# Test of the nonrelativistic $c\bar{c}$ potential

Ulugbek Yakhshiev<EMAIL_ADDRESS>Department of Physics, Inha
University, Incheon 22212, Korea  Theoretical Physics Department, National
University of Uzbekistan, Tashkent 700174, Uzbekistan

###### Abstract

We analyze the charmonium states by testing a phenomenological nonrelativistic
potential and propose a new set of parameters. This new set of parameters are
fixed using only the lowest lying S-wave states of charmonia where the spin-
orbit and tensor interactions will not contribute. After fitting the
parameters we analyze the whole fine structure of charmonium states taking
into account the spin-orbit and tensor interactions too. Calculations showed
that the nonrelativistic potential model with the phenomenologically defined
parameters is indeed well approximation for describing the charmonium states.

Heavy-quark potential, charmonium.

###### pacs:

12.38.Lg, 12.39.Pn, 14.40.Pq

##  I Introduction

An applicability of any nonrelativistic potential model during the studies of
heavy hadrons spectra can be well checked by reproducing the heavy quarkonium
states. The best candidate for this role is the charmonium consisting one
heavy quark $Q$ and one heavy antiquark $\bar{Q}$ Aubert:1974js ;
Augustin:1974xw . Analyzing the charmonium spectra one can establish some
interesting features. For that purpose one keeps in mind some facts in
formulating the better approach. First of all, one can note the large value of
the current quark mass in the characteristic energy scale
$m_{Q}\gg\Lambda_{\rm QCD}$. The second, the smallness of heavy-quark velocity
$v_{Q}\ll c$ inside the charmonia which can be crudely estimated from the
radial excitation energy differences corresponding to the given quantum
numbers. These two facts indicate that the relativistic effects could be taken
into account as the systematic corrections (for example, see review
Voloshin:2007dx ).

From other side, one can also note the smallness of nonperturbative effects in
the spectra of charmonia. The authors of Ref. Diakonov:1989un  estimated
contribution to the spin-independent heavy quark potential due to the
nonperturbative dynamics in the framework of instanton vacuum model of quantum
chromodynamics (QCD) Diakonov:1985eg ; Diakonov:2002fq . They also found the
relatively small value of difference between the current and constituent quark
masses in comparison with the constituent or current quark mass
Diakonov:1989un . The recent calculations Yakhshiev:2018juj ;
Musakhanov:2020hvk , based on the estimations above, showed that the
contributions due to the nonperturbative dynamics also can be considered as
the small corrections. In particular, the authors of Ref. Musakhanov:2020hvk
showed that the instanton effects are the first order perturbative
corrections. Nevertheless, in the Ref. Yakhshiev:2018juj  it was discussed
that the instantons may shed some light on the origin of parameters of the
potentials used in the phenomenological approaches Eichten:1978tg ;
Eichten:1979ms . The studies performed in Refs. Diakonov:1989un ;
Yakhshiev:2018juj  may lead to the conclusion that, although it is very
nontrivial at low energies, at the high energies the nonperturbative dynamics
seems to be tightly hidden behind the confinement mechanism which is not yet
fully understood.

All said above more or less explains the success of phenomenological potential
model Barnes:2005pb  where the complicated and unknown dynamics is expressed
in terms of the effective values of phenomenological parameters. Therefore,
one has readily a nonrelativistic Schrödinger approach for describing the
energy spectra of heavy quarkonium. Technically, the quarkonium is very
similar to the Hydrogen atom, i.e. one can solve the one body problem in the
given external potential field instead of considering the two body
relativistic system. The difference from the Hydrogen atom problem is only due
to the nature of interactions and its corresponding range. Consequently, due
to the strong nature of interactions the excitation energies of charmonia will
be much lager than the electron excitation energies in the Hydrogen atom. The
size of a charmonium also will be much less in comparison with the Hydrogen
atom size due to the short range nature of interactions. One can use these
obvious facts in applying a numerical method to the charmonium problem and
easily find the appropriate variational parameters of the model.

Starting the discussions of the heavy quark potential one can note that the
basic spin-independent central interaction between the quark and antiquark can
be well separated into two parts. The first scalar exchange part is fully
phenomenological because of an unknown confinement mechanism. The most popular
choice for this interaction is expressed as a linearly increasing potential
due to the area law of Wilson loop Wilson:1974sk  for the heavy-quark
potential. The second, the vector exchange part is due to the perturbative one
gluon exchange mechanism at short distances and in the lowest order has the
Coulomb interaction like form with the corresponding running coupling
constant. The spin-dependent parts of interactions can be reproduced from the
central potential in the framework of nonrelativistic expansions
Eichten:1980mw . The corresponding model is called a nonrelativistic
constituent quark model.

In the present work, we discuss the interesting features of the
nonrelativistic constituent quark model and propose the new set of parameters
for describing the charmonium states on a basis of updated experimental data
Tanabashi:2018oca . While we are doing that, as an input we concentrate only
to the minimal part of spectrum instead of considering the whole spectrum.
After fitting the parameters in a most compact way we concentrate to the whole
spectrum and analyze the applicability of nonrelativistic potential model
approach.

The paper is organized in the following way. In the next section II, we
briefly repeat the main features of the model and describe very shortly a
variational approach to the problem. In the section III, the results from
calculations will be presented and discussed. In the last section IV, we
summarize our results and make the corresponding conclusions.

##  II $Q\bar{Q}$ potential and variational method

In the simple constituent quark model, the total $Q\bar{Q}$ potential has the
following standard form

$\displaystyle V_{Q\bar{Q}}(\bm{r})$
$\displaystyle=V_{C}(r)+V_{SS}(r)(\bm{S}_{Q}\\!\cdot\\!\bm{S}_{\bar{Q}})$ (1)
$\displaystyle+V_{LS}(r)(\bm{L}\cdot\bm{S})$ (2)
$\displaystyle+V_{T}(r)\left[3(\bm{S}_{Q}\\!\cdot\\!\bm{n})(\bm{S}_{\bar{Q}}\\!\cdot\\!\bm{n})-\bm{S}_{Q}\cdot\bm{S}_{\bar{Q}}\right],$
(3)

where ${\bm{S}}_{Q}$ (${\bm{S}}_{\bar{Q}}$) spin of the quark (antiquark),
${\bm{L}}$ is relative orbital momentum,
${\bm{S}}={\bm{S}}_{Q}+{\bm{S}}_{\bar{Q}}$ is total spin of the quarkonium
system. We work in the center of mass frame and, therefore, a radius vector
${\bm{r}}$ is given in terms of the relative coordinates
${\bm{r}}={\bm{r}}_{Q}-{\bm{r}}_{\bar{Q}}$ and ${\bm{n}}={\bm{r}}/r$ defines
the unit vector in direction of the radius vector. In Eq. (3) $V_{C}(r)$,
$V_{SS}(r)$, $V_{LS}(r)$ and $V_{T}(r)$ are central, spin-spin, spin-orbit and
tensor potentials depending on the relative distance between the quark and
antiquarks.

The central part of the potential in a nonrelativistic reduction employs the
following “Coulomb+linear” form

$V_{C}(r)=\kappa r-\frac{4\alpha_{s}}{3r},$ (4)

where $\kappa$ is parameter of string tension and

$\displaystyle\alpha_{\mathrm{s}}(\mu)=\frac{1}{\beta_{0}\ln(\mu^{2}/\Lambda_{\mathrm{QCD}}^{2})}$
(5)

is the strong running coupling constant at the one-loop level. Its value is
determined from the characteristic energy scale $\mu$ corresponding to the
problem. Further, $\beta_{0}=(33-2N_{f})/(12\pi)$ is the beta function at the
one-loop level and $\Lambda_{\mathrm{QCD}}$ is the dimensional transmutation
parameter. The nonrelativistic expansion of $Q\bar{Q}$ interactions allows to
relate the spin-dependent parts of the potential to the central part
Eichten:1980mw . So, the spin-dependent interactions corresponding to the
“vector one-gluon-exchange+scalar confinement” are given as

$\displaystyle V_{SS}^{\rm(P)}(r)$ $\displaystyle=$
$\displaystyle\frac{32\pi\alpha_{s}}{9m_{Q}^{2}}\delta_{\sigma}(r),$ (6)
$\displaystyle V_{LS}^{\rm(P)}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{Q}^{2}}\left(\frac{4\alpha_{s}}{r^{3}}-\frac{\kappa}{r}\right),$
(7) $\displaystyle V_{T}^{\rm(P)}(r)$ $\displaystyle=$
$\displaystyle\frac{4\alpha_{s}}{m_{Q}^{2}r^{3}},$ (8)

where $m_{Q}$ is heavy quark mass. In practical calculations, the pointlike
spin-spin interaction in Eq. (6) is “smeared” by using an exponential function
of the form

$\delta_{\sigma}(r)=\left(\frac{\sigma}{\sqrt{\pi}}\right)^{3}e^{-\sigma^{2}r^{2}},$
(9)

where $\sigma$ is smearing parameter. In such a way $Q\bar{Q}$ potential is
described in terms of only four parameters, $\kappa$, $\alpha_{s}$, $m_{Q}$
and $\sigma$. Usually, these parameters are found by fitting the whole
charmonium spectrum. In the present work we will follow the phenomenological
approach but find those parameters by fitting only some minimal part of S-wave
spectrum instead of considering the whole spectrum.

