Loading [MathJax]/jax/output/SVG/jax.js
Review Special Issues

Recent advances in mechanism/data-driven fault diagnosis of complex engineering systems with uncertainties

  • Received: 26 August 2024 Revised: 23 September 2024 Accepted: 09 October 2024 Published: 21 October 2024
  • MSC : 65C20, 68T10

  • The relentless advancement of modern technology has given rise to increasingly intricate and sophisticated engineering systems, which in turn demand more reliable and intelligent fault diagnosis methods. This paper presents a comprehensive review of fault diagnosis in uncertain environments, focusing on innovative strategies for intelligent fault diagnosis. To this end, conventional fault diagnosis methods are first reviewed, including advances in mechanism-driven, data-driven, and hybrid-driven diagnostic models and their strengths, limitations, and applicability across various scenarios. Subsequently, we provide a thorough exploration of multi-source uncertainty in fault diagnosis, addressing its generation, quantification, and implications for diagnostic processes. Then, intelligent strategies for all stages of fault diagnosis starting from signal acquisition are highlighted, especially in the context of complex engineering systems. Finally, we conclude with insights and perspectives on future directions in the field, emphasizing the need for the continued evolution of intelligent diagnostic systems to meet the challenges posed by modern engineering complexities.

    Citation: Chong Wang, Xinxing Chen, Xin Qiang, Haoran Fan, Shaohua Li. Recent advances in mechanism/data-driven fault diagnosis of complex engineering systems with uncertainties[J]. AIMS Mathematics, 2024, 9(11): 29736-29772. doi: 10.3934/math.20241441

    Related Papers:

    [1] Yeyang Jiang, Zhihua Liao, Di Qiu . The existence of entire solutions of some systems of the Fermat type differential-difference equations. AIMS Mathematics, 2022, 7(10): 17685-17698. doi: 10.3934/math.2022974
    [2] Hong Li, Keyu Zhang, Hongyan Xu . Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables. AIMS Mathematics, 2021, 6(11): 11796-11814. doi: 10.3934/math.2021685
    [3] Zhenguang Gao, Lingyun Gao, Manli Liu . Entire solutions of two certain types of quadratic trinomial q-difference differential equations. AIMS Mathematics, 2023, 8(11): 27659-27669. doi: 10.3934/math.20231415
    [4] Minghui Zhang, Jianbin Xiao, Mingliang Fang . Entire solutions for several Fermat type differential difference equations. AIMS Mathematics, 2022, 7(7): 11597-11613. doi: 10.3934/math.2022646
    [5] Wenju Tang, Keyu Zhang, Hongyan Xu . Results on the solutions of several second order mixed type partial differential difference equations. AIMS Mathematics, 2022, 7(2): 1907-1924. doi: 10.3934/math.2022110
    [6] Jingjing Li, Zhigang Huang . Radial distributions of Julia sets of difference operators of entire solutions of complex differential equations. AIMS Mathematics, 2022, 7(4): 5133-5145. doi: 10.3934/math.2022286
    [7] Nan Li, Jiachuan Geng, Lianzhong Yang . Some results on transcendental entire solutions to certain nonlinear differential-difference equations. AIMS Mathematics, 2021, 6(8): 8107-8126. doi: 10.3934/math.2021470
    [8] Hong Yan Xu, Zu Xing Xuan, Jun Luo, Si Min Liu . On the entire solutions for several partial differential difference equations (systems) of Fermat type in C2. AIMS Mathematics, 2021, 6(2): 2003-2017. doi: 10.3934/math.2021122
    [9] Zheng Wang, Zhi Gang Huang . On transcendental directions of entire solutions of linear differential equations. AIMS Mathematics, 2022, 7(1): 276-287. doi: 10.3934/math.2022018
    [10] Hua Wang, Hong Yan Xu, Jin Tu . The existence and forms of solutions for some Fermat-type differential-difference equations. AIMS Mathematics, 2020, 5(1): 685-700. doi: 10.3934/math.2020046
  • The relentless advancement of modern technology has given rise to increasingly intricate and sophisticated engineering systems, which in turn demand more reliable and intelligent fault diagnosis methods. This paper presents a comprehensive review of fault diagnosis in uncertain environments, focusing on innovative strategies for intelligent fault diagnosis. To this end, conventional fault diagnosis methods are first reviewed, including advances in mechanism-driven, data-driven, and hybrid-driven diagnostic models and their strengths, limitations, and applicability across various scenarios. Subsequently, we provide a thorough exploration of multi-source uncertainty in fault diagnosis, addressing its generation, quantification, and implications for diagnostic processes. Then, intelligent strategies for all stages of fault diagnosis starting from signal acquisition are highlighted, especially in the context of complex engineering systems. Finally, we conclude with insights and perspectives on future directions in the field, emphasizing the need for the continued evolution of intelligent diagnostic systems to meet the challenges posed by modern engineering complexities.



    Consider the following sum of linear ratios optimization problem defined by

    (FP):{min G(x)=pi=1nj=1cijxj+finj=1dijxj+gis. t.  xD={xRn|Axb, x0},

    where p2, A is a m×n order real matrix, b is a m dimension column vector, D is a nonempty bounded polyhedron set, cij,fi,dij, and giR,i=1,2,,p,j=1,2,,n, and for any xD, nj=1dijxj+gi0.

    The problem (FP) has attracted the attention of many researchers and practitioners for decades. One reason is that the problem (FP) and its special form have a wide range of applications in computer vision, portfolio optimization, information theory, and so on [1,2,3]. Another reason is that the problem (FP) is a global optimization problem, which generally has multiple locally optimal solutions that are not globally optimal. In the past several decades, many algorithms have been proposed for globally solving the problem (FP) and its special form. According to the characteristics of these algorithms, they can be classified into the following categories: Parametric simplex algorithm [4], image space analysis method [5], monotonic optimization algorithm [6], branch-and-bound algorithms [7,8,9,10,11], polynomial-time approximation algorithm [12], etc. Jiao et al. [13,14] presented several branch-and-bound algorithms for solving the sum of linear or nonlinear ratios problems; Huang, Shen et al. [15,16] proposed two spatial branch and bound algorithms for solving the sum of linear ratios problems; Jiao et al. [17] designed an outer space methods for globally solving the min-max linear fractional programming problem; Jiao et al. [18,19,20,21] proposed several outer space methods for globally solving the generalized linear fractional programming problem and its special forms. In addition, several novel optimization algorithms [22,23,24] are also proposed for the fractional optimization problems. However, the above-reviewed methods are difficult to solve the problem (FP) with large-size variables. So it is still necessary to put forward a new algorithm for the problem (FP).

    In this paper, based on the branch-and-bound framework, the new linearizing technique, and the image space region reduction technique, an image space branch-and-bound algorithm is proposed for globally solving the problem (FP). Compared with some methods, the algorithm has the following advantages. First, the branching search takes place in the image space Rp of ratios, than in space Rn of variable x, and n usually far exceeds p, this will economize the required computations. Second, based on the characteristics of the problem (EP1) and the structure of the algorithm, an image space region reduction technique is proposed for improving the convergence speed of the algorithm. Third, the computational complexity of the algorithm is analyzed and the maximum iterations of the algorithm are estimated for the first time, which are not available in other articles. In addition, numerical results indicate the computational superiority of the algorithm. Finally, a practical application problem in education investment is solved to verify the usefulness of the proposed algorithm.

    The structure of this paper is as follows. In Section 2, we give the equivalent problem (EP1) of problem (FP) and its linear relaxation problem (LRP). In Section 3, an image space branch-and-bound algorithm is presented, the convergence of the algorithm is proved, and its computational complexity is analysed. Numerical results are reported in Section 4. A practical application from education investment problem is solved to verify the usefulness of the algorithm in Section 5. Finally, some conclusions are given in Section 6.

    To find a global optimal solution of the problem (FP), we need to transform the problem (FP) into the equivalent problems (EP) and (EP1). Next, the fundamental assignment is to globally solve the problem (EP1). To this end, for each i=1,2,,p, we need to compute the minimum value α0i=minxDnj=1cijxj+finj=1dijxj+gi and the maximum value β0i=maxxDnj=1cijxj+finj=1dijxj+gi of each linear ratio nj=1cijxj+finj=1dijxj+gi. Next, we first consider the following linear fractional programs:

    α0i=minxDnj=1cijxj+finj=1dijxj+gi,i=1,2,,p. (1)

    Since any linear ratio is quasi-convex, the problem (1) can attain the minimum value at some vertex of D. Since nj=1dijxj+gi0, without losing generality, we can suppose that nj=1dijxj+gi>0. Thus, for solving the problem (1), for any i{1,2,,p}, let ti=1nj=1dijxj+gi and zj=tixj, then the problem (1) can be converted into the following linear programming problems:

    {min  nj=1cijzj+fitis.t.   nj=1dijzj+giti=1      Azbti. (2)

    Obviously, if x is a global optimal solution of the problem (1), then if and only if (z,ti) is a global optimal solution of the problem (2) with z=tix, and the problems (1) and (2) have the same optimal value. Therefore, α0i can be obtained by solving a linear programming problem (2). Similarly, we can compute the maximum value β0i of each linear ratio over D.

    Let Ω0={ωRpα0iωiβ0i, i=1,2,,p} be the initial image space rectangle, so we can get the equivalent problem (EP) of the problem (FP) as follows:

    (EP):{min Ψ(x,ω)=pi=1ωi,s.t.ωi=nj=1cijxj+finj=1dijxj+gi, i=1,2,,p,xD, ωΩ0.

    Obviously, let ωi=nj=1cijxj+finj=1dijxj+gi, i=1,2,,p, if x is a global optimal solution to the problem (FP), then (x,ω) is a global optimal solution to the problem (EP); conversely, if (x,ω) is a global optimal solution to the problem (EP), then x is a global optimal solution to the problem (FP). Furthermore, from nj=1dijxj+gi0, the problem (EP) can be reformulated as the following equivalent problem (EP1).

    (EP1):{min Ψ(x,ω)=pi=1ωis.t.ωi(nj=1dijxj+gi)=nj=1cijxj+fi, i=1,2,,p,xD, ωΩ0.

    In the following, for globally solving the problem (EP1), we need to construct its linear relaxation problem, which can offer a reliable lower bound in the branch-and-bound searching process. The detailed deriving process of the linear relaxation problem is given as follows.

    For any xD and ωΩ={ωRpαiωiβi, i=1,2,,p}Ω0, we have

    ωi(nj=1dijxj+gi)nj=1,dij>0dijαixj+nj=1,dij<0dijβixj+giωi

    and

    ωi(nj=1dijxj+gi)nj=1,dij>0dijβixj+nj=1,dij<0dijαixj+giωi.

    Consequently, we can construct the linear relaxation problem (LPΩ) of the problem (EP1) over Ω as follows, which is a linear programming problem.

    (LPΩ):{min Ψ(x,ω)=pi=1ωi,s.t.nj=1,dij>0dijαixj+nj=1,dij<0dijβixj+giωinj=1cijxj+fi, i=1,2,,p,nj=1,dij>0dijβixj+nj=1,dij<0dijαixj+giωinj=1cijxj+fi, i=1,2,,p,xD, ωΩ.

