WHY YULE-WALKER SHOULD NOT BE USED FOR AUTOREGRESSIVE MODELLING

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1 WHY YULE-WALKER SHOULD NOT BE USED FOR AUTOREGRESSIVE MODELLING M.J.L. DE HOON, T.H.J.J. VAN DER HAGEN, H. SCHOONEWELLE, AND H. VAN DAM Inerfaculy Reacor Insiue, Delf Universiy of Technology Mekelweg 5, 2629 JB Delf, The Neherlands Absrac - Auoregressive modelling of noise daa is widely used for sysem idenificaion, surveillance, malfuncioning deecion and diagnosis. Several mehods are available o esimae an auoregressive model. Usually, he so-called Yule-Walker mehod is emloyed. The various esimaion mehods generally yield comarable arameer esimaes. In some secial cases however, involving nearly eriodic signals, he Yule-Walker aroach may lead o incorrec arameer esimaes. Burg s mehod offers he bes alernaive o Yule-Walker. In his aer a heoreical exlanaion of his henomenon is given, while he 994 IAEA Benchmark es is resened as a racical examle of Yule-Walker yielding oor arameer esimaes. I. INTRODUCTION Auoregressive modelling of noise daa was inroduced in nuclear engineering in he mid sevenies and gained oulariy during he decades hereafer. A hisorical survey of he gradual acceaion and he diversiy of is alicaions can be found in he so-called SMORN-roceedings (SMORN-III, SMORN-IV, SMORN-V). Nowadays, auoregressive modelling is a widely used means for erforming sysem idenificaion, surveillance, malfuncioning deecion and diagnosis. Is araciveness sems, among ohers, from he fac ha he numerical algorihms involved are raher simle. An auoregressive model deends on a limied number of arameers, which are esimaed from measured noise daa. Several mehods exis o esimae he auoregressive arameers, such as leassquares, Yule-Walker and Burg s mehod. I can be shown ha for large daa samles hese esimaion echniques should lead o aroximaely he same arameer esimaes. Mainly for hisorical reasons, mos eole use eiher he Yule-Walker or he leas-squares mehod. This aer will show, however, ha in some secial cases he Yule-Walker esimaion mehod leads o oor arameer esimaes, even for moderaely sized daa samles. Leas squares should no be used eiher, as i may lead o an unsable model. Burg s mehod is referable. In secion II, we will resen an overview of he basics of auoregressive modelling. The mahemaical circumsances causing oor arameer esimaes in case of he Yule-Walker echnique are described in secion III, while some simulaions of auoregressive rocesses are discussed ha suor our hyoheses. Finally, in secion IV, we will illuminae our findings wih he alicaion of auoregressive modelling for anomaly deecion in he 994 IAEA Benchmark noise daa (Journeau, 994).

2 II. THEORY OF AUTOREGRESSIVE MODELLING The successive samles y of an auoregressive rocess linearly deend on heir redecessors: y + ay - + a2 y a y - = h, () in which a i are he auoregressive arameers and he innovaions h are a saionary urely random rocess wih zero mean. I can be shown ha he auocovariance funcion R for delays o is relaed o he auoregressive arameers a i hrough he Yule-Walker equaion for he auoregressive rocess (Priesley, 994): Ê R R R R R R Ë R R R An esimaed auoregressive model of he same order can be wrien as ˆÊ a ˆ Ê R ˆ a2 R2 = -. (2) Ëa Ë R y + a y + a y + + a - - y- = 2 2 h, (3) in which â i are he auoregressive-arameer esimaes and ĥ are he esimaed innovaions. A clear disincion should be made beween he auoregressive rocess (Eq. ()) and he corresonding auoregressive model (Eq. (3)) (Broersen and Wensink, 993). Using Eq. (3), each daa samle can be rediced from is redecessors: y  = - a y -. (4) i i i= As he samles y canno be rediced exacly, a residue is inroduced, which is defined as he difference beween he measured value and he esimaed value: residue y - y = ĥ, (5) which means ha he residue is equal o he esimaed innovaion, as inroduced in Eq. (3). I is assumed in hese equaions ha he auoregressive model order is known. In racice, he model order has o be esimaed as well, which is usually done using Akaike s crierion (Priesley, 994). Suose ha he esimaion realisaion y consiss of N daa oins (an esimaion realisaion