Research article

ARU-DGAN: A dual generative adversarial network based on attention residual U-Net for magneto-acousto-electrical image denoising


  • Received: 28 August 2023 Revised: 08 October 2023 Accepted: 16 October 2023 Published: 26 October 2023
  • Magneto-Acousto-Electrical Tomography (MAET) is a multi-physics coupling imaging modality that integrates the high resolution of ultrasound imaging with the high contrast of electrical impedance imaging. However, the quality of images obtained through this imaging technique can be easily compromised by environmental or experimental noise, thereby affecting the overall quality of the imaging results. Existing methods for magneto-acousto-electrical image denoising lack the capability to model local and global features of magneto-acousto-electrical images and are unable to extract the most relevant multi-scale contextual information to model the joint distribution of clean images and noise images. To address this issue, we propose a Dual Generative Adversarial Network based on Attention Residual U-Net (ARU-DGAN) for magneto-acousto-electrical image denoising. Specifically, our model approximates the joint distribution of magneto-acousto-electrical clean and noisy images from two perspectives: noise removal and noise generation. First, it transforms noisy images into clean ones through a denoiser; second, it converts clean images into noisy ones via a generator. Simultaneously, we design an Attention Residual U-Net (ARU) to serve as the backbone of the denoiser and generator in the Dual Generative Adversarial Network (DGAN). The ARU network adopts a residual mechanism and introduces a linear Self-Attention based on Cross-Normalization (CNorm-SA), which is proposed in this paper. This design allows the model to effectively extract the most relevant multi-scale contextual information while maintaining high resolution, thereby better modeling the local and global features of magneto-acousto-electrical images. Finally, extensive experiments on a real-world magneto-acousto-electrical image dataset constructed in this paper demonstrate significant improvements in preserving image details achieved by ARU-DGAN. Furthermore, compared to the state-of-the-art competitive methods, it exhibits a 0.3 dB increase in PSNR and an improvement of 0.47% in SSIM.

    Citation: Shuaiyu Bu, Yuanyuan Li, Wenting Ren, Guoqiang Liu. ARU-DGAN: A dual generative adversarial network based on attention residual U-Net for magneto-acousto-electrical image denoising[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19661-19685. doi: 10.3934/mbe.2023871

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  • Magneto-Acousto-Electrical Tomography (MAET) is a multi-physics coupling imaging modality that integrates the high resolution of ultrasound imaging with the high contrast of electrical impedance imaging. However, the quality of images obtained through this imaging technique can be easily compromised by environmental or experimental noise, thereby affecting the overall quality of the imaging results. Existing methods for magneto-acousto-electrical image denoising lack the capability to model local and global features of magneto-acousto-electrical images and are unable to extract the most relevant multi-scale contextual information to model the joint distribution of clean images and noise images. To address this issue, we propose a Dual Generative Adversarial Network based on Attention Residual U-Net (ARU-DGAN) for magneto-acousto-electrical image denoising. Specifically, our model approximates the joint distribution of magneto-acousto-electrical clean and noisy images from two perspectives: noise removal and noise generation. First, it transforms noisy images into clean ones through a denoiser; second, it converts clean images into noisy ones via a generator. Simultaneously, we design an Attention Residual U-Net (ARU) to serve as the backbone of the denoiser and generator in the Dual Generative Adversarial Network (DGAN). The ARU network adopts a residual mechanism and introduces a linear Self-Attention based on Cross-Normalization (CNorm-SA), which is proposed in this paper. This design allows the model to effectively extract the most relevant multi-scale contextual information while maintaining high resolution, thereby better modeling the local and global features of magneto-acousto-electrical images. Finally, extensive experiments on a real-world magneto-acousto-electrical image dataset constructed in this paper demonstrate significant improvements in preserving image details achieved by ARU-DGAN. Furthermore, compared to the state-of-the-art competitive methods, it exhibits a 0.3 dB increase in PSNR and an improvement of 0.47% in SSIM.



    Weighted distributions (WDs) provide an approach to deal with model specification and data interpretations problems. They adjust the probabilities of the actual occurrence of events to arrive at a specification of the probabilities when those events are recorded. Reference [1] extended the basic ideas of the methods of ascertainment upon the estimation of frequencies in [2]. The author defined a unifying concept of the WDs and described several sample conditions that the WDs can model. The usefulness and applications of the WDs in various areas, including medicine, ecology, reliability, and branching processes, can also be seen in [3,4,5]. Important findings on the WDs have been reported by several research. For examples, reference [6] suggested a weighted x-gamma distribution, reference [7] derived a new generalized weighted Weibull distribution, reference [8] introduced the weighted exponential-Gompertz distribution, reference [9] studied the new weighted inverse Rayleigh distribution, reference [10] introduced a weighted version of the generalized inverse Weibull distribution, reference [11] proposed a bounded weighted exponential distribution, reference [12] derived a new weighted exponential distribution, reference [13] proposed a weighted power Lomax distribution, reference [14] derived a new generalized weighted exponential distribution, reference [15] introduced a new version of the weighted Weibull distribution, reference [16] proposed the modified weighted exponential distribution, and reference [17] proposed a weighted Nwikpe distribution, reference [18] introduced a new version of the double weighted quasi Lindley distribution and reference [19] proposed the modified length-biased weighted Lomax distribution.

    In contrast, statistical models have the capacity to depict and predict real-world phenomena. Over the past few decades, numerous extended distributions have been extensively utilized in data modeling. Recent progress has been centered on the development of novel distribution families that not only enhance existing distributions but also offer significant versatility in practical data modeling. Engineering, economics, biology, and environmental science are particular examples of this. Regarding this, a number of writers suggested some of the created families of continuous distributions, (see for example [20,21,22]). Our interest here is in the same scheme used for the beta-G (B-G) family prepared in [23]. The following is the cumulative distribution function (cdf) for the B-G family:

    F(x)=G(x)0r(t)dt, (1.1)

    where G(x) is a cdf of a continuous distribution and r(t) is the probability density function (pdf) of the beta distribution. Naturally, any new family can be created by taking another pdf for r(t) with support [0,1] (see reference [23]).

    As a matter of fact, few works about the weighted-G family have been proposed in the literature. For example, reference [24] studied the weighted exponential-G family, reference [25] introduced the weighted exponentiated family, reference [26] proposed a weighted general family, and reference [27] developed a weighted Topp-Leone-G family.

    The primary purpose of this study is to introduce the length-biased truncated Lomax-G (LBTLo-G) family. The following arguments give enough motivation to study it:

    1) The LBTLo-G family is very flexible and simple.

    2) The LBTLo-G family contains some new distributions.

    3) The shapes of the pdfs of the generated distributions can be unimodal, decreasing, bathtub, right-skewed, and symmetric. Also, the hazard rate function (hrf) shapes for these distributions can be increasing, decreasing, U-shaped, upside-down-shaped, or J- shaped.

    After emphasizing these important aspects, some statistical and mathematical properties of the newly suggested family are discussed. The maximum likelihood (ML) method of estimation is used to estimate the LBTLo Weibull (LBTLoW) model parameters based on complete and type Ⅱ censoring (T2C).

    The variability of the LBTLoW distribution is demonstrated through four authentic data sets. The first data set describes age data on rest times (in minutes) for analgesic patients. The second data set shows the percentage of natural gas reserves in 44 countries in 2020. The third authentic data set listed the top 20 countries by oil reserves. Proven reserves refer to the quantities of petroleum that can be predicted as commercially recoverable from known reservoirs, based on the analysis of geological and engineering data. These estimates are made considering existing economic conditions and are projected from a specific period onwards. The fourth data set displays the top 100 central banks in terms of gold reserves. This gold reserve data, collected from IMF IFS figures, tracks central banks' reported gold purchases and sales as a percentage of their international reserves. The application results show that the LBTLoW distribution can indeed match the data better than other competing distributions.

    The following is the structure for this article: Section 2 defines the crucial functions of the LBTLo-G family and provides four special distributions of the family. In Section 3, some statistical properties of the LBTLo-G family are provided. Section 4 deals with the ML estimates (MLEs) of the unknown parameters. A simulation study to examine the theoretical performance of MLEs for the LBTLoW distribution is studied in Section 5. Section 6 presents the applicability and goodness of fit of the proposed models using four real data sets. The paper ends with a few last observations, as may be seen in Section 7.

    Here, we suggest a new weighted family based on the weighted version of the truncated Lomax distribution, which is called the LBTLo distribution [28]. The cdf and pdf of the LBTLo distribution are, respectively, given by

    G(t;α)=Λ(α)[(1+t)α(1+αt)1],0<t<1,α>0, (2.1)
    g(t;α)=α(1α)Λ(α)t(1+t)(α+1), (2.2)

    where Λ(α)=[2α(1+α)1]1. For these functions, it is assumed the standard complementary values for t0 and t1.

    As mentioned in [28], the following advantages of the LBTLo distribution are outlined: (ⅰ) It depends on only one parameter; (ⅱ) the pdf has only one maximum point with a relatively sharp peak and a heavy tail; (ⅲ) the hrf has increasing behavior or is N-shaped; and (ⅳ) it outperforms some other competing models in real-world applications to medical data and the percentage of household spending on education out of total household expenditure from the Household Income, Expenditure, and Consumption Survey data for North Sinai Governorate.

    In light of these merits, the LBTLo distribution is a great choice to use in various fields. As a consequence, we present a novel generated family that is based on the LBTLo distribution. In order to define the LBTLo-G family, let G(x;ζ) and g(x;ζ) be the baseline cdf and pdf, respectively, of a continuous distribution, and ζ is a vector of parameters. The generalized B-G generator specified in (1.1) and the LBTLo distribution (2.2) are combined to generate the cdf of the LBTLo-G family:

    F(x;α,ζ)=α(1α)Λ(α)G(x;ζ)0t(1+t)α1dt=Λ(α)[(1+G(x;ζ))α(1+αG(x;ζ))1],xR,α>0, (2.3)

    where α is a shape parameter. Therefore, the pdf of the LBTLo-G family is given by

    f(x;α,ζ)=α(1α)Λ(α)g(x;ζ)G(x;ζ)(1+G(x;ζ))α1,xR,α>0. (2.4)

    A random variable X with the pdf (2.4) is designated as X LBTLo-G from here on out. The complementary cdf (ccdf), and hrf, are, provided by

    S(x;α,ζ)=1Λ(α)[(1+G(x;ζ))α(1+αG(x;ζ))1],
    h(x;α,ζ)=α(1α)Λ(α)g(x)G(x)(1+G(x))α11Λ(α)[(1+G(x;ζ))α(1+αG(x;ζ))1].

    We create four new LBTLo-G family sub-distributions in the subsections that follow: LBTLo-inverse exponential, LBTLo-uniform, LBTLo-Weibull, and LBTLo-Kumaraswamy distributions.

    The cdf and pdf of the LBTLo-inverse exponential (LBTLoIE) distribution are obtained from (2.3) and (2.4) for G(x;β)=e(β/x),β,x>0, as follows:

    F(x;α,β)=Λ(α)[(1+e(β/x))α(1+αe(β/x))1],x>0,α,β>0,
    f(x;α,β)=α(1α)Λ(α)βx2e2(β/x)(1+e(β/x))α1.