    For any Ω={ωRpαiωiβi,i=1,2,,p}Ω0, by the constructing method of the problem (LPΩ), all feasible points of the problem (EP1) over Ω are always feasible to the problem (LPΩ), and the optimal value of the problem (LPΩ) is less than or equal to that of the problem (EP1) over Ω. Thus, the optimal value of the problem (LPΩ) can provide a valid lower bound for that of the problem (EP1) over Ω.

    Without losing generality, for any Ω={ωRpαiωiβi, i=1,2,,p}Ω0, define

    ψi(x,ωi)=ωi(nj=1dijxj+gi)=nj=1dijωixj+giωi,ψ_i(x,ωi)=nj=1,dij>0dijαixj+nj=1,dij<0dijβixj+giωi,¯ψi(x,ωi)=nj=1,dij>0dijβixj+nj=1,dij<0dijαixj+giωi,

    then we have the following Theorem 1.

    Theorem 1. For any i{1,2,,p}, let ψi(x,ωi),ψ_i(x,ωi) and ¯ψi(x,ωi) be defined in the former, and let Δωi=βiαi. Then, we have:

    ψi(x,ωi)ψ_i(x,ωi)0  and  ¯ψi(x,ωi)ψi(x,ωi)0 as  Δωi0.

    Proof. By the definitions of the ¯ψi(x,ωi), ψ_i(x,ωi) and ψi(x,ωi), we can get that

    ψi(x,ωi)ψ_i(x,ωi)=ωi(nj=1dijxj+gi)[nj=1,dij>0dijαixj+nj=1,dij<0dijβixj+giωi]=nj=1,dij>0(ωiαi)dijxjnj=1,dij<0(βiωi)dijxj(βiαi)×(nj=1,dij>0dijxjnj=1,dij<0dijxj),

    which implies that

    ψi(x,ωi)ψ_i(x,ωi)0  as  Δωi0.

    Similarly, we also have

    ¯ψi(x,ωi)ψi(x,ωi)=nj=1,dij>0dijβixj+nj=1,dij<0dijαixj+giωiωi(nj=1dijxj+gi)=nj=1,dij>0(βiωi)dijxjnj=1,dij<0(ωiαi)dijxj(βiαi)×(nj=1,dij>0dijxjnj=1,dij<0dijxj),

    which implies that

    |¯ψi(x,ωi)ψi(x,ωi)|0  as  Δωi0.

    The proof is completed.

    From Theorem 1, the functions ψ_i(x,ωi) and ¯ψi(x,ωi) will infinitely approximate the function ψi(x,ωi) as βα0, which ensures that the problem (LPΩ) will infinitely approximate the problem (EP1) over Ω as βα0.

    In this section, based on the branch-and-bound framework, the linear relaxation problem, and the image space region reduction technique, we propose an image space branch-and-bound algorithm for globally solving the problem (FP).

    To improve the convergence speed of the algorithm, for any investigated image space rectangle Ωk, without losing the global optimal solution of the problem (EP1), the region reduction technique aims at replacing Ωk by a smaller rectangle ˉΩk or judging that the rectangle Ωk does not contain the global optimal solution of problem (EP1). For this purpose, let ˆΦk=pi=1αki, then the smaller rectangle ˉΩk can be derived by the following theorem.

    Theorem 2. Let UBk be the best currently known upper bound at the kth iteration, for any rectangle Ωk=[αk,βk]Ω0, we have the following conclusions:

    (ⅰ) If ˆΦk>UBk, then there exists no global optimal solution to the problem (EP1) over Ωk.

    (ⅱ) If ˆΦkUBk and αkρτkρβkρ for any ρ{1,2,,p}, then there is no global optimal solution to the problem (EP1) over ˆΩk where

    ˆΩk={ωRp|τkρ<ωρβkρ,αkiωiβki,i=1,2,,p,iρ},

    with

    τkρ=UBkˆΦk+αkρ, ρ{1,2,,p}.

    Proof. For any Ωk=[αk,βk]Ω0, we consider the following two cases:

    (ⅰ) If ˆΦk>UBk, then for any feasible solution (ˇx,ˇω) to the problem (EP1) over Ωk, the corresponding target function value Ψ(ˇx,ˇω) to the problem (EP1) over Ωk satisfies that

    Ψ(ˇx,ˇω)=pi=1ˇωipi=1αki=ˆΦk>UBk.

    Thus, there is no global optimal solution to the problem (EP1) over Ωk.

    (ⅱ) If ˆΦkUBk and αkρτkρβkρ for any ρ{1,2,,p}, then for any feasible solution (ˇx,ˇω) of the problem (EP1) over ˆΩk, we have

    Ψ(ˇx,ˇω)=pi=1ˇωi>pi=1,iραki+τkρ=ˆΦkαkρ+τkρ=ˆΦkαkρ+UBkˆΦk+αkρ=UBk.

    Thus, there exists no global optimal solution to the problem (EP1) over ˆΩk.

    From Theorem 2, the investigated image space rectangle Ωk can be replaced by a smaller rectangle ˉΩk=ΩkˆΩk or judged that the rectangle Ωk does not contain the global optimal solution of the problem (EP1).

    Definition 1. Denote xk as a known feasible solution for problem (FP), and denote v as the global optimal value for problem (FP), if G(xk)vε, then xk is called as a global εoptimum solution for problem (FP).

    The basic steps of the proposed image space branch-and-bound algorithm are given as follows.

    Step 0. Given the termination error ε>0 and the initial rectangle Ω0. Solve the problem (LP(Ω0)) to obtain its optimal solution (x0,ˆω0) and optimal value Ψ(x0,ˆω0). Set LB0=Ψ(x0,ˆω0), let ω0i=nj=1cijx0j+finj=1dijx0j+gi,i=1,2,,p,UB0=Ψ(x0,ω0). If UB0LB0ε, then stops, and x0 is a global ε -optimal solution to the problem (FP). Otherwise, let F={(x0,ω0)} be the set of feasible points, and let k=0, T0={Ω0} is the set of all active nodes.

    Step 1. Using the maximum edge binding method of rectangles to subdivide Ωk into two new sub-rectangles Ωk1 and Ωk2, and let H={Ωk1,Ωk2}.

    Step 2. For each rectangle Ωkσ(σ=1,2), use the image space region reduction technique proposed in Section 3.1 to compress its range, and still denote the remaining region of Ωkσ as Ωkσ, if Ωkσ, then solve the problem (LP(Ωkσ)) to obtain its optimal solution (xΩkσ,ˆωΩkσ) and optimal value Ψ(xΩkσ,ˆωΩkσ). Let LB(Ωkσ)=Ψ(xΩkσ,ˆωΩkσ), ωΩkσi=nj=1cijxΩkσj+finj=1dijxΩkσj+gi,i=1,2,,p, and F=F{(xΩkσ,ωΩkσ)}. If UBk<LB(Ωkσ), then let H=HΩkσ,F=F{(xΩkσ,ωΩkσ)} and Tk=Tk{Ωkσ}. Update the upper bound by UBk=min(x,ω)FΨ(x,ω) and denote by (xk,ωk)=argmin(x,ω)FΨ(x,ω). Let Tk=(TkΩk)H and LBk=min{LB(Ω)ΩTk}.

    Step 3. Set Tk+1={ΩUBkLB(Ω)>ε,ΩTk}. If Tk+1=, then the algorithm stops with that xk is a global ε -optimal solution to the problem (FP). Otherwise, select the rectangle Ωk+1 satisfying that Ωk+1=argminΩTk+1LB(Ω), set k=k+1, and return to Step 1.

    In the following, we will discuss the global convergence of the algorithm.

    Theorem 3. Let v be the global optimal value of the problem (FP), Algorithm ISBBA either ends at the global optimal solution of the problem (FP) or generates an infinite sequence of feasible solutions so that its any accumulation point is the global optimal solution of the problem (FP).

    Proof. If Algorithm ISBBA is terminated finitely after k iterations, then when Algorithm ISBBA is terminated, we obtain a better feasible solution xk of the problem (FP) and a better feasible solution (xk,ωk) of the problem (EP) with ωki=nj=1cijxkj+finj=1dijxkj+gi,i=1,2,,p, respectively. By the termination conditions, the updating method of the upper bound, and the steps of Algorithm ISBBA, we can get the following inequalities:

    LBkv, vΨ(xk,ωk), G(xk)=Ψ(xk,ωk)=vk, vkεLBk.

    By combining the above equalities and inequalities, we can get

    G(xk)ε=Ψ(xk,ωk)εLBkvΨ(xk,ωk)=G(xk).

    Therefore, we can get that xk is an ϵglobal optimal solution of the problem (FP).

    If Algorithm ISBBA produces an infinite sequence of feasible solutions {xk} for the problem (FP) and an infinite sequence of feasible solutions {(xk,ωk)} for the problem (EP) with ωki=nj=1cijxkj+finj=1dijxkj+gi,i=1,2,,p, respectively. Without losing generality, let x be an accumulation point of {xk}, we can get that limkxk=x.

    By the continuity of the nj=1cijxj+finj=1dijxj+gi, nj=1cijxkj+finj=1dijxkj+gi=ωki[αki,βki],i=1,2,,p, and the exhaustiveness of the partitioning method, we can get that

    nj=1cijxj+finj=1dijxj+gi=limknj=1cijxkj+finj=1dijxkj+gi=limkωki=limk[αki,βki]=limkk[αki,βki]=ωi.

    Thus, (x,ω) is a feasible solution for the problem (EP). Also since {LBk} is an increasing lower bound sequence satisfying that LBkv, we can follow that

    Ψ(x,ω)vlimkLBk=limkΨ(xk,ˆωk)=Ψ(x,ω). (3)

    Hence, by the method of updating upper bound and the continuity of G(x), we can get that

    limkvk=limkpi=1ωki=limkΨ(xk,ωk)=Ψ(x,ω)=G(x)=limkG(xk). (4)

    From (3) and (4), we can get that

    limkvk=v=G(x)=limkG(xk)=Ψ(x,ω)=limkLBk.

    Therefore, any accumulation point x of the infinite sequence {xk} of feasible solutions is a global optimal solution for the problem (FP), and the proof of the theorem is completed.

    In this subsection, by analysing the computational complexity of Algorithm ISBBA, we estimate the maximum iteration times of Algorithm ISBBA. For convenience, we first define the size of a rectangle

    Ω={ωRp|αiωiβi,i=1,2,,p}Ω0

    as

    δ(Ω):=max{βiαi,i=1,2,,p}.

    Theorem 4. For any given termination error ε>0, if there exists a rectangle Ωk, which is formed by Algorithm ISBBA at the kth iteration, and which is satisfied with δ(Ωk)εp, then we have that

    UBkLB(Ωk)ε,

    where LB(Ωk) represents the optimal value for the problem (LP(Ωk)), and UBk represents the currently known best upper bound of the global optimal value of the problem (EP).

    Proof. Without loss of generality, assume that (xk,ˆωk) is the optimal solution of the linear relaxation programming (LP(Ωk)), and let ωki=nj=1cijxkj+finj=1dijxkj+gi,i=1,2,,p, then (xk,ωk) must be a feasible solution to the problem (EP(Ωk)).

    By utilizing the definitions of UBk and LB(Ωk), we have that

    Ψ(xk,ωk)UBkLB(Ωk)=Ψ(xk,ˆωk).

