Modelling with temporal fuzzy chains.
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Conference Paper: Induction of Temporal Fuzzy Chains.
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ABSTRACT: The aim of this paper is to present an algorithm to induce the Temporal Fuzzy Chains (TFCs) (J. Moreno, 2002b). TFCs are used to model the dynamic systems in a linguistic manner. TFCs make use of two different concepts: the traditional method to represent the dynamic systems named state vectors (Ogata, 1998), and the linguistic variables (Zadeh, 1975) used in fuzzy logic (Tanaka, 1998). Thus, TFCs are qualitative and represents the "temporal zones" using linguistic states and linguistic transitions between the linguistic states.01/2003  01/1997; Springer., ISBN: 9780387948072

Conference Paper: Temporal Fuzzy Model for Dynamic Systems.
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ABSTRACT: Abstract The aim of this paper is to present the Temporal Fuzzy Models (TFMs). These models are used to represent the systems that change in time, i. e., dynamic systems. Temporal fuzzy rules of TFMs are ordered, so each rule represents a ”temporal zone”. An algorithm is presented to obtain the TFM. This algorithm needs as input a fuzzy model based in linguistic labels. An inference method is presented too. Finally, a fuzzy model of a shot put of the Spanish athlete Manuel Martinez is obtained with the direct linguistic induction algorithm [7] that is used as input. Keywords: Models induction, linguistic models,Proceedings of the International Conference on Artificial Intelligence, ICAI '02, June 24  27, 2002, Las Vegas, Nevada, USA, Volume 1; 01/2002
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Modelling with Temporal Fuzzy Chains
Juan Moreno
U. de CastillaLa Mancha
E.U.I.T.Industrial Toledo
Juan.Moreno@uclm.es
L. Jim´ enez, J.J. CastroSchez
U. de CastillaLa Mancha
E.S.Inform´ atica Ciudad Real
{Luis.Jimenez,JoseJesus.Castro}@uclm.es
L. Rodr´ ıguez
U. de CastillaLa Mancha
E.U.P. Almad´ en
Luis.Rodriguez@uclm.es
Abstract
The aim of this paper is to present the Tem
poral Fuzzy Chains (TFCs) [3] to model
the dynamic systems in a linguistic manner.
TFCs make use of two different concepts:
the traditional method to represent the dy
namic systems named state vectors [6], and
the linguistic variables [8] used in fuzzy
logic [7]. Thus, TFCs are qualitative and
represents the ”temporal zones” using lin
guistic states and linguistic transitions be
tween the linguistic states.
Keywords:
Temporal model, linguistic
model, dynamic systems, fuzzy logic.
1Introduction
Dynamic systems (DS) are systems whose perfor
mance change throughout the time. A DS is described
by means of a set of relevant features (input and out
put variables) and a set of relations among input and
output variables, which represent the modi£cations of
the output variables when the input variables are mod
i£ed throughout the time. The values of the system
variables at the time t depend on the variables values
at the times t −1...1.
The DSs with continuous physical magnitudes are
continuous at the time, that is, at a time t +1 the vari
able value vt+1is similar to the variable value vt at
the time t. This property is formally represented as
vt−vt−1 < ε with ε being a small constant. This hy
pothesis is supposed when we de£ne the TFCs.
The next section recalls the de£nition of TFCs, the
formal de£nition can be found in [3]. The induction
algorithm is shown in section 3. In section 4 is mod
elled a shot put of Manuel Mart´ ınez. Finally, the con
clusions and future works are exposed in section 5.
2De£ning the TFCs
We suggest to represent the temporal side of a DS
making use of the TFCs. A TFC is formed by linguis
ticstatesandlinguistictransitions. Alinguisticstateis
de£nedtorepresentthesystematatime. Betweentwo
consecutive linguistic states is established a linguis
tic transition that indicates the conditions necessary
to enters into the next linguistic state. The change of
state is described in a linguistic way (using linguistic
labels).
Let Ξ be a DS MISO with a set of m real input vari
ables X1,X2...Xmand an output real variable S. The
behavior of the system is given by means of a set of
examples E = {e1,e2...en} with ei= (xi
where xij∈Xj, si∈S andtiis the time in which occurs
the example i.
1...xim,si,ti)
TFCs work with linguistic variables [8]. These vari
ables have de£ned an ordered set of linguistic labels
over its domain named continuous linguistic vari
ables, from now on variables. The linguistic labels
(from now on labels) associated to these variables are
de£ned before the TFC will be obtained. Thus, an
ordered set of labels SAjis de£ned for each input
variable Xj. Its structure is SAj= {SA1
where i is the position of SAijin SAjand ijis the num
ber of linguistic labels in SAj, that is ij= SAj. An
ordered set of labels SC is de£ned for the output vari
able S. Its structure is SC = {SC1,SC2...SCiy} where
i is the position of SCiin SC and iyis the number of
linguistic labels in SC, that is iy= SC.
j,SA2
j...SAij
j},
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Figure 1: Linguistic interval
Our variable takes linguistic interval as value. A
linguistic interval (from now on interval) LIc
{SAp
as a subset of the ordered set of labels SAjthat begins
in the label p and has c labels (Figure 1). Its mem
bership function is the sum of the membership grade
of a value ajto each label belonging to the interval
(Equation 1).
j,p=
j,SAp+1
j
...SAp+(c−1)
j
} for a variable Xjis de£ned
µLIc
j,p(aj) = ∑
SAz
jεLIc
j,p
µSAz
j(aj)
(1)
where z ∈ [p..c−1].
A set of m intervals de£ned on m variables is
an ordered set of m intervals for each one of
the input variable, and is represented as SLIm=
{LIc1
a SLImis calculated applying a tnorm to the member
ship grade of the intervals in the SLIm(Equation 2).
1,p1,LIc2
2,p2...LIcm
m,pm}. The membership function of
µSLIm(ei) = ∗(µLI
cj
j,pj(xj))
(2)
where ei=(xi
longing to the set E, jε[1..m] and * is a t
norm.
1...xim,si,ti) is an example be
SLImis used to represent linguistically the range of
values of the m input variables.
Finally, a linguistic state i (from now on state) is de
£ned as a tuple esti=< Aim,SEi> where Aimis an or
dered set of m intervals of the state i corresponding to
the m input variables of the DS, and SEiis the output
label of the state i corresponding to the output variable
of the DS.
A linguistic transition i (from now on transition) is a
tuple transi=< Ti
of m intervals of the transition i corresponding to the
m input variables of the DS, and STiis the output label
of the transition i corresponding to the output variable
of the DS.
m,STi> where Ti
mis an ordered set
A TFC is a tuple CHAIN =< EST,TRANS > where
EST = {est1...estns} is an ordered set of ns states,
and TRANS = {trans1...transns−1} is an ordered set
of ns−1 transitions. Transition i re¤ects the condi
tions to change from estito esti+1.
In order to reproduce the DSs with TFCs, we offer
an inference method in [3]. The inference algorithm
needs a set of examples E as input and is based on
the de£nition of a state estcur named current state.
estcurindicates the state in which the DS is, and al
lows to calculate the output at this time. The inference
method begin selecting est1as the £rst current state.
Next µAcur
ship function µTcur
are calculated. If µAcur
the obtained output s is SEcur(corresponding to estcur)
and there isn’t state change. In other case, if µAcur
is less than or equal to µTcur
change: the obtained output s is STcur(corresponding
to transcur) and the new current state is the next in the
TFC, i.e., estcur+1. This process is repeated for each
example in E.
m(ei) of eito the state estcurand the member
m(ei) of eito the transition transcur
m(ei) is greater than µTcur
m(ei) then
m(ei)
m(ei), then there is a state
3Inducing the TFCs
In this section we show brie¤y the suggested algo
rithm to induce TFCs [5]. Firstly, some necessary
concepts are shown.
De£nition 3.1 Let SAj= {SA1
ordered set of labels de£ned on Xj and a set of
examples ESAj.
{SAp
which veri£es that: Its £rst label SAp
that veri£es the equation 3; and its last label SAu
the last one that veri£es the equation 4.
j,SA2
j...SAij
j} be an
A simpli£ed interval LIu−p+1
...SAu
j,p
=
j,SAp+1
j
j} depending on ESAjis an interval
jis the £rst one
jis
∃eiεESAj/µSAf
j(xi
j) > 0(3)
∃eiεESAj/µSAlj(xi
j) > 0(4)
where xijis the real value in the position j in
the example ei.
In short, a simpli£ed interval LIu−p+1
whereits£rstlabelisthe£rstoneofSAjthathassome
example of ESAjwith membership grade greater than
zero, and its last label is the last one of SAjthat has
j,p
is an interval
Page 3
some example of ESAjwith membership grade greater
than zero.
De£nition 3.2 Let LIcj1
vals de£ned on Xj, its union is de£ned as another in
terval LIu−p+1
j,p
label of LIcj1
is the greatest label of LIcj1
j,pj1,LIcj2
j,pj2...LIcjn
j,pjnbe n inter
where the £rst label SAp
j,pj1,LIcj2
jis the smallest
j,pj2...LIcjn
j,pjn; and the last label SAu
j,pj1,LIcj2
j
j,pj2...LIcjn
j,pjn.
De£nition 3.3 Let SLI1
sets of m intervals, its union is de£ned as another
SLIm, where each interval LIcj
makingtheunionofnintervalsLIcj1
of SLI1
m,SLI2
m...SLIn
mbe n ordered
j,pjof SLImis obtained
j,pj1,LIcj2
j,pj2...LIcjn
j,pjn
m,SLI2
m...SLIn
m.
De£nition 3.4 Let est1,est2...estn be n states and
EA1m,EA2m...EAnmn sets of examples associated to the
n ordered sets of intervals, the total union of the n
states is another state estunionwhere:
1. Aunion
tervals A1
(de£nition 3.3).
m
isobtainedbymakingtheunionofthenin
m,A2
m...An
mof the states est1,est2...estn
2. The output label is calculated using the equa
tion:
maxSCwµSCw(v)
(5)
The set of examples needs in this equation is
Eunion=?EA1m,EA2m...EAnm.
The total union of the states is used when a state that
represents the successive examples from eato ebhas
the same output label than the state that represents the
examples from eb+1to ec.
De£nition 3.5 Let LIc1j
with its labels de£ned in SAj, the difference of LIc2j
respect to LIc1j
j,p1jand LIc2j
j,p2jbe two intervals
j,p2j
j,p1jis the following set of labels:
LIc2j
j,p2j−LIc1j
In short, the difference between two intervals LIc2j
and LIc1j
that belong to LIc2j
j,p1j= {SAij∈ LIc2j
j,c2j\SAijnot ∈ LI1j
j,p1j}
j,p2j
j,p1jis a set of labels compounded by the labels
j,p2jand do not belong to LIc1j
j,c1j.
De£nition 3.6 Let
...,LIc1m
SLI1
m= {LIc21
m
=
{LIc11
2,p22,...,LIc2m
1,p11,
LIc12
2,p12,
m,p2m}
m,p1m} and SLI2
1,p21,LIc22
be two ordered sets of m labels with the labels of each
variable de£ned on SA1,SA2...SAm, the difference
between SLI2
labels formed by:
mrespect to SLI1
mis the set of sets of
SLI2
m−SLI1
LIc12
m= {LIc21
2,p12...LIc2m
1,p21−LIc11
m,c2m−LIc1m
1,p11,LIc22
m,p1m}
2,p22−
The difference between two SLIm is used to know
when the state change can be detected between two
successive states.
Algorithm 1 Direct Induction Algorithm of TFCs
EST ← θ
TRANS ← θ
Nstate← 2
EAcur
Acur
SEcur←CalculateConsequent(EAcur
for i = 2 to E do
EANstate
m
← {ei}
ANstate
m
← Simplify(SA1...SAm)
SENstate←CalculateConsequent(EANstate
if SENstate= SEcuror Acur
estcur← TotalUnion(estcur,estNstate)
EAcur
else
EST ← EST +estcur
if Nstate−2 > 0 then
TNstate−2
m
← Acur
STNstate−2←CL(SENstate−2,SEcur)
TRANS ← TRANS+transNstate−2
end if
estcur← estNstate
EAcur
m
Nstate← Nstate+1
end if
end for
EST ← EST +estNcur
TNstate−2
m
← Acur
STNstate−2←CL(SENstate−2,SEcur)
TRANS ← TRANS+transNstate−2
m← {e1}
m← Simplify(SA1...SAm)
m)
m
= θ then
)
m−ANstate−2
m
m←Union(EAcur
m,EANstate
m
)
m
m← EANstate
m
De£nition 3.7 LetSAj={SA1
dered set of labels and two labels SAp
p+u
2
j
is named the central label. The expo
nentp+u
if SAp
j,SA2
j...SAij
jand SAu
j}beanor
jof SAj,
the label SA
2is rounded by excess if SAp
j> SAu
j<SAu
j, and down
j.
Page 4
We make use of the central label concept for calcu
lating the output associated to each transition between
two states i and i+1. Thus, when SEiand SEi+1are
consecutive the output label STiof the transition i is
SEi+1, in other case STiwill be the label in the mid
dle of SEiand SEi+1. The central label between SEi
and SEi+1is represented as CL(SEi,SEi+1) in the al
gorithm.
To £nish this section, the algorithm 1 is shown. This
algorithm is used to induce the TFCs and is named Di
rect Induction Algorithm of TFCs. Its inputs are a set
of examples E, the ordered sets of labels SA1...SAm
for the m input variables and the ordered set of labels
SC for the output variable.
In the algorithm 1, EST and TRANS are the ordered
set of states and transitions respectively; estcuris the
current state, that is, the state that is going to be in
serted as the last state of the ordered set of states EST;
and Nstateis the number of the next state to estcurand
is used to detect when estcuris completely constructed
(with all its examples in EAcur
Firstly, EST, TRANS and Nstateare initialized to θ,
θ and 2 respectively, and the £rst current state estcur
is created: each interval LIcj
is initialized to the simpli£cation of the ordered set
of labels SAj using as set of examples associated
EAcur
ordered sets of labels SA1,SA2...SAmdepending on
the set of examples ESAjis a set of m intervals SLIm,
i.e., Acur
estcuris calculated using the equation 5.
m).
j,pjof the Acur
m
of estcur
m={e1} (de£nition 3.1). The simpli£cation of the
m. The output label of the £rst current state
where w=1..iyand v =∑
ESAj
i=1
ESAj
si
where ESAj is the number of examples in
ESAj, and siis the output real value in the
example eiwhich belongs to ESAj.
In brief, the selected label for the output of the state is
the one that has the maximum grade of membership to
the medium value of the output values of the examples
in ESAj. Others possibilities are given in [7].
Next, the loop for is used to examine from e2 to
the last example en in E.
calculated the state estNstatewhere: ANstate
to the simpli£cation of the ordered sets of labels
SA1,SA2...SAmusing EANstate
m
For each example ei is
m
is assigned
= {ei} as set of exam
ples associated; and the output label SENstateis calcu
lated by using of the equation 5.
If the output label of estNstateis equal to the output
label estcuror if Acur
m
states (that represent set of examples consecutive at
time) have the same output or the change can’t be de
tected (because Acur
tively, thus, estcuris assigned to the total union of the
states estcurand estNstate(de£nition 3.4) because repre
sent the same output. In other case, estcuris added to
the ordered set EST, and if estcurisn’t the £rst state
then the transition transNstate−2is created, i.e., the pre
vious transition to the last inserted state estcurwhere:
TNstate−2
m
isassignedtoAcur
label between the output label of the states SEstate−2
and SEcur(de£nition 3.7), that is, the central label be
tween the two last states of the TFC.
m−ANstate−2
= θ means that both
m is contained in ANstate−2
m
) respec
m; andSTNstate−2isthecentral
Next, the new transition transNstate−2 is added to
TRANS. Finally, estcurtakes the value of estNstate, the
set EAcur
m
When the loop for is £nished the last state and transi
tion are added to EST and TRANS.
mis now EANstate
and Nstateis incremented in 1.
4 TFC of a shot put
Algorithm 1 is evaluated using a shot put of the Span
ish athlete Manuel Martinez, the current world cham
pion of shot put. The set of examples E was captured
during the thesis [2]. This data corresponds to a shot
put of 19.43 meters. For more information about the
capture process see [1, 2].
The input variables of E are: pelvisscapular angle
(PSA), Elbow angle (EA), rightleft axis (RLA) and
backwardforwardaxis (BFA). The output variable is
the height of the weight (H).
Figure 2: Sequence of 7 labels
A sequence of 7 labels with domain equally spaced
is used for all variables. The structure of this sets is
shown in Figure 2 with VN being Very Negative, N:
Negative, FN: Few Negative, NR: Norm, FP: Few
Positive, P: Positive andVP: Very Positive.
The obtained TFC is shown in Figure 3. Formally the
obtained TFC is represented as a tuple CHAIN =<
Page 5
Figure 3: TFC of shot put
EST,TRANS > where EST =< est1,est2,...est7>
and TRANS =< trans1,trans2,trans3...trans6 >.
Figure 4 shows the temporal zone of each state and
transition.The squares with En and Tn represent
the temporal zone of the states and transitions respec
tively, where n is the number of the state or transition.
Figure 4: Temporal zone for each state
Finally, a inference process is made by using the infer
ence algorithm, the set of example E is used as input.
The obtained error is 0.0662o. Figure 4 shows graph
ically the comparison between the real output and the
obtained output. The obtained line is very similar to
the real line and the error is small.
5 Conclusions
TFCs are a new method to represent the DSs. TFCs
are qualitative making use of linguistic labels de£ned
Figure 5: Inference using E
a priori, and represent the performance of the DS.
Thus, TFCs are a good approach to model the DS. The
temporality is represented in the order of the states
and transitions, that is an important difference with
the fuzzy models. The obtained output is improved
using the central label in the transitions [5]. The cen
tral label represents the temporal zones in which the
output evolution is faster than the evolution of the in
put variables. This is a problem in the fuzzy models
that no consider the time.
We will also study the relation between TFCs and tra
ditional systems of fuzzy rules and the relation be
tween TFCs and Temporal Fuzzy Models [4]. We will
design a new induction algorithm that uses more than
one set of examples as input. We will develop an al
gorithm to study the best number of labels and its do
main for each input and output variables. Finally, we
will work in two different aspects: (1) A method to
covert the intervals to expressions like ”quick incre
ment”, ”quick decrement”, etc (2) Adding to the TFC
the ”time linguistic variable” to represent the ”dura
tion of the state”, ”time per label”, etc.
Acknowledgements
This work has been £nanced by the project TIC2000
1362C0202 of the Ministry of Science and Technol
ogy of Spanish state.
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