    Further, the hrf is as follows:

    h(x;α,β)=α(1α)Λ(α)βx2e2(β/x)(1+e(β/x))α11Λ(α)[(1+e(β/x))α(1+αe(β/x))1].

    The cdf and pdf of the LBTLo-uniform (LBTLoU) distribution are derived from (2.3) and (2.4) by taking G(x;β)=β1x,0<x<β, as follows:

    F(x;α,β)=[(1+αβ1x)(1+β1x)α1]Λ(α),0<x<β,α,β>0,
    f(x;α,β)=αβ2x(1+β1x)α1(1α)Λ(α).

    Further, the hrf is as follows:

    h(x;α,β)=αβ2x(1+β1x)α1(1α)Λ(α)1Λ(α)[(1+αβ1x)(1+β1x)α1].

    The cdf and pdf of the LBTLoW distribution are derived from (2.3) and (2.4) taking G(x;β,γ)=1eβxγ,x,β,γ>0, as follows:

    F(x;α,β,γ)=[(2eβxγ)α(1+ααeβxγ)1]Λ(α),x>0,α,β,γ>0, (2.5)
    f(x;α,β,γ)=αβγ(1α)xγ1eβxγ(1eβxγ)(2eβxγ)α1Λ(α). (2.6)

    Further, the hrf is:

    h(x;α,β,γ)=αβγ(1α)xγ1eβxγ(1eβxγ)(2eβxγ)α1Λ(α)1Λ(α)[(2eβxγ)α(1+ααeβxγ)1].

    The cdf and pdf of the LBTLo- Kumaraswamy (LBTLoKw) distribution are obtained from (2.3) and (2.4) by taking G(x;μ,b)=1(1xμ)b,0<x<1,b,μ>0, as follows:

    F(x;α,μ,b)=[(2(1xμ)b)α(1+αα(1xμ)b)1]Λ(α),0<x<1,α,μ,b>0,
    f(x;α,μ,b)=αμb(1α)xμ1(1xμ)b1(1(1xμ)b)×(2(1xμ)b)α1Λ(α).

    Further, the hrf is as follows:

    h(x;α,μ,b)=αμb(1α)xμ1(1xμ)b1(1(1xμ)b)Λ(α)(2(1xμ)b)α11Λ(α)[(2(1xμ)b)α(1+αα(1xμ)b)1].

    The plots of pdf and hrf for the LBTLoIE, LBTLoU, LBTLoW and LBTLoKw distributions are given in Figures 1 and 2, respectively.

    Figure 1.  The pdfs of the (a) LBTLoIE, (b) LBTLoU, (c) LBTLoW, and (d) LBTLoKw distributions.
    Figure 2.  The hrfs of the (a) LBTLoIE, (b) LBTLoU, (c) LBTLoW, and (d) LBTLoKw distributions.

    The pdfs of the investigated distributions can have a variety of forms, including right- and left-skewed, bathtub, uni-modal, declining, and symmetric shapes, as shown in Figure 1. The corresponding hrf can take any form, including U, J, reverse J, growing, or decreasing, as seen in Figure 1.

    In this part, we give some statistical properties of the LBTLo-G family.

    The LBTLo-G family representations in pdf and cdf format are displayed here. The generalized binomial theorem says that

    (1+z)β=i=0(1)i(β+i1i)zi, (3.1)

    for |z|<1. Hence, by using (3.1) in (2.4), the pdf of the LBTLo-G family can be written as follows:

    f(x;α,ζ)=i=0ϑig(x;ζ)G(x;ζ)i+1,xR, (3.2)

    where ϑi=(1)iα(1α)Λ(α)(α+ii). For example, the expansion of pdf of the LBTLoW distribution is derived from (3.2) as follows:

    f(x;α,β,γ)=βγi=0ϑixγ1eβxγ(1eβxγ)i+1,x>0,α,β,γ>0. (3.3)

    But, in the special case where b is a positive integer, the standard generalized binomial theorem says that

    (1z)b=bν=0(1)ν(bν)zν. (3.4)

    Then using the binomial expansion (3.4) in (3.3), we get

    f(x;α,β,γ)=i=0i+1ν=0ϑi,νxγ1eβ(ν+1)xγ, (3.5)

    where ϑi,ν=βγϑi(1)ν(i+1ν). In what follows, an expansion for F(x;α,ζ)his derived, for h is an integer, again, the exponential and the binomial expansions are worked out:

    F(x;α,ζ)h=Λ(α)h[(1+G(x;ζ))α(1+αG(x;ζ))1]h. (3.6)

    Using the binomial expansion (3.4) in (3.6), we get

    F(x;α,ζ)h=Λ(α)hhj=0(1)hj(hj)(1+G(x;ζ))αj(1+αG(x;ζ))j. (3.7)

    Using the binomial expansion (3.1), we obtain

    F(x;α,ζ)h=Λ(α)hd=0hj=0(1)d+hj(hj)(αj+d1d)×G(x;ζ)d(1+αG(x;ζ))j. (3.8)

    By using (3.4) in (3.8), we obtain

    F(x;α,ζ)h=d=0ϖd,j,mG(x;ζ)d+m, (3.9)

    where ϖd,j,m=hj=0jm=0(1)d+m+hjαm(hj)(jm)(αj+d1d)Λ(α)h.

    For example, the expansion of the cdf of the LBTLoW distribution is derived from (3.9), where G(x,ζ)=1eβxγ, as follows:

    F(x;α,β,γ)h=d=0ϖd,j,m(1eβxγ)d+m.

    By using the binomial expansion (3.4) in the last term of the previous equation, we get

    F(x;α,β,γ)h=d=0d+ml=0(1)l(d+ml)ϖd,j,meβlxγ. (3.10)

    The above representations are of interest to express various important moment measures as series. By truncating the index of summation, we can have a precise approximation with a reasonable computation cost.

    As a special class of moments, the probability weighted moments (PWMs) have been proposed in [29]. This class is used to derive estimates of the parameters and quantiles of distributions expressible in inverse form. Let X be a random variable with pdf and cdf f(x) and F(x), respectively, and r and q be non-negative integers. Then, the (r,q)th PWM of X, denoted by πr,q, can be calculated through the following relation:

    πr,q=E[XrF(X)q]=xrf(x)F(x)qdx. (3.11)

    On this basis, the (r,q)th PMW of X with pdf and cdf of the LBTLo-G family is obtained by substituting (3.2) and (3.9) into (3.11), as follows:

    πr,q=E[XrF(X;α,ζ)q]=i,d=0ϑiϖd,j,mxrg(x;ζ)[G(x;ζ)]i+d+m+1dx.

    Then, provided that the interchange of the integral and sum is valid, depending on the definitions of g(x;ζ) and G(x;ζ), we have

    πr,q=i,d=0ϑiϖd,j,mρr,i+d+m+1,

    where

    ρr,i+d+m+1=xrg(x;ζ)[G(x;ζ)]i+d+m+1dx.

    For example, the (r,q)th of a random variable X that follows the LBTLoW distribution can be obtained by substituting (3.5) and (3.10) into (3.11), and replacing h with q. We thus obtain

    πr,q=i,d=0i+1v=0d+ml=0(1)lϑi,vϖd,j,m(β(ν+l+1))rγ+1(d+ml)Γ(rγ+1),

    where Γ(.) stands for gamma function.

    In this part, for any non-negative integer r, the rth moment associated with the LBTLo-G family is derived.

    Let X be a random variable having the pdf of the LBTLo-G family. Then, the rth moment of X is obtained as follows:

    μr=E(Xr)=i=0ϑixrg(x;ζ)[G(x;ζ)]i+1dx=i=0ϑiυr,i+1,

    where υr,i+1 is the (r,i+1)th PWM of the baseline distribution. For example, after some developments, the rth moment associated with LBTLoW distribution is given by

    μr=i=0i+1ν=0ϑi,ν(β(ν+1))rγ+1Γ(rγ+1).

    Tables 13 show the numerical values of the first four moments μ1, μ2, μ3, μ4, also the numerical values of variance (σ2), coefficient of skewness (CS), coefficient of kurtosis (CK) and coefficient of variation (CV) associated with the LBTLoW and LBTLoIE distribution.

    Table 1.  Results of some moments, σ2, CS, CK, and CV associated with the LBTLo-W distribution at β=1.8.
    γ α μ1 μ2 μ3 μ4 σ2 CS CK CV
    0.4 0.2 2.629 8.323 30.531 126.387 1.413 0.729 3.596 0.452
    0.6 0.5 1.558 2.937 6.430 15.922 0.508 0.746 3.627 0.457
    0.8 0.8 1.184 1.700 2.847 5.403 0.300 0.765 3.660 0.463
    1.1 1.2 0.925 1.044 1.380 2.074 0.189 0.793 3.717 0.470
    1.3 1.5 0.805 0.794 0.920 1.216 0.146 0.814 3.758 0.475
    1.7 1.8 0.705 0.615 0.634 0.748 0.117 0.856 3.853 0.486
    1.9 2.0 0.655 0.532 0.513 0.568 0.103 0.878 3.906 0.491
    2.4 2.3 0.582 0.425 0.371 0.374 0.086 0.937 4.058 0.503
    2.7 2.6 0.530 0.355 0.285 0.266 0.073 0.974 4.162 0.511
    3.2 3.0 0.469 0.280 0.203 0.172 0.060 1.038 4.359 0.522

     | Show Table
    DownLoad: CSV
    Table 2.  Results of some moments, σ2, CS, CK, and CV associated with the LBTLo-W distribution at β=2.5.
    γ α μ1 μ2 μ3 μ4 σ2 CS CK CV
    0.4 0.2 1.964 4.277 10.154 25.945 0.420 0.380 3.017 0.330
    0.6 0.5 1.347 2.017 3.298 5.813 0.202 0.395 3.026 0.334
    0.8 0.8 1.105 1.360 1.829 2.658 0.139 0.411 3.037 0.337
    1.1 1.2 0.924 0.954 1.081 1.325 0.100 0.435 3.058 0.343
    1.3 1.5 0.836 0.783 0.805 0.897 0.084 0.453 3.075 0.346
    1.7 1.8 0.760 0.649 0.612 0.627 0.072 0.489 3.116 0.353
    1.9 2.0 0.720 0.584 0.524 0.511 0.066 0.508 3.140 0.356
    2.4 2.3 0.661 0.494 0.411 0.374 0.058 0.557 3.213 0.365
    2.7 2.6 0.617 0.433 0.339 0.290 0.052 0.588 3.265 0.369
    3.2 3.0 0.565 0.364 0.263 0.209 0.045 0.641 3.366 0.377