    Thus, by steps of Algorithm ISBBA, we can follow that

    UBkLB(Ωk)Ψ(xk,ωk)Ψ(xk,ˆωk)=pi=1ωkipi=1ˆωkipi=1(βkiαki)pi=1δ(Ωk)=pδ(Ωk).

    Furthermore, from the above formula and δ(Ωk)εp, we can get that

    UBkLB(Ωk)pi=1(δ(Ωk))=pδ(Ωk)ε,

    and the proof of the theorem is completed.

    According to Step 3 of Algorithm ISBBA, from Theorem 4, if δ(Ωk)εp, then it can be seen easily that the rectangle Ωk will be deleted. Thus, if the sizes of all sub-rectangles Ω generated by Algorithm ISBBA meet δ(Ω)εp, then Algorithm ISBBA will stop. The maximum iteration times of Algorithm ISBBA can be estimated by using Theorem 4, see Theorem 5 for details.

    Theorem 5. Given the termination error ε>0, Algorithm ISBBA can find an ε-global optimal solution to the problem (FP) after at most

    Λ=2pi=1log2p(β0iα0i)ε1

    iterations, where Ω0={ωRp|α0iωiβ0i,i=1,2,,p}.

    Proof. Without losing generality, we assume that the ith edge of the rectangle Ω0 is continuously selected for dividing γi times, and suppose that after γi iterations, there exists a sub-interval Ωγii=[αγii,βγii] of the interval Ω0i=[α0i,β0i] such that

    βγiiαγiiεp,  for every i=1,2,,p. (5)

    From the partitioning process of Algorithm ISBBA, we have that

    βγiiαγii=12γi(β0iα0i),  for every i=1,2,,p. (6)

    From (5) and (6), we can get that

    12γi(β0iα0i)εp,  for every i=1,2,,p,

    i.e.,

    γilog2p(β0iα0i)ε,  for every i=1,2,,p.

    Next, we let

    ˉγi=log2p(β0iα0i)ε,   i=1,2,,p.

    Let Λ1=pi=1ˉγi, then after Λ1 iterations, Algorithm ISBBA will generate at most Λ1+1 sub-rectangles, denoting these sub-rectangles as Ω1,Ω2,,ΩΛ1+1, which must meet

    δ(Ωt)=2Λ1tδ(ΩΛ1)=2Λ1tδ(ΩΛ1+1), t=Λ1,Λ11,,2,1,

    where δ(ΩΛ1)=δ(ΩΛ1+1)=max{βˉγiiαˉγii,i=1,2,,p} and

    Ω0=Λ1+1tΩt. (7)

    Furthermore, put these Λ1+1 sub-rectangles into the set TΛ1+1, i.e.,

    TΛ1+1={Ωt,t=1,2,,Λ1+1}.

    By (5), we have that

    δ(ΩΛ1)=δ(ΩΛ1+1)εp. (8)

    Thus, by (8), Theorem 4, and Step 3 of Algorithm ISBBA, the sub-rectangles ΩΛ1 and ΩΛ1+1 have been examined clearly, which should be discarded from the partitioning set TΛ1+1. Next, the remaining sub-rectangles are placed in the set TΛ1, where

    TΛ1=TΛ1+1{ΩΛ1,ΩΛ1+1}={Ωt,t=1,,Λ11},

    and the remaining sub-rectangles Ωt (t=1,,Λ11) will be examined further.

    Next, consider the sub-rectangle ΩΛ11, by using the branching rule, we can subdivide the sub-rectangle ΩΛ11 into two sub-rectangles ΩΛ11,1 and ΩΛ11,2, which satisfies that

    ΩΛ11=ΩΛ11,1ΩΛ11,2

    and

    δ(ΩΛ11)=2δ(ΩΛ11,1)=2δ(ΩΛ11,2)=2δ(ΩΛ1)=2δ(ΩΛ1+1)εp.

    Therefore, after Λ1+(211) iterations, the sub-rectangle ΩΛ11 has been examined clearly. By (8), Theorem 4, and Step 3 of Algorithm ISBBA, ΩΛ11 should be discarded from the partitioning set TΛ1. Furthermore, the remaining sub-rectangles will be placed in the set TΛ11, where

    TΛ11=TΛ1{ΩΛ11}=TΛ1+1{ΩΛ11,ΩΛ1,ΩΛ1+1}={Ωt,t=1,,Λ12}.

    Similarly, after Algorithm ISBBA executed Λ1+(211)+(221) iterations, the sub-rectangle ΩΛ12 has been examined clearly, and which should be discarded from the partitioning set TΛ11. Furthermore, the remaining sub-rectangles will be put into the set TΛ12, where

    TΛ12=TΛ11{ΩΛ12}=TΛ1+1{ΩΛ12,ΩΛ11,ΩΛ1,ΩΛ1+1}={Ωt,t=1,,Λ13}.

    Reduplicate the above procedures, until all sub-rectangles Ωt(t=1,2,,Λ1+1) are completely eliminated from Ω0. Thus, by (7), after at most

    Λ=Λ1+(211)+(221)+(231)++(2Λ111)=2Λ11=2pi=1log2p(β0iα0i)ε1

    iterations, Algorithm ISBBA will stop, and the proof of the theorem is completed.

    Remark 1. By Theorem 5, from the above complexity analysis of Algorithm ISBBA, the running time of Algorithm ISBBA is bounded by 2ΛT(m+2p,n+p) for finding an ε-global optimal solution for the problem (FP), where T(m+2p,n+p) represents the time taken to solve a linear programming problem with (n+p) variables and (m+2p) constraints.

    In this section, we numerically compare Algorithm ISBBA with the software "BARON" [25] and the branch-and-bound-algorithm presented in Jiao & Liu [10], denoted by Algorithm BBA-J. All used algorithms are coded in MATLAB R2014a, all test problems are solved on the same microcomputer with Intel(R) Core(TM) i5-7200U CPU @2.50GHz processor and 16 GB RAM. We set the maximum time limit for all algorithms to 4000 seconds. These test problems and their numerical results are listed as follows.

    Test Problem 1 is a problem with large-size variables, with the given termination error ϵ=102, numerical comparisons among Algorithm ISBBA, BBA-J, and BARON are listed in Table 1, respectively. Test Problem 2 is a problem with the large-size number of ratios, with the given termination error ϵ=103, numerical comparisons among Algorithm ISBBA and BARON are listed in Table 2, respectively. For test Problems 1 and 2, we solved arbitrary ten independently generated test examples and recorded their best, worst, and average results among these ten test examples, and we highlighted in bold the winners of average results in their numerical comparisons. What needs to be pointed out here is that "" represents that the used algorithm failed to terminate in 4000s. From the numerical results for Problem 1 in Table 1, first, we see that the software BARON is more time-consuming than Algorithm ISBBA, though the number of iterations for BARON is smaller than Algorithm ISBBA. Second, in terms of computational performance, Algorithm ISBBA outperforms Algorithm BBA-J. Especially, when we fixed m=100, let p=2 and n=8000,10000 or 20000, or let p=3 and n=8000, BARON failed to terminate in 4000s for all arbitrary ten independently generated test examples; when we fixed m=100, let p=3 and n=8000, Algorithm BBA-J and BARON all failed to terminate in 4000s for all arbitrary ten independently generated test examples, but in all cases, Algorithm ISBBA can globally solve all arbitrary ten independently generated test examples.

    Table 1.  Numerical comparisons among Algorithm ISBBA, BBA-J, and BARON on Problem 1.
    (p,m,n) algorithms iteration of algorithm CPU time in seconds
    min. ave. max. min. ave. max.
    (2,100, 5000) BBA-J 40 104.8 244 186.21 530.14 1244.53
    BARON 1 1.2 3 920.05 1083.93 1408.27
    ISBBA 30 37.5 46 139.76 194.13 253.43
    (2,100, 8000) BBA-J 32 84.9 139 276.25 802.90 1323.32
    BARON
    ISBBA 29 38.2 48 261.68 355.27 487.13
    (2,100, 10000) BBA-J 35 76.6 112 405.80 933.54 1414.22
    BARON
    ISBBA 31 36 45 355.57 405.93 510.31
    (2,100, 20000) BBA-J 41 69.4 105 1239.04 2216.69 3495.84
    BARON
    ISBBA 32 36.2 41 950.13 1075.0 1233.14
    (3,100, 5000) BBA-J
    BARON 3 9.8 31 1320.47 2310.83 3113.8
    ISBBA 65 249.9 482 398.56 1815.11 3600.96
    (3,100, 8000) BBA-J
    BARON
    ISBBA 95 217.9 338 1030.95 2564.92 3985.77

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical comparisons among Algorithm ISBBA and BARON on Problem 2.
    (p,m,n) algorithms iteration of algorithm CPU time in seconds
    min. ave. max. min. ave. max.
    (10,100,300) BARON 3 9.2 13 8.28 12.66 17.64
    ISBBA 9 13.6 19 5.28 8.87 12.7
    (10,100,400) BARON 9 35.8 93 22.28 30.86 42.33
    ISBBA 10 16 25 6.90 12.65 20.66
    (10,100,500) BARON
    ISBBA 10 17.4 30 8.07 15.89 26.52
    (15,100,400) BARON 11 34 157 36.14 47.92 79.81
    ISBBA 50 121.6 201 46.78 118.75 201.66
    (15,100,500) BARON
    ISBBA 49 118.1 258 54.57 137.92 303.49
    (20,100,300) BARON 5 14 17 22.53 36.14 51.11
    ISBBA 157 321.2 861 126.19 255.46 694.20
    (20,100,400) BARON
    ISBBA 99 399.9 1134 99.06 425.77 1199.2

     | Show Table
    DownLoad: CSV

    From the numerical results for Problem 2 in Table 2, we can see that when we fixed p=10 and n=500, or p=15 and n=500, or p=20 and n=400, the software BARON failed to terminate in 4000s for all arbitrary ten independently generated examples, but Algorithm ISBBA can successfully find the globally optimal solutions of all arbitrary ten independently generated tests. It should be noted that, when p is larger for Problem 2, Algorithm BBA-J failed to solve all arbitrary ten tests in 4000s. Therefore, we just report the computational comparisons among Algorithm ISBBA and BARON in Table 2, this indicates the robustness and stability of Algorithm ISBBA.

    Problem 1.

    {min  pi=1nj=1cijx+finj=1dijx+gis.t.   Axb,  x0,

    where cij,dij,fi, and giR,i=1,2,,p; ARm×n, bRm; cij, dij, and all elements of A are all randomly generated from [0,10]; all elements of b are equal to 10; fi and gi,i=1,2,,p, are all randomly generated from [0,1].

    Problem 2.

    {min  pi=1nj=1γijxj+ξinj=1δijxj+ηis.t.   Axb,  x0,

    where γij,ξi,δij,ηiR, i=1,2,,p,j=1,2,,n; ARm×n, bRm; all γij and δij are randomly generated from [0.1,0.1]; all elements of A are randomly generated from [0.01,1]; all elements of b are equal to 10; all ξi and ηi satisfies that nj=1γijxj+ξi>0 and nj=1δijxj+ηi>0.

