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

A hybrid model combining variational mode decomposition and an attention-GRU network for stock price index forecasting

  • Received: 22 July 2020 Accepted: 10 October 2020 Published: 21 October 2020
  • In this paper we introduce a new hybrid model based on variational mode decomposition (VMD) and Gated Recurrent Units (GRU) network improved by attention mechanism to enhance the accuracy of stock price indices forecasting. In the process of establishing the model, VMD is made a use to decompose the primary series into some almost orthogonal subsequences. The attention mechanism is introduced into GRU to assign different weights to the input elements in advance so that better predictive results can be achieved for each component. In empirical experiment, London FTSE Index (FTSE) and Nasdaq Index (IXIC) are adopted to examine the performance of VMD-AttGRU model. Empirical results report that the developed hybrid model outperforms the single models and indeed raises the accuracy of stock price indices forecasting. In addition, the introduction of attention mechanism can increase the level predictive accuracy but decrease the correctness of direction forecasting.

    Citation: Hongli Niu, Kunliang Xu. A hybrid model combining variational mode decomposition and an attention-GRU network for stock price index forecasting[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7151-7166. doi: 10.3934/mbe.2020367

    Related Papers:

    [1] N. E. Cho, G. Murugusundaramoorthy, K. R. Karthikeyan, S. Sivasubramanian . Properties of λ-pseudo-starlike functions with respect to a boundary point. AIMS Mathematics, 2022, 7(5): 8701-8714. doi: 10.3934/math.2022486
    [2] Pinhong Long, Huo Tang, Wenshuai Wang . Functional inequalities for several classes of q-starlike and q-convex type analytic and multivalent functions using a generalized Bernardi integral operator. AIMS Mathematics, 2021, 6(2): 1191-1208. doi: 10.3934/math.2021073
    [3] Sadaf Umar, Muhammad Arif, Mohsan Raza, See Keong Lee . On a subclass related to Bazilevič functions. AIMS Mathematics, 2020, 5(3): 2040-2056. doi: 10.3934/math.2020135
    [4] Mohammad Faisal Khan, Jongsuk Ro, Muhammad Ghaffar Khan . Sharp estimate for starlikeness related to a tangent domain. AIMS Mathematics, 2024, 9(8): 20721-20741. doi: 10.3934/math.20241007
    [5] Wenzheng Hu, Jian Deng . Hankel determinants, Fekete-Szegö inequality, and estimates of initial coefficients for certain subclasses of analytic functions. AIMS Mathematics, 2024, 9(3): 6445-6467. doi: 10.3934/math.2024314
    [6] Hava Arıkan, Halit Orhan, Murat Çağlar . Fekete-Szegö inequality for a subclass of analytic functions defined by Komatu integral operator. AIMS Mathematics, 2020, 5(3): 1745-1756. doi: 10.3934/math.2020118
    [7] Pinhong Long, Xing Li, Gangadharan Murugusundaramoorthy, Wenshuai Wang . The Fekete-Szegö type inequalities for certain subclasses analytic functions associated with petal shaped region. AIMS Mathematics, 2021, 6(6): 6087-6106. doi: 10.3934/math.2021357
    [8] K. R. Karthikeyan, G. Murugusundaramoorthy, N. E. Cho . Some inequalities on Bazilevič class of functions involving quasi-subordination. AIMS Mathematics, 2021, 6(7): 7111-7124. doi: 10.3934/math.2021417
    [9] Muhammad Ghaffar Khan, Sheza.M. El-Deeb, Daniel Breaz, Wali Khan Mashwani, Bakhtiar Ahmad . Sufficiency criteria for a class of convex functions connected with tangent function. AIMS Mathematics, 2024, 9(7): 18608-18624. doi: 10.3934/math.2024906
    [10] Ahmad A. Abubaker, Khaled Matarneh, Mohammad Faisal Khan, Suha B. Al-Shaikh, Mustafa Kamal . Study of quantum calculus for a new subclass of q-starlike bi-univalent functions connected with vertical strip domain. AIMS Mathematics, 2024, 9(5): 11789-11804. doi: 10.3934/math.2024577
  • In this paper we introduce a new hybrid model based on variational mode decomposition (VMD) and Gated Recurrent Units (GRU) network improved by attention mechanism to enhance the accuracy of stock price indices forecasting. In the process of establishing the model, VMD is made a use to decompose the primary series into some almost orthogonal subsequences. The attention mechanism is introduced into GRU to assign different weights to the input elements in advance so that better predictive results can be achieved for each component. In empirical experiment, London FTSE Index (FTSE) and Nasdaq Index (IXIC) are adopted to examine the performance of VMD-AttGRU model. Empirical results report that the developed hybrid model outperforms the single models and indeed raises the accuracy of stock price indices forecasting. In addition, the introduction of attention mechanism can increase the level predictive accuracy but decrease the correctness of direction forecasting.