     | Show Table
    DownLoad: CSV
    Table 3.  Results of some moments, σ2, CS, CK, and CV associated with the LBTLoIE distribution.
    β α μ1 μ2 μ3 μ4 σ2 CS CK CV
    1.5 0.2 0.029 0.026 0.023 0.021 0.025 5.273 29.151 5.386
    0.5 0.032 0.028 0.025 0.023 0.027 5.014 26.463 5.138
    0.8 0.035 0.031 0.028 0.025 0.030 4.772 24.069 4.906
    1.2 0.039 0.035 0.031 0.028 0.033 4.472 21.275 4.622
    1.5 0.043 0.038 0.033 0.03 0.036 4.264 19.441 4.424
    1.8 0.046 0.041 0.036 0.033 0.038 4.069 17.801 4.241
    2 0.049 0.043 0.038 0.034 0.040 3.946 16.806 4.125
    2.3 0.053 0.046 0.041 0.037 0.043 3.771 15.445 3.961
    2.6 0.056 0.049 0.044 0.039 0.046 3.607 14.225 3.809
    3 0.062 0.054 0.048 0.043 0.050 3.405 12.794 3.621
    2.5 0.2 0.006 0.005 0.005 0.004 0.005 12.420 156.328 12.403
    0.5 0.007 0.006 0.006 0.005 0.006 11.672 138.195 11.665
    0.8 0.007 0.007 0.006 0.006 0.007 10.981 122.434 10.984
    1.2 0.009 0.008 0.007 0.007 0.008 10.140 104.548 10.156
    1.5 0.01 0.009 0.008 0.007 0.009 9.563 93.121 9.589
    1.8 0.011 0.010 0.009 0.008 0.010 9.030 83.140 9.066
    2 0.012 0.011 0.010 0.009 0.010 8.697 77.190 8.739
    2.3 0.013 0.012 0.011 0.010 0.012 8.227 69.194 8.279
    2.6 0.014 0.013 0.012 0.011 0.013 7.792 62.179 7.853
    3 0.017 0.015 0.014 0.013 0.015 7.261 54.128 7.334

     | Show Table
    DownLoad: CSV

    It can be seen from Tables 13 that, when the value of α,γ increases for a fixed value of β, the first four moments and σ2 decrease, while the CS, CK, and CV measures increase. When the value of β increases for a fixed value of α and γ, we observe that the first four moments and σ decrease and then increase, while the CS, CK, and CV measures increase. The LBTLoW distribution is skewed to the right by leptokurtic curves.

    Furthermore, if X is a random variable having the pdf of the LBTLo-G family, then the rth incomplete moment of X is obtained as follows:

    φr(t)=E(XrI{Xt})=txrf(x;α,ζ)dx=ti=0ϑixrg(x;ζ)G(x;ζ)i+1dx.

    For example, after some developments, the rth incomplete moment associated with the LBTLoW distribution is given by

    φr(t)=i=0i+1ν=0ϑi,ν[β(ν+1)]rγ+1Γ(rγ+1,β(ν+1)tγ),

    where Γ(.,x) is the lower incomplete gamma function.

    Here, some uncertainty measures of the LBTLo-G family are derived. Then, these measures are specialized to the LBTLoW distribution. To begin, the Rényi entropy (RE), presented in [30], associated with a distribution with pdf f(x), is defined by

    IR(ε)=11εlog[f(x)εdx],ε1,ε>0.

    A numerical study with integral calculus is possible; here, we focus on a series expansion. In what follows, an expansion for f(x;α,ζ)ε is derived, for ε is a non-integer (again, the generalized binomial expansion is worked out):

    f(x;α,ζ)ε=i=0Δig(x;ζ)εG(x;ζ)i+ε,

    where

    Δi=(1)i[α(1α)Λ(α)]ε(ε(α+1)+i1i).

    Then, the RE associated with the LBTLo-G family is given by

    IR(ε)=(1ε)1log{i=0Δig(x;ζ)εG(x;ζ)i+εdx}.

    For example, the RE associated with the LBTLoW distribution can be obtained as follows:

    IR(ε)=(1ε)1log{i,j=0Δi,jγ[β(ε+j)]ε+(ε/γ)(1/γ)Γ(εεγ+1γ)}.

    The Havrda and Charvát entropy (HaCE) (see [31]) associated with a distribution with pdf f(x) is defined by

    HCR(ε)=121ε1({f(x)εdx}1/ε1),ε1,ε>0.

    Hence, the HaCE of the LBTLo-G family is given by

    HCR(ε)=121ε1({i=0Δig(x;ζ)εG(x;ζ)i+εdx}1/ε1).

    For example, the HaCE of the LBTLoW distribution can be obtained as follows:

    HCR(ε)=121ε1({i,j=0Δi,jγ[β(ε+j)]ε+(ε/γ)(1/γ)Γ(εεγ+1γ)}1/ε1).

    The Arimoto entropy (ArE) (see [32]) associated with a distribution with pdf f(x) is defined by

    AR(ε)=ε1ε({f(x)εdx}1/ε1),ε1,ε>0.

    Hence, the ArE of the LBTLo-G family is given by

    AR(ε)=ε1ε({i=0Δig(x;ζ)εG(x;ζ)i+εdx}1/ε1).

    For example, the ArE of the LBTLoW distribution can be obtained as follows:

    AR(ε)=ε1ε({i,j=0Δi,jγ[β(ε+j)]ε+(ε/γ)(1/γ)Γ(εεγ+1γ)}1/ε1).

    The Tsallis entropy (TsE) (see [33]) associated with a distribution with pdf f(x), is defined by

    TR(ε)=1ε1{1f(x)εdx},ε1,ε>0.

    Hence, the TsE of the LBTLo-G family is obtained as follows:

    TR(ε)=1ε1{1i=0Δig(x;ζ)εG(x;ζ)i+εdx}.

    For example, the TsE of the LBTLoW distribution can be obtained as follows:

    TR(ε)=1ε1{1i,j=0Δi,jγ[β(ε+j)]ε+(ε/γ)(1/γ)Γ(εεγ+1γ)}.

    Some numerical values for the proposed entropy measures are obtained for the LBTLoW and LBTLoIE distribution in Tables 4 and 5.

    Table 4.  Numerical values of entropy measures of the LBToW distribution.
    ε β α γ RE HaCE ArE TsE
    1.5 0.25 0.2 0.4 3.331 3.340 2.767 1.957
    0.5 0.6 3.252 3.333 2.753 1.953
    0.8 0.8 3.099 3.318 2.722 1.944
    1.2 1.1 2.875 3.290 2.670 1.927
    1.5 1.3 2.716 3.264 2.627 1.912
    1.8 1.7 2.524 3.227 2.568 1.891
    2.0 1.9 2.415 3.203 2.53 1.876
    2.3 2.4 2.229 3.152 2.458 1.846
    2.6 2.7 2.084 3.104 2.394 1.818
    3.0 3.2 1.889 3.026 2.296 1.773
    0.5 0.2 0.4 1.930 3.044 2.318 1.783
    0.5 0.6 1.924 3.042 2.315 1.782
    0.8 0.8 1.824 2.996 2.260 1.755
    1.2 1.1 1.660 2.909 2.161 1.704
    1.5 1.3 1.541 2.835 2.081 1.661
    1.8 1.7 1.392 2.726 1.969 1.597
    2.0 1.9 1.308 2.657 1.901 1.556
    2.3 2.4 1.161 2.517 1.770 1.475
    2.6 2.7 1.051 2.397 1.661 1.404
    3.0 3.2 0.903 2.207 1.500 1.293
    2.0 0.25 0.2 0.4 2.180 1.987 1.837 0.993
    0.5 0.6 2.167 1.986 1.835 0.993
    0.8 0.8 2.053 1.982 1.812 0.991
    1.2 1.1 1.876 1.973 1.769 0.987
    1.5 1.3 1.753 1.965 1.734 0.982
    1.8 1.7 1.595 1.949 1.681 0.975
    2.0 1.9 1.510 1.938 1.648 0.969
    2.3 2.4 1.358 1.912 1.581 0.956
    2.6 2.7 1.247 1.887 1.524 0.943
    3.0 3.2 1.098 1.840 1.435 0.920
    0.5 0.2 0.4 1.210 1.877 1.503 0.938
    0.5 0.6 1.279 1.895 1.541 0.947
    0.8 0.8 1.210 1.877 1.503 0.938
    1.2 1.1 1.080 1.834 1.423 0.917
    1.5 1.3 0.987 1.794 1.358 0.897
    1.8 1.7 0.861 1.725 1.258 0.862
    2.0 1.9 0.795 1.679 1.199 0.840
    2.3 2.4 0.672 1.574 1.077 0.787
    2.6 2.7 0.587 1.482 0.982 0.741
    3.0 3.2 0.470 1.323 0.836 0.661

     | Show Table
    DownLoad: CSV
    Table 5.  Numerical values of entropy measures of the LBToIE distribution.
    ε β α RE HaCE ArE TsE
    1.5 0.25 0.2 7.001 3.311 2.709 1.94
    0.5 7.075 3.315 2.716 1.942
    0.8 7.156 3.319 2.724 1.944
    1.2 7.274 3.324 2.735 1.947
    1.5 7.372 3.329 2.743 1.95
    1.8 7.476 3.333 2.752 1.952
    2 7.55 3.336 2.758 1.954
    2.3 7.667 3.34 2.767 1.957
    2.6 7.792 3.345 2.777 1.959
    3 7.971 3.351 2.79 1.963
    0.4 0.2 6.441 3.278 2.65 1.92
    0.5 6.452 3.279 2.651 1.921
    0.8 6.469 3.28 2.653 1.921
    1.2 6.503 3.282 2.657 1.923
    1.5 6.536 3.284 2.66 1.924
    1.8 6.577 3.287 2.665 1.925
    2 6.608 3.289 2.668 1.927
    2.3 6.661 3.292 2.674 1.928
    2.6 6.721 3.296 2.681 1.931
    3 6.813 3.301 2.69 1.934
    2.0 0.25 0.2 4.376 1.975 1.776 0.987
    0.5 4.429 1.976 1.782 0.988
    0.8 4.487 1.977 1.788 0.989
    1.2 4.57 1.979 1.796 0.99
    1.5 4.639 1.981 1.803 0.99
    1.8 4.713 1.982 1.81 0.991
    2 4.765 1.983 1.815 0.991
    2.3 4.847 1.984 1.823 0.992
    2.6 4.934 1.986 1.83 0.993
    3 5.058 1.987 1.841 0.994
    0.4 0.2 3.975 1.962 1.726 0.981
    0.5 3.987 1.963 1.728 0.981
    0.8 4.003 1.963 1.73 0.982
    1.2 4.031 1.965 1.734 0.982
    1.5 4.058 1.965 1.737 0.983
    1.8 4.09 1.967 1.741 0.983
    2 4.114 1.967 1.744 0.984
    2.3 4.154 1.969 1.749 0.984
    2.6 4.199 1.97 1.755 0.985
    3 4.267 1.972 1.763 0.986

     | Show Table
    DownLoad: CSV

    We can see from these tables that, as the value of ε rises, all entropy values decrease, providing more information. For a fixed value of β, as the values of α and γ rise, we infer that all entropy metrics decrease, indicating that there is less fluctuation. Additionally, we deduce that all entropies have less variability as the values of α, γ and β increase. When compared to other measures, the TsE measure values typically have the smallest values.

    Let x(1)x(2)x(n) be a T2C of size r resulting from a life test on n items whose lifetimes are described by the LBTLo-G family with a given set of parameters α and ζ, see [34,35,36,37]. The log-likelihood function of r failures and (nr) censored values, is given by

    logL(α,ζ)=rlogα+rlog(1α)+rlogΛ(α)+ri=1logg(xi;ζ)+ri=1logG(xi;ζ)(α+1)ri=1log(1+G(xi;ζ))+(nr)log[Ar(α,ζ)],

    where Ar(α,ζ)=1Λ(α)[(1+G(xr;ζ))α(1+αG(xr;ζ))1], and we write x(i)=xi for simplified form.

