A Semiquan i a i e App oach o S udy Semiquali a i e
Sys ems
Juan An onio O ega, Ra ael M. Gasca, Miguel To o, and Jesús To es
Depa amen o de Lenguajes y Sis emas In o má icos
Uni e si y o Se ille
A da. Reina Me cedes s/n – 41012 – Se illa (Spain)
{o ega,gasca,m o o,j o es}@lsi.us.es
Abs ac . In his pape is p oposed a semiquan i a i e me hodology o s udy
models o dynamic sys ems wi h quali a i e and quan i a i e knowledge. This
quali a i e in o ma ion may be composed by: ope a o s, en elope unc ions,
quali a i e labels and quali a i e con inuous unc ions. A o malism is also
desc ibed o inco po a e his quali a i e knowledge in o hese models. The
me hodology allows us o s udy all he s a es ( ansien and s a iona y) o a
semiquan i a i e dynamic sys em. I also helps o ob ain i s beha iou s pa e ns.
The me hodology is applied o a logis ic g ow h model wi h a delay.
1 In oduc ion
Models o dynamic sys ems s udied in science and enginee ing a e no mally
composed o quan i a i e, quali a i e, and semiquan i a i e knowledge. Di e en
app oxima ions ha e been p oposed when he quali a i e knowledge is aken in o
accoun : ans o ma ion o non-linea o piecewise linea ela ionships, Mon e Ca lo
me hod, cons ain logic p og amming, p obabili y dis ibu ions, causal ela ions,
uzzy se s, and combina ion o all le els o quali a i e and quan i a i e abs ac ion
[5], [9].
We a e in e es ed in he s udy o dynamic sys ems wi h quan i a i e and quali a i e
knowledge. All his knowledge should be aken in o accoun when hese models a e
s udied. Di e en le els o nume ic abs ac ion ha e been p oposed in he li e a u e:
pu ely quali a i e [6], semiquan i a i e [5] [8], nume ic in e al [14] and quan i a i e.
The p oposed me hodology ans o ms a semiquan i a i e model in o a amily o
quan i a i e models. A semiquan i a i e model may be composed o quali a i e
knowledge, a i hme ic and ela ional ope a o s, p ede ined unc ions (log,exp,sin,...),
numbe s and in e als.
A b ie desc ip ion o he p oposed me hodology is as ollows: a semiquan i a i e
model is ans o med in o a se o quan i a i e models. The simula ion o e e y
quan i a i e model gene a es a ajec o y in he phase space. A da abase is ob ained
wi h hese quan i a i e beha iou s o ajec o ies. Techniques o Knowledge
Disco e y in Da abases (KDD) a e applied by means o a language o ca y ou
que ies abou he quali a i e p ope ies o his ime-se ies da abase. This language is
also in ended o classi y he di e en quali a i e beha iou s o ou model. This
classi ica ion will help us o desc ibe he semiquan i a i e beha iou o a sys em by
means o a se o hie a chical ules ob ained by means o machine lea ning
algo i hms.
The e m KDD [1] is used o e e o he o e all p ocess o disco e ing use ul
knowledge om da a. The p oblem o knowledge ex ac ion om da abases in ol es
many s eps, anging om da a manipula ion and e ie al o undamen al
ma hema ical and s a is ical in e ence, sea ch and easoning. Al hough he p oblem o
ex ac ing knowledge om da a (o obse a ions) is no new, au oma ion in he
con ex o da abases opens up many new unsol ed p oblems.
KDD has e ol ed, and con inues o e ol e, om he con luence o esea ch in
such ields as da abases, machine lea ning, pa e n ecogni ion, a i icial in elligence
and easoning wi h unce ain y, knowledge acquisi ion o expe sys ems, da a
isualiza ion, so wa e disco e y, in o ma ion e ie al, and high-pe o mance
compu ing. KDD so wa e sys ems inco po a e heo ies, algo i hms, and me hods
om all o hese ields.
The e m da a mining is used mos by s a is icians, da abase esea che s and mo e
ecen ly by he business communi y. Da a mining is a pa icula s ep in he KDD
p ocess. The addi ional s eps in KDD p ocess a e da a p epa a ion, da a selec ion, da a
cleaning, inco po a ion o app op ia e p io knowledge and p ope in e p e a ion o
he esul s o mining ensu e he use ul knowledge is de i ed om he da a [11]. A
de ailed desc ip ions o hese s eps may be ound in [10].
The o iginali y o ou app oach is ha i combines in a p ope way quali a i e
easoning wi h machine lea ning echniques. This app oach is app op ia e o s udy all
he s a es ( ansien and s a iona y) o a semiquan i a i e dynamic sys em. I also
app op ia ed o ob ain i s beha iou s pa e ns. Howe e , some beha iou s maybe no
ound wi h his app oach, mainly, hose beha iou s ob ained wi h na owed domains
o he pa ame e s.
2 The Me hodology
The e has been a g ea deal o p e ious esea ch s udying he s a iona y s a e o a
sys em, howe e , i is also necessa y o s udy ansien s a es. Fo example, i is e y
impo an in p oduc ion indus ial sys ems o imp o e hei e iciency. Bo h s a es o a
semiquan i a i e dynamic sys em may be s udied wi h he p oposed me hodology.
The me hodology is shown in Figu e 1.
S a ing om a dynamic sys em wi h quali a i e knowledge, a semiquan i a i e
model S is ob ained. A amily o quan i a i e models F is ob ained om S by means o
he applica ion o some ans o ma ion echniques which a e bellow desc ibed.
S ochas ic echniques a e applied o choose a model M ∈ F. E e y model M is
quan i a i ely simula ed ob aining a ajec o y, which is composed by he alues o all
a iables om i s ini ial alue un il i s inal alue, and he alues o he pa ame e s.
The e o e, i con ains he alues o hese a iables in he ansien and s a iona y
s a es o he sys em.
A da abase o quan i a i e ajec o ies T is ob ained wi h hese quan i a i e
beha iou s. A language is p oposed o ca y ou que ies abou he quali a i e
p ope ies o he se o ajec o ies included in he da abase. A labelled da abase is
ob ained wi h he classi ica ion o hese ajec o ies in acco ding o a c i e ion.
Quali a i e beha iou pa e ns o he sys em may be au oma ically ob ained om
his da abase by applying machine lea ning based on gene ic algo i hms. These
algo i hms a e desc ibed in [2].
3 Semiquan i a i e Models
A semiquan i a i e model S is ep esen ed by
Φ
(dx/d ,x,q, ), x( 0) = x0,
Φ
0(q,x0)(1)
being x
∈ ℜ
n he se o s a e a iables o he sys em, q he pa ame e s, he ime,
dx/d he a ia ion o he s a e a iables wi h he ime,
Φ
cons ain s depending on
dx/d ,x,q, and
Φ
0 he se o cons ain s wi h ini ial condi ions.
I he me hodology is applied, he equa ions o he dynamic sys em (1) a e
ans o med in o a se o cons ain s among a iables, pa ame e s and in e als. In his
pape , we a e in e es ed in hose sys ems ha may be exp essed as (2) when he
ans o ma ion ules a e applied
dx/d = (x,p, ), x( 0) = x0, p
∈
Ip, x0
∈
I0(2)
whe e pincludes he pa ame e s o he sys em and new pa ame e s ob ained by means
o he ans o ma ion ules, is a unc ion ob ained by applying he ans o ma ion
Classi ica ion
Que y
Lea ning
Da abase
Gene a ion
Dynamic
Sys em
Semiquan i a i e
Model
Labelled
Da abase
T ajec o ies
Da abase
Quan i a i e
Model M
Answe
Sys em
Beha iou
Fig. 1. P oposed me hodology
ules, and Ip,I0 a e eal in e als. The equa ion (2) is a amily F o dynamic sys ems
depending on p and x0.
3.1 Quali a i e Knowledge
Ou a en ion is ocused on hose dynamic sys ems whe e he e may be quali a i e
knowledge in hei pa ame e s, ini ial condi ions and/o ec o ield. They cons i u e
he semiquan i a i e di e en ial equa ions o he sys em.
The ep esen a ion o he quali a i e knowledge is ca ied ou by means o
ope a o s, which ha e associa ed eal in e als. This ep esen a ion acili a es he
in eg a ion o quali a i e and quan i a i e knowledge in a simple way, and he
inco po a ion o knowledge om he expe s [4].
Quali a i e knowledge may be composed o quali a i e ope a o s, quali a i e
labels, en elope unc ions and quali a i e con inuous unc ions. This quali a i e
knowledge and i s ans o ma ion echniques a e now de ailed.
Quali a i e Ope a o s
These ope a o s a e used o ep esen quali a i e pa ame e s and ini ial condi ions.
They may be una y U and bina y B ope a o s. E e y quali a i e ope a o op is de ined
by means o an in e al Iop, which is supplied by he expe s.
Each quali a i e magni ude o he sys em has i s own una y ope a o s. Le Ux be
he una y ope a o s o a quali a i e a iable x, i.e. Ux={VNx, MNx, LNx,
AP0x,LPx,MPx,VPx }. They deno e o x he quali a i e labels e y nega i e, mode a ely
nega i e, sligh ly nega i e, app oxima ely ze o, sligh ly posi i e, mode a ely posi i e,
e y posi i e espec i ely. Le be a new gene a ed a iable and le Iu be an in e al
de ined in acco dance wi h [13], hen he ans o ma ion ule o a una y ope a o is as
ollows
opu(e) ≡ { ∈ Iu , e − = 0} (3)
Le e1,e2be wo a i hme ic exp essions, and le opbbe a bina y ope a o . The
exp ession opb(e1,e2) deno es a quali a i e ela ionship be ween e1 and e2. Bina y
quali a i e ope a o s a e classi ied in o:
ÿOpe a o s ela ed o he di e ence
≥
,=,
≤
, being hei ans o ma ion ules:
e1 = e2≡ { e1−e2=0 }
e1
≤
e2≡ { e1−e2− =0, ∈ [− ∞,0] }
e1
≥
e2≡ { e1−e2− =0, ∈ [0,+∞] }
(4)
ÿOpe a o s ela ed o he quo ien {«,−<,~,≈,»,Vo, Ne,...}. The ollowing ans-
o ma ion ule is applied:
opb(e1,e2) ≡ {e1−e2* =0 , ∈ Ib(5)
whe e Ib is an in e al de ined acco ding o [7].
In o de o main ain he consis ency o he model, i is necessa y o add cons ain s
o gua an ee he ela ion among he absolu e and ela i e o de o magni ude
ope a o s. in he gene al case [12].
En elope Func ions
An en elope unc ion y=g(x) ep esen s he amily o unc ions included be ween wo
de ined eal unc ions, a uppe one U:
ℜ ⇒ ℜ
and a lowe one L:
ℜ ⇒ ℜ
.
〈
L(x),U(x),I
〉
,
∀
x
∈
I: L(x)
≤
U(x) (6)
whe e I is he de ini ion domain o g, and x is he independen . The ans o ma ion
ule applied o (6) is
g(x) =
α
L(x) + (1 −
α
) U(x) wi h
α ∈
[0,1] (7)
whe e
α
is a new a iable. I
α
=0
⇒
g(x)=U(x) and i
α
=1
⇒
g(x)=L(x) and any
o he alue o
α
in (0,1) s ands o any included alue be ween L(x) and U(x).
Quali a i e Con inuous Func ions
A quali a i e con inuous unc ion y=h(x) ep esen s a se o cons ain s among he
alues o y and x acco ding o he p ope ies o h. I is deno ed by
y=h(x), h
≡
{P1,s1,P2,,..,.sk−1,Pk}(8)
being Pi he poin s o he unc ion. E e y Pi is de ined by means o (di,ei) whe e di is
he quali a i e landma k associa ed o he a iable x and ei o y. These poin s a e
sepa a ed by he sign si o he de i a i e in he in e al be ween wo consecu i e
poin s. A mono onous quali a i e unc ion is a pa icula case o hese unc ions
whe e he sign is always he same s1=... =sk−1.
The ans o ma ion ules o a quali a i e con inuous unc ion a e applied in h ee
s eps:
1. No maliza ion:
The de ini ion o he unc ion is comple ed and homogenised using hese
con inui y p ope ies:
ÿa unc ion ha changes, i s sign be ween wo consecu i e landma ks passes
h ough a landma k whose alue in he unc ion is ze o
ÿa unc ion whose de i a i e changes, i s sign be ween wo consecu i e
landma ks passes h ough a landma k whose de i a i e is ze o
The de ini ion o any unc ion (Equa ion 8) is always comple ed wi h: he ex eme
poin s (−
∞
, +
∞
), he poin s ha deno e he cu poin s wi h he axes, and whe e he
sign o he de i a i e changes (a maximum o a minimum o h).
2. Ex ension:
The de ini ion o hese unc ions is en iched by means o an au oma ic p ocess,
which inco po a es new landma ks o quali a i e labels. This ex ension is ca ied
ou o diminish he unce ain y in he de ini ion o he unc ion.
The numbe o new landma ks included be ween each wo consecu i e o iginal
landma ks may be always he same. Wi h his conside a ion, we don’ loose he
s a is ical ep esen a i i y o he selec ed quan i a i e samples ob ained o his
unc ion.
3. T ans o ma ion
A quali a i e unc ion h is ans o med in o a se o quan i a i e unc ions H.
The algo i hm ChooseH is applied o ob ain H.

ChooseH (h)
o each mono onous egion in h
segmen ={Pm,...,Pn}
choose a alue o e e y Pi in he segmen
e i ying he cons ain s o h
This algo i hm di ides h in o i s segmen s. A segmen is a sequence o consecu i e
poin s {Pm,...,Pn} sepa a ed by means o hose poin s whose landma k ei
=0 o whe e
si
≠
si+1. The segmen s di ide he unc ion in o hei mono onous egions whe e hei
landma ks ei ha e he same sign. The algo i hm applies s ochas ic echniques o
choose e e y quan i a i e unc ion o H. These echniques a e simila o he Mon e
Ca lo me hod, howe e , he alues ob ained mus sa is y he cons ain s o h. We use
aheu is ic ha applies a andom uni o m dis ibu ion o ob ain he alues o e e y
landma k o Pi.
4 Da abase Gene a ion
A amily F o quan i a i e models has been ob ained when he ans o ma ion ules
desc ibed in sec ion 3.1 ha e been applied o he semiquan i a i e model S. This
amily depends on a se o in e al pa ame e s p and unc ions H de ined by means o
a se o quan i a i e poin s. E e y pa icula model M o F is selec ed by means o
s ochas ic echniques, and i is quan i a i ely simula ed. This simula ion gene a es a
ajec o y ha is s o ed in o he da abase T.
The ollowing algo i hms a e applied o ob ain T.
ChooseModel (F)
o each in e al pa ame e o a iable o F
choose a alue in i s in e al o i
o each unc ion h o F
H:=ChooseH(h)
Da abase gene a ion T
T:={ }
o i=1 o N
M:=ChooseModel(F)
:= Quan i a i eSimula ion(M)
T:=T ∪
being N he numbe o simula ions o be ca ied ou , and i is de ined in acco dance
wi h he sec ion 7. The e o e, N is he numbe o ajec o ies o T.
5 Que y/Classi ica ion Language
In his sec ion, we p opose a language o ca y ou que ies o he ajec o ies da abase.
I is also possible o assign quali a i e labels o he ajec o ies wi h his language.
5.1 Abs ac Syn ax
Le T be he se o all ajec o ies s o ed in he da abase. A que y Q is: a quan i ie
ope a o
∀
,
∃
,
ℵ
applied on T, o a basic que y [ ,P] ha e alua es ue when he
ajec o y e i ies he p ope y P.
The p ope y Pmay be o mula ed by means o he composi ion o o he p ope ies
using he Boolean ope a o s
∧
,
∨
,
¬
.
Table 1. Abs ac Syn ax o he Language
Q :
∀
∈
T
•
[ ,P] P: PbPb: Pd
|
∃
∈
T
•
[ ,P] | P
∧
P | (L(F))
|
ℵ
∈
T
•
[ ,P] | P
∨
P |
∀
:F
•
F
| [ ,p] |
¬
P |
∃
:F
•
F
Pd:EQ F: FbFb: eb
|CL | F & F | e
∈
I
|F | F | u(e)
|! F | b(e,e)
A basic p ope y Pb may be: a p ede ined p ope y Pd, a Boolean unc ion applied o
a lis Lo poin s o in e als ha e i ies he o mula F, o a quan i ie
∀
,
∃
applied o
he alues o a pa icula ajec o y o a ime . This ime may be: an ins an o ime, a
una y ime ope a o (i.e. a ange o ime), a p ede ined ime landma k, o he lis o
imes whe e he o mula F is e i ied.
A de ined p ope y Pd is he one whose o mula ion is au oma ic. They a e que ies
commonly used in dynamic sys ems. The e a e wo p ede ined: EQ is e i ied when
he ajec o y ends up in a s able equilib ium; and CL when i ends up in a cycle limi .
A o mula Fmay be composed o o he o mulas combined by means o Boolean
ope a o s &,|,!.
A basic o mula Fb may be: a Boolean exp ession eb, o i a nume ic exp ession e
belongs o an in e al, o a una y u o bina y b quali a i e ope a o .
5.2 Seman ics
The seman ics o e e y ins uc ion o his language is ansla ed in o a que y on he
da abase. The echniques applied o ca y ou his ans o ma ion come om he
de elopmen o compile s o language p og amming. A que y [ ,P] is ue when
ajec o y e i ies he p ope y P. Seman ics o a que y wi h a quan i ie depends on
i s ela ed quan i ie . I i is
∀
, a Boolean alue ue is e u ned when all he
ajec o ies
∈
T e i y P. I i is
∃
hen ue is e u ned when he e is a leas one
ajec o y
∈
T ha e i ies he p ope y P. I he quan i ie is
ℵ
hen e u ns he
numbe o ajec o ies o T ha e i ies P.
Le
∀
: F
1
•
F
2
be a basic p ope y which is ue i du ing he ime ha F1 is
sa is ied, all he alues o e i y F2. Fo
∃
quan i ie is ue when a leas a alue o
ha sa is ies F1, also sa is ied F2. In o de o e alua e a o mula F, i is necessa y o
subs i u e i s a iables o hei alues. These alues a e ob ained om T.
5.3 Classi ica ion
A classi ica ion ule is o mula ed as a se o basic que ies wi h labels, and possibly
o he exp essions
[ ,PA] ⇒ A,en1,... [ ,Pb] ⇒ B,en2,... ... (9)
A ajec o y is classi ied wi h a label
η
i i e i ies he p ope y P
η
.
Le [ ,PA]
⇒
A,eA1 be a classi ica ion ule. A ajec o y
∈
T is classi ied wi h he
label A i i e i ies p ope y PA. The esul o e alua ing eA1 o his ajec o y is also
s o ed in o he da abase.
6 A Logis ic G ow h Model wi h a Delay
I is e y common o ind g ow h p ocesses whe e an ini ial phase o exponen ial
g ow h is ollowed by ano he phase o asymp o ic app oach o a sa u a ion alue.
The ollowing gene ic names a e gi en: logis ic, sigmoid, and s-shaped p ocesses.
This g ow h appea s in hose sys ems whe e he exponen ial expansion is unca ed by
he limi a ion o he esou ces equi ed o his g ow h. They abound in he e olu ion
o bac e ia, in mine al ex ac ion, in wo ld popula ion g ow h, in epidemics, in
umou s, in economic de elopmen , he lea ning cu es, e c.
In he bibliog aphy, hese models ha e been p o usely s udied. The e is a bimodal
beha iou pa e n a ac o : A s ands o no mal g ow h, and O o decay (Figu e 5.b).
Di e en ial equa ions o he model S a e
Φ ≡
dx/d =x(n -m), y=delayτ(x), >0, =h(y),
h ≡ {(-∞,-∞),+,(d0,0),+, (0,1),+, (d1,e0), -,(1,0), - (+∞,-∞)} (10)
being n he inc easing ac o , m he dec easing ac o , and h a quali a i e unc ion
wi h a maximum poin a (x1,y0). The ini ial condi ions a e
Φ
0
≡
{ x0
∈
[LPx,MPx], LPx(m),LPx(n),
τ ∈
MP
τ
,VP
τ
}(11)
whe e LP,MP,VP a e quali a i e una y ope a o s o x,
τ
a iables.
We would like o know:
1. i an equilib ium is always eached
2. i he e is an equilib ium whose alue is no ze o
3. i all he ajec o ies wi h alue ze o a he equilib ium a e eached wi hou
oscilla ions.
4. To classi y he da abase acco ding o he beha iou s o he sys em.
We apply ou app oach o his model. Fi s ly, he ans o ma ion ules a e applied,
dx/d =x(n -m), y=delayτ(x), x>0, =H(y),
H, x0∈ [0,3], m,n ∈ [0,1], τ ∈ [0.5,10]
(12)
whe e H has been ob ained by applying Choose H o h, and he in e als a e de ined
in acco dance wi h he expe s’ knowledge. The algo i hm Da abase gene a ion T
e u ns he ajec o ies da abase.
The p oposed que ies a e o mula ed as ollows:
1. ∈ T • [ ,EQ]
2. ∈ T • [ , EQ ∧ ∃ : ≈ • !AP0x(x)]
3. ∀ ∈ T • [ , EQ ∧ ∃ : ≈ • AP0x(x) ∧ leng h(dx/d =0)=0 ]
4. being AP0xa una y ope a o o x. The lis o poin s whe e dx/d =0 is he lis
wi h he maximum and minimum poin s. I leng h is 0 hen he e a e no
oscilla ions.
We classi y he da abase by means o he labels:
[ ,EQ
∧
leng h(dx/d =0)>0
∧ ∃
:
≈
•
!AP0x(x)]
⇒
eco e ed,
[ ,EQ
∧
leng h(dx/d =0)>0
∧ ∃
:
≈
•
AP0x(x)]
⇒
e a ded,
[ ,EQ
∧ ∃
:
≈
•
AP0x(x)]
⇒
ex inc ion,
They co espond o he h ee possible beha iou pa e ns o he sys em (Fig. 6). They
a e in acco dance wi h he ob ained beha iou s when a ma hema ical easoning is
ca ied ou [3].
7 Conclusions and Fu he Wo k
In his pape , a me hodology is p esen ed in o de o au oma e he analysis o dynamic
sys ems wi h quali a i e and quan i a i e knowledge. This me hodology is based on a
ans o ma ion p ocess, applica ion o s ochas ic echniques, quan i a i e simula ion,
gene a ion o ajec o ies da abase and de ini ion o a que y/classi ica ion language.
The e is enough bibliog aphy ha s udies s a iona y s a es o dynamic sys ems.
Howe e , he s udy o ansien s a es is also necessa y. These s udies a e possible
wi h he p oposed language.
The simula ion is ca ied ou by means o s ochas ic echniques. The esul s a e
s o ed in a quan i a i e da abase. I may be classi ied by means o he p oposed
language. Once he da abase is classi ied, gene ic algo i hms may be applied o ob ain
conclusions abou he dynamic sys em.
In he u u e, we a e going o en ich he que y/classi ica ion language wi h:
ope a o s o compa ing ajec o ies among hem, empo al logic among se e al imes
o a ajec o y, mo e ype o equa ions, e c.
Φ