{2} $ 0.9475541 0.04503070 0.047554140
    $ \varphi_{1} $ 0.7174831 0.00469112 0.017483127
    60 $ \delta $ 2.1504600 3.14461417 0.750459996
    $ \varphi_{2} $ 0.9382455 0.03002095 0.038245459
    $ \varphi_{1} $ 0.7139450 0.00303585 0.013945008
    80 $ \delta $ 2.0166510 2.71268772 0.616651319
    $ \varphi_{2} $ 0.9172897 0.02141037 0.017289700
    $ \varphi_{1} $ 0.7118858 0.00250170 0.011885794
    100 $ \delta $ 1.9587070 2.39162374 0.558707085
    $ \varphi_{2} $ 0.9215668 0.01819720 0.021566823
    $ \varphi_{1} $ 0.7071069 0.00095725 0.007106915
    200 $ \delta $ 1.6315320 1.20339825 0.231531527
    $ \varphi_{2} $ 0.8963361 0.00907772 -0.003663932
    $ \varphi_{1} $ 0.7059804 0.00056999 0.005980353
    300 $ \delta $ 1.4947460 0.59116842 0.094745813
    $ \varphi_{2} $ 0.8942843 0.00510254 -0.005715678
    $ \varphi_{1} $ 0.7046970 0.00042009 0.004696986
    400 $ \delta $ 1.4758080 0.42345175 0.075807616
    $ \varphi_{2} $ 0.8949554 0.00295727 -0.005044638
    $ \varphi_{1} $ 0.7026197 0.00015114 0.002619658
    500 $ \delta $ 1.4191470 0.17256813 0.019147385
    $ \varphi_{2} $ 0.8958860 0.00155967 -0.004113986
    $ \varphi_{1} $ 0.7021186 0.00011673 0.002118624
    600 $ \delta $ 1.3973690 0.05930217 -0.002630857
    $ \varphi_{2} $ 0.8964089 0.00084446 -0.003591110

     | Show Table
    DownLoad: CSV
    Table 2.  The results of the SiSt of the NEExp-Weibull model for $ \varphi_{1} = 0.6, \delta = 1.5, \varphi_{2} = 0.8 $.
    n Parameters MLEs MSEs Biases
    $ \varphi_{1} $ 0.6538964 1.61446e-02 0.053896399
    20 $ \delta $ 2.3444560 3.753281849 0.844456065
    $ \varphi_{2} $ 0.8422871 0.074129355 0.042287133
    $ \varphi_{1} $ 0.6335936 6.57586e-03 0.033593572
    40 $ \delta $ 2.0966990 2.835106550 0.596698837
    $ \varphi_{2} $ 0.8114326 0.027465620 0.011432631
    $ \varphi_{1} $ 0.6207884 3.08374e-03 0.020788424
    60 $ \delta $ 1.9743140 2.393837599 0.474313588
    $ \varphi_{2} $ 0.8107312 0.022889902 0.010731190
    $ \varphi_{1} $ 0.6195547 2.83132e-03 0.019554745
    80 $ \delta $ 1.8076620 1.762573786 0.307661602
    $ \varphi_{2} $ 0.7912179 0.016246953 -0.008782086
    $ \varphi_{1} $ 0.6135883 1.52055e-03 0.013588293
    100 $ \delta $ 1.7689760 1.451201967 0.268976369
    $ \varphi_{2} $ 0.7923297 0.011392565 -0.007670316
    $ \varphi_{1} $ 0.6062087 4.83673e-04 0.006208661
    200 $ \delta $ 1.5338360 0.437971185 0.033836372
    $ \varphi_{2} $ 0.7870588 0.003783507 -0.012941210
    $ \varphi_{1} $ 0.6050252 2.87229e-04 0.005025202
    300 $ \delta $ 1.4799110 0.134430960 -0.020089413
    $ \varphi_{2} $ 0.7911718 0.001925689 -0.008828233
    $ \varphi_{1} $ 0.6021233 1.08930e-04 0.002123257
    400 $ \delta $ 1.5034740 0.104238258 0.003473527
    $ \varphi_{2} $ 0.7954403 0.000832362 -0.004559674
    $ \varphi_{1} $ 0.6014035 6.62443e-05 0.001403539
    500 $ \delta $ 1.4982590 0.048978064 -0.001741474
    $ \varphi_{2} $ 0.7974116 0.000411811 -0.002588439
    $ \varphi_{1} $ 0.6009386 3.77904e-05 0.000938590
    600 $ \delta $ 1.4899270 0.005002637 -0.010072511
    $ \varphi_{2} $ 0.7976296 0.000242969 -0.002370384

     | Show Table
    DownLoad: CSV
    Figure 2.  A visual display of the SiSt for $ \varphi_{1} = 0.7, \delta = 1.4, \varphi_{2} = 0.9 $.
    Figure 3.  A visual display of the SiSt for $ \varphi_{1} = 0.6 $, $ \delta = 1.5 $, $ \varphi_{2} = 0.8 $.

    The primary aim of the introduction of the proposed distribution is its implementation for data analysis in the health and other related sectors. This section illustrates the respective fact by analyzing two data sets. The first data set (Data 1) represents the mortality rates of the COVID-19 infected persons in Mexico. Whereas, the second illustration is based on taking another COVID-19 data from Holland.

