en opy
A icle
New Fas ApEn and SampEn En opy Algo i hms
Implemen a ion and Thei Applica ion o
Supe compu e Powe Consump ion
Jiˇ í Tomˇcala
IT4Inno a ions, VSB—Technical Uni e si y o Os a a, 17.lis opadu 2172/15, 70833 Os a a-Po uba,
Czech Republic; [email p o ec ed]
Recei ed: 20 June 2020; Accep ed: 31 July 2020; Published: 5 Augus 2020


Abs ac :
App oxima e En opy and especially Sample En opy a e ecen ly equen ly used
algo i hms o calcula ing he measu e o complexi y o a ime se ies. A lesse known ac is ha
he e a e also accele a ed modi ica ions o hese wo algo i hms, namely Fas App oxima e En opy
and Fas Sample En opy. All hese algo i hms a e e ec i ely implemen ed in he R so wa e package
TSEn opies. This pape con ains no only an explana ion o all hese algo i hms, bu also he p inciple
o hei accele a ion. Fu he mo e, he pape con ains a desc ip ion o he unc ions o his so wa e
package and hei pa ame e s, as well as simple examples o using his so wa e package o calcula e
hese measu es o complexi y o an a i icial ime se ies and he ime se ies o a complex eal-wo ld
sys em ep esen ed by he cou se o supe compu e in as uc u e powe consump ion. These ime
se ies we e also used o es he speed o his package and o compa e i s speed wi h ano he R
package p acma. The esul s show ha TSEn opies is up o 100 imes as e han p acma and
ano he impo an esul is ha he compu a ional imes o he new Fas App oxima e En opy and
Fas Sample En opy algo i hms a e up o 500 imes lowe han he compu a ional imes o hei
o iginal e sions. A he e y end o his pape , he possible use o his so wa e package TSEn opies
is p oposed.
Keywo ds:
en opy; measu e o complexi y; app oxima e en opy; sample en opy;
as app oxima e en opy; as sample en opy; benchma king; so wa e compa ison; supe compu e
powe consump ion
1. In oduc ion
Nowadays, i is necessa y in many scien i ic ields o ind ou whe he a ce ain ime se ies is
chao ic and wha is he deg ee o i s chao ic beha io . The e a e se e al es s o de ec chao ic dynamics
o ime se ies such as a 0–1 es o chaos [
1
] o Shilniko chaos condi ion [
2
]. Howe e , hese es s
can only be used o dis inguish be ween egula and chao ic dynamics. Howe e , i i is necessa y
o de e mine he le el o de e minism in he analyzed ime se ies, i would be mos app op ia e o
calcula e i s en opy.
In gene al, he en opy o a sys em de e mines he deg ee o i s diso de [
3
]. This s a e a iable is
he highe he mo e diso de ed he sys em is. This le el o diso de hen p opo ionally a ec s he
deg ee o p edic abili y o such a sys em. The cha ac e is ics o his sys em a e e lec ed in he ime
se ies c ea ed by his sys em, so wi h some simpli ica ion i can be a gued ha he en opy o his ime
se ies would be a measu e o he unp edic abili y o his sys em and also a measu e o i s complexi y.
In ecen yea s, he o en used algo i hms o calcula ion o a measu e o ime se ies complexi y
a e App oxima e En opy (ApEn) [
4
] and Sample En opy (SampEn) [
5
]. A his poin , howe e , i is
En opy 2020,22, 863; doi:10.3390/e22080863 www.mdpi.com/jou nal/en opy
En opy 2020,22, 863 2 o 10
impo an o say ha hese algo i hms do no calcula e en opy in a ma hema ical sense, bu essen ially
de e mine he deg ee o complexi y o he analyzed ime se ies in a ious ways.
The App oxima e En opy was o iginally de eloped o analyze medical da a, such as hea a e,
and la e i s applica ion was ex ended in inance, psychology, e c. I is de ined as ollows:
ApEn(x,m, ) =
=1
N−m+1"N−m+1
∑
i=1
log |ji|
N−m+1#−1
N−m"N−m
∑
i=1
log |ki|
N−m#, (1)
whe e
ji={ξ| kyi−yξk ≤ ∧ξ∈ h1, N−m+1i},
ki={ξ| kzi−zξk ≤ ∧ξ∈ h1, N−mi},
yi= [xi,xi+1, ..., xi+m−1],zi= [xi,xi+1, ..., xi+m],N=|x|.
This Equa ion (1) looks o simila sub-sequences
yi
esp.
zi
o leng hs
m
esp.
m+
1. I i
is assumed ha he e alua ion o
kyi−yξk ≤
is one elemen a y ope a ion, hen
N−m+
1
ope a ions a e equi ed o calcula e each
|ji|
. The o al numbe o ope a ions would hen be
2N2+N(6−4m) + 2m2−6m+7, so he ime complexi y o he App oxima e En opy is O(N2).
Ano he algo i hm used o ime se ies complexi y analysis is he Sample En opy, which is
sligh ly simple han he App oxima e En opy algo i hm. Al hough i equi es ewe ope a ions o
calcula e, i will be shown below ha i has he same ime complexi y. This algo i hm was p oposed in
2000 by Richman and Moo man o assess he complexi y o physiological ime se ies.
The de ini ion o Sample En opy is:
SampEn(x,m, ) = log ∑N−m+1
i=1|bi|
∑N−m
i=1|ai|!, (2)
whe e
bi={ξ| kyi−yξk ≤ ∧ξ∈ h1, N−m+1i i},
ai={ξ| kzi−zξk ≤ ∧ξ∈ h1, N−mi i},
yi= [xi,xi+1, ..., xi+m−1],zi= [xi,xi+1, ..., xi+m],N=|x|.
No e ha se s
bi
and
ai
a e di e en om se s
ji
and
ki
o App oxima e En opy. They do no
con ain index
i
, so calcula ing
|bi|
esp.
|ai|
akes only
N−m
esp.
N−m−
1 ope a ions. Howe e ,
i also b ings wi h i he possibili y ha he sum o all
|ai|
can be ze o. The o al numbe o ope a ions
is hen 2N2+N(2−4m) + 2m2−2m+1 and he ime complexi y is O(N2)again.
2. New Fas Algo i hms
Quad a ic ime complexi y o he App oxima e and Sample En opy algo i hms was he main
eason why hei accele a ed modi ica ions Fas App oxima e En opy and Fas Sample En opy ha e
been p oposed in [
6
]. The o iginal App oxima e En opy algo i hm looks o mu ually simila
sub-sequences by compa ing all possible sub-sequences wi h each o he . This me hod is appa en ly an
ex emely ime-consuming and also he same pai o sub-sequences is unnecessa ily compa ed wice.
The accele a ion o he modi ied Fas App oxima e En opy algo i hm is ha , once i inds wo simila
sub-sequences, all o he sub-sequences in he same neighbo hood will be ma ked as al eady included
in some neighbo hood. An impo an ac is ha hese sub-sequences a e no longe aken in o accoun
when sea ching o o he simila sub-sequences, he eby speeding up hese u he sea ches.
The o mula o he Fas App oxima e En opy hen eads as ollows:
En opy 2020,22, 863 3 o 10
Fas ApEn(x,m, ) =
=1
Nm"Nm
∑
i=1
log |si,m|
Nm#−1
Nm+1"Nm+1
∑
i=1
log |si,m+1|
Nm+1#, (3)
whe e
si,m={ξ|(kyi−yξk ≤ )∧(ξ/∈sj,m,j<i)},yi= [xi,xi+1, ..., xi+m−1],
si,m
is a se o sub-sequences o leng h
m
belonging o he
i
- h neighbo hood, and
Nm
is numbe o
hese neighbo hoods.
The same p inciple is used in he modi ied Fas Sample En opy algo i hm:
Fas SampEn(x,m, ) = log ∑Nm
i=1|si,m|
∑Nm+1
i=1|si,m+1|!, (4)
whe e
si,m={ξ|(kyi−yξk ≤ ,ξ6=i)∧(ξ/∈sj,m,j<i)},
yi= [xi,xi+1, ..., xi+m−1],
si,m
is a se o sub-sequences o leng h
m
belonging o he
i
- h neighbo hood, and
Nm
is numbe o
hese neighbo hoods.
In his way, he sea ch ime o simila sub-sequences and hus he o al calcula ion imes can be
signi ican ly educed. The na u e o his modi ica ion implies ha he numbe o ope a ions is di e en
o di e en ime se ies o he same leng h, bu he numbe o ope a ions o he bes case and o
he wo s can be de e mined. Fo he bes case, he numbe o ope a ions equi ed o calcula e bo h
accele a ed algo i hms is equal o 2
N−
2
m+
7. In he wo s case, he numbe o ope a ions equi ed
o compu e Fas App oxima e En opy is equal o
N2+N(
5
−
2
m) + m2−
5
m+
7 and o compu e Fas
Sample En opy equals N2+N(2−2m) + m2−2m+1.
Thus, i can be seen om he abo e ha he ime complexi y o bo h accele a ed algo i hms anges
om
O(N)
o
O(N2)
. An in e es ing ac is ha he abo e-men ioned bes case co esponds o a ime
se ies wi h a low deg ee o diso de , which also implies a low Fas ApEn o Fas SampEn alue o
analyzed ime se ies. On he con a y, he wo s case co esponds o a ime se ies wi h a high deg ee o
diso de and hence a high Fas ApEn o Fas SampEn alue. The amoun o compu a ional ope a ions
and he ime complexi y o hese accele a ed algo i hms is hus in e es ingly dependen on he alue
hey calcula e.
3. Supe compu e Powe Consump ion Time Se ies
Fo he pu poses o his a icle, access o he ime se ies ep esen ing he elec ical consump ion
o i s in as uc u es [
7
] was ob ained. The powe consump ion o his supe compu e depends on
many ac s ha may depend nonlinea ly on each o he , and he numbe o such ac s is so la ge ha i
is in ac able o modeling, so i can be said ha his supe compu e in as uc u e is so called complex
sys em in e ms o i s powe consump ion. Thus, he eco ded ime se ies o elec ical consump ion is a
ime se ies wi h complex dynamics, o simply a complex ime se ies.
Fo his eason, his ime se ies is a good ep esen a i e o eal-wo ld ime se ies o es ing and
compa ing he pe o mance o complexi y deg ee analyze s. The esul s o hese es s and pe o mance
compa ison a e shown in Sec ion 4.
A his poin , i should be men ioned why an analysis o a complexi y deg ee o his ime se ies
o powe consump ion is use ul. This analysis is usually used in medicine whe e he cou se o an
elec oca diog am (ECG) o elec oencephalog am (EEG) is analyzed. Fo example, based on EEG
complexi y deg ee, he beginning, end, and he ype o epilep ic seizu e can be iden i ied.
En opy 2020,22, 863 4 o 10
Howe e , as shown in [
8
], he ime se ies complexi y can also be an indica o o i s p edic abili y
in o ecas ing i s u u e cou se. A high deg ee o he complexi y in such a case sugges s ha any
p edic ion a a gi en poin in he ime se ies is e y likely o be a mo e e oneous han a p edic ion
a ano he poin in he analyzed ime se ies whe e he complexi y deg ee is lowe . One could e en
say ha , om a ce ain deg ee o ime se ies complexi y, any o ecas is wo hless and can easily be
eplaced by a simple a i hme ic mean.
By analyzing he complexi y o his ime se ies o powe consump ion, i is possible o de e mine
in ad ance whe he he p edic ion o i s de elopmen can be c edible. This is e y impo an i he
o ecas indica es he possibili y o o e loading he powe sys em.
Accele a ed e sions o en opy algo i hms such as Fas App oxima e En opy and Fas Sample
En opy om he so wa e package
TSEn opies
[
9
] may be help ul in eally quickly de e mining he
plausibili y o u u e ime se ies p edic ions. See Appendix A o ins alla ion ins uc ions.
The calcula ed wa e o ms o ApEn, SampEn, Fas ApEn, and Fas SampEn alues oge he wi h
he cou se o he analyzed powe consump ion ime se ies a e shown in he g aph in Figu e 1. As can
be seen om his image, he alues o he new accele a ed algo i hms a e a di e en le els han hei
o iginal e sions. This ac is discussed in de ail in [
6
], whe e i is also shown ha hei abili y o de ec
an inc ease o dec ease in complexi y o he ime se ies is oughly he same as in hei o iginal e sions.
A mo e de ailed analysis o his phenomenon goes beyond he scope o his a icle.
Figu e 1. The no malized supe compu e powe consump ion ime se ies, which is he subjec o he
analysis, along wi h calcula ed alues o App oxima e En opy (ApEn), Sample En opy (SampEn),
Fas App oxima e En opy (Fas ApEn), and Fas Sample En opy (Fas SampEn). A loa ing window
wi h a wid h o 2880 min was used o he calcula ion.
He e is an example o a simple sou ce code, which can be used o calcula e he cou se o he
Sample En opy o powe consump ion, he measu ed alues o which a e s o ed in he ime se ies
powe TS. The loa ing calcula ion window in his case is 2880 in leng h:
lib a y(TSEn opies)
SampEn_powe TS <- nume ic()
o (i in 1:7120) {
SampEn_powe TS[i] <- SampEn(powe TS[i:(i+2880)])
}
plo (powe TS[1:10000], ype="l")
lines(x = 1440 + (1:leng h(SampEn_powe TS)), y = SampEn_powe TS, ype = "l",
l y = 3, col = "da kg een")
En opy 2020,22, 863 5 o 10
# x = 1440 + ... is he o se o he middle o he calcula ion window
This chunk o code assumes ha he powe consump ion ime se ies powe TS is al eady
no malized, which should be done in ad ance. The displayed cou se o he calcula ed SampEn
alue is shi ed by hal a calcula ion window so ha i s alues in he g aph a e loca ed in he middle o
he a ea om which hey we e calcula ed.
4. Benchma ks and Compa ison
Ano he so wa e p og am ha can calcula e bo h App oxima e En opy and Sample En opy is
he R package
p acma
[
10
]. The e o e, his R package was chosen o compa ison wi h he
TSEn opies
package. The
p acma
package is a e sa ile so wa e ha allows one o calcula e many di e en
p ac ical ea u es. I p o ides a la ge numbe o unc ions om nume ical analysis and linea
algeb a, nume ical op imiza ion, di e en ial equa ions, ime se ies, plus some well-known special
ma hema ical unc ions.
Th ee ypes o ime se ies we e used in he compa a i e es s. Fi s ly, he eal-wo ld ime se ies
powe TS ep esen ed by he powe consump ion o he supe compu e in as uc u e. Fu he mo e,
an a i icial ime se ies sinTS whose cou se is de e mined by he sine unc ion. This ime se ies
is assumed o be a low deg ee o complexi y. I will be in e es ing o obse e he di e ence in
compu a ional imes compa ed o he las ype o ime se ies no mTS wi h a high deg ee o complexi y,
which is a andom signal wi h no mal p obabili y dis ibu ion.
F om he poin o iew o he ocus o his pape , he mos impo an a e calcula ions o ApEn,
SampEn, Fas ApEn, and Fas SampEn alues o he supe compu e powe consump ion ime se ies
powe TS om he eal wo ld and he e o e he a en ion will be gi en especially on hem. The de ailed
esul s o calcula ions o a i icially c ea ed ime se ies as sinTS and no mTS ha e al eady been
published in [6].
The measu ed un- imes o ime se ies powe TS a e depic ed in he g aphs in Figu e 2, whe e he
package unc ions implemen ed in C we e used, and, in Figu e 3, whe e he package unc ions
implemen ed in R we e used. The un- imes achie ed by he package
p acma
can also be ound in
hese wo igu es, which allows an immedia e g aphical compa ison o he pe o mance wi h he
TSEn opies package.
As men ioned in Sec ion 2, he numbe o compu a ional ope a ions and hence he compu a ional
ime equi ed o new modi ied algo i hms depends on he deg ee o diso de in he analyzed ime
se ies. This is one o he easons why all h ee ypes o hese ime se ies we e used o es he
pe o mance o his so wa e. Since he lowes le el o diso de is assumed o he sinTS, i is expec ed
ha i s un- imes will also be he lowes .
A summa y o measu ed un- imes equi ed by bo h accele a ed algo i hms o all h ee ypes o
ime se ies using he
TSEn opies
package is shown in Figu e 4, which p esen s he esul ing un- imes
o he Fas App oxima e En opy algo i hm, and Figu e 5, which p esen s he esul ing un imes o
he Fas Sample En opy algo i hm. These calcula ions we e pe o med o a ious leng hs o ime
se ies om 103 o 107samples.
Du ing he calcula ions, he de aul alues o he pa ame e s we e se o he execu ed unc ions.
Thei se ings can be seen in Appendix B.
The calcula ions we e pe o med on a compu e wi h an In el Co e i5-6200U CPU @ 2.30 GHz
p ocesso , 8 GB RAM SO-DIMM DDR3 1600 MHz and SSD Li eOn L8H-256V2G-HP. The ope a ing
sys em ins alled was Ubun u 18.04.4 (64-bi ).
This es ing could, o cou se, be done on one o he supe compu e clus e s (Salomon, Anselm,
o Ba bo a) and su ely sho e compu a ional imes would be achie ed, bu he e was an e o o
show ha his R package
TSEn opies
and also new algo i hms can handle any ime se ies e en on a
egula compu e .

En opy 2020,22, 863 6 o 10
Figu e 2.
Compa ison o he ime needed o un a ious ypes o algo i hms using he TSEn opies
package unc ions implemen ed in C and he p acma package unc ions. The analyzed ime se ies is
he powe consump ion o he supe compu e .
Figu e 3.
Compa ison o he ime needed o un a ious ypes o algo i hms using he TSEn opies
package unc ions implemen ed in R and he p acma package unc ions. The analyzed ime se ies is
he powe consump ion o he supe compu e .
En opy 2020,22, 863 7 o 10
Figu e 4.
Compa ison o Fas App oxima e En opy compu a ion imes o ime se ies wi h a ious
le els o diso de .
Figu e 5.
Compa ison o Fas Sample En opy compu a ion imes o ime se ies wi h a ious le els
o diso de .
5. Conclusions and Fu u e Wo k
As can be seen om he g aphs o un- imes in Figu es 2–5, he
TSEn opies
package calcula es
bo h App oxima e and Sample En opy much as e han package
p acma
. This is ue no only o
unc ions implemen ed in C (Figu e 2), which we e app oxima ely 100 imes as e , bu e en in R
(Figu e 3), which we e abou 20 imes as e .
I can also be obse ed in Figu es 4and 5 ha he un- imes o he new accele a ed algo i hms
con i med he assump ion o he dependence o compu a ion ime on he analyzed ime se ies diso de
le el. In e es ingly, wi h hese algo i hms, he mul iplica i e di e ence be ween sinTS and no mTS
un- imes o C unc ions was much g ea e (150 imes) han o R unc ions (only 30 imes) o he
same TSEn opies package.
The compu ed alues o all algo i hms as well as hei un- imes we e sligh ly lowe o powe TS
han o no mTS, sugges ing ha he e a e some egula i ies in he powe consump ion wa e o m
ha dis inguish i om a comple ely andom signal. This p o es ha an e en ual p edic ion o i s
de elopmen would make sense. Al hough his inding is no he pu pose o his pape , i ne e heless
indica es he possible use o his so wa e.
Pe haps he mos impo an esul o es ing his
TSEn opies
package seems o be ha he
compu a ional imes o he new Fas App oxima e En opy and Fas Sample En opy algo i hms a e up
o 500 imes lowe in he ime se ies om he eal wo ld han he compu a ional imes o hei o iginal
En opy 2020,22, 863 8 o 10
e sions. This makes his so wa e a eally powe ul ool o he as sea ching o any egula i ies in
huge amoun s o da a. Such ype o sea ch is needed in a wide ange o ields. F om he de ec ion o
epilep ic seizu es in he ECG, h ough he p edic ion o engine gea ailu e by mechanical ib a ion
analysis, o such exo ic issues as he sea ch o in elligen li e mani es a ions in adio signals om he
su ounding uni e se. In all hese cases (and many o he s), he e is a need o apid analysis o huge
amoun s o da a. The new accele a ed Fas App oxima e En opy and Fas Sample En opy algo i hms
implemen ed in he TSEn opies package a e pe ec o his.
As a u u e de elopmen o his so wa e, i seems app op ia e o c ea e a ully pa allelized
e sion, which would be in ended mainly o p ocessing big da a on supe compu e s. Ano he possible
di ec ion ha his so wa e de elopmen could ake is o add a g aphical ou pu o he cou se o he
calcula ed alues along wi h he cou se o he analyzed ime se ies, simila o hose in Figu e 1.
Funding: This esea ch ecei ed no ex e nal unding.
Acknowledgmen s:
This wo k was suppo ed by he Minis y o Educa ion, You h and Spo s om he La ge
In as uc u es o Resea ch, Expe imen al De elopmen , and Inno a ions p ojec “e-INFRA CZ—LM2018140”
and by SGC G an No. SP2020/137 “Dynamic sys em heo y and i s applica ion in enginee ing”, VSB—Technical
Uni e si y o Os a a, Czech Republic.
Con lic s o In e es : The au ho decla es no con lic o in e es .
Appendix A. Ins alla ion o TSEn opies Package
The
TSEn opies
package [
9
] is a ailable om he Comp ehensi e R A chi e Ne wo k (CRAN) a
h ps://c an. -p ojec .o g/package=TSEn opies. The e o e, he TSEn opies package can be ins alled
in he s anda d way using he command:
R> ins all.packages("TSEn opies")
This package does no use any o he R packages, so he e is no need o wo y abou se ing
dependencies. Howe e , he e sion o R i sel should be a leas 3.4.0 [11].
I he ins alla ion was success ul, he package can be loaded by:
R> lib a y(TSEn opies)
Appendix B. TSEn opies Package Con en and Implemen a ion
The package con ains unc ions o calcula ion o App oxima e En opy and Sample En opy
as well as hei modi ied e sions: Fas App oxima e En opy and Fas Sample En opy. All hese
algo i hms a e implemen ed in wo ways.
Fi s , hey a e implemen ed as unc ions only in R. These unc ions include he su ix _R in hei
name. Then, hese algo i hms a e implemen ed again in R, bu , in his implemen a ion, R se es only
as a w appe o hidden in e nal unc ions w i en in C. As i will be seen in he nex chap e , using his
way a signi ican accele a ion o he calcula ion is achie ed. Func ions c ea ed in his way a e indica ed
by an _C su ix.
The package also includes unc ions wi hou any su ix in hei name. These unc ions only igge
unc ions o he same name, bu wi h he su ix _C and pass hem he e y same pa ame e s. These alias
unc ions a e he e o make i easie o use hose as e e sions o implemen a ion. I someone wan s o
use unc ions w i en pu ely in R, hen hey mus explici ly use unc ions wi h he su ix _R. Howe e ,
in e ms o speed, his op ion is no ecommended.
The s uc u e o he pa ame e s is essen ially he same o all implemen ed unc ions. The only
di e ence is in he case o unc ions implemen ing modi ied algo i hms in he de aul alue o he las
pa ame e
. This pa ame e de e mines he highes alue o he dis ance be ween sub-sequences ha
a e s ill conside ed as simila . Fo he o iginal algo i hms, i is ecommended o se his pa ame e
o a alue co esponding o 0.2 imes he s anda d de ia ion o he analyzed ime se ies. Howe e ,
in nume ical expe imen s wi h modi ied algo i hms, i has been ound ha i is mo e app op ia e o
hem o se his pa ame e o a alue equal o 0.15 imes his s anda d de ia ion.
En opy 2020,22, 863 9 o 10
The emaining pa ame e s, as well as hei de aul alues, a e he same o all package unc ions.
The analyzed ime se ies, deno ed as
TS
, is always he i s and manda o y pa ame e . I should be in
he o m o some nume ic R ec o .
An impo an pa ame e is he dimension
dim
, which is op ional. This pa ame e co esponds o
he a iable
m
in all he abo e o mulas. As i can be seen om he package lis ing below, i s de aul
alue is 2.
Fo cla i y, he e is a lis o all he unc ions o his package and hei syn ax usage:
ApEn(TS, dim = 2, lag = 1, = 0.2 * sd(TS))
ApEn_C(TS, dim = 2, lag = 1, = 0.2 * sd(TS))
ApEn_R(TS, dim = 2, lag = 1, = 0.2 * sd(TS))
SampEn(TS, dim = 2, lag = 1, = 0.2 * sd(TS))
SampEn_C(TS, dim = 2, lag = 1, = 0.2 * sd(TS))
SampEn_R(TS, dim = 2, lag = 1, = 0.2 * sd(TS))
Fas ApEn(TS, dim = 2, lag = 1, = 0.15 * sd(TS))
Fas ApEn_C(TS, dim = 2, lag = 1, = 0.15 * sd(TS))
Fas ApEn_R(TS, dim = 2, lag = 1, = 0.15 * sd(TS))
Fas SampEn(TS, dim = 2, lag = 1, = 0.15 * sd(TS))
Fas SampEn_C(TS, dim = 2, lag = 1, = 0.15 * sd(TS))
Fas SampEn_R(TS, dim = 2, lag = 1, = 0.15 * sd(TS))
All o hese unc ions e u n a alue o ype nume ic. Examples o use may be as ollows.
A simple example o calcula ing he App oxima e En opy o a andom ime se ies wi h a no mal
p obabili y dis ibu ion using a unc ion implemen ed in C:
R> lib a y(TSEn opies)
R> no mTS <- no m(1000)
R> ApEn( no mTS)
[1] 1.61967
R> ApEn( no mTS, = 0.1*sd( no mTS))
[1] 1.07962
R> ApEn( no mTS, dim = 4, = 0.5*sd( no mTS))
[1] 0.8241541
o an example o calcula ing he Fas Sample En opy o a ime se ies whose alues a e de i ed om
he cou se o a sine wa e using a unc ion implemen ed in R:
R> lib a y(TSEn opies)
R> sinTS <- sin(seq(0,100*pi,pi/10))
R> Fas SampEn_R(sinTS)
[1] 0.003059666
R> Fas SampEn_R(sinTS, = 0.3*sd(sinTS))
[1] 0.003047234
R> Fas SampEn_R(sinTS, dim = 5, = 0.4*sd(sinTS))
[1] 0.001019888
I is simila o ApEn_C(),ApEn_R(),SampEn(),SampEn_C(),SampEn_R(),Fas ApEn(),
Fas ApEn_C(),Fas ApEn_R(),Fas SampEn(), and Fas SampEn_C() unc ions.