Citation: Herbert F. Jelinek, Jemal H. Abawajy, David J. Cornforth, Adam Kowalczyk, Michael Negnevitsky, Morshed U. Chowdhury, Robert Krones, Andrei V. Kelarev. Multi-layer Attribute Selection and Classification Algorithm for the Diagnosis of Cardiac Autonomic Neuropathy Based on HRV Attributes[J]. AIMS Medical Science, 2015, 2(4): 396-409. doi: 10.3934/medsci.2015.4.396
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Let $ p $ be a prime, $ f $ be a polynomial with k variable and $ \mathbf{F}_p = \mathbf{Z}/(p) $ be the finite field, where $ \mathbf{Z} $ is the integer ring, and let
| $ N(f;p) = \#\{(x_1, x_2, \cdots, x_k)\in\mathbf{F}_p^k|f(x_1, x_2, \cdots, x_k) = 0\}. $ |
Many scholars studied the exact formula (including upper bound and lower bound) for $ N(f; p) $ for many years, it is one of the main topics in the finite field theory, the most elementary upper bounds was given as follows (see [14])
| $ N(f;p)\leq p^{k-1}deg f. $ |
Let $ ord_p $ denote the p-adic additive valuation normalized such that $ ord_pp = 1 $. The famous Chevalley-Warning theorem shows that $ ord_pN(f; p) > 0 $ if $ n > deg f $. Let [x] denote the least integer $ \geq x $ and $ e $ denote the extension degree of $ \mathbf{F}_q /\mathbf{F}_p $. Ax (see [2]) showed that
| $ ord_pN(f;q)\geq e\left[\frac{n-deg f}{deg f}\right]. $ |
In 1977, S. Chowla et al. (see [7]) investigated a problem about the number of solutions of a equation in finite field $ \mathbf{F}_p $ as follow,
| $ x_1^3+x_2^3+\cdot\cdot\cdot+x_k^3\equiv0, $ |
where $ p $ is a prime with $ p\equiv1\bmod\ 3 $ and $ x_i\in \mathbf{F}_p $, $ 1\leq i\leq k $.
Let $ M_k $ denotes the number of solutions of the above equation. They proved that
| $ M3=p2+d(p−1),M4=p2+6(p2−p),∞∑s=1Msxs=x1−px+x2(p−1)(2+dx)1−3px2−pdx3, $ |
where $ d $ is uniquely determined by $ 4p = d^2+27y^2 $ and $ d\equiv1\ mod\ 3 $.
Myerson [12] extended the result in [2] to the field $ \mathbf{F}_q $ and first studied the following equation over $ \mathbf{F}_q $,
| $ x_1^3+x_2^3+\cdot\cdot\cdot+x_k^3\equiv0. $ |
Recently J. Zhao et al. (see [17]) investigated the following equations over field $ \mathbf{F}_p $,
| $ f1=x41+x42+x43,f2=x41+x42+x43+x44. $ |
And they give exact value of $ N(f_1;p) $ and $ N(f_2;p) $. For more general problem about this issue interested reader can see [6,9,10,11].
In this paper, let $ A(k, p) $ denotes the number of solutions of the following equation in $ \mathbf{F}_p $,
| $ x_1^6+x_2^6+\cdot\cdot\cdot+x_k^6\equiv0, $ |
where $ p $ is a prime with $ p\equiv1\bmod\ 3 $ and $ x_i\in \mathbf{F}_p $, $ 1\leq i\leq k $, and for simplicity, in the rest of this paper, we assume there exists an integer $ z $ such that $ z^3\equiv2\bmod\ p $, we use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid a lot of complicated calculations. We proved the following:
Theorem 1. For any positive integer $ k\geq1 $, we have the recurrence formula
| $ A(k+6,p)=5pA(k+4,p)+10dpA(k+3,p)+(46p2+5d2p+dp)A(k+2,p)+(2p2+120dp2+3d3p+d2p+dp)A(k+1,p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)A(k,p)+pk+5−pk+4−(10dp+2d2)pk+3−64pk+5/2−(429+121d+5d2)pk+2−2d2pk+3/2−(3d3+130d2+12d+6)pk+1−d4pk, $ |
with the initial condition
| $ A(1,p)=1,A(2,p)=4(p−1)+p,A(3,p)=10d(p−1)+p2,A(4,p)=56p(p−1)+10d2(p−1)+p3,A(5,p)=188dp(p−1)+5d3(p−1)+16dC(p)(p−1)+p4,A(6,p)=p5+1400p2(p−1)+(388d2+8d−576)p(p−1)+d2p−d2, $ |
where $ d $ is uniquely determined by $ 4p = d^2+27y^2 $ and $ d\equiv1\ mod\ 3 $, and $ C(p) = \sum\limits_{a = 1}^{p}e_p(a^3) $.
Remark. Our method is suitable to calculus the number of solutions of the following equation in $ \mathbf{F}_p $,
| $ x_1^t+x_2^t+\cdot\cdot\cdot+x_k^t\equiv0, $ |
where $ p $ satisfied a certain congruence conditions, and $ t $ is any nature number.
Our Theorem 2 can be deduced from Theorem 1 and the theory of the Difference equations.
Theorem 2. Let $ t_i $ ($ 1\leq i\leq k $) be the real root of the below equation with multiplicity $ s_i $ ($ 1\leq i\leq k $) respectively, and $ \rho_je^{\pm iw_j} $ ($ 1\leq j\leq h $) be the complex root of the below equation with multiplicity $ r_j $ ($ 1\leq j\leq h $) respectively,
| $ x6=5px4+10dpx3+(46p2+5d2p+dp)x2+(2p2+120dp2+3d3p+d2p+dp)x+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p). $ |
We have
| $ A(n,p)=k∑i=1si∑a=1Ciansi−atni+h∑j=1rj∑b=1Djbnrj−bρnjcosnwj+h∑j=1rj∑b=1Ejbnrj−bρnjsinnwj, $ |
where $ C_{ia}, D_{jb}, E_{jb} $, are determined by
| $ A(6,p)=k∑i=1si∑a=1Cia6si−at6i+h∑j=1rj∑b=1Djb6rj−bρ6jcos6wj+h∑j=1rj∑b=1Ejb6rj−bρ6jsin6wj,A(5,p)=k∑i=1si∑a=1Cia5si−at5i+h∑j=1rj∑b=1Djb5rj−bρ5jcos5wj+h∑j=1rj∑b=1Ejb5rj−bρ5jsin5wj,A(4,p)=k∑i=1si∑a=1Cia4si−at4i+h∑j=1rj∑b=1Djb4rj−bρ4jcos4wj+h∑j=1rj∑b=1Ejb4rj−bρ4jsin4wj,A(3,p)=k∑i=1si∑a=1Cia3si−at3i+h∑j=1rj∑b=1Djb3rj−bρ3jcos3wj+h∑j=1rj∑b=1Ejb3rj−bρ3jsin3wj,A(2,p)=k∑i=1si∑a=1Cia2si−at2i+h∑j=1rj∑b=1Djb2rj−bρ2jcos2wj+h∑j=1rj∑b=1Ejb2rj−bρ2jsin2wj,A(1,p)=k∑i=1si∑a=1Ciati+h∑j=1rj∑b=1Djbρjcoswj+h∑j=1rj∑b=1Ejbρjsinwj. $ | (1.1) |
Before we prove these lemmas, we give some notations, $ \chi_2 $ denotes the second-order character of $ \mathbf{F}_p $, $ \chi $ denotes the third-order character of $ \mathbf{F}_p $, $ \psi $ denotes the sixth order character of $ \mathbf{F}_p $.
| $ ep(x)=e2πixp,τ(χ)=p∑m=1χ(m)ep(m),G(χ,m)=p∑a=1χ(a)ep(am). $ |
We call $ G(\chi, m) $ the Gauss sum, and we have the following:
| $ G(χ,m)=τ(χ)¯χ(m),(m,p)=1. $ | (2.1) |
And also we have
| $ |τ(χ)|=√p, $ | (2.2) |
where $ \chi $ is a primitive character of $ \mathbf{F}_p $. And let $ G(m, 6;p) = \sum\limits_{a = 0}^{p-1}e_p(ma^6) $. For the property of the exponential sum and the general Gauss sum, interested readers can see [1,4,5,8,13,15].
Lemma 1. Let $ p $ be a prime with $ p\equiv1\bmod\ 3 $. Then for any third-order character $ \chi $ of $ \mathbf{F}_p $, we have the identity
| $ \tau^3(\chi)+\tau^3(\overline{\chi}) = dp, $ |
where $ d $ is uniquely determined by $ 4p = d^2+27y^2 $ and $ d\equiv1\bmod\ 3 $.
Proof. For the proof of this lemma see [3].
Lemma 2. Let $ \chi $ be a third-order character of $ \mathbf{F}_p $ with $ p\equiv1\bmod\ 3 $, and $ C(p) = \tau(\chi)+\tau(\overline{\chi}) $, then $ C(p) = \sum\limits_{a = 1}^{p}e_p(a^3) $.
Proof.
| $ A = \tau(\chi)+\tau(\overline{\chi}) = \sum\limits_{a = 1}^p(1+\chi(a)+\overline{\chi}(a))e(\frac{a}{p}) = \sum\limits_{a = 1}^pe(\frac{a^3}{p}). $ |
Lemma 3. Let $ p\equiv1\bmod\ 6 $, $ 2\equiv z^3\bmod\ p $ for some $ z $, and let $ \chi $ be a third-order character of $ \mathbf{F}_p $, $ \psi $ be a sixth-order character of $ \mathbf{F}_p $, then we have the identity
| $ \tau(\psi) = \frac{\tau^2(\chi)}{\sqrt{p}}.\\ $ |
Proof. This is Lemma 3 in [16].
Lemma 4. As the definition above, we have the identity
| $ G(m, 6;p) = \sqrt{p}\chi_2(m)+\frac{\overline{x^2}}{\sqrt{p}}\psi(m)+\frac{x^2}{\sqrt{p}}\overline{\psi}(m)+\overline{x}\chi(m)+x\overline{\chi}(m), $ |
where $ (m, p) = 1 $ and $ x = \tau(\chi) $.
Proof. Firstly we have the identity
| $ 1+χ2(m)+χ(m)+¯χ(m)+ψ(m)+¯ψ(m)={6, if m≡a6 mod p;0,otherwise. $ |
So we have
| $ G(m,6;p)=p−1∑a=0(1+χ2(a)+χ(a)+¯χ(a)+ψ(a)+¯ψ(a))ep(ma)=G(χ2,m)+G(ψ,m)+G(¯ψ,m)+G(χ,m)+G(¯χ,m) $ |
By (2.1) and Lemma 3, we have
| $ G(m,6;p)=τ(χ2)χ2(m)+τ(¯ψ)ψ(m)+τ(ψ)¯ψ(m)+τ(¯χ)χ(m)+τ(χ)¯χ(m)=√pχ2(m)+¯x2√pψ(m)+x2√p¯ψ(m)+¯xχ(m)+x¯χ(m). $ | (2.3) |
By (2.3), we complete the proof of our lemma.
Next we let,
| $ Gn(m,6;p)=an+bnχ2(m)+cnψ(m)+dn¯ψ(m)+enχ(m)+fn¯χ(m). $ | (2.4) |
Then we have following Lemma 5.
Lemma 5. Let $ a_n, b_n, c_n, d_n, e_n, f_n $ are defined as above, then we have that $ a_n, b_n, c_n, d_n, e_n, f_n $ are uniquely determined by $ n $, where $ n\geq1 $.
Proof. By the orthogonality of characters of $ \mathbf{F}_p $, we have
| $ p−1∑a=1χ(a)={p−1, if χ=χ0;0,otherwise. $ | (2.5) |
By (2.4) and (2.5) we have
| $ p−1∑m=1Gn(m,6;p)=(p−1)an+bnp−1∑m=1χ2(m)+cnp−1∑m=1ψ(m)+dnp−1∑m=1¯ψ(m)+enp−1∑m=1χ(m)+fnp−1∑m=1¯χ(m)=(p−1)an. $ |
So we have
| $ an=1p−1p−1∑m=1Gn(m,6;p). $ | (2.6) |
By the same method, we have
| $ bn=1p−1p−1∑m=1χ2(m)Gn(m,6;p),cn=1p−1p−1∑m=1¯ψ(m)Gn(m,6;p),dn=1p−1p−1∑m=1ψ(m)Gn(m,6;p),en=1p−1p−1∑m=1¯χ(m)Gn(m,6;p),fn=1p−1p−1∑m=1χ(m)Gn(m,6;p). $ |
So now it is easy to see the conclusion of the lemma.
Lemma 6. The sequences $ \{a_n\} $, $ \{b_n\} $, $ \{c_n\} $, $ \{d_n\} $, $ \{e_n\} $, $ \{f_n\} $ are defined above, then they satisfied the following recurrence formulae ($ n\geq0 $):
| $ an+1=√pbn+¯x2√pdn+x2√pcn+xen+¯xfn, $ | (2.7) |
| $ bn+1=√pan+¯x2√pen+x2√pfn+xdn+¯xcn, $ | (2.8) |
| $ cn+1=√pfn+¯x2√pan+x2√pen+xbn+¯xdn, $ | (2.9) |
| $ dn+1=√pen+¯x2√pfn+x2√pan+xcn+¯xbn, $ | (2.10) |
| $ en+1=√pdn+¯x2√pcn+x2√pbn+xfn+¯xan, $ | (2.11) |
| $ fn+1=√pcn+¯x2√pbn+x2√pdn+xan+¯xen, $ | (2.12) |
with the initial condition
| $ a_0 = 1, b_0 = c_0 = d_0 = e_0 = f_0 = 0. $ |
Proof. We only prove (2.7), the rest can be proved in the same way. By Lemma 5, we know $ a_n $ is unique determined by $ n $. We can compare the coefficient of the equation
| $ G^{n+1}(m, 6;p) = G^{n}(m, 6;p)G(m, 6;p). $ |
We have
| $ a_{n+1} = \sqrt{p}b_n+\frac{\overline{x^2}}{\sqrt{p}}d_n+\frac{x^2}{\sqrt{p}}c_n+xe_n+\overline{x}f_n. $ |
So we complete the proof of the lemma.
Lemma 7. Let $ a_n $ is defined as above, then we have
| $ a0=1,a1=0,a2=5p,a3=10dp,a4=56p2+10d2p,a5=188dp2+5d3p+16dpC(p). $ |
Proof. By Lemma 4 and after some elementary calculations we have
| $ G2(m,6;p)=5p+2dp1/2χ2(m)+4p1/2xψ(m)+4p1/2¯x¯ψ(m)+(p−1¯x4+3x2)χ(m)+(p−1x4+3¯x2)¯χ(m),G3(m,6;p)=10dp+(16p3/2+dp1/2)χ2(m)+(15p¯x+2dx2+p−1x5)χ(m)+(15px+2d¯x2+p−1¯x5)¯χ(m)+(4p−1/2x4+12p1/2¯x2+2dp1/2x)ψ(m)+(4p−1/2¯x4+12p1/2x2+2dp1/2¯x)¯ψ(m),G4(m,6;p)=60p2+9d2p+dp+48dp3/2χ2(m)+(p−2x8+17¯x4+46px2+16dp)χ(m)+(p−2¯x8+17x4+46p¯x2+16dp)¯χ(m)+(56p3/2x+4dp−1/2x4+12dp1/2¯x2+8p−1/2¯x5)ψ(m)+(56p3/2¯x+4dp−1/2¯x4+12dp1/2x2+8p−1/2x5)¯ψ(m),G5(m,6;p)=188dp2+5d3p+16dpC(p)+(52d2p3/2+208p5/2+16dp1/2(x2+¯x2))χ2(m)+(p−2/5x10+p−3/2¯x8+4dp−1/2¯x5+71p1/2x4+(46p3/2+16p1/2)x2+(129p3/2+10d2p1/2)¯x2+60dp3/2x+16dp3/2)ψ(m)+(p−2/5¯x10+p−3/2x8+4dp−1/2x5+71p1/2¯x4+(46p3/2+16p1/2)¯x2+(129p3/2+10d2p1/2)x2+60dp3/2¯x+16dp3/2)¯ψ(m)+(8p−1¯x7+p−1x7+25x5+52dpx2+(28dp+46p2)x+16d¯x4+112p2¯x)χ(m)+(8p−1x7+p−1¯x7+25¯x5+52dp¯x2+(28dp+46p2)¯x+16dx4+112p2x)¯χ(m), $ |
and comparing the above formulae with (2.6), we have
| $ a_0 = 1, a_1 = 0, a_2 = 5p, a_3 = 10dp, a_4 = 60p^2+9d^2p+dp, a_5 = 188dp^2+5d^3p+16dpC(p).\\ $ |
Lemma 8. Let $ a_n $, $ b_n $, $ c_n $, $ d_n $, $ e_n $, $ f_n $ are defined as above, then we have
| $ a6=5pa4+10dpa3+(46p2+5d2p+dp)a2+(2p2+120dp2+3d3p+d2p+dp)a1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)a0b6=5pb4+10dpb3+(46p2+5d2p+dp)b2+(2p2+120dp2+3d3p+d2p+dp)b1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)b0c6=5pc4+10dpc3+(46p2+5d2p+dp)c2+(2p2+120dp2+3d3p+d2p+dp)c1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)c0d6=5pd4+10dpd3+(46p2+5d2p+dp)d2+(2p2+120dp2+3d3p+d2p+dp)d1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)d0e6=5pe4+10dpe3+(46p2+5d2p+dp)e2+(2p2+120dp2+3d3p+d2p+dp)e1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)e0f6=5pf4+10dpf3+(46p2+5d2p+dp)f2+(2p2+120dp2+3d3p+d2p+dp)f1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)f0 $ |
Proof. We only proof the first formula, the rest can be proof in the same way. By Lemma 6, we have
| $ a6=√pb5+¯x2√pd5+x2√pc5+xe5+¯xf5=5pa4+2dp1/2b4+4p1/2¯xc4+4p1/2xd4+(3¯x2+p−1x4)e4+(3x2+p−1¯x4)f4=5pa4+10dpa3+(d2p1/2+12p3/2)b3+(2dp1/2¯x+8p1/2x2+p−1/2¯x4)c3+(2dp1/2x+8p1/2¯x2+p−1/2x4)d3+(11px+¯x2+p−1¯x5)e3+(11p¯x+x2+p−1x5)f3=5pa4+10dpa3+(46p2+5d2p+dp)a2+(25dp3/2+2p3/2)b2+(p−3/2¯x7+2p−1/2x5+p−1/2¯x4+42p3/2¯x+2dp1/2x2+(d2+1)p1/2¯x)c2+(p−3/2x7+2p−1/2¯x5+p−1/2x4+42p3/2x+2dp1/2¯x2+(d2+1)p1/2x)d2+(10x4+(32p+d2)¯x2+(4dp+p)x)e2+(10¯x4+(32p+d2)x2+(4dp+p)¯x)f2=5pa4+10dpa3+(46p2+5d2p+dp)a2+(2p2+120dp2+3d3p+d2p+dp)a1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)a0. $ |
So we complete the proof of this lemma.
Lemma 9. Let $ a_n $ is defined as above, then for any integer $ n\geq0 $, we have
| $ an+6=5pan+4+10dpan+3+(46p2+5d2p+dp)an+2+(2p2+120dp2+3d3p+d2p+dp)an+1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)an. $ |
Proof. By (2.4) and Lemma 8, we have
| $ G6(m,6;p)=5pG4(m,6;p)+10dpG3(m,6;p)+(46p2+5d2p+dp)G2(m,6;p)+(2p2+120dp2+3d3p+d2p+dp)G(m,6;p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p). $ |
We multiple $ G^{n}(m, 6;p) $ to the both side of the above formula, we have
| $ Gn+6(m,6;p)=5pGn+4(m,6;p)+10dpGn+3(m,6;p)+(46p2+5d2p+dp)Gn+2(m,6;p)+(2p2+120dp2+3d3p+d2p+dp)Gn+1(m,6;p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)Gn(m,6;p). $ |
By Lemma 5, we can compare the coefficient of the above equation, we have
| $ an+6=5pan+4+10dpan+3+(46p2+5d2p+dp)an+2+(2p2+120dp2+3d3p+d2p+dp)an+1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)an. $ |
In the formula below, we always let $ k\geq1 $. By the following formula,
| $ p−1∑a=0ep(ma)={p, if p∣m;0, otherwise, $ |
we have
| $ A(k,p)=1pp−1∑m=0p−1∑x1=0,x2=0,⋅⋅⋅,xk=0ep(m(x61+x62+⋅⋅⋅+x6k))=1pp−1∑m=0Gk(m,6;p). $ | (3.1) |
By (8), we have
| $ A(k,p)=1pp−1∑m=0Gk(m,6;p)=1p(p−1∑m=1Gk(m,6;p)+pk)=1p((p−1)ak+pk)=p−1pak+pk−1. $ | (3.2) |
So by Lemma 9, we have
| $ A(k+6,p)−pk+5=5p(A(k+4,p)−pk+3)+10dp(A(k+3,p)−pk+2)+(46p2+5d2p+dp)(A(k+2,p)−pk+1)+(2p2+120dp2+3d3p+d2p+dp)(A(k+1,p)−pk)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)(A(k,p)−pk−1). $ |
So we have
| $ A(k+6,p)=5pA(k+4,p)+10dpA(k+3,p)+(46p2+5d2p+dp)A(k+2,p)+(2p2+120dp2+3d3p+d2p+dp)A(k+1,p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)A(k,p)+pk+5−pk+4−(10dp+2d2)pk+3−64pk+5/2−(429+121d+5d2)pk+2−2d2pk+3/2−(3d3+130d2+12d+6)pk+1−d4pk. $ |
And by Lemma 7 and (3.2), we have the initial conditions
| $ A(1,p)=1,A(2,p)=4(p−1)+p,A(3,p)=10d(p−1)+p2,A(4,p)=56p(p−1)+10d2(p−1)+p3,A(5,p)=188dp(p−1)+5d3(p−1)+16dC(p)(p−1)+p4.A(6,p)=p5+1400p2(p−1)+(388d2+8d−576)p(p−1)+d2p−d2. $ |
So we complete the proof of the theorem.
The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. And we give an expression of the number of solutions of the above equation by the root of sixth degree polynomial. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.
The author thanks to referees for very important recommendations and warnings which improved the paper.
The author declares that there is no competing interest.
| [1] | Colagiuri S, Colagiuri R, Ward J (1998) National diabetes strategy and implementation plan. Canberra: Paragon Printers. |
| [2] | Pop-Busui R, Evans GW, Gerstein HC, et al. (2010) The ACCORD Study Group. Effects of cardiac autonomic dysfunction on mortality risk in the Action to Control Cardiovascular Risk in Diabetes (ACCORD) Trial. Diab Care 33: 1578-1584. |
| [3] | Spallone V, Ziegler D, Freeman R, et al. (2011) Cardiovascular autonomic neuropathy in diabetes: clinical impact, assessment, diagnosis, and management. Diab Metab Res Rev 27: 639-653. |
| [4] |
Dimitropoulos G, Tahrani AA, Stevens MJ (2014) Cardiac autonomic neuropathy in patients with diabetes mellitus. World J Diab 5: 17-39. doi: 10.4239/wjd.v5.i1.17
|
| [5] |
Vinik AI, Erbas T, Casellini CM (2013) Diabetic cardiac autonomic neuropathy, inflammation and cardiovascular disease. J Diabetes Investig 4: 4-18. doi: 10.1111/jdi.12042
|
| [6] |
Task Force of The European Society of Cardiology and The North American Society of Pacing and Electrophysiology (1996) Special report: heart rate variability standards of measurement, physiological interpretation, and clinical use. Circulation 93: 1043-1065. doi: 10.1161/01.CIR.93.5.1043
|
| [7] | Jelinek HF, Abawajy JH, Kelarev AV, et al. (2014) Decision trees and multi-level ensemble classifiers for neurological diagnostics. AIMS Med Sci 1: 1-12. |
| [8] | Dietrich DF, Schindler C, Schwartz J, et al. (2006) Heart rate variability in an ageing population and its association with lifestyle and cardiovascular risk factors: results of the SAPALDIA study. Europace 8: 521-529. |
| [9] | Lake DE, Richman JS, Griffin MP, et al. (2002) Sample entropy analysis of neonatal heart rate variability. Am J Physiol 283: 789-797. |
| [10] |
La Rovere MT, Pinna GD, Maestri R, et al. (2003) Short-term heart rate variability strongly predicts sudden cardiac death in chronic heart failure patients. Circulation 107: 565-570. doi: 10.1161/01.CIR.0000047275.25795.17
|
| [11] |
Huikuri HV, Linnaluoto MK, Seppänen T, et al. (1992) Circadian rhythm of heart rate variability in survivors of cardiac arrest. Am J Cardiol 70: 610-615. doi: 10.1016/0002-9149(92)90200-I
|
| [12] | Cornforth DJ, Tarvainen MP, Jelinek HF (2014) Visualization methods for assisting detection of cardiovascular neuropathy. Engineering in Medicine and Biology Society (EMBC2014) 36th Annual International Conference of the IEEE, 26-30 Aug. 2014, 6675-6678. |
| [13] | Tarvainen MP, Cornforth DJ, Jelinek HF (2014) Principal component analysis of heart rate variability data in assessing cardiac autonomic neuropathy. Engineering in Medicine and Biology Society (EMBC2014), 36th Annual International Conference of the IEEE, 26-30 Aug. 2014, 6667-6670. |
| [14] | Abawajy J, Kelarev A, Chowdhury M (2013) Multistage approach for clustering and classification of ECG data. Comp Meth Pro Biomed 112: 720-730. |
| [15] | Abawajy J, Kelarev A, Chowdhury M, et al. (2013) Predicting cardiac autonomic neuropathy category for diabetic data with missing values, Comp Bio Med 43: 1328-1333. |
| [16] |
Stranieri A, Abawajy J, Kelarev A, et al. (2013) An approach for Ewing test selection to support the clinical assessment of cardiac autonomic neuropathy. Art Intel Med 58: 185-193. doi: 10.1016/j.artmed.2013.04.007
|
| [17] | Jelinek HF, Yatsko A, Stranieri A, et al. (2015) Diagnostic with incomplete nominal/discrete data. Art Intel Med 4: 22-35. |
| [18] | Cornforth D, Jelinek HF (2007) Automated classification reveals morphological factors associated with dementia, App Soft Compu 8: 182-190. |
| [19] | Ewing DJ, Campbell JW, Clarke BF (1980) The natural history of diabetic autonomic neuropathy. Q J Med 49: 95-100. |
| [20] |
Ewing DJ, Martyn CN, Young RJ, et al. (1985) The value of cardiovascular autonomic functions tests: 10 years experience in diabetes. Diab Care 8: 491-498. doi: 10.2337/diacare.8.5.491
|
| [21] |
Khandoker AH, Jelinek HF, Palaniswami M (2009) Identifying diabetic patients with cardiac autonomic neuropathy by heart rate complexity analysis. BioMed Engine Online 8: 1-12. doi: 10.1186/1475-925X-8-1
|
| [22] | Thayer JF, Yamamoto SS, Brosschot JF (2010) The relationship of autonomic imbalance, heart rate variability and cardiovascular disease risk factors, Int J Cardiol 141: 122-131. |
| [23] |
Karmakar CK, Khandoker AH, Jelinek HF, et al. (2013) Risk stratification of cardiac autonomic neuropathy based on multi-lag Tone-Entropy. Med Bio Engine Comp 51: 537-546. doi: 10.1007/s11517-012-1022-5
|
| [24] | Tan CO (2013) Heart rate variability: are there complex patterns? Front Physiol 4: 1-3. |
| [25] | Imam MH, Karmakar C, Khandoker AH, et al. (2014) Analysing cardiac autonomic neuropathy in diabetes using electrocardiogram derived systolic-diastolic interval interactions. Compu Cardiol 41: 85-88. |
| [26] |
Spallone V, Menzinger G (1997) Diagnosis of cardiovascular autonomic neuropathy in diabetes. Diabetes 46: 67-76. doi: 10.2337/diab.46.2.S67
|
| [27] | Jelinek HF, Pham P, Struzik ZR, et al. (2007) Short term ECG recording for the identification of cardiac autonomic neuropathy in people with diabetes mellitus. Proceedings of the 19th International Conference on Noise and Fluctuations, Tokyo, Japan, pp. 683-686. |
| [28] | Khandoker AH, Weiss DN, Skinner JE, et al. (2011) PD2i heart rate complexity measure can detect cardiac autonomic neuropathy: an alternative test to Ewing battery. Compu Cardiol 38: 525-528. |
| [29] | TFESC/NASPE (1996) Heart rate variability. Standards of measurement, physiological interpretation, and clinical use. Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology. Euro Heart J 17: 354-381. |
| [30] |
Goldberger AL, Amaral LAN, Hausdorff JM, et al. (2002) Fractal dynamics in physiology: Alterations with disease and aging. Pro Nat Aca Sci USA 99: 2466-2472. doi: 10.1073/pnas.012579499
|
| [31] |
Peng CK, Havlin S, Stanley HE, et al. (1995) Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 5: 82-87. doi: 10.1063/1.166141
|
| [32] | Ho YL, Lin C, Lin YH, et al. (2011). The prognostic value of non-linear analysis of heart rate variability in patients with congestive heart failure—a pilot study of multiscale entropy. PLoS ONE 6: 1-6. |
| [33] | Sturmberg JP, Bennett JM, Picard M, et al. (2015) The trajectory of life. Decreasing physiological network complexity through changing fractal patterns. Front Physiol 6: 1-11. |
| [34] |
Oida ET, Moritani KT, Yamori Y (1999) Diabetic alteration of cardiac vago-sympathetic modulation assessed with tone-entropy analysis. Acta Physiol Scandi 165: 129-135. doi: 10.1046/j.1365-201x.1999.00494.x
|
| [35] |
Lake DE, Moorman JR (2011) Accurate estimation of entropy in very short physiological time series: the problem of atrial fibrillation detection in implanted ventricular devices. Am J Physiol Heart Cicul Physiol 300: H319-325. doi: 10.1152/ajpheart.00561.2010
|
| [36] | Jelinek HF, Khandoker A, Palaniswami M, et al. (2010) Tone-entropy analysis as a cardiac risk stratification tool. Compu Cardiol 37: 955-958. |
| [37] | Quinlan R (1993) C4.5: Programs for Machine Learning. San Mateo, CA: Morgan Kaufmann Publishers. |
| [38] |
Breiman L (2001) Random Forests. Machine Learning 45:5-32. doi: 10.1023/A:1010933404324
|
| [39] | Williams G (2011) Data mining with Rattle and R: the art of excavating data for knowledge discovery (use R!). New York, Dordrecht, Heidelberg, London: Springer. |
| [40] | Platt J (1998) Fast training of support vector machines using sequential minimal optimization. In: Schoelkopf B, Burges C, Smola A, editors, Advances in Kernel Methods—Support Vector Learning 41-64. |
| [41] | Witten H, Frank E, Hall MA (2011) Data mining: practical machine learning tools and techniques with java implementations. 3ed, New York, Sydney: Morgan Kaufmann, 2011. |
| [42] | Bouckaert, RR, Frank E, Hall M, et al. WEKA manual for version 3-7-13, http://www.cs.waikato.ac.nz/ml/weka/, viewed 15 July 2015. |
| [43] | Hall M, Frank E, Holmes G, et al. (2009) The WEKA data mining software: an update. SIGKDD Explor 11: 10-18. |
| [44] | Negnevitsky M (2011) Artificial intelligence: a guide to intelligent systems. 3rd eds., New York: Addison Wesley. |
| [45] | Williams GJ (2009) Rattle: a data mining GUI for R, The R J 1: 45-55. |
| [46] |
Kohavi R, John GH (1997) Wrappers for feature subset selection. Art Intell 97:273-324. doi: 10.1016/S0004-3702(97)00043-X
|
| [47] | Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. New York: Addison-Wesley. |
| [48] | Cornforth DJ, Jelinek HF, Teich MC, et al. (2004) Wrapper subset evaluation facilitates the automated detection of diabetes from heart rate variability measures. Proceedings of the International Conference on Computational Intelligence for Modelling Control and Automation (CIMCA'2004), University of Canberra, Australia, pp. 446-455. |
| [49] | Moraglio A, Di Chio C, Poli R (2007) Geometric Particle Swarm Optimisation. Proceedings of the 10th European Conference on Genetic Programming, Berlin, Heidelberg, 125-136. |
| [50] | Demsar J (2006) Statistical comparisons of classifiers over multiple data sets. J Machine Learning Res 7: 1-30. |
| [51] |
Dietterich TG (1998) Approximate statistical tests for comparing supervised classification learning algorithms. Neural Compu 10: 1895-1924. doi: 10.1162/089976698300017197
|
| [52] | Cornforth D, Tarvainen M, Jelinek HF (2014) How to calculate Rényi entropy from Heart Rate Variability, and why it matters for detecting cardiac autonomic neuropathy. Front Bioeng Biotecho 2: 1-7. |
| [53] |
Ziegler DA, Rathmann VW, Strom A, et al. (2015) Increased prevalence of cardiac autonomic dysfunction at different degrees of glucose intolerance in the general population: the KORA S4 survey. Diabetologia 58:1118-1128. doi: 10.1007/s00125-015-3534-7
|
Herbert F. Jelinek, Jemal H. Abawajy, David J. Cornforth, Adam Kowalczyk, Michael Negnevitsky, Morshed U. Chowdhury, Robert Krones, Andrei V. Kelarev. Multi-layer Attribute Selection and Classification Algorithm for the Diagnosis of Cardiac Autonomic Neuropathy Based on HRV Attributes[J]. AIMS Medical Science, 2015, 2(4): 396-409. doi: 10.3934/medsci.2015.4.396
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