    Consider finding the optimal solution of the education investment problem, whose mathematical modelling can be given as below:

    {minG(x)=pj=1ni=1cjixini=1djixi=pj=1cTjxdTjxs.t.ni=1xi1,Axb, x0,

    where cji (j=1,2,,p,i=1,2,,n) is the ith invested fund of the jth education investment, xi (i=1,2,,n) is the investment percentage of the ith education investment, dji (j=1,2,,p,i=1,2,,n) is the ith output fund of the jth education investment.

    The parameters of an education investment problem are given as below:

    p=2; n=3; c=[0.1,0.2,0.4;0.1,0.1,0.2]; d=[0.1,0.1,0.1;0.1,0.3,0.1];A=[1,1,1;1,1,1;12,5,12;12,12,7;6,1,1]; b=[1;1;34.8;29.1;4.1].

    Using the presented algorithm in this article to solve the above problem, the global optimal solution can be obtained as below:

    x=(0.7286,0.0000,0.2714).

    It is to say, the optimal investment percentage of these three kinds of education investment is 0.7286,0.0000,0.2714, respectively.

    We study the problem (FP). Based on the image space search, the new linearizing technique, and the image space region reduction technique, we propose an image space branch-and-bound algorithm. In contrast to the existing algorithm, the proposed algorithm can find an ϵ-approximate global optimal solution of the problem (FP) in at most (2pi=1log2p(β0iα0i)ε1) iterations. Numerical results show the computational superiority of the algorithm.

    A potential field for future research lies in investigating the existence of analogous linear or convex relaxation problems with closed-form solutions in cases where both the numerators and denominators are nonlinear functions. Furthermore, there is also need to design an efficient algorithm for globally solving generalized nonlinear ratios optimization problems with non-convex feasible region, as well as for more general non-convex ratios optimization problems under uncertain variable environments.

    All authors of this article have been contributed equally. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This paper is supported by the National Natural Science Foundation of China (12001167), the China Postdoctoral Science Foundation (2018M642753), the Key Scientific and Technological Research Projects in Henan Province (232102211085; 202102210147), the Research Project of Nanyang Normal University(2024ZX014).

    The author declares no conflict of interest.