    By analyzing these two COVID-19 data sets, the numerical results of the proposed model is compared with the

    ● Weibull model with SF given by

    $ S\left(v; \varphi_{1}, \varphi_{2} \right) = e^{-\varphi_{2} v^{\varphi_{1}}}, {\;\;\;\;\;\;} v, \varphi_{1}, \varphi_{2} > 0, $

    ● Logarithmic Weibull (L-Weibull) model with SF given by

    $ S\left(v; \varphi_{1}, \varphi_{2}, \theta, \sigma \right) = \left(1- \frac{\sigma \left[1-e^{-\varphi_{2} v^{\varphi_{1}}}\right]}{\sigma - \left[\log \left(1-e^{-\varphi_{2} v^{\varphi_{1}}}\right) \right]} \right)^{\alpha}, {\;\;\;\;\;\;} v, \varphi_{1}, \varphi_{2}, \alpha, \sigma > 0, $

    ● Novel exponent power-Weibull (NEP-Weibull) model with SF given by

    $ S\left(v;\varphi_{1}, \varphi_{2}, \theta \right) = \left(1- \frac{1 - e^{-\varphi_{2} v^{\varphi_{1}}}}{e^{e^{- \varphi_{2} v^{\varphi_{1}}}}} \right)^{\theta} , {\;\;\;\;\;\;} v, \varphi_{1}, \varphi_{2}, \theta > 0, $

    and

    ● New modified Weibull (NM-Weibull) model with SF given by

    $ S\left(v;\varphi_{1}, \varphi_{2}, \sigma \right) = 1- \frac{\left(1-e^{- \varphi_{2} v^{\varphi_{1}}}\right)}{\sigma} \left[\sigma - e^{- \varphi_{2} v^{\varphi_{1}}} \right] , {\;\;\;\;\;\;} v, \varphi_{1}, \varphi_{2}, \sigma \geq 1, \sigma \leq -1. $

    To figure out the best competitive model for the COVID-19 data, certain statistical tests such as the (i) AD (Anderson-Darling) test statistic given by $ AD = -p-\frac{1}{p}\sum\limits_{a = 1}^{p} {\left({2a-1} \right)\left[{\log M\left({v_{a} } \right)+\log \left\{ {1-M\left({v_{p-a+1} } \right)} \right\}} \right]}, $ (ii) CM (Cramer-von Mises) test statistic expressed by $ CM = \frac{1}{12p}+\sum\limits_{a = 1}^{p} {\left[{\frac{2a-1}{2p}-M\left({v_{a} } \right)} \right]^{2}}, $ and (iii) KS (Kolmogorov-Smirnov (KS) test statistic derived by $ KS = sup_{v} \left[{M_{p} \left(v\right)-M\left(v\right)} \right], $ are considered.

    Here, we analyze the mortality rates of the patients infected by the COVID-19 pandemic in Mexico; see https://covid19.wh. It is also studied by Almongy et al. [23] using a new extension of the Rayleigh distribution. This data set consists of 106 observations and is recorded from March 4, 2020, to July 20, 2020. The data set is given by: 4.4130, 3.0525, 4.6955, 7.4810, 5.1915, 3.6335, 6.6100, 8.2490, 5.8325, 3.0075, 5.4275, 3.0610, 3.3280, 1.7200, 2.9270, 5.3425, 5.0175, 2.6210, 2.1720, 2.5715, 3.8150, 7.3020, 3.9515, 3.1850, 1.7685, 3.1635, 2.3650, 1.6075, 4.6420, 6.4390, 4.4065, 5.0215, 3.6300, 2.9925, 3.2060, 1.6975, 2.2120, 4.9675, 3.9200, 4.7750, 1.7495, 1.8755, 3.4840, 1.6430, 5.0790, 4.0540, 3.3485, 3.5755, 3.2800, 1.0385, 1.8890, 1.4940, 1.6680, 3.4070, 4.1625, 3.9270, 4.2755, 1.6140, 3.7430, 3.3125, 3.0700, 2.4545, 2.3305, 2.6960, 6.0210, 4.3480, 0.9075, 1.6635, 2.7030, 3.0910, 0.5205, 0.9000, 2.4745, 2.0445, 1.6795, 1.0350, 1.6490, 2.6585, 2.7210, 2.2785, 2.1460, 1.2500, 3.2675, 2.3240, 2.3485, 2.7295, 2.0600, 1.9610, 1.6095, 0.7010, 1.2190, 1.6285, 1.8160, 1.6165, 1.5135, 1.1760, 0.6025, 1.6090, 1.4630, 1.3005, 1.0325, 1.5145, 1.0290, 1.1630, 1.2530, 0.9615.

    Corresponding to the COVID-19 data (mortality rate of the COVID-19 infected persons) of Mexico, the summary measures are: minimum = 0.5205, $ 1^{st} $ quartile = 1.6445, median = 2.6397, mean = 2.9112, $ 3^{rd} $ quartile = 3.7970, maximum = 8.2490, variance = 2.640433, range = 7.7285, standard deviation = 1.624941, skewness = 0.9732453, and kurtosis = 3.666136.

    Corresponding to the mortality rate of the COVID-19 infected persons in Mexico, some basic plots are presented in Figure 4. The plots in Figure 4, show that the first data set is right-skewed and possesses the increasing failure rate behavior.

    Figure 4.  Basic plots of the Mexico data.

    Corresponding to the Mexico data, the values of $ \hat{\varphi_{1}} $, $ \hat{\varphi_{2}} $, and $ \hat{\delta} $ are presented in Table 3. The standard errors (SEs) (numerical values in the parentheses) of $ \hat{\varphi_{1}} $, $ \hat{\varphi_{2}} $, and $ \hat{\delta} $ are also presented in Table 3.