conains hose daa oins ha are used for arameer esimaion). Three mehods of auoregressive-arameer esimaion from hese daa samles shall be considered here, being he leas-squares aroach (LS), he Yule-Walker aroach (YW) and Burg s mehod (Burg): LS: The oal squared residue over he daa samles + o N is minimised, leading o a sysem of linear equaions: Ê c c2 c c2 c22 c2 Ëc c c ˆÊ a ˆ a 2 Ëa 2 Ê c ˆ c2 = -, (6) Ëc in which he marix elemens c N - y y ij -i - j = + N  (7) form an unbiased esimae of he auocovariance funcion for delay i - j. YW: The firs and las daa oins are also included in he summaion of Eq. (7), resuling in

3 Ê R R R R R R Ë R R R ˆÊ a ˆ Ê R ˆ a R 2 2 = -, (8) Ëa Ë R in which he marix elemens R consiue a biased esimae of he auocovariance funcion (Parzen, 96): R N  y y - N = +. (9) The Levinson-Durbin algorihm rovides a fas soluion of a sysem of linear equaions conaining a Toeliz-syle marix as in Eq. (8). Boh Eqs. (6) and (8) are in fac aroximaions o he Yule-Walker rocess equaion (Eq. (2)). Burg: The arameer esimaion aroach ha is nowadays regarded as he mos aroriae, is known as Burg s mehod. In conras o he leas-squares and Yule-Walker mehod, which esimae he auoregressive arameers direcly, Burg s mehod firs esimaes he reflecion coefficiens, which are defined as he las auoregressive-arameer esimae for each model order. From hese, he arameer esimaes are deermined using he Levinson-Durbin algorihm. The reflecion coefficiens consiue unbiased esimaes of he arial correlaion coefficiens. Usually, hese esimaion mehods lead o aroximaely he same resuls for he auoregressive arameers. Once hese have been esimaed from he ime series y, he auoregressive model can be alied o an indeenden redicion realisaion x of he same sochasic rocess. In erms of x, he auoregressive rocess (Eq. ()) can be wrien as x + ax - + a2x a x- = e, () in which he innovaion rocess e is saisically idenical o he innovaion rocess h. The corresonding auoregressive model can be wrien as in Eq. (3): x + a x + a x + + a - - x- = 2 2 e, () in which â i are he auoregressive arameers esimaed from realisaion y and e are he esimaed innovaions. As in Eq. (4), each daa samle can be esimaed from is redecessors: x  = - a x -. (2) i i i= The difference beween he measured value and he esimaed value is now defined as he redicion error: redicion error x - x = ê. (3) The redicion error is herefore equal o he esimaed innovaion, as inroduced in Eq. (). Each redicion error can be calculaed once he acual value of he daa oin is measured. A clear disincion should be made beween he residue and he redicion error and heir variances (Broersen and Wensink, 993). The residual variance var( h ) is a measure for he fi of he auoregressive model o hose daa ha have been used for he esimaion of he auoregressive arameers, and can be esimaed from he realisaion y, which is used for he arameer esimaion: vâr N 2 ( h ) = Â( y - y ) N - = +. (4) For he redicion of fuure daa, insead of he residual variance, he variance of he redicion error var ê is essenial. If he indeenden redicion realisaion x conains N daa samles, he redicion ( ) error variance can be esimaed from he samle variance:

4 ( e ) = Â( x - x ) vâr N 2 N '- = +. (5) The LS arameer esimaion is based on he minimisaion of he residual variance. Such a minimisaion however does no imly ha he variance of he redicion error is minimised as well. As usually he minimisaion of he redicion error variance is our goal, he LS esimaion of he auoregressive arameers is no necessarily suerior o YW or Burg s mehod. III. STABILITY, POLE LOCATION AND PARAMETER ESTIMATION For large daa samles, he difference beween he esimaes obained by he various mehods will be small (Priesley, 994). In some secial cases however, subsanial differences may arise beween hese aroaches even for daa samles of moderae size. In he resen aer i will be shown ha YW should always be avoided. The behaviour of he auoregressive rocess as described by is ole locaions is essenial in his resec. Using he backward shif oeraor z - -, which is defined as z x = x -, and defining a, a realisaion of an auoregressive rocess can be exressed in erms of he innovaions sequence e as: Ê x = z Âaiz Ë i= -i ˆ - e, (6) ignoring he so-called comlemenary funcion (Priesley, 994). The auoregressive oeraor - Ê ˆ -i z Âaiz obviously conains a -fold zero a z =, as well as oles deermined by he Ë i= characerisic equaion of he auoregressive rocess -i  a z =. (7) i= The roos of characerisic equaion should lie inside he uni circle o ensure he auoregressive rocess o be sable. If he roos lie on he uni circle, he auoregressive rocess will only be saionary in case of e being idenical o zero. In ha case a harmonic rocess will resul, consising of a sum of cosine funcions. One migh wonder wha will haen if he auoregressive rocess has oles in he neighbourhood of he uni circle. As oles on he uni circle reresen a harmonic rocess, an auoregressive rocess wih oles near he uni circle can be execed o demonsrae some kind of seudo-eriodic behaviour (Priesley, 994). In his case he auocovariance funcion can be described as a sum of weakly damed eriodic funcions. Furhermore, as he noise erms e are sill resen, he auoregressive rocess may exhibi a kind of almos non-saionary behaviour. Finally, he arial auocorrelaion coefficiens will be close o uniy in absolue value. In he conex of linear filering heory, his means ha he ransfer funcion relaing x o e will be close o insabiliy in he filering sense (Oenheim, 978). The ole locaions will also affec he reliabiliy of he various arameer esimaion echniques. I is claimed by Priesley (994) ha YW may lead o oor arameer esimaes, even for moderaely large daa samles, if he auoregressive oeraor has a ole near he uni circle. This is he more remarkable since LS and YW only differ in heir reamen of he firs and las daa oins. Since his limied number of daa oins is relaively small comared o he oal number of daa oins used for arameer esimaion, i would be execed ha LS and YW lead o almos he same resuls. In his aer, insead of he ole locaions, he oor condiion of he auocovariance marix in Eq. (2) is regarded as he cause of oor YW esimaes. Side-effecs of he oor auocovariance marix condiion are an almos non-saionary, seudo-eriodic behaviour of he auoregressive rocess as well as oles locaed closely o he uni circle and arial auocorrelaion coefficiens close o uniy. These feaures can be used o deec he ossibiliy of oor YW esimaes, bu should no be regarded as is cause. I should be noed i

5 ha auoregressive rocesses are ossible having oles near he uni circle while he auocovariance marix is well condiioned. In hose cases YW will sill rovide correc resuls for he auoregressive arameers. To inroduce he conce of marix condiioning, we consider a general sysem of linear equaions: Ax = B, (8) in which A is a marix of order and B is a vecor of dimension. A well-known resul from linear algebra saes ha Eq. (8) canno have one single soluion if he marix A is singular: de(a) =. (9) In cases in which marix A is almos singular, he soluion of Eq. (8) will be exremely sensiive o erurbaions in eiher marix A or vecor B. The sensiiviy o hese erurbaions can be measured wih he so-called condiion number, which is defined as - k( A) = A A, (2) where is some marix norm (Cybenko, 98). In our case he -norm will be considered: Ê ˆ A = max  Aij : j Œ {, 2,..., }, (2) Ë i= in which A ij denoe he marix elemens. The larger he condiion number, he more sensiive he soluion of Eq. (8) will be o erurbaions. Roughly seaking, if k( A) =, we may exec o lose abou significan figures in invering an aroximaion o A. I should be noed ha in order o calculae he auoregressive arameers from Eq. (2), an inversion of he marix on he lef hand side is required. A deailed reamen of marix comuaions, norms and condiion numbers is given by Sewar (973). The oor coefficien esimaes in case of YW can be exlained in erms of he condiion of he auocovariance marix in Eq. (2) (Cybenko, 98). If he auocovariance marix is oorly condiioned, he soluion of Eq. (2) will srongly deend on erurbaions in he coefficiens R. The bias in he YW auocovariance esimaes R is one of hese erurbaions. Alhough his bias is usually oo small o jeoardise he arameer esimaion, in case of a oorly condiioned auocovariance marix i will be magnified, as a resul of which he YW arameer esimaes will be useless. In case of a firs-order auoregressive rocess, he condiion number reduces o uniy for all ole locaions. Therefore, YW will always yield a Condiion number (I) (II) Disance o he uni circle Fig. Calculaed condiion number of he auocovariance marix in case of auoregressive rocesses (I) and (II). correc resul for he arameer esimae in case of a firs-order rocess. In case of second-order auoregressive rocesses, oor YW esimaes may occur deending on he exac ole locaions. Two second-order auoregressive rocesses are considered here, one having has is oles on he osiive real axis (I), while he second one (II) has is oles on he osiive imaginary axis. Simulaions were made using LS, YW and Burg for oles aroaching he uni circle. The condiion number of he auocovariance marix can be calculaed heoreically as a funcion of he disance o he uni circle, which is shown in Fig.. As he condiion number in case of auoregressive rocess (I) increases dramaically, i is execed ha YW will erform oorly if he oles are locaed near he uni circle. In case of auoregressive rocess (II), he condiion

6 Residual variance Predicion error variance YW YW LS, Burg LS, Burg Disance o he uni circle number equals uniy for all ole disances o he uni circle. Therefore, we exec YW o erform well in his case. Each simulaion consised of 24 daa oins, using a normally disribued urely random innovaion rocess having uni variance. To reven he occurrence of close o non-saionary behaviour, he recorded simulaions were receded wih 24 dummy simulaions. Each simulaion was carried ou 25 imes. In each simulaion, he residual and he redicion error variance as well as he firs and second auoregressive arameer were esimaed, which were hereuon averaged over he number of simulaions. The residual variance was esimaed from Eq. (4). The redicion error variance was esimaed using a second series of 24 daa samles. The firs series served as he esimaion realisaion, while he second series rovided a redicion realisaion in order o esimae he redicion error variance from Eq. (5). The simulaion resuls for he residual variance as well as he redicion error variance in case of auoregressive rocess (I) are shown in Fig. 2. While LS and Burg sill yield a residual variance close o he acual value (being uniy) as he oles aroach he uni circle, YW is no longer able o describe he auoregressive rocess correcly. Even for oles locaed a. from he uni circle, he residual variance in case of YW is almos weny imes as large as in case of LS. Furhermore, i was found ha he auoregressive-arameer esimaes were no accurae in case of YW. The firs and second auoregressivearameer esimaes and heir acual values for auoregressive rocess (I) are loed in Fig. 3. The Yule- Walker echnique acually esimaes a firs-order auoregressive model, since he second auoregressivearameer esimae aroaches zero, while he firs auoregressive-arameer is esimaed o be is value in a firs-order model. In case of auoregressive rocess (II), no such effecs were found. All of he esimaion echniques, including YW, rovided correc resuls for he residual and redicion error variance as well as for he esimaed arameers, due o he marix condiion remaining bounded. These resuls agree wih our execaions. We can conclude ha YW should no be used o esimae auoregressive arameers if he auocovariance marix is almos singular. LS should no be used eiher for reasons of sabiliy of he esimaed model. The sabiliy of an esimaed auoregressive model can be verified by subsiuing he esimaed auoregressive arameers ino Eq. (7). If here urns ou o be a roo lying ouside he uni circle, he esimaed auoregressive model becomes invalid as he heory of auoregressive modelling is alicable o saionary sochasic rocesses only. Forunaely, YW as well as Burg guaranees he esimaed model o be sable, in conrary o LS. I can furhermore be shown ha sligh deviaions in he - Disance o he uni circle Fig. 2 Simulaion resuls for he residual and he redicion error variance in case of auoregressive rocess (I), having oles on he osiive real axis, using he various esimaion echniques.

7 Firs auoregressive arameer YW Second auoregressive arameer Acual, LS, Burg Acual, LS, Burg Disance o he uni circle auoregressive-arameer esimaes can resul in large deviaions in he esimaed ole locaions if he oles of an auoregressive rocess are locaed near he uni circle (Treer, 976). Therefore, each sligh deviaion is he arameer esimaes can resul in an unsable auoregressive model if he LS aroach is emloyed. Burg is he only reliable auoregressive-arameer esimaion echnique, yielding accurae arameer esimaes as well as an auoregressive model guaraneed o be sable. We will now urn o an acual auoregressive analysis in which oor YW arameer esimaion occurred, having derimenal effecs on is conclusions..2 Disance o he uni circle Fig. 3 Simulaion resuls for he firs and second auoregressive-arameer esimae in case of auoregressive rocess (I), having oles on he osiive real axis, using he various esimaion echniques (Acual = he acual value of he auoregressive arameer). YW IV. AUTOREGRESSIVE ANALYSIS OF THE 994 IAEA BENCHMARK One of he alicaions of auoregressive modelling in nuclear reacor analysis is he deecion of anomalies during he reacor oeraion. The basic idea is o deermine an auoregressive model of measured signals in a nuclear reacor during normal oeraion. I is assumed ha he auoregressive model hus found will no longer be alicable in case of a disurbance of he reacor oeraion. This will lead o a large redicion error variance during he anomaly, which can hen be deeced. In his secion, we will discuss he auoregressive analysis of he 994 IAEA Benchmark es daa aimed a he deecion of anomalies. A deailed descriion of his Benchmark es is rovided by Journeau (994). Noise measuremens during normal reacor oeraion were available, as well as synhesised noise daa ha conained an anomaly during he reacor oeraion. As he samling inerval is no relevan for our discussion on he erformance of YW, we will use a discree ime axis. Hoogenboom and Schoonewelle emloyed auoregressive analysis as described reviously o deermine he onse of he anomaly (994a, 994b). The noise daa during normal oeraion were used for deermining he auoregressive model of he seady-sae rocess, while he synhesised noise daa were used for anomaly deecion by soing sudden increases in he redicion error variance. Hoogenboom and Schoonewelle (994b) concluded ha increases in he redicion error variance due o anomalies occurred only if he auoregressive arameers were esimaed using LS. Emloying YW, here is hardly any increase in he redicion error variance during he anomaly. These resuls are due o he naure of he noise daa used for arameer esimaion. Figure 4 shows, in arbirary unis, a secion of 3 consecuive daa oins of he noise signal during normal reacor oeraion

8 Noise daa samle (in arbirary unis) Time (in arbirary unis) Fig. 4 Secion of he noise signal during normal reacor oeraion conaining 3 consecuive noise daa oins in arbirary unis. Power secral densiy (in arbirary unis) Frequency ordinae Fig. 5 Esimaed secrum of he noise daa in arbirary unis as a funcion of he discree frequency ordinaes. o illusrae is almos eriodic behaviour. I shows ha a srong cyclical comonen is resen, having a eriod of abou 6 imes he samling inerval, as well as eriodic comonens a higher frequencies. The almos eriodic behaviour of he background noise can also be demonsraed by esimaing is ower secral densiy, which is shown semi-logarihmically in Fig. 5. Since he secrum conains large eaks a secific frequency ordinaes, i can be concluded ha srong harmonic comonens are resen. Finally, he seudo-eriodic behaviour of he noise daa is shown by he esimaed ole locaions (Fig. 6). Since he oles are all locaed closely o he uni circle, he auoregressive rocess will behave seudoeriodically. In his case, he seudo-eriodic behaviour of he noise daa leads o a oor condiion of he auocovariance marix. Because he auocovariance funcion is no available heoreically as in he revious chaer, he marix condiion can only be esimaed. The bes esimae can be calculaed from he LS auocovariance marix, because all of is esimaes are unbiased. This resuls in k = (22) This condiion number is exremely large. The marix condiion is exremely oor (comare o Fig. ), as a resul of which oor YW arameer esimaes can be execed. The auoregressive model was esimaed from he firs 248 daa oins alying LS, YW and Burg. Using Akaike s Informaion Crierion (AIC), model order = 32 was seleced. The auoregressive-arameer esimaes are given in he Aendix. LS and Burg lead o comarable arameer esimaes, while hose esimaed by YW deviae srongly. The oles of he esimaed models were always locaed inside he uni circle, hereby fulfilling he condiion for sabiliy. Imaginary axis Real axis Fig. 6 Poles of he esimaed auoregressive model (LS).

9 The residual variance var( ĥ ) was esimaed for each esimaion rocedure from he esimaion realisaion using Eq. (4), while he var ê was esimaed redicion error variance ( ) using Eq. (5) from a redicion realisaion conaining 248 daa oins. Table I shows he resuls. As he residual variance and he redicion error variance in case of YW are abou wice as large as he resecive figures for LS and Burg, i can be concluded ha YW does no yield a correc auoregressive model. Alying he auoregressive model o he anomaly noise daa, he anomaly can be deeced from he redicion error variance only if LS or Burg s arameer esimaes are emloyed. In case of YW, he anomaly canno be deeced from he redicion error variance due o he oor arameer esimaes. Fig. 7 shows he redicion error for he noise secion conaining he anomaly in each of he cases LS, YW and Burg. In case of LS and Burg, visual insecion of Fig. 7 enables us o locae he sar of he anomaly a he increase he redicion error variance, as indicaed by he arrow. No such conclusion can be made if he YW arameer esimaes are used, which means ha YW is unusable for anomaly deecion. Predicion error (LS) Time Table I Residual and redicion error variance in case of he various esimaion rocedures Residual variance Predicion error variance LS YW Burg Predicion error (YW) 5 Time Fig. 7 Predicion error for he anomaly noise daa using LS, YW and Burg resecively. Predicion error (Burg) 5 Time V. CONCLUSION The Yule-Walker mehod should no be used as a means of auoregressive-arameer esimaion if he auocovariance marix is oorly condiioned. In ha case he relaively small covariance esimae bias can lead o a large deviaion in he esimaed arameers, resuling in an invalid model. A oor auocovariance marix condiion also involves ole locaions near he uni circle, as a resul of which he auoregressive rocess exhibis a kind of almos non-saionary, seudo-eriodic behaviour. The variance of he sochasic rocess will be large due o he innovaion rocess no being idenically zero, which is he case for a harmonic rocess. The leas-squares aroach as well as Burg s mehod are sill able o esimae he auoregressive model correcly. Leas squares should be used wih cauion hough, as i does no guaranee he esimaed auoregressive model o be sable, as a resul of which a small deviaion in he arameer esimaes may cause he esimaed oles o move ouside he uni circle. In ha case he esimaed auoregressive model will be invalid. This leaves Burg s mehod as he mos reliable esimaion echnique, as i rovides reliable arameer esimaes as well as an esimaed model guaraneed o be sable. The receding conclusions were obained for univariae auoregressive analysis only. However, due o he mahemaical similariy beween univariae and mulivariae auoregressive analysis, we exec similar resuls for he mulivariae case.

10 ACKNOWLEDGEMENTS These sudies were made as a final reor of suden research work a he Dearmen of Reacor Physics, Inerfaculy Reacor Insiue, Delf Universiy of Technology. Secial hanks are due o dr. Pie Broersen (Delf Universiy of Technology, Faculy of Alied Physics), who inroduced me o he oic of auoregressive modelling and is ifalls. REFERENCES Broersen, P.M.T. and Wensink H.E. (993) IEEE Transacions on Signal Processing, 4, Cybenko, G. (98) Sociey for Indusrial and Alied Mahemaics, Journal on Scienific and Saisical Comuing,, 3, Hoogenboom, J.E. and Schoonewelle H. (994a) Inerfaculy Reacor Insiue Hoogenboom, J.E. and Schoonewelle H. (994b) Inerfaculy Reacor Insiue /. Journeau, C. (994) Inernaional Aomic Energy Agency IWGFR Exended Co-ordinaed Research Program on Acousic Signal Processing for he Deecion of Sodium Boiling or Sodium/Waer Reacion in LMFBR. Daa for 994 Benchmark Tes. Commissaria à l Energie Aomique, Cadarache, France. Oenheim, A.V. (978) Alicaions of Digial Signal Processing. Prenice-Hall, Englewood Cliffs. Parzen, E. (96) Technomerics, 3, Priesley, M.B. (994) Secral Analysis and Time Series. Academic Press, London. SMORN-III (982), Proc. of he Third Secialis Meeing on Reacor Noise, Tokyo, Jaan, 26-3 Ocober 98, Progress in Nuclear Energy, 9. SMORN-IV (985), Proc. of he Fourh Secialis Meeing on Reacor Noise, Dijon, France, 5-9 Ocober 984, Progress in Nuclear Energy, 5. SMORN-V (988), Proc. of he Fifh Secialis Meeing on Reacor Noise, Munich, F.R.G., 2-6 Ocober 987, Progress in Nuclear Energy, 2. Sewar, G. W. (973) Inroducion o Marix Comuaions. Academic Press, New York. Treer, S.A. (976) Inroducion o Discree Time-Signal Processing. Wiley, New York. APPENDIX: AUTOREGRESSIVE-PARAMETER ESTIMATES The following able conains he auoregressive-arameer esimaes for LS, YW and Burg. The arameers were esimaed using 248 daa samles from he 994 IAEA benchmark (secion IV). Auoregressive LS YW Burg Auoregressive LS YW Burg arameer arameer