    [1] L. Jia, Q. Gao, Z. P. Liu, H. B. Tan, L. W. Zhou, Multidisciplinary fault diagnosis of complex engineering systems: A case study of nuclear power plants, Int. J. Ind. Ergon., 80 (2020), 103060. https://doi.org/10.1016/j.ergon.2020.103060 doi: 10.1016/j.ergon.2020.103060
    [2] Y. B. Li, B. Li, J. C. Ji, H. Kalhori, Advanced fault diagnosis and health monitoring techniques for complex engineering systems, Sensors, 22 (2022), 10002. https://doi.org/10.3390/s222410002 doi: 10.3390/s222410002
    [3] C. Wang, H. G. Matthies, Random model with fuzzy distribution parameters for hybrid uncertainty propagation in engineering systems, Comput. Meth. Appl. Mech. Eng., 359 (2020), 112673. https://doi.org/10.1016/j.cma.2019.112673 doi: 10.1016/j.cma.2019.112673
    [4] F. Villecco, A. Pellegrino, Evaluation of uncertainties in the design process of complex mechanical systems, Entropy, 19 (2017), e19090475. https://doi.org/10.3390/e19090475 doi: 10.3390/e19090475
    [5] E. Hüllermeier, W. Waegeman, Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods, Mach. Learn., 110 (2021), 457–506. https://doi.org/10.1007/s10994-021-05946-3 doi: 10.1007/s10994-021-05946-3
    [6] H. R. Fan, C. Wang, S. H. Li, Novel method for reliability optimization design based on rough set theory and hybrid surrogate model, Comput. Meth. Appl. Mech. Eng., 429 (2024), 117170. https://doi.org/10.1016/j.cma.2024.117170 doi: 10.1016/j.cma.2024.117170
    [7] M. Mansouri, R. Fezai, M. Trabelsi, M. Hajji, M.-F. Harkat, H. Nounou, et al., A novel fault diagnosis of uncertain systems based on interval Gaussian process regression: Application to wind energy conversion systems, IEEE Access, 8 (2020), 219672–219679. https://doi.org/10.1109/access.2020.3042101 doi: 10.1109/access.2020.3042101
    [8] X. X. Liu, Y. T. Ju, X. H. Liu, S. Miao, W. G. Zhang, An imu fault diagnosis and information reconstruction method based on analytical redundancy for autonomous underwater vehicle, IEEE Sens. J., 22 (2022), 12127–12138. https://doi.org/10.1109/jsen.2022.3174340 doi: 10.1109/jsen.2022.3174340
    [9] D. Yu, Fault diagnosis for a hydraulic drive system using a parameter-estimation method, Control Eng. Practice, 5 (1997), 1283–1291. https://doi.org/10.1016/s0967-0661(97)84367-5 doi: 10.1016/s0967-0661(97)84367-5
    [10] G. C. Zhang, L. Chen, K. K. Liang, Fault monitoring and diagnosis of aerostat actuator based on pca and state observer, Int. J. Model. Identif. Control, 32 (2019), 145. https://doi.org/10.1504/ijmic.2019.102367 doi: 10.1504/ijmic.2019.102367
    [11] Y. Song, M. Y. Zhong, J. Chen, Y. Liu, An alternative parity space-based fault diagnosability analysis approach for linear discrete time systems, IEEE Access, 6 (2018), 16110–16118. https://doi.org/10.1109/access.2018.2816970 doi: 10.1109/access.2018.2816970
    [12] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, A review of process fault detection and diagnosis, Comput. Chem. Eng., 27 (2003), 313–326. https://doi.org/10.1016/s0098-1354(02)00161-8 doi: 10.1016/s0098-1354(02)00161-8
    [13] S. W. Pan, D. Xiao, S. T. Xing, S. S. Law, P. Y. Du, Y. J. Li, A general extended kalman filter for simultaneous estimation of system and unknown inputs, Eng. Struct., 109 (2016), 85–98. https://doi.org/10.1016/j.engstruct.2015.11.014 doi: 10.1016/j.engstruct.2015.11.014
    [14] E. Walker, S. Rayman, R. E. White, Comparison of a particle filter and other state estimation methods for prognostics of lithium-ion batteries, J. Power Sources, 287 (2015), 1–12. https://doi.org/10.1016/j.jpowsour.2015.04.020 doi: 10.1016/j.jpowsour.2015.04.020
    [15] S. Nolan, A. Smerzi, L. Pezzè, A machine learning approach to Bayesian parameter estimation, npj Quantum Inform., 7 (2021), 169. https://doi.org/10.1038/s41534-021-00497-w doi: 10.1038/s41534-021-00497-w
    [16] R. Tarantino, F. Szigeti, E. Colina-Morles, Generalized luenberger observer-based fault-detection filter design: An industrial application, Control Eng. Practice, 8 (2000), 665–671. https://doi.org/10.1016/s0967-0661(99)00181-1 doi: 10.1016/s0967-0661(99)00181-1
    [17] L. A. Rusinov, N. V. Vorobiev, V. V. Kurkina, Fault diagnosis in chemical processes and equipment with feedbacks, Chemometrics Intell. Lab. Syst., 126 (2013), 123–128. https://doi.org/10.1016/j.chemolab.2013.03.015 doi: 10.1016/j.chemolab.2013.03.015
    [18] F. Pierri, G. Paviglianiti, F. Caccavale, M. Mattei, Observer-based sensor fault detection and isolation for chemical batch reactors, Eng. Appl. Artif. Intell., 21 (2008), 1204–1216. https://doi.org/10.1016/j.engappai.2008.02.002 doi: 10.1016/j.engappai.2008.02.002
    [19] H. M. Odendaal, T. Jones, Actuator fault detection and isolation: An optimised parity space approach, Control Eng. Practice, 26 (2014), 222–232. https://doi.org/10.1016/j.conengprac.2014.01.013 doi: 10.1016/j.conengprac.2014.01.013
    [20] C. J. Duan, Z. Y. Fei, J. C. Li, A variable selection aided residual generator design approach for process control and monitoring, Neurocomputing, 171 (2016), 1013–1020. https://doi.org/10.1016/j.neucom.2015.07.042 doi: 10.1016/j.neucom.2015.07.042
    [21] P. Zhang, S. X. Ding, Disturbance decoupling in fault detection of linear periodic systems, Automatica, 43 (2007), 1410–1417. https://doi.org/10.1016/j.automatica.2007.01.005 doi: 10.1016/j.automatica.2007.01.005
    [22] Q. Wang, C. Taal, O. Fink, Integrating expert knowledge with domain adaptation for unsupervised fault diagnosis, IEEE Trans. Instrum. Meas., 71 (2022), 1–12. https://doi.org/10.1109/tim.2021.3127654 doi: 10.1109/tim.2021.3127654
    [23] P. Zhao, X. D. Mu, Z. R. Yin, Z. X. Yi, An approach of fault diagnosis for system based on fuzzy fault tree, 2008 International Conference on MultiMedia and Information Technology, Three Gorges, China, 2008,697–700. https://doi.org/10.1109/mmit.2008.142
    [24] Z. N. Lin, Y. X. Wang, H. Q. Xu, F. R. Wei, A novel reduced-order analytical fault diagnosis model for power grid, IEEE Access, 12 (2024), 59521–59532. https://doi.org/10.1109/access.2024.3392905 doi: 10.1109/access.2024.3392905
    [25] C. Cheng, X. Y. Qiao, H. Luo, W. X. Teng, M. L. Gao, B. C. Zhang, et al., A semi-quantitative information based fault diagnosis method for the running gears system of high-speed trains, IEEE Access, 7 (2019), 38168–38178. https://doi.org/10.1109/access.2019.2906976 doi: 10.1109/access.2019.2906976
    [26] J. P. Shi, W. G. Tong, D. L. Wang, Design of the transformer fault diagnosis expert system based on fuzzy reasoning, 2009 International Forum on Computer Science-Technology and Applications, Chongqing, China, 2009,110–114. https://doi.org/10.1109/ifcsta.2009.34
    [27] A. R. Sahu, S. K. Palei, A. Mishra, Data-driven fault diagnosis approaches for industrial equipment: A review, Expert Syst., 41 (2024), 13360. https://doi.org/10.1111/exsy.13360 doi: 10.1111/exsy.13360
    [28] G. Wang, J. Y. Zhao, J. H. Yang, J. F. Jiao, J. L. Xie, F. Feng, Multivariate statistical analysis based cross voltage correlation method for internal short-circuit and sensor faults diagnosis of lithium-ion battery system, J. Energy Storage, 62 (2023), 106978. https://doi.org/10.1016/j.est.2023.106978 doi: 10.1016/j.est.2023.106978
    [29] Z. Zhang, X. He, Active fault diagnosis for linear systems: Within a signal processing framework, IEEE Trans. Instrum. Meas., 71 (2022), 1–9. https://doi.org/10.1109/tim.2022.3150889 doi: 10.1109/tim.2022.3150889
    [30] R. N. Liu, B. Y. Yang, E. Zio, X. F. Chen, Artificial intelligence for fault diagnosis of rotating machinery: A review, Mech. Syst. Signal Proc., 108 (2018), 33–47. https://doi.org/10.1016/j.ymssp.2018.02.016 doi: 10.1016/j.ymssp.2018.02.016
    [31] Y. Q. Liu, B. Liu, X. J. Zhao, M. Xie, A mixture of variational canonical correlation analysis for nonlinear and quality-relevant process monitoring, IEEE Trans. Ind. Electron., 65 (2018), 6478–6486. https://doi.org/10.1109/tie.2017.2786253 doi: 10.1109/tie.2017.2786253
    [32] G. Lee, C. H. Han, E. S. Yoon, Multiple-fault diagnosis of the tennessee eastman process based on system decomposition and dynamic pls, Ind. Eng. Chem. Res., 43 (2004), 8037–8048. https://doi.org/10.1021/ie049624u doi: 10.1021/ie049624u
    [33] G. Yu, C. N. Li, J. Sun, Machine fault diagnosis based on Gaussian mixture model and its application, Int. J. Adv. Manuf. Technol., 48 (2010), 205–212. https://doi.org/10.1007/s00170-009-2283-5 doi: 10.1007/s00170-009-2283-5
    [34] W. Deng, S. J. Zhang, H. M. Zhao, X. H. Yang, A novel fault diagnosis method based on integrating empirical wavelet transform and fuzzy entropy for motor bearing, IEEE Access, 6 (2018), 35042–35056. https://doi.org/10.1109/access.2018.2834540 doi: 10.1109/access.2018.2834540
    [35] J. B. Guo, Fault diagnosis method of flexible converter valve equipment based on ensemble empirical mode decomposition and temporal convolutional networks, J. Electr. Syst., 20 (2024), 344–352. https://doi.org/10.52783/jes.2386 doi: 10.52783/jes.2386
    [36] D. J. Yu, M. Wang, X. M. Cheng, A method for the compound fault diagnosis of gearboxes based on morphological component analysis, Measurement, 91 (2016), 519–531. https://doi.org/10.1016/j.measurement.2016.05.087 doi: 10.1016/j.measurement.2016.05.087
    [37] L. Ciabattoni, F. Ferracuti, A. Freddi, A. Monteriu, Statistical spectral analysis for fault diagnosis of rotating machines, IEEE Trans. Ind. Electron., 65 (2018), 4301–4310. https://doi.org/10.1109/tie.2017.2762623 doi: 10.1109/tie.2017.2762623
    [38] W. E. Sanders, T. Burton, A. Khosousi, S. Ramchandani, Machine learning: At the heart of failure diagnosis, Curr. Opin. Cardiol., 36 (2021), 227–233. https://doi.org/10.1097/hco.0000000000000833 doi: 10.1097/hco.0000000000000833
    [39] Y. G. Lei, B. Yang, X. W. Jiang, F. Jia, N. P. Li, A. K. Nandi, Applications of machine learning to machine fault diagnosis: A review and roadmap, Mech. Syst. Signal Proc., 138 (2020), 106587. https://doi.org/10.1016/j.ymssp.2019.106587 doi: 10.1016/j.ymssp.2019.106587
    [40] Z. N. An, F. Wu, C. Zhang, J. H. Ma, B. Sun, B. H. Tang, et al., Deep learning-based composite fault diagnosis, IEEE Jour. Emer. Select. Top. Circu. Syste., 13 (2023), 572–581. https://doi.org/10.1109/jetcas.2023.3262241 doi: 10.1109/jetcas.2023.3262241
    [41] D. T. Hoang, H. J. Kang, A survey on deep learning based bearing fault diagnosis, Neurocomputing, 335 (2019), 327–335. https://doi.org/10.1016/j.neucom.2018.06.078 doi: 10.1016/j.neucom.2018.06.078
    [42] X. Y. Fan, J. Li, H. Hao, Review of piezoelectric impedance based structural health monitoring: Physics-based and data-driven methods, Adv. Struct. Eng., 24 (2021), 3609–3626. https://doi.org/10.1177/13694332211038444 doi: 10.1177/13694332211038444
    [43] Q. Ni, X. M. Li, Z. W. Chen, Z. L. Zhao, L. L. Lai, A mechanism and data hybrid-driven method for main circuit ground fault diagnosis in electrical traction system, IEEE Trans. Ind. Electron., 70 (2023), 12806–12815. https://doi.org/10.1109/tie.2023.3260356 doi: 10.1109/tie.2023.3260356
