244
Modeling Uncertain Temporal Evolutions
in Model-Based Diagnosis
Luigi Portinale
Dipartimento di Informatica- Universita' di Torino
C.so Svizzera 18.5- 10149 Torino (Italy)
Abstract
with the observations, restore the consistency in
the model). Alternatively, the use of causal mod­
els has been proposed to model the faulty behavior
Although the notion of diagnostic prob­
of the system where an abductive approach is used
lem has been extensively invest igated in
the context of static systems, in most
practical applications the behavior of the
[19] (the dia gnos is must cover the observations).
!3oth approaches use a logical model of the sys­
tem to be diagnosed and it has been shown that
they are just the extremes of a spectrum of logical
definitions of diagnosis f6). On the other hanJ, in
the probabilistic paradigm, diagnostic kuowleJge
is usually represent.ed by means of associations be­
tween symptoms and disorders; per for mi ng a J.iag­
nosis means finding the most probable set of dis­
orders, given the sy m pto ms
The major part of
the work in probabilistic diagnosis has been con­
cerned with the use of Bayesian methods baseJ. on
belief networks [17] as in IVIUNIN [1], in Q!IIR-BI\"
[22] and in [12]. However, other approaches have
been p ro pose d for instance, r ely ing on Dempster­
Sha fer theory [13] or on the integrat ion of B ayesian
classification with the set covering moJel [18].
modeled system is significantly variable
during time. The goal of the paper is to
a nmel approach to the modeling
uncertainty about temporal evolutions
of time-varying systems and a charact.er­
propose
of
ization of model based temporal diagno­
-
sis. Since in most real world cases knowl­
edge about the temporal evolution of the
system to be diagnos ed is uncertain, we
.
consider the case when probabilistic tem­
poral knowledge is available for each com­
ponent of the system and we choose to
model it by means of Markov chains. In
,
fact, we aim at exploiting the statistical
assumptions underlying reliability theory
in the context of the diagnosis of time
These two basic paradigms to diagnosis seem to be
com p lemen tary for certain aspects (for example,
probabilistic d iagnosti c reasoning well a<.hlrcsses
­
varying systems. lVe finally show how to
exploit 1\Iarkov chain theory in order to
discard, in the diagnos tic process, very
unlikely diagnoses
.
1
-
INTRODUCTION
The not ion of diagnostic reasoning and in par t ic u
lar of model-hased diagnosis, has been extensively
investigated in the past and two basic p a rad igms
,
­
the logic and the probabilistic one. In
paradigm it is often the case that the
mo de l represents the fun ction and structure of a
emerged:
the logic
device; in this case the model usually represents
the normal behavior of a system and the di agnostic
reasoning
the problem of minimizing the c ost of I he com­
putation, whi le logic b ased reasoning is in gene r al
more flexible in dealing with multiple disorders or
contextual information); for this reason some <It­
tempts have been made for combining them. In
fact, probabilistic inf ormation has been integrated
in logical diagnostic framework either for defin­
ing criteria for choosing the best next. measure­
ment [5,8] or for extending probabilistic diagnostic
fr ameworks to non propositional for m [20]. How­
ever, the context in which s uch approac!Jes ha,·e
been proposed is mainly that of static system> in
which the system behavior can be thought a:> fixed
during the diagnostic process and s o time indepen­
dent. C learly in importan t applications this is not
sufficient, because the modeled system is int.rinsi-
follows a consistency-based approach [ 21]
(the diagnosis is a set of abnormality assum pt ions
on the components of the system that, together
,
Modeling Uncertain Temporal Evolutions in Model-based Diagnosis
cally d ynamic and its behavior is significantly vari­
amount of time required for a t ransit i on be­
tween two behavioral modes, or the amount
s t udy of the behavior of time­
varying systems in the attempt of either extend­
of time e lapsing between two consecutive time
points with observations ) ;
a ble d uring time. For this reason there is a grow­
ing interest in the
ing static diagnostic techniques [14] or finding new
appr oaches more suitable for time-dependent be­
k n ow ledge about the temporal evol ution of a
component is uncertain and has to he modeled
at some level of abstraction.
•
havio r [11). Diagnosing systems with time-varying
behavior requi res the ability of dealin g at least
with the fo ll owing important aspect s : observ ation s
ac ross diffe re nt time instant s and exp lici t de s crip
tion of state changes of the system w i th r espe c t to
time. One of the immediate consequences of these
aspects of the problem is the n eed of having some
form of abstraction in the model representin g the
­
system to be diagno sed ; in fact, alth ough some of
the earlier approach es to diagnosis of dynamic sys­
te ms tried to deal with sin g le level description of
the system [24), most of the recent pro p osals are
foc use d on the possibility of having a hierar chical
model in which multiple layers of abs t r actions can
be used in order
[14,1G].
to perform the diagnostic task
In previo us works we concentrated on ab ductiv e
r ea!; o n ing as an useful framewo rk for diagn osing
static systems, even if s ome attention has to be
paid for mitigat i ng t.he comput a t i onal comp lexity
of the a b ducti ve process [2]. However, exten din g
tlwse mechanisms in the direction of time-varying
systems can lead to sever al problems. F irs t of all,
The aim of the paper is to propose, given this
kind of ass ump t ions, a. characterization of tempo­
ral d iagnosi s in such a way that di a gnost i c tech­
niques, developed for static systems, could be used
as much
as
p ossi ble in the diagno si s of systems ex­
hi biting time varyin g behavior.
-
This is pursuE;d
having in mind the fact that tem p oral information
is abstracted from ot h er
system s uch
as
behavioral features of the
the rel ationships bel\Yeen beh av
ioral modes of the c omp on ents and their observable
,
­
manifestations; this means that such relationships
do not take into account time which is a dded 1.0
the
model as an o rthogo nal dimension ( see also [15]).
The kind of abstraction we use throughout the pa­
have at
the tempo­
per is a probabili s tic one. We assume to
dis p o s a l probabilistic knowledge about
ral behavior of the components of the system to be
diagnosed in such a way that such behavior ca.n be
modeled as a stochastic process; in particular, we
aim at exploiting th e theory of markovian stochas­
tic processes in a diagn ostic setting by adopling
in mo s t real world cases temporal i n formatio n is
the usual assumptions followed in reliability theory
[23]. In fact, th is ki n d o f probabilis tic know ledge
be
can be supposed to be available from the statistics
un c ertain and not al\\" ays easy to encode in the
model of the system to
d iag nosed. The use of
propagation techniques for fuzzy temporal inter­
vals has been proposed in[4] and [ l
can
O]
,
but even if in
determine the temporal evolutions
of the system in a mo 1 e precise way, serious prob­
practice we
·
lems arise from the computational p oin t of v iew
[I]: this means that some kind of approximation
has
are i nteres ted in analyzing ho w
to explo it in a. component-oriented mo del p r oba­
bilistic kn o wledge about possi ble temporal evolu­
tions of the component.->. In particular we assume
that.:
,
,
declarative characterization of the prob­
lem, without co n sidering re as o ning issues.
just the
2
STOCHASTIC PROCESSES
AND RELIABILITY THEORY
,
components have a di scr ete set of different
be­
h a t'tOT'al modts [D] (a behavior al mode repre­
sents a p articula r state of a component; the set.
of behavioral modes consists of o n e "correct"
mode and in general, several abn orm al m o de s
,
representing fault.y conditions of the compo­
nent).
•
card temporal evolutious which are very unlikely,
however, for the lack of space we discussed here
to be considered.
In particular, we
•
about the behavior of the system. We will s how
how th is approac h can be used in orJer to dis­
f:'ach component ca n
c han ge i t s
mo de
d ur in g
time. while the manifestations of a behavioral
mode are inst.<1nt.aneous (with res pec t to the
In this section, we briefly recall some
basic no t i on s
relative to stochastic processes and the proba bi lis­
tic assumptions usually adopted in r eliability the­
ory concerning the life cycle of a comp onent of a
,
physical sy s t em.
Definition 1 A stochastic process is a ja111ily
of random variables {X(t)jt E T} defn
i ed Ot'U lhe
same probability space, taking values in a stl Sand
indexed
by a pam meter t E T.
The values assumed by the v a riables of the stoch as­
tic process are ca lled states and the setS is called
the stale space of the process. Usually the p;name-
245
24o
Portinale
t.er t represen ts time, so a stochastic process can be
thought as the model of the evolution of a system
crete time parameter and in particular on Markov
phases: the infant mortality phase where the fail­
ure probability pis decreasing with time, the us11al
life ph a se with constant failure probability anll the
wear-out phase where the failu re probability is in­
creasing with age. If we consider the lifetime X of
Definition 2 A Markov
bution with paramet.er p.
across time1. In the following we will concentrate
on chains (i.e. discrete-state processes) with dis­
chains.
chain
is a discrete­
state stochastic process {X(t)/t E T} such that for
any to < t1 < . . . t11 the conditional distribution of
for given values X(to) . ..X(tn-d depends
on X(t11_t) that is:
X(tn)
only
P(X(t,)
Xn].X(t,_t) = Xn-1 . . . X(to)
P[X(f71) = Xn]X(ln-d = Xn-d
=
=
xo]
=
We will be concerned only with Discrete-Time
l\Iarkov Chains
(DTl\IC).
In a 1\Iarkov cha in , the
component to be a discrete random variable:?, in
the usual life phase X follows a geometric di�tri­
a
3
DIAGNOSTIC FRAMEWORK
In the introduction , we discussed problems con­
cerning the management of temporal information
in diagnostic systems; one of the basic difficulty is
relative to the approach to be followed in model­
ing such a k ind of information.
probability of transition from one state to another
depends only from the current state and from the
current time instan t . If the conditio n a l probabil­
ity showed in the above definition is invariant with
ing such assumptions in a
framework.
time
3.1
respect to the
that
time origin , then we have a so called
- hom:ogeneorts
Markov chain; this means
P[X(tn) = x, ]X(tn-1) Xn-1] =
P[X(tn- ln-1) Xn]X(O) = Xn- d
=
=
In this case, the past history of the chain is com­
pletely summarized in the current state; with the
assumption of time-homogeneity, it can be shown
that , in the case of a DTMC, the sojoum time
in a given state follows a geometric distribution.
This is the only distribution satisfying the memo­
ryless property, for a discrete r ando m variable (the
corresponding me moryless distribution for a. con­
tinuous variable is the exponential distribution). A
discrete random variable X is geometrically dis­
tributed with p a r ameter p if its probability mass
function is p mf(l)
= p(l- p) 1- 1 •
= P(X
t)
We say that X has the memory less property if and
only if P(X = t + niX > t) = P(X = n); this
means that we need not remember how lo n g the
=
process has been spending time in a given state
to determine the probabilities of the next possi­
ble transitions (i.e. we can arbitrarily choose the
origin of the time axis ) . In the following we will
consider only timc-lwmogeneous DT�IC.
Reliability theory copes with the application of
particular probability distributions to the analysis
of the life cycle of t.he components of a. physical sys­
tem. In particular, it. has been recognized that the
a comp onent can be subdivided into three
life of
1
In t.he following we will refer t.o t as t.he t.ime
pa ra.met er.
Since
a
lot of \\'ork
in reliability theory has lead to well defined statis ­
tical assumptio ns about the temporal belJavior of
t.he components of a system, we aims at extend­
model based diagno_,;tic
REPRESENTATIONAL ISSUES
Following the approach proposed in
[3],
we
choose
to decompose the model of the system to be diag­
nosed into two parts:
•
•
a logical static behavioral model ( wi t h no
representation of ti me);
a
mode transition model showing
the pos­
sible temporal evolutions of the behaYioral
modes of each component of the system.
However, differently from
[3],
\\'e concentra te here
on a different characterization of the mode tran­
sition model concerning uncertain temporal infor­
mation. 'Ve assume time to be discrete, allCl use
natural numbers to denote time p oints. �lore im­
portantly , we extend the basic assumption con­
ceming the p r obability distribution of lite liklime
of a component in the usual life phase to the dif­
ferent beh avioral modes of the compoucnt. Tl1is
means that we assume a memoryless Jistribution
of the time spent by a component in a giveu ll!ude,
so the state transition model will be a J)'L\IC. Let.
us discuss in more detail the represe11tationa.l is­
su es; we assume that each component of the sys­
tem has a set of possi ble behavioral mo des, (one
of which is the "conect" mode). The modes of a
component are mutually exclusive with respect to a
give n time point. The fact that a compone 1 1 t. c is in
2It. is possible to generalize to the coutinuous case
foilure role
instead of t.l1e failure probability.
(exponential lifetime) by considering the
l\11�tl1·1iu1•, 1 111,, •• 1.1111 l•·•npor.tll:volutions in Model-based Diagnosis
behavioral modem at t ime
atom m(c, t).
l i;; l"l'lll'l'>'l'lll<'d hy l l w
The relationships betll'<'<'ll lwliav­
•
[3] for
model is atemp o r al (src
more details). Temporal information is ab­
st r acte d from the logical mo del and is represented
in a st ochas tic way. Let CO AI P S be the set of
components of
t.he system
Definition 3
Tlte Associated Markov Chain
a
llfarkov
chain 1those states represent behavioral modes ofc.
dis c rete each
A.. M C will be a DT�IC. Furthermore, if p; is the
probability of being in the current mode m; at the
next t ime instant, then the sojourn time 5; in m; is
It is clea r that, assuming time to be
geometrically distributed with parameter
(1 - p;)
i.e. P(Si = t) = p;-1(1- p;). Notice that, assum­
that. a component has several fault modes and
iug
modeling the t ru sitions among su ch modes as a
n
�[arkov chain allows us to gener alize the con cept
of fail me probability to the concept of transiti on
probability fr om one mode to anot.her.
Definition
Marko1:
4 Ltt AMC(c)
a
Chain of
be
component
c
tf1e
Associated
E COM PS rwd
p,,,,1(c) be the transition probability from m o de
m;
Pc
to mode mj
= [Pm,m;(c)]
of the component c; the matnr
i� called the transition proba­
bility matrix of the elwin.
The matrix Pc rep resen ts the transition probabi li­
tiP.s fro m one mock to ano t her in onP. time im;tant.
The probabilistic b e ha v ior of a M ar kov chain is
completely determinct..! by its transition probability
mat.rix and the iui t.i<�l pro ba bil ity distribution. In
fact, let
iTc(n)
=
{p;;,,(c),p�,�(c), .. . }he
the vector
representing the probabilities of the states ( m odes )
of the chain A.MC( c) at the ins tan t 11 (p�,, (c) is the
the comp one n t c bein g in mode .n�;
at t ime n); th is distribution depends on the mJ­
tial distribution of modes, indeed the fundamen­
probability of
tal rel a t. ion among mode probability clist.ribut.ion
is given by
\Yhere
iTc(ll)
=
.,
iT0(0)Pcn
[p:;,.,m/c)J
fro1;1 !�10cle in time instants of th e com­
p�nent �1oreO\'Cr, an interesting characteristic
Ill·
c.
m;
n
of \linkov chains is the possibility of classifyin g its
states in a rigorous way.
•
an
ugodic
sd
of st.at.es is a set. in "·hich every
reached from every other state
which cannot be left once entered; eaclJ
state can be
of states
is a set in which C\·cry
be reached from every
other state
the set
•
a b s orb ing state is a state wh i ch once en­
tered is never left ( i .e . the probability of re­
an
maining in t.his state once enteret..! is 1);
An inter est ing way of exploiting this cla�>sification
is to use
cation of
the AMC(c)
faults3.
faul t mo de o f c:
•
m;
is
a
to give an a-priori cl assi fi­
Given the AMC(c), let m; be a
permanent fault iff it corresponds to
an absorbing state of AMC(c);
•
is a transient fault iff it corresponds to a
m;
tran sie nt. state of A.MC(c);
•
•
3.2
m;
is
some
a
n
ret•ersible
and
m0
fault iff p;�.mo (c)
is t he correct mode;
m; is an irrevers ible fault iff p�,,�(c)
_ the correct mode.
every 11 and m0 IS
>
=
0 for
0 for
CHARACTERIZATION OF A
TEMPORAL DIAGNOSTIC
PROBLEM
In order to show how stochastic infonnatioll can
be i nte grated in a logical model based diagnos­
tic framework, in this section we briefly di:-;cuss
a possible characte-rization of a temp o r al di<�giiOS­
tic pro blem. \\'e can d e f me a temporal dtagnosltc
TPD as composed by a b eh aYior a l model
B M, a set of components COM P S and a set of
observat.ions at. different t ime instants OBS to l>,;
problem
explained\ we can extract from
a
TPD
a
corre­
atemporal diagu ostic problt m APD by
co nsiderin g a set of obs e r vatio ns 0 BS(l) at a gi V\:11
sponding
time inst an t
t.
Solving an atemporal diagnvstic
problem at the instant
t
means to clet ermine an
assignment at timet, !V(t), to each componem in
COM PS explaining the observations in OIJS(t).
This as sig nment is
an
atemporal diagnosis.
Ob­
viouslv the time instants which are of interest in
power) matrix Pen =
(c) is the probability of reaching mode
the (nth
and p;;,
can
is called transient state;
to be diagnosed.
AMC(c) of a component c E COM PSis
lnnt.�inll sd
and which can be left ; each element of
ioral modes and t heir manifestations a re c-xpr<'S:'<'d
as Horn clau ses and the
a
sl.at.f'
am!
t:•lE"ment of the set is called ergodic ,�tate:
�
the di� nostic process
are just those
for whid1 we
such inst<mts
rdtvant time insta11ts. Let us assuu1e, as iu the
most part of the previous work on diagnosis, t !I at
each c ompon ent is independent of e at h other (i.e.
have some observations; we w ill call
J A similar proposal is presented in [11] hy introduc­
ing an O·Jloslerior·i classilica.tion.
4 The suitable notion of explanation can !Je t:X­
traded from the �pectrum of definitions ill [G]; more­
oYer, notice
for the sake of simplicity, we tlo nol
discuss the role of contextual informat.ioH in a diagllos­
tic problem (see [G] for an analysis of this pro hlc·lll ) .
that,
247
2--1�
Portinale
t.he b eha v ior of a component c ann o t influence
behavior of the others). \'Ve indicate as mtv(!)
mo d e
that W(t) assigns
to c; because of
dependence of the comp one nts 1 if ti <
relevant time instant, we have
=
P[lF(tJ·)!W(t,·)]
tj
the
the
the in­
are two
IT
pm�l'(t;)miV(Ij)
c
(c)
IJ
P;n� . (c)
, ,,)
cEC0ft.1 PS
Pump P
and c learly
P[TV(t)]
,,·here
t.ions
over,
=
cECOMPS
p�1c (c) d�pends on the initial distribuw(t)
iTc(O) gi ve n for each component c. More­
we are also interested in the joint probability
P[W(O), W(l), .. . W(t)] at time t of a wh ole evolution {W(O) . .. Tl'(t)}. Since the analysis of the
4/5
Container C
prob3bilities at time t requires knowledge abo u t
the initial distribution of modes, in the absence of
specific information we can reasonably assume
uniform distributio11.
an
1: let us sup p os e to have a simple sys­
tem composed of a water pump P and a container
C receiving the pumped \\·ater. For the sake of
1:
Figure
ACMs of
Example
bre vit y we do
not describe the logic a l behavioral
model of this simple system, so we will suppose to
have directly at disp os a l the atemporal diagnoses
at the relevant instants. The Ar.ICs of the two
components and the
ces
:2.
transition probability matri­
are s ho wn respectively in figure 1 and in figure
S u p p ose to have the following hypothesis at
t. i me t
H"1(0)
TT':? (0)
1V3(0)
=
0:
=
{cm·reci(P,O),correct(C,O)}
{partial! y_occluded( P, 0), cor1'eci( C, 0)}
{occlud(d(P, 0), coiTect(C, 0)}
=
=
Dy
assumi n g
tion
on
an
uniform
distribution
we
get
�·
P[TFdO)] = P[TF:!{O)] = P[W3(0)] =
This
means that we have the fo llo wing initial distribu­
t.he
com p o u cn
ts (we
assume the probabil­
ities given in the following order : pun ct-und, {tak­
ing. correct for C and broken, occluded, leaki11g,
correct for P).
Jlartially_occlucl€d,
rrc(O)
rrp(O)
=
=
(0, 0.1)
(0, �� 0, �
Let. us introduce t.he follo w in g abbreviations we
11·i l l use in the examples: punctured=zm, !eak­
irlg = le . correct=co.
b ro k en = b r, occluded=oc,
S u pp ose that <�t ti m e t
par­
1
the logical model concludes the atemporal diag­
=
nosis rr( 1) = {ocdttded(P), coJTect(C)}, then
ha1·e the foll o w i n g conditional probabilities
P[ll'( l)!Jrl(O)j
=
Pc",o�(P) Pco,co(C)
and
=
0
we
=
=
Pro,oc(P) Pco,co(C) =
Po<,oc(P) Pco,co(C) =
the follow ing joint
P[W2(0 ), W(l))
=
P[Wz(O)] P[IV(l)\vV2(0)]
P[W3(0), lV(l)] =
P[W3(0)]P(W(l)\Wa(O)]
It is now
diagnosis
��
{o
proba bilit.ies
P[W1(0)1 W(l)] =
P[W1(0)]P[W(l)\W1(0)]
possible to define
a
0
=
�0
=
� :iqs
=
1�
=
� 190
=
2u
notion
of
temporal
=
on the basis of the con c ep t of soluLion
for an atemp or al problem.
Definition 5 Given an atemporal diagno�lic
problem APD, an assignment W(t) of beharioral
modes to e a ch c E COMP S is a temporal di­
agnosis for the corresponding temporal diagnostic
problem iff:
1.
5)
twlliJ-OCcluded=po.
P[W(l)\W2(0)]
P(IY(l)\H'3(0)]
the components P an<l C
W(t) is a solution for ADP for evuy n:.leuud
time instant t;
relevant tir!le iils/a,tb
fJ wheH 0 :S <J :S 1
is a predefin ite threshold named the plausi­
bility threshold for· the temporal diagnostic
problem.
2. for
t; <
et,ery consecutive
ti, P[W(t1)\W(t;)]2':
s econd part of the deftJJit ion co u ld
be different if we sup p osed to have inform<ttion
Notice t hat the
on t he
plausibility threshold d ire ctly
on the
com-
Modeling Uncertain Temporal Evolutions in Model-based Diagnosis
and P C 2 , i t i s easy t o see t h at. , i f the s ame d i a g­
noses were concluded at time
i>rol<.en occ\udeo \eal<.ing pan_occ\. coneCI
0
broken
0
occluded
leaking
correct
Pp :
0
0
0
0
0
0
0
215
315
0
1 150
0
1/25
1/25
9110
31 \ 0
0
cor rect
711 0
0
1/10
9110
F i gu r e 2 :
is
4
C
C
cou ld check the pl au­
sibility of the transitions of each component c
from ;. = m�V(t,) t.o s = m�Y(ti J " Obviously, i n
c as e the corresponding t h resh ol d should be
different; indeed , giv en the same threshold CT , if
this
u
and
n =
tj - l ; , t h e n
but not vice versa.
The i nformat ion in t. h e lVI arkov
with
c h a in s
P�s �
u
associated
the comp onen ts can th en b e used to di sc a rd
very i mp r o b ab le di ag n os t i c hypotheses
from
the
t e m p or al p o i n t of view .
Example
1,
2:
s up p ose
g i v C' n t. h e same system of
m�
h y p o t h e ses :
I T '! ( l )
W:d l )
! 1 '3( 1 )
=
=
=
{occludul( P, 1 ) , correct (C, l ) }
{b1·oke n ( P, l ) , conect(C, 1)}
{C07'1'ect ( P, 1 ) , puncttiTed( C' , 1 ) }
Let t h e plausibil i t y threshold b e
P(l l "d l ) ! lF ( O)]
P[I F� ( l ) I IF ( O)]
=
=
u =
P[TV3 ( l ) I W(O)]
}0 1�
temporal
P [IF(O)]
the mat r i c es P c k . For i nst an ce , i n
1, the most probable t em p or a l di agnosis
{ W3(0) , W( l ) } .
DIS CU SSION
In t h is p a p e r , a n a p p roach i ntegrating stoclws­
tic i nfor ma t ion in logical model- b ased diagnos i s is
proposed, i n the attempt to characterize proJ,Iems
deriving from t h e i n t r o d u c tio n of t em p oral infor­
mat i o n in t h e model of the system to be dia gnose d .
Howeve r , a n import a n t problem s t il l to be consid­
ered c oncer n s the role of t h e A M Cs of the compo­
nents with re sp ect to the u n d erl y i n g static logi c a l
mod e l . In fac t , using the approach described in
t h i s paper , we assume to mod el t h e temporal b e­
h avior of the components at a very abstract p oi n t
of v i e w i n wh i ch a transition from one mo de: to
an o t her is just a r a n d om function of t ime . The
problem a r i s es when stochastic information con­
t r asts w i t h i nfor m a t i o n gat h ere d
by means of n e w
it is renson­
observations on the sys tem. Actua lly,
ab le to assume t h at new observations supply new
exam­
get the atemporal d i agnosis
T T ' ( O) = {con·eci ( P, 0) , correct(C, 0 ) } . C onsi d er
now the case in w h ich new observations a l low us
t.o con c l ud e , at time t = 1 the fol lowing diagnostic
ple
of ran king tl1e
b y m eans of
p onen t s ; in this c as e we
P[IV ( tj ) I W(t; )] �
clear t h at an interesting as pect of t h i s
exampl e
Transit ion p ro b a b i l i t y matri ces of the
P and
be
fm;t one .
where the first fac tor is the joint p robability at
t he previous step and the second can b e computed
:transition probability matrix for com ponent
components
t h e n t.he on ly
P [!V(O), . . . W(t - k ) , H'(t)] =
P[W(O) , . . . W(t - k )] P[!V(t) I W(t - k)]
P
0
0
2,
comp u t e d , g i v e n the a-priori dist ributions
by the fol lowing formu l a :
leaking correct
punctured
=
di agn ose s we get; indeed , it is easy to show t h a t
t h e jo i n t probabilities at time t c a n be recursi ,·cly
0
transition probability matrix for component
leaking
It should
a p p roach is the p ossi bi l ity
415
punctu red
Pc
0
0
liS
part_occl.
0
t
non-admissib1e diagnosis would he the
=
560
> (T
=
1�0 .
ev i d e n c e for the a.t emporal diagnostic hypotheses
a t a gi v e n instant t, ev en if t he pred ictet.l prob­
figure out
t he
the
stoch astic mo del ; this mea ns that we h a ve t o per­
form a rev is i o n of the probabilities <�t time I in
order to t a ke i n to a c co u n t the new i u fon n a t ion .
Howe ver , we h ave to p a id a t t e n t i on to the cout.ex.t
abi l i t ies
were low.
in which such
We have
0
The only ad mi ssi b l e d i agnosis at t i me t = 1 i�
l r-:> ( 1 ) . By com p u t i n g the s q u are matrices P p J
The
problem
i ::> to
how to comb i ne this new evidence obt ai ned by
logi ca l step wit h the pred ic tions ga thered from
a
re v is i o n is reasonable .
The
as­
s u m pt io n we make is th at the reason i n g step on
the log i c al mo de l c a n not lo ose any possible d i <� g­
n osis ; so if H'1 (t) . . . W, (t ) a r e the atem p oral di<tg­
n ose s at ti me t , there not exis ts a n y other p oss i ble
d i a g no s i s a t that t i me , even if so me of them coult.l
not ac t u a l ly b e d i agnoses ( the obser n t ions n 1 ight
not allow us to d iscriminate b e t ween t.he1 1 1 ) . I11
249
250
Portinale
t h is contex t , the probabilities of the temporal di­
a g n oses at t ime t c an be normal ized to s u m 1 . If
the n umber of joint p rob ab i lit ies at t ime t is N and
we i n d i c ate as P1 (t) the i t h of such pr ob ab ili t i es ,
the normalization factor at time t will be
l
F(t) -
- L�t P;(t)
\Ve can now compute the
revised joint probabilities
RP;( t ) P; (t)F(t ) , the revised conditional ones
RP[W(t ) i vV(t- k )] = P[W(tWF(t- k)]F(t) and
\\'e can check the p lausibility threshold on these
as
as
=
new values. Notice that if we c heck the plausibil­
ity threshold on the t ran sition probabilities of the
we have to compute, at each relevant
for each com p o nen t c , the d istribu t io n
component s ,
instant t ,
;r,, (t) :::: il'c(t - k)Pck which is not nee ded in the
previous case. In de ed , in th is case, we need to
n o rm a l ize the p rob abili t ies of each c om p on en t in a
g i ven mode u si n g a fa c t or f(c, t ) . In this way we
c a n get the revised t r an si t io n probabilit ies from
m o de m ; to 111j of
=
c as 7'P�1 , m ;
P�n , m ; f(c, t)
checking the pl a usi b ility thres hol d
and
ones, at disposal . I n such a case , the assumption
of getting from the logical model all the p ossi ble
atemp oral diagnostic hypot hesis can be considered
su itable and reasonable .
vVe did not dis cussed h ere p roble ms conceming;
re as oning issues, however c on si dera tion s similar to
those discussed in [3] can be applied to the fr ame­
work described i n the p resent p a p er .
Acknowledgements
The author would like to thank P. Torasso and
L. Console for their v alu able help and usefu l dis­
cussions about the topic p r esent e d in the pa"
This work has been partially support(:d by
Project under grant. n .
9 1 .0235 l . CT 1 2 .
per.
CNR Tem po r al Re ason i ng
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