    Let A denote the class of functions of the form

    f(z)=z+a2z2+a3z3+a4z4+, (1.1)

    which are analytic in the open unit disk D=(z:∣z∣<1) and normalized by f(0)=0 and f(0)=1. Recall that, SA is the univalent function in D=(z:∣z∣<1) and has the star-like and convex functions as its sub-classes which their geometric condition satisfies Re(zf(z)f(z))>0 and Re(1+zf(z)f(z))>0. The two well-known sub-classes have been used to define different subclass of analytical functions in different direction with different perspective and their results are too voluminous in literature.

    Two functions f and g are said to be subordinate to each other, written as fg, if there exists a Schwartz function w(z) such that

    f(z)=g(w(z)),zϵD (1.2)

    where w(0) and w(z)∣<1 for zϵD. Let P denote the class of analytic functions such that p(0)=1 and p(z)1+z1z, zϵD. See [1] for details.

    Goodman [2] proposed the concept of conic domain to generalize convex function which generated the first parabolic region as an image domain of analytic function. The same author studied and introduced the class of uniformly convex functions which satisfy

    UCV=Re{1+(zψ)f(z)f(z)}>0,(z,ψA).

    In recent time, Ma and Minda [3] studied the underneath characterization

    UCV=Re{1+zf(z)f(z)>|zf(z)f(z)|},zϵD. (1.3)

    The characterization studied by [3] gave birth to first parabolic region of the form

    Ω={w;Re(w)>∣w1}, (1.4)

    which was later generalized by Kanas and Wisniowska ([5,6]) to

    Ωk={w;Re(w)>kw1,k0}. (1.5)

    The Ωk represents the right half plane for k=0, hyperbolic region for 0<k<1, parabolic region for k=1 and elliptic region for k>1 [30].

    The generalized conic region (1.5) has been studied by many researchers and their interesting results litter everywhere. Just to mention but a few Malik [7] and Malik et al. [8].

    More so, the conic domain Ω was generalized to domain Ω[A,B], 1B<A1 by Noor and Malik [9] to

    Ω[A,B]={u+iv:[(B21)(U2+V2)2(AB1)u+(A21)]2
    >[2(B+1)(u2v2)+2(A+B+C)u2(A+1)]2+4(AB)2v2}

    and it is called petal type region.

    A function p(z) is said to be in the class UP[A,B], if and only if

    p(z)(A+1)˜p(z)(A1)(B+1)˜p(z)(B1), (1.6)

    where ˜p(z)=1+2π2(log1+z1z)2.

    Taking A=1 and B=1 in (1.8), the usual classes of functions studied by Goodman [1] and Kanas ([5,6]) will be obtained.

    Furthermore, the classes UCV[A,B] and ST[A,B] are uniformly Janoski convex and Starlike functions satisfies

    Re((B1)(zf(z))f(z)(A1)(B+1)(zf(z))f(z)(A+1))>|(B1)(zf(z))f(z)(A1)(B+1)(zf(z))f(z)(A+1)1| (1.7)

    and

    Re((B1)zf(z)f(z)(A1)(B+1)zf(z)f(z)(A+1))>|(B1)zf(z)f(z)(A1)(B+1)zf(z)f(z)(A+1)1|, (1.8)

    or equivalently

    (zf(z))f(z)UP[A,B]

    and

    zf(z)f(z)UP[A,B].

    Setting A=1 and B=1 in (1.7) and (1.8), we obtained the classes of functions investigated by Goodman [2] and Ronning [10].

    The relevant connection to Fekete-Szegö problem is a way of maximizing the non-linear functional |a3λa22| for various subclasses of univalent function theory. To know much of history, we refer the reader to [11,12,13,14] and so on.