    [44] D. An, N. H. Kim, J. H. Choi, Practical options for selecting data-driven or physics-based prognostics algorithms with reviews, Reliab. Eng. Syst. Saf., 133 (2015), 223–236. https://doi.org/10.1016/j.ress.2014.09.014 doi: 10.1016/j.ress.2014.09.014
    [45] J. Guo, Z. Y. Li, M. Y. Li, A review on prognostics methods for engineering systems, IEEE Trans. Reliab., 69 (2020), 1110–1129. https://doi.org/10.1109/tr.2019.2957965 doi: 10.1109/tr.2019.2957965
    [46] L. Kou, C. Liu, G. W. Cai, J. N. Zhou, Q. D. Yuan, S. M. Pang, Fault diagnosis for open-circuit faults in npc inverter based on knowledge-driven and data-driven approaches, IET Power Electron., 13 (2020), 1236–1245. https://doi.org/10.1049/iet-pel.2019.0835 doi: 10.1049/iet-pel.2019.0835
    [47] X. X. Xiao, C. H. Li, J. Huang, T. Yu, Fault diagnosis of rolling bearing based on knowledge graph with data accumulation strategy, IEEE Sens. J., 22 (2022), 18831–18840. https://doi.org/10.1109/JSEN.2022.3201839 doi: 10.1109/JSEN.2022.3201839
    [48] K. Sachin, M. Torres, Y. C. Chan, M. Pecht, A hybrid prognostics methodology for electronic products, 2008 IEEE International Joint Conference on Neural Networks, Hong Kong, China, 2008, 3479–3485. https://doi.org/10.1109/IJCNN.2008.4634294
    [49] S. F. Cheng, M. Pecht, A fusion prognostics method for remaining useful life prediction of electronic products, 2009 IEEE International Conference on Automation Science and Engineering, Bangalore, India, 2009,102–107. https://doi.org/10.1109/COASE.2009.5234098
    [50] H. G. Zhang, R. Kang, M. Pecht, A hybrid prognostics and health management approach for condition-based maintenance, 2009 IEEE International Conference on Industrial Engineering and Engineering Management, Hong Kong, China, 2009, 1165–1169. https://doi.org/10.1109/ieem.2009.5372976
    [51] M. A. Chao, C. Kulkarni, K. Goebel, O. Fink, Fusing physics-based and deep learning models for prognostics, Reliab. Eng. Syst. Saf., 217 (2022), 107961. https://doi.org/10.1016/j.ress.2021.107961 doi: 10.1016/j.ress.2021.107961
    [52] T. T. Li, Y. Zhao, C. B. Zhang, J. Luo, X. J. Zhang, A knowledge-guided and data-driven method for building hvac systems fault diagnosis, Build. Environ., 198 (2021), 107850. https://doi.org/10.1016/j.buildenv.2021.107850 doi: 10.1016/j.buildenv.2021.107850
    [53] L. H. Ye, X. Ma, C. L. Wen, Rotating machinery fault diagnosis method by combining time-frequency domain features and cnn knowledge transfer, Sensors, 21 (2021), 8168. https://doi.org/10.3390/s21248168 doi: 10.3390/s21248168
    [54] W. Xu, Y. Wan, T. Y. Zuo, X. M. Sha, Transfer learning based data feature transfer for fault diagnosis, IEEE Access, 8 (2020), 76120–76129. https://doi.org/10.1109/ACCESS.2020.2989510 doi: 10.1109/ACCESS.2020.2989510
    [55] X. P. Niu, R. Z. Wang, D. Liao, S. P. Zhu, X. C. Zhang, B. Keshtegar, Probabilistic modeling of uncertainties in fatigue reliability analysis of turbine bladed disks, Int. J. Fatigue, 142 (2021), 105912. https://doi.org/10.1016/j.ijfatigue.2020.105912 doi: 10.1016/j.ijfatigue.2020.105912
    [56] M. Valdenegro-Toro, D. S. Mori, A deeper look into aleatoric and epistemic uncertainty disentanglement, 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), New Orleans, LA, USA, 2022, 1508–1516. https://doi.org/10.1109/cvprw56347.2022.00157
    [57] C. Wang, H. G. Matthies, M. H. Xu, Y. L. Li, Dual interval-and-fuzzy analysis method for temperature prediction with hybrid epistemic uncertainties via polynomial chaos expansion, Comput. Meth. Appl. Mech. Eng., 336 (2018), 171–186. https://doi.org/10.1016/j.cma.2018.03.013 doi: 10.1016/j.cma.2018.03.013
    [58] A. D. Kiureghian, O. Ditlevsen, Aleatory or epistemic? Does it matter?, Struct. Saf., 31 (2009), 105–112. https://doi.org/10.1016/j.strusafe.2008.06.020 doi: 10.1016/j.strusafe.2008.06.020
    [59] M. E. Paté-Cornell, Uncertainties in risk analysis: Six levels of treatment, Reliab. Eng. Syst. Saf., 54 (1996), 95–111. https://doi.org/10.1016/s0951-8320(96)00067-1 doi: 10.1016/s0951-8320(96)00067-1
    [60] C. Wang, H. R. Fan, X. Qiang, A review of uncertainty-based multidisciplinary design optimization methods based on intelligent strategies, Symmetry-Basel, 15 (2023), 1875. https://doi.org/10.3390/sym15101875 doi: 10.3390/sym15101875
    [61] C. Wang, X. Qiang, M. H. Xu, T. Wu, Recent advances in surrogate modeling methods for uncertainty quantification and propagation, Symmetry-Basel, 14 (2022), 1219. https://doi.org/10.3390/sym14061219 doi: 10.3390/sym14061219
    [62] D. Di Francesco, M. Girolami, A. B. Duncan, M. Chryssanthopoulos, A probabilistic model for quantifying uncertainty in the failure assessment diagram, Struct. Saf., 99 (2022), 102262. https://doi.org/10.1016/j.strusafe.2022.102262 doi: 10.1016/j.strusafe.2022.102262
    [63] P. Manfredi, Probabilistic uncertainty quantification of microwave circuits using Gaussian processes, IEEE Trans. Microw. Theory Tech., 71 (2023), 2360–2372. https://doi.org/10.1109/TMTT.2022.3228953 doi: 10.1109/TMTT.2022.3228953
    [64] J. S. Wu, G. E. Apostolakis, D. Okrent, Uncertainties in system analysis: Probabilistic versus nonprobabilistic theories, Reliab. Eng. Syst. Saf., 30 (1990), 163–181. https://doi.org/10.1016/0951-8320(90)90093-3 doi: 10.1016/0951-8320(90)90093-3
    [65] B. Hu, Q. M. Gong, Y. Q. Zhang, Y. H. Yin, W. J. Chen, Characterizing uncertainty in geotechnical design of energy piles based on Bayesian theorem, Acta Geotech., 17 (2022), 4191–4206. https://doi.org/10.1007/s11440-022-01535-3 doi: 10.1007/s11440-022-01535-3
    [66] K. Yao, J. Gao, Law of large numbers for uncertain random variables, IEEE Trans. Fuzzy Syst., 24 (2016), 615–621. https://doi.org/10.1109/TFUZZ.2015.2466080 doi: 10.1109/TFUZZ.2015.2466080
    [67] C. Zhang, Q. Liu, B. Zhou, C. Y. Chung, J. Li, L. Zhu, et al., A central limit theorem-based method for dc and ac power flow analysis under interval uncertainty of renewable power generation, IEEE Trans. Sustain. Energy, 14 (2023), 563–575. https://doi.org/10.1109/TSTE.2022.3220567 doi: 10.1109/TSTE.2022.3220567
    [68] C. Wang, Z. K. Song, H. R. Fan, Novel evidence theory-based reliability analysis of functionally graded plate considering thermal stress behavior, Aerosp. Sci. Technol., 146 (2024), 108936. https://doi.org/10.1016/j.ast.2024.108936 doi: 10.1016/j.ast.2024.108936
    [69] C. Wang, Evidence-theory-based uncertain parameter identification method for mechanical systems with imprecise information, Comput. Meth. Appl. Mech. Eng., 351 (2019), 281–296. https://doi.org/10.1016/j.cma.2019.03.048 doi: 10.1016/j.cma.2019.03.048
    [70] F. Arévalo, M. P. C. Alison, M. T. Ibrahim, A. Schwung, Adaptive information fusion using evidence theory and uncertainty quantification, IEEE Access, 12 (2024), 2236–2259. https://doi.org/10.1109/ACCESS.2023.3348270 doi: 10.1109/ACCESS.2023.3348270
    [71] H. R. Bae, R. V. Grandhi, R. A. Canfield, Uncertainty quantification of structural response using evidence theory, AIAA J., 41 (2003), 2062–2068. https://doi.org/10.2514/2.1898 doi: 10.2514/2.1898
    [72] Y. He, M. Mirzargar, R. M. Kirby, Mixed aleatory and epistemic uncertainty quantification using fuzzy set theory, Int. J. Approx. Reasoning, 66 (2015), 1–15. https://doi.org/10.1016/j.ijar.2015.07.002 doi: 10.1016/j.ijar.2015.07.002
    [73] C. Wang, H. G. Matthies, Hybrid evidence-and-fuzzy uncertainty propagation under a dual-level analysis framework, Fuzzy Sets Syst., 367 (2019), 51–67. https://doi.org/10.1016/j.fss.2018.10.002 doi: 10.1016/j.fss.2018.10.002
    [74] R. M. Rodríguez, L. Martínez, V. Torra, Z. S. Xu, F. Herrera, Hesitant fuzzy sets: State of the art and future directions, Int. J. Intell. Syst., 29 (2014), 495–524. https://doi.org/10.1002/int.21654 doi: 10.1002/int.21654
    [75] S. H. Khairuddin, M. H. Hasan, M. A. Hashmani, M. H. Azam, Generating clustering-based interval fuzzy type-2 triangular and trapezoidal membership functions: A structured literature review, Symmetry-Basel, 13 (2021), 239. https://doi.org/10.3390/sym13020239 doi: 10.3390/sym13020239
    [76] C. Wang, H. R. Fan, T. Wu, Novel rough set theory-based method for epistemic uncertainty modeling, analysis and applications, Appl. Math. Model., 113 (2023), 456–474. https://doi.org/10.1016/j.apm.2022.09.002 doi: 10.1016/j.apm.2022.09.002
    [77] X. Y. Zhang, Y. Y. Yao, Tri-level attribute reduction in rough set theory, Expert Syst. Appl., 190 (2022), 116187. https://doi.org/10.1016/j.eswa.2021.116187 doi: 10.1016/j.eswa.2021.116187
    [78] F. Y. Li, Z. Luo, G. Y. Sun, N. Zhang, An uncertain multidisciplinary design optimization method using interval convex models, Eng. Optimiz., 45 (2013), 697–718. https://doi.org/10.1080/0305215x.2012.690871 doi: 10.1080/0305215x.2012.690871
    [79] H. Lü, K. Yang, X. T. Huang, W.-B. Shangguan, K. G. Zhao, Uncertainty and correlation propagation analysis of powertrain mounting systems based on multi-ellipsoid convex model, Mech. Syst. Signal Proc., 173 (2022), 109058. https://doi.org/10.1016/j.ymssp.2022.109058 doi: 10.1016/j.ymssp.2022.109058
    [80] X. Qiang, C. Wang, H. R. Fan, Hybrid interval model for uncertainty analysis of imprecise or conflicting information, Appl. Math. Model., 129 (2024), 837–856. https://doi.org/10.1016/j.apm.2024.02.014 doi: 10.1016/j.apm.2024.02.014
    [81] C. Wang, X. Qiang, H. R. Fan, T. Wu, Y. L. Chen, Novel data-driven method for non-probabilistic uncertainty analysis of engineering structures based on ellipsoid model, Comput. Meth. Appl. Mech. Eng., 394 (2022), 114889. https://doi.org/10.1016/j.cma.2022.114889 doi: 10.1016/j.cma.2022.114889
    [82] C. Wang, H. G. Matthies, A modified parallelepiped model for non-probabilistic uncertainty quantification and propagation analysis, Comput. Meth. Appl. Mech. Eng., 369 (2020), 113209. https://doi.org/10.1016/j.cma.2020.113209 doi: 10.1016/j.cma.2020.113209
    [83] C. Wang, L. Hong, X. Qiang, M. H. Xu, Novel numerical method for uncertainty analysis of coupled vibro-acoustic problem considering thermal stress, Comput. Meth. Appl. Mech. Eng., 420 (2024), 116727. https://doi.org/10.1016/j.cma.2023.116727 doi: 10.1016/j.cma.2023.116727
    [84] L. X. Cao, J. Liu, L. Xie, C. Jiang, R. G. Bi, Non-probabilistic polygonal convex set model for structural uncertainty quantification, Analog Integr. Circuits Process., 89 (2021), 504–518. https://doi.org/10.1016/j.apm.2020.07.025 doi: 10.1016/j.apm.2020.07.025
    [85] L. P. Zhu, I. Elishakoff, J. H. Starnes, Derivation of multi-dimensional ellipsoidal convex model for experimental data, Math. Comput. Model., 24 (1996), 103–114. https://doi.org/10.1016/0895-7177(96)00094-5 doi: 10.1016/0895-7177(96)00094-5
    [86] C. Jiang, X. Han, G. Y. Lu, J. Liu, Z. Zhang, Y. C. Bai, Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique, Comput. Meth. Appl. Mech. Eng., 200 (2011), 2528–2546. https://doi.org/10.1016/j.cma.2011.04.007 doi: 10.1016/j.cma.2011.04.007
    [87] J. Liu, Z. B. Yu, D. Q. Zhang, H. Liu, X. Han, Multimodal ellipsoid model for non-probabilistic structural uncertainty quantification and propagation, Int. J. Mech. Mater. Des., 17 (2021), 633–657. https://doi.org/10.1007/s10999-021-09551-z doi: 10.1007/s10999-021-09551-z
    [88] Z. Kang, W. B. Zhang, Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data, Comput. Meth. Appl. Mech. Eng., 300 (2016), 461–489. https://doi.org/10.1016/j.cma.2015.11.025 doi: 10.1016/j.cma.2015.11.025
    [89] L. Wang, J. X. Liu, Dynamic uncertainty quantification and risk prediction based on the grey mathematics and outcrossing theory, Appl. Sci.-Basel, 12 (2022), 5389. https://doi.org/10.3390/app12115389 doi: 10.3390/app12115389
    [90] Y. H. Yan, X. J. Wang, Y. L. Li, Non-probabilistic credible set model for structural uncertainty quantification, Structures, 53 (2023), 1408–1424. https://doi.org/10.1016/j.istruc.2023.05.011 doi: 10.1016/j.istruc.2023.05.011
    [91] T. Zhang, J. Y. Jiao, J. Lin, H. Li, J. D. Hua, D. He, Uncertainty-based contrastive prototype-matching network towards cross-domain fault diagnosis with small data, Knowledge-Based Syst., 254 (2022), 109651. https://doi.org/10.1016/j.knosys.2022.109651 doi: 10.1016/j.knosys.2022.109651
    [92] J. Chen, D. Zhou, Z. Guo, J. Lin, C. Lyu, C. Lu, An active learning method based on uncertainty and complexity for gearbox fault diagnosis, IEEE Access, 7 (2019), 9022–9031. https://doi.org/10.1109/ACCESS.2019.2890979 doi: 10.1109/ACCESS.2019.2890979
    [93] H. Ma, C. Ekanayake, T. K. Saha, Power transformer fault diagnosis under measurement originated uncertainties, IEEE Trns. Dielectr. Electr. Insul., 19 (2012), 1982–1990. https://doi.org/10.1109/tdei.2012.6396956 doi: 10.1109/tdei.2012.6396956
    [94] X. J. Shi, H. B. Gu, B. Yao, Fuzzy Bayesian network fault diagnosis method based on fault tree for coal mine drainage system, IEEE Sens. J., 24 (2024), 7537–7547. https://doi.org/10.1109/jsen.2024.3354415 doi: 10.1109/jsen.2024.3354415
    [95] R. X. Duan, Y. N. Lin, Y. N. Zeng, Fault diagnosis for complex systems based on reliability analysis and sensors data considering epistemic uncertainty, Eksploat. Niezawodn., 20 (2018), 558–566. https://doi.org/10.17531/ein.2018.4.7 doi: 10.17531/ein.2018.4.7
    [96] J. Wang, H. Peng, W. P. Yu, J. Ming, M. J. Pérez-Jiménez, C. Y. Tao, et al., Interval-valued fuzzy spiking neural p systems for fault diagnosis of power transmission networks, Eng. Appl. Artif. Intell., 82 (2019), 102–109. https://doi.org/10.1016/j.engappai.2019.03.014 doi: 10.1016/j.engappai.2019.03.014
    [97] A. Hoballah, D. E. A. Mansour, I. B. M. Taha, Hybrid grey wolf optimizer for transformer fault diagnosis using dissolved gases considering uncertainty in measurements, IEEE Access, 8 (2020), 139176–139187. https://doi.org/10.1109/access.2020.3012633 doi: 10.1109/access.2020.3012633
    [98] K. Zhou, J. Tang, Probabilistic gear fault diagnosis using Bayesian convolutional neural network, IFAC-PapersOnLine, 55 (2022), 795–799. https://doi.org/10.1016/j.ifacol.2022.11.279 doi: 10.1016/j.ifacol.2022.11.279
    [99] H. T. Zhou, W. H. Chen, L. S. Cheng, J. Liu, M. Xia, Trustworthy fault diagnosis with uncertainty estimation through evidential convolutional neural networks, IEEE Trans. Ind. Inform., 19 (2023), 10842–10852. https://doi.org/10.1109/TⅡ.2023.3241587 doi: 10.1109/TⅡ.2023.3241587
    [100] S. Huang, R. Duan, J. He, T. Feng, Y. Zeng, Fault diagnosis strategy for complex systems based on multi-source heterogeneous information under epistemic uncertainty, IEEE Access, 8 (2020), 50921–50933. https://doi.org/10.1109/ACCESS.2020.2980397 doi: 10.1109/ACCESS.2020.2980397
    [101] S. X. Liu, S. Y. Zhou, B. Y. Li, Z. H. Niu, M. Abdullah, R. R. Wang, Servo torque fault diagnosis implementation for heavy-legged robots using insufficient information, ISA Transactions, 147 (2024), 439–452. https://doi.org/10.1016/j.isatra.2024.02.004 doi: 10.1016/j.isatra.2024.02.004
    [102] T. Zhang, S. He, J. Chen, T. Pan, Z. Zhou, Toward small sample challenge in intelligent fault diagnosis: Attention-weighted multidepth feature fusion net with signals augmentation, IEEE Trans. Instrum. Meas., 71 (2022), 1–13. https://doi.org/10.1109/TIM.2021.3134999 doi: 10.1109/TIM.2021.3134999
    [103] A. Kulkarni, J. Terpenny, V. Prabhu, Sensor selection framework for designing fault diagnostics system, Sensors, 21 (2021), 6470. https://doi.org/10.3390/s21196470 doi: 10.3390/s21196470
    [104] C. Herrojo, F. Paredes, J. Mata-Contreras, F. Martín, Chipless-rfid: A review and recent developments, Sensors, 19 (2019), 3385. https://doi.org/10.3390/s19153385 doi: 10.3390/s19153385
    [105] T. Kalsoom, N. Ramzan, S. Ahmed, M. Ur-Rehman, Advances in sensor technologies in the era of smart factory and industry 4.0, Sensors, 20 (2020), 6783. https://doi.org/10.3390/s20236783 doi: 10.3390/s20236783
    [106] A. Leal, J. Casas, C. Marques, M. J. Pontes, A. Frizera, Application of additive layer manufacturing technique on the development of high sensitive fiber bragg grating temperature sensors, Sensors, 18 (2018), 4120. https://doi.org/10.3390/s18124120 doi: 10.3390/s18124120
    [107] G. D. Lewis, P. Merken, M. Vandewal, Enhanced accuracy of cmos smart temperature sensors by nonlinear curvature correction, Sensors, 18 (2018), 4087. https://doi.org/10.3390/s18124087 doi: 10.3390/s18124087
    [108] H. Landaluce, L. Arjona, A. Perallos, F. Falcone, I. Angulo, F. Muralter, A review of iot sensing applications and challenges using rfid and wireless sensor networks, Sensors, 20 (2020), 2495. https://doi.org/10.3390/s20092495 doi: 10.3390/s20092495
    [109] S. L. Wei, W. B. Qin, L. W. Han, F. Y. Cheng, The research on compensation algorithm of infrared temperature measurement based on intelligent sensors, Cluster Comput., 22 (2019), 6091–6100. https://doi.org/10.1007/s10586-018-1828-5 doi: 10.1007/s10586-018-1828-5
    [110] M. Tessarolo, L. Possanzini, E. G. Campari, R. Bonfiglioli, F. S. Violante, A. Bonfiglio, et al., Adaptable pressure textile sensors based on a conductive polymer, Flex. Print. Electron., 3 (2018), 034001. https://doi.org/10.1088/2058-8585/aacbee doi: 10.1088/2058-8585/aacbee
    [111] K. A. Mathias, S. M. Kulkarni, Investigation on influence of geometry on performance of a cavity-less pressure sensor, IOP Conf. Ser.: Mater. Sci. Eng., 417 (2018), 012035. https://doi.org/10.1088/1757-899x/417/1/012035 doi: 10.1088/1757-899x/417/1/012035
    [112] W. P. Eaton, J. H. Smith, Micromachined pressure sensors: Review and recent developments, Smart Mater. Struct., 6 (1997), 30–41. https://doi.org/10.1117/12.276606 doi: 10.1117/12.276606
    [113] M. Mousavi, M. Alzgool, S. Towfighian, A mems pressure sensor using electrostatic levitation, IEEE Sens. J., 21 (2021), 18601–18608. https://doi.org/10.1109/JSEN.2021.3091665 doi: 10.1109/JSEN.2021.3091665
    [114] A. P. Cherkun, G. V. Mishakov, A. V. Sharkov, E. I. Demikhov, The use of a piezoelectric force sensor in the magnetic force microscopy of thin permalloy films, Ultramicroscopy, 217 (2020), 113072. https://doi.org/10.1016/j.ultramic.2020.113072 doi: 10.1016/j.ultramic.2020.113072
    [115] A. Nastro, M. Ferrari, V. Ferrari, Double-actuator position-feedback mechanism for adjustable sensitivity in electrostatic-capacitive mems force sensors, Sens. Actuator A-Phys., 312 (2020), 112127. https://doi.org/10.1016/j.sna.2020.112127 doi: 10.1016/j.sna.2020.112127
    [116] M. L. Gödecke, C. M. Bett, D. Buchta, K. Frenner, W. Osten, Optical sensor design for fast and process-robust position measurements on small diffraction gratings, Opt. Lasers Eng., 134 (2020), 106267. https://doi.org/10.1016/j.optlaseng.2020.106267 doi: 10.1016/j.optlaseng.2020.106267
    [117] Y. J. Chan, A. R. Carr, S. Charkhabi, M. Furnish, A. M. Beierle, N. F. Reuel, Wireless position sensing and normalization of embedded resonant sensors using a resonator array, Sens. Actuator A-Phys., 303 (2020), 111853. https://doi.org/10.1016/j.sna.2020.111853 doi: 10.1016/j.sna.2020.111853
    [118] J. A. Kim, J. W. Kim, C. S. Kang, J. Y. Lee, J. Jin, On-machine calibration of angular position and runout of a precision rotation stage using two absolute position sensors, Measurement, 153 (2020), 107399. https://doi.org/10.1016/j.measurement.2019.107399 doi: 10.1016/j.measurement.2019.107399
    [119] L. E. Helseth, On the accuracy of an interdigital electrostatic position sensor, J. Electrost., 107 (2020), 103480. https://doi.org/10.1016/j.elstat.2020.103480 doi: 10.1016/j.elstat.2020.103480
    [120] K. Palmer, H. Kratz, H. Nguyen, G. Thornell, A highly integratable silicon thermal gas flow sensor, J. Micromech. Microeng., 22 (2012), 065015. https://doi.org/10.1088/0960-1317/22/6/065015 doi: 10.1088/0960-1317/22/6/065015
    [121] A. Moreno-Gomez, C. A. Perez-Ramirez, A. Dominguez-Gonzalez, M. Valtierra-Rodriguez, O. Chavez-Alegria, J. P. Amezquita-Sanchez, Sensors used in structural health monitoring, Arch. Comput. Method Eng., 25 (2018), 901–918. https://doi.org/10.1007/s11831-017-9217-4 doi: 10.1007/s11831-017-9217-4
    [122] A. M. Shkel, Smart mems: Micro-structures with error-suppression and self-calibration control capabilities, Proceedings of the 2001 American Control Conference, Arlington, VA, USA, 2001, 1208–1213. https://doi.org/10.1109/ACC.2001.945886
    [123] X. Insausti, M. Zárraga-Rodríguez, C. Nolasco-Ferencikova, J. Gutiérrez-Gutiérrez, In-network algorithm for passive sensors in structural health monitoring, IEEE Signal Process. Lett., 30 (2023), 952–956. https://doi.org/10.1109/lsp.2023.3298279 doi: 10.1109/lsp.2023.3298279
    [124] B. Jeon, J. S. Yoon, J. Um, S. H. Suh, The architecture development of industry 4.0 compliant smart machine tool system (smts), J. Intell. Manuf., 31 (2020), 1837–1859. https://doi.org/10.1007/s10845-020-01539-4 doi: 10.1007/s10845-020-01539-4
    [125] M. H. Zhu, J. Li, W. B. Wang, D. P. Chen, Self-detection and self-diagnosis methods for sensors in intelligent integrated sensing system, IEEE Sens. J., 21 (2021), 19247–19254. https://doi.org/10.1109/JSEN.2021.3090990 doi: 10.1109/JSEN.2021.3090990
    [126] J. Chen, P. Li, G. B. Song, Z. Ren, Y. Tan, Y. J. Zheng, Feedback control for structural health monitoring in a smart aggregate based sensor network, Int. J. Struct. Stab. Dyn., 18 (2017), 1850064. https://doi.org/10.1142/S0219455418500645 doi: 10.1142/S0219455418500645
    [127] C. Wang, Z. M. Peng, R. Liu, C. Chen, Research on multi-fault diagnosis method based on time domain features of vibration signals, Sensors, 22 (2022), 8164. https://doi.org/10.3390/s22218164 doi: 10.3390/s22218164
    [128] Z. F. Du, R. H. Zhang, H. Chen, Characteristic signal extracted from a continuous time signal on the aspect of frequency domain, Chin. Phys. B, 28 (2019), 090502. https://doi.org/10.1088/1674-1056/ab344a doi: 10.1088/1674-1056/ab344a
    [129] Y. Lu, J. Tang, On time-frequency domain feature extraction of wave signals for structural health monitoring, Measurement, 114 (2018), 51–59. https://doi.org/10.1016/j.measurement.2017.09.016 doi: 10.1016/j.measurement.2017.09.016