    By maximizing the previous likelihood function, the MLEs of unknown parameters are determined. To achieve this, we can first compute the first derivative of the score function (Uα,Uζk), given as follows:

    Uα=rαr1α+rΛ(α)(αΛ(α))ri=1log(1+G(xi;ζ))+(nr)Ar(α,ζ)(αAr(α,ζ)),
    Uζk=(α+1)ri=111+G(xi;ζ)ζk(G(xi;ζ))+(nr)Ar(α,ζ)ζkAr(α,ζ),

    where

    αΛ(α)=[Λ(α)]22α[(1+α)log21],
    αAr(α,ζ)=αΛ(α)[(1+G(xr;ζ))α(1+αG(xr;ζ))1]+Λ(α)[(1+G(xr;ζ))α{(1+αG(xr;ζ))log(1+G(xr;ζ))G(xr;ζ)}],

    and

    ζk(Ar(α,ζ))=α(α1)Λ(α)G(xr;ζ)(1+G(xr;ζ))α1ζkG(xr;ζ).

    By putting Uα and Uζk equal to zero and solving these equations simultaneously, the MLEs of the LBTLo-G family are found. These equations are not amenable to analytical solution, however they are amenable to numerical solution by iterative techniques utilizing statistical software.

    The confidence interval (CI) of the vector of the unknown parameters ξ=(α,ζ) could be obtained from the asymptotic distribution of the MLEs of the parameters as (ˆξMLEξ)N2(0,I1(ˆξMLE)), where I(ξ) is the Fisher information matrix. Under particular regularity conditions, the two-sided 100(1v), asymptotic CI for the vector of unknown parameters ξ can be acquired in the following ways: ˆξMLE±zv/2var(ˆξ), where var(ˆξ) is the element of the main diagonal of the asymptotic variance-covariance matrix I1(ˆξMLE) and zv/2 is the upper vth/2 percentile of the standard normal distribution.

    This section includes a simulation study to evaluate the performance of the MLEs for the LBTLoW model (α,β,γ), for complete and T2C. The Mathematica 9 package is used to get the mean squared error (MSE), lower bound (LB) of CI, upper bound (UB) of CI, average length (AL) of 95%, and coverage probability (CP) of 95% of the estimated values of α, β and γ. The algorithm is developed in the way described below:

    1) From the LBTLoW distribution, 5000 random samples of sizes n = 50,100,150, and 200 are created.

    2) Values of the unknown parameters (α,β,γ) are selected as Set 1 =(α=0.5,β=0.5,γ=0.5), Set 2 =(α=0.7,β=0.5,γ=0.25), Set 3 =(α=0.7,β=0.7,γ=0.5), and Set 4 =(α=0.6,β=0.3,γ=0.5).

    3) Three levels of censorship are chosen: r = 70%, 80% (T2C), and 100% (complete sample).

    4) The MLEs, Biases, and MSEs for all sample sizes and for all selected sets of parameters are computed. Furthermore, the LB, UB, AL, and CP with a confidence level of 0.95 for all sample sizes and for all selected sets of parameters are calculated.

    5) Numerical outcomes are reported in Table 6. Based on complete and T2C samples, we can detect the following about the performance of the estimated parameters.

    Table 6.  Accuracy measures of the LBTLoW estimates under T2C and complete samples.
    n r Set1 (α = 0.5, β = 0.5, γ = 0.5)
    MLE Bias MSE LB UB AL CP
    50 70% α 0.4204 0.0796 0.0064 0.0019 0.839 0.8370 97.4%
    β 0.7041 0.2041 0.0471 0.5036 0.9046 0.4010 96.9%
    γ 0.4201 0.0799 0.0069 0.3023 0.5379 0.2356 96.0%
    80% α 0.4218 0.0782 0.0061 0.0191 0.8245 0.8053 94.8%
    β 0.6382 0.1382 0.0242 0.4508 0.8256 0.3748 95.8%
    γ 0.4386 0.0614 0.0053 0.3282 0.5490 0.2208 97.1%
    100% α 0.4234 0.0766 0.0059 0.0357 0.8111 0.7754 95.4%
    β 0.5177 0.0177 0.0056 0.3661 0.6694 0.3033 95.5%
    γ 0.5316 0.0316 0.0027 0.4303 0.6328 0.2025 96.0%
    100 70% α 0.4213 0.0787 0.0062 0.0844 0.7583 0.6740 96.2%
    β 0.6750 0.1750 0.0312 0.5375 0.8125 0.2750 95.9%
    γ 0.4237 0.0763 0.0065 0.3389 0.5084 0.1694 96.0%
    80% α 0.4230 0.0770 0.0061 0.2099 0.6360 0.4262 96.2%
    β 0.6099 0.1099 0.0127 0.4819 0.7379 0.2560 96.1%
    γ 0.4487 0.0513 0.0033 0.3652 0.5321 0.1669 97.3%
    100% α 0.4238 0.0762 0.0058 0.2501 0.5975 0.3473 95.6%
    β 0.4683 0.0317 0.0027 0.3558 0.5807 0.2249 95.8%
    γ 0.4967 0.0033 0.0025 0.4199 0.5734 0.1535 96.0%
    150 70% α 0.4217 0.0783 0.0061 0.2710 0.5725 0.3015 95.2%
    β 0.6626 0.1626 0.0281 0.5577 0.7675 0.2097 95.6%
    γ 0.4277 0.0723 0.0058 0.3571 0.4983 0.1412 97.3%
    80% α 0.4236 0.0764 0.0059 0.3005 0.5466 0.2461 95.7%
    β 0.5977 0.0977 0.0113 0.4957 0.6997 0.2040 96.2%
    γ 0.4649 0.0351 0.0022 0.3972 0.5325 0.1353 97.0%
    100% α 0.4238 0.0762 0.0058 0.3010 0.5467 0.2457 95.6%
    β 0.4766 0.0234 0.0023 0.3784 0.5749 0.1965 96.4%
    γ 0.5277 0.0277 0.0015 0.4659 0.5894 0.1236 96.9%
    200 70% α 0.4219 0.0781 0.0061 0.3154 0.5285 0.2132 96.1%
    β 0.6592 0.1592 0.0268 0.5675 0.7510 0.1835 96.3%
    γ 0.4375 0.0625 0.0046 0.3789 0.4962 0.1173 96.7%
    80% α 0.4239 0.0761 0.0058 0.3236 0.5242 0.2006 96.3%
    β 0.5912 0.0912 0.0099 0.5074 0.6750 0.1676 97.0%
    γ 0.4667 0.0333 0.0020 0.4101 0.5233 0.1132 97.5%
    100% α 0.4240 0.0760 0.0058 0.3372 0.5109 0.1737 96.5%
    β 0.4905 0.0095 0.0009 0.4167 0.5642 0.1475 96.7%
    γ 0.5035 0.0035 0.0006 0.4524 0.5546 0.1022 97.1%
    n r Set2 (α = 0.7, β = 0.5, γ = 0.25)
    MLE Bias MSE LB UB AL CP
    50 70% α 0.4206 0.2794 0.0782 0.0055 0.8358 0.8304 97.7%
    β 0.7016 0.2016 0.0462 0.5015 0.9016 0.4001 96.5%
    γ 0.2172 0.0328 0.0017 0.1568 0.2776 0.1208 100%
    80% α 0.4214 0.2786 0.0776 0.2084 0.6345 0.4261 97.9%
    β 0.6361 0.1361 0.0243 0.4490 0.8231 0.3741 98.5%
    γ 0.2297 0.0203 0.0010 0.1719 0.2874 0.1155 100%
    100% α 0.4234 0.2766 0.0765 0.2497 0.5970 0.3473 98.3%
    β 0.5161 0.0161 0.0062 0.3649 0.6673 0.3025 97.6%
    γ 0.2565 0.0065 0.0007 0.2044 0.3086 0.1043 100%
    100 70% α 0.4210 0.2790 0.0779 0.0208 0.8212 0.8004 96.4%
    β 0.7006 0.2006 0.0431 0.5593 0.8419 0.2826 98.0%
    γ 0.2141 0.0359 0.0016 0.1720 0.2562 0.0842 100%
    80% α 0.4215 0.2785 0.0776 0.2708 0.5721 0.3013 97.2%
    β 0.6357 0.1357 0.0214 0.5035 0.7680 0.2645 97.7%
    γ 0.2270 0.0230 0.0008 0.1866 0.2673 0.0807 100%
    100% α 0.4234 0.2766 0.0765 0.3006 0.5462 0.2456 97.3%
    β 0.5158 0.0158 0.0033 0.4088 0.6227 0.2140 98.2%
    γ 0.2540 0.0040 0.0003 0.2176 0.2905 0.0729 100%
    150 70% α 0.4212 0.2788 0.0778 0.0330 0.8093 0.7763 97.7%
    β 0.7000 0.2000 0.0419 0.5847 0.8153 0.2306 97.7%
    γ 0.2122 0.0378 0.0016 0.1781 0.2463 0.0682 100%
    80% α 0.4215 0.2785 0.0776 0.2985 0.5445 0.2460 98.8%
    β 0.6350 0.1350 0.0203 0.5271 0.7430 0.2159 98.1%
    γ 0.2259 0.0241 0.0008 0.1931 0.2587 0.0656 96.0%
    100% α 0.4234 0.2766 0.0765 0.3232 0.5237 0.2005 97.2%
    β 0.5151 0.0151 0.0023 0.4278 0.6024 0.1746 97.0%
    γ 0.2529 0.0029 0.0002 0.2232 0.2825 0.0593 95.4%
    200 70% α 0.4209 0.2791 0.0779 0.0849 0.7569 0.6720 100%
    β 0.6981 0.1981 0.0405 0.5984 0.7978 0.1994 97.2%
    γ 0.2118 0.0382 0.0016 0.1823 0.2412 0.0589 97.3%
    80% α 0.4215 0.2785 0.0776 0.3150 0.5280 0.2131 100%
    β 0.6331 0.1331 0.0191 0.5398 0.7265 0.1867 98.2%
    γ 0.2256 0.0244 0.0007 0.1973 0.2540 0.0567 98.0%
    100% α 0.4234 0.2766 0.0765 0.3366 0.5103 0.1737 100%
    β 0.5136 0.0136 0.0016 0.4381 0.5891 0.1510 98.8%
    γ 0.2523 0.0023 0.0002 0.2267 0.2779 0.0512 100%
    n r Set3 (α = 0.7, β = 0.7, γ = 0.5)
    MLE Bias MSE LB UB AL CP
    50 70% α 0.4178 0.2822 0.0797 0.1937 0.6419 0.4482 96.2%
    β 0.8994 0.1994 0.0477 0.6064 1.1923 0.5859 95.9%
    γ 0.6151 0.1151 0.0239 0.3763 0.8540 0.4776 95.0%
    80% α 0.4193 0.2807 0.0788 0.2255 0.6131 0.3875 95.9%
    β 0.8238 0.1238 0.0218 0.5471 1.1006 0.5535 95.9%
    γ 0.5695 0.0695 0.0182 0.3447 0.7943 0.4496 96.7%
    100% α 0.4211 0.2789 0.0778 0.2480 0.5942 0.3461 96.8%
    β 0.7612 0.0612 0.0104 0.5012 1.0213 0.5201 97.0%
    γ 0.5425 0.0425 0.0163 0.3395 0.7456 0.4061 95.0%
    100 70% α 0.4174 0.2826 0.0798 0.2439 0.5910 0.3470 95.0%
    β 0.8787 0.1787 0.0353 0.6426 1.1148 0.4722 96.3%
    γ 0.5696 0.0696 0.0201 0.3697 0.7696 0.3998 96.0%
    80% α 0.4191 0.2809 0.0789 0.2690 0.5691 0.3001 95.5%
    β 0.7802 0.0802 0.0134 0.5600 1.0004 0.4404 95.7%
    γ 0.5479 0.0479 0.0091 0.3814 0.7144 0.3330 96.0%
    100% α 0.4206 0.2794 0.0781 0.2866 0.5546 0.2680 95.6%
    β 0.7146 0.0146 0.0068 0.5069 0.9223 0.4154 95.7%
    γ 0.5597 0.0597 0.0074 0.4063 0.7131 0.3067 96.0%
    150 70% α 0.4176 0.2824 0.0798 0.2949 0.5403 0.2454 95.8%
    β 0.8697 0.1697 0.0305 0.7056 1.0338 0.3282 96.2%
    γ 0.5924 0.0924 0.0174 0.4449 0.7399 0.2950 97.1%
    80% α 0.4193 0.2807 0.0788 0.3131 0.5254 0.2122 96.2%
    β 0.8023 0.1023 0.0124 0.6492 0.9555 0.3063 96.1%
    γ 0.5524 0.0524 0.0058 0.4332 0.6716 0.2384 97.0%
    100% α 0.4209 0.2791 0.0779 0.3261 0.5156 0.1895 95.8%
    β 0.7374 0.0374 0.0032 0.5932 0.8817 0.2885 96.3%
    γ 0.5507 0.0507 0.0041 0.4447 0.6568 0.2121 96.9%
    200 70% α 0.4175 0.2825 0.0798 0.3173 0.5177 0.2004 96.1%
    β 0.8406 0.1406 0.0265 0.7075 0.9736 0.2661 97.2%
    γ 0.5683 0.0683 0.0069 0.4569 0.6798 0.2230 96.9%
    80% α 0.4193 0.2807 0.0788 0.3327 0.5059 0.1733 96.3%
    β 0.8043 0.1043 0.0123 0.6790 0.9295 0.2505 96.6%
    γ 0.5502 0.0502 0.0041 0.4533 0.6471 0.1938 97.0%
    100% α 0.4209 0.2791 0.0779 0.3435 0.4983 0.1548 96.1%
    β 0.7411 0.0411 0.0030 0.6229 0.8592 0.2362 97.0%
    γ 0.5463 0.0463 0.0041 0.4589 0.6338 0.1749 96.2%
    n r Set4 (α = 0.6, β = 0.3, γ = 0.5)
    MLE Bias MSE LB UB AL CP
    50 70% α 0.4197 0.1803 0.0325 0.2456 0.5939 0.3482 98.1%
    β 0.5719 0.2719 0.0744 0.3742 0.7696 0.3954 97.0%
    γ 0.2990 0.2010 0.0406 0.1769 0.4211 0.2442 98.0%
    80% α 0.4221 0.1779 0.0317 0.2716 0.5725 0.3009 98.4%
    β 0.4934 0.1934 0.0384 0.3126 0.6742 0.3616 97.4%
    γ 0.3593 0.1407 0.0200 0.2383 0.4803 0.2421 98.2%
    100% α 0.4246 0.1754 0.0308 0.2903 0.5589 0.2686 97.9%
    β 0.4198 0.1198 0.0150 0.2576 0.5820 0.3244 97.4%
    γ 0.4172 0.0828 0.0078 0.2968 0.5376 0.2408 98.4%
    100 70% α 0.4198 0.1802 0.0325 0.2966 0.5429 0.2463 97.2%
    β 0.5674 0.2674 0.0717 0.4278 0.7069 0.2791 97.7%
    γ 0.3137 0.1863 0.0351 0.2278 0.3995 0.1717 98.0%
    80% α 0.4222 0.1778 0.0316 0.3157 0.5286 0.2128 97.9%
    β 0.4857 0.1857 0.0350 0.3584 0.6130 0.2546 97.9%
    γ 0.3669 0.1331 0.0179 0.2818 0.4519 0.1701 98.7%
    100% α 0.4248 0.1752 0.0307 0.3242 0.5253 0.2011 98.1%
    β 0.4082 0.1082 0.0125 0.2944 0.5221 0.2277 97.8%
    γ 0.4171 0.0829 0.0074 0.3349 0.4993 0.1644 98.3%
    150 70% α 0.4198 0.1802 0.0325 0.3248 0.5148 0.1899 97.2%
    β 0.5653 0.2653 0.0706 0.4518 0.6788 0.2270 98.3%
    γ 0.3140 0.1860 0.0350 0.2421 0.3860 0.1439 99.3%
    80% α 0.4222 0.1778 0.0316 0.3353 0.5091 0.1738 97.7%
    β 0.4832 0.1832 0.0338 0.3797 0.5866 0.2069 98.2%
    γ 0.3677 0.1323 0.0178 0.2996 0.4358 0.1363 99.0%
    100% α 0.4248 0.1752 0.0307 0.3377 0.5119 0.1742 97.3%
    β 0.4018 0.1018 0.0107 0.3032 0.5004 0.1972 97.9%
    γ 0.4248 0.0752 0.0061 0.3608 0.4888 0.1280 99.7%
    200 70% α 0.4198 0.1802 0.0325 0.3423 0.4973 0.1551 99.0%
    β 0.5650 0.2650 0.0705 0.4738 0.6561 0.1823 99.1%
    γ 0.3256 0.1744 0.0309 0.2637 0.3875 0.1239 98.7%
    80% α 0.4222 0.1778 0.0316 0.3470 0.4975 0.1505 99.6%
    β 0.4781 0.1781 0.0321 0.3882 0.5679 0.1798 99.3%
    γ 0.3733 0.1267 0.0173 0.3129 0.4336 0.1206 99.5%
    100% α 0.4250 0.1750 0.0306 0.3578 0.4921 0.1343 98.7%
    β 0.3904 0.0904 0.0088 0.3106 0.4703 0.1596 99.6%
    γ 0.4262 0.0738 0.0060 0.3681 0.4843 0.1162 100%

     | Show Table
    DownLoad: CSV

    A. For almost all the true values, the MSE of all the estimates decreases as the sample sizes and the censoring level r increase, demonstrating that the various estimates are consistent (see Table 6 and Figure 3).

    Figure 3.  MSE of the estimates at the true value of Set1.

    B. For all true parameter values, the ALs of all the estimates decrease as the sample sizes and the censoring level r increase (see Table 6 and Figure 4).

    Figure 4.  AL of the estimates at the true value of Set2.

    C. For all true parameter values, the CP of all the estimates increases as the sample sizes and the censoring level r increase (see Table 6).

    D. The MSE of the estimate of α at the true value of Set1 yields the lowest values in comparison to the other actual parameter values for all sample sizes (see Table 6 and Figure 5).

    Figure 5.  MSE of ˆα for all sets.

    E. At all actual values, the MSE of the estimate of β produces the largest results for all sample sizes (see Table 6 and Figure 6). Also, it is evident that except for n=50 and 200, the MSE of β estimates obtains the smallest values for the actual value of Set1 compared to the other actual sets at the censoring level 70%. At the censoring level 80%, the MSE of β estimates gets the smallest values at all sets of parameters except at n=50.

    Figure 6.  MSE of ˆβ for all sets.

    F. The MSE of the estimate of γ at the true value of Set2 gets the smallest values in comparison to the other actual parameter values for all sample sizes (see Table 6 and Figure 7).

    Figure 7.  MSE of ˆγ for all sets.

    G. The MSEs, biases, and ALs of γ are smaller than the other estimates of α and β in almost all of the cases.

    H. As n rises, the CI's lengths get shorter.

    I. As n increases, parameter estimates grow increasingly accurate, suggesting that they are asymptotically unbiased.

    J. For the parameter values examined, the CI's overall performance is fairly strong.

    Here, we provide applications to four real data sets to illustrate the importance and potentiality of the LBTLoW distribution. The goodness-of-fit statistics for these distributions and other competitive distributions are compared, and the MLEs of their parameters are provided.

    The first real data set [38] on the relief times of twenty patients receiving an analgesic is 1.1, 1.4, 3, 1.7, 2.3, 1.4, 1.3, 1.7, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.9, 1.8, 1.6, 1.2, 1.6, 2.

    The second dataset illustrates the proportion of global reserves of natural gas in various countries as of the year 2020. In contrast to other nations, Russia possesses the largest natural gas reserves globally and maintains its position as the leading exporter of natural gas. Iran, on the other hand, ranks second in terms of natural gas reserves worldwide. Qatar, although holding slightly over 13% of the total global natural gas reserves, also plays a significant role in the natural gas market. Lastly, Saudi Arabia possesses the fifth-largest natural gas reserves globally. The electronic address from which it was taken is as follows: https://worldpopulationreview.com/. The data set is reported in Table 7.

    Table 7.  The percent Global Reserves Natural Gas of the Countries (2020).
    Rank Country % Global Reserves Rank Country % Global Reserves
    1 Russia 19.9 23 Ukraine 0.6
    2 Iran 17.1 24 Malaysia 0.5
    3 Qatar 13.1 25 Uzbekistan 0.4
    4 Turkmenistan 7.2 26 Oman 0.4
    5 United States 6.7 27 Vietnam 0.3
    6 China 4.5 28 Israel 0.3
    7 Venezuela 3.3 29 Argentina 0.2
    8 Saudi Arabia 3.2 30 Pakistan 0.2
    9 United Arab Emirates 3.2 31 Trinidad 0.2
    10 Nigeria 2.9 32 Brazil 0.2
    11 Iraq 1.9 33 Myanmar 0.2
    12 Canada 1.3 34 United Kingdom 0.1
    13 Australia 1.3 35 Thailand 0.1
    14 Azerbaijan 1.3 36 Mexico 0.1
    15 Algeria 1.2 37 Bangladesh 0.1
    16 Kazakhstan 1.2 38 Netherlands 0.1
    17 Egypt 1.1 39 Bolivia 0.1
    18 Kuwait 0.9 40 Brunei 0.1
    19 Norway 0.8 41 Peru 0.1
    20 Libya 0.8 42 Syria 0.1
    21 Indonesia 0.7 43 Yemen 0.1
    22 India 0.7 44 Papua New Guinea 0.1

     | Show Table
    DownLoad: CSV

    The third dataset pertains to the Top 20 Countries with the Largest Oil Reserves, measured in thousand million barrels. Crude oil serves as the predominant fuel source globally and is the primary source of energy on a wide scale. In the year 2020, global oil consumption reached around 88.6 million barrels per day, or 30.1% of the overall primary energy consumption. Venezuela possesses the largest oil reserves globally, over 300 billion barrels in total. Saudi Arabia holds the world's second-largest oil reserves, with 297.5 billion barrels. The United States is the world's leading producer of oil as well as the world's greatest user of oil, necessitating additional imports from dozens of other oil-producing countries. Despite having the world's highest oil production, the United States is only 9th in the world in terms of available oil reserves. It was obtained from the following electronic address: https://worldpopulationreview.com/. The data set is reported in Table 8.

    Table 8.  Top 20 Countries with the Largest Oil Reserves (in thousand million barrels).
    Rank Country reserves2020 Rank Country reserves2020
    1 Venezuela 303.8 11 Nigeria 36.9
    2 Saudi Arabia 297.5 12 Kazakhstan 30
    3 Canada 168.1 13 China 26
    4 Iran 157.8 14 Qatar 25.2
    5 Iraq 145 15 Algeria 12.2
    6 Russia 107.8 16 Brazil 11.9
    7 Kuwait 101.5 17 Norway 7.9
    8 United Arab Emirates 97.8 18 Angola 7.8
    9 United States 68.8 19 Azerbaijan 7
    10 Libya 48.4 20 Mexico 6.1

     | Show Table
    DownLoad: CSV

    The fourth data set represents the Top 100 central banks that owned the largest gold Reserves (in thousand tons). Because of its safety, liquidity, and return qualities-the three major investment objectives for central banks-gold is an essential component of central bank reserves. As such, they are significant gold holders, accounting for around one-fifth of all gold extracted throughout history. They present gold reserve data derived using IMF IFS figures to help comprehend this sector of the gold market, which records central banks' (and other official institutions, when appropriate) reported purchases and sales of gold as a percentage of their international reserves. It was obtained from the following electronic address: https://www.gold.org/. The data set is reported in Table 9.

    Table 9.  Top 100 central bank owned the largest gold Reserves (in thousand tons).
    Rank Country Reserves of Gold Rank Country Reserves of Gold Rank Country Reserves of Gold
    1 USA 8.1335 35 LBY 0.1166 68 CYP 0.0139
    2 DEU 3.3585 36 GRC 0.1141 69 CUW 0.0131
    3 IMF 2.814 37 ROK 0.1045 70 MUS 0.0124
    4 ITA 2.4518 38 ROU 0.1036 71 IRL 0.012
    5 FRA 2.4365 39 BIS 0.102 72 CZE 0.0109
    6 RUS 2.2985 40 IRQ 0.0964 73 KGZ 0.0102
    7 CHN 1.9483 41 HUN 0.0945 74 GHA 0.0087
    8 CHE 1.04 42 AUS 0.0798 75 PRY 0.0082
    9 JPN 0.846 43 KWT 0.079 76 NPL 0.008
    10 IND 0.7604 44 IDN 0.0786 77 MNG 0.0076
    11 NLD 0.6125 45 DNK 0.0666 78 MMR 0.0073
    12 ECB 0.5048 46 PAK 0.0647 79 GTM 0.0069
    13 TUR 0.4311 47 ARG 0.0617 80 MKD 0.0069
    14 TAI 0.4236 48 ARE 0.0553 81 TUN 0.0068
    15 PRT 0.3826 49 BLR 0.0535 82 LVA 0.0067
    16 KAZ 0.3681 50 QAT 0.0513 83 LTU 0.0058
    17 UZB 0.3375 51 KHM 0.0504 84 COL 0.0047
    18 SAU 0.3231 52 FIN 0.049 85 BHR 0.0047
    19 GBR 0.3103 53 JOR 0.0435 86 BRN 0.0046
    20 LBN 0.2868 54 BOL 0.0425 87 GIN 0.0042
    21 ESP 0.2816 55 BGR 0.0408 88 MOZ 0.0039
    22 AUT 0.28 56 MYS 0.0389 89 SVN 0.0032
    23 THA 0.2442 57 SRB 0.0378 90 ABW 0.0031
    24 POL 0.2287 58 WAEMU 0.0365 91 BIH 0.003
    25 BEL 0.2274 59 PER 0.0347 92 ALB 0.0028
    26 DZA 0.1736 60 SVK 0.0317 93 LUX 0.0022
    27 VEN 0.1612 61 UKR 0.0271 94 HKG 0.0021
    28 PHL 0.1563 62 SYR 0.0258 95 ISL 0.002
    29 SGP 0.1537 63 MAR 0.0221 96 TTO 0.0019
    30 BRA 0.1297 64 ECU 0.0219 97 HTI 0.0018
    31 SWE 0.1257 65 AFG 0.0219 98 YEM 0.0016
    32 ZAF 0.1254 66 NGA 0.0215 99 SUR 0.0015
    33 EGY 0.125 67 BGD 0.014 100 SLV 0.0014
    34 MEX 0.1199

     | Show Table
    DownLoad: CSV

    The descriptive analysis of all the data sets is reported in Table 10.

    Table 10.  Some descriptive analysis of all data sets.
    n Mean Median Skewness Kurtosis Range Min Max Sum
    Data1 20 1.900 1.700 1.860 4.185 3.000 1.100 4.100 38.000
    Data2 44 2.248 0.650 2.990 8.864 19.800 0.100 19.900 98.900
    Data3 20 83.375 42.650 1.430 1.420 297.700 6.100 303.800 1667.500
    Data4 100 0.347 0.050 5.590 38.257 8.130 0.001 8.133 34.676

     | Show Table
    DownLoad: CSV

    These real data sets are utilized to assess the goodness of fit of the LBTLoW distribution. The suggested model is compared with exponentiated transmuted generalized Rayleigh (ETGR) [39], beta Weibull (BW) [40], transmuted Lindley (T-Li) [41], McDonald log-logistic (McLL) [42], new modified Weibull (NMW) [43], weighted exponentiated inverted Weibull (WEIW) [44], transmuted complementary Weibull geometric (TCWG) [45], transmuted modified Weibull (TMW) [46], exponentiated Kumaraswamy Weibull (EKW) [47] and Weibull (W) models.

    The maximum likelihood estimators (MLEs) and standard errors (SEs) of the model parameters are computed. In order to assess the distribution models, various criteria are taken into account, including the Akaike information criterion (AIC), correct AIC (CAIC), Bayesian IC (BIC), Hannan-Quinn IC (HQIC), Kolmogorov-Smirnov (KS) test, and p-value (PV) test. In contrast, the broader dissemination is associated with reduced values of AIC, CAIC, BIC, HQIC, KS, and the highest magnitude of PV. The maximum likelihood estimators (MLEs) of the competitive models, along with their standard errors (SEs) and values of AIC, CAIC, BIC, HQIC, PV, and KS for the suggested data sets, are displayed in Tables 11-18. It has been observed that the LBTLoW distribution, characterized by three parameters, exhibits superior goodness of fit compared to alternative models. This distribution exhibits the lowest values of AIC, CAIC, BIC, HQIC, and KS, and the highest value of PV among the distributions under consideration in this analysis. Furthermore, Figures 8-15 exhibit the graphical representations of the estimated pdf, cdf, ccdf, and probability-probability (PP) plots for the competitive model applied to the given data sets.

    Table 11.  MLEs and SEs for the first data set.
    Distributions MLE and SE
    α β γ λ θ
    LBTLoW 8.648 3.074 0.042
    (3.545) (0.474) (0.025)
    ETGR 0.103 0.692 23.539 -0.342
    (0.436) (0.086) (105.137) (1.971)
    BW 0.831 0.613 29.947 11.632
    (0.954) (0.340) (40.414) (21.900)
    T-Li 0.665 0.359
    (0.332) (0.048)
    McLL 0.881 2.070 1.926 19.225 32.033
    (0.109) (3.693) (5.165) (22.341) (43.081)
    NMW 0.121 2.784 2.787 0.003 0.008
    (0.056) (20.370) (0.428) (0.025) (0.002)
    W 0.122 2.787
    (0.056) (0.427)

     | Show Table
    DownLoad: CSV
    Table 12.  Measures of fitting for the first data set.
    Distributions AIC CAIC BIC HQIC KS PV
    LBTLoW 40.140 41.640 38.040 40.720 0.146 0.790
    ETGR 44.860 47.520 42.060 45.630 0.190 0.465
    BW 42.400 45.060 39.600 43.170 0.160 0.683
    T-Li 65.730 66.440 64.330 66.120 0.380 0.006
    McLL 43.850 48.140 40.360 44.830 0.147 0.734
    NMW 51.170 55.460 47.680 52.150 0.190 0.501
    W 45.170 45.880 43.780 45.560 0.180 0.509

     | Show Table
    DownLoad: CSV
    Table 13.  MLEs and SEs for the global reserves natural gas data set.
    Distributions MLE and SE
    α β γ λ θ
    LBTLoW 6.268 0.623 0.484
    (2.631) (0.066) (0.210)
    ETGR 0.055 0.071 8.773 0.947
    (0.027) (0.029) (7.043) (0.081)
    TCWG 34.076 0.802 0.005 1.12
    (81.023) (0.021) (0.013) (0.285)
    EKW 0.221 400.298 5.215 1 3.823
    (0.038) (718.99) (0.649) (0.004) (3.036)
    TMW 0.851 1.159 -0.554 0.519
    (0.163) (1.026) (0.985) (0.379)
    BW 2.861 0.075 78.550 42.576
    (69.095) (0.090) (167.320) (187.300)
    T-Li 0.604 0.671
    (0.155) (0.074)
    McLL 0.181 1.565 1.286 21.234 28.124
    (0.193) (9.254) (5.432) (34.701) (45.757)
    NMW 6.8 x 108 0.680 0.223 0.015 0.806
    (0.623) (0.110) (617.48) (0.015) (0.418)
    W 0.799 0.621
    (0.136) (0.068)

     | Show Table
    DownLoad: CSV
    Table 14.  Measures of fitting for the global reserves natural gas data set.
    Distributions AIC CAIC BIC HQIC KS PV
    LBTLoW 132.210 132.810 131.140 134.200 0.130 0.425
    ETGR 143.470 144.490 142.040 146.110 0.180 0.118
    TCWG 137.690 138.710 136.260 140.330 0.150 0.251
    EKW 133.890 135.470 132.110 140.330 0.140 0.355
    TMW 140.900 142.480 139.120 144.210 0.150 0.276
    BW 133.180 134.200 131.750 135.820 0.130 0.408
    T-Li 174.360 174.660 173.650 175.690 0.200 0.057
    McLL 134.830 136.410 133.040 138.130 0.130 0.419
    NMW 143.780 145.360 142.000 147.090 0.160 0.243
    W 138.650 138.940 137.940 139.970 0.170 0.139

     | Show Table
    DownLoad: CSV
    Table 15.  MLEs and SEs for the global oil reserves data set.
    Distributions MLE and SE
    α β γ λ θ
    LBTLoW 2.515 0.756 0.040
    (3.877) (0.166) (0.048)
    WEIW 0.909 0.871 7.225
    (106700) (0.152) (384700)
    TMW 0.998 0.459 -0.443 0.202
    (0.081) (18.537) (18.537) (0.769)
    T-Li 0.021 0.384
    (0.345) (0.004)
    McLL 0.208 93.978 1.279 24.759 32.815
    (0.499) (1721) (19.272) (142.806) (161.611)
    NMW 10.7 x 108 0.930 0.859 7.46 x 108 0.017
    (0.001) (0.250) (1.216) (0.002) (0.017)
    EKW 0.167 261.64 45.725 1.201 2.138
    (0.079) (1709) (219.725) (0.741) (7.209)

     | Show Table
    DownLoad: CSV
    Table 16.  Measures of fitting for the global oil reserves data set.
    Distributions AIC CAIC BIC HQIC KS PV
    LBTLoW 221.690 223.190 219.600 222.280 0.135 0.857
    WEIW 223.400 224.900 221.300 223.980 0.157 0.708
    TMW 226.410 230.690 222.910 227.380 0.153 0.734
    T-Li 230.480 231.180 229.080 230.870 0.265 0.120
    McLL 225.990 230.280 222.500 222.500 0.146 0.789
    NMW 226.570 230.860 223.080 227.540 0.140 0.826
    EKW 226.290 230.570 222.790 229.300 0.148 0.776

     | Show Table
    DownLoad: CSV
    Table 17.  MLEs and SEs for the global gold reserves data set.
    Distributions MLE and SE
    α β γ λ θ
    LBTLoW 6.498 0.482 1.490
    (2.301) (0.034) (0.573)
    EKW 0.221 1096 4.424 1 1.717
    (0.030) (1376) (1.817) (0.001) (0.901)
    TMW 0.596 2.612 0.588 -0.523
    (0.057) (0.689) (0.256) (0.346)
    BW 134.832 0.073 49.149 22.930
    (956.622) (0.060) (74.497) (46.500)
    WEIW 27.512 0.549 0.094
    (3272000) (0.042) (856.967)
    W 2.648 0.489
    (0.281) (0.035)

     | Show Table
    DownLoad: CSV
    Table 18.  Measures of fitting for the global gold reserves data set.
    Distributions AIC CAIC BIC HQIC KS PV
    LBTLoW –170.510 –170.260 –170.510 –167.350 0.070 0.704
    ETGR –167.710 –167.070 –167.710 –157.710 0.078 0.584
    BW –158.180 –157.540 –158.180 –152.910 0.077 0.593
    T-Li –170.220 –169.800 –170.220 –166.010 0.071 0.703
    McLL –170.200 –169.950 –170.200 –167.040 0.091 0.383
    W –157.390 –157.260 –157.390 –155.280 0.100 0.269

     | Show Table
    DownLoad: CSV
    Figure 8.  Estimated pdf, cdf and ccdf plots of the competitive models for the first data set.
    Figure 9.  The PP plots of the fitted models for the first data set.
    Figure 10.  Estimated pdf, cdf and ccdf plots of the competitive models for global reserves natural gas data set.
    Figure 11.  The PP plots of the fitted models for the global reserves natural gas data set.
    Figure 12.  Estimated pdf, cdf and ccdf plots of the competitive models for global oil reserves data set.
    Figure 13.  The PP plots of the fitted models for the global oil reserves data set.
    Figure 14.  Estimated pdf, cdf and ccdf plots of the competitive models for global gold reserves data set.
    Figure 15.  The PP plots of the fitted models for global gold reserves data set.

    From the previous figures, we conclude that the LBTLoW model clearly gives the best overall fit and so may be picked as the most appropriate model for explaining data.

    The LBTLo-G family of distributions is explored in this article. The LBTLo-G family of probability distributions has a number of desirable characteristics, including being very flexible and simple, containing a number of new distributions, the ability for the generated distributions' pdfs to be unimodal, decreasing, bathtub-shaped, right-skewed, and symmetric, and the ability for their hrf shapes to be increasing, decreasing, U-shaped, upside-down-shaped, or J-shaped. These include discussion of the characteristics of the LBTLo-G family, including expansion for the density function, moments, incomplete moments, and certain entropy metrics. Estimating the model parameters is done using the ML technique. A simulation study demonstrated that the estimates of the model parameters are not far from their true values. Also, the biases and mean squared errors of estimates based on censored samples are larger than those based on complete samples. As the censoring levels and sample sizes increase, the coverage probability of estimates increases in approximately most cases.

    As one distribution of the LBTLo-G family, the real datasets for global reserves of oil, gold, and natural gas were chosen to fit the LBTLoW distribution. The first data set proposed was the lifetime data relating to relief times (in minutes) of patients receiving an analgesic. The second data set provides the percent of global reserves of natural gas for 44 countries. We have considered the third real data analysis of the countries with the largest oil reserves in 20 countries. We consider another real-data analysis of the central bank owning the largest gold reserves in 100 countries. This gold reserve data, compiled using international monetary funds and international financial statistics, tracks central banks' reported purchases and sales of gold as a percentage of their international reserves. The LBTLoW model typically provides superior fits in comparison to certain other alternative models, as shown by real-world data applications.

    The authors declare that they have not used artificial intelligence tools in the creation of this article.

    Researchers Supporting Project number (RSPD2023R548), King Saud University, Riyadh, Saudi Arabia.

    The authors declare that there are no conflicts of interest.



    [1] P. Grasland-Mongrain, C. Lafon, Review on biomedical techniques for imaging electrical impedance, IRBM, 39 (2018), 243–250. https://doi.org/10.1016/j.irbm.2018.06.001 doi: 10.1016/j.irbm.2018.06.001
    [2] H. Wen, R. S. Balaban, The potential for hall effect breast imaging, Breast Dis., 10 (1998), 191–195. https://doi.org/10.3233/BD-1998-103-418 doi: 10.3233/BD-1998-103-418
    [3] Y. Zhou, Z. Yu, Q. Ma, G. Guo, J. Tu, D. Zhang, Noninvasive treatment-efficacy evaluation for HIFU therapy based on magneto-acousto-electrical tomography, IEEE Trans. Biomed. Eng., 66 (2018), 666–674. https://doi.org/10.1109/TBME.2018.2853594 doi: 10.1109/TBME.2018.2853594
    [4] Y. Li, G. Liu, H. Xia, Z. Xia, Numerical simulations and experimental study of magneto-acousto-electrical tomography with plane transducer, IEEE Trans. Magn., 54 (2017), 1–4. https://doi.org/10.1109/TMAG.2017.2771564 doi: 10.1109/TMAG.2017.2771564
    [5] G. Guo, J. Wang, Q. Ma, J. Tu, D. Zhang, Non-invasive treatment efficacy evaluation for high-intensity focused ultrasound therapy using magnetically induced magnetoacoustic measurement, J. Appl. Phys., 123 (2018), 154901. https://doi.org/10.1063/1.5024735 doi: 10.1063/1.5024735
    [6] M. S. Gözü, R. Zengin, N. G. Gençer, Numerical implementation of magneto-acousto-electrical tomography (MAET) using a linear phased array transducer, Phys. Med. Biol., 63 (2018), 35012. https://doi.org/10.1088/1361-6560/aa9f3b doi: 10.1088/1361-6560/aa9f3b
    [7] P. Grasland-Mongrain, F. Destrempes, J. Mari, R. Souchon, S. Catheline, J. Chapelon, Acousto-electrical speckle pattern in electrical impedance tomography, in 2014 IEEE International Ultrasonics Symposium, (2014), 221–223. https://doi.org/10.1109/ULTSYM.2014.0056
    [8] L. Guo, G. Liu, H. Xia, Magneto-acousto-electrical tomography with magnetic induction for conductivity reconstruction, IEEE Trans. Biomed. Eng., 62 (2014), 2114–2124. https://doi.org/10.1109/TBME.2014.2382562 doi: 10.1109/TBME.2014.2382562
    [9] L. Kunyansky, C. P. Ingram, R. S. Witte, Rotational magneto-acousto-electric tomography (MAET): theory and experimental validation, Phys. Med. Biol., 62 (2017), 3025. https://doi.org/10.1088/1361-6560/aa6222 doi: 10.1088/1361-6560/aa6222
    [10] L. Kunyansky, A mathematical model and inversion procedure for magneto-acousto-electric tomography, Inverse Probl., 28 (2012), 035002. https://doi.org/10.1088/0266-5611/28/3/035002 doi: 10.1088/0266-5611/28/3/035002
    [11] H. Ammari, P. Grasland-Mongrain, P. Millien, L. Seppecher, J. Seo, A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography, J. Math. Pures Appl., 103 (2015), 1390–1409. https://doi.org/10.1016/j.matpur.2014.11.003 doi: 10.1016/j.matpur.2014.11.003
    [12] Y. Li, J. Song, H. Xia, G. Liu, The experimental study of mouse liver in magneto-acousto-electrical tomography by scan mode, Phys. Med. Biol., 65 (2020), 215024. https://doi.org/10.1088/1361-6560/abb4bb doi: 10.1088/1361-6560/abb4bb
    [13] Z. Sun, G. Liu, H. Xia, S. Catheline, Lorentz force electrical-impedance tomography using linearly frequency-modulated ultrasound pulse, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 65 (2018), 168–177. https://doi.org/10.1109/TUFFC.2017.2781189 doi: 10.1109/TUFFC.2017.2781189
    [14] M. Dai, X. Chen, T. Sun, L. Yu, M. Chen, H. Lin, A 2D magneto-acousto-electrical tomography method to detect conductivity variation using multifocus image method, Sensors, 18 (2018), 2373. https://doi.org/10.3390/s18072373 doi: 10.3390/s18072373
    [15] E. Renzhiglova, V. Ivantsiv, Y. Xu, Difference frequency magneto-acousto-electrical tomography (DF-MAET): application of ultrasound-induced radiation force to imaging electrical current density, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 57 (2010), 2391–2402. https://doi.org/10.1109/TUFFC.2010.1707 doi: 10.1109/TUFFC.2010.1707
    [16] A. Montalibet, J. Jossinet, A. Matias, Scanning electric conductivity gradients with ultrasonically-induced lorentz force, Ultrason. Imaging, 23 (2001), 117–132. https://doi.org/10.1177/016173460102300204 doi: 10.1177/016173460102300204
    [17] Y. Jin, H. Zhao, G. Liu, H. Xia, Y. Li, The application of wavelet filtering method in magneto-acousto-electrical tomography, Phys. Med. Biol., 68 (2023), 145014. https://doi.org/10.1088/1361-6560/ace09c doi: 10.1088/1361-6560/ace09c
    [18] S. Anwar, N. Barnes, Real image denoising with feature attention, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), (2019), 3155–3164. https://doi.org/10.1109/ICCV.2019.00325
    [19] Z. Yue, H. Yong, Q. Zhao, D. Meng, L. Zhang, Variational denoising network: toward blind noise modeling and removal, in Advances in Neural Information Processing Systems, 32 (2019). Available from: https://proceedings.neurips.cc/paper_files/paper/2019/file/6395ebd0f4b478145ecfbaf939454fa4-Paper.pdf.
    [20] K. Zhang, W. Zuo, Y. Chen, D. Meng, L. Zhang, Beyond a gaussian denoiser: residual learning of deep CNN for image denoising, IEEE Trans. Image Process., 26 (2017), 3142–3155. https://doi.org/10.1109/TIP.2017.2662206 doi: 10.1109/TIP.2017.2662206
    [21] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, et al., Generative adversarial nets, in Advances in Neural Information Processing Systems, 27 (2014). Available from: https://proceedings.neurips.cc/paper_files/paper/2014/file/5ca3e9b122f61f8f06494c97b1afccf3-Paper.pdf.
    [22] J. Chen, J. Chen, H. Chao, M. Yang, Image blind denoising with generative adversarial network based noise modeling, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2018), 3155–3164. https://doi.org/10.1109/CVPR.2018.00333
    [23] D. W. Kim, J. R. Chung, S. W. Jung, Grdn: Grouped residual dense network for real image denoising and gan-based real-world noise modeling, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), (2019), 2086–2094. https://doi.org/10.1109/CVPRW.2019.00261
    [24] P. Grasland-Mongrain, J. M. Mari, J. Y. Chapelon, C. Lafon, Lorentz force electrical impedance tomography, IRBM, 34 (2013), 357–360. https://doi.org/10.1016/j.irbm.2013.08.002 doi: 10.1016/j.irbm.2013.08.002
    [25] Y. Li, S. Bu, X. Han, H. Xia, W. Ren, G. Liu, Magneto-acousto-electrical tomography with nonuniform static magnetic field, IEEE Trans. Instrum. Meas., 72 (2023), 1–12. https://doi.org/10.1109/TIM.2023.3244814 doi: 10.1109/TIM.2023.3244814
    [26] H. Lin, Y. Chen, S. Xie, M. Yu, D. Deng, T. Sun, et al., A dual-modal imaging method combining ultrasound and electromagnetism for simultaneous measurement of tissue elasticity and electrical conductivity, IEEE Trans. Biomed. Eng., 69 (2022), 2499–2511. https://doi.org/10.1109/TBME.2022.3148120 doi: 10.1109/TBME.2022.3148120
    [27] M. Aharon, M. Elad, A. Bruckstein, K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process., 54 (2006), 4311–4322. https://doi.org/10.1109/TSP.2006.881199 doi: 10.1109/TSP.2006.881199
    [28] M. Elad, M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), 3736–3745. https://doi.org/10.1109/TIP.2006.881969 doi: 10.1109/TIP.2006.881969
    [29] S. Gu, L. Zhang, W. Zuo, X. Feng, Weighted nuclear norm minimization with application to image denoising, in 2014 IEEE Conference on Computer Vision and Pattern Recognition, (2014), 2862–2869. https://doi.org/10.1109/CVPR.2014.366
    [30] A. Buades, B. Coll, J. M. Morel, A non-local algorithm for image denoising, in 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), 2 (2005), 60–65. https://doi.org/10.1109/CVPR.2005.38
    [31] K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080–2095. https://doi.org/10.1109/TIP.2007.901238 doi: 10.1109/TIP.2007.901238
    [32] J. Portilla, V. Strela, M. J. Wainwright, E. P. Simoncelli, Image denoising using scale mixtures of Gaussians in the wavelet domain, IEEE Trans. Image Process., 12 (2003), 1338–1351. https://doi.org/10.1109/TIP.2003.818640 doi: 10.1109/TIP.2003.818640
    [33] L. I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259–268. https://doi.org/10.1016/0167-2789(92)90242-F doi: 10.1016/0167-2789(92)90242-F
    [34] A. Barbu, Training an active random field for real-time image denoising, IEEE Trans. Image Process., 18 (2009), 2451–2462. https://doi.org/10.1109/TIP.2009.2028254 doi: 10.1109/TIP.2009.2028254
    [35] K. G. G. Samuel, M. F. Tappen, Learning optimized MAP estimates in continuously-valued MRF models, in 2009 IEEE Conference on Computer Vision and Pattern Recognition, (2009), 477–484. https://doi.org/10.1109/CVPR.2009.5206774
    [36] J. Sun, M. F. Tappen, Learning non-local range Markov Random field for image restoration, CVPR 2011, (2011), 2745–2752. https://doi.org/10.1109/CVPR.2011.5995520 doi: 10.1109/CVPR.2011.5995520
    [37] U. Schmidt, Half-quadratic Inference and Learning for Natural Images, Ph.D thesis, Technische University, 2017. Available from: https://tuprints.ulb.tu-darmstadt.de/id/eprint/6044.
    [38] U. Schmidt, S. Roth, Shrinkage fields for effective image restoration, in 2014 IEEE Conference on Computer Vision and Pattern Recognition, (2014), 2774–2781. https://doi.org/10.1109/CVPR.2014.349
    [39] Y. Chen, T. Pock, Trainable nonlinear reaction diffusion: a flexible framework for fast and effective image restoration, IEEE Trans. Pattern Anal. Mach. Intell., 39 (2017), 1256–1272. https://doi.org/10.1109/TPAMI.2016.2596743 doi: 10.1109/TPAMI.2016.2596743
    [40] V. Jain, S. Seung, Natural image denoising with convolutional networks, in Advances in Neural Information Processing Systems, 21 (2008). Available from: https://proceedings.neurips.cc/paper_files/paper/2008/file/c16a5320fa475530d9583c34fd356ef5-Paper.pdf.
    [41] H. C. Burger, C. J. Schuler, S. Harmeling, Image denoising: can plain neural networks compete with BM3D? in 2012 IEEE Conference on Computer Vision and Pattern Recognition, (2012), 2392–2399. https://doi.org/10.1109/CVPR.2012.6247952
    [42] X. Mao, C. Shen, Y. B. Yang, Image restoration using very deep convolutional encoder-decoder networks with symmetric skip connections, in Advances in Neural Information Processing Systems, 29 (2016). Available from: https://proceedings.neurips.cc/paper_files/paper/2016/file/0ed9422357395a0d4879191c66f4faa2-Paper.pdf.
    [43] D. Liu, B. Wen, Y. Fan, C. C. Loy, T. S. Huang, Non-local recurrent network for image restoration, in Advances in Neural Information Processing Systems, 31 (2018). Available from: https://proceedings.neurips.cc/paper_files/paper/2018/file/fc49306d97602c8ed1be1dfbf0835ead-Paper.pdf.
    [44] T. Plötz, S. Roth, Neural nearest neighbors networks, in Advances in Neural Information Processing Systems, 31 (2018). Available from: https://proceedings.neurips.cc/paper_files/paper/2018/file/f0e52b27a7a5d6a1a87373dffa53dbe5-Paper.pdf.
    [45] Z. Yue, Q. Zhao, L. Zhang, D. Meng, Dual adversarial network: toward real-world noise removal and noise generation, in Computer Vision – ECCV 2020, (2020), 41–58. https://doi.org/10.1007/978-3-030-58607-2_3
    [46] N. Mu, Z. Lyu, M. Rezaeitaleshmahalleh, J. Tang, J. Jiang, An attention residual u-net with differential preprocessing and geometric postprocessing: learning how to segment vasculature including intracranial aneurysms, Med. Image Anal., 84 (2023), 102697. https://doi.org/10.1016/j.media.2022.102697 doi: 10.1016/j.media.2022.102697
    [47] X. Liu, D. Zhang, J. Yao, J. Tang, Transformer and convolutional based dual branch network for retinal vessel segmentation in OCTA images, Biomed. Signal Process. Control, 83 (2023), 104604. https://doi.org/10.1016/j.bspc.2023.104604 doi: 10.1016/j.bspc.2023.104604
    [48] M. Versaci, G. Angiulli, P. Crucitti, D. de Carlo, F. Laganà, D. Pellicanò, et al., A fuzzy similarity-based approach to classify numerically simulated and experimentally detected carbon fiber-reinforced polymer plate defects, Sensors, 22 (2022), 4232. https://doi.org/10.3390/s22114232 doi: 10.3390/s22114232
    [49] Z. Yue, H. Yong, D. Meng, Q. Zhao, Y. Leung, L. Zhang, Robust multiview subspace learning with nonindependently and nonidentically distributed complex noise, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 1070–1083. https://doi.org/10.1109/TNNLS.2019.2917328 doi: 10.1109/TNNLS.2019.2917328
    [50] D. P. Kingma, M. Welling, Auto-encoding variational bayes, preprint, arXiv: 1312.6114.
    [51] C. Li, T. Xu, J. Zhu, B. Zhang, Triple generative adversarial nets, in Advances in Neural Information Processing Systems, 30 (2017). Available from: https://proceedings.neurips.cc/paper_files/paper/2017/file/86e78499eeb33fb9cac16b7555b50767-Paper.pdf.
    [52] M. Arjovsky, S. Chintala, L. Bottou, Wasserstein generative adversarial networks, in Proceedings of the 34th International Conference on Machine Learning, 70 (2017), 214–223. Available from: https://proceedings.mlr.press/v70/arjovsky17a.html.
    [53] O. Ronneberger, P. Fischer, T. Brox, U-Net: Convolutional networks for biomedical image segmentation, in Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015, (2015), 234–241. https://doi.org/10.1007/978-3-319-24574-4_28
    [54] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 770–778. https://doi.org/10.1109/CVPR.2016.90
    [55] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, et al., Attention is all you need, in Advances in Neural Information Processing Systems, 30 (2017). Available from: https://proceedings.neurips.cc/paper_files/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf.
    [56] I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, A. C. Courville, Improved training of wasserstein GANs, in Advances in Neural Information Processing Systems, 30 (2017). Available from: https://proceedings.neurips.cc/paper_files/paper/2017/file/892c3b1c6dccd52936e27cbd0ff683d6-Paper.pdf.
    [57] A. Radford, L. Metz, S. Chintala, Unsupervised representation learning with deep convolutional generative adversarial networks, preprint, arXiv: 1511.06434.
    [58] P. Isola, J. Y. Zhu, T. Zhou, A. A. Efros, Image-to-image translation with conditional adversarial networks, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2017), 1125–1134. https://doi.org/10.1109/CVPR.2017.632
    [59] J. Y. Zhu, T. Park, P. Isola, A. A. Efros, Unpaired image-to-image translation using cycle-consistent adversarial networks, in Proceedings of the IEEE International Conference on Computer Vision (ICCV), (2017), 2223–2232.
    [60] S. Guo, Z. Yan, K. Zhang, W. Zuo, L. Zhang, Toward convolutional blind denoising of real photographs, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2019), 1712–1722. https://doi.org/10.1109/CVPR.2019.00181
    [61] J. Korhonen, J. You, Peak signal-to-noise ratio revisited: is simple beautiful? in 2012 Fourth International Workshop on Quality of Multimedia Experience, (2012), 37–38. https://doi.org/10.1109/QoMEX.2012.6263880
    [62] Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600–612. https://doi.org/10.1109/TIP.2003.819861 doi: 10.1109/TIP.2003.819861
    [63] X. Cao, Y. Chen, Q. Zhao, D. Meng, Y. Wang, D. Wang, et al., Low-rank matrix factorization under general mixture noise distributions, in 2015 IEEE International Conference on Computer Vision (ICCV), (2015), 1493–1501. https://doi.org/10.1109/ICCV.2015.175
    [64] D. P. Kingma, J. Ba, Adam: a method for stochastic optimization, preprint, arXiv: 1412.6980.
    [65] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, et al., PyTorch: an imperative style, high-performance deep learning library, in Advances in Neural Information Processing Systems, 32 (2019). Available from: https://proceedings.neurips.cc/paper_files/paper/2019/file/bdbca288fee7f92f2bfa9f7012727740-Paper.pdf.
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