    Table 3.  The numerical values of $ \hat{\varphi_{1}}, \hat{\varphi_{2}}, \hat{\delta}, \hat{\alpha}, \hat{\sigma} $, and $ \hat{\theta} $ using the mortality rate data.
    Model $ \hat{\varphi_{1}} $ $ \hat{\varphi_{2}} $ $ \hat{\delta} $ $ \hat{\alpha} $ $ \hat{\sigma} $ $ \hat{\theta} $
    NEExp-Weibull 2.53401 (0.31876) 0.01723 (0.01562) 0.26201 (0.24135) - - -
    Weibull 1.92738 (0.14019) 0.09976 (0.02220) - - - -
    L-Weibull 1.57117 (0.01001) 1.20372 (0.00879) - 0.16022 (0.01580) 0.45036 (0.01773) -
    NEP-Weibull 1.71906 (0.03634) 0.95340 (0.03634) - - - 0.15201 (0.01503)
    NM-Weibull 1.89237 (0.15734) 0.11004 (0.03088) - - 12.06058 (20.5532) -

     | Show Table
    DownLoad: CSV

    For the Mexico data, the values of the selected tests CM, AD, and KS of the fitted models are reported in Table 4. The associated p-value of fitted models is also provided in Table 4. From the numerical illustration in Table 4, we can see that the proposed model has the smallest values of CM, AD, and KS, and the largest p-value. These facts show that the proposed model is the best competitor. Besides the numerical illustration, a visual display of the performances of the proposed model is also provided in Figure 5.

    Table 4.  The analytical measures of the fitted models using Data 1.
    Model CM AD KS p-value
    NEExp-Weibull 0.05834 0.33674 0.06266 0.79940
    Weibull 0.10264 0.66004 0.07147 0.65100
    L-Weibull 0.06002 0.36130 0.07091 0.66060
    NEP-Weibull 0.07017 0.43993 0.07862 0.52880
    NM-Weibull 0.10723 0.68969 0.07024 0.67240

     | Show Table
    DownLoad: CSV
    Figure 5.  For the first data, the plots of the fitted DFs of the proposed and all the fitted models.

    Here, we provide a second illustration of the proposed model by analyzing another COVID-19 data taken from Holland; see Almongy et al. [23]. This data set consists of 30 observations and is recorded between March 31, 2020, and April 30, 2020. The second data set is given by: 7.4590, 3.7490, 3.4700, 5.3280, 1.4285, 1.1270, 6.1370, 5.1445, 5.4160, 3.5495, 1.7305, 1.8235, 2.9640, 6.6055, 3.9840, 3.7920, 2.6535, 2.5240, 2.7155, 2.7775, 3.0135, 2.0485, 1.8055, 2.4800, 2.2310, 1.9415, 0.9870, 0.6365, 0.7080, 2.1175.

    The summary measures of the mortality rate of the COVID-19 infected persons in Holland are: minimum = 0.6365, $ 1^{st} $ quartile = 1.8530, median = 2.6845, mean = 3.0783, $ 3^{rd} $ quartile = 3.7812, maximum = 7.4590, variance = 3.121073, range = 6.8225, standard deviation = 1.766656, skewness = 0.8339708, and kurtosis = 2.953478.

    In link to the mortality rate of the COVID-19 infected persons in Holland, some basic plots are sketched in Figure 6. From the plots in Figure 6, it is obvious that the second data set, related to the mortality rate of the COVID-19 infected persons, is skewed to the right and has increasing failure rate behavior.

    Figure 6.  Basic plots of the Holland data.

    Based on the Holland's COVID-19 data, the numerical values of estimators $ \hat{\varphi_{1}}, \hat{\varphi_{2}}, $ and $ \hat{\delta} $ are obtained in Table 5. Furthermore, the SEs of these estimators are also reported in Table 5.

    Table 5.  The numerical values of $ \hat{\varphi_{1}}, \hat{\varphi_{2}}, \hat{\delta}, \hat{\alpha}, \hat{\sigma} $, and $ \hat{\theta} $ using the mortality rate data.
    Model $ \hat{\varphi_{1}} $ $ \hat{\varphi_{2}} $ $ \hat{\delta} $ $ \hat{\alpha} $ $ \hat{\sigma} $ $ \hat{\theta} $
    NEExp-Weibull 2.20197 (0.48765) 0.03704 (0.04876) 0.75896 (0.10983) - - -
    Weibull 1.92738 (0.14019) 0.09976 (0.02220) - - - -
    L-Weibull 1.42252 (0.51193) 0.12239 (0.05844) - 2.85548 (0.97654) 0.89686 (2.15028) -
    NEP-Weibull 1.77603 (0.07305) 1.03756 (0.07304) - - - 0.11297 (0.02890)
    NM-Weibull 2.00489 (0.28946) 0.06377 (0.04170) - - -2.41345 (3.47713)

     | Show Table
    DownLoad: CSV

    Corresponding to the Holland data, the p-value and values of CM, AD, and KS are reported in Table 6. From the numerical comparison of the fitted models in Table 6, it is obvious that the proposed model performs better than the other competitors as it has the largest p-value and smallest CM, AD, and KS values. In support of Table 6, the performances of the proposed model are also illustrated visually by plotting the estimated DF, PP, PDF, QQ, and SF; see Figure 7.

    Table 6.  The analytical measures of the fitted models using Data 2.
    Model CM AD KS p-value
    NEExp-Weibull 0.03088 0.20437 0.08557 0.96710
    Weibull 0.04794 0.29434 0.10236 0.88030
    L-Weibull 0.05165 0.31630 0.10144 0.88660
    NEP-Weibull 0.03423 0.22143 0.10418 0.86740
    NM-Weibull 0.03926 0.24960 0.08981 0.95100

     | Show Table
    DownLoad: CSV
    Figure 7.  For the first data, the plots of the fitted DFs of the proposed and all the fitted models.

