txt/2203.10094.txt

K. López de Ipiña  et al., "Selection of entropy based features for the analysis of the Archimedes' spiral applied to 
essential tremor,"  2015 4th International Work Conference on Bioinspired Intelligence (IWOBI ), 2015, pp. 157 -
162, doi: 10.1109/IWOBI.2015.7160160 . 
_________________________________________________________________________________________  
Selection  of entropy  based features  for the  analysis  of 
the Archimedes’  spiral  applied to essential  tremor  
 
K. López  de Ipiña,  M. Iturrate,  P. Calvo,  B. Beitia,  J. 
Garcia -Melero  
Universidad  del País  Vasco/Euskal Herriko  Unibertsitatea  
{karmele.ipina, mikel.iturrate, itziar.gurruchaga,  
mariablanca.beitia,  joseba.garcia}@ehu.eus  
 
A. Bergareche,  P. De  la Riva,  J.F. Marti -Masso,  
BioDonostia  Health  Institute,  Donostia,  Spain  
{jesusalberto.bergarecheyarza,  patricia.delarivajuez,  
josefelix.martimasso}@osakidetza.eus  M. Faundez -Zanuy,  E. Sesa -Nogueras,  J.Roure  
Escola  Universitaria  Politècnica  de Mataró  (UPF),  
Tecnocampus  
{faundez,  sesa, roure  }@tecnocampus.cat  
 
J. Solé-Casals  
Data  and Signal  Processing  Group.  University  of Vic – 
Central University of Catalonia  
[email protected]  
 
Abstract—: Biomedical systems are regulated by interacting  
mechanisms that operate across multiple spatial and temporal  
scales  and produce  biosignals  with  linear  and non-linear  
information inside. In this sense entropy could provide a useful  
measure about disorder in the system, lack of information in  
time -series and/or irregularity of the signals. Essential tremor  
(ET) is t he most common movement  disorder,  being 20 times  
more  common  than  Parkinson’s  disease,  and 50-70%  of this 
disease cases are estimated to be genetic in origin. Archimedes  
spiral drawing is one of the most used standard tests for clinical  
diagnosis. This work, on selection of nonlinear biomarkers from  
drawings and handwriting, is part of a wide -ranging cross study  
for the diagnosis  of essential  tremor  in BioDonostia  Health  
Institute. Several entropy algorithms are used to generate non - 
linear feayures. The automatic analysis system consists of several  
Machine Learning  paradigms.  
 
Keywords — Permutation entropy; Essential tremor; Automatic  
drawing  analysis;  Archimedes’  spiral;  Non-linear  features;  
automatic  selection  of features  
I. INTRODUCTION  
Biomedical  systems  are regulated  by interacting  
mechanisms that operate across multiple spatial and temporal  
scales  and produce  biosignals  with linear  and non-linear  
information  inside.  Output  variables  of real systems  often  have  
complex fluctuations that are not only due to noise but also  
contain  information  about  the intrinsic  dynamics  and the 
underlying system. In all cases the dynamics’ global aspects  
can be somehow captured by classic linear methods, but the  
different  approaches  are not equivalent  to discern  all the 
relevant physical details [1,2]. In this sense the measurement  
of non -linear features such as the system entropy are essential  
and useful tools to analyse the system stage. The analysis of  
system entropy provides not only the probability distributions  
of the possible  state of a system but  also the information  
encoded in it [1]. However the applicability of entropy based  
methodologies  depends  on particular  characteristics  of the 
data,  such as stationarity,  time series  length,  variation  of the parameters, level of noise contamination, etc., and important  
information  may be codified  also in the temporal  dynamics,  an 
aspect which is not usually taken into account [1,3]. Time  
series generated by biological and biomedical systems most  
likely contain  deterministic and stochastic components [4].  
Classical methods of  signal  and noise  analysis  can quantify  the 
degree  of regularity  of a time series  by evaluating  the 
appearance of repetitive patterns, but most such methods only  
model linear components without introducing any information  
about non -linearity, irregularities or stochastic components.  
This complex  information  could  be essential  when  subtle  
changes are anal ysed. Massimiliano Zanin et al [1] present a  
review  based  on biomedical  applications  which  includes  
analysis  about  EEG,  anesthesia,  cognitive neuroscience  or 
heart rhythms. Among biomedical applications, the related to  
neurological diseases are a challenge due to their variability  
and impact in society. Essential tremor is one of the most  
common.  
Essential  tremor  is a condition  that affects  individuals  
worldwide, being 20 times more common than Parkinson’s  
disease.  The prevalence  of essential  tremor  (ET)  in the 
western world is of about 0.3 -4.0%, 40 years of old males and  
females  are affected  more  or less equally  with an incidence  of 
23.7 per 100,000  people  per year.  Studies  in the elderly  
suggest that prevalence in these  patients ranges between 3.9%  
and 14.0%. 50 -70% of essential tremor cases are estimated to  
be genetic in origin [5]. Essential tremor presents itself as a  
rhythmic tremor (4 –12 Hz) that occurs only when the affected  
muscle  is exerting  effort.  The amplitude  of the tremor  
increases its variability with regard to age  but there is no  
gender predilection. Physical or mental stress could make the  
tremor worse and the prevalence of Parkinson's disease, in  
people with essential tremor is greater than in the general  
population.  Parkinson's  disease  and parkinsonism can  also 
occur simultaneously with essential tremor. With regard to  
symptoms  hand  tremor  predominates  (as it does in Parkinson’s  
disease), and  occurs in nearly all cases,  followed  by head  
tremor,  voice  tremor, neck,  face, leg,  tongue  and trunk  tremor.  K. López de Ipiña  et al., "Selection of entropy based features for the analysis of the Archimedes' spiral applied to 
essential tremor,"  2015 4th International Work Conference on Bioinspired Intelligence (IWOBI ), 2015, pp. 157 -
162, doi: 10.1109/IWOBI.2015.7160160 . 
_________________________________________________________________________________________  
ET is characterized  by postural  and kinetic  tremor  which  often  
maximally  affects  the hands.  PD and ET can appear  in 
individuals  of the same  family  [5]. 
The clinical  hallmark and  earliest  manifestation  of the 
disorder is essential to manage and palliate the symptoms. All  
these symptoms lead to impaired performance in everyday  
activities. Approaches to the early diagnosis of ET have in the  
past few years made significant advances in the deve lopment  
of reliable  clinical  biomarkers.  Despite  the usefulness  of 
biomarkers, the cost and technology requirements involved  
make it impossible to apply such tests to all patients with  
motor  troubles.  Given  these  problems,  non-invasive  intelligent  
techniques of diagnosis may become valuable tools for early  
detection  of disorders.  Non-technical  staff in the habitual  
environments of the patient could use these methodologies,  
without altering or blocking the patients' abilities, as speech  
analysis, han dwriting or drawing analysis involved in these  
techniques is not perceived as a stressful test by the patient.  
Moreover,  these  techniques  are very low -cost and do not 
require extensive infrastructure or the availability of medical  
equipment.  They  are thus  capable  of yielding information  
easily, quickly, and inexpensively [6 -8]. It is well established  
that handwritten tasks can be used for diagnosis of essential  
tremor. In this sense Archimedes’s spiral is one of the most  
used standard  tests in clinical  diagnosis  [14].  
In the past, the analysis  of handwriting  had to be 
performed  in an offline  manner.  Only  the writing  itself  
(strokes on a paper) were available for analysis. Nowadays,  
modern capturing devices, such as digitizing tablets and pens  
(with  or without  ink) can gather  data without  losing  its 
temporal  dimension.  When  spatiotemporal  information  is 
available, its analysis is referred as online. Modern digitizing  
tablets  not only gather  the x and y coordinates  that describe  the 
movement of the writing device as it changes its position, but  
it can also collect other data, such as the pressure exerted by  
the writing device on the writing surface, to the azimuth, the  
angle of the pen in the horizontal plane, the altitude, the angle  
of the pen with respect the vertical axis [9]. This gives the  
possibility to  analyze  not only static  (“off -line”.)  but also 
dynamic  (“on-line”)  features  [10].  Figure 1. . The Archimedes’ spiral drawing performed by an  
individual  with essential  tremor.  
This work is part of a wide -ranging cross study for the  
diagnosis of essential tremor.  The general transversal study is  
focused to characterize ET (Biodonostia Health Institute) in a  
study  based  on families  with identified  genetics  loci. 
Archimedes’s spiral h as been selected for the evaluation of  
nonlinear  biomarkers  from drawings and handwriting.  The 
presence of integrated features of other diseases such as stress  
is also analyzed. In the next sections not only classical linear  
features  static  and dynamics  but also non-linear  features  based  
on several  entropy  algorithms  will be analyzed.  In that 
biomarker selection, automatic methodologies will be used.  
Finally  an automatic  analysis  system  based  on Machine  
Learning  paradigms  measures  the quality  of the selec ted 
features.  
II. MATERIALS  
The acquisition is carried out using an Intuos Wacom 4  
digitizing  tablet.  The pen tablet  USB  [11] captures  the 
following information. The tablet acquired 100 samples per  
second including the spatial coordinates ( x, y), the pressure,  
and azimuth and altitude angles. Using this set of dynamic  
data, further information can be inferred, such as acceleration,  
velocity,  instantaneous  trajectory  angle,  instantaneous  
displacement,  tangential  acceleration,  curvature  radius,  
centripetal accele ration, etc [12].  The database BIODARW  
consists of 21 control people (CR) and 29 ET people with  
identified  genetics  loci and register  of electrophysiological  test 
(EPT) and fMRI. The test consists of a line, the Archimedes’  
spiral drawing and handwriting w ith dominant hand and non - 
dominant hand. Table 1 summarizes the features of the group  
with ET with regard to EPT, diagnosis and demography [13].  
In this work  Archimedes’  spiral  is used (Figure  1). The 
database presents variability with regard to: tremor frequency,  
amplitude and pattern, diagnosis scale and demography data  
(age and gender).  From the original database, a subset of the  
samples  of Archimedes’s  spiral  is selected.  Moreover  from  the 
group wit h ET, only the samples of the hand with essential  
tremor  are selected.  Thus  this sub-database  BIODARWO  
consists of 51 samples:  27 samples of the control group and  
24 samples  of the group  with ET. 
III. METHODS  
A. Feature  extraction  
The research  presented  here is in the nature  of a 
preliminary experiment; its aim is to define thresholds for a  
number of biomarkers related to handwriting. It forms part of  
a broader study focused on early ET detection. Feature search  
in this work aims at pre clinical evaluation so as to formulate  
useful  tests for ET diagnosis  [5,13,14].  
1) Linear  features  
In this study, we aim at automatically distinguishing of  
handwriting between an ET patient and a healthy subject by  
analyzing  different  linear  features  (LF) and their variants  
(max, min, mean and median) in handwriting: time, spatial  
components, pressure, speed, acceleration, zero crossing rate,  
and spectral components  for on-surface  and in-air signals.  
K. López de Ipiña  et al., "Selection of entropy based features for the analysis of the Archimedes' spiral applied to 
essential tremor,"  2015 4th International Work Conference on Bioinspired Intelligence (IWOBI ), 2015, pp. 157 -
162, doi: 10.1109/IWOBI.2015.7160160 . 
_________________________________________________________________________________________  
i 2) Non-linear  feature:  entropy  
Entropy is  a measure  of disorder  in physical  systems,  and n−m  
Am(r) = (n − m)−1 ∑ Am(r), (5) 
also a basic  quantity  with multiple  field-specific  
interpretations.  It has  been  associated with disorder, state - 
space volume, or lack of information [1,2,15,16]. When try to  
analyze information  content,  the Shannon entropy is  often  
considered as the classic, foundational and most natural one  
[3,4,17]. Richman et al., analyze that entropy, as it relates to  
dynamical systems, is the rate of information production [18].  
On the one hand some authors points that the calculation of  
entropy  usually  requires  very long data sets that can be 
difficult or impossible to obtain mainly for biomedical signal.  
On the other  hand methods for estimation of the entropy of a  
system represented  by a time series are not well suited to  
analysis  of the short  and noisy  data sets encountered  in 
biomedical studies [1,18]. Several proposals for calculating  
entropy used in  this work  are presented.  
The entropy H(X) of a single discrete random variable  is 
a measure of its average uncertainty. Shannon entropy [17] is  
calculated by  the equation:  
 
H(X) = − ∑ p(xi) logp (xi) = − E[log p(xi)]          (1) 
Xi∈Θ 
Where  X represents  a random  variable  with a set of values  
             and probability  mass  function  
p(xi) = P r {X = x i}, xi ∈ Θ, and E represents  the expectation  
operator.  Note  that p logp  = 0 if p = 0. 
The Approximate Entropy (ApEn) is a statistical measure  
that smooth  transient  interference  and can suppress  the 
influence  of noise  by properly  setting  of the algorithms  
parameters.  It can be employed  in the analysis  of both 
stochastic and deterministic signals [19,20]. This is crucial in  
the case of biological signals, which are outputs of complex  
biological networks and may be deterministic or stochastic, or  
both.  ApEn  provides a  model -independent  measure  of the  
irregularity of the signals. The algorithm summarizes a time  
series  into a non-negative  number,  with higher  values  
representing  more  irregular  systems  [19,20].  The method  
examines time series for similar epochs [21]: more frequent  
and more  similar  epochs  lead to lower  values  of ApEn.  
ApEn (m, r, n ) measures  for n points,  sequences  that are 
similar for m points remain similar, within a tolerance r, at the  
next point. Thus a low value of ApEn reflects a high degree of  
regularity. ApEn algorithm counts each sequence as matching  
itself.  In order  to reduce  this bias, Sample  Entropy  
SampEn (m, r, n ) was developed,  which  does not count  self- 
matches.  
The sample  entropy  statistic  SampEn (m, r, n)is defined  as: 
SampEn (m, r, n) = lim{−ln(Am(r)/Bm(r))} i=1  
The scalar  is the tolerance  for accepting  matches.  In the 
present  investigation,  we used the parameters  recommended  in 
[22],  with m = 2  and r = 0.2  (standard  deviation  of the 
sources is normalized to 1). SampEn is a robust quantifier of  
complexity as for instance EEG signals [23], and can be used  
to detect  of artifacts  in EEG  recordings  [24].  
Permutation  entropy directly accounts  for the temporal  
information contained in the time series; furthermore, it has  
the quality  of simplicity,  robustness  and very low 
computational cost [1,3,4]. Bandt and Pompe [25] introduce a  
simple  and robust  method  based  on the Shannon  entropy  
measurement  that takes  into account  time causality  by 
comparing  neighboring  values  in a time series.  The 
appropriate symbol sequence arises naturally from the time  
series, with no prior knowledge assumed [1]. The Permutation  
entropy is calculated for a given time series {x1, x2, … , x n} as a 
function of the scale factor . In order to be able to compute  
the     permutation     of      a      new      time      vector      Xj, 
St = [Xt, X t+1, … , X t+m−1 ] is generated  with the embedding  
dimension  and then arranged  in an increasing  order:  
[Xt+j1−1 ≤ X t+j2−1 ≤ ⋯ ≤ X t+jn−1]. Given  different values,  
there   will   be   m! possible   patterns   , also known  as 
permutations. If f(π) denotes its frequency in the time series,  
its relative  frequency  is p(π) = f(π)⁄(L⁄s − m + 1 ). The 
permutation entropy  is then defined as:  
m! 
PE = − ∑ p(πi) ln p(πi) (6) 
i=1  
B. Automatic  classification  
The main goal of the present work is feature search in  
handwriting aiming at preclinical evaluation in order to define  
tests for ET diagnosis. These features will define the control  
group (CR) and the essential tremor group (ET). A secondary  
goal is the optimization of computational cost with the aim of  
making  these techniques useful for real -time applications in  
real environments.  Thus,  automatic  classification  will be 
modeled  with this in mind.  We used five different  classifiers:  
 A Support Vector Machine (SVM) with polynomial  
kernel  
 A multi layer perceptron (MLP) with neuron number  
in hidden  layer  (NNHL)  = 
max(Attribute/Number+Classes/Number) and training  
step (TS) = NNHL*10  
 k-NN Algorithm.  
The WEKA  software  suite  [26] has  been  used to carry  out 
 
 
with  n→∞  
= −ln(A/B ), (2) the experiments.  The results  were  evaluated  using  
Classification  Error  Rate (CER,  %). For training  and 
validation  steps  we used k-fold cross -validation  with k=10.  
A = [(n − m − 1)(n − m)/2]Am(r), (3) 
and 
B = [(n − m − 1)(n − m)/2]Bm(r). (4) 
Bm(r) is the probability  that  two sequences  match  for 
   points.  Similarly,  Am(r) is the probability  that  two  
sequences match  for m + 1 points:  Cross -validation is a robust validation method  for variable  
selection [27]. Repeated cross -validation (as calculated by the  
WEKA environment) allows robust statistical tests. We also  
use the measurement  provided  automatically  by WEKA  
“Coverage  of cases”  (0.95  level)  that is the confidence  interval  
at 95% level.  
K. López de Ipiña  et al., "Selection of entropy based features for the analysis of the Archimedes' spiral applied to 
essential tremor,"  2015 4th International Work Conference on Bioinspired Intelligence (IWOBI ), 2015, pp. 157 -
162, doi: 10.1109/IWOBI.2015.7160160 . 
_________________________________________________________________________________________  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 2. CER (%) for paradigms with linear and non -linear  
features sets based on entropy: Shannon, ApEn, SmEn and  
Permutation Entropy.  
IV. RESULTS  AND  DISCUSSION  
The experimentation  has been  carried  out with the 
balanced subset BIODARWO. The goal of these experiments  
was to examine  the potential  of entropy  algorithms  and 
selected  features  for automatic  measurement  of the 
degradation of Archimedes’s spiral drawing with ET. Thus,  
previously  defined  feature  sets have  been  evaluated  in order  to 
properly define control and ET groups. In a first stage linear  
and non-linear  features  have  been  extracted  by several  
methods described in section III: Shannon entropy, ApEn and  
SmEn.  Automatic  classification  by described  (section  III) 
paradigms was performed over the database. The results of  
CER (%) for paradigms with linear and non -linear features  
sets are summarized in Figure 2. For both algorithms m=2,3  
and tolerance  r=0.2.  
 Non Linear Features (NLF) sets increase about 7% 
the features  number  
 Non-linear  features  sets improve  the results  for all the 
paradigms  
 Shannon entropy outperforms LF (LFSE) for MLP  
and k-NN 
 ApEn  improve  system  performance  for m=3 with 
MLP and SVM.  k-NN has better performance for  
m=2.  
 SmEn with m=3 appears as the best option for all  
paradigms.  
 The best option is SmEn with m=3 and MLP with a  
CER  of 15.69%.  
 
In a second  stage  permutation  entropy  is evaluated  for 
different orders m and time delay. In our particular case and  
due the signals  are composed  from  5000 -1000  samples,  and  
parameter was fixed until m=7. The results are also compared  
with SmEn with m=3 and tolerance of r=0.2. The results are  
shown  in Figure  3.  PE improves  the results  in most  of the cases  
 The best results  are obtained  with PE and m=7,  t=7. 
 MLP  obtains  the best results  for this last option  
 The best option  is PE-m7t7  with MLP  and CER  of 
15.65%.  
 Good  results  are achieved  even  with k-NN with less 
computational  cost. 
V. CONCLUSIONS  
This work,  on selection  of nonlinear  biomarkers  from  
drawings  and handwriting, is part of a wide -ranging cross  
study for the diagnosis of essential tremor which is developed  
in Biodonostia Health Institute. Specifically the main goal of  
the present work is the analysis of features in Archimed es’s 
spiral drawing, one of the most used standard tests for clinical  
diagnosis of ET.  In this sense entropy based features have  
been  add to a set of classical  linear  features  (static  and 
dynamics).  Several  entropy  algorithms  have  been  evaluated  by 
an automatic  analysis  system  consists  of several  Machine  
Learning. The best option is MLP with permutation entropy  
and good results are obtained even with k-NN. Then these  
new biomarkers will be integrated in future works with those  
obtained in the Biodonosti a study.  It should be highlighted  
that the use of this technology  could  provide  undoubted  
benefits towards the development of more sustainable, low  
cost, high quality, non -invasive technologies. These systems  
are easily adaptable to the user and environment, and from a  
social and economic point of view can be very useful in real  
complex  environments.  In future  works  new non-linear  
features,  entropy  algorithms  and automatic  selection  
methodologies  will be used.  
 
 
Figure 3. CER (%) for the paradigms with the references  
Linear Features (LF) and Shannon Entropy (SE) and other  
entropy  paradigms  
K. López de Ipiña  et al., "Selection of entropy based features for the analysis of the Archimedes' spiral applied to 
essential tremor,"  2015 4th International Work Conference on Bioinspired Intelligence (IWOBI ), 2015, pp. 157 -
162, doi: 10.1109/IWOBI.2015.7160160 . 
_________________________________________________________________________________________  
Acknowledgments  
This work has been partially supported by the University  
of the Basque  Country  by UPV/EHU —58/14  project,  
SAIOTEK from the Basque Government, University of Vic - 
Central  University  of Catalonia  under  the research  grant  
R0904, and the Spanish Ministerio de Ciencia e Innovación  
TEC2012 -38630 -C04-03. 
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