After fitting the form of potential, in order to evaluate the energy states of
quarkonia in a nonrelativistic potential approach, one needs to solve the
Schrödinger equation

$\displaystyle(\hat{H}-E)|\Psi_{JJ_{3}}\rangle=0.$ (10)

Here $\hat{H}$ is Hamilton operator and $|\Psi_{JJ_{3}}\rangle$ represents the
state vector with the total angular momentum $J$ and its third component
$J_{3}$. The coordinate space projection of the state vector
$\langle\bm{r}|\Psi_{JJ_{3}}\rangle$ will reproduce the coordinate space
representation of the Hamiltonian

$\displaystyle\hat{H}(\bm{r})=-\frac{\hbar^{2}}{m_{Q}}\nabla^{2}+V_{Q\bar{Q}}(\bm{r}),$
(11)

where $m_{Q}$ arises from the doubled reduced mass of the quarkonium system.
The matrix elements of $Q\bar{Q}$ potential in the standard basis
$|{}^{2S+1}L_{J}\rangle$, which is given in terms of the total spin $S$, the
orbital angular momentum $L$, and the total angular momentum $J$ satisfying
the relation $\bm{J}=\bm{L}+\bm{S}$, has the following form

$\displaystyle V_{Q\bar{Q}}(r)$
$\displaystyle=\langle{}^{2S+1}L_{J}|V_{Q\bar{Q}}(\bm{r})|{}^{2S+1}L_{J}\rangle$
(12) $\displaystyle=V(r)+\left[\frac{1}{2}S(S+1)-\frac{3}{4}\right]V_{SS}(r)$
(13) $\displaystyle+\langle{\bm{L}}\cdot{\bm{S}}\rangle V_{LS}(r)\
+\left\\{-\frac{\langle{\bm{L}}\cdot{\bm{S}}\rangle\left(\langle{\bm{L}}\cdot{\bm{S}}\rangle+2\right)}{(2L-1)(2L+3)}\right.$
(14)
$\displaystyle\left.+\frac{S(S+1)L(L+1)}{3(2L-1)(2L+3)}\right\\}V_{T}(r),$
(15)

where $\langle{\bm{L}}\cdot{\bm{S}}\rangle$ is defined as

$\langle{\bm{L}}\cdot{\bm{S}}\rangle=\frac{1}{2}\left[J(J+1)-L(L+1)-S(S+1)\right].$

The corresponding radial part of the wave function for a given orbital
momentum $L$ is a solution of the Schrödinger equation

$\left(-\frac{\hbar^{2}}{m_{Q}}\nabla^{2}+V_{Q\bar{Q}}(r)-E\right)\psi_{LL_{3}}(\bm{r})=0,$
(16)

where an angular part of the wave function $\psi_{LL_{3}}(\bm{r})$ is
represented in terms of the standard spherical harmonics
$Y_{LL_{3}}(\hat{\bm{r}})$. In order to solve Eq. (16) numerically, we will
follow the gaussian expansion method (for details, see review Hiyama:2003cu ),
where the state vector $|\psi_{LL_{3}}\rangle$ is expanded in terms of a set
of basis vectors $\\{|\phi_{nLL_{3}}\rangle;\,n=1,2,\dots,n_{\mathrm{max}}\\}$
as

$\displaystyle|\psi_{LL_{3}}\rangle=\sum_{n=1}^{n_{\rm
max}}C_{n}^{(L)}|\phi_{nLL_{3}}\rangle.$ (17)

Here $n$ is a radial quantum number. So, the radial excitations corresponding
to the given angular momentum value will be reproduced naturally. In the
gaussian expansion method, the radial part $\phi^{G}_{nL}(r)$ of the total
eigenfunction in the spherical coordinate basis

$\phi_{nLL_{3}}(\bm{r})=\phi^{G}_{nL}(r)Y_{LL_{3}}(\hat{\bm{r}})$ (18)

is expressed in terms of gaussian trial functions

$\displaystyle\phi^{G}_{nL}(r)=\left(\frac{2^{2L+{7}/{2}}r_{n}^{-2L-3}}{\sqrt{\pi}(2L+1)!!}\right)^{1/2}r^{L}e^{-(r/r_{n})^{2}}.$
(19)

For the given set $n$ runs the values $n=1,2,\dots,n_{\rm max}$ and the
corresponding $r_{n}$’s are playing the role of variational parameters. The
variational parameters could be optimized using a geometric progression
Hiyama:2003cu

$r_{n}=r_{1}a^{n-1},\quad n=1,2,\dots,n_{\rm max}$ (20)

and, therefore, the actual number of parameters is reduced to the three (e.g.
$r_{1}$, $r_{n_{\rm max}}$ and $n_{\rm max}$) for the given values of the
orbital quantum number $L$, the spin $S$ and the total angular momentum $J$.

The expansion coefficients $C_{n}^{(L)}$ in Eq. (17) and the eigenenergies
$E_{n}^{(L)}$ are determined by employing Rayleigh-Ritz variational principle.
This leads to a generalized matrix eigenvalue problem

$\displaystyle\sum_{n=1}^{n_{\rm
max}}\left(K_{mn}^{(L)}+V_{mn}^{(L)}-E_{n}^{(L)}N_{mn}^{(L)}\right)$
$\displaystyle C_{n}^{(L)}=0,$ (21) $\displaystyle m=1,2,\dots,n_{\rm max},$

where the corresponding matrix elements are defined in the following way

$\displaystyle K_{mn}^{(L)}$
$\displaystyle=\langle\phi_{mLL_{3}}\left|\frac{\hat{p}^{2}}{m_{Q}}\right|\phi_{nLL_{3}}\rangle,$
(22) $\displaystyle V_{mn}^{(L)}$
$\displaystyle=\langle\phi_{mLL_{3}}|V_{Q\bar{Q}}|\phi_{nLL_{3}}\rangle,$ (23)
$\displaystyle N_{mn}^{(L)}$
$\displaystyle=\langle\phi_{mLL_{3}}|\phi_{nLL_{3}}\rangle.$ (24)

##  III Results and discussions

As we mentioned above in the phenomenological approaches the parameters of the
model, $\kappa$, $\alpha_{s}$, $\sigma$ and $m_{c}$, are fitted to the spectra
of experimentally known charmonium states. For example, the authors of Ref.
Barnes:2005pb  proposed the set of potential parameters given in Table 1 (see
the model referred as NR).

Table 1: Parameters of the nonrelativistic potential models. NR corresponds to the potential model in Ref. Barnes:2005pb  where eleven charmonium states are used as an input, NR4 describes the present work with the potential parameters corresponding to the four charmonium states as an input, respectively.  The | $m_{c}$ | $\alpha_{s}$ | $\kappa$ | $\sigma$
---|---|---|---|---
model | [GeV] | [GeV] | [GeV2] | [GeV]
NR | 1.4794 | 0.5461 | 0.1425 | 1.0946
NR4 | 1.4796 | 0.5426 | 0.1444 | 1.1510

Using that set of parameters they calculated all allowed E1 radiative partial
width and some important M1 width. As an input for the fitting of parameters
they used 11 meson states: 6 states corresponding to the S-wave, 3 states
corresponding to the P-wave and 2 states corresponding to the D-wave,
respectively. The input values of these energy states are given in Table 2
(see the 2nd column). The results of their calculations showed that the
nonrelativistic potential model with the certain set of parameters describes
the charmonium spectrum very well.

Table 2: Experimental and calculated spectrum of $c\bar{c}$ states. All energy states are given in MeV and the output results are rounded up to 1 MeV. Authors of NR model in Ref. Barnes:2005pb  used 11 input states and their values are shown in the second column. In the present work in order to reproduce NR4 results only 4 states are used as an input, and their values are shown in the fourth column.  State | Ref. Barnes:2005pb  | This work | Exp. Tanabashi:2018oca
---|---|---|---
| Input | NR | Input | NR4 |
$J/\psi(1^{3}{\rm S}_{1})$ | 3097 | 3090 | 3097 | 3098 |  $3096.900\pm 0.006$
$\eta_{c}(1^{1}{\rm S}_{0})$ | 2979 | 2982 | 2984 | 2984 | $2983.9\pm 0.5$
$\psi(2^{3}{\rm S}_{1})$ | 3686 | 3672 | 3686 | 3682 |  $3686.097\pm 0.025$
$\eta_{c}(2^{1}{\rm S}_{0})$ | 3638 | 3630 | 3638 | 3638 | $3637.6\pm 1.2$
$\psi(3^{3}{\rm S}_{1})$ | 4040 | 4072 |  | 4084 |  $4039\pm 1$
$\eta_{c}(3^{1}{\rm S}_{0})$ |  | 4043 |  | 4055 |
$\psi(4^{3}{\rm S}_{1})$ | 4415 | 4406 |  | 4422 |  $4421\pm 4$
$\eta_{c}(4^{1}{\rm S}_{0})$ |  | 4384 |  | 4397 |
$\chi_{c2}(1^{3}{\rm P}_{2})$ | 3556 | 3556 |  | 3559 |  $3556.17\pm 0.07$
$\chi_{c1}(1^{3}{\rm P}_{1})$ | 3511 | 3505 |  | 3505 | $3510.67\pm 0.05$
$\chi_{c0}(1^{3}{\rm P}_{0})$ | 3415 | 3424 |  | 3415 | $3414.71\pm 0.30$
$h_{c}(1^{1}{\rm P}_{1})$ |  | 3516 |  | 3524 | $3525.38\pm 0.11$
$\chi_{c2}(2^{3}{\rm P}_{2})$ |  | 3972 |  | 3978 |  $3927.2\pm 2.6$
$\chi_{c1}(2^{3}{\rm P}_{1})$ |  | 3925 |  | 3937 |
$\chi_{c0}(2^{3}{\rm P}_{0})$ |  | 3852 |  | 3864 | $3862^{+26+40}_{-32-13}$
$h_{c}(2^{1}{\rm P}_{1})$ |  | 3934 |  | 3945 |
$\chi_{c2}(3^{3}{\rm P}_{2})$ |  | 4317 |  | 4325 |
$\chi_{c1}(3^{3}{\rm P}_{1})$ |  | 4271 |  | 4293 |
$\chi_{c0}(3^{3}{\rm P}_{0})$ |  | 4202 |  | 4227 |
$h_{c}(3^{1}{\rm P}_{1})$ |  | 4279 |  | 4293 |
$\psi_{3}(1^{3}{\rm D}_{3})$ |  | 3806 |  | 3816 |
$\psi_{2}(1^{3}{\rm D}_{2})$ |  | 3800 |  | 3807 | $3822.2\pm 1.2$
$\psi(1^{3}{\rm D}_{1})$ | 3770 | 3785 |  | 3794 | $3778.1\pm 1.2$
$\eta_{c2}(1^{1}{\rm D}_{2})$ |  | 3799 |  | 3809 |
$\psi_{3}(2^{3}{\rm D}_{3})$ |  | 4167 |  | 4179 |
$\psi_{2}(2^{3}{\rm D}_{2})$ |  | 4158 |  | 4167 |
$\psi(2^{3}{\rm D}_{1})$ | 4159 | 4142 |  | 4153 | $4191\pm 5$
$\eta_{c2}(2^{1}{\rm D}_{2})$ |  | 4158 |  | 4170 |
$\chi_{4}(1^{3}{\rm F}_{4})$ |  | 4021 |  | 4033 |
$\chi_{3}(1^{3}{\rm F}_{3})$ |  | 4029 |  | 4039 |
$\chi_{2}(1^{3}{\rm F}_{2})$ |  | 4029 |  | 4041 |
$h_{c3}(1^{1}{\rm F}_{3})$ |  | 4026 |  | 4037 |
$\chi_{4}(2^{3}{\rm F}_{4})$ |  | 4348 |  | 4362 |
$\chi_{3}(2^{3}{\rm F}_{3})$ |  | 4352 |  | 4365 |
$\chi_{2}(2^{3}{\rm F}_{2})$ |  | 4351 |  | 4365 |
$h_{c3}(2^{1}{\rm F}_{3})$ |  | 4350 |  | 4364 |

However, nowadays the experimental data is improved and some new states where
fixed in the particle data Tanabashi:2018oca . The values of charmonium states
extracted from the current experimental data are given in the last column of
the Table 2. The natural question arises – “How the parameters of the
nonrelativistic potential model will change if one concentrates to the updated
experimental data?” Partially, our aim in the present work is to answer this
question. However, our main aim in the present work is not only fitting the
updated experimental data by means of the new set of parameters. In addition
to the fitting process we want to check “How well does a nonrelativistic
expansion work in the potential approaches?”

As we said above the authors of Ref. Barnes:2005pb  fitted the parameters of
the potential to the all eleven, experimentally known at that time, states of
the charmonium spectrum. Therefore, a beauty of nonrelativistic expansion
seems remained to be hidden behind. We want to emphasize that, in principle,
one can concentrate to the part of spectrum in order to fit the parameters of
model. For example, one can concentrate to the S-wave part of spectrum for
fitting the parameters of model. During this process the spin-orbit and tensor
interactions are not contributing to the total interaction. One can also act
in opposite form by concentrating to the part of spectra where the spin-orbit
and tensor interactions are important. The reason for the possibility of such
choices is due to the fact that the central and spin dependent parts of the
potential are related to each-other in the nonrelativistic expansion and
described by the same set of parameters. In an ideal case, only four input
states are enough to fit _four_ “arbitrary” parameters of the potential model.

Consequently, as a possible test of nonrelativistic expansion, in the present
work we consider the “ideal case” and fit the parameters of model according to
some part of S-wave charmonia. In such a way we ignore the spin-orbit and
tensor interactions during the fitting process. More specifically, we propose
a potential model where the parameters are fitted using the four lowest S-wave
spin 0 and spin 1 states. On top of that we will fit the lowest spin zero
$1^{1}{\rm S}_{0}$ and $2^{1}{\rm S}_{0}$ states exactly. The parameters of
the corresponding interaction potential are also given in Table 1 (see the
model referred as NR4) and the values of corresponding input states are given
in Table 2, respectively (see the 4th column).

In order to keep a good accuracy of numerical calculations, during the fitting
process we used a basis set with 30 to 50 gaussian functions corresponding to
the given values of $L$, $S$ and $J$. So, the value of the first parameter
$n_{\rm max}$ from the three variational parameters is free input and equals
to the definite integer number belonging to the interval $n_{\rm
max}\in[30,50]$. The values of other two corresponding variational parameters,
$r_{1}$ and $r_{n_{\rm max}}$, are found by minimizing not only the ground
state energy $E_{1}$ but also minimizing simultaneously the lowest 10 to 20
radial excitation energies $\sum_{i=n}^{n_{\rm min}}E_{n}$ (i.e. $n_{\rm
min}\in[5,20]$) from the possible 30 to 50 energy states. The convergence of
the results are checked by increasing the number of total radial excitations
starting from $n_{\rm max}\sim 10$ with $n_{\rm min}\sim 5$ to the above
mentioned final values, $n_{\rm max}\in[30,50]$ and $n_{\rm min}\in[10,20]$.

Now let us discuss our results. Analyzing the S-wave results of NR4 model one
can note that the values of the first two spin 0 and the first two spin 1
S-wave states are obviously reproduced very well. Naturally, these energy
states are input and reproduced better in comparison with the results of NR
model. Quick look for the calculated $3^{3}{\rm S}_{1}$-state energy value for
NR4 and comparing it with the corresponding experimental value shows the large
difference, around 45 MeV. Nevertheless, this problem seams to be unavoidable
if one fits the parameters to the whole spectrum, e.g. compare the
corresponding NR result where the difference from the experimental value is
around 33 MeV. However, the next excited $4^{3}{\rm S}_{1}$-state energy for
NR4 is reproduced at almost its averaged experimental value while NR model
gives the relatively different result. One can conclude that, in general,
S-wave states are reproduced better in NR4 model in comparison with NR model.

The power of nonrelativistic expansion becomes more obvious when we include
the spin-orbit and tensor interactions for the analysis of whole spectrum.
While the input parameters are already fixed we do not need to play with them
anymore. Therefore, the calculations of $L\geq 1$ states are straightforward
and does not require any fitting process.

The beauty of the nonrelativistic expansion is realized when we analyze the
P-wave states. One can see that, although we are not fitting them, among the
six experimentally known states three of them $1^{3}{\rm P}_{0}$, $1^{1}{\rm
P}_{1}$ and $2^{3}{\rm P}_{0}$ are reproduced at their experimental value for
NR4 model. Two states, $1^{3}{\rm P}_{2}$ and $1^{3}{\rm P}_{1}$, among the
remaining three P-wave states in the table are also reproduced quite well with
the differences 3 MeV and 6 MeV from the experiment, respectively. Only one
state $2^{3}{\rm P}_{2}$ is far from its experimental value, the difference is
51 MeV. For comparison, the general fit using NR model reproduces only one
state $1^{3}{\rm P}_{2}$ at its experimental value. Another state $1^{3}{\rm
P}_{1}$ is same as in the case of NR4 and three ($1^{3}{\rm P}_{0}$,
$1^{1}{\rm P}_{1}$ and $2^{3}{\rm P}_{0}$) from the remaining four states are
reproduced approximately with 10 MeV differences from the experimental values,
respectively. The last state $2^{3}{\rm P}_{2}$ is very far from the
experimental value and difference 47 MeV is almost same as NR4 case. One could
also conclude, that the concentration, respectively, to S-wave (e.g.
$1^{1}{\rm S}_{0}$ and $2^{1}{\rm S}_{0}$) and P-wave (e.g. $1^{3}{\rm P}_{0}$
and $1^{1}{\rm P}_{1}$) states as an input will lead to more or less similar
values of the potential parameters in comparison with the values in Table 2.
Summarizing analysis of P-wave states one can conclude that NR4 model much
better reproduces the experimental data in comparison with NR model.

It is also interesting to analyze more higher energy states corresponding to
NR and NR4 models and compare them with the available experimental data.
Coming to the D-wave states, one can note that $1^{3}{\rm D}_{1}$-state energy
value is reproduced relatively better in NR model. However, the situation
becomes opposite if we analyze $1^{3}{\rm D}_{2}$-state energy. Here NR4 model
gives relatively better result in comparison with NR model. Finally, an
experimentally available the highest energy state $2^{3}{\rm D}_{1}$ is again
reproduced relatively better in NR4 model. Again and in general, D-wave states
are also better reproduced in NR4 model.

Consequently, one can conclude that, although the number of input parameters
in NR4 model are chosen in a maximum compact form it gives better result than
the NR model. From the Table 2 one can also make a general conclusion that the
fitting to the S-wave part of the spectrum is completely satisfactory. In a
such way we see that the nonrelativistic expansion in the potential models for
describing the fine structure of charmonium states indeed works pretty well.

##  IV Summary

In the present work we aimed at testing the nonrelativistic potential model
and reparametrization of the nonrelatistic potential based on the currently
available experimental data. In particular, we investigated the applicability
of nonrelativistic expansion in the potential approaches to the charmonium
spectrum. For that purpose we concentrated only on the four lowest S-wave
states of charmonium spectrum. By doing that we demonstrated, that the
concentration to the minimal part of charmonium spectrum is enough during the
fitting process of the values of potential parameters. The model quite
satisfactorily described the whole spectrum of charmonia. From our studies,
one can make a general conclusion that the nonrelativistic potential approach
is indeed good approximation in describing the spectrum including the fine-
structure of charmonium states.

## Acknowledgments

This work is supported by the Basic Science Research Program through the
National Research Foundation (NRF) of Korea funded by the Korean government
(Ministry of Education, Science and Technology, MEST), Grant Number
2020R1F1A1067876.

## References

  * (1) J. J. Aubert et al. [E598 Collaboration], Phys. Rev. Lett. 33, 1404 (1974).
  * (2) J. E. Augustin et al. [SLAC-SP-017 Collaboration], Phys. Rev. Lett. 33, 1406 (1974).
  * (3) M. B. Voloshin, Prog. Part. Nucl. Phys. 61, 455 (2008)
  * (4) D. Diakonov, V. Y. Petrov and P. V. Pobylitsa, Phys. Lett. B 226, 372 (1989).
  * (5) D. Diakonov and V. Y. Petrov, Nucl. Phys. B 272, 457 (1986).
  * (6) D. Diakonov, Prog. Part. Nucl. Phys. 51, 173 (2003)
  * (7) U. T. Yakhshiev, H. C. Kim and E. Hiyama, Phys. Rev. D 98, 114036 (2018).
  * (8) M. Musakhanov, N. Rakhimov and U. T. Yakhshiev, Phys. Rev. D 102, 076022 (2020).
  * (9) E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Phys. Rev. D 17, 3090 (1978) Erratum: [Phys. Rev. D 21, 313 (1980)].
  * (10) E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Phys. Rev. D 21, 203 (1980).
  * (11) T. Barnes, S. Godfrey and E. S. Swanson, Phys. Rev. D 72, 054026 (2005).
  * (12) K. G. Wilson, Phys. Rev. D 10, 2445 (1974).
  * (13) E. Eichten and F. Feinberg, Phys. Rev. D 23, 2724 (1981).
  * (14) E. Hiyama, Y. Kino and M. Kamimura, Prog. Part. Nucl. Phys. 51 (2003) 223.
  * (15) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, 030001 (2018).


# Using Machine Learning for Anomaly Detection on a System-on-Chip under Gamma
Radiation

Eduardo Weber Wächter, Server Kasap, Şefki Kolozali, Xiaojun Zhai, Shoaib
Ehsan, and Klaus McDonald-Maier, E. Wachter is with the School of Computer
Science and Electronic Engineering, University of Essex, Colchester, U.K. and
also with the School of Engineering, University of Warwick, Coventry, UKS.
Kasap is with the School of Computing, Electronics and Mathematics, Coventry
University, Coventry, UKŞ. Kolozali, X. Zhai, S. Ehsan and K. McDonald-Maier
are with the School of Computer Science and Electronic Engineering, University
of Essex, Colchester, U.K.

###### Abstract

The emergence of new nanoscale technologies has imposed significant challenges
to designing reliable electronic systems in radiation environments. A few
types of radiation like Total Ionizing Dose (TID) effects often cause
permanent damages on such nanoscale electronic devices, and current state-of-
the-art technologies to tackle TID make use of expensive radiation-hardened
devices. This paper focuses on a novel and different approach: using machine
learning algorithms on consumer electronic level Field Programmable Gate
Arrays (FPGAs) to tackle TID effects and monitor them to replace before they
stop working. This condition has a research challenge to anticipate when the
board results in a total failure due to TID effects. We observed internal
measurements of the FPGA boards under gamma radiation and used three different
anomaly detection machine learning (ML) algorithms to detect anomalies in the
sensor measurements in a gamma-radiated environment. The statistical results
show a highly significant relationship between the $\gamma$ radiation exposure
levels and the board measurements. Moreover, our anomaly detection results
have shown that a One-Class Support Vector Machine with Radial Basis Function
Kernel has an average Recall score of 0.95. Also, all anomalies can be
detected before the boards stop working.

###### Index Terms:

Machine Learning, Anomaly Detection, Gamma radiation, Field Programmable Gate
Arrays, TID

##  I Introduction

One of the biggest challenges in the European Union is the cleaning of nuclear
waste [1]. This task involves handling and moving extreme toxic material
contaminated with different types of ionizing radiation. Unfortunately, higher
doses of radiation harm the human body; therefore, the cleaning process should
be taken with much precaution. Luckily, most of this work can be done by
robots, but sadly the electronic devices controlling them are also susceptible
to radiation. Remarkably, many artificial intelligence algorithms used in
robots heavily rely on high-speed processors or graphics processing units
(GPUs) which leads to significant growth in reliability issues due to the
nature of the nanoscale technologies used in those chips,

One can categorize the radiation consequences on contaminated sites into two
broad types: permanent and transient effects. One example of the permanent
effects is the Total Ionizing Dose (TID), while Single Event Upset (SEU) is an
example of transient effects. TID effect is a phenomenon that causes permanent
damage, which inevitably, at some point, is going to result in total failure
of the electronics. TID effects can only be minimized to extend the system
life. The literature has come up with special hardened devices specifically
targeting this effect [2, 3]. Typically, these devices are much more expensive
than unhardened ones. On the other hand, SEUs are transient effects that cause
bit-flips on memory elements. For example, one of the most employed solutions
for radiation environments, especially for space applications, is Field
Programmable Gate Arrays (FPGAs) due to their reconfiguration capability and
performance aspects. FPGAs are predominantly used for these applications [4,
5, 6] as they allow reconfiguration and thus hardware adaptability to deal
with most transient effects.

The underlying idea of this paper is that instead of using an approach with
radiation-hardened devices, we adopt consumer electronic level COTS
(commercial off-the-shelf) devices and then monitor, manage and replace them
with healthy ones before they stop working. This condition brings us a
research challenge to anticipate when the board results in a total failure due
to radiation, specifically TID effects.

In this paper, we first employ a statistical analysis of the collected data to
prove that the measured values of the board under radiation show statistical
significance compared to a scenario without radiation. Later, the paper tries
to answer whether it is possible to predict when the board will stop working
due to radiation with reasonable accuracy. For this purpose, we tested a low-
cost COTS unhardened consumer electronic level 28nm FPGA used in many fault-
tolerant techniques for transient radiation effects. We employ this board
under gamma ($\gamma$) radiation and log its behaviour.

The monitoring of these boards needs to be executed with an enhanced approach.
The paper shows that using simple techniques to observe voltage and
temperatures may not indicate that the board will stop working. For example,
one widely used data analysis tool is R control charts. This type of chart,
popularly known as a control chart, monitors the mean and range of normally
distributed variables simultaneously when samples are collected at regular
intervals. Such a technique uses upper and lower control limits (UCL and LCL)
to monitor the behaviour of the variables. Still, it may not be sufficient to
indicate that a given board is behaving abnormally. A few data points outside
the operational limits caused by radiation do not suggest that the board will
stop operating. This paper shows that this might take a few minutes or more
than one hour. For this reason, we employ state-of-the-art machine learning
(ML) techniques to understand and try to predict when the board is behaving
abnormally. Authors in [7] reviews works that used ML for failure prediction
in industrial mechanical systems for the last decade and identify
opportunities for future research, although there is no focus on electronics.

The novelty of this works is three-folded: First, this is the first study to
monitor and measure voltages and temperatures on a consumer electronic level
SRAM-based FPGA SoC (System on Chip) under $\gamma$ radiation. Second, this is
the first study performing a quantitative/statistical analysis of the effects
of gamma radiation on voltages and temperatures of an FPGA and then compare to
an environment without radiation. Finally, this is the first work using
machine learning algorithms trying to predict when the board will be rendered
in-operational by gamma radiation through the observation of temperature and
voltage values only.

The paper is organized as follows. Section III details the hardware setup to
monitor and test the boards under experiment, later presenting an example
where the proposed technique might be used. Section II reviews the state-of-
the-art of related techniques. Section IV explains the statistical analysis
while Section V details the employed machine learning techniques and how to
measure its accuracy. Section VI details the experiments carried out under
$\gamma$ radiation and how the data was organized to feed the machine learning
algorithms. Finally, Section VII evaluates, compares and discuss the results,
and Section VIII draws conclusions and future plans.

##  II Related Work

As the sophistication of embedded systems grows, their vulnerability to errors
is adversely affected due to an increase in critical points of failure.
Adopting fault mitigation or fault tolerance techniques is vital if FPGAs are
used in radiation environments. Fault tolerance techniques that enhance
embedded processor reliability can be categorized as hardware-, software- and
hybrid-based techniques [8].

The hardware-based techniques, which mainly rely on spatial redundancy,
provide two or more instances of a hardware component, such as processors,
memories, buses or power supplies, for protection against soft errors. This
class of techniques include Triple Modular Redundancy (TMR) [9], Duplication
with Comparison (DWC) [10] and hardware monitors [11] which incorporate
watchdog or checker modules to monitor the system and detect errors by
verifying the control-flow related memory accesses of the target processor.
These techniques can protect the system from errors in the computation
outputs, i.e. SDCs, as exemplified in [12].

Software-implemented hardware fault tolerance (SIHFT) approaches handle
hardware malfunctions by merely shielding the software without any hardware
alteration. These techniques rely on adding redundant software code for
comparison to detect errors. However, they exhibit a high-performance
overhead, which may not be viable for some real-time systems. These kinds of
techniques, such as ABFT [13], HETA [14] and S-SETA [15], detect control-flow
faults leading to FIs, which manifest themselves as hangs or crashes, and then
place the system into a fail-safe state. Note that both hardware- and
software-based techniques are not capable of correcting 100% of the errors,
but rather detecting them to avoid a failure that would have adverse effects
on the entire mission. Furthermore, they protect the system either from SDCs
or FIs, but not both.

The hybrid techniques are the ones that use a SIHFT method combined with a
hardware intellectual property (IP), which performs consistency checks in the
processor, making them effective against both SDCs and FIs. For instance, the
lockstep technique is a hybrid fault-tolerance technique based on software and
hardware redundancy. It employs the concepts of checkpointing and recovery
mechanisms (e.g. roll-back recovery, roll-forward recovery) at the software
level, and processor replication and checker circuits at the hardware level,
as explained in the following sections. Therefore, it is capable of both error
detection and correction. The lockstep technique’s most significant merit is
its ability to detect and correct both SDCs and FIs, unlike many other fault
tolerance techniques. Several researchers have developed and implemented their
lockstep technique version, such as those in [16, 17, 18, 19, 20], to make a
range of processors resistant to radiation-induced soft errors, extensively
analyzed and compared in [21].

The Authors in [22] evaluated a (${}^{60}C$) gamma-ray radiation testing of a
space application FPGA, namely the RT4G150 from Microsemi. Microsemi is a
qualified manufacturers list (QML)-certified manufacturer of high-reliability
FPGAs for space applications, while RTG4 is the 4th generation family of
radiation-tolerant flash-based FPGAs. The work assesses the degradation of the
flash cell through its threshold-voltage ($V_{T}$) shift. For space
applications, dynamic burn-in (DBI) testing is used to evaluate the long term
reliability of the device. Among all product screening tests employed by many
business categories, including automotive, aerospace, and defence, the burn-in
(BI) test is one of the most effective tests for early failure detection. The
work indicates that RTG4 shows a shift of the programmed Pflash cell $V_{T}$
post-DBI is observed. The programmed Pflash VTshift is due to voltage
degradation, resulting from approximately 1.75% degradation of the DAC’s
output. It is essential to emphasize that this work does not monitor the
temperature of the FPGA, this being out of the scope of the project.

In [23] Authors evaluated a COTS FPGA, namely Microsemi ProASIC3E A3PE1500.
Despite their low reliability, the authors state that this FPGA has been
considered a promising alternative to replace radiation-hardened ones. The
paper analyses the Single-Event Upset (SEU) sensitivity of the FPGA for a
combined set of Electromagnetic Interference (EMI) and TID tests. This
component was under the pre-qualification process for use in some satellites
of the Brazilian Space Program. The TID test was performed by exposing the
FPGA to a 10-keV effective energy X-ray beam. The device was roughly exposed
to the TID expected to be cumulated on satellite electronics after operation
for a period of 4 or 5 years in a given orbit, as specified by the Brazilian
National Institute for Space Research. The conclusion is that FFs present a
lower SEU-immunity degradation when exposed to conducted-EMI (5.6%) than SRAM
cells (resp. 6.3%); the latter memory elements are intrinsically more robust
to EMI since they present a much lower cross-section than FFs. Note that this
work does not monitor the temperature of the FPGA as well.

Tarrilho et al. [24] analyze the behaviour of flash-based FPGA from Actel
under TID. In this work, a design with an embedded system composed of a MIPS
microprocessor hardened with fault-tolerant techniques is employed on a COTS
flash-based FPGA ProASIC3E family from Actel. The TID experiment with no
reconfiguration monitored the power supply current during radiation and the
FPGA temperature. They reported that the Icc started to change after 45
krad(Si) when some modules stopped working. The current increases promptly and
reaches 1.5 times the original current just before 65 krad(Si). The
temperature and current drop abruptly when most modules fail around 65
krad(Si). The paper primarily showed the failure dose for some internal
modules and did not present when the current starts behaving abnormally.

In [25] evaluate TID effects on an SRAM-based COTS FPGA. They combine hardware
and software techniques to perform on-chip irradiation via a
${}^{90}Sr$/${}^{90}Y$ electron source and assess the degradation of the
system. The experiment consists of two executions of Zynq XC7Z020T chips
hosted by two distinct Zedboards. The analysis focuses on ring oscillators
(ROs) implemented on the programmable logic of the FPGA for
estimating/predicting the performance degradation due to the TID effects. The
authors show specifically the RO frequency according to temperature, current
and accumulated dose. The authors indicate that specifically for COTS 28-nm
Zynq-7000 chips, the results show increased TID tolerance beyond the Mrad
level. The tests were conducted with the FPGAs surviving up to 2 Mrad(Si). In
summary, this work shows that COTS FPGAs can survive significant amounts of
ionizing radiation, although not displaying the limit, since it was not the
objective of their work.

These papers [24, 23, 22] summarize most related work for TID effects in
FPGAs. They either focus on flash-based devices[24, 22] or point out [23] that
they could be a promising alternative for radiation-hardened ones. Also, only
one paper monitors current and temperature, although it is not the main
objective of that paper to correlate or predict the FPGA stop working.

##  III Experimental Design

This section explains the setup, shown in Fig. 1, built to measure and log all
the tested experiments, later detailed in Section VI. The experimental setup
is composed of (i) a laptop to control, collect and log all data; (ii) a
Design Under Test (DUT) board and (iii) a special monitoring board to collect
the data from it. The only electronic device exposed to radiation is the DUT,
that being the reason for the monitoring board, which is outside the radiation
cavity and connected to the DUT with voltage and temperature probes.

Figure 1: Block diagram of the experimental setup. Only the DUT is under
radiation.

The DUT is a MiniZed board, depicted in Fig. 2, a widely available and, most
used, equipped with a low-cost ($\approx$£100) Xilinx 28nm Zynq FPGA. The
monitoring board is an Arduino board coupled with temperature and voltage
sensors, which during the experiments are located outside the radiation
chamber; therefore, it can be reused.

Figure 2: Employed a Minized board with connectors attached to monitor voltage
and two thermocouple pairs to measure temperatures.

The MiniZed board [26] (Fig. 2) is a development board containing a Xilinx
Zynq single-core SoC XC7Z007S-1CLG225C [27] with 512MB DDR3L micron storage, a
128MB QSPI flash and 8GB eMMC. The FPGA contains a programmable logic and a
Programmable Logic (PL)(FPGA) and Processing Subsystem(PS) (ARM Cortex-A9).
The FPGA chip is fed by a power management integrated circuit (PMIC) Dialog
DA9062 [28] which controls five different voltages[29]: 0.675, 1.0, 1.35, 1.8
and 3.3 volts, detailed in Table I.

For example, $V_{DDR3}$ supplies the voltage to the DDR3 memory of the board.
The monitoring board is organized to monitor these voltages. We also use a
thermocouple Wire to monitor the temperature on the surface of both FPGA and
PMIC. This setup is arranged so that only the DUT and wires connecting to the
monitoring board are inside the radiation chamber; therefore, the radiation
does not interfere with the monitoring electronics.

TABLE I: Temperature and voltage monitored values Symbol | Expected value | Description
---|---|---
$T_{FPGA}$ | - | FPGA temperature
$T_{PMIC}$ | - | PMIC temperature
$V_{core}$ | 1 V | Core supply
$V_{aux}$ | 1.8 V | Auxiliary supply
$V_{ddr3}$ | 1.35 V | DDR3
$V_{tt}$ | 0.675 V | Vtt
$V_{cco}$ | 3.3 V | Vcco / board voltage
Figure 3: The monitoring board composed of an Elegoo R3 board (1) coupled with
voltage (2) and temperature (3) sensors.

The monitoring board is an Elegoo R3 board [30], which is completely
compatible with the official Arduino R3 version. The board is composed of an
Atmel ATMEGA328P chip [31] and coupled with five DC0-25V voltage sensors and
two Digilent Pmod TC1 K-Type thermocouple modules [32]. The thermocouple
sensor module works with a temperature range of -73°C to 482°C with a 0.25°C
resolution. The voltage sensor works with a range of DC 0 to 25 V with a
resolution of 0.01 V. The board collects one reading each second, operating in
a 1 Hz frequency. The data is sent to a laptop and stored for later
computation. In parallel, the MiniZed board runs a compute an intense set of
operations whose outputs are sent to the host laptop through a serial
interface for a sanity check aiming to determine if the board is still
operational. When the board stops sending data, it is considered to be dead.
Later, all the boards used on experiments were tested individually outside the
radiation chamber to ensure they were inoperational. We tried to download the
provided sanity check, and none of the boards responded to the cable
connection attempts, confirming that the boards were dead.

###  III-A Motivational example

Figure 4 shows the measurements of the Minized board under normal operation.
We monitor the power controlling IC and the FPGA temperature ($T_{PMIC}$ and
$T_{FPGA}$ respectively) and five voltages ($V_{ddr3}$, $V_{aux}$, $V_{core}$,
$V_{tt}$ and $V_{cco}$) supplied to the FPGA. One can see a normal fluctuation
in the temperature and voltages within some bounds. On the other hand, Figure
5 shows the same type of board under gamma radiation, where the black lines on
voltages are the average of the initial readings. The temperature starts to
rise faster, and voltages operate outside their normal bounds and eventually,
the board stops working. These two simple examples clearly show that the board
is not behaving normally under radiation, as expected. Later sections try to
correlate these behaviours to predict when the board will stop working.
Furthermore, we would use state-of-the-art machine learning techniques to
evaluate whether it is possible to indicate when a board will fail based on
voltages and temperature sensor readings.

Figure 4: Voltage and temperature sensor reading in an environment without
radiation. Temperature and voltage values operate inside certain bounds, and
there are few deviations from the mean. Figure 5: Voltage and temperature
sensor reading on an environment with $\gamma$ radiation. Temperature and
voltage values operate outside bounds compared to a mean without radiation.

##  IV Understanding the statistical effect of gamma radiation on Minized
boards

To examine the effects of different levels of $\gamma$ radiation on our
boards, we conducted a statistical analysis using measurements for boards that
stopped working after being exposed to $\gamma$ radiation and for boards used
in a non-radiation environment. To test whether any significant differences
occurred in these measurements, we conducted one-way Multivariate Analyses of
Variance (MANOVA). The first MANOVA test compared the $\gamma$ radiation
effects on the boards (i.e. 0: the measurements collected from boards working
under $\gamma$ radiation; 1: measurements obtained from boards on a non-
radiation site) as independent variable, and sensor measurements, two
temperatures ($T_{PMIC}$ and $T_{FPGA}$) and five voltages($V_{ddr3}$,
$V_{aux}$, $V_{core}$, $V_{tt}$ and $V_{cco}$) as dependent variables. The
second MANOVA test involved different $\gamma$ radiation levels (to which
boards were exposed) as an independent variable (i.e. $\gamma$ radiation
levels detailed in Table IV) and temperature and voltage as dependent
variables.

The statistical results are presented in Table II and Table III. The partial
eta squared ($\eta^{2}$) represents the effect size, determining how much the
relationship will affect the values. On the other hand, the F-value is the
test statistic used to determine how much one variable is associated with the
response. The factors and interaction effects were analyzed with one-way
analysis using the partial eta squared index of effect size. The Bonferroni
procedure was used here. The definitions in [33] have been adopted to discuss
the effect sizes: small effect size ($\eta^{2}\leq.01$), medium effect size
($.01\leq\eta^{2}\leq.06$) and large effect size ($.06\leq\eta^{2}\leq.14$).
The MANOVA levels of significance are reported using the F-statistics and
probability $p$. A risk of $\alpha$ of .05 was used in all statistical tests.

TABLE II: Results of One-Way Multivariate Analyses of Variance to discover the sensory observation difference between functioning and non-functioning sensor boards used in radiation and non-radiation environments. $\eta^{2}$ is the partial eta squared measure of effect size. ${}^{\star}p<.05,^{\star\star}p<.01,^{\star\star\star}p<.001$. The table demonstrates the statistical effect of the main factor. The error degrees of freedom was the same for each dependent variable.  Source | Dependent Variable | $df$ | $F$ | $\eta^{2}$
---|---|---|---|---
|  $\gamma$ Radiation
---
Functioning
$T_{PMIC}$ | $1$ | $33051.493^{\star\star\star}$ | $.214$
| $T_{FPGA}$ | $1$ | $18569.785^{\star\star\star}$ | $.133$
| $V_{aux}$ | $1$ | $1040594.28^{\star\star\star}$ | $.896$
| $V_{ddr3}$ | $1$ | $463097.020^{\star\star\star}$ | $.793$
| $V_{core}$ | $1$ | $30362.611^{\star\star\star}$ | $.200$
| $V_{tt}$ | $1$ | $19258.336^{\star\star\star}$ | $.137$
| $V_{cco}$ | $1$ | $87822.015^{\star\star\star}$ | $.420$
Error |  | 121226
TABLE III: Results of One-Way Multivariate Analyses of Variance to discover the sensory observation difference for sensor boards at different gamma radiation levels. ${}^{\star}p<.05,^{\star\star}p<.01,^{\star\star\star}p<.001$. The table demonstrates the statistical effect of the main factor. The error degrees of freedom was the same for each dependent variable.  Source | Dependent Variable | $df$ | $F$ | $\eta^{2}$
---|---|---|---|---
$\gamma$ Radiation | $T_{PMIC}$ | $6$ | $9721.465^{\star\star\star}$ | $.325$
| $T_{FPGA}$ | $6$ | $4066.381^{\star\star\star}$ | $.168$
| $V_{aux}$ | $6$ | $197999.152^{\star\star\star}$ | $.907$
| $V_{ddr3}$ | $6$ | $94615.951^{\star\star\star}$ | $.824$
| $V_{core}$ | $6$ | $6617.718^{\star\star\star}$ | $.247$
| $V_{tt}$ | $6$ | $4006.448^{\star\star\star}$ | $.165$
| $V_{cco}$ | $6$ | $148.701^{\star\star\star}$ | $.577$
Error |  | 121221

###  IV-A Comparison of boards behaviour on radiation and non-radiation sites

There was a highly significant effect of the functioning ($Functioning$) of
the boards deployed in radiation site when compared to a non-radiation sites
in Table II. Overall, there was highly significant effect on the functioning
of the sensor boards with very large effect sizes when the measurements
collected from radiation and non-radiation sites:
$F(1,121226)=33051.493,p<.001$, $\eta^{2}=.214$ for temperature 1;
$F(1,121226)=18569.785,p<.001$, $\eta^{2}=.133$ for temperature 2;
$F(1,121226)=1040594.28,p<.001$, $\eta^{2}=.896$ for voltage 1;
$F(1,121226)=463097.020,p<.001$, $\eta^{2}=.793$ for voltage 2; ;
$F(1,121226)=30362.611,p<.001$, $\eta^{2}=.200$ for voltage 3; ;
$F(1,121226)=19258.336,p<.001$, $\eta^{2}=.137$ for voltage 4; ;
$F(1,121226)=87822.015,p<.001$, $\eta^{2}=.420$ for voltage 5.

Our results indicate a significant difference in the obtained measurements
between the radiation and non-radiation sites. This was an expected result,
but this is the first work reporting these findings to the best of the
authors’ knowledge. It is important to emphasize that since it corroborates
the underlying assumption of the proposed analysis of this paper.

###  IV-B The effect of different $\gamma$ radiation levels

There was a highly significant effect of $\gamma$ radiation levels ($\gamma$
Radiation) with a very large effect sizes on the sensor board measurements in
Table III: $F(6,121221)=9721.465,p<.001$, $\eta^{2}=.325$ for temperature 1;
$F(6,121221)=4066.381,p<.001$, $\eta^{2}=.168$ for temperature 2;
$F(6,121221)=197999.152,p<.001$, $\eta^{2}=.907$ for voltage 1;
$F(6,121226)=94615.951,p<.001$, $\eta^{2}=.824$ for voltage 2; ;
$F(6,121221)=6617.718,p<.001$, $\eta^{2}=.247$ for voltage 3; ;
$F(6,121221)=4006.448,p<.001$, $\eta^{2}=.165$ for voltage 4; ;
$F(6,121221)=27547.584,p<.001$, $\eta^{2}=.577$ for voltage 5.

The posthoc analyses showed highly significant differences between most
sensors when exposed to different $\gamma$ radiation levels $p<.001$. However,
there were no significant differences in the following interactions. For
temperature 1 measurements, there was no significant difference between 2469
and 7707 $\gamma$ radiation levels. For temperature 2, there was no
significant difference between 0 and 5871 $\gamma$ radiation levels, and there
was only a significant difference between 7707 and 16966 $\gamma$ radiation
levels $p<.05$. For voltage 1, there was no significant difference between
2469 and 5137 and 5871 $\gamma$ radiation levels. Similar to temperature 2
results, there was only a significant difference between 7707 and 16966
$\gamma$ radiation levels for voltage 1 measurements. For voltage 2
measurements, there was a significant difference between 7707 and 16966
$\gamma$ radiation levels ($p<.05$). For voltage 3 measurements, there was no
significant difference between 1209 and 7707 $\gamma$ radiation levels and
between 5137 and 5871 $\gamma$ radiation levels. For voltage 4 measurements,
there was no significant difference between 1209 and 7707 and 16966 $\gamma$
radiation levels. For voltage 5 measurements, there was no significant
difference between 2469 and 5971 $\gamma$ radiation levels and between 7707
and 16966 $\gamma$ radiation levels.

These results indicate that it is impossible to correlate a given voltage to a
radiation level and create a relationship between them. Different sensor
inputs would have different weights depending on the radiation rate level. For
this reason, more elaborate approaches, such as machine learning algorithms,
would have better results since there is a tuning of inputs through
experience, in this case, the historical readings of temperature and voltage.

##  V Anomaly Detection with Machine Learning Models

Typically, anomalous data in this study are connected to problems or rare
events such as abnormal temperatures or voltages or malfunctioning components.
This connection may imply which data points can be considered anomalies to
identify these events that are typically useful for predicting the early
failure of the system. This section explores three machine learning models: 1)
Elliptical Envelope, 2) Local Outlier Factor, 3) One-Class Support Vector
Machine as our anomaly detectors.

###  V-A Machine Learning Models

####  V-A1 Elliptical Envelope

Elliptical Envelope is a Gaussian distribution-based method that forms the key
data parameters into an underlying multivariate Gaussian distribution
expression. In short, it attempts to identify a boundary ellipse that covers
most of the data. Therefore, any data not within the ellipse can be classified
as an anomaly. The FAST-minimum covariance determinant is used to estimate the
size of the ellipse, which selects non-overlapping samples of data and
computes the mean $u$ and covariance matrix $C$. Therefore, Mahalanobis
distance $d_{MH}$ for the input data vector $x$ can be calculated using the
following equation, and the data are then ordered ascendingly by $d_{MH}$
[34].

${d_{MH}}=\sqrt{{{(x-\mu)}^{T}}{C^{-1}}(x-\mu)}$

####  V-A2 Local Outlier Factor

Local Outlier Factor is one of the Nearest Neighbour based methods for anomaly
detection. In general, normal data are usually grouped in a neighbourhood that
seems dense compared to the abnormal data, which are far from their close
neighbours. To quantify this neighbourhood, these types of approaches
typically use distance-based or density-based methods, where both ways require
a similarity or a distance calculation to determine whether the data are on
the degree of abnormality or not. We use the Local Outlier Factor (LOF)
abnormal detector in this study [35].

####  V-A3 One-Class Support Vector Machine

One-Class Support Vector Machine (OCSVM) [36] is a classification-based
anomaly detection method. Depending on the availability of labels, it can be
divided into one-class and multi-class classification models. This approach is
similar to all other supervised learning techniques, has two phases: 1)
Training phase and 2) Testing phase. In the training phase, the classifier is
trained using the labelled data, and then the data are classified as normal or
abnormal using the trained model in the testing phase. In OCSVM, the
classification rule for the linear decision boundary is given as follows:

$f(x)=\left\langle{\vec{w},\vec{x}}\right\rangle+b$

where $\vec{w}$ and $b$ are the normal vector and bias, respectively. The
algorithm is trying to find the rule $f$ within the maximal geometric margin
and then assign a label to a test example $\vec{x}$. For example, if $f(x)>0$,
the label of $\vec{x}$ will be marked as normal; otherwise, it is labelled
anomaly. This optimization problem can be solved by

$\mathop{\min}\limits_{\alpha}\frac{1}{2}\sum\limits_{ij}{{\alpha_{i}}{\alpha_{j}}K({x_{i,}}{x_{j}})}$

where $\alpha_{i}$ is the $i^{th}$ weight, $0\leq\alpha_{i}\leq\frac{1}{vl}$
and $\sum\limits_{i}\alpha_{i}=1$. $v$ is a variable to control the maximizing
the distance between the origin and the number of data points contained in the
boundary. $l$ is the number of points in the training dataset.
$K(x_{i},x_{j})$ is the kernel function, and it is given as follows.

$K(x,y)=\left\langle{\phi(x),\phi(y)}\right\rangle$

where $\phi$ maps the training vectors from input space $X$ to a high
dimensional feature space.

A series of mathematical functions, known as the kernel, is used by SVM
algorithms. The kernel function takes data as input and translates it into the
appropriate form. Different SVM algorithms use various kernel function types.
The adopted version, OCSVM, uses the Radial Basis Function (RBF) kernel [37].

###  V-B Evaluation Metrics: Precision, Recall, F1 score

This paper employs three well-known metrics for pattern
recognition/classification to evaluate the machine learning models: Precision,
Recall and F1 score [38]. Precision is the fraction of relevance among the
retrieved instances, while Recall is the fraction of the total amount of
relevant instances. All these metrics would signify better results as they
approach the value of 1.

In the context of this paper, there are two kinds of data: annotated and not
annotated – data with an anomaly and without it. The employed models should
recognize if the data are an anomaly or not. Suppose we have a given dataset
with ten anomalies and the remaining data are not an anomaly. A given ML model
identifies eight data points as anomalies, of which five are anomalies (true
positive) while the rest are not (false positives). The ML model’s precision
is 5/8, while the Recall is 5/10. In this example, precision means how valid
the results are, while Recall shows how complete the results are.

F1 score is a metric for accuracy. It considers both the precision and the
Recall to compute the score. The F1 score is then calculated using the
harmonic mean of the Precision and Recall, where an F1 score has its best
value at 1 (perfect precision and Recall).

##  VI Radiation Experiments

The experiments took place on the Dalton Cumbrian Facility (DCF) laboratory,
where a $\gamma$ radiation source is available [39]. The source is composed of
a Cobalt (${}^{60}C$) self-shielded irradiator that can provide absorbed dose
rates of up to 20 kGy/h depending on the distance from the source to the DUT.
The radiation cavity contains three rods, as shown in Fig. 6, with different
dose rates for each one. Thus, different radiation rates are achieved using
various configurations of the available rods, including lead obstacles to
absorb the radiation and/or positioning the samples at different distances
from the sources.

Figure 6: Minized board inside radiation chamber. The three pipes on the back
contain three radiation rods that are lifted from the ground when the chamber
is closed. Figure 7: Voltages behaviour of experiments 0 and 5, respectively.

We employ six different experiments, each one using a separate board (DUT)
under radiation – each board will be referred by its experiment number. Table
IV shows each feature of the experiments where different radiation rates were
applied to see how the boards would behave. The DUT time is the time between
the start of the experiment until the FPGA board stops sending data through
the serial connection. This time should be considered the operational time of
the DUT. On the other hand, the Monitoring board time is the time from the
start of the experiment until the monitored voltages drop to zero, which means
that the board itself, including the PMIC, is entirely inoperative.  In each
experiment, the board is under a constant radiation rate and stays the same
distance from the rods. Before the actual radiation experiment begins, the
radiation rate is measured using a probe removed afterwards. For this reason,
the probe is inserted, the radiation is released and measured for one minute,
then the experiment is stopped, and the probe is removed. The radiation rate
measured during that period is assumed to be constant for the whole
experiment.

TABLE IV: Radiation experiment for the DUT Exp. # |  | Radiation
---
Rate (Gy/h)
DUT Time | Monitoring Board Time
0 | 1209 | 1:56:00 | 2:00:11
1 | 2469 | 0:39:14 | 0:41:31
2 | 5137 | 0:23:16 | 0:27:28
3 | 5871 | 0:19:49 | 0:23:23
4 | 7707 | 0:12:39 | 0:13:39
5 | 16966 | 0:06:58 | 0:07:38

After the experiments, all the boards stopped working. During the experiments,
one can see an increase in the voltage bounds on all experiments. The
interesting point is that the deviation from the normal bounds does not
indicate that the board will stop working right away. If we take, for example,
the two extremes for the radiation rates, i.e. experiments 0 and 5, one took
almost two hours while the other took less than ten minutes to stop working.
Fig. 7 shows that although experiment 5 exhibits an early change in the bounds
of the voltages, the board can take more than one hour to stop working. On the
other hand, the board might take a few minutes to stop working with a higher
radiation rate after the first values outside the normal bounds are observed.
This observation leads to a search for a more elaborate way of predicting when
the board will stop working than just watching voltages outside the normal
bounds of operation.

##  VII Evaluations

This Section compares the OCSVM method against the Local Outlier Factor and
Elliptical Envelope methods. OCSVM copes well with non-linear functions and
might be a more suitable approach for this problem. The section is divided
into four subsections. The first one shows how the data was preprocessed and
explains the methodology used in the second subsection, where the training set
is built. Then, the third subsection compares the OCSVM against the two other
methods using the training set. Finally, the last subsection further explores
how early we can detect anomalies with OCSVM.

###  VII-A Training and Testing Methodology

This subsection explains the steps to build the training data, later used to
compare different approaches. The objective here is to compare how the
training data would affect the results rather than comparing different ML
algorithms. Hence, we show the comparison using the OCSVM algorithm.

The first step is to remove as least from the collected data, all of which are
time-stamped. Unfortunately, a few points show the temperature as zero or
undefined (e.g. NaN). Therefore, all data points for that period were removed,
even if another sensor showed consistent data. These removals do not interfere
with the evaluation since all the data points are time-stamped.

Figure 8: Comparison using different features as input for the training data.
Respectively lines show the results using $i$) seven features (five voltages
and two temperatures) and $ii$) the same seven features plus the constant
radiation rate.

After the data trimming step, we compare two approaches: one using seven
features (five voltages and two temperatures) and one using the same features
plus a constant value for the radiation rate – the rates shown in Table IV.
Fig. 8 shows the precision of the two models with the six experiments. Using
the constant radiation rate as input weakens the results significantly.
Therefore, for further experiments, only voltage and temperature values are
considered.

###  VII-B Exploring the best training and testing strategy in a radiation
site

At this point, the number of features is defined, we came up with three
training and testing strategies to evaluate, compute or organize the measured
data. The target is to find the right balance of data points in the training
set. The following strategies evaluate trade-offs between a set of experiments
employed and the number of data points. Each strategy represents a data
processing approach, and it is detailed next as follows:

  * •

Strategy 1: Train the model with the first minutes of a given board and then
test on the remaining measurements of the same board.

  * •

Strategy 2: Train the model with the first minutes of a set of different
experiments and test on the remaining measurements of all boards.

  * •

Strategy 3: Choose the number of data points that gives the best results in
terms of F1 score.

The first strategy was employed using the first few minutes of each board and
then test on the remaining measurements on the same board. Fig. 9 shows the
results for this strategy. Even using the same board data and comparing it
with the remaining data does not provide good results. Only board from
experiment 5 shows good scores for F1 and precision.

Figure 9: Results using each experiment separately as training data.

Because of the weaker results of strategy 1, we decided to include more boards
on the training data. Strategy 2 evaluates different sets of boards as
training data to check how many we need to include to get the best results.
For this strategy, we trained all combinations sets of two up to all six
boards, i.e. sets of two boards {{0,1}, {0,2}, … {4,5}}, sets of three boards
{{0,1,2}, {0,1,3}, … {3,4,5}} and so on. Later we compared the F1 score for
each board separately. Results shown in Fig. 10 include the worst and best set
of boards only for the sake of space. One can assume that we get better
results as we feed more information to the model (different boards). For
example, the worst results for sets with two, three and four boards are much
lower than results using five or all six boards.

Figure 10: Using sets of 2, 3, 4, 5 and all boards as training data. Figure
showing only best and worst results for each set.

On the third strategy, we evaluate the effects of the amount of training data
fed into the model. Fig. 11 shows the comparison of using the first 300, 360,
420, 480, 520 and all data points as input for the model. Interestingly, as we
feed more data to the training set, it does not mean we would get better
results. Adding more than 420 points does not increase the F1 score, and as it
shows better average results, we keep this number of data points as a training
set for all experiments.

Figure 11: Comparison of training data for input data. Using the first 420
data points from all experiments show better results. 420 data points
represent roughly seven minutes of data.

###  VII-C Comparison of OCSVM with other anomaly detection techniques

This subsection compares three state-of-the-art machine learning techniques:
$i$) Elliptical Envelope, $ii$) Local Outlier Factor and $iii$) One Class
Support Vector Machine (OCSVM). These techniques are trained using the
proposed dataset, discussed previously, as inputs.

TABLE V: Window times where voltages exhibit values outside usual bounds for each experiment. Exp. # | Window Time
---|---
0 | 74 min
1 | 18 min
2 | 07 min
3 | 07 min
4 | 06 min
5 | 03 min

Figure 12: Comparison of machine learning models for the dataset.

To compare the anomaly detection results, we first need to annotate the data
showing where the anomalies happened and then compare with the ML algorithms
outputs. We have calculated the size of the time window where each board
exhibits values outside usual bounds during radiation experiments, visually
observing the measured voltage (we could not observe such values for
temperature). Table shows the size of this window, and associated calculations
are displayed for each board in Table V. As can be seen, the minimum window
size observed is approximately 3 minutes. That is to say that the last 3
minutes of each experiment is the time where each board exhibits values
outside normal bounds. Therefore, after adding a safety margin of 2 minutes,
we annotate the last 5 minutes of each experiment as anomalies for use in the
training dataset fed into the machine learning algorithms. Five minutes is
roughly equivalent to 300 data points; therefore, we annotate this number of
points as an anomaly. Besides the observed window size, 300 data points also
give the developer a reasonable amount of time to take action before the board
stops working, e.g., move the computed data to safe storage, to another
computing node in the system or even move it away from the radiation
environment. We can also consider a different amount of time as annotation,
but it would result in the retraining of the model. Then finally, we can
summarize the training set with the annotation as follows:

  * •

Remove all the inconsistent sensor readings;

  * •

Collect the first 420 data points from all boards (as justified in Fig. 11);

  * •

We annotate the last 300 data points as an anomaly.

Using that training set, we then employ a multivariate analysis with seven
variables (including five voltage and two temperature values) to feed the ML
algorithms. Fig. 12 summarizes the results for Elliptical Envelope, LOF and
OCSVM algorithms where the F1, Precision, and Recall scores are shown for each
model. OCSVM demonstrates better outcomes for all scores. It is essential to
highlight that the Recall score for OCSVM is the one for all experiments
except number two, which is a remarkable result, showing an average Recall
score of 0.95 and strong evidence that the model can detect the relevant
results. The recall score in experiment 2 is not the maximum one but shows a
high value (0.842) and will be discussed next.

###  VII-D Exploring how early OCSVM can detect anomalies

As the OCSVM showed the best F1, Precision and Recall results, this subsection
details each experiment separately, as illustrated in Fig. 13. Each experiment
shows in the rows: (i) the five observed voltages; (ii) two temperatures;
(iii) anomaly annotation; and (iv) the OCSVM model output (i.e. 0 represents
normal data and 1 an anomaly both for annotation and model output).

|  (a) - Exp. 0 |  (b) - Exp. 1 |  (c) - Exp. 2
---|---|---|---

|  (d) - Exp. 3 |  (e) - Exp. 4 |  (f) - Exp. 5
---|---|---|---

Figure 13: Results for each experiment using One-Class Support Vector Machine
to predict the anomaly before the board stops working.

Table VI shows that in all experiments except one, the model marks the
abnormal behaviour of the board before the annotation. This is an exceptional
result, indicating that the model can be used as a suitable indicator/warning
before the board stops working. Note that the ML output for experiment 2 is
delayed by only 12 seconds; nevertheless, there still would be enough time for
the board to be saved before it is permanently damaged.

TABLE VI: Time showing how earlier the OCSVM marks the annotation (around 5 minutes before the board stops working). The model prediction is calculated with (Annotation - Prediction) from Fig. 13. Exp. # | Experiment duration |  | Model prediction
---
before the annotation
0 | 2:00:11 | 1:43:53
1 | 0:41:31 | 0:25:56
2 | 0:27:28 | -0:00:12
3 | 0:23:23 | 0:07:04
4 | 0:13:39 | 0:06:42
5 | 0:07:38 | 0:00:34

Experiment 0 is the longest in time under radiation mainly because it has the
lowest radiation rate among the experiments. The board took almost two hours
to stop working, and it is the worst result for the model output. Although the
model could point to the anomaly, it has done it earlier, more than one hour
before the board stops working. Experiment 1 has a shorter execution time, and
the model was also capable of predicting before the anomaly, i.e. 25 minutes
before the annotation. Experiment 2 has one particular behaviour different
from the others. The model output shows one unique point marking as an anomaly
in the first minutes. As this was the only point and not a continuous trend,
we can disregard this result - one should always observe the trend rather than
particular points. The model then marks only points 12 seconds after the
anomaly annotation. We still consider this a good result since a few seconds
later would still allow time to take action.

In experiments 3, 4 and 5, the model behaves similarly, marking the anomaly 7,
6 and 0.5 minutes before the annotation, which is a remarkable result. To
summarize, all boards would have been saved since the model can point out
before a board dies completely, thus allowing a few minutes for the designer
to save or transfer the processed data to a safe environment.

It is essential to point out that the model has a trade-off. As we gathered
more experiments with high radiation rates, five of the experiments have more
than 2000 Gy/h; the model has been trained with more data for these
environments. Therefore it also performs better, i.e. predicts near the
annotation, on experiments with high radiation rates. As we feed more
information, that is, more experiments in different radiation rates, the model
should perform better.

##  VIII Conclusion and Future Works

This paper proposed an anomaly detection machine-learning algorithm to predict
when a COTS FPGA would stop working due to gamma radiation. The OCSVM
algorithm showed the best results in terms of Recall score and was capable of
pointing out 100% of the anomalies before the board stopped working.
Annotating the anomaly before it stops working and giving time to the designer
to take actions allowed to detect the anomaly before the board dies,
considered an extraordinary achievement since it will enable the designer to
use this as an assumption for future works. This work employed six boards that
were inoperable after the experiments. Using more boards with different
radiation rates can improve the model results.

Future works include using the OCSVM algorithm at run-time experiments. The
proposed approach with DUT and a monitoring board can also be modified to
execute on a self-contained solution. The Arduino board of the DUT was used to
remove the monitoring board and sensors from the radiation environment. In a
real-case scenario, this would not be possible. To tackle that approach, the
Minized board (DUT) could use the sensor boards directly and execute the OCSVM
algorithm itself.

## Acknowledgment

This work is supported by the U.K. Engineering and Physical Sciences Research
Council through grants EP/R02572X/1 and EP/P017487/1.

## References

  * [1] “Nuclear Provision: explaining the cost of cleaning up Britain’s nuclear legacy,” https://www.gov.uk/government/.
  * [2] O. Gonçalves, G. Prenat, and B. Dieny, “Radiation hardened mram-based fpga,” _IEEE Transactions on Magnetics_ , vol. 49, no. 7, pp. 4355–4358, 2013.
  * [3] S. Ramaswamy, L. Rockett, D. Patel, S. Danziger, R. Manohar, C. W. Kelly, J. L. Holt, V. Ekanayake, and D. Elftmann, “A radiation hardened reconfigurable fpga,” in _2009 IEEE Aerospace conference_ , 2009, pp. 1–10.
  * [4] A. Lindoso, L. Entrena, M. García-Valderas, and L. Parra, “A Hybrid Fault-Tolerant LEON3 Soft Core Processor Implemented in Low-End SRAM FPGA,” _IEEE Transactions on Nuclear Science_ , vol. 64, no. 1, pp. 374–381, Jan 2017.
  * [5] H. Quinn, W. H. Robinson, P. Rech, M. Aguirre, A. Barnard, M. Desogus, L. Entrena, M. Garcia-Valderas, S. M. Guertin, D. Kaeli, F. L. Kastensmidt, B. T. Kiddie, A. Sanchez-Clemente, M. S. Reorda, L. Sterpone, and M. Wirthlin, “Using Benchmarks for Radiation Testing of Microprocessors and FPGAs,” _IEEE Transactions on Nuclear Science_ , vol. 62, no. 6, pp. 2547–2554, 2015.
  * [6] S. M. Trimberger, “Three ages of fpgas: A retrospective on the first thirty years of fpga technology,” _Proceedings of the IEEE_ , vol. 103, no. 3, pp. 318–331, 2015.
  * [7] J. Leukel, J. González, and M. Riekert, “Adoption of machine learning technology for failure prediction in industrial maintenance: A systematic review,” _Journal of Manufacturing Systems_ , vol. 61, pp. 87–96, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0278612521001849
  * [8] S. Kasap, E. Weber Wächter, X. Zhai, S. Ehsan, and K. Mcdonald-Maier, “Survey of Soft Error Mitigation Techniques Applied to LEON3 Soft Processors on SRAM-Based FPGAs,” _IEEE Access_ , vol. 8, pp. 28 646–28 658, 2020.
  * [9] C. Carmichael, “Triple Module Redundancy Design Techniques for Virtex FPGAs,” Xilinx Inc., San Jose, CA, USA, XAPP197 Application Note, 2006.
  * [10] D. K. Pradhan and N. H. Vaidya, “Roll-Forward and Rollback Recovery: Performance-Reliability Trade-off,” _IEEE Transactions on Computers_ , vol. 46, no. 3, pp. 372–378, March 1997.
  * [11] L. Parra, A. Lindoso, M. Portela-Garcia, L. Entrena, B. Du, M. S. Reorda, and L. Sterpone, “A new hybrid nonintrusive error-detection technique using dual control-flow monitoring,” _IEEE Transactions on Nuclear Science_ , vol. 61, no. 6, pp. 3236–3243, Dec 2014.
  * [12] H. Quinn, Z. Baker, T. Fairbanks, J. L. Tripp, and G. Duran, “Robust Duplication With Comparison Methods in Microcontrollers,” _IEEE Transactions on Nuclear Science_ , vol. 64, no. 1, pp. 338–345, Jan 2017.
  * [13] Kuang-Hua Huang and J. A. Abraham, “Algorithm-Based Fault Tolerance for Matrix Operations,” _IEEE Transactions on Computers_ , vol. C-33, no. 6, pp. 518–528, June 1984.
  * [14] J. R. Azambuja, M. Altieri, J. Becker, and F. L. Kastensmidt, “HETA: Hybrid Error-Detection Technique Using Assertions,” _IEEE Transactions on Nuclear Science_ , vol. 60, no. 4, pp. 2805–2812, Aug 2013.
  * [15] E. Chielle, G. S. Rodrigues, F. L. Kastensmidt, S. Cuenca-Asensi, L. A. Tambara, P. Rech, and H. Quinn, “S-SETA: Selective Software-Only Error-Detection Technique Using Assertions,” _IEEE Transactions on Nuclear Science_ , vol. 62, no. 6, pp. 3088–3095, Dec 2015.
  * [16] H. H. Ng, “PPC405 Lockstep System on ML310,” Xilinx Inc., San Jose, CA, USA, XAPP564 Application Note, 2007.
  * [17] F. Abate, L. Sterpone, C. A. Lisboa, L. Carro, and M. Violante, “New Techniques for Improving the Performance of the Lockstep Architecture for SEEs Mitigation in FPGA Embedded Processors,” _IEEE Transactions on Nuclear Science_ , vol. 56, no. 4, pp. 1992–2000, Aug 2009.
  * [18] M. Violante, C. Meinhardt, R. Reis, and M. Sonza Reorda, “A Low-Cost Solution for Deploying Processor Cores in Harsh Environments,” _IEEE Transactions on Industrial Electronics_ , vol. 58, no. 7, pp. 2617–2626, July 2011\.
  * [19] H. Pham, S. Pillement, and S. J. Piestrak, “Low-Overhead Fault-Tolerance Technique for a Dynamically Reconfigurable Softcore Processor,” _IEEE Transactions on Computers_ , vol. 62, no. 6, pp. 1179–1192, June 2013.
  * [20] A. B. de Oliveira, G. S. Rodrigues, F. L. Kastensmidt, N. Added, E. L. A. Macchione, V. A. P. Aguiar, N. H. Medina, and M. A. G. Silveira, “Lockstep Dual-Core ARM A9: Implementation and Resilience Analysis Under Heavy Ion-Induced Soft Errors,” _IEEE Transactions on Nuclear Science_ , vol. 65, no. 8, pp. 1783–1790, Aug 2018.
  * [21] E. W. Wächter, S. Kasap, X. Zhai, S. Ehsan, and K. McDonald-Maier, “Survey of lockstep based mitigation techniques for soft errors in embedded systems,” in _Computer Science and Electronic Engineering Conference (CEEC 2019)_ , 2019, pp. 124–127.
  * [22] N. Rezzak, J. Wang, M. Traas, A. Zerrouki, G. Bakker, F. Xue, A. Cai, F. Hawley, J. McCollum, and E. Hamdy, “Investigation of tid and dynamic burn-in-induced $v_{{T}}$ shift on rtg4 flash-based fpga,” _IEEE Transactions on Nuclear Science_ , vol. 65, no. 1, pp. 64–70, 2018\.
  * [23] P. Villa, E. Bezerra, R. Goerl, L. Poehls, F. Vargas, N. Medina, N. Added, V. de Aguiar, E. Macchione, F. Aguirre, and M. da Silveira, “Analysis of cots fpga seu-sensitivity to combined effects of conducted-emi and tid,” in _2017 11th International Workshop on the Electromagnetic Compatibility of Integrated Circuits (EMCCompo)_ , 2017, pp. 27–32.
  * [24] J. Tarrillo, J. R. Azambuja, F. L. Kastensmidt, E. C. P. Fonseca, R. Galhardo, and O. Goncalez, “Analyzing the effects of tid in an embedded system running in a flash-based fpga,” _IEEE Transactions on Nuclear Science_ , vol. 58, no. 6, pp. 2855–2862, 2011.
  * [25] G. Lentaris, K. Maragos, D. Soudris, F. Di Capua, L. Campajola, M. Campajola, A. Costantino, G. Furano, A. Tavoularis, and L. Santos, “Tid evaluation system with on-chip electron source and programmable sensing mechanisms on fpga,” _IEEE Transactions on Nuclear Science_ , vol. 66, no. 1, pp. 312–319, 2019.
  * [26] “Minized board product brief,” http://zedboard.org/.
  * [27] “Zynq-7000 SoC Data Sheet,” https://www.xilinx.com/.
  * [28] “DA9062 Data Sheet,” https://www.dialog-semiconductor.com/.
  * [29] “MiniZed Power Architecture,” zedboard.org.
  * [30] “Elegoo R3 board,” https://www.elegoo.com/.
  * [31] “ATmega Data Sheet,” http://ww1.microchip.com/.
  * [32] “PmodTC1 Board Reference Manual,” https://reference.digilentinc.com/.
  * [33] J. Cohen, _Statistical power analysis for the behavioral sciences (2nd ed.)._ Hillsdale, NJ: Lawrence Erlbaum Associates., 1977.
  * [34] B. Hoyle, M. M. Rau, K. Paech, C. Bonnett, S. Seitz, and J. Weller, “Anomaly Detection for Machine Learning Redshifts applied to SDSS Galaxies,” _Monthly Notices of the Royal Astronomical Society_ , vol. 452, no. 4, pp. 4183–4194, 2015.
  * [35] M. Alshawabkeh, B. Jang, and D. Kaeli, “Accelerating the Local Outlier Factor Algorithm on a GPU for Intrusion Detection Systems,” in _Proceedings of the 3rd Workshop on General-Purpose Computation on Graphics Processing Units_ , ser. GPGPU-3. New York, NY, USA: Association for Computing Machinery, 2010, p. 104–110.
  * [36] K. A. Heller, K. M. Svore, A. D. Keromytis, and S. J. Stolfo, “One Class Support Vector Machines for Detecting Anomalous Windows Registry Accesses,” in _Proceedings of the ICDM Workshop on Data Mining for Computer Security_ , 2003.
  * [37] D. S. Broomhead and D. Lowe, “Radial basis functions, multi-variable functional interpolation and adaptive networks,” Royal Signals and Radar Establishment Malvern (United Kingdom), Tech. Rep., 1988.
  * [38] D. Powers, “Evaluation: from precision, recall and f-measure toroc, informedness, markedness and correlation,” _Journal of Machine Learning Technologies_ , vol. 2, no. 1, pp. 37–63, 2011.
  * [39] “Dalton Cumbrian Facility,” https://www.dalton.manchester.ac.uk/research/facilities/cumbria-facilities/.

|  Eduardo Weber Wachter received the B.Eng. degree in computer engineering
from the State University of Rio Grande do Sul, Guaíba, Brazil, in 2009 and
the PhD degree in computer science from the Pontifical Catholic University of
Rio Grande do Sul, Porto Alegre, Brazil, in 2015\. His research interests are
Many Cores, NoC, reconfigurable architectures, Fault Tolerance and Reliability
of such systems. He is currently a Teaching Fellow with the University of
Warwick, U.K.
---|---
|  Server Kasapr (M’10) received the B.Sc. (Hons.) degree in electrical and
electronic engineering from Middle East Technical University, Ankara,
Turkey,and the M.Sc.(Distinction) and PhD degrees in electronic engineering
from the System-Level Integration Research Group, University of Edinburgh,
Edinburgh, U.K., in 2006, 2007, and 2010, respectively. His current research
interests include reconfigurable dataflow computing for finite-difference
time-domain simulations, FPGAhardware design and implementation for digital
signal processing applications, and high-performance scientific computing in
general. He is currently a Lecturer at Coventry University, U.K.
---|---
|  Şefki Kolozali received the B.Sc. degree in computer engineering from Near
East University, Nicosia, Cyprus, in 2005, the M.Sc. degree from the
University of Essex, Colchester, U.K., and the PhD. degree from the Queen Mary
University of London, London, U.K. His main research interests include signal
processing, machine learning, semantic Web technologies, and the Internet of
Things. He is currently a Lecturer with the University of Essex, U.K.
---|---
|  Xiaojun Zhai received the B.Sc. degree from the North China University of
Technology, China, in 2006, and the M.Sc. degree in embedded intelligent
systems and the PhD degree from the University of Hertfordshire, U.K., in 2009
and 2013, respectively. He is currently a Lecturer with the School of Computer
Science and Electronic Engineering, University of Essex. His research
interests mainly include the design and implementation of the digital image
and signal processing algorithms, custom computing using FPGAs, embedded
systems, and hardware/software co-design.
---|---
|  Shoaib Ehsan received the B.Sc. degree in electrical engineering from the
University of Engineering and Technology, Taxila, Pakistan, in 2003, and the
PhD degree in computing and electronic systems (with specialization in
computer vision)from the University of Essex, Colchester, U.K., in 2012. His
current research interests are in intrusion detection for embedded systems,
local feature detection and description techniques, and image feature matching
and performance analysis of vision systems.
---|---
|  Klaus D. McDonald-Maier (S’91–SM’06)is currently the Head of the Embedded
and Intelligent Systems Laboratory, University of Essex, Colchester, U.K. He
is also the Chief Scientist with UltraSoC Technologies Ltd., the CEO of
Metrarc Ltd., and a Visiting Professor with the University of Kent. His
current research interests include embedded systems and system-on-chip design,
security, development support and technology, parallel and energy-efficient
architectures, computer vision, data analytics, and the application of soft
computing and image processing techniques for real-world problems.
---|---