# Phase transitions in ferroelectric domain walls

E. A. Eliseev  $|$ ${}^{\textsf{\footnotesize{\mbox{?}}}}$, M. D. Glinchuk
$|$ ${}^{\textsf{\footnotesize{\mbox{?}}}}$, L. P. Yurchenko  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$, P. Maksymovych  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$, A. N. Morozovska  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$ Corresponding author:
<EMAIL_ADDRESS>
(Received June 21, 2022, in final form September 12, 2022)

###### Abstract

Despite multiple efforts, there exist many unsolved fundamental problems
related with detection and analysis of internal polarization structure and
related phase transitions in ferroelectric domain walls. Their solution can be
very important for the progress in domain wall nanoelectronics and related
applications in advanced memories and other information technologies. Here, we
theoretically study the features of phase transitions in the domain walls,
which are potentially detectable by the scanning probe capacitance microwave
microscopy. The finite element modelling based on the Landau-Ginzburg-
Devonshire theory is performed for the capacitance changes related with the
domain wall motion in a multiaxial ferroelectric BaTiO3.

Key words: ferroelectrics, domain walls, local phase transition

###### Abstract

Íåçâàæàþè íà çíàí çóñèëëÿ, ñíó áàãàòî íåâèðøåíèõ ôóíäàìåíòàëüíèõ ïðîáëåì,
ïîâÿçàíèõ ç âèÿâëåííÿì òà àíàëçîì âíóòðøíüî¿ ñòðóêòóðè ïîëÿðèçàö¿ òà âäïîâäíèõ
ôàçîâèõ ïåðåõîäâ ó ñåãíåòîåëåêòðèíèõ äîìåííèõ ñòíêàõ. õ ðøåííÿ ìîæå áóòè äóæå
âàæëèâèì äëÿ ïðîãðåñó çàñòîñóâàííÿ äîìåííèõ ñòíîê ó íàíîåëåêòðîíö òà
ïîâÿçàíîãî ç íèì çàñòîñóâàííÿ ó åëåìåíòàõ ïàìÿò òà íøèõ íôîðìàöéíèõ
òåõíîëîãÿõ. Ó ðîáîò òåîðåòèíî âèâàþòüñÿ îñîáëèâîñò ôàçîâèõ ïåðåõîäâ ó äîìåííèõ
ñòíêàõ, ÿê ïîòåíöéíî ìîæíà âèÿâèòè çà äîïîìîãîþ ñêàíóþî¿ çîíäîâî¿ ìíñíî¿
ìêðîõâèëüîâî¿ ìêðîñêîï¿. Ñêíåííî-åëåìåíòíå ìîäåëþâàííÿ íà îñíîâ òåîð¿ Ëàíäàó-
Ãíçáóðãà-Äåâîíøèðà âèêîíàíî äëÿ çìíè ìíîñò, ïîâ’ÿçàíî¿ ç ðóõîì äîìåííî¿ ñòíêè
â áàãàòîîñüîâîìó ñåãíåòîåëåêòðèêó BaTiO3.

Ключов слова: ñåãíåòîåëåêòðèêè, äîìåíí ñòíè, ëîêàëüí ôàçîâ ïåðåõîäè

##  1 Introduction

Since its appearance and until now, nanoscale ferroics (ferromagnets,
ferroelectrics, ferroelastics) have been the main object of fundamental
research on the physical nature of long-range order of polar, magnetic, and
structural properties [1, 2, 3]. The leading role is played by the emergence
of a domain structure of long-range order parameters, such as electric
polarization, magnetization, and (anti)ferrodistortion, and its interaction
with the surface of nanoferroic [4, 5]. In nanoferroics, the role of the
surface increases significantly with a decrease in their size and very often
begins to dominate in comparison with the bulk contribution [6]. Moreover,
this trend is not only due to the distinctive behavior of all properties near
the surface but also due to the presence of the developed gradients of order
parameters from the surface to the bulk of the nanoferroic [1]. The
development of the gradients largely generates and is generated by the
appearance of internal electric and elastic fields near the surface of a
ferroic.

Recently, it was predicted theoretically that a decrease in the correlation-
gradient polarization energy leads to a significant increase in the
polarization gradient, which in turn leads to spontaneous bending of uncharged
domain walls in thin multiferroic films [7] and in ferroelectric nanoparticles
[8]. Such walls can form meandering [7] and/or labyrinthine [8] structures.
Later, similar structures were discovered experimentally in thin BiFeO3 and
Pb(Zr0.4Ti0.6)O3 films by the high-resolution scanning transmission electron
microscopy (HR-STEM) [9] and piezoresponse force microscopy (PFM) [10]
methods, respectively, and corroborated by ab initio calculations [10]. HR-
STEM, PFM and conductive atomic force microscopy (C-AFM) experiments
registered an enhanced conductivity of nominally uncharged domain walls in
ferroelectrics and multiferroics [11, 12, 13]. Non-Ising and chiral
ferroelectric domain walls were predicted theoretically [14] and were revealed
by nonlinear optical microscopy [15]. Despite multiple efforts, there exist
fundamental problems related with the observation of the internal polarization
structure and the related phase transitions in the domain walls. Their
solution can be very important for the progress in domain wall nanoelectronics
[16, 17] and in the related advanced applications.

To contribute to the problems solution, here we theoretically study the
features of phase transitions in ferroelectric domain walls, which are
potentially observable using the scanning probe capacitance microwave
microscopy [18, 19, 20]. The finite element modelling (FEM) and analytical
calculations based on the Landau-Ginzburg-Devonshire (LGD) theory are
performed for the capacitance changes related with the domain wall motion in a
multiaxial ferroelectric multiaxial ferroelectric BaTiO3.

##  2 Problem statement

Let us consider the features of the capacitance of a thin-film ferroelectric
capacitor in a scanning probe geometry, with or without a domain structure.
The control parameter is the temperature of the ferroelectric, which changes
leads to the changes of the domain structure period, the width and orientation
of the walls. Specifically, the magnitude and angles between the various
components of polarization at the walls significantly depends on temperature
and undergoes phase transitions. Typical parameters of the probe are the
contact radius $\sim$50–100 nm (in fact, this is a free parameter), and the
thickness of the ferroelectric film $\sim$10–20 nm [see figure 1 (a)].

Figure 1: (Colour online) Ferroelectric thin film with a domain structure in
the probe (a) and planar capacitor geometry (b).

If the probe is modelled by a flat electrode (for example by a disk), then an
ultrathin (less than 0.1 nm thick) dielectric gap between the PFM probe and
the ferroelectric surface is required to stabilize the domain structure [see
figure 1 (b)]. The domain structure is unstable without the gap, and the
system transits into a single-domain state at rather low voltages. However,
even the presence of an ultrathin gap leads to the fact that a significant
part of the applied voltage drops across it. To minimize this effect, the gap
can be filled with a liquid dielectric having a high dielectric permittivity
or can be replaced with a dead layer.

The electric polarization of the film is calculated from three coupled
nonlinear Euler-Lagrange (EL) equations for each of the three components of
the vector $P_{i}$, taking into account the model lattice potential (in fact,
lattice pinning). These equations are derived from the variation of LGD
thermodynamic potential, with respect to $P_{i}$. These EL equations must be
accomplished with electrostatic and elastic problems. The pinning value
determines the critical electric field required for the walls to move, and
different models of pinning exist. As a rule, the critical field can be
estimated analytically using several models. Suzuki-Ishibashi (S-I) model [21]
can be used for the threshold field determination. The activation filed, that
determines the nucleation kinetics, can be defined within Miller-Weinreich
[22] or Burtsev-Chervonobrodov (B-C) models [23] modified by Rappe et al. [24]
and Aravind et al. [25], respectively, for consideration of the phenomena
related with polarization gradient and depolarization effects at the wall.

Note that the critical field of the wall motion depends on many factors [26,
27], such as the structure (single-component and multicomponent) of the wall
[28], which in turn is determined by the temperature and thickness of the
film, by the bending of the wall and its mechanical clamping determined by the
gradient of the applied electric field, electrostriction and flexoelectric
couplings [29, 30].

For numerical modelling, we implement the simplest situation, when the lattice
pinning can be modelled by a 3D quasi-harmonic potential that leads to the
internal field
${E_{i}}\cong{E_{0}}\sin\bigl{(}{\vec{k}\vec{x}+{\psi_{0}}}\bigr{)}$ in the
right-hand side of the EL equations. Here, $\vec{k}$ is the reciprocal lattice
vector, or its integral multiple.

The linear capacitance of the system can be calculated in two ways: as the
second derivative of its electrostatic energy with respect to the applied
electric voltage, or as the derivative of the total charge at the electrodes
of the capacitor with respect to the voltage applied between the electrodes.
In the case of an exact solution and linear capacity, as can be seen, both
methods give the same answer. The first method turns out to be easier for
numerical modelling, while the second method is sometimes more useful for
analytical calculations.

##  3 Capacitance of a single domain film

For the model situation shown in figure 1 (b), except in the case of a single-
domain polarization distribution $P_{3}(z)$, electric potential distribution
$\varphi\left(z\right)$, charge density on the electrodes $\sigma$ and total
system capacitance $C$ are equal to

$\displaystyle{\varphi_{FE}}\left(z\right)$ $\displaystyle=$
$\displaystyle\frac{U}{{1+({\varepsilon_{b}}/{\varepsilon_{g}})(d/h)}}\left({1-\frac{z}{h}}\right)+\frac{1}{{{\varepsilon_{0}}{\varepsilon_{b}}}}\Bigg{[}{{\overline{P}}_{3}}\frac{{h-z}}{{1+({\varepsilon_{b}}/{\varepsilon_{g}})(d/h)}}\Bigg{.}$
(3.1) $\displaystyle-$
$\displaystyle\Bigg{.}\mathop{\int}_{z}^{h}{P_{3}}\left({{z^{\prime}}}\right)\mathrm{d}{z^{\prime}}\Bigg{]},\quad
0<z<h,$
$\displaystyle{\varphi_{g}}\left(z\right)=\frac{U}{{1+({\varepsilon_{b}}/{\varepsilon_{g}})(d/h)}}\left({1-\frac{{{\varepsilon_{b}}}}{{{\varepsilon_{g}}}}\frac{z}{h}}\right)-\frac{{{{\overline{P}}_{3}}}}{{{\varepsilon_{0}}{\varepsilon_{g}}}}\frac{{z+d}}{{1+({\varepsilon_{b}}/{\varepsilon_{g}})(d/h)}},\quad-d<z<0,$
(3.2) $\displaystyle\sigma\left({-d}\right)$
$\displaystyle=\frac{U}{{1+({\varepsilon_{b}}d/{\varepsilon_{g}}h)}}\frac{{{\varepsilon_{0}}{\varepsilon_{b}}}}{h}+\frac{{{{\overline{P}}_{3}}}}{{1+({\varepsilon_{b}}d/{\varepsilon_{g}}h)}},$
$\displaystyle\sigma\left(h\right)$
$\displaystyle=-\frac{U}{{1+({\varepsilon_{b}}d/{\varepsilon_{g}}h)}}\frac{{{\varepsilon_{0}}{\varepsilon_{b}}}}{h}-\frac{{{{\overline{P}}_{3}}}}{{1+({\varepsilon_{b}}d/{\varepsilon_{g}}h)}},$
(3.3) $\displaystyle
C=S\frac{{\mathrm{d}\sigma}}{{\mathrm{d}U}}=S\left(\frac{{{\varepsilon_{0}}{\varepsilon_{b}}{\varepsilon_{g}}}}{{{\varepsilon_{g}}h+{\varepsilon_{b}}d}}+\frac{{{\varepsilon_{g}}h}}{{{\varepsilon_{g}}h+{\varepsilon_{b}}d}}\frac{{\mathrm{d}{{\overline{P}}_{3}}}}{{\mathrm{d}U}}\right),$
(3.4)

where $h$ is the thickness of the ferroelectric film located at $0<z<h$, $d$
is the thickness of the ultra-thin gap located at $-d<z<0$. $S$ is the surface
area of the electrodes. The upper surface of the film corresponds to $z=0$,
and the probe electrode under the applied voltage $U$ is located in the plane
$z=-d$. The bottom electrode $z=h$ is grounded. The relative background
permittivity of the ferroelectric film is $\varepsilon_{b}$, and the relative
dielectric constant of the gap is $\varepsilon_{g}$. The average value of the
spontaneous polarization normal component is defined as
${\overline{P}_{3}}=({1}/{h})\int_{0}^{h}{P_{3}}\left({{z^{\prime}}}\right)\mathrm{d}{z^{\prime}}$.
In the approximate equality (3.4), $C\approx
S({\varepsilon_{0}}{\varepsilon_{b}}{\varepsilon_{g}})/({\varepsilon_{g}}h+{\varepsilon_{b}}d)$,
and the value ${\overline{P}_{3}}$ is regarded voltage-independent, and in
this case, the capacitance is indifferent to structural phase transitions.
Thus, only the behavior of the second term, which is
${\varepsilon_{g}}hS/({\varepsilon_{g}}h+{\varepsilon_{b}}d)\left(\mathrm{d}{{\overline{P}}_{3}}/\mathrm{d}U\right)$,
can be an indicator of the transition from the paraelectric phase to the
tetragonal, then to the orthorhombic and rhombohedral phases, which occurs
under the temperature decrease. These transitions will correspond to the jumps
in the relative dielectric susceptibility of a single-domain ferroelectric
film ${\chi_{f}}\left(T\right)$, because
$\mathrm{d}{{\overline{P}}_{3}}/\mathrm{d}U\sim{\chi_{f}}\left(T\right)$ in
the linear approximation.

The case of a single-domain BaTiO3 film was used to debug the code. In this
case, the thickness of the gap tends to zero, since the single-domain state
may be unstable at a thickness above the critical one (less than 0.1 nm in the
considered case). The dependence of the capacitance of the single-domain state
on temperature is shown by dashed curves in figure 2, figure 3. BaTiO3
parameters are listed in table 1.

Figure 2: (Colour online) The temperature dependence of the surface charge at
the electrodes (a), the effective dielectric permittivity (b) and the
capacitance (c) of the BaTiO3 film placed between the electrodes calculated
for different voltages $U$ indicated at the plot legends. Solid and dashed
curves correspond to different initial conditions, namely poly- and single-
domain states of the film, respectively. The curves are calculated for a dead
layer thickness $d=0.2$ nm and permittivity $\varepsilon_{g}=81$. Film
thickness is 20 nm, mismatch strain is $-0.05$%.

Figure 3: (Colour online) Distribution of the polarization components $P_{i}$
(a–c) in the cross-section of the film at 300 K, and $U=0$. (d) One-component
Ising domain wall at 370 K. Other parameters are the same as in figure 2.

###  3.1 Influence of a single domain wall on the capacitance of the
ferroelectric film

The result (3.4) is obvious and precise for an arbitrary distribution of
$P_{3}(z)$. In the case of a domain wall, the polarization depends on the
transverse coordinates, $P_{3}(x,z)$. Therefore, the charge density on the
electrodes depends on the transverse coordinates, $\sigma(x,-d)$, primarily on
the width of the wall and on the distribution of the normal polarization
component on the wall. In fact, the wall-surface contact area contributes to
the capacitance through these two contributions. Let us roughly estimate their
influence on the capacitance of the system, based on the results of numerical
modelling.

Table 1: LGD coefficients and other material parameters of BaTiO3. Coefficient | Numerical value
---|---
|
$\varepsilon_{b,e}$ |  $\varepsilon_{b}=7$ (core background), $\varepsilon_{e}=10$ (surrounding)
$a_{i}$ (in mJ/C2) | $a_{1}=3.34(T-381)\cdot{10^{5}},\alpha_{T}=3.34\cdot{10^{5}}$
|  $(a_{1}=-2.94\cdot{10^{7}}$ at 298 K)
$a_{ij}$ (in m5J/C4) |  $a_{11}=4.69(T-393)\cdot{10^{6}}-2.02\cdot{10^{8}},a_{12}=3.230\cdot{10^{8}}$,
| ($a_{11}=-6.71\cdot{10^{8}}$ at 298 K)
$a_{ijk}$ (in m9J/C6) |  $a_{111}=-5.52(T-393)\cdot{10^{7}}+2.76\cdot{10^{9}}$, $a_{112}=4.47\cdot{10^{9}}$,
|  $a_{123}=4.91\cdot{10^{9}}$ (at 298 K $a_{111}=82.8\cdot{10^{8}}$,
| $a_{112}=44.7\cdot{10^{8}},a_{123}=49.1\cdot{10^{8}})$
$Q_{ij}$ (m4/C2) | $Q_{11}=0.11,Q_{12}=-0.043,Q_{44}=0.059$
$s_{ij}$ (in $10^{-12}$ Pa-1) | $s_{11}=8.3,s_{12}=-2.7,s_{44}=9.24$
$g_{ij}$ (in $10^{-10}$ m3J/C2) | $g_{11}=5.0,g_{12}=-0.2,g_{44}=0.2$
$F_{ij}$ (in $10^{-11}$ m3/C); $f_{ij}$ (in V) | $F_{11}=2.4,F_{12}=0.5,F_{44}=0.06$
| (these values are recalculated from the values
|  $f_{11}=5.1,f_{12}=3.3,f_{44}=0.065$ V).
| The equality $F_{44}=F_{11}-F_{12}$ is valid in the isotropic case.

As it follows from the FEM results at $U=0$, the domain wall significantly
broadens when approaching an electrically open surface $z=0$. However, this
broadening occurs in a relatively thin sub-surface layer near the top surface
due to the depolarization field, and it is absent near the bottom surface due
to its perfect electrical contact with the electrode [31]. In the middle of
the film and near the bottom electrode, which is in a perfect electrical
contact with the film without any gap, a 180-degree Ising domain wall is
characterized by a profile,
${P_{3}}\left({x,z}\right)\approx{P_{b}}\tanh\left(x/L_{c}\right)$, where
$L_{c}$ is the correlation length that determines the width of the wall. The
same domain wall, broadened near the surface due to the depolarization field,
has a different profile, ${P_{3}}\left({x,z}\right)\approx
P_{s}\left(x/L_{s}\right)/\left[(x/L_{s})^{2}+1\right]$, where $L_{s}$ is a
characteristic length, which exceeds (and sometimes significantly) the value
of $L_{c}$. It should be noted that the value of $L_{c}$ significantly depends
on the type of the wall and on the orientation of polarization vector.

The change in the charge density on electrodes associated with the presence of
a 180-degree wall with a plane $x=0$ can be approximated in the following way:

$\displaystyle\Delta\sigma\left({x,-d}\right)\approx\frac{{{{\overline{P}}_{3}}}}{{1+({\varepsilon_{b}}/{\varepsilon_{g}})(d/h)}}f\left({\frac{x}{{{L_{c}}}}}\right).$
(3.5)

Here, we regard that $U=0$, $f(x/L_{c})$ is an odd function, equal to either
$+1$ or $-1$ far away from the wall, i.e., at $\left|x\right|\gg{L_{c}}$. In
the interval $\left|x\right|\gg{L_{c}}$, the function $f(x/L_{c})$ changes its
sign. Taking into account that the charge density contains an alternating
contribution, which dominates at low voltages $U$ (when the wall is
stationary), the contribution to the capacitance in the presence of a domain
structure is actually made only by the difference in polarization values taken
to the left and to the right of the domain wall caused by the voltage
dependence of polarization, which is found from the nonlinear EL equations.
These equations correspond to minimization of the LGD functional.

Under the increase of $U$, the main contribution to capacitance is made by the
shift of the domain walls. Due to an increase in the effective capacitance of
the domain wall region, this contribution can significantly exceed the
contribution from single-domain regions. It can be estimated as follows:

$\displaystyle
C=\frac{{\mathrm{d}Q}}{{\mathrm{d}U}}=\frac{{{\varepsilon_{0}}{\varepsilon_{b}}{\varepsilon_{g}}S}}{{{\varepsilon_{g}}h+{\varepsilon_{b}}d}}+\frac{{{\varepsilon_{g}}hS}}{{{\varepsilon_{g}}h+{\varepsilon_{b}}d}}\frac{{2{L_{c}}}}{L}{\chi_{DW}}\left(T\right).$
(3.6)

Here, $L$ is the film length along $x$-axis, which is perpendicular to the
wall plane. The function $\chi_{DW}(T)$ is the effective dielectric
susceptibility of the domain wall, which, in principle, contains information
about all the changes in the wall structure, including phase transitions.
However, this information is indirect, and, as our FEM shows, it is not
trivial to separate it from other contributions, such as transformation to a
single in the external electric field (actually, departure of domain walls).

The charge at the electrodes, effective dielectric permittivity and
capacitance of the BaTiO3 film with thickness 20 nm are shown in figure 2 (a),
(b) and (c), respectively. A very small compressive strain $u_{m}=-0.05$% is
applied to the ferroelectric film, leading to the disappearance of the region
of meta-stable orthorhombic phase and to the shift of the temperatures of
structural and polar transitions [32], [33]. The transition from rhombohedral
(Rh) to tetragonal (Te) phase occurs at a temperature of about 360 K, and the
transition from the Te to paraelectric (PE) phase occurs just above 410 K.
Herein, the Te-PE transition is smeared due to the presence of internal
electric field. In the case of a single-domain film, the Rh-Te phase
transition appearing at 360 K is very close to the second order transition.
That is why one could see a subtle kink on the dashed curves. In the case of a
polydomain film, a sharp jump is seen at this temperature, that is very close
to the first order phase transition. It is caused by the movement of the
domain wall, since in the vicinity of a phase transition the critical field is
very small and the walls move even under a tiny electric field. Thus, the
ferroelectric film tends to become a single-domain in this case. The
permittivity is calculated based on the capacitance of the system, obtained by
differentiation of the charge density under the voltage increase. Similarly,
the application of a very small electric field destroys the poly-domain
structure near the point of transition to the PE phase.

In the Rh phase, that is stable below 350 K, all three components of
polarization are non-zero and change their signs when coming across the
180-degree wall (see figure 3). In the Te phase, that is stable above 360 K,
the walls become of Ising type, i.e., only one component of the polarization
exists. One of the periods of the polydomain structure is shown.

At the temperatures above 360 K, but this depends on the voltage, a giant
dielectric susceptibility is associated with the movement of the domain walls
under the influence of the electric field (see figure 4). Only one period of
the polydomain structure is shown to demonstrate the trend towards the
development of a single domain state (“the run-away” or departure of the
walls). Herein, the critical field of movement of the walls depends on the
temperature, and its definition is a separate task.

Figure 4: (Colour online) Distribution of polarization component $P_{3}$ in
the cross-section of the film calculated for different voltage values $U=0$,
1, 2 and 3 mV (from left to right) at 370 K. Other parameters are the same as
in figure 2.

##  4 Conclusion

Using LGD theory and FEM, we analytically and numerically study the influence
of the domain walls on the capacitance of the multiaxial ferroelectric BaTiO3
film. We analyze the distinct features of the phase transitions in the domain
walls, which are potentially detectable using the scanning probe capacitance
microwave microscopy.

In the Rh phase, that is stable below 350 K, all three components of
polarization are non-zero and change their signs when coming across the
180-degree wall. In the Te phase, that is stable above 360 K, the walls become
of Ising type, i.e., only one component of the polarization exists.
Furthermore, we observe the trend towards the development of a single domain
state (“the run-away” or departure of walls). Herein, the critical field of
movement of the walls depends on the temperature, and its definition is a
separate task.

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# Consecutive primes and IP sets

William D. Banks  Department of Mathematics, University of Missouri, Columbia
MO, USA<EMAIL_ADDRESS>
###### Abstract.

For an infinite set $\mathcal{M}\subset\mathbb{N}$, let ${\tt
FS}(\mathcal{M})$ be the set of all nonzero finite sums of distinct numbers in
$\mathcal{M}$. An _IP set_ is any set of the form ${\tt FS}(\mathcal{M})$. Let
$p_{n}$ denote the $n$-th prime number for each $n\geqslant 1$. A _de Polignac
number_ is any number $m$ such that $p_{n+1}-p_{n}=m$ for infinitely many $n$.
In this note, we show that every IP set $\mathcal{A}\subset 2\mathbb{N}$
contains infinitely many de Polignac numbers.

MSC Primary: 11N05; Secondary: 11N36, 11B75.

Keywords: Consecutive primes, Maynard sieve, IP sets.

##  1\. Introduction

A classical result of Hindman [6] (answering a question posed by Graham and
Rothschild [5]) relates to _finite sums sets_ , which are those of the form

${\tt FS}(\mathcal{M})\mathrel{\vbox{\hbox{\scriptsize.}
\hbox{\scriptsize.}}}=\Big{\\{}\sum_{m\in\mathcal{M}^{\prime}}m:\varnothing\neq\mathcal{M}^{\prime}\subset\mathcal{M},~{}|\mathcal{M}^{\prime}|<\infty\Big{\\}}$

for some infinite set $\mathcal{M}$ of natural numbers. Hindman showed that
for any finite coloring of $\mathbb{N}$, there is an infinite set
$\mathcal{M}$ such that ${\tt FS}(\mathcal{M})$ is monochromatic. In their
treatise on topological dynamics and combinatorial number theory, Furstenberg
and Weiss [3] coined the term _IP set_ to mean any finite sums set ${\tt
FS}(\mathcal{M})$. There are numerous interesting examples of IP sets.
Bergelson and Rusza [2] gave a general method for constructing a variety of IP
sets; they showed that if $\mathcal{S}$ is the set of squarefree natural
numbers and $s\in\mathbb{N}$, then $\mathcal{S}-s$ contains an IP set if and
only if $s\in\mathcal{S}$.

Let $p_{1}\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=2$,
$p_{2}\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=3$,
$p_{3}\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=5$, etc., be
the sequence of all primes. Our aim in this note is to establish a connection
between IP sets and the sequence $(p_{n+1}-p_{n})_{n\in\mathbb{N}}$ of
consecutive prime gaps.

######  Theorem 1.

For every IP set $\mathcal{A}\subset 2\mathbb{N}$, there exist infinitely many
$a\in\mathcal{A}$ with the property that $p_{n+1}-p_{n}=a$ for infinitely many
$n$.

######  Corollary 2.

For every $s\in\mathcal{S}$, there are infinitely many $m\in\mathcal{S}-s$
such that $p_{n+1}-p_{n}=m$ for infinitely many $n$.

######  Corollary 3.

For every $c>0$, there are infinitely many numbers $m$ in the set

$\\{m\in\mathbb{N}:p\mid m\Longrightarrow p>c\\}$

with the property that $\tfrac{1}{2}(p_{n+1}-p_{n})=m-1$ for infinitely many
$n$.

######  Corollary 4.

There are infinitely many $m\in\mathbb{N}$ that have only the digits $0$ and
$2$ in their decimal expansion and satisfy $p_{n+1}-p_{n}=m$ for infinitely
many $n$.

Theorem 1 is a simple consequence of our main result (Theorem 8) whose
formulation uses terminology given in the next section; the corollaries follow
at once by choosing appropriate IP sets. The proof of Theorem 8 is
straightforward and short, but it relies on several deep results from number
theory and additive combinatorics. Thus, the “heavy lifting” has largely been
accomplished thanks to the efforts of various mathematicians; their
contributions are outlined in the next section.

##  2\. Background Material

We briefly review background material associated with our main theorem and
some results needed in the proof.

###  2.1. de Polignac numbers

The notorious _twin prime conjecture_ asserts the existence of infinitely many
natural numbers $n$ for which $p_{n+1}-p_{n}=2$. This conjecture first
appeared in an 1849 monograph of Alphonse de Polignac [8], who also speculated
that for every even natural number $2m$, there are infinitely many prime gaps
of size $2m$, i.e., that $p_{n+1}-p_{n}=2m$ holds for infinitely many $n$. Let
Pol denote the set of even integers $2m$ with this property. The elements of
Pol are called _de Polignac numbers_ , and de Polignac’s hypothesis that ${\bf
Pol}=2\mathbb{N}$ is known as _Polignac’s conjecture_.111Another equally
famous conjecture of Alphonse de Polignac is that every odd integer
$k\geqslant 3$ can be expressed as the sum of a power of two and a prime
number. Thus, in the literature, the term _de Polignac number_ often refers to
odd integers $k$ of this form.

The _bounded gap conjecture_ (that is, the assertion that ${\bf
Pol}\neq\varnothing$) was first proved in the celebrated 2013 paper of Zhang
[11], who employed a modification of the Goldston-Pintz-Yıldırım sieve [4] to
prove the existence of a de Polignac number not exceeding $7\times 10^{7}$.
The Polymath8 project sought to understand and improve Zhang’s arguments and
ultimately showed that ${\bf Pol}\cap[2,4680]\neq\varnothing$. Later in 2013,
Maynard [7] showed that for every $m\in\mathbb{N}$ one has222The same result,
albeit with a slightly weaker bound, had been independently proven by Tao
around the same time; the latter work remains unpublished.

$\liminf_{n}(p_{n+m}-p_{m})\ll m^{3}\mathrm{e}^{4m}.$

His breakthrough paper also proved the existence of infinitely many prime gaps
of size at most $600$; in other words, it showed that ${\bf
Pol}\cap[2,600]\neq\varnothing$. Following Maynard’s announcement, a new
polymath project was launched (Polymath8b) and eventually achieved the result
that ${\bf Pol}\cap[2,246]\neq\varnothing$, currently the strongest known
result in this direction.

###  2.2. The Maynard-Tao theorem

An ordered tuple $\mathcal{H}$ of distinct nonnegative integers is said to be
_admissible_ if it avoids at least one residue class mod $p$ for every prime
$p$. Following Tao and Ziegler [10], we say that a finite admissible tuple
$\mathcal{H}=(h_{1},\ldots,h_{k})$ is _prime-producing_ if there are
infinitely many $n\in\mathbb{N}$ such that $\\{n+h_{1},\ldots,n+h_{k}\\}$ are
simultaneously prime. The _Dickson-Hardy-Littlewood conjecture_ asserts that
every such tuple $\mathcal{H}$ is prime-producing. This conjecture remains one
of the great unsolved problems in number theory, and the strongest
unconditional result in this direction is the following theorem of Maynard [7]
and Tao.

######  Theorem 5 (Maynard–Tao).

For every integer $m\geqslant 2$, there is a number $k_{m}$ for which the
following holds. If $(h_{1},\ldots,h_{k})$ is admissible with $k\geqslant
k_{m}$, then the set $\\{n+h_{1},\ldots,n+h_{k}\\}$ contains at least $m$
primes for infinitely many $n\in\mathbb{N}$.

Shortly after this theorem was announced, a version of the Maynard-Tao theorem
producing _consecutive_ primes in tuples was given by Banks, Freiberg and
Turnage-Butterbaugh [1], who used the result to solve an old problem of Erdős.
The following is a slightly weakened version of their theorem.

######  Theorem 6 (Banks–Freiberg–Turnage-Butterbaugh).

Fix an integer $m\geqslant 2$, and let $k_{m}$ have the property stated in
Theorem 5. If $(h_{1},\ldots,h_{k})$ is admissible with $k\geqslant k_{m}$,
then there is a set
$\\{h^{\prime}_{1},\ldots,h^{\prime}_{m}\\}\subset\\{h_{1},\ldots,h_{k}\\}$
such that the set $\\{n+h^{\prime}_{1},\ldots,n+h^{\prime}_{m}\\}$ consists of
$m$ consecutive primes for infinitely many $n\in\mathbb{N}$.

###  2.3. Ramsey’s theorem

We recall the well known result of Ramsey [9].

######  Theorem 7 (Ramsey).

For any $r,s\in\mathbb{N}$, there is a least natural number $R(r,s)$ such that
every blue–red edge coloring of the complete graph on $R(r,s)$ vertices
contains either a blue clique on $r$ vertices or a red clique on $s$ vertices.

##  3\. Statement of the main theorem

The present paper was inspired by recent work of Tao and Ziegler [10], who
study infinite restricted sumsets in the shifted primes and ascending chains
of prime-producing tuples. Although our results revolve around the study of
sums of de Polignac numbers, we do not use any results from [10] but instead
rely only on the theorems discussed in §2 and an auxiliary result (see Lemma 9
below).

######  Theorem 8.

Let $\mathcal{A}\subset 2\mathbb{N}$ be an IP set. Then, for any $k\geqslant
1$, there is a sequence $a_{1},a_{2},\ldots,a_{k}$ in $\mathcal{A}$ such that

$\sum_{i\leqslant n\leqslant j}a_{n}\in{\bf Pol}\qquad\text{whenever}\quad
1\leqslant i\leqslant j\leqslant k.$ (1)

Moreover, the $a_{j}$’s can be chosen so that all of the sums in (1) are
distinct from one another.

Note that Theorem 8 implies that $|\mathcal{A}\cap{\bf Pol}|=\infty$, and
Theorem 1 follows.

##  4\. Proof of the main theorem

We need the following lemma.

######  Lemma 9.

Every infinite set $\mathcal{B}\subset\mathbb{N}$ contains a strictly
increasing sequence $(b_{n})_{n\in\mathbb{N}}$ for which the tuple
$(b_{1},b_{2},\ldots)$ is admissible.

###### Proof.

Put $b_{0}\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=0$ and
$\mathcal{B}_{0}\mathrel{\vbox{\hbox{\scriptsize.}
\hbox{\scriptsize.}}}=\mathcal{B}$. Using induction, we construct a sequence
$(b_{n})_{n\in\mathbb{N}}$ contained in $\mathcal{B}$ and a family of sets
$\\{\mathcal{B}_{n}\\}_{n\in\mathbb{N}}$ satisfying

$b_{n}>b_{n-1},\qquad\mathcal{B}_{n}=\mathcal{B}_{n-1}\cap(p_{n}\mathbb{Z}+b_{n})\qquad\mbox{and}\qquad|\mathcal{B}_{n}|=\infty.$
(2)

Indeed, let $n\geqslant 1$, and suppose $b_{j}$ and $\mathcal{B}_{j}$ have
been determined for all $j<n$. Since $\mathcal{B}_{n-1}$ is infinite, the
pigeonhole principle shows that
$\mathcal{B}_{n}\mathrel{\vbox{\hbox{\scriptsize.}
\hbox{\scriptsize.}}}=\mathcal{B}_{n-1}\cap(p_{n}\mathbb{Z}+h)$ is infinite
for at least one integer $h\in[0,p_{n})$. Replacing $h$ by any number
$b_{n}\in\mathcal{B}_{n}$ larger than $b_{n-1}$, we obtain (2), completing the
induction.

To finish the proof, we show that $(b_{1},b_{2},\ldots)$ is admissible.
Indeed, let $p_{n}$ be an arbitrary prime, and observe that
$\mathcal{B}_{m}\subset\mathcal{B}_{n}\subset p_{n}\mathbb{Z}+b_{n}$ for all
integers $m\geqslant n$; consequently,

$\big{|}\\{b_{1},b_{2},\ldots\\}\bmod
p_{n}\big{|}=\big{|}\\{b_{1},\ldots,b_{n}\\}\bmod p_{n}\big{|}\leqslant
n<p_{n},$

and thus $(b_{1},b_{2},\ldots)$ avoids at least one residue class mod $p_{n}$.
∎

###### Proof of Theorem 8.

As $\mathcal{A}$ is an IP set, $\mathcal{A}={\tt FS}(\mathcal{M})$ for some
infinite set $\mathcal{M}\subset\mathbb{N}$. We order the elements
$\\{m_{1},m_{2},\ldots\\}$ of $\mathcal{M}$ with $m_{1}<m_{2}<\cdots$ and
denote by $\mathcal{B}$ the set of all partial sums of the form
$\sum_{k\leqslant\ell}m_{k}$. According to Lemma 9, the set $\mathcal{B}$
contains a strictly increasing sequence $(b_{n})_{n\in\mathbb{N}}$ for which
the tuple $(b_{1},b_{2},\ldots)$ is admissible. Note that, for any pair
$(i,j)$ with $1\leqslant i<j$, there are numbers
$\ell^{\prime}_{i}<\ell^{\prime}_{j}$ such that

$b_{j}-b_{i}=\sum_{\ell^{\prime}_{i}<k\leqslant\ell^{\prime}_{j}}m_{k}\in\mathcal{A}.$

Next, we extract from $(b_{n})_{n\in\mathbb{N}}$ a lacunary subsequence
$(v_{n})_{n\in\mathbb{N}}$ such that $v_{n+1}/v_{n}>10$ (say) for each $n$;
this ensures that the differences $v_{j}-v_{i}$ are all distinct from one
another. The resulting tuple $(v_{1},v_{2},\ldots)$ is also admissible. As
before, for any pair $(i,j)$ with $1\leqslant i<j$, there are numbers
$\ell_{i}<\ell_{j}$ such that

$v_{j}-v_{i}=\sum_{\ell_{i}<k\leqslant\ell_{j}}m_{k}\in\mathcal{A}.$ (3)

Now, in the notation of Theorems 6 and 7, we set

$r\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=L+1,\qquad
s\mathrel{\vbox{\hbox{\scriptsize.}
\hbox{\scriptsize.}}}=k_{2},\qquad\mbox{and}\qquad
N\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=R(r,s).$

We now view the finite collection $\\{v_{1},\ldots,v_{N}\\}$ as the vertex set
$\mathcal{V}$ of a complete graph $\mathcal{G}$ with edge set is
$\mathcal{E}\mathrel{\vbox{\hbox{\scriptsize.}
\hbox{\scriptsize.}}}=\\{e_{i,j}:1\leqslant i<j\leqslant N\\}$; each edge
$e_{i,j}$ connects vertex $v_{i}$ to vertex $v_{j}$. Define a blue–red
coloring of the edges via the map

$\mathscr{C}:\mathcal{E}\to\\{{\tt blue},{\tt
red}\\},\qquad\mathscr{C}(e_{i,j})\mathrel{\vbox{\hbox{\scriptsize.}
\hbox{\scriptsize.}}}=\begin{cases}{\tt blue}&\quad\hbox{if
$v_{j}-v_{i}\in{\tt Pol}$},\\\ {\tt red}&\quad\hbox{if $v_{j}-v_{i}\not\in{\tt
Pol}$}.\\\ \end{cases}$

By Theorem 7, the graph $\mathcal{G}$ contains either a blue clique on $r=L+1$
vertices or a red clique on $s=k_{2}$ vertices.

We claim that the latter possibility is untenable, i.e., no such red clique
exists. Indeed, suppose on the contrary that there is a complete subgraph
$\mathcal{G}^{\flat}\subset\mathcal{G}$ determined by vertices
$\mathcal{V}^{\flat}\mathrel{\vbox{\hbox{\scriptsize.}
\hbox{\scriptsize.}}}=\\{h_{1},\ldots,h_{k_{2}}\\}$ in $\mathcal{V}$ such that
$\mathcal{G}^{\flat}$ is monochromatic and colored red by the map
$\mathscr{C}$. By Theorem 6, there is a set
$\\{h^{\prime}_{1},h^{\prime}_{2}\\}\subset\mathcal{V}^{\flat}$ such that
$n+h^{\prime}_{1}$ and $n+h^{\prime}_{2}$ are consecutive primes for
infinitely many $n\in\mathbb{N}$; in other words,
$|h^{\prime}_{2}-h^{\prime}_{1}|\in{\bf Pol}$. But this is not possible since
$h^{\prime}_{1}$ and $h^{\prime}_{2}$ are distinct vertices in
$\mathcal{V}^{\flat}$ and hence the corresponding edge is colored red by
$\mathscr{C}$.

Consequently, within the graph $\mathcal{G}$ there is blue clique on $r=L+1$
vertices, i.e., there is a complete subgraph
$\mathcal{G}^{\sharp}\subset\mathcal{G}$ determined by vertices
$\mathcal{V}^{\sharp}\mathrel{\vbox{\hbox{\scriptsize.}
\hbox{\scriptsize.}}}=\\{h_{1},\ldots,h_{L+1}\\}$ in $\mathcal{V}$ such that
$\mathcal{G}^{\sharp}$ is monochromatic and colored blue by $\mathscr{C}$.
Ordering $\mathcal{V}^{\sharp}$ so that $h_{1}<\cdots<h_{L+1}$, we have

$\\{h_{j}-h_{i}:1\leqslant i<j\leqslant m+1\\}\subset{\bf Pol}$

since the edge determined by $h_{i}$ and $h_{j}$ is colored blue by
$\mathscr{C}$ for $1\leqslant i<j\leqslant m+1$. By construction, all of the
differences $h_{j}-h_{i}$ are distinct from one another. Since $h_{i}$ and
$h_{j}$ are distinct vertices in $\mathcal{V}$, we have
$h_{n+1}-h_{n}\in\mathcal{A}$ for each $n\in[1,L]$ in view of (3). Defining
$a_{n}\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=h_{n+1}-h_{n}$,
we obtain a sequence $a_{1},\ldots,a_{L}$ with the properties stated in the
theorem, since

$\sum_{i\leqslant n\leqslant j}a_{n}=\sum_{i\leqslant n\leqslant
j}(h_{n+1}-h_{n})=h_{j+1}-h_{i}\in{\bf Pol}.$

This completes the proof of Theorem 8. ∎

## Acknowledgement

The author thanks Terry Tao for various helpful conversations.

## References

  * [1] W. D. Banks, T. Freiberg, and C. Turnage-Butterbaugh, Consecutive primes in tuples. _Acta Arith._ 167 (2015), no. 3, 261–266.
  * [2] V. Bergelson and I. Ruzsa, Squarefree numbers, IP sets and ergodic theory. _Paul Erdős and his mathematics, I_ (Budapest, 1999), 147–160.
  * [3] H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory. _J. Analyse Math._ 34 (1978), 61–85.
  * [4] D. A. Goldston, J. Pintz and C. Y. Yıldırım, Primes in tuples. I. _Ann. of Math. (2)_ 170 (2009), no. 2, 819–862.
  * [5] R. L. Graham and B. L. Rothschild, Ramsey’s theorem for $n$-parameter sets. _Trans. Amer. Math. Soc._ 159 (1971), 257–292.
  * [6] N. Hindman, Finite sums from sequences within cells of a partition of N. _J. Combinatorial Theory Ser. A_ 17 (1974), 1–11.
  * [7] J. Maynard, Small gaps between primes. _Ann. of Math. (2)_ 181 (2015), no. 1, 383–413.
  * [8] A. de Polignac, _Recherches nouvelles sur les nombres premiers._ Comptes Rendus des Séances de l’Académie des Sciences (1849).
  * [9] F. P. Ramsey, On a problem of formal logic. _Proc. London Math. Soc. (2)_ 30 (1929), no. 4, 264–286.
  * [10] T. Tao and T. Ziegler, Infinite partial sumsets in the primes. _J. Analyse Math._ 151 (2023), no. 1, 375–389.
  * [11] Y. Zhang, Bounded gaps between primes. _Ann. of Math. (2)_ 179 (2014), no. 3, 1121–1174.


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[1,2]

[2]This paper supported in part …

[type=editor, auid=000,bioid=1, orcid=0000-0001-9344-6658] [2] [1]

Conceptualization of this study, Methodology, Software, Data curation, Writing
- Original draft preparation

[1] [1,2]

MS.c. thesis supervision, project administration, funding acquisition

Conceptualization, formal analysis, review & editing [cor1]Corresponding
author [cor2]Principal corresponding author [fn1]S. A. Salman has completed
this research during his MSc study at NJUST, PR. China in 2021 Current PhD
Advisor

Lab-mate assists with software and data

# Classification of ADHD Patients Using Kernel Hierarchical Extreme Learning
Machine

Sartaj Ahmed Salman<EMAIL_ADDRESS>Department of Informatics, The
University of Electro-Communications (UEC), Chofu, Tokyo, Japan.  School of
Information Management, Nanjing University (NJU), Nanjing, P.O. Box 210023
P.R, China.  School of Computer Science and Engineering, Nanjing University of
Science and Technology (NJUST), Nanjing, P.O. Box 210094 P.R, China.  Zhichao
Lian<EMAIL_ADDRESS>Milad Taleby Ahvanooey<EMAIL_ADDRESS>Hiroki Takahashi
<EMAIL_ADDRESS>Yuduo Zhang<EMAIL_ADDRESS>
###### Abstract

Recently, the application of deep learning models to diagnose neuropsychiatric
diseases from brain imaging data has received more and more attention.
However, in practice, the exploration of interactions in brain functional
connectivity based on operational magnetic resonance imaging data is critical
for studying mental illness. Since Attention-Deficit and Hyperactivity
Disorder (ADHD) is a type of chronic disease that is very difficult to
diagnose in the early stages, it is necessary to improve the diagnosis
accuracy of such illness using machine learning models treating patients
before the critical condition. In this study, we utilize the dynamics of brain
functional connectivity to model features from medical imaging data, which can
extract the differences in brain function interactions between Normal Control
(NC) and ADHD. To meet that requirement, we employ the Bayesian connectivity
change-point model to detect brain dynamics using the local binary encoding
approach and kernel hierarchical extreme learning machine for classifying
features. To verify our model, we experimented with it on several real-world
children datasets, and our results achieved superior classification rates
compared to the state-of-the-art models.

###### keywords:

fMRI BCCPM Local features Hierarchical Extreme Learning Machine ADHD Sparse-
auto encoder

##  1 Introduction

These days, the Attention-Deficit and Hyperactivity Disorder (ADHD) is one of
the most common neurodevelopmental disorders that influence up to 10% of
children all over the world, i.e., ADHD’s symptoms often remain throughout the
whole lifespan and affect to ”comorbidities” same as the Major Depressive
Disorder (MDD) [1]. Nevertheless, significant fluctuation of prevalence was
appeared across different nations, considering diagnostic measures that are
likely based on behavioral symptoms rather than impartial biomarkers.
Moreover, because of the pathognomonic for multiple disorders, the ADHD’s lack
of biomarkers is particularly baleful since the overlap of main symptoms with
other conventional psychiatric disorders such as personality, mood, and
anxiety disorders. In addition, the diagnosis of ADHD for adults is obstructed
by retrospective evaluation of symptoms during their childhood. Also, there
exist disagreements between scientists/experts on ADHD as well as debates on
misuse of ADHD treatment (e.g., Methylphenidate (MPH)) that have attracted
research based on objective ADHD over subjective predictors with modest
success so far [2].

In the literature, there are three types of ADHD: combined,
hyperactive/impulsive, and inattentive that have various symptoms when a
patient is infected. The existing studies showed that ADHD is a
disorder/disease, which happens in childhood and is classified based on
symptoms such as difficulty in concentration, hyperactivity, short attention
span, or impulsivity compared with children of the same age [3]. This
diagnosis problem involves preliminary knowledge about an emergent disease and
is challenging to identify because of insufficient information and physical
symptoms. Hence, diagnosing whether the patient has ADHD or not was a hot
topic for bioinformatics researchers.

Over the past decades, physicians or medical experts have gathered information
from different sources such as ADHD checklists [5]. Moreover, the Neuro Bureau
experts provide pre-processed data to contribute to the potential research for
further improvements regarding the diagnosis of such disease (nitrc.org).
Clinical studies have suggested that there are several illnesses, which alter
the functional connectivity of the brain, and ADHD is also one of these
disorders [6]. In general, the human brain is the most complex system amongst
all the other organs. While coordinating with the body, the brain regions
continuously change their formation, and the areas that generate temporary
correlation are assumed to be connected functionally [7]. Hence, brain
function connection modeling can better understand the pathological basis of
neurological disorders. Building a network of brain function connections can
describe the interactions between various functions of the brain [8]. In the
current literature, researchers have suggested several other neural network-
based models for extracting disorder features from medical imaging data. But
they mainly suffer from a low diagnosis accuracy rate due to the lack of
insufficient considered features and variation of the functional connectivity
changes in the brain.

Below, we summarize our contributions twofold.

  * •

We employ the bayesian connectivity change point Model to find the time of
dynamic change of brain function interactions, enhancing the accuracy of
extracted features from medical imaging data.

  * •

We develop a deep learning model by applying the Local Binary Encoding Method
for feature extraction and Kernel Hierarchical Extreme Learning Machine
(KHELM) for classification.

##  2 Related Works

Recently, biomedical scientists have suggested several automatic diagnosis
approaches for extracting mass features from Functional Magnetic Resonance
Imaging (fMRI) data. For example, Waqas Majeed developed a new approach to
assess that the reproducible spatiotemporal pattern of BOLD fluctuations is
consistent with previous research and may have vital information about brain
activity at rest. It indicates that in the resting state, the brain is active.
Moreover, several studies have considered the dynamics of brain function
connections [10]. Lindquist[11] have proposed a modified exponentially
weighted moving average (EWMA) model, which can be applied to FMRI data, and
then used it to analyze the change point of time series . Chang et al [12]
have investigated the dynamic connectivity of the brain signals by applying
the sliding-window methodology [12]. In [15], Ren et al have recommended a
dynamic graph metrics strategy to characterize temporal changes of functional
brain networks. In [9], Atif Riaz et al have suggested a Hybrid fMRI framework
that utilizes affinity propagation clustering and density peak for functional
connectivity. In [36], Ahmed et al. presented a classification model that and
non-ADHD using ELM with different datasets. We have extracted different FC’s
using different time atlases and use the FC’s for classification from our
previous work we conclude that the accuracy of our model increases with the
number of extracted FC. During the past few years, researchers have introduced
different types of models for the classification of ADHD. As this illness is a
medically brain disorders that physicians usually diagnose it by assessing
some symptoms through a subjective observation process. Therefore,
bioinformatial studies have identified this disorder as a two-category
problem, such as ADHD and non-ADHD. Gülay Çıçek as well as created two
separate datasets, including gray level co-occurrence matrix and Haralick
texture features for classification purposes using the machine learning
algorithms [14].

Recently, there have been many advances in this field, in [37] Jie Wang uses
fNIRS signals for functional connectivity and interval features for
classification of ADHD and non-ADHD. Shuiqi Lui proposed a novel algorithm for
ADHD classification based on (CDAE) convolutional denoising auto-encoder and
(AdaDT) adaptive boosting decision trees[38]. Yibin Tang et al. [39] have
introduced a self-encoding network with non-imaging fusion for ADHD
classification, which achieves quite high accuracy. However, it has some
limitations such as it does not work well with different datasets and is also
not able to extract the required features from the fMRI data. Miao and Zhang
[15] have suggested a relief and VA -relief based feature extraction approach
to achieve high precision classification. Later on, in 2017 Sudha et al. in
[16] have suggested a model to extract the gait signal characteristics of ADHD
children from the video signals, which provides the disease diagnosis and
strengthen the cognition of sick children [16]. Chang et al.[17] have
introduced a feature extraction method based on a texture point of view
considering the isotropic local binary patterns on three orthogonal planes
(LBP-TOP) that employ the support vector machine (SVM) to classify the
identified features. Athena Taymourtash, use sparse based representation
method by extracting the feature by cluster ICs and uses k-nn classifier to
find out the EEG source differences between adults with ADHD and healthy
controls [18]. Later in [19], Zhang has introduced a the dual diagnosis model,
which recognizes the feature space separation by applying sparse
representation.. Juan L. Lopez Marcano explained that the United States allows
using the $\theta/\beta$ power ratio (TBPR) as a diagnostic feature of ADHD
[20]. F.M. Grisales-Franco uses a Dynamic Sparse Coding (DSC) method based on
physiologically motivated Spatio-temporal constraints to construct non-
stationary brain activity, they search the difference between ADHD and control
groups using statistical results [21].

However, state-of-the-art models have studied ADHD diagnoses using different
machine learning and deep learning models. Still, the literature suffers from
an insufficient diagnosis rate which is a challenging task for scientists.
Therefore, in this article, we propose a novel classification framework for
enhancing the diagnosis accuracy the ADHD from the brain imaging data as a
novel contribution to classifying children with and without ADHD. In
particular, we apply the BCCPM to detect change points, and the local binary
encoding method to extract local information from FIPs of the brain as well as
employ the KH-ELM for classification.

Figure 1: The data processing flow chart of the classification framework
proposed in our paper

##  3 Proposed Models

In this section, we describe two proposed models in details.

###  3.1 Bayesian Connectivity Change Point Model

Previous studies have shown that even when the brain is at resting status, it
undergoes a series of dynamic changes, which affect the interaction of its
functionalit connections. Iny this subsection, the proposed Bayesian
connectivity change point model (BCCPM) for detecting dynamic changes in brain
activities is explained. When the imaging data are classified directly without
change point detection, the performance is not satisfactory. Therefore, we
refer to the [9] method to dynamically divide the time period into multiple
time blocks, i.e., this process can actually identify the time points of
dynamic brain change activity. Next, we will briefly introduce BCCPM.

With BCCPM, the data matrix $\ Z=(z_{1},z_{2},...,z_{T})$ can be divided into
several time blocks. The matrix $Z$ consists of $Tm$-dimensional vectors,
where each vector represents the value of m ROIs (Regions of Interest) at time
$t$. If each vector in $Z$ satisfies the independent and identical
distribution, and $z_{t}\sim N(\mu,\sum)t=1,2,...,T.$

Here, an indication vector$L^{\rightarrow}=(L_{1},L_{2},...,L_{T})$ is used,
where $L_{t}=1$ represents the change point and the indication
$L_{1}=1,L_{t}=0$ means unchanged. Then, the probability of $Z$ can be
expressed as follows:

$p(Z\mid L^{\rightarrow})=\prod_{b=1}^{\sum L_{i}}p(Z_{b})$ (1)

where $\sum L_{i}$ is the number of divided time blocks, and $p(Z_{b})$ can be
obtained by the following equation (2). [22]:

$\begin{split}p(z_{1},z_{2},...,z_{T})=\frac{p(z_{1},z_{2},...,z_{T};\mu,\sum)}{p(\mu,\sum|z_{1},z_{2},...,z_{T})}=\\\
(\frac{1}{2\pi})^{\frac{mT}{2}}(\frac{k_{0}}{k_{T}})^{\frac{m}{2}}\frac{\Gamma_{m}(\frac{v_{T}}{2})(det(\Lambda_{0}))^{\frac{v_{0}}{2}}}{\Gamma_{m}(\frac{v_{0}}{2})(det(\Lambda_{T}))^{\frac{v_{T}}{2}}}2^{\frac{mT}{2}}\end{split}$
(2)

where $\Gamma_{m}$ is the multivariate gamma function.

Since $Z$ is divided into independent blocks of time, the posterior
distribution of $p(L^{\rightarrow}|Z)$ is as follows:

$p(L^{\rightarrow}\mid Z)\propto p(L^{\rightarrow})\cdot p(Z\mid
L^{\rightarrow})$ (3)

and $p(L^{\rightarrow})=\sum_{t=1}^{T}p(L_{t})$, where $p(L_{t})\sim
Bern(0.5)$.Eventually, we employ the Markov chain Monte-Carlo [23] with random
$L^{\rightarrow}$ to generate samples with posterior distribution
$p(L^{\rightarrow}\mid Z)$. See [9] for details.

###  3.2 Local Binary Encoding Method (LBEM)

In order to extract the local information of brain function interaction
patterns (FIPs), we proposed in our previous research that LBEM. [24] can be
applied to local feature extraction from fMRI data.

The LBEM processes the data object DICCCOL-fMRI and the DICCCOL-fMRI is the
data set matrix

$D=(D_{1},D_{2},...,D_{T})\in R^{358*T}$

, where each vector $D_{t}=(d_{1},d_{2},...,d_{358}),1\leq t\leq T$ represents
the value of 358 ROIs at time $t$.

First, we consider a re-encoding process in each column vector $D_{t}$ within
the data matrix $D$. If each element in $D_{t}$ is compared with its
neighbouring elements, we have a new vector $F_{t}=(f_{1},f_{2},...,f_{7}14)$,
$1<t<T$ that can be denoted by the equations (4-6):

$f_{2(i-1)-1}=\\{^{1,d_{i}\leq d_{(}i-1)}_{0,d_{i}>d_{(}i-1)},2\leq i\leq 358$
(4) $f_{2(i-1)}=\\{^{1,d_{i}\leq d_{(}i+1)}_{0,d_{i}>d_{(}i+1)},2\leq i\leq
357$ (5) $f_{724}=\\{^{1,d_{358}\leq d_{1}}_{0,d_{358}>d_{1}}$ (6)

Then, we get a matrix of binary, which contains only 0 and 1 elements
$F=(F_{1},F_{2},...,F_{T})\in R^{714*T}$. Figure 2 shows an example of a
conversion to binary matrix form. As shown, the
vector$A=(a_{1},a_{2},a_{3},a_{4})$ is encoded to obtain the vector
$B=(b_{1},b_{2},...,b_{6})$. Since $a_{2}<a_{1}$ assigns $b1$ to $1$, and
$a_{2}>a_{3}$ assigns $b2$ to $0$. The values of $b_{3},b_{4},b_{5}$ can be
determined in the same way, $b_{6}$ is equal to $1$, because of $a_{4}<a_{1}$
and the process of encoding into a binary would continue until we obtain the
remaining element of vector $F$. As a final assignment, all the Binary values
of vector $B$ was converted in to decimal form. Each of the six elements of
the vector $F_{t}$ is assigned into a group. Next, each group number is
converted from a 6-bit binary form to a decimal form, and the range of the
converted decimal number is $[0,63]$. After transforming the $F4$, we have the
data matrix $E=(E_{1},E_{2},...,E_{T})\in R^{119*T}$, which represents the
local information between each ROI and its neighbouring ROIs. Figure 2 depicts
the example of encoding process.

Third, we perform the histogram equalisation processing for each row vector of
the data matrix $E$ obtained in the second step. Thereby, the final output
matrix is $O\in R^{119*64}$. As shown in Figure 3(c) (a,b), the sample object
is processed to obtain 119 histograms.

Figure 2: Encoding data from A to binary pattern B and then to decimal pattern
C

(a)

(b)

(c)

Figure 3: (a,b) Number of discovered histograms (119), and (c) The r-fMRI
signal of five DICCCOLs in one sample.

###  3.3 Kernel Extreme Learning Machine Algorithm (KELM)

To extend the understanding of the KELM [25], we briefly explain the ELM [26]
theory, which only has one layer $H$ containing $L$ hidden nodes. Let us
assume that the G is the classification problem, given a training set
$\\{(x_{j},z_{j})\mid x_{j}\in R^{d},z_{j}\in R^{G}\\}$, where $x_{j}$ is the
vector, $z_{j}$ is the label, and $j=1,2,...,N$. The ELM model can be denoted:

  1. 1.

Calculate the hidden layer matrix H.

$H=\left[\matrix{h}\left(x_{1}\right)\\\ \vdots\\\ h\left(x_{N}\right)\\\
\right]\left[\matrix{h}\left(w_{1}x_{1}+b_{1}\right)&\cdots
h\left(w_{L}x_{1}+b_{L}\right)\\\ \vdots\ddots\vdots\\\
h\left(w_{1}x_{N}+b_{1}\right)\cdots h\left(w_{L}x_{N}+b_{L}\right)\\\
\right]$ (7)

where h ($\cdot$ ) denotes the nonlinear activation function, the input
weights wi and biases bi are randomly generated, $i=1,2,\ldots,L.$

  2. 2.

The output weight vector $\beta$ can be calculate by equation (8).

$\beta=H^{\dagger}Z$ (8)

where $H^{\dagger}$ is the Moore-Penrose generalized inverse of matrix $H$,
and $Z$ is the training label matrix.

$Z=\left[\matrix{z}_{1}^{T}\\\ \vdots\\\ z_{N}^{T}\\\
\right]=\left[\matrix{z}_{11}&\cdots z_{1G}\\\ \vdots\ddots\vdots\\\
z_{N1}\cdots z_{NG}\\\ \right]$ (9)

The corresponding result of ELM can be obtained as follow:

$f\left(x_{j}\right)=h\left(x_{j}\right)\beta$ (10)

In KELM, a kernel matrix

$\Omega_{ELM}=HH^{T}:\Omega_{ELM}=h\left(x_{p}\right)\cdot
h\left(x_{q}\right)=K\left(x_{p},x_{q}\right)$

is applied. Then, the kernel case of ELM output function is:

$(f\left(x_{j}\right)=\left[\matrix{K}\left(x_{j},x_{1}\right)\\\ \vdots\\\
K\left(x_{j},x_{N}\right)\\\
\right]^{T}\left(\frac{1}{\rho}+\Omega_{ELM}\right)^{-1}Z$ (11)

In this work, a radial basis function as $\Omega_{ELM}$ is used to achieve
better experimental results.

###  3.4 Kernel Hierarchical Extreme Learning Machine Algorithm (KH-ELM)

The KH-ELM has a more complex structure than KELM, the training algorithm is
different from the greedy lay-wise [27]. Figure 4 depicts the architecture of
KH-ELM, which consists of two parts: i) the unsupervised hierarchical feature
extraction based on the ELM sparse-auto encoder, and ii) the supervised
classification based on KELM.

For the G-classification problem, given a training set.
${(x_{j},z_{j})|x_{j}\in R^{d},z_{j}\in R^{G}}$ where $x_{j}$ is the training
vector, $z_{j}$ is the training label and $j=1,2,...,N$. Before unsupervised
hierarchical feature extraction, the data-set should simply be scaled to
$[0,1]$ and the labels are adjusted to the G dimension, i.e. $-1$ or $1$. Then
the input data is mapped into the ELM random feature space and extracted by
n-layer unsupervised learning. For the KH-ELM learning algorithm, the input
layer is taken as layer 0, the first layer is a hidden layer, the weights of
the first layer $\beta$ are learned by the ELM sparse auto encoder (see Figure
4 (a)), and the other weights of the hidden layer are also learned by the ELM
sparse-auto-encoder. The result of each hidden layer can be calculated as
follows:

$H_{i}=g\left(H_{i-1}\cdot\beta^{T}\right),1\leq i\leq n$ (12)

Where Hi is the output of the ith layer, Hi-1 is the output of the (i-1)th
layer, g( $\cdot$ ) is the activation function of the hidden layers, and
$\beta$ is the weight of the ith hidden layer (output weight) [28]. At KH-ELM,
each hidden layer is an independent module. Once the feature of the previous
hidden layer is extracted, the weights of the current hidden layer are set
without fine-tuning [29]. Here, the L1 regularisation norm is used to create a
ELM auto-encoder so that the resulting ELM sparse auto-encoder provides better
generalisation performance for the parameters. In addition, the use of penalty
terms can constrain our model features. As a result, the learned model may
have sparser features. The ELM sparse-auto encoder equation for this
optimisation model is expressed as follows:

$(O_{\beta}=argmin_{\beta}\\{||A\beta-X||^{2}+||\beta||_{l1}\\}$ (13)

Here, $A$ denotes the random mapping output, $\beta$ which is the weight of
the hidden layer to be obtained, X represents the input data. In the proposed
auto-encoder, A is a randomly initialised output mapped by a random weight
matrix $b=\left[b_{1},b_{2},\ldots,b_{n}\right]$ that does not require any
optimisation. The structure of the ELM sparse-auto-encoder is shown in Figure
4 (a,b). As shown, this not only saves training time but also improves
learning accuracy.

Next, we describe the ELM optimisation algorithm based on the sparse auto-
encoder. In addition, we applied the Fast Iterative Shrinkage Threshold
algorithm (FISTA) to solve the constraint minimisation problem for
continuously differential functions. We use a FISTA type with constant steps.
The specific algorithm is express as follows:

  1. 1.

Calculate the Lipschitz constant $\gamma$ of the gradient $\nabla_{p}$ of the
function $||A\beta-X||^{2}$

  2. 2.

Calculate $\beta$ iteratively. Begin the iteration by taking
$y_{1}=\beta_{0}\in R^{n}$, $t1=1$ as the initial points. Then, for $i(i\geq
1)$

Step 1. $\beta_{i}=S_{\gamma}\left(y_{i}\right)$ , Where

$S_{\gamma}=arg\min_{\beta}\\{\frac{\gamma}{2}||\beta-\left(\beta_{i-1}-\frac{1}{\gamma}\left(\beta_{i-1}\right)\right)||^{2}+||\beta||_{l1}\\}$

Step 2. $t_{i+1}=\frac{1+\sqrt[]{1+4t_{i}^{2}}}{2}$

Step 3.
$y_{i+1}=\beta_{i}+\left(\frac{t_{i-1}}{t_{i+1}}\right)\left(\beta_{i}-\beta_{i-1}\right)$

The second part of KH-ELM functions based on the supervised feature
classification employing the KELM in which the input is the matrix $H_{n}$,
and the output is the n-layer ELM sparse autoencoder.

Figure 4: (a)The ELM Sparse Autoencoder structure of the first hidden layer.
(b)The ELM Sparse Autoencoder structure of the ith hidden layer. (c)The
architecture of the KH-ELM learning algorithm [35]

##  4 Experimental Results and Analysis

In this section, we conducted a comparative analysis by performing our model
on the r-fMRI dataset, i.e., it [8] contains 23 ADHD patients and 45 NC. To
verify our proposed model, first, we selected 358 landmarks [30] as nodes of
the network. Second, we determined the DICCCOL for 68 subjects exploiting a
commonly available tool in [31], and extracted the r-fMRI signal corresponding
to it. Figure 3(c) shows the r-fMRI signal from five in one sample. After
preprocessing, we obtain a $358*270$ time series matrix for each sample. Then,
the r-fMRI time series of each sample was dynamically detected using BCCPM to
obtain a $358*230$ time series matrix.

In this paper, 5-fold cross-validation is used to evaluate the proposed
method. The experiment was repeated 30 times and then the average
classification accuracy of each stage was calculated. To analogize via
different approaches, the experimental results of three distinct methods were
used: our proposed method along KH-ELM and KELM with the Jinli Ou [8], the
KELM with different kernels and of KH-ELM with multiple layers. First, it
shows that our methodology with KELM achieves the better performance (the
average of ADHD is 95.51% and NC is 97.57%), compared to the results of Ous
[8] (ADHD 92% and NC 95%). In addition, KELM has achieved more stable results
than the Jinli Ou’s technique. The comparison flash that it is effective to
extract local features with LBEM. Then, it shows that the precision of KH-ELM,
which is shown in the first column of table 1 table. (97.17% for ADHD and
99.76% for NC) is higher than that of KELM. This expose that by using the ELM
sparse auto-encoder to extract features from the training and the test dataset
before KELM improves performance of the model.

Table 1: Accuracy comparison of our proposed methods and Jinli Ou’s method Class |  KHELM |  KELM |  KELM without BCCPM |  Jinli Ou’s Method
---|---|---|---|---
|  NC |  ADHD |  NC |  ADHD |  NC |  ADHD |  NC |  ADHD
Fold1 |  0.9973 |  1 |  0.9747 |  0.9522 |  0.7111 |  0.44 |  0.8889 |  1
Fold2 |  0.9907 |  1 |  0.9779 |  0.9449 |  0.7778 |  0.4 |  0.8889 |  0.8
Fold3 |  1 |  0.9974 |  0.983 |  0.959 |  0.6222 |  0.4 |  1 |  0.8
Fold4 |  1 |  0.9564 |  0.9711 |  0.9463 |  0.6667 |  0.55 |  1 |  1
Fold5 |  1 |  0.9048 |  0.972 |  0.9731 |  0.5556 |  0.3 |  1 |  1
Average |  99.76$\%$ |  97.17$\%$ |  97.57$\%$ |  95.51$\%$ |  66.67$\%$ |  41.80$\%$ |  95.00$\%$ |  92.00$\%$

To verify the need for dynamic detection under r-fMRI data for each subject
after pre-treatment. We carry out a controlled experiment with data processed
by BCCPM and with out it. In both sets,a five-fold cross-validation using LBEM
to extract local features were commit, which then classified using KELM. The
procedure was repeated 30 times and the average classification result for each
fold was put down. The experimental outcomes are shown in Table 1. It shows
that after dynamic detection, the same classification method bring off better
classification accuracy and proves that the brain has temporal dynamics even
at rest.

Finally, we assimilate the effects of the different parameters on
classification accuracy. We analyse the outcomes with two types of parameters:
a) Repeated the whole process with Linearl and RBF kernels. While using these
two different kernel functions, for this experiment RBF achieves the higher
accuracy. The experimental results are shown in Table 2.

Table 2: Average accuracy comparison using different kernels Class |  KHELM Using RBF kernel |  KHELM Using Linear kernel
---|---|---
|  NC |  ADHD |  NC |  ADHD
Fold1 |  1 |  0.8462 |  0.9615 |  0.6154
Fold2 |  0.9600 |  1 |  0.9600 |  0.6154
Fold3 |  1 |  1 |  0.9600 |  0.6429
Fold4 |  1 |  1 |  0.9615 |  0.6923
Fold5 |  1 |  1 |  0.9600 |  0.4615
Average |  99.2$\%$ |  96.924$\%$ |  96.06$\%$ |  60.55$\%$

As depicted in Table 2, that the linear kernel function average ADHD is
60.55%, average accuracy of NC is 96.06%), the calculation result of the RBF
kernel function (average accuracy of ADHD is 96.92%, average accuracy of NC is
99.2%) is more accurate. Here we get better results using RBF kernel, which
has been proved by different researchers by comparing different works using
different kernels [32][33]. b) the effect of increasing number of hidden
layers on classification accuracy using KH-ELM. The same method was used for
HELM to perform a comparative experiment and the results are shown in Table 3.
(59.19% for ADHD and 85.97% for NC). The experiments show that the
classification results of HELM [32] are very unstable compared to the high
classification accuracy of KH-ELM. Table 3 shows the experimental results of
KH-ELM with three hidden layers (41.80% for ADHD and 66.67% for NC) and with
one hidden layer (97.17% for ADHD and 99.76% for NC).

Table 3: The comparison of classification accuracy in different parameters Class |  KHELM with Three Hidden Layer |  KHELM with One Hidden Layer |  HELM
---|---|---|---
|  NC |  ADHD |  NC |  ADHD |  NC |  ADHD
Fold1 |  0.7111 |  0.44 |  0.9973 |  1 |  0.8667 |  0.5974
Fold2 |  0.7778 |  0.4 |  0.9907 |  1 |  0.892 |  0.4692
Fold3 |  0.6222 |  0.4 |  1 |  0.9974 |  0.8267 |  0.6359
Fold4 |  0.6667 |  0.55 |  1 |  0.9564 |  0.8667 |  0.6
Fold5 |  0.5556 |  0.3 |  1 |  0.9048 |  0.8462 |  0.6571
Average |  66.67$\%$ |  41.80$\%$ |  99.76$\%$ |  97.17$\%$ |  85.97$\%$ |  59.19$\%$

We have found that the accuracy of the classification results decreases as the
number of hidden levels increases. To illustrate more, identical operation was
carry out with up to six hidden layers. Same protocol has being followed as
above. The experiment was repeated 30 times and record the average result of
each convolution, gradually increasing the number of layers. The details are
shown in Table 4 and 5.

Table 4: Accuracy comparison of KH-ELM with different no of hidden layers (1-3) Class |  KH-ELM with One Hidden Layer |  KH-ELM with second Hidden Layer |  KH-ELM with Third Hidden Layer
---|---|---|---
|  NC |  ADHD |  NC |  ADHD |  NC |  ADHD
Fold1 |  0.9973 |  1 |  0.9423 |  0.9025 |  0.7111 |  0.44
Fold2 |  0.9907 |  1 |  0.972 |  0.7974 |  0.7778 |  0.4
Fold3 |  1 |  0.9974 |  0.976 |  0.9642 |  0.6222 |  0.4
Fold4 |  1 |  0.9564 |  0.9667 |  0.9359 |  0.6667 |  0.55
Fold5 |  1 |  0.9048 |  0.9586 |  0.8846 |  0.5556 |  0.3
Average |  99.76$\%$ |  97.17$\%$ |  96.31$\%$ |  89.69$\%$ |  66.67$\%$ |  41.80$\%$
Table 5: Accuracy comparison of KH-ELM with different no of hidden layers (4-6 Class |  KH-ELM with Forth Hidden Layer |  KH-ELM with Fifth Hidden Layer |  KH-ELM with Sixth Hidden Layer
---|---|---|---
|  NC |  ADHD |  NC |  ADHD |  NC |  ADHD
Fold1 |  0.8307 |  0.3538 |  0.7474 |  0.3743 |  0.7397 |  0.3512
Fold2 |  0.7693 |  0.5230 |  0.688 |  0.5025 |  0.6973 |  0.5051
Fold3 |  0.7213 |  0.6278 |  0.7026 |  0.5380 |  0.7346 |  0.5166
Fold4 |  0.8359 |  0.5256 |  0.7538 |  0.4461 |  0.7667 |  0.4410
Fold5 |  0.724 |  0.5820 |  0.672 |  0.5615 |  0.644 |  0.4897
Average |  77.62$\%$ |  52.24$\%$ |  71.27$\%$ |  48.45$\%$ |  71.64$\%$ |  46.07$\%$

As the deep learning methods generally produce better results when the number
of hidden layers increases. But in our experiments, we obtained worse results
when we increased the number of layers. The reason for this conclusion is that
in experiments with multiple layers, some features are extracted after each
layer, which leads to the repetition of attributes, and more features are
generated simultaneously. These features are incompatible with the proposed
algorithm and have the most significant impact on the outputs when the number
of layers increases because the extension layer is limited. After that, we no
longer extract features but tend to ”overfit” the data, and overfitting
reduces the accuracy of the experiment. Therefore, the KH-ELM classification
with only one hidden layer for feature extraction offers more efficiency than
the one with multiple hidden layers.

##  5 Conclusion

In this paper, we propose a novel classification scheme to improve the
diagnosis accuracy from brain imaging data to distinguish the ADHD in
comparison to the NC from the brain imaging data of patients. The innovation
of our work is to use a hierarchical ELM sparse autoencoder to extract
features prior to KELM, which outperformed other classifiers concerning the
accuracy rate while requiring fewer hidden layers. Our experiments proved that
the proposed framework offers superior efficiency regarding the classification
metrics; however, the dynamic detection and selection of the number of layers
for classifier feature extraction is not as good as possible. We try to extent
our study by conducting further experiments in the future to improve the
average accuracy of fMRI classification for discriminating between the ADHD
and NC diseases.

##  6 Acknowledgment

The authors would like to thank Prof.Tianming Liu and his group CAID for
providing the ADHD data.

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