    The error function was defined because of the normal curve, and shows up anywhere the normal curve appears. Error function occurs in diffusion which is a part of transport phenomena. It is also useful in biology, mass flow, chemistry, physics and thermomechanics. According to the information at hand, Abramowitz [15] expanded the error function into Maclaurin series of the form

    Erf(z)=2πz0et2dt=2πn=0(1)nz2n+1(2n+1)n! (1.9)

    The properties and inequalities of error function were studied by [16] and [4] while the zeros of complementary error function of the form

    erfc(z)=1erf(z)=2πzet2dt, (1.10)

    was investigated by [17], see for more details in [18,19] and so on. In recent time, [20,21,22] and [23] applied error functions in numerical analysis and their results are flying in the air.

    For f given by [15] and g with the form g(z)=z+b2z2+b3z3+ their Hadamard product (convolution) by fg and at is defined as:

    (fg)(z)=z+n=2anbnzn (1.11)

    Let Erf be a normalized analytical function which is obtained from (1.9) and given by

    Erf=πz2erf(z)=z+n=2(1)n1zn(2n1)(n1)! (1.12)

    Therefore, applying a notation (1.11) to (1.1) and (1.12) we obtain

    ϵ=AErf={F:F(z)=(fErf)(z)=z+n=2(1)n1anzn(2n1)(n1)!,fA}, (1.13)

    where Erf is the class that consists of a single function or Erf. See concept in Kanas et al. [18] and Ramachandran et al. [19].

    Babalola [24] introduced and studied the class of λpseudo starlike function of order β(0β1) which satisfy the condition

    Re(z(f(z))λf(z))>β, (1.14)

    where λ1(zD) and denoted by λ(β). We observed from (1.14) that putting λ=2, the geometric condition gives the product combination of bounded turning point and starlike function which satisfy

    Ref(z)(z(f(z))f(z))>β

    Olatunji [25] extended the class λ(β) to βλ(s,t,Φ) which the geometric condition satisfy

    Re((st)z(f(z))λf(sz)f(tz))>β,

    where s,tC,st,λ1,0β<1,zD and Φ(z) is the modified sigmoid function. The initial coefficient bounds were obtained and the relevant connection to Fekete-Szegö inequalities were generated. The contributions of authors like Altinkaya and Özkan [26] and Murugusundaramoorthy and Janani [27] and Murugusundaramoorthy et al. [28] can not be ignored when we are talking on λ-pseudo starlike functions.

    Inspired by earlier work by [18,19,29]. In this work, the authors employed the approach of [13] to study the coefficient inequalities for pseudo certain subclasses of analytical functions related to petal type region defined by error function. The first few coefficient bounds and the relevant connection to Fekete-Szegö inequalities were obtained for the classes of functions defined. Also note that, the results obtained here has not been in literature and varying of parameters involved will give birth to corollaries.

    For the purpose of the main results, the following lemmas and definitions are very necessary.

    Lemma 1.1. If p(z)=1+p1z+p2z2+ is a function with positive real part in D, then, for any complex μ,

    |p2μp21|2max{1,|2μ1|}

    and the result is sharp for the functions

    p0(z)=1+z1zorp(z)=1+z21z2(zD).

    Lemma 1.2. [29] Let pUP[A,B],1B<A1 and of the form p(z)=1+n=1pnzn. Then, for a complex number μ, we have

    |p2μp21|4π2(AB)max(1,|4π2(B+1)23+4μ(ABπ2)|). (1.15)

    The result is sharp and the equality in (1.15) holds for the functions

    p1(z)=2(A+1)π2(log1+z1z)2+22(B+1)π2(log1+z1z)2+2

    or

    p2(z)=2(A+1)π2(log1+z1z)2+22(B+1)π2(log1+z1z)2+2.

    Proof. For hP and of the form h(z)=1+n=1cnzn, we consider

    h(z)=1+w(z)1w(z)

    where w(z) is such that w(0)=0 and |w(z)|<1. It follows easily that

    w(z)=h(z)1h(z)+1=12z+(c22c214)z2+(c32c2c12+c318)z3+ (1.16)

    Now, if ˜p(z)=1+R1z+R2z2+, then from (1.16), one may have,

    ˜p(w(z))=1+R1w(z)+R2(w(z))2+R3(w(z))3 (1.17)

    where R1=8π2,R2=163π2, and R3=18445π2, see [30]. Substitute R1,R2 and R3 into (1.17) to obtain

    ˜p(w(z))=1+4c1π2z+4π2(c2c216)z2+4π2(c3c1c23+2c3145)z3+ (1.18)