    In this section, we implement three different machine learning algorithms, namely, SVR, NNAR, and RF to forecast the data set analyzed in Section 5. Before modeling, we split the data set into two parts; 80 percent as training data and 20 percent as testing data, followed by Qi and Zhang [24]. We apply all the models to training data and compare their forecasting performances using the testing data. To assess the out-of-sample (also known as post-sample) prediction accuracy, multistep ahead forecasts with RMSE and MAE are calculated.

    The SVR is a popular ML approach, which is used for regression as well as classification problems. The SVR was first developed by Cortes and Vapnik [25], and to date, it is one of the most widely used supervised learning methods that is based on structured risk minimization rule and statistical theory. The structured risk minimization rule maximizes prediction accuracy and mitigates the likelihood of over fitting.

    In practice, it can effectively approximate linear and nonlinear problems and work well for numerous problems. The SVR uses various kernel functions to figure out the similarity between two data points to tackle the non-linear situations. The main advantage of SVR lies in its potential to capture the predictors' nonlinearity and then utilize it to enhance forecasting accuracy. In our case, the set of predictors contains the lagged values. The SVR assists to detect the margin of error which is bearable in the model; see Ribeiro et al. [26] and Bibi et al. [27]. The SVR equation with kernel function can be expressed as

    $ Ft=hk=1(λkλk)N(ck,c)+μ,
    $
    (6.1)

    where $ F_{t} $ is the outcome variable. The kernel function, $ N \left(c_{k}, c \right) $ denotes the inner product, while $ \mu $ is accommodated within the kernel function. In the literature, several kernel functions have been developed. Among them, the radial basis function (RBF) is the most popular, which can be illustrated as

    $ N(ck,c)=exp{||ckcN||22τ2},
    $
    (6.2)

    where the Euclidean distance between the two predictors squared vectors is represented by $ ||c_{k} - c_{N}||^{2} $ and $ \tau^{2} $ is basically the width of RBF; see Lu et al. [28]. Hence, in this study, we focus on the RBF kernel function for the SVR. Tuning the SVR model enable us to arrive at optimal parameters.

    Another ML approach is RF which is also known as the random decision forest. It falls within the supervised learning category. The RF is a very effective algorithm that is used for both regression and classification problems as proposed by Breiman [29]. Dietterich [30] argued that RF is considered the most efficient ensemble technique appearing in ML and fulfills good forecasting properties. The RF approach is employed in different areas, including stock trading, finance, e-commerce, and health care. It provides a forest out of a collection of decision trees that are usually estimated (trained) by employing the bagging approach.

    The RF approach discovers the output based on the decision trees' forecasts. The forecasts are computed by averaging the output of several trees. The improvement is achieved in prediction by expanding the number of trees. In other words, as the number of trees in the forest grows, a more accurate forecast is obtained and circumvents the issue of over fitting as well. To estimate the RF, we use three hyper parameters i.e., the number of trees, number of nodes, and sample repetition. The number of nodes and trees are utilized as 3 and 500, respectively.

    In general, neural networks (NNs) are basically a network or circuit of neurons. The artificial neural network (ANNs) is composed of nodes or artificial neurons. NNs are highly flexible computing frameworks for analyzing a wide range of nonlinear problems. A key advantage of such networks is that they have not required prior information regarding the functional form in the model establishing process, rather highly determined by the characteristics of data; see Peng et al. [31].

    A feedback NN is established with lagged realization as predictors and hidden layer(s) with dimension nodes. The neural network autoregression (NNAR) contains three layers such as (i) the input layer, (ii) the hidden layer, and (iii) the output layer. The NNAR model is fitted to forecast a time series by utilizing its past information as inputs $ F_{t}, F_{t-1}, ..., F_{t-m}, $ the entire process refers to feedback delay, where $ t $ indicates the time delay parameter. The NNAR (m, n) entails that the hidden layer consists of m delayed inputs and n nodes. The mathematical form of NNAR can be illustrated as

    $ Ft=ω0+mc=1θωc(c+zn=1cnFti)+μ,
    $
    (6.3)

    where $ \aleph_{cn} \left(c = 1, 2, 3, ..., m, {\; \; } n = 1, 2, 3, ..., z \right) $ and $ \omega_{c} \left(c = 1, 2, 3, ..., m \right) $ are the weights of interconnection, and z is the length of input layers, and $ m $ is the length of hidden layers.

    The prediction accuracy of all ML techniques is quantified by using two statistical accuracy criteria computed from a testing data set. Statistically, the forecast errors are more appropriate criteria to evaluate the predictive power and to finalize the best approach. In general, the most popular measures are MAE and RMSE. Hence, we compare the forecasting performance of ML techniques using these two measures in our study. Their mathematical expressions are given by

    $ MAE = mean \left(|F_{t} - \hat{F_{t}} |\right), $

    and

    $ RMSE = \sqrt{mean \left(F_{t} - \hat{F_{t}}\right)^{2}}, $

    respectively.

    This subsection is further divided into two parts. In the first part, we deal with the mortality rates of the patients infected by the COVID-19 pandemic in Mexico. In the second part, we deal with the death rates of the patients infected by the COVID-19 pandemic in Holland.

    From Figure 8, it can be observed that the entire trajectory of the mortality rate has experienced many ups and downs in the last few months, but the trend of data is decreasing over time. The mortality rate series is plotted in Figure 8, where the vertical blue dotted line splits the estimation and post-sample forecasting periods. The histogram plot and box plot are also provided in Figure 8, which reveal that the underlying time series is right skewed.

    Figure 8.  A virtual display of the mortality rate data in Mexico.

    The two statistical accuracy measures for the Mexico data set are given in Table 7. The MAE and RMSE are calculated for ML algorithms such as SVR, RF, and NNAR. It is inferred that the RF outperformed the other competitor counterparts. For the RF method, the RMSE and MAE values are given by 0.066 and 0.039, respectively. Whereas, for the SVR, the values of the RMSE and MAE are given by 0.073 and 0.043, respectively. On the other hand, the RMSE and MAE values for the NNAR are, respectively, given by 0.198 and 0.149. From the values of the RMSE and MAE for the NNAR approach, it is clear that the values of NNAR are higher than the values of the RMSE and MAE computed for RF and SVR.

    Table 7.  The error metrics.
    Criteria SVR RF NNAR
    RMSE 0.073 0.066 0.198
    MAE 0.043 0.039 0.149

     | Show Table
    DownLoad: CSV

    Furthermore, the visual illustration of forecast errors is porvided in Figure 9. The plots in Figure 9 show that the ML algorithms, specifically, the RF remained an efficient tool in forecasting the mortality rates of COVID-19 patients.

    Figure 9.  The schematic representation of forecast errors for the mortality rate data in Mexico.

    We also plot the prediction curves for COVID-19 deaths under the three machine learning algorithms to get a more intuitive picture of the prediction accuracy and results. In this regard, the line chart has been constructed, for the representation of predicted and actual values; see Figure 10. Some predicted values of the NNAR approach are very close, but few are substantially far away from the actual data. On the other hand, SVR prediction is highly stable over time and therefore beat the rival algorithms in predictive modeling.

    Figure 10.  Forecasting performance of ML algorithms for the mortality rate data in Mexico.

    Here, we estimate and predict the daily death data of COVID-19 for Holland. It can be observed from Figure 11 that the entire trajectory of mortality rate has experienced numerous episodes in several months, but the times series is declining over time. The mortality rate series is plotted in Figure 11, where the vertical blue dotted line represents the estimation and post-sample forecasting periods. The histogram plot and box plot are also presented in Figure 10, which demonstrates that the underlying series is right skewed.

    Figure 11.  A virtual display of the mortality rate data in Holland.

    The two considered statistical accuracy measures for the Holland data set are reported in Table 8. The MAE and RMSE are computed for ML algorithms, namely SVR, RF, and NNAR. From Table 8, we can observe that the SVR showed superior forecast performance as compared to the rival counterparts. For the SVR method, the RMSE and MAE values are 0.16 and 0.118, respectively. On the other hand, for the RF approach, these values are given by 0.191 and 0.139, respectively. Whereas, the forecast errors (RMSE and MAE) of NNAR are given by 0.444 and 0.398, respectively. From the above discussion, it is obvious that the forecast errors of the NNAR approach are substantially higher than the forecast errors of RF and SVR.

    Table 8.  The error metrics.
    Criteria SVR RF NNAR
    RMSE 0.160 0.191 0.444
    MAE 0.118 0.139 0.398

     | Show Table
    DownLoad: CSV

    The graphical comparison of forecast errors is also depicted in Figure 12. It manifests that the ML algorithms, specifically SVR remained an effective tool in forecasting the post-sample trajectory of the mortality rate of COVID-19 patients.

    Figure 12.  The schematic representation of forecast errors for the mortality rate data in Holland.

    We portray the prediction curves for COVID-19 deaths using these three ML algorithms to get a more clear picture of the models performances. In this regard, we use the line chart for the representation of forecasted and observed data; see Figure 13. From the plots in Figure 13, it can be seen that the observed test series is highly volatile and noisy. Despite this volatility, SVR and RF have shown good results, particularly the RF performance is highly satisfactory.

    Figure 13.  Forecasting Performance of ML algorithms for the mortality rate data in Holland.

    The efforts in this paper added another useful approach to the literature on statistical methodologies by introducing a new family of distributions. The new family was named a NEExp family of distributions. Based on a NEExp family, an updated version of the Weibull model called, a NEExp-Weibull model was studied. The MLEs of the NEExp-Weibull model were obtained. The evaluation of the MLEs of the NEExp-Weibull distribution was carried out through a brief SiSt. The usefulness of the NEExp-Weibull model was shown by analyzing two data sets taken from the healthcare sector. The first data set was representing the mortality rate of COVID-19 patients in Mexico. Whereas, the second data was also related to the COVID-19 events taken from Holland. Using these two COVID-19 data sets, the NEExp-Weibull distribution was compared with the Weibull and three other well-known models such as the L-Weibull, NEP-Weibull, and NM-Weibull distributions. Based on four well-known comparative tools, it is observed that the NEExp-Weibull distribution was the best competitive model as compared to the Weibull and other well-known modified forms of the Weibull distribution. Therefore, based on the numerical results and findings of this study, it is observed that the NEExp-Weibull distribution may be the best suitable choice to use for analyzing the medical and other related data sets. Besides the statistical modeling, for prediction purposes, we further implemented three ML methods including SVR, RF, and NNAR using the same data sets. To compare their forecasting performances, two well-known statistical accuracy quantities such as the RMSE and MAE were computed. We found that the RF algorithm was very efficient in forecasting using the first data. However, for the second data set, the SVR showed superior performance in contrast to other methods.

    Besides the certain advantages of the NEExp-Weibull distribution over the Weibull and other competitive distributions, the NEExp-Weibull distribution has also some certain limitations, for example

    ● The proposed NEExp-Weibull model is a continuous probability distribution, and it is employed to analyze the mortality rates of COVID-19 patients. Therefore, the NEExp-Weibull model could not be implemented to analyze other forms of the COVID-19 data that are discrete in nature, for example, (i) the number of deaths, (ii) the number of confirmed cases, or (iii) the number of recovered cases, etc.

    ● Due to the introduction of the additional parameter, the NEExp-Weibull distribution has a complicated form of its PDF. Therefore, the expressions for the estimators of its parameters are not in explicit forms. Therefore, computer software must be used to obtain the estimated values of the parameters.

    ● Since, the PDF of the NEExp family has a complicated form, therefore, more computational efforts are needed to derive the key mathematical properties.

    In the future, we are motivated to obtain the discrete version of the proposed NEExp-Weibull distribution to counter the discrete data sets. In this work, we only used the maximum likelihood estimation method to estimate the parameters of the NEExp-Weibull distribution. In the future, we are intended to estimate the parameters of the NEExp-Weibull distribution using other classical methods such as ordinary least square, weighted ordinary least square, maximum product spacing methods, etc. Neutrosophic statistics is a generalization of classical statistics and is implemented when the data sets are generated from a complex process. In the future, we are also motivated to study the neutrosophic extension of the NEExp-Weibull distribution.

    [1] Ioannidi E, Kalamaki MS, Engineer C, et al. (2009) Expression profiling of ascorbic acid-related genes during tomato fruit development and ripening and in response to stress conditions. J Exp Bot 60: 663-678. doi: 10.1093/jxb/ern322
    [2] Petro-Turza M (1987) Flavor of tomato and tomato products. Food Rev Int 2: 309-351.
    [3] Buttery R, Takeoka G, Teranishi R, et al. (1990) Tomato aroma components: identification of glycoside hydrolysis volatiles. J Agr Food Chem 38: 2050-2053. doi: 10.1021/jf00101a010
    [4] Klee HJ, Giovannoni JJ (2011) Genetics and control of tomato fruit ripening and quality attributes. Annu Rev Genet 45: 41-59. doi: 10.1146/annurev-genet-110410-132507
    [5] Zanor MI, Rambla JL, Chaib J, et al. (2009) Metabolic characterization of loci affecting sensory attributes in tomato allows an assessment of the influence of the levels of primary metabolites and volatile organic contents. J Exp Bot 60: 2139-2154. doi: 10.1093/jxb/erp086
    [6] Goff SA, Klee HJ (2006) Plant volatile compounds: sensory cues for health and nutritional value? Science (New York, N Y ) 311: 815-819. doi: 10.1126/science.1112614
    [7] Dudareva N, Pichersky E, Gershenzon J (2004) Biochemistry of plant volatiles. Plant Physiol 135: 1893-1902. doi: 10.1104/pp.104.049981
    [8] Tikunov Y, Lommen A, De Vos R, et al. (2005) A novel approach for non targeted data analysis for metabolomics: large-scale profiling of tomato fruit volatiles. Plant Physiology: 1125-1137.
    [9] Vogt T (2010) Phenylpropanoid biosynthesis. Mol Plant 3: 2-20. doi: 10.1093/mp/ssp106
    [10] Tzin V, Galili G, Aharoni A (2012) Shikimate pathway and aromatic amino acid biosynthesis. eLS ohn Wiley & Sons Ltd.
    [11] Tzin V, Galili G (2010) New insights into the shikimate and aromatic amino acids biosynthesis pathways in plants. Mol Plant 3: 956-972. doi: 10.1093/mp/ssq048
    [12] Oliva M, Ovadia R, Perl A, et al. (2015) Enhanced formation of aromatic amino acids increases fragrance without affecting flower longevity or pigmentation in Petunia × hybrida. Plant Biotechnology Journal 13: 125-136. doi: 10.1111/pbi.12253
    [13] Colquhoun TA, Clark DG (2011) Unraveling the regulation of floral fragrance biosynthesis. Plant Signal Behav 6: 378-381. doi: 10.4161/psb.6.3.14339
    [14] Pratelli R, Pilot G (2014) Regulation of amino acid metabolic enzymes and transporters in plants. J Exp Bot 65: 5535-5556. doi: 10.1093/jxb/eru320
    [15] Voll LM, Allaire EE, Fiene G, et al. (2004) The Arabidopsis phenylalanine insensitive growth mutant exhibits a deregulated amino acid metabolism. . Plant Physiology 136: 3058-3069. doi: 10.1104/pp.104.047506
    [16] Pratelli R, Voll LM, Horst RJ, et al. (2010) Stimulation of nonselective amino acid export by glutamine dumper proteins. Plant Physiol 152: 762-773. doi: 10.1104/pp.109.151746
    [17] Galili G, Amir R (2013) Fortifying plants with the essential amino acids lysine and methionine to improve nutritional quality. Plant Biotechnol J 11: 211-222. doi: 10.1111/pbi.12025
    [18] Vivancos PD, Driscoll SP, Bulman CA, et al. (2011) Perturbations of amino Acid metabolism associated with glyphosate-dependent inhibition of shikimic Acid metabolism affect cellular redox homeostasis and alter the abundance of proteins involved in photosynthesis and photorespiration. Plant Physiology 157: 256-268. doi: 10.1104/pp.111.181024
    [19] Tzin V, Malitsky S, Ben Zvi MM, et al. (2012) Expression of a bacterial feedback-insensitive 3-deoxy-D-arabino-heptulosonate 7-phosphate synthase of the shikimate pathway in Arabidopsis elucidates potential metabolic bottlenecks between primary and secondary metabolism. New Phytol 194: 430-439. doi: 10.1111/j.1469-8137.2012.04052.x
    [20] Baldwin GS, Davidson BE (1981) A kinetic and structural comparison of chorismate mutase/prephenate dehydratase from mutant strains of Escherichia coli K 12 defective in the PheA gene. Arch Biochem Biophys 211: 66-75. doi: 10.1016/0003-9861(81)90430-6
    [21] Tzin V, Rogachev I, Meir S, et al. (2013) Tomato fruits expressing a bacterial feedback-insensitive 3-deoxy-D-arabino-heptulosonate 7-phosphate synthase of the shikimate pathway possess enhanced levels of multiple specialized metabolites and upgraded aroma. J Exp Bot 64: 4441-4452. doi: 10.1093/jxb/ert250
    [22] Langer KM, Jones CR, Jaworski EA, et al. (2014) PhDAHP1 is required for floral volatile benzenoid/phenylpropanoid biosynthesis in Petunia × hybrida cv ‘Mitchell Diploid’. Phytochemistry 103: 22-31. doi: 10.1016/j.phytochem.2014.04.004
    [23] Maeda H, Shasany AK, Schnepp J, et al. (2010) RNAi Suppression of Arogenate Dehydratase1 Reveals That Phenylalanine Is Synthesized Predominantly via the Arogenate Pathway in Petunia Petals. Plant Cell 22: 832-849. doi: 10.1105/tpc.109.073247
    [24] Tzin V, Malitsky S, Aharoni A, et al. (2009) Expression of a bacterial bi-functional chorismate mutase/prephenate dehydratase modulates primary and secondary metabolism associated with aromatic amino acids in Arabidopsis. Plant J 60: 156-167. doi: 10.1111/j.1365-313X.2009.03945.x
    [25] Yoo H, Widhalm JR, Qian YC, et al. (2013) An alternative pathway contributes to phenylalanine biosynthesis in plants via a cytosolic tyrosine:phenylpyruvate aminotransferase. Nature Communications 4.
    [26] Adato A, Mandel T, Mintz-Oron S, et al. (2009) Fruit-surface flavonoid accumulation in tomato is controlled by a SlMYB12-regulated transcriptional network. PLoS Genetics 5: e1000777. doi: 10.1371/journal.pgen.1000777
    [27] Deikman J, Kline R, Fischer RL (1992) Organization of Ripening and Ethylene Regulatory Regions in a Fruit-Specific Promoter from Tomato (Lycopersicon esculentum). Plant Physiology 100: 2013-2017. doi: 10.1104/pp.100.4.2013
    [28] Zhao L, Lu L, Zhang L, et al. (2009) Molecular evolution of the E8 promoter in tomato and some of its relative wild species. J Biosciences 34: 71-83. doi: 10.1007/s12038-009-0010-x
    [29] Shaul O, Galili G (1993) Concerted regulation of lysine and threonine synthesis in tobacco plants expressing bacterial feedback-insensitive aspartate kinase and dihydrodipicolinate synthase. Plant Molecular Biology 23: 759-768. doi: 10.1007/BF00021531
    [30] McCormick S (1991) Transformation of tomato with Agrobacterium tumefaciens; Lindsey K, editor: Kluwer.
    [31] Filati J, Kiser J, Rose R, et al. (1987) Efficient transfer of a glyphosphate tolerance gene into tomato using a binary Agrobacterium tumefaciens vector. Biotechnology 5: 726-730. doi: 10.1038/nbt0787-726
    [32] Mintz-Oron S, Mandel T, Rogachev I, et al. (2008) Gene expression and metabolism in tomato fruit surface tissues. Plant Physiology 147: 823-851. doi: 10.1104/pp.108.116004
    [33] Smith CA, Want EJ, O'Maille G, et al. (2006) XCMS: processing mass spectrometry data for metabolite profiling using nonlinear peak alignment, matching, and identification. Anal Chem 78: 779-787. doi: 10.1021/ac051437y
    [34] Saeed AI, Sharov V, White J, et al. (2003) TM4: a free, open-source system for microarray data management and analysis. Biotechniques 34: 374-378.
    [35] Scholz M, Gatzek S, Sterling A, et al. (2004) Metabolite fingerprinting: detecting biological features by independent component analysis. Bioinformatics 20: 2447-2454. doi: 10.1093/bioinformatics/bth270
    [36] Spitzer-Rimon B, Marhevka E, Barkai O, et al. (2010) EOBII, a Gene Encoding a Flower-Specific Regulator of Phenylpropanoid Volatiles' Biosynthesis in Petunia. Plant Cell 22: 1961-1976. doi: 10.1105/tpc.109.067280
    [37] Davidovich-Rikanati R, Sitrit Y, Tadmor Y, et al. (2007) Enrichment of tomato flavor by diversion of the early plastidial terpenoid pathway. Nat Biotechnol 25: 899-901. doi: 10.1038/nbt1312
    [38] Hu C, Jiang P, Xu J, et al. (2003) Mutation analysis of the feedback inhibition site of phenylalanine-sensitive 3-deoxy-D-arabino-heptulosonate 7-phosphate synthase of Escherichia coli. J Basic Microb 43: 399-406. doi: 10.1002/jobm.200310244
    [39] Zhang S, Pohnert G, Kongsaeree P, et al. (1998) Chorismate mutase-prephenate dehydratase from Escherichia coli. Study of catalytic and regulatory domains using genetically engineered proteins. J Biol Chem 273: 6248-6253.
    [40] Carrari F, Baxter C, Usadel B, et al. (2006) Integrated analysis of metabolite and transcript levels reveals the metabolic shifts that underlie tomato fruit development and highlight regulatory aspects of metabolic network behavior. Plant Physiology 142: 1380-1396. doi: 10.1104/pp.106.088534
    [41] Osorio S, Vallarino JG, Szecowka M, et al. (2013) Alteration of the interconversion of pyruvate and malate in the plastid or cytosol of ripening tomato fruit invokes diverse consequences on sugar but similar effects on cellular organic acid, metabolism, and transitory starch accumulation. Plant Physiol 161: 628-643. doi: 10.1104/pp.112.211094
    [42] Rogachev I, Aharoni A (2012) UPLC-MS-based metabolite analysis in tomato. Methods Mol Biol (Clifton, N J ) 860: 129-144.
    [43] Baldwin E, Scott J, Einstein M, et al. (1998) Relationship between sensory and instrumental analysis for tomato flavor. J Am Soc Hortic Sci 906-915.
    [44] Dal Cin V, Tieman DM, Tohge T, et al. (2011) Identification of Genes in the Phenylalanine Metabolic Pathway by Ectopic Expression of a MYB Transcription Factor in Tomato Fruit. Plant Cell 23: 2738-2753. doi: 10.1105/tpc.111.086975
    [45] Kaminaga Y, Schnepp J, Peel G, et al. (2006) Plant phenylacetaldehyde synthase is a bifunctional homotetrameric enzyme that catalyzes phenylalanine decarboxylation and oxidation. J Biol Chem 281: 23357-23366. doi: 10.1074/jbc.M602708200
    [46] Zvi MMB, Shklarman E, Masci T, et al. (2012) PAP1 transcription factor enhances production of phenylpropanoid and terpenoid scent compounds in rose flowers. New Phytol 195: 335-345. doi: 10.1111/j.1469-8137.2012.04161.x
    [47] Baldwin EA, Scott JW, Shewmaker CK, et al. (2000) Flavor trivia and tomato aroma: Biochemistry and possible mechanisms for control of important aroma components. Hortscience 35: 1013-1022.
  • This article has been cited by:

    1. Broderick Oluyede, Thatayaone Moakofi, The Gamma-Topp-Leone-Type II-Exponentiated Half Logistic-G Family of Distributions with Applications, 2023, 6, 2571-905X, 706, 10.3390/stats6020045
    2. Guang Lu, Osama Abdulaziz Alamri, Badr Alnssyan, Mohammed A. Alshahrani, A new probabilistic model: Its implementations to time duration and injury rates in physical training, sports, and reliability sector, 2024, 108, 11100168, 839, 10.1016/j.aej.2024.09.049
    3. Cicero Eduardo Walter, Manuel Au-Yong-Oliveira, Marcos Ferasso, Measuring the Importance of Innovation in Portuguese Economic Development, 2024, 1868-7873, 10.1007/s13132-024-02446-2
    4. Liangyu Li, Jing Yang, Lip Yee Por, Mohammad Shahbaz Khan, Rim Hamdaoui, Lal Hussain, Zahoor Iqbal, Ionela Magdalena Rotaru, Dan Dobrotă, Moutaz Aldrdery, Abdulfattah Omar, Enhancing lung cancer detection through hybrid features and machine learning hyperparameters optimization techniques, 2024, 10, 24058440, e26192, 10.1016/j.heliyon.2024.e26192
    5. Shiv Kumar Sharma, Abhishek Thakur, 2024, Software Reliability Growth Modeling Based on Generalized Lindley Distribution, 979-8-3503-7523-7, 1, 10.1109/ISCS61804.2024.10581215
    6. Sanaa Al-Marzouki, Afaf Alrashidi, Christophe Chesneau, Mohammed Elgarhy, Rana H. Khashab, Suleman Nasiru, On improved fitting using a new probability distribution and artificial neural network: Application, 2023, 13, 2158-3226, 10.1063/5.0176715
    7. Shahid Akbar, Ali Raza, Tamara Al Shloul, Ashfaq Ahmad, Aamir Saeed, Yazeed Yasin Ghadi, Orken Mamyrbayev, Elsayed Tag-Eldin, pAtbP-EnC: Identifying Anti-Tubercular Peptides Using Multi-Feature Representation and Genetic Algorithm-Based Deep Ensemble Model, 2023, 11, 2169-3536, 137099, 10.1109/ACCESS.2023.3321100
    8. Hualong Zhong, Yuanjun Xue, Tmader Alballa, Wafa F. Alfwzan, Somayah Abdualziz Alhabeeb, Hamiden Abd El-Wahed Khalifa, A new probabilistic model with properties and Monte Carlo simulation: Its explorations in dance education and music engineering, 2025, 112, 11100168, 461, 10.1016/j.aej.2024.10.095
    9. Hang Yuan, A new statistical model with simulation study and reliability analysis of an audio and podcast tool, 2025, 116, 11100168, 548, 10.1016/j.aej.2024.12.044
    10. Yaxi Li, Haihua Fang, Ningjun Wang, A new probabilistic approach with properties and statistical evaluation using the teacher’s rating and reliability data, 2025, 117, 11100168, 451, 10.1016/j.aej.2024.12.093
    11. Xu Li, Peng Xue, The role of social work in enhancing social governance: Policy, law, practice, and integration of machine learning for improved outcomes, 2025, 118, 11100168, 208, 10.1016/j.aej.2025.01.002
  • Reader Comments
  • © 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(8164) PDF downloads(1377) Cited by(15)

Figures and Tables

Figures(6)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog