# Beyond SMEFT with $b\to c\,\tau^{-}{\bar{\nu}}$

C.P. Burgess<EMAIL_ADDRESS>Physics & Astronomy, McMaster
University, Hamilton, ON, Canada, L8S 4M1  Perimeter Institute for Theoretical
Physics, Waterloo, ON, Canada, N2L 2Y5  CERN, Theoretical Physics Department,
Genève 23, Switzerland  Serge Hamoudou<EMAIL_ADDRESS>Physique
des Particules, Université de Montréal, Montréal, QC, Canada H2V 0B3  Jacky
Kumar<EMAIL_ADDRESS>Physique des Particules, Université de Montréal,
Montréal, QC, Canada H2V 0B3  Institute for Advanced Study, Technical
University Munich, Lichtenbergstr. 2a, D-85747 Garching, Germany  David London
<EMAIL_ADDRESS>Physique des Particules, Université de Montréal,
Montréal, QC, Canada H2V 0B3

(August 27, 2024)

###### Abstract

Electroweak interactions assign a central role to the gauge group
$SU(2)_{L}\times U(1)_{Y}$, which is either realized linearly (SMEFT) or
nonlinearly (e.g., HEFT) in the effective theory obtained when new physics
above the electroweak scale is integrated out. Although the discovery of the
Higgs boson has made SMEFT the default assumption, nonlinear realization
remains possible. The two can be distinguished through their predictions for
the size of certain low-energy dimension-6 four-fermion operators: for these,
HEFT predicts $O(1)$ couplings, while in SMEFT they are suppressed by a factor
$v^{2}/\Lambda_{\rm NP}^{2}$, where $v$ is the Higgs vev. One such operator,
$O_{V}^{LR}\equiv({\bar{\tau}}\gamma^{\mu}P_{L}\nu)\,({\bar{c}}\gamma_{\mu}P_{R}b)$,
contributes to $b\to c\,\tau^{-}{\bar{\nu}}$. We show that present constraints
permit its non-SMEFT coefficient to have a HEFTy size. We also note that the
angular distribution in ${\bar{B}}\to D^{*}(\to
D\pi^{\prime})\,\tau^{-}(\to\pi^{-}\nu_{\tau}){\bar{\nu}}_{\tau}$ contains
enough information to extract the coefficients of all new-physics operators.
Future measurements of this angular distribution can therefore tell us if non-
SMEFT new physics is really necessary.

††preprint: UdeM-GPP-TH-21-290

Introduction — The Standard Model (SM) of particle physics provides a
spectacular description of the physics so far found in the Large Hadron
Collider (LHC). But it also cannot be complete because it leaves several
things unexplained (like neutrino masses, dark matter and dark energy, etc.),
and it makes some of cosmology’s initial conditions (such as primordial
fluctuations and baryon asymmetry) seem unlikely. To have hitherto escaped
detection, any new particles must either couple extremely weakly or be very
massive (or possibly both).

This – together with the eventual need for something to unitarize gravity at
high energies – underpins the widespread belief that the SM is the leading
part of an effective field theory (EFT) describing the low-energy limit of
something more fundamental. EFTs are largely characterized by their particle
content and symmetries (see, e.g., Refs. EFTBook ; Burgess:2007pt ). Since the
discovery of the Higgs boson, the known particle content at energies above the
top-quark mass, $m_{t}$, suffices to linearly realize the electroweak gauge
group $SU(2)_{L}\times U(1)_{Y}$. Whether the known particles actually do
linearly realize this symmetry is what distinguishes SMEFT, which lineary
realizes it (see, e.g., Refs. Buchmuller:1985jz ; Brivio:2017vri ) from
alternatives like HEFT, which do not, despite also including a ‘Higgs’ scalar
(see, e.g., Refs. Feruglio:1992wf ; Bagger:1995mk ; Grinstein:2007iv ;
Alonso:2012px ; Alonso:2012pz ; Burgess:1999ha ; Buchalla:2012qq ;
Buchalla:2013rka ; Cohen:2020xca ).

The question of whether the symmetry is realized linearly or nonlinearly can
only be answered experimentally. One proposal for doing this Banta:2021dek
seeks new particles whose presence requires nonlinear realization. In the
present paper, we show how to use indirect $b$-physics signals to extract
evidence for nonlinearly-realized new physics.

How symmetries are realized in an EFT comes up when power-counting how
effective interactions are suppressed at low energies. For instance, an
effective interaction like $g_{z}\,Z_{\mu}(\bar{u}\gamma^{\mu}P_{R}u)\in{\cal
L}_{\rm eff}$, which describes a non-standard $Z{\bar{u}}_{R}u_{R}$ coupling,
naively arises at mass-dimension 4 when $SU(2)_{L}\times U(1)_{Y}$ is
nonlinearly realized Burgess:1993vc ; Bamert:1996px , but instead arises at
dimension-6 through an operator
$\Lambda_{h}^{-2}(H^{\dagger}D_{\mu}H)(\bar{u}\gamma^{\mu}P_{R}u)$ when it is
linearly realized, implying a coupling $g_{z}\sim v^{2}/\Lambda_{h}^{2}$ that
is suppressed by the ratio of Higgs vev $v$ to a UV scale $\Lambda_{h}$.

The assumption underlying SMEFT is that the scale $\Lambda_{h}$ appearing here
is the same order of magnitude as the scale $\Lambda$ that suppresses all
other dimension-6 operators If $\Lambda_{h}\sim\Lambda$ then the lower bound
on $\Lambda$ required to have generic dimension-6 SMEFT operators not be
detected also implies an upper bound on the effective dimension-4 non-SMEFT
coupling $g_{z}$. While this assumption is not unreasonable, it is an
assumption, since nothing in the power-counting of EFTs requires the scale
$\Lambda_{h}$ that accompanies powers of a field like $H$ to be the same as
the scale $\Lambda$ that appears with derivatives EFTBook ; Burgess:2007pt .
(For example, these scales are very different in supergravity theories, and
this is why it is consistent to have complicated target-space metrics
appearing in the kinetic energies of fields while working only to two-
derivative order. Similar observations have also been made for SMEFT
Helset:2020yio .)

Because it is an assumption, it should be tested. It is ultimately an
experimental question which kind of symmetry realization provides a better
description of Nature. Our purpose in this paper is to identify how to do so
using a class of $B$-physics measurements. Despite being at relatively low
energies, $B$-meson properties suggest themselves for this purpose because
they can be precisely studied and because there are at present several
observables that seem to disagree with the predictions of the SM.

SMEFT vs LEFT at low energies — A complicating issue arises when using $B$
physics to distinguish SMEFT from non-SMEFT effective interactions because the
EFT relevant at such low-energies necessarily already integrates out many of
the heavier SM particles ($W^{\pm}$, $Z^{0}$, $H$, $t$). But once these
particles are removed the remaining EFT necessarily nonlinearly realizes
$SU(2)_{L}\times U(1)_{Y}$, while linearly realizing its $U(1)_{em}$ subgroup.
This is why heavy top-quark loops can generate otherwise SM-forbidden
effective interactions such as $\delta{\cal L}_{\rm eff}\ni\delta
M^{2}_{W}W_{\mu}^{*}W^{\mu}+\delta M_{Z}^{2}Z_{\mu}Z^{\mu}$ that violate the
SM condition $M_{W}=M_{Z}\cos\theta_{W}$, or more broadly contribute to
oblique corrections or modification of gauge couplings Burgess:1993vc .

The exercise of separating these more mundane sources of symmetry breaking
from those coming from higher energies has been studied in the literature. For
instance, the theory obtained below the $W$ mass has been called LEFT (low-
energy effective field theory) or WET (weak effective field theory), and in
Ref. Jenkins:2017jig , Jenkins, Manohar and Stoffer (JMS) present a complete
and non-redundant basis of operators in this theory up to dimension 6. For the
particularly interesting class of dimension-6 four-fermion operators that
conserve baryon and lepton number, they also identify how these effective
interactions can be obtained (at tree level) from the similarly complete and
non-redundant list of operators given for SMEFT in Ref. Grzadkowski:2010es .
(For a fuller discussion of the relationships amongst these various EFTs see
Ref. Aebischer:2017ugx  and references therein.)

JMS find that most dimension-6 LEFT operators can be generated in this way
starting from dimension-6 operators in SMEFT. However, a handful of
dimension-6 LEFT operators are not invariant under $SU(2)_{L}\times U(1)_{Y}$,
and so are not contained amongst dimension-6 SMEFT operators. Tree graphs can
also generate these ‘non-SMEFT’ operators, but in this case only do so
starting from SMEFT operators with mass dimension greater than 6. It is these
non-SMEFT operators that interest us in our applications to $B$ physics.

The existence of non-SMEFT operators affects the search for new physics at low
energies, such as when analyzing discrepancies from the SM in the decay $b\to
c\,\tau^{-}{\bar{\nu}}$ using four-fermion $b\to c\,\tau^{-}{\bar{\nu}}$
effective operators in LEFT. Assuming only left-handed neutrinos, five such
operators are possible:

$\displaystyle
O_{V}^{LL,LR}\equiv({\bar{\tau}}\gamma^{\mu}P_{L}\nu)\,,({\bar{c}}\gamma_{\mu}P_{L,R}b)\,,$
$\displaystyle
O_{S}^{LL,LR}\equiv({\bar{\tau}}P_{L}\nu)\,({\bar{c}}P_{L,R}b)\leavevmode\nobreak\
,$ $\displaystyle
O_{T}\equiv({\bar{\tau}}\sigma^{\mu\nu}P_{L}\nu)\,({\bar{c}}\sigma_{\mu\nu}P_{L}b)\leavevmode\nobreak\
,$ (1)

where $P_{L,R}$ are the left-handed and right-handed projection operators. As
we see below, $O_{V}^{LR}$ is a non-SMEFT operator: it is generated at tree
level starting from a dimension-8 SMEFT operator. Because of this, the
coefficient of $O_{V}^{LR}$ would naively be suppressed by the small factor
$v^{2}/\Lambda^{4}$ if SMEFT were true at UV scales. It is usually excluded
when seeking new physics in $b\to c\,\tau^{-}{\bar{\nu}}$ (see, e.g., Refs.
Blanke:2018yud ; Blanke:2019qrx ).

But to test how the gauge symmetries are realized one must measure the
coefficients of such non-SMEFT operators, and see if their size is consistent
with SMEFT power counting. If the SMEFT-predicted suppression in the
coefficients is not present it would point to a more complicated realization
of $SU_{(}2)_{L}\times U(1)_{Y}$ in the UV than is usually assumed.

The first step in performing such an analysis is to identify all the non-SMEFT
dimension-6 operators in LEFT. We list these in Table 1, along with the
higher-dimension SMEFT operators from which they can be obtained at tree
level. Operators appearing in the ‘LEFT operator’ column are denoted by
$\mathcal{O}$ and are as defined in Ref. Jenkins:2017jig . Operators appearing
in the ‘Tree-level SMEFT origin’ column are denoted by $Q$. The one with
dimension 6 (the operator $Q_{Hud}$) is as defined in Ref. Grzadkowski:2010es
, and the nomenclature for the dimension-8 operators is taken from Ref.
Murphy:2020rsh . JMS also identified these non-SMEFT operators, simply saying
they had no direct dimension-6 SMEFT counterpart, and our list agrees with
their findings.

LEFT operator | Tree-level SMEFT origin | Dims.
---|---|---
Semileptonic operators
$\mathcal{O}_{\nu edu}^{V,LR}:(\overline{\nu}_{Lp}\gamma^{\mu}e_{Lr})(\overline{d}_{Rs}\gamma_{\mu}u_{Rt})+\mathrm{h.c.}$ | $Q_{Hud}:i(\tilde{H}^{\dagger}D_{\mu}H)(\overline{u}_{p}\gamma^{\mu}d_{r})+\mathrm{h.c.}$ | $6\to 6$
| $Q_{\ell^{2}udH^{2}}:(\overline{\ell}_{p}d_{r}H)(\tilde{H}^{\dagger}\overline{u}_{s}l_{t})+\mathrm{h.c.}$ | $6\to 8$
$\mathcal{O}_{ed}^{S,RR}:(\overline{e}_{Lp}e_{Rr})(\overline{d}_{Ls}d_{Rt})$ | $Q_{\ell eqdH^{2}}^{(3)}:(\overline{\ell}_{p}e_{r}H)(\overline{q}_{s}d_{t}H)$ | $6\to 8$
$\mathcal{O}_{eu}^{S,RL}:(\overline{e}_{Rp}e_{Lr})(\overline{u}_{Ls}u_{Rt})$ | $Q_{\ell equH^{2}}^{(5)}:(\overline{\ell}_{p}e_{r}H)(\tilde{H}^{\dagger}\overline{q}_{s}u_{t})$ | $6\to 8$
$\mathcal{O}_{ed}^{T,RR}:(\overline{e}_{Lp}\sigma^{\mu\nu}e_{Rr})(\overline{d}_{Ls}\sigma_{\mu\nu}d_{Rt})$ | $Q_{\ell eqdH^{2}}^{(4)}:(\overline{\ell}_{p}\sigma_{\mu\nu}e_{r}H)(\overline{q}_{s}\sigma^{\mu\nu}d_{t}H)$ | $6\to 8$
Four-lepton operators
$\mathcal{O}_{ee}^{S,RR}:(\overline{e}_{Lp}e_{Rr})(\overline{e}_{Ls}e_{Rt})$ | $Q_{\ell^{2}e^{2}H^{2}}^{(3)}:(\overline{\ell}_{p}e_{r}H)(\overline{\ell}_{s}e_{t}H)$ | $6\to 8$
Four-quark operators
$\mathcal{O}_{uddu}^{V1,LR}:(\overline{u}_{Lp}\gamma^{\mu}d_{Lr})(\overline{d}_{Rs}\gamma_{\mu}u_{Rt})+\mathrm{h.c.}$ | $Q_{Hud}:i(\tilde{H}^{\dagger}D_{\mu}H)(\overline{u}_{p}\gamma^{\mu}d_{r})+\mathrm{h.c.}$ | $6\to 6$
| $Q_{q^{2}udH^{2}}^{(5)}:(\overline{q}_{p}d_{r}H)(\tilde{H}^{\dagger}\overline{u}_{s}q_{t})+\mathrm{h.c.}$ | $6\to 8$
$\mathcal{O}_{uddu}^{V8,LR}:(\overline{u}_{Lp}\gamma^{\mu}T^{A}d_{Lr})(\overline{d}_{Rs}\gamma_{\mu}T^{A}u_{Rt})+\mathrm{h.c.}$ | $Q_{q^{2}udH^{2}}^{(6)}:(\overline{q}_{p}T^{A}d_{r}H)(\tilde{H}^{\dagger}\overline{u}_{s}T^{A}q_{t})+\mathrm{h.c.}$ |
$\mathcal{O}_{uu}^{S1,RR}:(\overline{u}_{Lp}u_{Rr})(\overline{u}_{Ls}u_{Rt})$ | $Q_{q^{2}u^{2}H^{2}}^{(5)}:(\overline{q}_{p}u_{r}\tilde{H})(\overline{q}_{s}u_{t}\tilde{H})$ | $6\to 8$
$\mathcal{O}_{uu}^{S8,RR}:(\overline{u}_{Lp}T^{A}u_{Rr})(\overline{u}_{Ls}T^{A}u_{Rt})$ | $Q_{q^{2}u^{2}H^{2}}^{(6)}:(\overline{q}_{p}T^{A}u_{r}\tilde{H})(\overline{q}_{s}T^{A}u_{t}\tilde{H})$ |
$\mathcal{O}_{dd}^{S1,RR}:(\overline{d}_{Lp}d_{Rr})(\overline{d}_{Ls}d_{Rt})$ | $Q_{q^{2}d^{2}H^{2}}^{(5)}:(\overline{q}_{p}d_{r}H)(\overline{q}_{s}d_{t}H)$ | $6\to 8$
$\mathcal{O}_{dd}^{S8,RR}:(\overline{d}_{Lp}T^{A}d_{Rr})(\overline{d}_{Ls}T^{A}d_{Rt})$ | $Q_{q^{2}d^{2}H^{2}}^{(6)}:(\overline{q}_{p}T^{A}d_{r}H)(\overline{q}_{s}T^{A}d_{t}H)$ |
Table 1: Non-SMEFT four-fermion operators in LEFT and the dimension-8 SMEFT
operators to which they are mapped at tree level. In the ‘LEFT operator’
column, the subscripts $p,r,s,t$ are weak-eigenstate indices; they are
suppressed in the operator labels. In the ‘Tree-level SMEFT origin’ column,
$\ell$ and $q$ denote left-handed $SU(2)_{L}$ doublets, while $e$, $u$ and $d$
denote right-handed $SU(2)_{L}$ singlets. Here, $\tilde{H}=i\sigma_{2}H^{*}$
denotes the conjugate of the Higgs doublet $H$.

Of course, there is nothing sacred about tree level, and in principle loops
can also generate effective operators as one evolves down to lower energies
(as the example of non-SM gauge-boson masses generated by top-quark loops
mentioned above shows). Whether such loops are important in any particular
instance depends on the size of any loop-suppressing couplings and the masses
that come with them. As the top-quark example also shows, generating non-SMEFT
operators from loops involving SMEFT operators necessarily involves a
dependence on $SU(2)_{L}\times U(1)_{Y}$-breaking masses, implying a
suppression (and a lowering of operator dimension) when these masses are
small. The authors of Ref. Dekens:2019ept  have computed how SM loops dress
individual SMEFT operators, and show that such loops do not generate non-SMEFT
operators in LEFT at the one-loop level.

Ref. Jenkins:2017dyc  computes the running of the LEFT operators that are
unsuppressed by such factors, arising due to dressing by photon and gluon
loops, and shows that non-SMEFT dimension-6 operators of this type also can
arise from the mixing of dimension-5 dipole operators of the form
$(\bar{\psi}\sigma^{\mu\nu}\psi)X_{\mu\nu}$ in LEFT, where
$X_{\mu\nu}=G_{\mu\nu},F_{\mu\nu}$ are gauge field strengths. The two required
insertions of these dipole operators ensure that they do not change the tree-
level counting of powers of $1/\Lambda$, in their coefficients.

Applications to $B$ physics — Although the Table shows quite a few non-SMEFT
operators that can, in principle, be used to search for non-SMEFT new physics,
one of these is particularly interesting: the operator $O_{V}^{LR}$ of Eq.
(1),

$\mathcal{O}_{\nu\tau
bc}^{V,LR}\equiv({\bar{\tau}}_{L}\gamma^{\mu}\nu_{L})({\bar{c}}_{R}\gamma_{\mu}b_{R})+\mathrm{h.c.}\leavevmode\nobreak\
,$ (2)

that contributes to the decay $b\to c\,\tau^{-}{\bar{\nu}}$.

Notice that Table 1 offers two possible SMEFT operators from which this
operator can be obtained at tree level, one of which is the dimension-6 SMEFT
operator $Q_{Hud}$. Naively this seems to imply that $\mathcal{O}_{\nu\tau
bc}^{V,LR}$ is actually a SMEFT operator after all. But there is a subtlety
here: $Q_{Hud}$ is a lepton-flavour-universal operator that generates equal
effective couplings for the operators $\mathcal{O}_{\nu ebc}^{V,LR}$,
$\mathcal{O}_{\nu\mu bc}^{V,LR}$ and $\mathcal{O}_{\nu\tau bc}^{V,LR}$. An
effective operator that generates only $\mathcal{O}_{\nu\tau bc}^{V,LR}$
without the other two violates lepton-flavour universality, and this can only
come from the dimension-8 operator given in the Table. (A similar reasoning
applies also to $\mathcal{O}_{uddu}^{V1,LR}$ and $\mathcal{O}_{uddu}^{V8,LR}$
, where superscripts ‘1’ and ‘8’ give the colour transformation of the quark
pairs.) Furthermore, at the 1-loop level, $\mathcal{O}_{\nu\tau bc}^{V,LR}$
does not mix with any other LEFT operators Jenkins:2017dyc .

The five four-fermion operators given in Eq. (1) imply – assuming only left-
handed neutrinos – that the most general LEFT effective Hamiltonian describing
$b\to c\,\tau^{-}{\bar{\nu}}$ decay is

$\displaystyle{\cal H}_{eff}$ $\displaystyle=$
$\displaystyle\frac{4G_{F}}{\sqrt{2}}\,V_{cb}\,O_{V}^{LL}-\frac{C_{V}^{LL}}{\Lambda^{2}}\,O_{V}^{LL}\,-\frac{C_{V}^{LR}}{\Lambda^{2}}\,O_{V}^{LR}\,,$
(3) $\displaystyle-\leavevmode\nobreak\
\frac{C_{S}^{LL}}{\Lambda^{2}}\,O_{S}^{LL}\,-\frac{C_{S}^{LR}}{\Lambda^{2}}\,O_{S}^{LR}-\frac{C_{T}}{\Lambda^{2}}\,O_{T}\leavevmode\nobreak\
\,.$

The first term is the SM contribution; the remaining five terms are the
various new-physics contributions. Within LEFT, these are all dimension-6
operators and so, in the absence of other information, for a given new-physics
scale $\Lambda$, their dimensionless coefficients (the $C$s) are all at most
$O(1)$. By contrast, the coefficient $C_{V}^{LR}$ is instead proportional to
$v^{2}/\Lambda_{h}^{2}$ if the new physics is described at higher energies by
SMEFT (since $\mathcal{O}_{V}^{LR}$ then really descends from a Higgs-
dependent interaction with dimension 8), and so is predicted to be small if
$\Lambda_{h}\sim\Lambda$.

The beauty of $b\to c\,\tau^{-}{\bar{\nu}}$ decays is that, in principle, they
provide sufficiently many observables to measure each of the couplings in Eq.
(3) separately, thereby allowing a test of the prediction that $C_{V}^{LR}$
should be negligible (assuming that the presence of new physics is confirmed).
If the effective couplings do not follow the SMEFT pattern, non-SMEFT new
physics must be involved.

What is currently known about $C_{V}^{LR}$? At present several observables
have been measured that involve the decay $b\to c\,\tau^{-}{\bar{\nu}}$. These
include

$\displaystyle{\cal R}(D^{(*)})$ $\displaystyle\equiv$
$\displaystyle\frac{{\cal B}(B\to D^{(*)}\tau\nu)}{{\cal B}(B\to
D^{(*)}\ell\nu)}\,,{\cal R}(J/\psi)\equiv\frac{{\cal B}(B_{c}\to
J/\psi\tau\nu)}{{\cal B}(B_{c}\to J/\psi\mu\nu)}\,,$ $\displaystyle
F_{L}(D^{*})$ $\displaystyle\equiv$ $\displaystyle\frac{\Gamma(B\to
D^{*}_{L}\tau\nu)}{\Gamma(B\to D^{*}\tau\nu)}\,,\leavevmode\nobreak\
P_{\tau}(D^{*})\equiv\frac{\Gamma^{+1/2}-\Gamma^{-1/2}}{\Gamma^{+1/2}+\Gamma^{-1/2}}\leavevmode\nobreak\
,$ (4)

where $\Gamma^{\lambda}\equiv\Gamma(B\to D^{*}\tau^{\lambda}\nu)$.
$P_{\tau}(D^{*})$ measures the $\tau$ polarization asymmetry while
$F_{L}(D^{*})$ measures the longitudinal $D^{*}$ polarization. These
observables are useful for distinguishing new-physics models with different
Lorentz structures and (interestingly) the measurements of most of these
observables seem to be in tension with the predictions of the SM. Refs.
Blanke:2018yud ; Blanke:2019qrx ) perform fits to the data using the
interactions of Eq. (3) (though with a different operator normalization than
is used here), but with $O_{V}^{LR}$ assumed not to be present (precisely
because it is a non-SMEFT operator).

We make two observations about how to use these measurements to probe the size
of $\mathcal{O}_{V}^{LR}$, one using existing data and one using new
observables – proposed elsewhere Bhattacharya:2020lfm  – to exploit future
data to access more information about the effective coefficients appearing in
Eq. (3).

Fits to current data — We have repeated the fit of Refs. Blanke:2018yud ;
Blanke:2019qrx ), though this time including $\mathcal{O}_{V}^{LR}$ for
comparison. The values of the experimental observables used in the fit are
those found in Ref. Blanke:2019qrx . One observable that is not used is ${\cal
B}(B_{c}\to\tau\nu)$. This decay has not yet been measured, but it has been
argued that its branching ratio has an upper limit in order to be compatible
with the $B_{c}$ lifetime. Unfortunately this upper bound varies enormously in
different analyses, from 10% Akeroyd:2017mhr  to 60% Blanke:2018yud . Because
of this uncertainty, we do not use this upper bound as a constraint, but
simply compute the prediction for ${\cal B}(B_{c}\to\tau\nu)$ in each new
physics scenario. For the theoretical predictions of the observables in the
presence of new physics, we use the program flavio Straub:2018kue  and the fit
itself is done using MINUIT James:1975dr ; James:2004xla ; James:1994vla .

Because the data is not yet rich enough to permit an informative simultaneous
fit to all five effective couplings111Fits involving the other new-physics
coefficients were performed in Ref. Blanke:2019qrx . We have redone these fits
in order to verify that we reproduce the results of this paper. we instead
perform fits in which only one or two of the effective couplings are nonzero.
We choose $\Lambda=5$ TeV and consider the following three scenarios for
nonzero new-physics coefficients: either $C_{V}^{LL}$ or $C_{V}^{LR}$ are
turned on by themselves, or both $C_{V}^{LL}$ and $C_{V}^{LR}$ are turned on
together. The results of fits using these three options are presented in Table
2, and Fig. 1 presents the (correlated) allowed values of $C_{V}^{LL}$ and
$C_{V}^{LR}$ for the joint fit.

New-physics coeff. Best fit $p$ value (%) pullSM $C_{V}^{LL}$ $-3.1\pm 0.7$ 51
4.1 $C_{V}^{LR}$ $2.8\pm 1.2$ 0.3 2.3 $(C_{V}^{LL},C_{V}^{LR})$ $(-3.0\pm
0.8,0.6\pm 1.2)$ 35 3.7

Table 2: Fit results for the scenarios in which $C_{V}^{LL}$, $C_{V}^{LR}$ or
both $C_{V}^{LL}$ and $C_{V}^{LR}$ are allowed to be nonzero. At the best-fit
point the prediction for ${\cal B}(B_{c}\to\tau\nu)$ is $\sim$2.8% for all
scenarios.

We see that the scenario that adds only $C_{V}^{LL}$ provides an excellent fit
to the data. On the other hand, the fit is poor when $C_{V}^{LR}$ alone is
added (though it is still much better than for the SM itself). The fit remains
acceptable when both $C_{V}^{LL}$ and $C_{V}^{LR}$ are allowed to be nonzero.
In all scenarios, ${\cal B}(B_{c}\to\tau\nu)$ is predicted to be $<3\%$, which
easily satisfies all constraints.

It is clear that the current data is insufficient to constrain the value for
$C_{V}^{LR}$ in a useful way. Both the SMEFT prediction $C_{V}^{LR}\sim
v^{2}/\Lambda^{2}=O(10^{-3})$ and $C_{V}^{LR}\sim\mathcal{O}(1)$ are
consistent with the joint fit with both $C_{V}^{LL}$ and $C_{V}^{LR}$ nonzero;
the best-fit value $C_{V}^{LR}=0.6\pm 1.2$ is consistent with both zero and
large $O(1)$ values222We note that the central values satisfy
$C_{V}^{LL}/C_{V}^{LR}\simeq-0.2$. In Ref. Gomez:2019xfw , it was assumed that
the $b\to c\,\tau^{-}{\bar{\nu}}$ anomaly could be explained by the addition
of a $W^{\prime}$ with general couplings. When they performed a fit with $LL$
and $LR$ couplings, they also found a ratio of $LL/LR\simeq-0.2$.. At present,
the data are consistent with the non-SMEFT coefficient $C_{V}^{LR}$ being much
larger than the SMEFT prediction.

Figure 1: (Correlated) allowed values of $C_{V}^{LL}$ and $C_{V}^{LR}$ at
$1\sigma$ (inner region) and $2\sigma$ (outer region).

It is worth noting that the same is not true for other non-SMEFT operators.
From the operators listed in Table 1, consider for example the specific
operators $({\bar{\mu}}_{L}\mu_{R})({\bar{s}}_{L}b_{R})$ and
$({\bar{\mu}}_{R}\mu_{L})({\bar{s}}_{R}b_{L})$ in the class
$\mathcal{O}_{ed}^{S,RR}$, or the $\mathcal{O}_{ed}^{T,RR}$ operators of the
type
$({\bar{\mu}}_{L}\sigma_{\mu\nu}\mu_{R})({\bar{s}}_{L}\sigma^{\mu\nu}b_{R})$
and
$({\bar{\mu}}_{R}\sigma_{\mu\nu}\mu_{L})({\bar{s}}_{R}\sigma^{\mu\nu}b_{L})$.
These all contribute in a chirally unsupressed way to the decay $b\to
s\mu^{+}\mu^{-}$ (unlike the case in the SM), and so the addition of any of
these operators can dramatically change the prediction for ${\cal
B}(B_{s}^{0}\to\mu^{+}\mu^{-})$. But the measured value ${\cal
B}(B_{s}^{0}\to\mu^{+}\mu^{-})=(2.9\pm 0.4)\times 10^{-9}$
ParticleDataGroup:2020ssz  is close to the SM prediction, so that the
coefficients of these operators cannot be larger than order $O(10^{-4})$,
consistent with SMEFT expectations. Things are similar for the analogous
operators contributing to $b\to se^{+}e^{-}$, for which the upper limit of
${\cal B}(B_{s}^{0}\to e^{+}e^{-})<9.4\times 10^{-9}$
ParticleDataGroup:2020ssz  constrains the coefficients of these operators to
be $<O(10^{-3})$, again consistent with SMEFT.

Future prospects — The above discussion shows that the non-SMEFT operator
$O_{V}^{LR}$ can have a large effective coupling, $C_{V}^{LR}\sim O(1)$
without causing observational difficulties with $b\to c\,\tau^{-}{\bar{\nu}}$
decays, though there are large errors. But even if the experimental errors on
the currently measured observables were to improve dramatically, the five
observables of Eq. (4) are never enough to measure all of these parameters in
the most general case. This is simply because these five measurements cannot
pin down all ten of the parameters that can appear in the five complex
couplings given in Eq. (3).

Fortunately, there are potentially many more observables whose measurement can
remedy this situation. Ref. Bhattacharya:2020lfm  has proposed to measure the
angular distribution in ${\bar{B}}\to D^{*}(\to
D\pi^{\prime})\,\tau^{-}(\to\pi^{-}\nu_{\tau}){\bar{\nu}}_{\tau}$. This decay
includes three final-state particles whose four-momenta can be measured: $D$,
$\pi^{\prime}$ and $\pi^{-}$. Using this information, the differential decay
rate can be constructed. This depends on two non-angular variables, $q^{2}$
and $E_{\pi}$, as well as a number of angular variables. Here, $q^{2}$ is the
invariant mass-squared of the $\tau^{-}{\bar{\nu}}_{\tau}$ pair and $E_{\pi}$
is the energy of the $\pi^{-}$ in the $\tau$ decay. The idea is then to
separate the data into $q^{2}$-$E_{\pi}$ bins, and then to perform an angular
analysis in each of these bins. Each angular distribution consists of twelve
different angular functions; nine of these terms are CP-conserving, and three
are CP-violating. There are therefore a large number of observables in this
differential decay rate; the exact number depends on how many
$q^{2}$-$E_{\pi}$ bins there are.

Eq. (1) lists five new-physics operators, but only four of these actually
contribute to ${\bar{B}}\to D^{*}\tau^{-}{\bar{\nu}}_{\tau}$. To see why,
consider the following linear combinations of the two scalar operators:

$\displaystyle O_{LS}$ $\displaystyle\equiv$ $\displaystyle
O_{S}^{LR}+O_{S}^{LL}=({\bar{\tau}}P_{L}\nu)\,({\bar{c}}\,b)\,,$
$\displaystyle O_{LP}$ $\displaystyle\equiv$ $\displaystyle
O_{S}^{LR}-O_{S}^{LL}=({\bar{\tau}}P_{L}\nu)\,({\bar{c}}\gamma_{5}b)\leavevmode\nobreak\
.$ (5)

Of these, only $O_{LP}$ contributes to ${\bar{B}}\to
D^{*}\tau^{-}{\bar{\nu}}_{\tau}$.

With complex coefficients, there are therefore eight unknown theoretical
parameters in the remaining four effective interactions. Observables are
functions of these parameters, as well as $q^{2}$ and $E_{\pi}$. Thus, if the
angular distribution in ${\bar{B}}\to D^{*}(\to
D\pi^{\prime})\,\tau^{-}(\to\pi^{-}\nu_{\tau}){\bar{\nu}}_{\tau}$ can be
measured, it may be possible to extract all of the new physics coefficients
from a fit to observations. If the real or imaginary part of $C_{V}^{LR}$ were
found to be much larger than the SMEFT expectation, it would suggest the
presence of non-SMEFT physics at higher energies.

Note also that the decay $b\to c\,\mu^{-}{\bar{\nu}}$ can also be analyzed in
a similar way (even though there is no hint of new physics in this reaction).
The angular distribution for $b\to c\,\mu^{-}{\bar{\nu}}$ described in Ref.
Bhattacharya:2019olg  provides enough observables to perform a fit for the
coefficients of all dimension-6 new physics operators, including the non-SMEFT
one.

In summary, we reproduce here the list of non-SMEFT four-fermion operators and
identify their provenance, assuming that they arise at tree level starting
from even-higher-dimension SMEFT operators, in order to pin down the SMEFT
estimate for the size of their effective couplings. We show that fits to
current observations allow one of these couplings – that of the semileptonic
$b\to c\,\tau^{-}{\bar{\nu}}$ operator $\mathcal{O}_{V}^{LR}$ – to be
$\mathcal{O}(1)/\Lambda^{2}$ for $\Lambda\sim 5$ TeV, which is consistent with
couplings that are several orders of magnitude larger than would be predicted
by SMEFT. We also identify a sufficiently large class of $b\to
c\,\tau^{-}{\bar{\nu}}$ observables whose measurement would in principle allow
all of the relevant effective couplings to be determined, including that of
$\mathcal{O}_{V}^{LR}$. There is a good prospect that these measurements can
be done in the future.

Finally, suppose it were eventually established that non-SMEFT new physics is
present in $b\to c\,\tau^{-}{\bar{\nu}}$. The obvious question then is: What
could this non-SMEFT new physics be? Although serious exploration of models
probably awaits evidence for such a signal, some preliminary attempts have
been made in the literature. One example is Ref. Cata:2015lta , which studies
the non-SMEFT operators in $b\to s\mu^{+}\mu^{-}$ and $b\to
c\,\tau^{-}{\bar{\nu}}$ in the context of HEFT, and argues that such operators
can be generated by a nonstandard Higgs sector containing additonal strongly-
interacting scalars. We regard a more systematic exploration of non-SMEFT
physics in the UV to be well worthwhile, and look forward to that happy day
when experimental results are what drives it.

Acknowledgments: We thank Mike Trott and B. Bhattacharya for helpful
discussions. We also thank Christopher Murphy for pointing out some oversights
in the first version of the paper. This work was partially financially
supported by funds from the Natural Sciences and Engineering Research Council
(NSERC) of Canada. Research at the Perimeter Institute is supported in part by
the Government of Canada through NSERC and by the Province of Ontario through
MRI. JK is financially supported by a postdoctoral research fellowship of the
Alexander von Humboldt Foundation.

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# A Generative Model to Synthesize EEG Data for Epileptic Seizure Prediction

Khansa Rasheed1, Junaid Qadir1, Terence J. O’Brien2, Levin Kuhlmann3 and Adeel
Razi4,5
1 Information Technology University (ITU)-Punjab, Lahore, Pakistan
2 Department of Neuroscience, Central Clinical School,Monash University,
Melbourne, Victoria, Australia
3 Faculty of Information Technology, Monash University, Clayton, Victoria,
Australia
4 Turner Institute for Brain and Mental Health, Monash University, Clayton,
Victoria, Australia
5 Wellcome Centre for Human Neuroimaging, UCL, London, United Kingdom

###### Abstract

Epilepsy, a group of neurological disorders, is characterized by recurrent and
unpredictable seizures, which can cause an unexpected loss of awareness for
patients resulting in a risk of life-threatening incidents. Prediction of
seizure before they occur is therefore vital for bringing normalcy to the
lives of patients. For decades researchers have been employing machine
learning (ML) techniques based on hand-crafted features for seizure
prediction. However, ML methods are too complicated to select the best ML
model or the best features. Deep Learning (DL) methods are beneficial in the
sense of automatic feature extraction. Lamentably despite decades of research,
epilepsy prediction remains an unresolved problem. One of the roadblocks for
accurate seizure prediction, among others, is the scarcity of good quality
epileptic seizure data. This paper addresses this problem by proposing a deep
convolutional generative adversarial network (DCGAN) to generate synthetic EEG
data samples. We use two methods to validate the synthesized data namely, one-
class SVM and a new proposal which we refer to as convolutional epileptic
seizure predictor (CESP). Another objective of our study is to evaluate the
performance of well-known deep learning models (e.g., VGG16, VGG19, ResNet50,
and Inceptionv3) by training the models on augmented EEG data using transfer
learning with average time of 10 min between true prediction and seizure
onset. Our results show that the CESP model achieves sensitivity of 78.11% and
88.21%, and false prediction rate of 0.27/h and 0.14/h for the training on
synthesized and testing on real Epilepsyecosystem and CHB-MIT datasets,
respectively. The impressive performance of CESP trained on synthesized data
shows that the synthetic data acquired the correlation between features and
labels very well. We also show that using the employment of the idea of
transfer learning and data augmentation in patient-specific manner provides
the highest accuracy with sensitivity of 90.03% and 0.03 FPR/h which was
achieved using Inceptionv3, and that augmenting the data with the samples
generated from DCGAN increased the prediction results of our CESP model and
Inceptionv3 by 4-5% as compared to the state-of-the-art traditional
augmentation techniques. Finally, we note that the prediction results of CESP
achieved by using the augmented data are better than the chance level for both
datasets.

###### Index Terms:

Epileptic Seizure, EEG, Machine Learning, Deep Learning, Transfer Learning,
Adversarial Networks

##  I Introduction

A sudden abnormal, self sustaining electrical discharge in the cerebral
networks of the brain is a cause of epileptic seizures (ES). The attack may
occur at any time on any day. The unpredictability of duration, seriousness,
and time of attack makes it very difficult for patients to perform everyday
chores and on occasions can be life-threatening. According to the World Health
Organization (WHO), 70 million people around the globe suffer from epilepsy
with around one-third of these patients resistant to anti-epileptic medication
[1]. The early prediction of these attacks before they occur will be helpful
for the patients to take precautionary measures and potentially allow the
implementation of preventative therapies.

Figure 1: Pipeline of the methodology used in the paper.

Electroencephalography (EEG) is used for measuring and monitoring brain
activity before, during, and after ES and is widely used to predict seizures.
Machine Learning (ML) based prediction algorithm uses the hand-crafted
features of EEG from the time-domain, frequency-domain, or time-frequency
domain to make predictions. Previously, researchers have evaluated various
features—such as Kolmogorov entropy [2], largest Lyapunov exponent (LLE) [3],
phase synchronization of different EEG channels [4], and correlation
density—to perform seizure prediction [5]. In 2014 and 2016, contests of
(epileptic) seizure prediction were held by the American Epilepsy Society and
Melbourne University. These competitions were open to the EEG feature (for
seizure prediction) computing algorithm or ML models trained on the extracted
features. However, the submitted algorithms were too complicated to select the
best features or the best ML model for seizure prediction [6].

Deep learning (DL) algorithms are beneficial in the sense of automatic feature
extraction from the data [7]. Over the past few years, researchers have
applied several DL methods to predict epileptic seizures [8], [9], [10].
However, these DL algorithms require an extensive amount of labeled data to
produce effective results. Researchers typically use scalp EEG, where signals
are collected from the wearable sensors placed on the scalp, or intracranial
EEG (iEEG), signals collected by placing the electrode on the exposed surface
of the brain through surgery, for the ES prediction. iEEG data gives the high
temporal resolution and frequency information of brain activity with high
signal to noise ratio as compared to scalp EEG. However, the acquisition of
iEEG data is challenging due to surgery and implantation risks.

Despite decades of research in the field of ES prediction, there are still
challenges related to the availability of a large amount of labeled EEG data
and computationally efficient prediction algorithms. Our major contributions
in this paper are as follows:

  1. 1.

We propose a deep convolutional generative adversarial network (DCGAN) to
resolve the EEG data scarcity problem. As a proof of principle, we generate
synthetic scalp EEG and iEEG data of each patient by training the DCGAN model
on the Epilepsyecosystem iEEG data and CHB-MIT scalp EEG data separately in a
patient-specific way.

  2. 2.

To evaluate the effectiveness of simulated data, we then use two methods:
classical ML method of one-class SVM, and secondly a CNN classifier—which we
refer to as convolutional epileptic seizure predictor (CESP)—for seizure
prediction in the last block of Figure 1). We train CESP on the real data and
test on synthesized data (iEEG data generated from DCGAN). The idea of
augmenting the real data with generated data also improved the performance of
CESP.

  3. 3.

We also evaluate data augmentation with the well-known technique of transfer
learning (TL). We trained the popular DL models ResNet50, Inceptionv3, VGG16,
and VGG19 on the large amount of already generated synthesized data. After the
training, we fine-tuned these pre-trained models on the real data to develop
the patient-specific prediction algorithm.

To the best of our knowledge, this is the first study on the generation of
synthetic scalp EEG and iEEG data and the use of transfer learning with data
augmentation performed by generative methods for seizure prediction. Figure 1
summarizes the method we have developed.

The remaining paper is organized as follows: Section II covers the previous
work on ES prediction using DL methods. Section III provides face validation,
i.e., the detailed methodology we propose along with the used dataset. Section
IV provides construct validation, which comprises the results obtained from
the proposed methods and comparison of the results with previous work. The
paper is concluded in Section VI.

##  II Related Work

EEG is a complex and challenging functional brain mapping modality to handle
due to the presence of noise and various measurement and physiological
artifacts. Pre-processing of EEG data for noise and artifact removal is an
involved time-consuming exercise that greatly compromises the utility of
online development of EEG based ES prediction solutions [11]. Furthermore,
most classical ML-based prediction techniques heavily rely on hand-crafted
features [12]. In contradistinction, the ability of DL-based algorithms to
automatically and robustly learn data features from partially or unprocessed
EEG data has fueled many new research ideas for accurate and time-efficient ES
prediction.

With minimal pre-processing of EEG data, Troung et al. [8] proposed a three-
layered convolutional neural network (CNN)-based prediction algorithm for the
first time. The auuthors used the spectrograms generated from the short-time
Fourier transform (STFT) of the Freiburg hospital iEEG database and the CHBMIT
scalp EEG database to train the CNN network. As compared to the interictal or
ictal class, the pre-ictal class samples were less, which caused the
imbalanced dataset. To overcome this problem of an imbalanced dataset, they
also generated preictal samples by sliding a 30sec window over the preictal
samples along the time axis. They achieved 89.8% sensitivity with 0.17 false
prediction rate (FPR/h) for 5 min of Seizure Prediction Horizon (SPH). SPH is
defined as the time interval between the alarm raised in the anticipation of
an impending seizure onset and the actual start of the ictal state [13].
Haider et al. [9] improved the results by achieving 87.8% sensitivity and
0.142 FPR for 10 min SPH with the CHBMIT and MSSM databases. They transformed
the EEG data into wavelet tensors before training the CNN classifier. Cook et
al. [14] demonstrated the feasibility of seizure prediction in clinical
setting with the success of implantable EEG recording system. Their work
initiated new avenues of research for seizure prediction. Building on the work
of Cook et al., Kiral-Kornek et al. [15] presented a portable seizure
prediction system with tunable parameters according to the patient’s need.
They transformed iEEG data into spectrograms and used frequency transformed
data as an input to the DL model for automatic pre-ictal feature learning.
Their prediction system performed at an average sensitivity of 69% and average
time in warning of 27%, significantly exceeding a comparable random predictor
for all patients by 42%.

Ramy Hussain et al. [16] implemented a downsampling technique to reduce the
dimension of EEG data but only used fraction of data gathered by Cook et al.
They demonstrated that feature extraction is not a viable methodology for
reliable ES prediction because the EEG measurements not only vary from patient
to patient but also vary for the same patient over time of data acquisition.
They applied STFT to transform the EEG data and train a CNN model on this
transformed data. They achieved an average of 87.85% sensitivity and 0.84 area
under the curve (AUC).

In the existing literature, ES prediction is posed as a supervised learning
problem. Proposed solutions, which we surveyed above, have been successful
however supervised learning requires an abundant amount of labeled data for
training process. Expert neurologists and physicians do the manual annotation
of the large EEG dataset which is time-consuming and prone to human errors. To
address the problem of unavailability of labeled EEG data. Troung et al. [10]
proposed a generative adversarial network (GAN) to perform unsupervised
training. They used the spectrogram of STFT of EEG as an input to the GAN.
Then, they used the features learned from the discriminator to predict the
seizure. To overcome the problem of an imbalanced dataset, they employed the
sliding window technique used in their previous work [8]. They measured the
area under the receiver operator characteristic (ROC) curve as a performance
measure with 5 min SPH and seizure occurrence period (SOP), time interval
during which seizure occurrence is expected, of 30 min. They achieved 77.68%
AUC for the CHBMIT scalp EEG data, 75.47% AUC on the Freiburg hospital data
and 65.05% AUC with the EPILEPSIAE database.

Data augmentation is a traditional solution to the intricacy of a small
dataset. Zhang et al. [17] extracted the time and frequency domain features
using wavelet packet decomposition and common spatial pattern (CSP). To
address the problem of an imbalanced dataset, they generated artificial pre-
ictal samples by randomly combining the segments of original pre-ictal samples
and augmented the real data with the generated samples. Then they fed the
extracted features to the shallow CNN classifier to predict the seizure. They
achieved a sensitivity of 92.2% and 0.12 FPR/h on the 23 patients of the CHB-
MIT dataset. In contrast to these techniques, we used a DL-based method for
artificial generation of seizure data.

##  III Methodology

In this section, we describe the datasets used in the study, pre-processing of
these datasets, and then we present the face validation of our proposed
methodology.

###  III-A Dataset

We are using two datasets for this work: the CHB-MIT dataset [18] and the
Epilepsyecosystem dataset [14] (summarized in Table I). We trained the DCGAN
on both datasets separately to generate the samples of scalp EEG and iEEG
signals. After the generation of synthetic data, we augmented the real data of
both datasets with the synthetic samples. We then employed the idea of
transfer learning (TL) on various DL models using augmented Epilepsyecosystem
dataset. We also evaluated the performance of the CESP model on both augmented
datasets.

The Epilepsyecosystem dataset recorded at St Vincent’s Hospital in Melbourne,
Australia is used for the experiments. The dataset contains the intracranial
EEG (iEEG) signals of three patients (all female). Data of each patient
contain signals from 16 electrodes sampled at 400Hz sampling frequency. Data
is segmented in 10 min long pre-ictal and interictal intervals. Pre-ictal
intervals are selected from the one hour earlier recordings of every seizure
with a five-minute seizure horizon while interictal intervals are segmented
from randomly selected one-hour recording blocks at least four hours away from
any seizure.

The CHB-MIT dataset consists of the 844h long continuous scalp EEG data of 23
patients with 163 episodes of seizure. Scalp EEG data were collected using 22
electrodes at a sampling rate of 256 Hz. We segmented the interictal periods
that are at least 4 hours away before a seizure onset and after a seizure
ends. We are interested in anticipating the leading seizures, therefore for
two or more consecutive seizures that are less than 30 min apart from each
other, we considered these as only one seizure. Moreover, we only considered
patients with no more than 10 seizures per day for the ES prediction because
it is not very crucial to predict ES for patients that have a seizure every 2
hours on average. With the preceding criteria, there are 13 patients with an
adequate amount of data, i.e., have at least 3 leading seizures and 3h of the
interictal period.

TABLE I: Summary of the datasets used in the paper. Dataset | EEG type |  | No. of
---
patients
| No. of
---
channels
| No. of
---
seizures
| Duration
---
average (hr)
CHB-MIT | scalp | 13 | 22 | 64 |  | 1-4
---
| Epilepsy-
---
-ecosystem
iEEG | 3 | 16 | 1362 |  | 10608
---

###  III-B Data Preparation

EEG data was contaminated by the power line noise at 60Hz for the CHB-MIT
dataset and 50Hz for the Epilepsyecosystem dataset. We removed the line noise
in both datasets using the Butterworth infinite impulse response. After
eliminating the line noise, short-time Fourier transform (STFT) is applied to
get the spectrogram of EEG signals. We applied STFT on each electrode of the
EEG signals of both datasets with 1 min window length with no overlaps.
Spectrograms of all electrodes (16 for Epilepsyecosystem and 22 electrodes for
CHB-MIT) were concatenated vertically to obtain the final spectrogram.

###  III-C Synthetic Data Generation

We use a DCGAN to generate synthetic iEEG and scalp EEG data. Here we only
describe it for the Epilepsyecosystem dataset. The same description is
implemented for the CHB-MIT dataset. The Generator takes a 100 dimensional
randomly generated samples from the standard Gaussian distribution of zero
mean and standard deviation of one as an input. The input layer is a dense
hidden layer. The output dimension of the first hidden layer is 4096 which is
reshaped to $4\times 4\times 256$. The dense layer is succeeded by 6 de-
convolutional layers with a stride size of $2\times 2$, filter size $5\times
5$, and the same padding. Number of filters in first de-convolutional layer
are 256 and 128 in all other de-convolutional layers. The output of the
generator is the same as the spectrograms generated by the STFT ($256\times
256\times 3$).

We configured the discriminator to distinguish the synthetic iEEG data from
the real data. The discriminator consists of 4 convolutional layers with 256,
128, 64, and 32 number of filters. The filter size in the convolutional layers
is 5 x 5, with a stride of 2x2, and the same padding. While training, the task
of the discriminator is to detect whether the spectrograms generated by the
Generator are real or fake. The Generator updates its parameters to generate
the spectrograms that are not distinctive from real spectrograms [19].

The equations of discriminator loss $D_{loss}$ and the Generator loss
$G_{loss}$ are defined as [19]:

$D_{loss}=\frac{1}{n}\sum_{j=1}^{n}\left[\log
D(x^{(j)})+\log(1-D(G(z^{(j)})))\right]$ (1)
$G_{loss}=\frac{1}{n}\sum_{j=1}^{n}\log(D(x^{(j)}))$ (2)

where $n$ is our batch size (32), $x$ is the real EEG spectrograms generated
from the STFT, and $z$ is a random sample generator from the distribution
$\mathcal{N}(0,1)$.

To overcome the problems of overfitting and convergence of discriminator, we
configured an early-stopping to have a check on $D_{loss}$ and $G_{loss}$. The
early-stopping monitoring stops the training of DCGAN if, over subsequent $k$
training batches, the $D_{loss}$ keeps getting larger than the $G_{loss}$. We
used a batch size of 32, $k=15$, Adam as an optimizer for gradient learning
with 0.5 value of $\beta_{1}$, and $1e^{-3}$ learning rate. The value of
$G_{loss}$ and $D_{loss}$ achieved the equilibrium point in around 3000 epochs
with the early-stopping monitoring.

We trained the DCGAN on the Epilepsyecosystem iEEG data and the CHB-MIT scalp
EEG data separately to obtain the synthetic iEEG and scalp EEG data. To
achieve the best results, we trained the DCGAN with three different dataset
settings: (i) training on all patients of the Epilepsyecosystem dataset; (ii)
training on all patients of the CHB-MIT dataset; and (iii) training of DCGAN
only on the pre-ictal class of data of all patients to generate the pre-ictal
synthetic samples for data augmentation.

Figure 2: Architecture of Convolutional Epileptic Seizure Predictor (CESP).

###  III-D One-class SVM for Data Validation

Schölkopf et al. [20] proposed the idea of one-class SVM, which is an
extension of a two-class SVM algorithm. One-class SVM is widely utilized to
identify the outliers and anomalies in the dataset. A one-class SVM algorithm
separates the data from the origin point by a wide margin in the higher
dimensional feature space. Then the algorithm computes the surface of a
hyperplane, which encloses the anomaly free data (+ve class). The data samples
which are out of the hyperplane are outliers/anomalies. The radius of the
hyperplane and the number of outliers/anomalies are hyperparameters to select
through multiple experiments.

Let $X$ be the samples of the positive class of dataset such that
$\left\\{x_{i}\in R^{n},i=1\cdots l\right\\}$, the optimization equation of
the algorithm is as follows:

$\displaystyle\underset{\xi,\rho,z,b}{\text{minimize}}$
$\displaystyle\frac{1}{2}z^{T}z-\rho+\frac{1}{\nu l}\sum_{i=1}^{l}\xi_{i}$ (3)
subject to $\displaystyle z^{T}\phi(x_{i})\geq\rho-\xi_{i},\;\xi_{i}\geq 0.$

In the optimization problem above, $\rho$ is the distance of hyperplane from
the origin, $\xi_{i}$ are the hyperparameters, and $\nu\in(0,1]$ selects the
fraction of outliers/anomalies outside the hyperplane. The decision function
is:

$\mathrm{sgn}(\sum_{i}^{l}\beta_{i}K(x_{i},x)-\rho).$

We used the Radial Basis Function (RBF) as a kernel function in our
experiments:

$K(x_{i},x_{j})=e^{-\gamma\left\|x_{i}-x_{j}\right\|^{2}}.$

###  III-E CESP Model

CNNs models have been widely used for predicting the seizures successfully in
literature [16], [9], due to their ability to learn local dependences of input
and the fewer number of trainable parameters due to weight sharing. Based on
the state-of-the-art performance of CNN in seizure prediction, we are using
the CNN classifier here to evaluate the effectiveness of synthetic data
generated from the above-mentioned architecture of DCGAN. The detailed
architecture of the proposed model is described in Figure 2.

In this work, we used a CNN network that consists of 3 convolutional blocks
followed by one fully connected (FC) layer. Each block contains a
convolutional layer, a rectified linear unit (ReLu), and a max-pooling layer.
The max-pooling technique enables the CNN model to learn temporal or spatially
invariant features. The convolutional layers have the filter size of $3\times
3$, stride $1\times 1$, $2\times 2$ size of max-pooling with 126, 64, and 64
number of filters respectively. The FC layers have a sigmoid activation
function with output sizes of 32 and 2. We designed this particular
architecture to achieve good performance with a simple model. We experimented
with a different number of layers of the model and chose the described model
of 3 convolutional layers providing good prediction results. To avoid the
overfitting for the simple model, we evaluated the training process on the
k-fold cross-validation. We used $k=10$ to split the training data into 90%
for training and 10% for validation. We trained the CESP model for the binary-
cross entropy loss on the Adam optimizer with a $1e^{-4}$ learning rate.

###  III-F Transfer Learning

With the rapidly growing applications of supervised learning in ML, a problem
arises when we do not have a sufficient amount of labeled data for training.
Transfer learning (TL) deals with this problem by leveraging the already
available labeled data of relevant or similar tasks. Since 1993, TL has been
used in discriminability-based transfer (DBT) algorithm [21], multi-tasking
learning [22], cognitive science [23], detection of cancer subtypes [24], text
classification [25], and spam filtration [26]. In another recent work, Bird et
al. used TL between EEG signal classification and Electromyographic (EMG)
signals [27].

We trained four well-known DL models: VGG16, VGG19, Inceptionv3, and ResNet50
on augmented EEG data. These are well-designed DL models intended to resolve
problems of convolutional networks, i.e., vanishing gradient, degradation,
long training time, and the large number of trainable parameters. We used the
weights of these models trained on the ImageNet dataset as initial weights
instead of training from random weights with learning rate of $1e^{-4}$.

###  III-G System Evaluation

Before evaluating the performance of the prediction algorithm, the SPH, and
the Seizure Occurrence Period (SOP) need to be defined. For our work, we are
using the definitions established in [13]. To make a correct prediction, a
seizure must transpire after the SPH and within the SOP. A false alarm will be
raised if the prediction algorithm gives a positive signal (seizure is going
to occur) but there is no seizure during the SOP. For the best clinical use,
the SPH must be long enough to give a patient sufficient time to take
precautionary measures after the alarm is raised. We use sensitivity, FPR/h,
specificity, and accuracy with SPH of 10 min and SOP of 30 min. For the
Epilepsyecosystem the value of SPH is fixed only for the training dataset. As
no information about the segmentation timing relative to seizures is provided
for the test set, we cannot determine the exact value of SPH, we only have the
information that we are 65 to 5 minutes away from the seizure.

We also compared the performance of CESP model trained on the augmented data
with a random predictor. Using the method proposed by Schelter et al. [28], we
computed the probability of alarm generation in the duration of SOP for a
given value of FPR:

$\displaystyle\textit{P}\approx 1-e^{-FPR\times SOP}.$ (4)

Then the probability to predict at least n out of N independent seizure events
at random can be calculated using the following equation:

$\displaystyle\textit{p}=\sum_{k\geq
n}\binom{N}{k}\textit{P}^{k}(1-\textit{P})^{N-k}.$ (5)

Using the FPR value of each patient and the number of true predictions using
CESP (n), we computed the p-value for each patient. We can infer that for the
significance level of $\alpha$ = 0.05, our approach performed better than a
chance level predictor.

##  IV Results

To generate the synthetic data, we trained the DCGAN on the iEEG data of
Epilepsyecosystem and the scalp EEG data of CHB-MIT datasets. In this section,
we test the effectiveness of synthetic data using the one-class SVM and CESP.
The selection of samples of generated data is based on the results of one-
class SVM and CESP model. These selected samples were then used for data
augmentation for ES prediction. For the CESP we applied different combinations
of real and synthetic data while testing and training. Detailed results are
described in Table II, III.

(a) Epilepsyecosystem

(b) CHB-MIT

Figure 3: Seizure prediction performance (AUC) of CESP using different combinations of real (iEEG data of Epilepsyecosystem, scalp EEG data of CHB-MIT) and synthetic (generated iEEG data from DCGAN trained on Epilepsyecosystem dataset, generated scalp EEG data from DCGAN trained on the CHB-MIT dataset) for testing and training. Legends: TRTR = train & test on real data, TSTR = train on synthesized data & test on real data, TRTS = train on real data & test on synthesized data, TSTS = train & test on synthesized data. TABLE II: Validation of synthesized pre-ictal samples of scalp EEG data using different combinations of synthesized and real data for training and testing on CESP. Synthesized data generated from the DCGAN trained on the CHB-MIT dataset. The comparison of results of i) test & train on real data (TRTR) and ii) train on synthetic data & test on real data (TSTR), shows that the synthetic data has fully captured the correlation between the features of data and the labels. Legends: TRTR = train & test on real data, TSTR = train on synthesized data & test on real data, TRTS = train on real data & test on synthesized data, TSTS = train & test on synthesized data, Sen = Sensitivity, Spec = Specificity, Acc = Accuracy. Patie- -nts | TRTR | TSTR | TRTS | TSTS
---|---|---|---|---
| Sen
---
(%)
FPR/h |  | Spec
---
(%)
| Acc
---
(%)
| Sen
---
(%)
FPR/h |  | Spec
---
(%)
| Acc
---
(%)
| Sen
---
(%)
FPR/h |  | Spec
---
(%)
| Acc
---
(%)
| Sen
---
(%)
FPR/h |  | Spec
---
(%)
| Acc
---
(%)
Pat1 | 91.59 | 0.19 | 81 | 92.03 | 88.15 | 0.24 | 76 | 90.18 | 88.09 | 0.25 | 75 | 90.85 | 90.58 | 0.18 | 82 | 91.57
Pat2 | 75.2 | 0.01 | 99 | 89.24 | 70.59 | 0.04 | 96 | 85.49 | 68.13 | 0.01 | 99 | 84.08 | 71.23 | 0.02 | 98 | 88.93
Pat3 | 95.11 | 0.14 | 86 | 96.11 | 96.36 | 0.17 | 83 | 96 | 95.21 | 0.15 | 85 | 95.55 | 95 | 0.14 | 86 | 94.89
Pat5 | 89.1 | 0.15 | 85 | 88.36 | 78.59 | 0.19 | 81 | 83.19 | 88.64 | 0.18 | 82 | 86.52 | 90.26 | 0.16 | 84 | 89.91
Pat9 | 65.23 | 0.01 | 99 | 78.51 | 70.51 | 0.02 | 98 | 80.26 | 62 | 0.00 | 100 | 89.97 | 71.87 | 0.00 | 100 | 89.99
Pat10 | 88.56 | 0.16 | 84 | 89.91 | 90.23 | 0.15 | 85 | 89.99 | 85.27 | 0.13 | 87 | 88.92 | 92.58 | 0.12 | 88 | 90.56
Pat13 | 90.14 | 0.22 | 78 | 91.15 | 89.36 | 0.19 | 81 | 91.05 | 88.06 | 0.20 | 80 | 90.18 | 92.41 | 0.18 | 82 | 91.07
Pat14 | 99.01 | 0.15 | 85 | 96.34 | 98.24 | 0.16 | 84 | 95.74 | 100 | 0.12 | 88 | 98.43 | 100 | 0.11 | 89 | 97.87
Pat18 | 95.41 | 0.21 | 79 | 89.37 | 93.36 | 0.25 | 75 | 88.43 | 96.28 | 0.24 | 76 | 92.09 | 98.99 | 0.20 | 80 | 91.99
Pat19 | 90.38 | 0.29 | 71 | 92.31 | 91 | 0.30 | 70 | 91.50 | 91.45 | 0.31 | 69 | 90.83 | 93.89 | 0.25 | 75 | 91.38
Pat20 | 87.58 | 0.18 | 82 | 90.84 | 88.45 | 0.14 | 86 | 89.09 | 89.63 | 0.16 | 84 | 90.44 | 91.56 | 0.15 | 85 | 90.59
Pat21 | 93.57 | 0.00 | 100 | 94.83 | 93.79 | 0.02 | 98 | 92.58 | 92.03 | 0.00 | 100 | 95.14 | 96.32 | 0.01 | 99 | 96.46
Pat23 | 100 | 0.05 | 95 | 97.62 | 98.1 | 0.03 | 97 | 97.32 | 99.56 | 0.07 | 93 | 92.21 | 100 | 0.02 | 98 | 91.17
Average | 89.28 | 0.13 | 87 | 91.24 | 88.21 | 0.139 | 86 | 90.03 | 88.02 | 0.14 | 86 | 91.17 | 90.9 | 0.11 | 89 | 92.02
TABLE III: Validation of synthesized pre-ictal samples of iEEG data using different combinations of synthesized and real data for training and testing on CESP. Synthesized data generated from the DCGAN trained on Epilepsyecosystem dataset. The comparison of results of: i) test & train on real data (TRTR) and ii) train on synthetic data & test on real data (TSTR), shows that the synthetic data has fully captured the correlation between the features of data and the labels. Data | Patient 1 | Patient 2 | Patient 3
---|---|---|---
| Accuracy
---
(%)
| Sensitivity
---
(%)
FPR/h |  | Specificity
---
(%)
| Accuracy
---
(%)
| Sensitivity
---
(%)
FPR/h |  | Specificity
---
(%)
| Accuracy
---
(%)
| Sensitivity
---
(%)
FPR/h |  | Specificity
---
(%)
TRTR | 72.59 | 78.01 | 0.4 | 60 | 73.00 | 82.16 | 0.31 | 69 | 69.81 | 75.00 | 0.22 | 78
TSTR | 70.12 | 76.59 | 0.38 | 62 | 74.21 | 81.46 | 0.25 | 75 | 71.24 | 76.30 | 0.20 | 80
TRTS | 68.98 | 75.30 | 0.39 | 61 | 71.18 | 79.23 | 0.28 | 72 | 70.00 | 72.39 | 0.23 | 77
TSTS | 72.00 | 79.18 | 0.39 | 61 | 72.99 | 82.00 | 0.22 | 78 | 70.98 | 75.11 | 0.21 | 79

(a) Receiver operating characteristics (ROC) curves without data augmentation

(b) Receiver operating characteristics (ROC) curves with data augmentation

Figure 4: Receiver operating characteristics (ROC) curves of ES prediction
performance testing for two datasets with and without augmentation: (a)
without augmenting the datasets (b) with data augmentation using the synthetic
samples from DCGAN. In these curves, each line represents a patient. Above the
black dash line: good prediction performance; above the red dash line; very
good prediction performance (adapted from [29]).

(a)

(b)

(c)

(d)

Figure 5: Comparison of performance of DL models trained on augmented
Epilepsyecosystem dataset using transfer learning.

For the selection of correct generated samples, we trained the one-class SVM
algorithm on the datasets of real EEG signals separately. We considered that
the real data is a positive class and the anomalous data is a negative class.
The algorithm learns the distribution of real EEG data and classifies the
generated data sample in a positive class or negative class. After training
one-class SVM on real data, we tested it for the synthesized samples. We
selected those synthesized samples which belonged to the positive class and
discarded the samples that belonged to the negative class (see Figure 1).

TABLE IV: Comparison of CESP performance trained on the augmented data with
the random predictor.

(a) For CHB-MIT dataset Patient | p-value
---|---
Pat1 | <.001
Pat2 | .029
Pat3 | <.001
Pat5 | <.001
Pat9 | <.001
Pat10 | .001
Pat13 | <.001
Pat14 | .002
Pat18 | <.001
Pat19 | 0.03
Pat20 | <.001
Pat21 | <.001
Pat23 | <.001

(b) For Epilepsyecosystem dataset. Patient | p-value
---|---
Patient1 | .008
Patient2 | <.001
Patient3 | .0035

TABLE V: Comparison of our work with previous works. Year | Authors |  | No. of seizure
---
in original
dataset
Features | Classifier |  | Sensitivity
---
(%)
FPR/h | Augmentation technique
CHB-MIT
2017 |  | Truong et
---
al. [8]
64 | STFT | CNN | 81.2 | 0.16 | Windowing
2020 |  | Zhang et
---
al. [17]
156 |  | Common spatial
---
features
CNN | 92 | 0.12 |  | Division of a data sample into 3
---
smaller segments and generation
of artificial sample from random
concatenation of these segments
2020 | This work | 64 | STFT | CESP | 96 | 0.05 |  | Samples generated form
---
DCGAN
Epilepsyecosystem
2020 |  | Stojanovic
---
et al. [30]
692 |  | Non-negative
---
matrix factorization
SVM | 69 | 0.76 | Synthetic monitory over-sampling
2020 |  | Ramy et
---
al. [31]
1326 |  | Continuous wavelet
---
transform
| Semi-Dilated
---
CNN
89.52 | N/M |  | Division of data into
---
smaller segments
2020 | This work | 1326 | STFT | CESP | 92.87 | 0.15 |  | Samples generated form
---
DCGAN

To further validate the selected samples, we performed the testing and
training of the CESP model for four data combinations. We train and test the
model on real data (TRTR) to check and compare the performance of the model
with generated data. Then, we test the model trained on real data for samples
of synthetic data (TRTS). We also trained the model on synthetic data and
evaluated the performance on the real data (TSTR) to validate the selected
samples of synthetic data. The results of these experiments for the
Epilepsyecosystem and CHB-MIT datasets are provided in Table II and III
respectively. Figure 3 depicts the AUC of ES prediction results for multiple
scenarios of testing and training. For the Epilepsyecosystem data, we achieved
an average sensitivity of 78.39% for TRTR and 77.56% for TSTR. It shows that
the generated samples selected from the one-class SVM are correct. Similar is
the case for the CHB-MIT dataset, we achieved an average of 89.02% sensitivity
for TRTR and 88.21% sensitivity for TSTR.

We experimented with the training of the CESP model on the $5\times$ and
$3\times$ augmented Epilepsyecosystem and the CHB-MIT dataset respectively.
Compared to the results achieved by using unaugmented data, data augmentation
using DCGAN increased the sensitivity $\sim 15\%$ and AUC $\sim 10\%$ for
Epilepsyecosystem dataset. For the CHB-MIT dataset, AUC increased $\sim 6\%$
for augmented data. Figure 4 demonstrates the overall ES prediction
performance for two datasets with and without augmentation. Table IV shows the
statistical comparison of CESP model trained on the augmented data using the
proposed approach of data generation with the chance level predictor. Results
indicate that the performance of CESP model for both datasets is better than
the chance level predictor. We also compared the results of our data
augmentation approach with the previous works in Table V. The comparison shows
a significant increase in the prediction results by augmenting the data with
synthetic samples for both datasets.

With the availability of augmented data, we evaluated the performance of four
widely used DL models: VGG16, VGG19, Inceptionv3, and ResNet50. These models
are used for image classification and weights of these models trained on the
ImageNet dataset are available in Keras. We twice trained these models on
augmented iEEG data with the pre-trained weights as initial weights. First we
trained all models on the augmented data of all patients and then we fine-
tuned the models in a patient-specific manner. Figure 5 depicts the results of
seizure prediction with the models trained on Epilepsyecosystem augmented
data. The performance of Inceptionv3 and ResNet50 is considerable as compared
to the VGG16 and VGG19. VGG16 and VGG19 have more trainable parameters and
required more training time. VGG16 and VGG19 overfit to the training data
after some time of training which leads to the poor testing performance.
However, the performance of other two models is good enough to use and explore
the idea of training these models on adequate amount of data for seizure
prediction in future.

Table VI shows the comparison of AUCs of TL models with the CESP. p-values for
the VGG16, VGG19, and ResNet50 indicate a significant difference between the
performance and AUC curves of prediction models. The performance of
Inceptionv3 is best among the TL algorithms and approximately equal to the
performance of the CESP model. However, the advantage of the CESP model is the
low computational cost and complexity.

TABLE VI: Statistical comparison (p-values) of prediction models. p-values are
derived from the single-tailed Hanley-McNeil test for comparing AUCs.

Patients |  | CESP
---
VS
VGG16
| CESP
---
VS
VGG19
| CESP
---
VS
Inceptionv3
| CESP
---
VS
ResNet50
Patient1 | <.000001* | <.000001* | .046 | .000021*
Patient2 | <.000001* | <.000001* | .085 | .005*
Patient3 | <.000001* | <.000001* | .059 | .0025*

* indicates significant p-values after penalizing for correction for multiple comparisons using Bonferroni for each patient at the level $\alpha$ = 0.05/4.

##  V Discussion

This work aimed to address the scarcity problem of good quality EEG data for
ES prediction. With the advancement of DL techniques, high-quality artificial
data generation is now possible. Deep generative models trained in an
adversarial manner can simulate complex data distributions. In this paper, we
presented a DL based generated model (DCGAN) that can generate the artificial
EEG samples of patients. After measuring the quality of data using a
traditional one-class SVM model and four different tests and training
experiments, we augmented the real data with synthetic pre-ictal samples. We
then trained CESP model on the augmented data and compared the performance of
the model for ES prediction with previous works that used traditional
augmentation techniques, i.e., SMOTE, moving windows, and data sampling. The
comparison shows that the prediction performance using synthetic pre-ictal
samples increased for both datasets. Figure 4 demonstrated that the ES
prediction performance for all patients of both datasets increased than a
chance level predictor by using the augmented data.

In contrary to previously used augmentation techniques, our technique
generates artificial samples of data, which is a solution to medical data
sharing problems. Besides data acquisition difficulties, medical data sharing
comes with the privacy-preserving issues. Researchers and hospitals cannot use
the data without the permission of patients and ethical approval [32]. The
synthetic data is not only used to augment the real data for performance
improvement but also can be shared with researchers without privacy issues.

Previous works [30], [31] using synthetic monitoring over-sampling (SMOTE) and
data division into smaller segments techniques for augmentation achieved the
sensitivity of 69% and 89.52% for the Epilepsyecosystem dataset. Authors in
[30] achieved the results by extracting 12 non-negative matrix features and
used 692 original pre-ictal samples while we trained DCGAN on the 1362
original samples without any feature extraction. The generalization of the
prediction technique is a major pitfall of many research works. The
generalization of the model requires the prediction performance for different
datasets. The work in [17] divided the original pre-ictal samples into 3
smaller samples and concatenated these samples with random selection to
generate new data samples that belonged to the distribution of real data.
Their augmentation technique provided significant results. However, they
tested the augmentation methods for only one dataset which had a small number
of seizures per patient. They also employed feature engineering techniques on
the data before feeding the data to CNN. Our data generation is more
generalized and applicable to both iEEG and scalp EEG data.

To date, researchers and developers are applying traditional ML and DL
techniques for ES prediction. However, with the availability of computational
resources and an ample amount of data, TL is an emerging technique to
implement for problem-solving. In this paper, we presented the use of famous
DL models, i.e., VGG16, VGG19, Inceptionv3, and ResNet50, for the first time
to predict seizure. With the availability of augmented data, the experiments
performed on these models showed significant results. More precisely, the
performance results of Inceptionv3 and ResNet50 were accurate enough to use
these pre-trained models for future works, e.g., use the model for new
patients with fine tunning, use the model for prediction of other diseases
with EEG signals, or use these pre-trained models for extracting significant
features of EEG data and make predictions based on extracted features. Besides
the promising results of Inceptionv3 and ResNet50, due to the computational
complexities of these models, a clinically implantable device will not be an
appropriate idea for these models. CESP model has less computational cost as
compared to these two models. So, another future direction of TL work can be
the reduction of complexities of these models while preserving the performance
efficiency, i.e., we can utilize the features extracted from selected layers
of models and predict seizures based on these features.

For the purpose of simplicity and ease of comparison, the research community
stated the ES prediction problem as a binary classification problem. We have
performed all the experiments with the binary classification assumption.
However, for the clinical application of these experiments, the formulation of
the problem will be complex because seizure highly depends on patient-specific
characteristics, i.e., seizure type, age and gender of the patient, and the
medication that the patient was taking during data acquisition. For evaluating
the prediction models reliably, we need continuous seizure data because the
pitfall of AUC performance metric is that it is typically calculated for the
balanced dataset (same number of pre-ictal and interictal samples). However,
the actual data contained more inter-ictal periods as compared to pre-ictal
events.

Unavailability of annotated data, privacy-preserving issues, and the ethical
problems regarding private data sharing come with the promising results of ML
and DL models. Artificial data generation is one solution to these problems.
However, the time-series, i.e., EEG data for seizure prediction, contains
information that appears many hours ago from the seizure event but is as
useful as the information appear one minute before the seizure. That is why
the generation of continuous data is one of the future extension of our work.
With the significant results of synthesized data samples for seizure
prediction, the generation of artificial patient’s data is also possible. In
this way, researchers can work with an ample amount of data of various
patients to address the generalization problem.

##  VI Conclusions

The main aim of ES predictions research is to provide an accurate seizure
warning system to patients to take precautionary measures ahead of seizure
onset. However, such a solution is not yet available due to scarcity of
suitable amount of seizure EEG data. In this paper, we proposed a deep
convolutional generative adversarial network (DCGAN) model to overcome the
hurdle of the unavailability of an extensive amount of EEG data. The proposed
DCGAN model showed good generalization for the generation of both iEEG and
scalp EEG data. Moreover, a convolutional epileptic seizure predictor (CESP),
was proposed to validate the synthetic data, is also generalized to work with
both types of EEG data. To measure the quality of synthetic data, we employed
one-class SVM and training and testing of the CESP model with four
combinations of real and synthetic data. The CESP model produced sensitivity
of 78.11%, 88.21% and FPR/h of 0.27, 0.14 for training on synthesized and
testing on real Epilepsyecosystem and CHB-MIT datasets respectively. These
results are higher than the training and testing of the CESP model on real
data. This shows that the synthetic samples fully captured the relationship
between the features of data and the labels of pre-ictal samples. We also
evaluated the performance of CESP, VGG16, VGG19, Inceptionv3, and ResNet50 on
the augmented dataset using the concept of transfer learning (TL). Using the
TL on augmented data, we showed that the Inceptionv3 performed very well with
highest accuracy of 90.03% and 89.50% sensitivity. With these significant
results, using TL, we can further explore this novel idea of employing TL
techniques for ES prediction in the future work.

##  VII Acknowledgments

Adeel Razi is funded by the Australian Research Council (Refs: DE170100128 and
DP200100757) and Australian National Health and Medical Research Council
Investigator Grant (Ref: 1194910).

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