    [130] M. Imani, Modified pca, lda and lpp feature extraction methods for polsar image classification, Multimed. Tools Appl., 83 (2024), 41171–41192. https://doi.org/10.1007/s11042-023-17269-7 doi: 10.1007/s11042-023-17269-7
    [131] Z. Xia, Y. Chen, C. Xu, Multiview pca: A methodology of feature extraction and dimension reduction for high-order data, IEEE T. Cybern., 52 (2022), 11068–11080. https://doi.org/10.1109/TCYB.2021.3106485 doi: 10.1109/TCYB.2021.3106485
    [132] Y. Aliyari Ghassabeh, F. Rudzicz, H. A. Moghaddam, Fast incremental lda feature extraction, Pattern Recognit., 48 (2015), 1999–2012. https://doi.org/10.1016/j.patcog.2014.12.012 doi: 10.1016/j.patcog.2014.12.012
    [133] E. Parsaeimehr, M. Fartash, J. A. Torkestani, Improving feature extraction using a hybrid of cnn and lstm for entity identification, Neural Process. Lett., 55 (2023), 5979–5994. https://doi.org/10.1007/s11063-022-11122-y doi: 10.1007/s11063-022-11122-y
    [134] P. Wang, X. M. Zhang, Y. Hao, A method combining cnn and elm for feature extraction and classification of sar image, J. Sens., 2019 (2019), 6134610. https://doi.org/10.1155/2019/6134610 doi: 10.1155/2019/6134610
    [135] O. İrsoy, E. Alpaydın, Unsupervised feature extraction with autoencoder trees, Neurocomputing, 258 (2017), 63–73. https://doi.org/10.1016/j.neucom.2017.02.075 doi: 10.1016/j.neucom.2017.02.075
    [136] Y. Y. Wang, D. J. Song, W. T. Wang, S. X. Rao, X. Y. Wang, M. N. Wang, Self-supervised learning and semi-supervised learning for multi-sequence medical image classification, Neurocomputing, 513 (2022), 383–394. https://doi.org/10.1016/j.neucom.2022.09.097 doi: 10.1016/j.neucom.2022.09.097
    [137] W. X. Sun, J. Chen, J. Q. Li, Decision tree and pca-based fault diagnosis of rotating machinery, Mech. Syst. Signal Proc., 21 (2007), 1300–1317. https://doi.org/10.1016/j.ymssp.2006.06.010 doi: 10.1016/j.ymssp.2006.06.010
    [138] N. R. Sakthivel, V. Sugumaran, S. Babudevasenapati, Vibration based fault diagnosis of monoblock centrifugal pump using decision tree, Expert Syst. Appl., 37 (2010), 4040–4049. https://doi.org/10.1016/j.eswa.2009.10.002 doi: 10.1016/j.eswa.2009.10.002
    [139] Y. Y. Li, L. Y. Song, Q. C. Sun, H. Xu, X. G. Li, Z. J. Fang, et al., Rolling bearing fault diagnosis based on quantum ls-svm, EPJ Quantum Technol., 9 (2022), 18. https://doi.org/10.1140/epjqt/s40507-022-00137-y doi: 10.1140/epjqt/s40507-022-00137-y
    [140] A. H. Zhang, D. L. Yu, Z. Q. Zhang, Tlsca-svm fault diagnosis optimization method based on transfer learning, Processes, 10 (2022), 362. https://doi.org/10.3390/pr10020362 doi: 10.3390/pr10020362
    [141] T. Huang, Q. Zhang, X. A. Tang, S. Y. Zhao, X. N. Lu, A novel fault diagnosis method based on cnn and lstm and its application in fault diagnosis for complex systems, Artif. Intell. Rev., 55 (2022), 1289–1315. https://doi.org/10.1007/s10462-021-09993-z doi: 10.1007/s10462-021-09993-z
    [142] H. Fang, H. Liu, X. Wang, J. Deng, J. An, The method based on clustering for unknown failure diagnosis of rolling bearings, IEEE Trans. Instrum. Meas., 72 (2023), 1–8. https://doi.org/10.1109/TIM.2023.3251406 doi: 10.1109/TIM.2023.3251406
    [143] A. Rodríguez-Ramos, A. J. da Silva Neto, O. Llanes-Santiago, An approach to fault diagnosis with online detection of novel faults using fuzzy clustering tools, Expert Syst. Appl., 113 (2018), 200–212. https://doi.org/10.1016/j.eswa.2018.06.055 doi: 10.1016/j.eswa.2018.06.055
    [144] L. K. Chang, S. H. Wang, M. C. Tsai, Demagnetization fault diagnosis of a pmsm using auto-encoder and k-means clustering, Energies, 13 (2020), 4467. https://doi.org/10.3390/en13174467 doi: 10.3390/en13174467
    [145] J. Du, S. P. Wang, H. Y. Zhang, Layered clustering multi-fault diagnosis for hydraulic piston pump, Mech. Syst. Signal Proc., 36 (2013), 487–504. https://doi.org/10.1016/j.ymssp.2012.10.020 doi: 10.1016/j.ymssp.2012.10.020
    [146] Y. Y. Li, J. D. Wang, H. Y. Zhao, C. Wang, Q. Shao, Adaptive dbscan clustering and gasa optimization for underdetermined mixing matrix estimation in fault diagnosis of reciprocating compressors, Sensors, 24 (2024), 167. https://doi.org/10.3390/s24010167 doi: 10.3390/s24010167
    [147] C. X. Jian, K. J. Yang, Y. H. Ao, Industrial fault diagnosis based on active learning and semi-supervised learning using small training set, Eng. Appl. Artif. Intell., 104 (2021), 104365. https://doi.org/10.1016/j.engappai.2021.104365 doi: 10.1016/j.engappai.2021.104365
    [148] S. Zheng, J. Zhao, A self-adaptive temporal-spatial self-training algorithm for semisupervised fault diagnosis of industrial processes, IEEE Trans. Ind. Inform., 18 (2022), 6700–6711. https://doi.org/10.1109/TⅡ.2021.3120686 doi: 10.1109/TⅡ.2021.3120686
    [149] J. Y. Long, Y. B. Chen, Z. Yang, Y. W. Huang, C. Li, A novel self-training semi-supervised deep learning approach for machinery fault diagnosis, Int. J. Prod. Res., 61 (2023), 8238–8251. https://doi.org/10.1080/00207543.2022.2032860 doi: 10.1080/00207543.2022.2032860
    [150] K. Yu, H. Z. Han, Q. Fu, H. Ma, J. Zeng, Symmetric co-training based unsupervised domain adaptation approach for intelligent fault diagnosis of rolling bearing, Meas. Sci. Technol., 31 (2020), 115008. https://doi.org/10.1088/1361-6501/ab9841 doi: 10.1088/1361-6501/ab9841
    [151] L. Wang, D. F. Zhou, H. Tian, H. Zhang, W. Zhang, Parametric fault diagnosis of analog circuits based on a semi-supervised algorithm, Symmetry-Basel, 11 (2019), 228. https://doi.org/10.3390/sym11020228 doi: 10.3390/sym11020228
    [152] C. X. Jian, Y. H. Ao, Imbalanced fault diagnosis based on semi-supervised ensemble learning, J. Intell. Manuf., 34 (2023), 3143–3158. https://doi.org/10.1007/s10845-022-01985-2 doi: 10.1007/s10845-022-01985-2
    [153] X. Li, F. L. Zhang, Classification of multi-type bearing fault features based on semi-supervised generative adversarial network (gan), Meas. Sci. Technol., 35 (2024), 025107. https://doi.org/10.1088/1361-6501/ad068e doi: 10.1088/1361-6501/ad068e
    [154] L. Wang, H. Tian, H. Zhang, Soft fault diagnosis of analog circuits based on semi-supervised support vector machine, Analog Integr. Circuits Process., 108 (2021), 305–315. https://doi.org/10.1007/s10470-021-01851-w doi: 10.1007/s10470-021-01851-w
    [155] P. Xu, L. X. Fu, K. Xu, W. B. Sun, Q. Tan, Y. P. Zhang, et al., Investigation into maize seed disease identification based on deep learning and multi-source spectral information fusion techniques, J. Food Compos. Anal., 119 (2023), 105254. https://doi.org/10.1016/j.jfca.2023.105254 doi: 10.1016/j.jfca.2023.105254
    [156] P. F. Zhang, T. R. Li, Z. Yuan, C. Luo, G. Q. Wang, J. Liu, et al., A data-level fusion model for unsupervised attribute selection in multi-source homogeneous data, Inf. Fusion, 80 (2022), 87–103. https://doi.org/10.1016/j.inffus.2021.10.017 doi: 10.1016/j.inffus.2021.10.017
    [157] M. B. Song, Y. F. Zhi, M. D. An, W. Xu, G. H. Li, X. L. Wang, Centrifugal pump cavitation fault diagnosis based on feature-level multi-source information fusion, Processes, 12 (2024), 196. https://doi.org/10.3390/pr12010196 doi: 10.3390/pr12010196
    [158] L. L. Liu, X. Wan, J. Y. Li, W. X. Wang, Z. G. Gao, An improved entropy-weighted topsis method for decision-level fusion evaluation system of multi-source data, Sensors, 22 (2022), 6391. https://doi.org/10.3390/s22176391 doi: 10.3390/s22176391
    [159] Y. W. Liu, Y. Q. Cheng, Z. Z. Zhang, J. J. Wu, Multi-information fusion fault diagnosis based on knn and improved evidence theory, J. Vib. Eng. Technol., 10 (2022), 841–852. https://doi.org/10.1007/s42417-021-00413-8 doi: 10.1007/s42417-021-00413-8
    [160] J. Xu, Y. Sui, T. Dai, A Bayesian network inference approach for dynamic risk assessment using multisource-based information fusion in an interval type-2 fuzzy set environment, IEEE Trans. Fuzzy Syst., 32 (2024), 5702–5713. https://doi.org/10.1109/TFUZZ.2024.3425495 doi: 10.1109/TFUZZ.2024.3425495
    [161] Y. C. Jie, Y. Chen, X. S. Li, P. Yi, H. S. Tan, X. Q. Cheng, Fufusion: Fuzzy sets theory for infrared and visible image fusion, In: Pattern recognition and computer vision, Singapore: Springer, 2024,466–478. https://doi.org/10.1007/978-981-99-8432-9_37
    [162] F. Y. Xiao, Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy, Inf. Fusion, 46 (2019), 23–32. https://doi.org/10.1016/j.inffus.2018.04.003 doi: 10.1016/j.inffus.2018.04.003
    [163] G. Koliander, Y. El-Laham, P. M. Djuric, F. Hlawatsch, Fusion of probability density functions, Proceedings of the IEEE, 110 (2022), 404–453. https://doi.org/10.1109/jproc.2022.3154399 doi: 10.1109/jproc.2022.3154399
    [164] Y. J. Pan, R. Q. An, D. Z. Fu, Z. Y. Zheng, Z. H. Yang, Unsupervised fault detection with a decision fusion method based on Bayesian in the pumping unit, IEEE Sens. J., 21 (2021), 21829–21838. https://doi.org/10.1109/jsen.2021.3103520 doi: 10.1109/jsen.2021.3103520
    [165] K. V. Kumar, A. Sathish, Medical image fusion based on type-2 fuzzy sets with teaching learning based optimization, Multimed. Tools Appl., 83 (2024), 33235–33262. https://doi.org/10.1007/s11042-023-16859-9 doi: 10.1007/s11042-023-16859-9
    [166] P. F. Zhang, T. R. Li, G. Q. Wang, C. Luo, H. M. Chen, J. B. Zhang, et al., Multi-source information fusion based on rough set theory: A review, Inf. Fusion, 68 (2021), 85–117. https://doi.org/10.1016/j.inffus.2020.11.004 doi: 10.1016/j.inffus.2020.11.004
    [167] Y. S. Wang, M. Y. He, L. Sun, D. Wu, Y. Wang, X. L. Qing, Weighted adaptive kalman filtering-based diverse information fusion for hole edge crack monitoring, Mech. Syst. Signal Proc., 167 (2022), 108534. https://doi.org/10.1016/j.ymssp.2021.108534 doi: 10.1016/j.ymssp.2021.108534
    [168] N. Guenther, M. Schonlau, Support vector machines, Stata J., 16 (2016), 917–937. https://doi.org/10.1177/1536867x1601600407 doi: 10.1177/1536867x1601600407
    [169] P. Cunningham, S. J. Delany, K-nearest neighbour classifiers-a tutorial, ACM Comput. Surv., 54 (2021), 128. https://doi.org/10.1145/3459665 doi: 10.1145/3459665
    [170] Z. Liu, S. B. Zhong, Q. Liu, C. X. Xie, Y. Z. Dai, C. Peng, et al., Thyroid nodule recognition using a joint convolutional neural network with information fusion of ultrasound images and radiofrequency data, Eur. Radiol., 31 (2021), 5001–5011. https://doi.org/10.1007/s00330-020-07585-z doi: 10.1007/s00330-020-07585-z
    [171] A. Y. Chen, F. Wang, W. H. Liu, S. Chang, H. Wang, J. He, et al., Multi-information fusion neural networks for arrhythmia automatic detection, Comput. Meth. Programs Biomed., 193 (2020), 105479. https://doi.org/10.1016/j.cmpb.2020.105479 doi: 10.1016/j.cmpb.2020.105479
  • This article has been cited by:

    1. Yihui Gong, Qi Yang, On Entire Solutions of Two Fermat-Type Differential-Difference Equations, 2025, 51, 1017-060X, 10.1007/s41980-024-00942-4
  • Reader Comments
  • © 2024 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(1725) PDF downloads(132) Cited by(1)

Figures and Tables

Figures(10)  /  Tables(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog