ANY TIME PROBABILISTIC REASONING FOR SENSOR VALIDATION

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1 266 ANY TIME PROBABILISTIC REASONING FOR SENSOR VALIDATION P.H. Ibargiiegoytia Istituto de Ivestigacioes Eh ctricas, A.P Cueravaca, Mor., 62001, Mexico L.E. Sucar Istituto Tecol6gico y de Estudios Superiores de Moterrey Campus Morelos, A.P. 99-C Cueravaca, Mor., 62050, Mexico esucar@campus.mor.itesm.mx S. Vadera Uiversity of Salford Dept. of Mathematics ad Computer Sciece Salford, M5 4WT, U.K. S.Vadera@mcs.salford.ac.uk Abstract For may real time applicatios, it is importat to validate the iformatio received from the sesors before eterig higher levels of reasoig. This paper presets a ay time probabilistic algorithm for validatig the iformatio provided by sesors. The system cosists of two Bayesia etwork models. The first oe is a model of the depedecies betwee sesors ad it is used to validate each sesor. It provides a list of potetially faulty sesors. To isolate the real faults, a secod Bayesia etwork is used, which relates the potetial faults with the real faults. This secod model is also used to make the validatio algorithm ay time, by validatig first the sesors that provide more iformatio. To select the ext sesor to validate, ad measure the quality of the results at each stage, a etropy fuctio is used. This fuctio captures i a sigle quatity both the certaity ad specificity measures of ay time algorithms. Together, both models costitute a mechaism for validatig sesors i a ay time fashio, providig at each step the probability of correct/faulty for each sesor, ad the total quality of the results. The algorithm has bee tested i the validatio of temperature sesors of a power plat. 1 Itroductio Artificial itelligece (AI) techiques are playig a icreasigly importat role i real applicatios. I idustry, differet techiques have bee proposed, for example, i diagosis, automatic cotrol, ad moitorig. Geerally, these applicatios require a overall model which usually, its iputs are maily sesors. Also, may of these real applicatios eed to maitai a real time behaviour, i.e., the correctess of the system depeds ot oly o the logical result of the computatio but also o the time at which the results are produced (Stakovic 1988]. Usually, real applicatios possess a time limit by which some actios must be performed. This paper presets a model for the validatio of the sesors used i real time applicatios. The proposed validatio is carried out i a separate module that works together with other fuctios i a system. I other words, it is assumed that a layered scheme is used i which the lowest level cocetrates o validatig the sigals trasmitted by the sesors as preseted i Fig. 1 (Yug & Clarke 1989]. The mai beefit of Figure 1: Layered diagosis architecture. usig a layered approach is that it eables the costructio of models i a modular fashio. That is, it is easier to costruct a model for sesor validatio ad the a model for the process tha it is to costruct a overall model i oe step. This separatio of the sesor validatio layer ca also result i simpler higher layer models ad leave the higher layers to utilize other techiques. Faults i the sesors readigs are detected i a decetralised ad hierarchical approach, so that they ca be easily isolated ad repaired. Additioally, suppose that the higher layers of the system represet other importat ad critical fuctios, e.g., the fault diagosis of a uclear plat. The itermediate layer (loop diagosis) may be usig model based reasoig to diagose a cotrol loop i the plat, whereas the system diagosis layer may be utilizig a differet approach. The validatio module preseted i this paper, utilizes a probabilistic model which cosiders oly the relatioships betwee the variables to be validated. This

2 Ay Time Probabilistic Reasoig for Sesor Validatio 267 probabilistic model is idepedet of the higher layers models, so it is easier to costruct ad matai whe ecessary. This paper presets the cotiuatio of the project described i a previous paper [Ibargiiegoytia et al. 1996]. I that paper, the authors described a probabilistic approach to sesor validatio that took advatage of a Markov blaket property to distiguish real faults from apparet faults. A Bayesia etwork was used as a basis for predictig a probability distributio for a sesor value based o other sesors. The predicted distributio ad the actual sesor readig was used i order to determie if there was a potetial fault. However, the sesor validatio process described i that paper works i batch mode, i.e., o itermediate results are available, ad o attempt is made to estimate the quality of the results. For a real time applicatio, these characteristics are iadequate. This paper presets the extesio of the sesor validatio algorithm, so it ca be applied i real time systems. This cosists i the use of ay time algorithms. Thus, the extesio of the sesor validatio algorithm cosists i the followig features. First, the use of a probabilistic causal etwork that relates the real ad apparet faults. Secod, i order to perform i ay time basis, the validatio algorithm selects the sesor which provides more iformatio whe validated. Fially, a quality fuctio is calculated i order to characterize the behaviour of the algorithm. The selectio of the most iformative sesor is made usig the etropy fuctio. To summarize, this paper presets a ay time probabilistic algorithm for validatig the iformatio provided by sesors. The system cosists of two Bayesia etwork models. The first oe is a model of the depedecies betwee sesors ad it is used to validate each sesor. It provides a list of potetially faulty sesors. To isolate the real faults, a secod Bayesia etwork is used, which relates the potetial faults with the real faults. This secod model is also used to make the validatio algorithm ay time, by validatig first the sesors that provide more iformatio. To select the ext sesor to validate, ad measure the quality of the results at each stage, a etropy fuctio is used. Together, both models costitute a mechaism for validatig sesors i a ay time fashio, providig at each step the probability of correct/faulty for each sesor, ad the total quality of the results. The ext sectio briefly describes the basis of ay time algorithms. 2 Ay Time Algorithms Ay time algorithms represet oe directio of work that aims to achieve the use of artificial itelligece techiques i real time systems. This term was iitially used by Dea i his research about time depedet plaig [Dea & Boddy 1988]. At the same time, Horvitz (1987) proposed the ame of flexible computatio for this mechaism. Ay time algorithms are those that ca be iterrupted at ay poit durig computatio, ad retur a aswer whose value icreases as it is allocated additioal time [Boddy & Dea 1994]. However, how ca this value be measured i a specific applicatio? The literature cotais descriptios of differet dimesios that have bee proposed as metrics [Zilberstei & Russell 1996]: certaity, accuracy ad specificity. Performace profiles represet the expected value of these metrics for a give procedure as a fuctio of time. I other words, performace profiles characterize the quality of a algorithm's output as a fuctio of computatio time. Figure 2 illustrates three cases of performace profiles [Zilberstei & Russell 1996], [Dea & Boddy 1988]: q (a) (b) (c) Figure 2: Examples of performace profiles. (a) a stadard or oe shot algorithm. (b) a ideal, expoetial precisio algorithm, ad (c) a more realistic profile for a ay time algorithm i practice. Clearly, all these types of performace profiles are special cases of a superclass that ca be defied as mootoic improvemet, i.e., the quality of its itermediate results does ot decrease as more time is spet to produce the result. The ext sectio explais the basis of the validatio algorithm, so that sectio 4 develops the ay time algorithm for the sesor validatio problem. 3 Sesor Validatio The probabilistic sesor validatio model utilizes Bayesia etworks. The odes represet the measures of the sesors. The structure of the etwork makes explicit the depedece relatios betwee the variables. The probabilistic sesor validatio icludes the diagosis of all sigle sesors i the etwork. The idea is to istatiate all the odes with the sesor readigs, except the oe beig validated. A probabilistic propagatio provides a distributio of the posterior probability of the estimatio of a sigal value based o the readigs of the most related sigals. The estimated value is the compared with the curret value i order to decide if the measuremet is correct. The most closely related variables for each sesor cosist of a Markov blaket of the sesor variable. A Markov

3 268 Ibargiiegoytia, Sucar, ad Vadera blaket is defied as the set of variables that makes a variable idepedet from the others. I a Bayesia etwork, the followig three sets of eighbours is sufficiet for formig a Markov blaket of a ode: the set of direct predecessors, direct successors, ad the direct predecessors of the successors (i.e. parets, childre, ad spouses) [Pearl 1988]. The set of variables that costitutes the Markov blaket of a variable ca be see as a protectio of this variable agaist chages of variables outside the blaket. This meas that, i order to aalyze a variable, it is oly ecessary to kow the value of all variables i its blaket [Ibargiiegoytia et al. 1996]. Additioally, the exteded Markov blaket (EMB) of a sesor is formed by its Markov blaket plus the variable itself. However, sice validatig a sesor based o a faulty oe results i a erroeous validatio, the probabilistic validatio oly provides a list of apparet faults. Thus, the probabilistic validatio provides a list of potetial correct ad potetial faulty sesors. The fault isolatio is carried out whe the list of potetial faulty sesors is compared with the list of EMB of each sesor. Whe a match exists, the the faulty sesor has bee distiguished. Otherwise, differet coditios exist for the isolatio of multiple failures [Ibargiiegoytia et al. 1996]. The ext sectio describes the extesios of the sesor validatio model i order to discrimiate faulty ad correct sesors i a ay time basis. will be explaied ext i order to clarify the rest of the algorithm. 4.1 Validatio The validatio step was briefly itroduced i sectio 3 ad more extesively i [Ibargiiegoytia et al. 1996]. The sesors are processed oe by oe by the validatio fuctio utilizig the followig algorithm: 1. Read the actual value of the variable provided by the sesor. 2. Read the value of all variables that appear i the Markov blaket of the selected variable. 3. Propagate the probabilities ad obtai the posterior probability distributio for the selected variable. 4. If the probability (obtaied i 3) of the value acquired i step 1 is lower tha a specified value, retur failure; else retur success For example, cosider the simplified model of a gas turbie show i Fig. 4. The validatio of m is carried 4 Ay Time Sesor Validatio Ay time sesor validatio algorithm implies that the kowledge about the state of the sesors (faulty or correct) becomes more certai ad complete as time progresses. Certaity about the state of a sesor refers to the degree of belief i the correctess of a sesor, ad completeess is characterized by the umber of sesors from which the state is kow. Thus, it is required to be able to moitor the state of the sesors durig all the validatio process. This is doe through a vector whose elemets PJ ( s;) represet the probabilities of failure for the sesors s;. Give that the ay time validatio process eeds to be cyclic, the top level of the algorithm ca take the form show i Fig Iitialize P1 ( s;) for all sesors s;. 2. While there are uvalidated sesors do: (a) choose the ext sesor to validate (b) validate it (c) update the probability of failure vector P1 (d) measure the quality of the partial respose Figure 3: Top level of the ay time sesor validatio algorithm. The probabilistic validatio of a sigle sesor (step b) Figure 4: A reduced Bayesia etwork of a gas turbie. out by calculatig the probability distributio of m give the measuremets oft ad p. If the real value of sesor m has a probability greater tha certai value, the the sesor is cosidered correct, ad faulty otherwise. However, if the fault is i sesor p, the the validatio of m will also result i apparet fault. Thus, the validatio step is carried out by this algorithm that receives as iput, the sesor that will be validated. As output, the algorithm returs a biary value { correct,faulty} with the apparet status of the sesor. 4.2 Selectio of ext sesor This sectio develops a mathematical model for choosig the best sesor to validate give the history of the validatio process ad the curret state of the system. Also, the model proposed here will be used for measurig the quality of the respose i order to obtai the performace profile of the validatio algorithm. The cetral idea is that the validatio of a sesor provides

4 Ay Time Probabilistic Reasoig for Sesor Validatio 269 some iformatio ad also, extra iformatio ca be iferred. Therefore, a measure of the iformatio that a sigle validatio produces is required. A defiitio of the expected amout of iformatio that a evet produces was first proposed by Shao ad used i commuicatio theory [Shao & Weaver 1949]. Shao proposed the followig defiitios. Defiitio 4.1 Give a fiite probability distributio Pi 0 for (i = 1,..., ), ad L: p; = 1 Shao's etropy measure is defied as H=H( Pl,, P)=-2:.,p;log2Pi (1) i=l Thus, the etropy measures the related umber of bits required to store the iformatio. Sice the validatio of a sesor s has two possible outcomes, the etropy fuctio H(s) is the defied as: { 0 if p = 0 or p = 1 H(s) = -plog2(p)- (1-p)log2(1- p) otherwise (2) where p represets the probability of failure of the sesor. Notice that the expressio plog2(p) = 0 whe p = 1 but it is udefied whe p = 0. However, sice plog2(p) teds to zero as p teds to zero, the values defied i equatio 2 ca be safely assumed. Notice that it has its maximum whe p = 'i.e., whe the igorace is maximum, ad it is zero whe either p = 0 or p = 1, i.e., whe the iformatio is maximum ad igorace is miimum. This fuctio ca be cosidered either as a measure of the iformatio provided by a experimet, or as a measure of the ucertaity i the experimet's outcome. Thus, cosiderig each sigle sesor validatio as a experimet, this fuctio ca be used to measure the amout of iformatio provided by that validatio. The, the average amout of iformatio E for the system ca be defied as follows: 1 E(s1,...,s) = - 2:.,H(s;) i=l 1 = -- L PJ(s;)log2PJ(s;) i=l + (1- P1(s;))log2(1- PJ(s;)) 2 = -- L PJ(s;)log2PJ(s;) (3) i=l where is the umber of sesors i the system S, ad P1(s;) represets the curret probability of failure value assiged to sesors;. Notice that the vector whose elemets are P1 ( s;) provides a measure of the certaity i the validatio while the sum of idividual etropies provides a specificity measure of the result. Give this measure, the ay time sesor validatio algorithm eeds to select a sesor X that gives the best improvemet i the average etropy of the system S. Hece the followig coditioal versio of equatio 3 ca be writte E(S I X) = E(S I x = ok) + E(S I x = flty) = (2:.,H( s; I x = ok) + 2:.,H(s; I x = flty)) (4) This fuctio ca be evaluated for each sesor ad the oe which gives the most iformatio (the miimum E(S I X;)) ca be selected as the ext sesor X; to be validated. The computatio suggested by the above formulae could be too expesive for a real time sesor validatio process. To overcome this problem, a pre compilatio of the sesor selectio mechaism is implemeted as follows. The above formulae are used to select the sesor, Sr which gives the most iformatio. This selected sesor forms the root of a biary decisio tree. A fault is simulated i this sesor ad the formulae are agai used to select the ext sesor Sr-. The, the root Sr is assumed to be correct, ad the formulae are used to select the sesor Sr+ i this case. This results i the partial decisio tree show i Fig. 5. This process is repeated recursively o the Figure 5: Partial decisio tree. odes Sr- ad Sr+ to obtai a complete decisio tree, so that each path i the tree icludes all the sesors. As a example, cosider the etwork show i Fig. 4. This process results i the decisio tree show i Fig. 6. Notice that this tree ca be reduced cosiderig Figure 6: Biary tree idicatig the order of validatio give the respose of the validatio step. oly the valid trajectories formed by the assumptio of, for example, sigle faults amog the set of sesors. See [Ibargiiegoytia 1997] for more details.

5 270 lbargiiegoytia, Sucar, ad Vadera This decisio tree ca be used to select the ext sesor more efficietly i real time tha by performig the calculatios. Thus, the selectio step of the algorithm of Fig. 3 cosists of simply traversig the tree oe level after every sigle sesor validatio. The cycle starts at the root, ad the decisio tree poits to the ext ode i the tree accordig to the result of validatig the curret sesor. gas turbie give i Fig. 4. Thus, the cosequeces of 4.3 Isolatio The validatio step provides oly a list of potetially faulty sesors. Thus, a compariso is made betwee the set of potetially faulty sesors with the table of exteded Markov blakets of all the sesors. Whe a match is foud, a real fault is determied. However, the set of potetially faulty sesors is obtaied after all the sesors have bee validated. Therefore, i order to exted that algorithm for ay time behaviour, a differet mechaism for distiguishig real faults from apparet oes is required. This ew mechaism provides, as the output of the isolatio phase, a vector with the probability of a real fault i all the sesors. This vector is refied icremetally i time, so the ay time behaviour ca be achieved. The ay time fault isolatio process is based o the relatioship betwee real ad apparet faults. There are two situatios that arise: (i) the existece of a real fault causes a apparet fault (as show i Fig. 7(a)), ad (ii) oe apparet fault is the maifestatio of several possible real faults (as show i Fig. 7(b)). (a) (b) Figure 7: Causal relatio betwee real faults (R) ad apparet (A) faults represeted as odes. I (a), oe real fault causes several apparet oes, while i (b), oe apparet fault is caused by oe or more real faults. I both figures, the relatio betwee root odes ad leaf odes is the same as the exteded Markov blaket (EMB) of a sesor. Cosiderig all the sesors, a causal model relatig the real ad apparet faults ca therefore be obtaied from the fault detectio Bayesia etwork (i fact, the EMB table is sufficiet to build this etwork). I the first level (roots), the odes represet the evets of real failure i every sesor. The, the secod level (leaves) is formed by odes represetig apparet failures i all the sesors. Arcs are icluded betwee every root ode, ad the correspodig odes of the exteded Markov blaket. For example, the causal etwork show i Fig. 8 ca be obtaied directly from the Bayesia etwork of the Figure 8: Probabilistic causal model for fault isolatio i the example of Fig. 4. Rt_ represets a real fault m sesor i while Aj represets a apparet fault i sesor j. observig a apparet fault ca be propagated i the causal etwork i order to obtai the probabilities of a real fault i all the sesors. The etwork of Fig. 8 is multiply coected. Hece, the propagatio method of trees of cliques is utilized [Lauritze & Spiegelhalter 1988). I geeral, 0(2) coditioal probabilities would be required (for a ode with parets). However, the oisy or model ca be adopted here. Two assumptios eed to hold i order to use this model: accoutability ad exceptio idepedece [Pearl1988). The accoutability assumptio holds by the way the model is costructed, i.e., a sesor is apparetly faulty oly if there is a fault i its MB. The exceptio idepedece assumptio is cocered about a rare situatio for this particular model. The relatioship betwee the real ad apparet faults is obtaied from a Bayesia etwork i which the depedecies are assumed to be strog. Hece, the probability of a real fault ot resultig i a apparet fault is small. Further, the mechaism by which a real fault i oe sesor does ot result i a apparet fault is eve less likely to be depedet o aother real fault. Hece, give that these assumptios are reasoable, the coditioal probability matrix ca be calculated by utilizig equatio 5. where d is the set of assigmets of the set of apparet faults, ad Td represet the set of all apparet faults actually preset. Thus, the oly parameter required is defied as: Cij = 1-% = P(Aj I R; oly). I the case of the sesor validatio problem, i a ideal case, all the parameters Cij 1. Of course, these values ca be obtaied by simulatio from the data if the problem is expected to depart from this ideal case. That is, accordig to the theory developed i Ibargiiegoytia et al. (1996), whe a real fault Rt_ is (5)

6 Ay Time Probabilistic Reasoig for Sesor Validatio 271 preset, it will always cause the apparet fault Aj (assumig that there is a arc from R; to Aj). The etwork of Fig. 8 is iitialized with the followig iformatio: (i) the prior probability of all the root odes i the model is 0.5 (assumig igorace at the begiig of a cycle), ad (ii) the parameters Cij = 0.99 for all 1 :::; i, j ::;umber of odes. Havig described how real ad apparet faults ca be related, the fault isolatio model ca ow be summarized. It receives as a iput, a validated sesor with its detected state (faulty or correct) ad updates the probability of failure of all the sesors. It does this by istatiatig the value of the correspodig apparet ode ad usig a propagatio algorithm to obtai the posterior probabilities of the real faulty odes. A vector Pf of these posterior probabilities represets the curret state of kowledge about the sesors, ad ca be viewed as the output of the system at ay time. For example, assumig a fault i g i the etwork of Fig. 4, produces the sequece of values of the probability vector as show i Table (a) time (b) Figure 9: Performace profile describig the combiatio of certaity ad specificity i oe parameter agaist time. (a) without failure, (b) with a simulated failure i sesor g. Mexico. A Bayesia etwork represetig the depedecies betwee the sesors of the plat is show i Fig. 10. The depedecy model was obtaied by utilizig a automatic learig program that uses real data from the start up phase of the turbie [Sucar et al. 1997]. Table 1: Example of the values of the probability vector Pf. Step Pr(m) Pr(t) Pr(P) Pr(g) P1(a) t =faulty m = correct g =faulty a= c orrect p =correct Quality measure A measure that is idepedet of the applicatio is the average etropy of the sesors give i equatio 3. That is, if the curret quality measure is: 2 Q(s1,..., s) == -- L Pj(si)log2Pj(si) (6) i=l the, the reported quality fuctio is calculated with the formula Q = 9maxQ9curret where Qmax is the maximum value of the q ;lity measure (i.e.,, the umber of odes). Notice that this measure captures both the certaity ad specificity measures of ay time algorithms. It captures certaity sice the probabilities of the sesors are used, ad specificity sice all the sesors are combied to give a average. Figure 9 shows the performace profile obtaied with this quality measure for the example of Fig 4. 5 Empirical Results The sesor validatio algorithm was evaluated by applyig it to the validatio of temperature sesors of the gas turbie at the Gomez Palacio power plat i Figure 10: Probabilistic tree of the applicatio. Nodes represet temperature sigals of a gas turbie. The data set was partitioed i two subsets: oe partitio for traiig the etwork, ad the other partitio for testig. The traiig/testig partitio used was 70-30% of the origial data set, i.e., 610 istaces for traiig the model (calculatig the prior ad coditioal probabilities), ad 260 istaces for testig. Theoretically, the system should always detect ad isolate sigle faults correctly. However, i reality, some errors may occur sice i practice it is ulikely that the depedecy model will be perfect. Cosequetly, two types of errors could occur: a correct readig might be cosidered faulty, ad a real fault might ot be detected. These two possible errors are called type I ad type II errors i the literature, ad defied as follows [Cohe 1995]: type I: rejectio of the ull hypothesis whe it is true, type II: acceptace of the ull hypothesis whe it is actually false. The ull hypothesis used refers to the hypothesis that a sesor is workig properly. Thus, i other words,

7 272 lbargiiegoytia, Sucar, ad Vadera type I errors occur whe a correct sesor is reported as faulty while type II errors occur whe faulty sesors are ot detected. The criteria for decidig if a readig is faulty or ot ca result i a trade off betwee these two types of errors. The criteria cosidered i this project are the followig: 1. Calculate the distace of the real value from the expected value, ad map it to faulty if it is beyod a specified threshold ad to correct if it is less tha a specified threshold. The threshold values cosidered were 2, 2.5 ad 3 times the stadard deviatio rr. 2. Assume that the sesor is workig properly ad establish a cofidece level at which this hypothesis ca be rejected, i which case it ca be cosidered faulty. This cofidece level is kow as the p value. The p values cosidered were 0.05 ad The accuracy of the model, i.e., the proportio of type I ad II errors, is evaluated by varyig the possible thresholds for each of these criteria. Two differet faults were simulated: Severe. The sesor value modified is the most distat extreme value, i.e., the real value is substituted by oe which differs by miimum 50 %. Mild. The real value is replaced by oe which differs by 25 %. A test procedure was used to evaluate the accuracy of the whole validatio process. Table 2 presets the fial evaluatio of the prototype with the percetage of type I ad II errors for severe ad mild faults. Criteria Type I Type II Type I Type II Type I errors imply that most of the sesors i a EMB preset apparet type I errors. This is more commo as it ca be see i Table 2. That is, there are cases where the existece of a ivalid apparet fault, together with the valid oes, completes the EMB of a misdiagosed sesor. Hece, a type I error is produced. O the cotrary, type II errors are detected at this stage whe most of the sesors of a EMB preset misdiagosed apparet faults. This is very improbable as the results of Table 2 cofirms. The percetages are obtaied comparig the average umber of errors, with the total umber of experimets. 6 Discussio Sectio 4.2 developed a ay time sesor validatio algorithm that utilizes a etropy fuctio as a criterio for selectig the ext sesor to validate. This etropy fuctio calculates the amout of iformatio that ay sigle validatio provides for diagosig all the sesors. Hece, to evaluate this criterio, this sectio compares the performace profile of the ay time sesor validatio algorithm as a fuctio of time whe the etropy based measure is used, ad whe a radom selectio scheme is used. Figure 11 shows the resultat performace profile of the ay time sesor validatio algorithm. That is, the 0.9 o.s (J Tme -- Radom -x- Etropy x-x Figure 11: Performace profile of the ay time sesor validatio algorithm (timex 10-2 sec.). quality of the respose as a fuctio of time. A experimet cosisted i the simulatio of a sigle fault. Thus, 21 idepedet experimets were ecessary to simulate a fault i all the sesors. I total, 260 experimets were carried out, so every oe of the 21 sesors was simulated faulty at least 12 times (12.6 times). The etropy graph represets the average of the resultat quality with the etropy based scheme for the 21 sesors of Fig. 10. The radom graph represets the average of the same experimet with a radom selectio scheme. The time axis is a qualitative compariso rather tha quatitative. Alteratively, the results ca also be evaluated by comparig the time required to reach differet levels of quality. For example i Fig. 11, whe the radom criterio reaches 60 % of quality, the etropy criterio has already reached more tha 80 %. The approach has bee implemeted ad is beig tested o the validatio of temperature sesors i a gas turbie of a combied cycle power plat. The results for the accuracy of the model were reported i terms of the type I ad type II errors ad with respect to detectig severe ad mild faults. The results showed, that for this particular test applicatio, more

8 Ay Time Probabilistic Reasoig for Sesor Validatio 273 striget criteria for detectig failures reduced type I errors but did ot sigificatly icrease type II errors. The results of the evaluatio of the validatio ad isolatio phases together are show i Table 2. Agai, with a p value of 0.01, there are 2.9% of type I errors, ad 0.4 % type II errors. Notice that, i geeral, the sesor validatio algorithm performs almost perfectly with respect to udetected faulty sesors, i.e., all the faults are detected. At the same time, the rate of icorrect detectio faults is satisfactory for most of the criteria aalyzed. Two complexity aspects eed to be discussed. The first oe is the size of the pre compliled decisio tree preseted i sectio 4.2. A biary tree for sesors cotais levels ad up to 2-l odes (1,048,575 odes for 21 sesors). However, if a sigle fault is assumed, the the decisio tree results i a prued tree with levels ad at most x ( + 1) odes. The secod oe is the complexity for probability propagatio i the fault isolatio etwork as i Fig. 8. The propagatio complexity (usig the closterig algorithm [Lauritze & Spiegelhalter 1988]), depeds o the the size of the largest etry of the EM B table, i.e., the largest clique. However, if a tree is assumed for the detectio Bayesia etwork, the umber of odes i the EM B table remais small, i.e., just oe paret ad the childre of a ode. 7 Coclusios This paper has preseted a ay time, probabilistic algorithm for sesor validatio. A layered approach is cosidered where the lowest layer performs the validatio. A Bayesia etwork is used to defie the relatioships betwee variables ad to estimate the expected value of a sesor. The expected value is the compared with the actual readig obtaied. If these measures differ the a faulty sesor is suspected. A faulty sesor is the distiguished from apparetly faulty sesors by the use of a property based o the Markov blaket. A ay time versio of the validatio algorithm, that improves the quality of its aswer icremetally, has also bee preseted. This ay time algorithm uses a causal etwork to distiguish the real fault from the apparet oes. The ay time behaviour is obtaied with the selectio of the sesor that provides more iformatio whe validated. The selectio is made with the etropy fuctio. The evaluatio of the ay time behaviour of the algorithm preseted i this paper was doe by carryig out experimets to obtai the performace profile of the etropy based selectio scheme ad comparig it with a radom selectio scheme. Future research will attempt to use the probabilities obtaied i the fault detectio Bayesia etwork, as the iput to the fault isolatio Bayesia etwork. At this stage, the output of the detectio etwork cosists i a biary value ({correct, faulty}). Ackowledgmets Thaks to the aoymous referees for their commets which improved this article. This research is supported by a grat from CONACYT ad lie uder the IIE/SALFORD/CONACYT doctoral programme. Refereces Boddy, M. & Dea, T. (1994), 'Decisio theoretic deliberatio schedullig for problem solvig i timecostraied eviromets', Artificial Itelligece 67(2), Cohe, P. (1995), Empirical methods for artificial itelligece, MIT press, Cambridge, Mass., U.S.A. Dea, T. & Boddy, M. (1988), A aalysis of time depedet plaig, i 'Proc. Seveth Natl. Cof. o AI', St. Paul, MN, U.S.A. Horvitz, E. (1987), Reasoig about beliefs ad actios uder computatioal resource costraits, i 'Proc. Third Coferece o Ucertaity i Artificial Itelligece', Seatle, WA, U.S.A., pp Ibargiiegoytia, P. (1997), Ay time probabilistic sesor validatio, PhD dissertatio, Uiversity of Salford, Computer ad Mathematical Scieces, Salford U.K. Ibargiiegoytia, P., Sucar, L. & Vadera, S. (1996), A probabilistic model for sesor validatio, i 'Proc. Twelfth Coferece o Ucertaity i Artificial Itelligece', Portlad, Orego, U.S.A., pp Lauritze, S. & Spiegelhalter, D. J. (1988), 'Local computatios with probabilities o graphical structures ad their applicatio to expert systems', Joural of the Royal Statistical Society series B 50(2), Pearl, J. (1988), Probabilistic reasoig i itelliget systems: etworks of plausible iferece, Morga Kaufma, Palo Alto, Calif., U.S.A. Shao, C. & Weaver, W. (1949), The mathematical theory of commuicatio, Uiversity of Illiois press, Urbaa, Ill., U.S.A. Stakovic, J. (1988), 'Miscoceptios about real time computig: a serious problem for ext geeratio systems', Computer 21(10), Sucar, L., Perez-Brito, J., Ruiz-Suarez, J. & Morales, E. (1997), 'Learig structure from data ad its applicatio to ozoe predictio', Applied Itelligece 7, Yug, S. & Clarke, D. (1989), 'Local sesor validatio', Measuremet f3 Cotrol22 (3), Zilberstei, S. & Russell, S. (1996), 'Optimal compositio of real-time systems', Artificial Itelligece 82 (1-2) '