    Since pUP[A,B], so from relations (1.16), (1.17) and (1.18), one may have,

    p(z)=(A+1)˜p(w(z))(A1)(B+1)˜p(w(z))(B1)=2+(A+1)4π2c1z+(A+1)4π2(c2c216)z2+2+(B+1)4π2c1z+(B+1)4π2(c2c216)z2+

    This implies that,

    p(z)=1+2(AB)c1π2z+2(AB)π2(c2c2162(B1)c21π2)z2+8(AB)π2[((B+1)2π4+B+16π2190)c21(B+1π2+112)c1c2+c34]z3+ (1.19)

    If p(z)=1+n=1pnzn, then equating coefficients of z and z2, one may have,

    p1=2π2(AB)c1

    and

    p2=2π2(AB)(c2c2162(B1)c21π2).

    Now for a complex number μ, consider

    p2μp21=2(AB)π2[c2c21(16+2(B+1)π2+2μ(AB)π2)]

    This implies that

    |p2μp21|=2(AB)π2|c2c21(16+2(B+1)π2+2μ(AB)π2)|.

    Using Lemma 1.1, one may have

    |p2μp21|=4(AB)π2max{1,|2v1|},

    where v=16+2(B+1)π2+2μ(AB)π2, which completes the proof of the Lemma.

    Definition 1.3. A function FϵA is said to be in the class UCV[λ,A,B], 1B<A1, if and only if,

    Re((B1)(z(F(z)λ))F(z)(A1)(B+1)(z(F(z)λ))F(z)(A+1))>|(B1)(z(F(z)λ))F(z)(A1)(B+1)(z(F(z)λ))F(z)(A+1)1|, (1.20)

    where λ1ϵR or equivalently (z(F(z)λ))F(z)ϵUP[A,B].

    Definition 1.4. A function FϵA is said to be in the class US[λ,A,B], 1B<A1, if and only if,

    Re((B1)z(F(z)λ)F(z)(A1)(B+1)z(F(z)λ)F(z)(A+1))>|(B1)z(F(z)λ)F(z)(A1)(B+1)z(F(z)λ)F(z)(A+1)1|, (1.21)

    where λ1ϵR or equivalently z(F(z)λ)F(z)ϵUP[A,B].

    Definition 1.5. A function FϵA is said to be in the class UMα[λ,A,B], 1B<A1, if and only if,

    Re((B1)[(1α)z(F(z)λ)F(z)+α(z(F(z)λ))F(z)](A1)(B+1)[(1α)z(F(z)λ)F(z)+α(z(F(z)λ))F(z)](A+1))>|(B1)[(1α)z(F(z)λ)F(z)+α(z(F(z)λ))F(z)](A1)(B+1)[(1α)z(F(z)λ)F(z)+α(z(F(z)λ))F(z)](A+1)1|,

    where α0 and λ1ϵR or equivalently (1α)z(F(z)λ)f(z)+α(z(f(z)λ))f(z)UP[A,B].

    In this section, we shall state and prove the main results, and several corollaries can easily be deduced under various conditions.

    Theorem 2.1. Let FUS[λ,A,B], 1B<A1, and of the form (1.13). Then, for a real number μ, we have

    |a3μa22|40(AB)|13λ|π2max{1,|4(B+1)π2132(AB)(12λ)2π2(2(2λ24λ+1)9μ(13λ)5)|}.

    Proof. If FUS[λ,A,B], 1B<A1, the it follows from relations (1.18), (1.19), and (1.20),

    z(F(z)λ)F(z)=(A+1)˜p(w(z))(A1)(B+1)˜p(w(z))(B1),

    where w(z) is such that w(0)=0 and w(z)∣<1. The right hand side of the above expression get its series form from (1.13) and reduces to

    z(F(z)λ)F(z)=1+2(AB)c1π2z+2(AB)π2(c2c2162(B1)c21π2)z2
    +8(AB)π2[((B+1)2π4+B+16π2190)c21(B+1π2+112)c1c2+c34]z3+. (2.1)

    If F(z)=z+n=2(1)n1anzn(2n1)(n1)!, then one may have

    z(F(z)λ)F(z)=1+12λ3a2z+(2λ24λ+19a2213λ10a3)z2+ (2.2)

    From (2.1) and (2.2), comparison of coefficient of z and z2 gives,

    a2=6(AB)(12λ)π2c1 (2.3)

    and

    2λ24λ+19a2213λ10a3=2(AB)π2(c216c212(B+1)π2c21).

    This implies, by using (2.3), that

    a3=20(AB)(13λ)π2[c216c212(B+1)π2c212(2λ24λ+1)(AB)(12λ)2π2c21].

    Now, for a real number μ consider

    |a3μa22|=
    |20(AB)(13λ)π2(c216c212(B+1)π2c21)+40(AB)2(2λ24λ+1)(12λ)2(13λ)π436μ(AB)2c21(12λ)2π4|
    =20(AB)(13λ)π2|c2c21(16+2(B+1)π22(AB)(2λ24λ+1)(12λ)2π2+9μ(AB)(13λ)5(12λ)2π2)|
    =20(AB)(13λ)π2|c2vc21|

    where v=16+2(B+1)π2(AB)(12λ)2π2(2(2λ24λ+1)9μ(13λ)5).

    Theorem 2.2. Let FUCV[λ,A,B], 1B<A1, and of the form (1.13). Then, for a real number μ, we have

    |a3μa22|40(AB)3|1+3λ|π2max{1,|4(B+1)π2132(1+3λ)(AB)(1+2λ)2π2(λ27μ20)|}

    Proof. If FUCV[λ,A,B], 1B<A1, then it follows from relations (1.18), (1.19), and (1.21),

    (zF(z)λ)F(z)=(A+1)˜p(w(z))(A1)(B+1)˜p(w(z))(B+1),

    where w(z) is such that w(0)=0 and w(z)∣<1. The right hand side of the above expression get its series form from (1.13) and reduces to,

    (zF(z)λ)F(z)=1+2(AB)c1π2z+2(AB)π2(c2c2162(B+1)π2c21)z2+8(AB)π2[(B+1π4+B+16π2+190)c31(B+1π2+112)c1c2+c34]z3+ (2.4)

    If F(z)=z+(1)n1anzn(2n1)(n1)!, then we have,

    (zF(z)λ)F(z)=12(1+2λ)3a2z+(1+3λ)(3a310+2λ9a22)z2+ (2.5)

    From (2.4) and (2.5), comparison of coefficients of z and z2 gives,

    a2=3(AB)c1(1+2λ)π2 (2.6)

    and

    (1+3λ)(3a310+2λ9a22)=2(AB)π2(c2c2162(B+1)c21π2)

    This implies, by using (2.6), that

    a3=103[2(AB)(1+3λ)π2(c2c2162(B+1)c21π2)+2λ(AB)2c21(1+2λ)2π4].

    Now, for a real number μ, consider

    |a3μa22|=|20(AB)3(1+3λ)π2(c216c12(B+1)π2c21)+20(AB)2c213(1+2λ)π49μ(AB)2c21(1+2λ)2π4|
    =20(AB)3(1+3λ)π2|c2c21(16+2(B+1)π2λ(1+3λ)(AB)(1+2λ)2π2+27μ(AB)(1+3λ)20(1+2λ)2π2)|
    =20(AB)3(1+3λ)π2|c2vc21|,

    where

    v=16+2(B+1)π2(1+3λ)(AB)(1+2λ)2π2(λ27μ20).

    Theorem 2.3. FMα[λ,A,B], 1B<A1, α0 and of the form (1.13). Then, for a real number μ, we have

    |a3μa22|40(AB)π2|3(λ+α+2αλ)+α1|max{1,|4(B+1)π2134(AB)[12λα(3+2λ)]2π2(2λ2(1+2α)+2λ(3α2)+1α9μ(3(λ+α+2αλ)+α1)10)|}.

    Proof. Let FMα[λ,A,B], 1B<A1, α0 and of the form (1.13). Then, for a real number μ, we have

    (1α)z(F(z))λF(z)+α(z(F(z))λ)F(z)=(A+1)˜p(w(z))(A1)(B+1)˜p(w(z))(B1), (2.7)

    where w(z) is such that w(z0)=0 and |w(z)|<1. The right hand side of the above expression get its series form from (2.7) and reduces to

    (1α)z(F(z))λF(z)+α(z(F(z))λ)F(z)=1+2(AB)Gπ2z+2(AB)π2(c2c2162(B+1)π2c21)z2+... (2.8)

    If F(z)=z+n=2(1)n1anzn(2n1)(n1)!, then one may have

    (1α)z(F(z))λF(z)+α(z(F(z))λ)F(z)=(1α)[1+12λ3a2z+(2λ24λ+19a2213λ10a3)z2+...]+α[12(1+2λ)3a2z+(1+3λ)(3a310+2λ9a22)z2+...] (2.9)

    from (2.8) and (2.9), comparison of coefficients of z and z2 gives

    a2=6(AB)c1[12λα(3+2λ)]π2 (2.10)

    and

    3(λ+α+2αλ)+α110a32λ2(1+2λ)+α19a22=2(AB)π2(c2c2162(B+1)π2c21)

    This implies, by using (2.10), that

    a3=103(λ+α+2αλ)+α1[2(AB)π2(c2c2162(B+1)π2c21)+4(AB)2[2λ2(1+2λ)+2λ(3α2)+1α][12λα(3+2λ)]2π4c21]

    Now, for a real number μ, consider

    |a3μa22|=|103(λ+α+2αλ)+α1[2(AB)π2(c2c2162(B+1)π2c21)+4(AB)2[2λ2(1+2λ)+2λ(3α2)+1α][12λα(3+2λ)]2π4c21]36(AB)2μG2[12λα(3+2λ)]2π4|
    =|20(AB)π(3(λ+α+2αλ)+α1)|c2c21[16+2(B+1)π22(AB)[2λ2(1+2α)+2λ(3α2)+1α](12λα(3+2λ))2π2+18μ(AB)[3(λ+α+2αλ)+α1]10[12λα(3+2λ)]2π2
    =20(AB)π(3(λ+α+2αλ)+α1)|c2vc21|,

    where

    v=16+2(B+1)π22(AB)[2λ2(1+2α)+2λ(3α2)+1α](12λα(3+2λ))2π2+18μ(AB)[3(λ+α+2αλ)+α1]10[12λα(3+2λ)]2π2.

    The force applied on certain subclasses of analytical functions associated with petal type domain defined by error function has played a vital role in this work. The results obtained are new and varying the parameters involved in the classes of function defined, these will bring new more results that has not been in existence.

    The authors would like to thank the referees for their valuable comments and suggestions.

    The authors declare that they have no conflict of interests.



    [1] C. Zhang, H. Pan, Y. Ma, X. Huang, Analysis of Asia Pacific stock markets with a novel multiscale model, Phys. A, 534 (2019), 120939.
    [2] A. L. D. Loureiro, V. L. Miguéis, L. F. M. da Silva, Exploring the use of deep neural networks for sales forecasting in fashion retail, Decis. Support Syst., 114 (2018), 81-93.
    [3] J. Wang, J. Wang, Forecasting stock market indexes using principle component analysis and stochastic time effective neural networks, Neurocomputing, 156 (2015), 68-78. doi: 10.1016/j.neucom.2014.12.084
    [4] Y. Xu, S. B. Cohen, Stock movement prediction from tweets and historical prices, In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics, 2018.
    [5] D.P. Mandic, J.A. Chambers, Exploiting inherent relationships in RNN architectures, Neural Networks, 12 (1999), 1341-1345. doi: 10.1016/S0893-6080(99)00076-3
    [6] T. Deng, X. He, Z. Zeng, Recurrent neural network for combined economic and emission dispatch, Applied Intelligence, Appl. Intell., 48 (2018), 2180-2198.
    [7] S. Hochreiter, J. Schmidhuber, Long short-term memory, Neural Comput., 9 (1997), 1735-1780.
    [8] K. Wang, X. Qi, H. Liu, Photovoltaic power forecasting based LSTM-Convolutional Network, Energy, 189 (2019), 116225.
    [9] Z. Karevan, J. A. K. Suykens, Transductive LSTM for time-series prediction: An application to weather forecasting, Neural Networks, 125 (2020), 1-9.
    [10] B. Zhao, Z. P. Wang, W. J. Ji, X. Gao, X. B. Li, A Short-term Power Load Forecasting Method Based on Attention Mechanism of CNN-GRU, Power Syst. Technol., 12 (2019).
    [11] Z. Y. Peng, S. Peng, L. D. Fu, B. C. Lu, J. J. Tang, K. Wang, et al., A novel deep learning ensemble model with data denoising for short-term wind speed forecasting, Energy Convers. Manage., 207 (2020), 112524.
    [12] W. Y. Wu, W. L. Liao, J. Miao, G. L. Du, Using Gated Recurrent Unit Network to Forecast Short-Term Load Considering Impact of Electricity Price, Energy Procedia, 158 (2019) 3369-3374.
    [13] J. Zhang, D. Li, Y. Hao, Z. Tan, A hybrid model using signal processing technology, econometric models and neural network for carbon spot price forecasting, J. Cleaner Prod., 204 (2018), 958-964.
    [14] J. Wang, L. Y. Tang, Y. Y. Luo, P. Ge, A weighted EMD-based prediction model based on TOPSIS and feed forward neural network for noised time series, Knowl. Based Syst., 132 (2017), 167-178. doi: 10.1016/j.knosys.2017.06.022
    [15] J. Cao, Z. Li, J. Li, Financial time series forecasting model based on CEEMDAN and LSTM, Phys. A, 519 (2019), 127-139. doi: 10.1016/j.physa.2018.11.061
    [16] K. Dragomiretskiy, D. Zosso, Variational mode decomposition, IEEE Trans. Signal Process., 62 (2014), 531-544.
    [17] S. Lahmiri, Intraday stock price forecasting based on variational mode decomposition, J. Comput. Sci., 12 (2016), 23-27. doi: 10.1016/j.jocs.2015.11.011
    [18] S. Lahmiri, A variational mode decomposition approach for analysis and forecasting of economic and financial time series, Expert Syst. Appl., 55 (2016), 268-273. doi: 10.1016/j.eswa.2016.02.025
    [19] S. Lahmiri, Comparing variational and empirical mode decomposition in forecasting day-ahead energy prices, IEEE Syst. J., 11 (2015), 1907-1910.
    [20] Q. Zhu, F. Zhang, S. Liu, Y. Wu, L. Wang, A hybrid VMD-BiGRU model for rubber futures time series forecasting, Appl. Soft Comput., 84 (2019), 105739.
    [21] C. Li, G. Tang, X. Xue, A. Saeed, X. Hu, Short-term wind speed interval prediction based on ensemble GRU model, IEEE Trans. Sustainable Energy, 11 (2020), 1370-1380. doi: 10.1109/TSTE.2019.2926147
    [22] R. Wang, C. Li, W. Fu, G. Tang, Deep learning method based on gated recurrent unit and variational mode decomposition for short-term wind power interval prediction, IEEE Trans. Neural Networks Learn. Syst., 31 (2019), 3814-3827.
    [23] S. Boyd, N. Parikh, E. Chu, B. Peleato. J. Eckstein, Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers, Now Foundations and Trends, 2011.
    [24] Y. Liu, C. Yang, K. Huang, W. Cui, Non-ferrous metals price forecasting based on variational mode decomposition and LSTM network, Knowl. Based Syst., 188 (2020), 105006.
    [25] J. W. E, J. M. Ye, L. L. He, H. H. Jin, Energy price prediction based on independent component analysis and gated recurrent unit neural network, Energy, 189 (2019), 116278.
    [26] S. Chen, L. Ge, Exploring the attention mechanism in LSTM-based Hong Kong stock price movement prediction, Quant. Finance, 19 (2019), 1507-1515. doi: 10.1080/14697688.2019.1622287
    [27] R. Desimon, J. Duncan, Neural mechanisms of selective visual attention, Annu. Rev. Neurosci., 18 (1995), 193-222. doi: 10.1146/annurev.ne.18.030195.001205
    [28] M. T. Luong, H. Pham, C. D. Manning, Effective approaches to attention-based neural machine translation, arXiv: 1508.04025.
    [29] L. Li, S. Tang, Y. Zhang, L. Deng, Q. Tian, GLA: global-local attention for image description, IEEE Trans. Multimedia, 20 (2017), 726-737.
    [30] S. Wang, X. Wang, S. Wang, D. Wang, Bi-directional long short-term memory method based on attention mechanism and rolling update for short-term load forecasting, Int. J. Electric. Power Energy Syst., 109 (2019), 470-479. doi: 10.1016/j.ijepes.2019.02.022
  • This article has been cited by:

    1. Sheza M. El-Deeb, Luminita-Ioana Cotîrlă, Coefficient Estimates for Quasi-Subordination Classes Connected with the Combination of q-Convolution and Error Function, 2023, 11, 2227-7390, 4834, 10.3390/math11234834
    2. Arzu Akgül, 2024, Chapter 8, 978-981-97-3237-1, 159, 10.1007/978-981-97-3238-8_8
  • Reader Comments
  • © 2020 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(5904) PDF downloads(258) Cited by(21)

Figures and Tables

Figures(8)  /  Tables(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog