Recursive Least-Squares Estimation in Case of Interval Observation Data

Transcription

1 Recusve Least-Squaes Estmaton n Case of Inteval Obsevaton Data H. Kuttee ), and I. Neumann 2) ) Geodetc Insttute, Lebnz Unvesty Hannove, D-3067 Hannove, Gemany, kuttee@gh.un-hannove.de 2) Insttute of Geodesy - Geodetc Laboatoy, Unvesty FAF Munch, D Neubbeg, Gemany, ngo.neumann@unbw.de Abstact: In the engneeng scences, obsevaton uncetanty often conssts of two man types: andom vaablty due to uncontollable extenal effects, and mpecson due to emanng systematc eos n the data. Inteval mathematcs s well-suted to teat ths second type of uncetanty n, e. g., ntevalmathematcal extensons of the least-squaes estmaton pocedue f the set-theoetcal oveestmaton s avoded (Schön and Kuttee, 2005). Oveestmaton means that the tue ange of paamete values epesentng both a mean value and mpecson s only quantfed by ough, meanngless uppe bounds. If ecusvely fomulated estmaton algothms ae used fo bette effcency, oveestmaton becomes a key poblem. hs s the case n state-space estmaton whch s elevant n eal-tme applcatons and whch s essentally based on ecusons. Hence, oveestmaton has to be analyzed thooughly to mnmze ts mpact on the ange of the estmated paametes. hs pape s based on pevous wok (Kuttee and Neumann, 2009) whch s extended egadng the patcula modelng of the nteval uncetanty of the obsevatons. Besdes a naïve appoach, obsevaton mpecson models usng physcally meanngful nfluence paametes ae consdeed; see, e. g., Schön and Kuttee (2006). he mpact of possble oveestmaton due to the espectve models s goously avoded. In addton, the ecuson algothm s efomulated yeldng an nceased effcency. In ode to llustate and dscuss the theoetcal esults a damped hamonc oscllaton s pesented as a typcal ecusve estmaton example n Geodesy. Keywods: Inteval mathematcs, mpecson, ecusve paamete estmaton, oveestmaton, leastsquaes, damped hamonc oscllaton.. Intoducton State-space estmaton s an mpotant task n many engneeng dscplnes. It s typcally based on a compact ecusve efomulaton of the classcal least-squaes estmaton of the paametes whch descbe the system state. hs efomulaton eflects the optmal combnaton of the most ecent paamete estmate and of newly avalable obsevaton data; t s equvalent to a least-squaes paamete estmaton whch uses all avalable data. Howeve, though the ecusve fomulaton t allows a moe effcent update of the estmated values whch makes t well-suted fo eal-tme applcatons. Conventonally, the eal-tme capablty of a pocess o algothm, espectvely, means that the esults ae avalable wthout any delay when they ae equed wthn the pocess. In a system-theoetcal famewok also physcal knowledge about the dynamc system state can be avalable n tems of a system of dffeental equatons. In ths case a state-space flte such as the well- 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200) Edted by Mchael Bee, Raf L. Muhanna and Robet L. Mullen Copyght 200 Pofessonal Actvtes Cente, Natonal Unvesty of Sngapoe. ISBN: Publshed by Reseach Publshng Sevces. do:0.3850/

2 H. Kuttee, and I. Neumann known Kalman flte s used whch extends the concept of state-space estmaton as t combnes pedcted system nfomaton fom the soluton of the set of dffeental equatons and addtonal, newly avalable obsevaton data (Gelb, 974). As a specal case of state-space flteng, state-space estmaton consdes the same paamete vecto though all ecuson steps; nevetheless the estmated values wll vay. Moeove, tme s not the elevant quantty but the obsevaton ndex. hs allows some convenent featues such as the effcent elmnaton of obsevaton data whch ae consdeed as outles. In any case, the state-space can compse paametes whch ae system-mmanent and not dectly obsevable. It s common pactce to assess the uncetanty of the obsevaton data n a stochastc famewok, only. hs means that the obsevaton eos ae modeled as andom vaables and vectos, espectvely. hs type of uncetanty s called andom vaablty. Classcal models n paamete estmaton efe to expectaton vectos and vaance-covaance matces as fst and second moments of the andom dstbuton of the obsevaton. Othe appoaches based on the Maxmum-Lkelhood estmaton take the complete andom dstbuton nto account. In case of non-nomal dstbuton numecal appoxmaton technques such as Monte-Calo samplng pocedues ae appled fo the devaton of the denstes of the estmated paametes as well as of deved quanttes and measues (Koch, 2007). Howeve, thee ae moe souces of uncetanty n the data than just andom eos. Actually, dependng on the patcula applcaton unknown detemnstc effects can ntoduce a sgnfcant level of uncetanty. Such effects ae also known as systematc eos whch ae typcally educed o even elmnated by a mxtue of dffeent technques f an adequate obsevaton confguaton was mplemented: () modfcaton of the obsevaton values usng physcal o geometcal coecton models, () lnea combnatons of the ognal obsevatons such as obsevaton dffeences whch can educe synchonzaton eos o atmosphecally nduced un-tme dffeences n dstance obsevatons, () dedcated paametezaton of the effect n the obsevaton equatons. Snce none of these technques s goously capable to elmnate an unknown detemnstc effect completely o to detemne ts value, ths effect has to be modeled accodngly. Hee, nteval mathematcs s used as theoetcal backgound ntoducng ntevals and nteval vectos as addtonal uncetan quanttes. hs second type of uncetanty s called mpecson. he jont assessment of andom vaablty and mpecson of obsevaton data n least-squaes estmaton has been teated n a numbe of publcatons. Howeve, the consdeaton of ecusve state-space estmaton has to teat the oveestmaton poblem of nteval-mathematcal evaluatons n a moe elaboated way than n classcal estmaton. Oveestmaton s caused by, e. g., (hdden) dependences between nteval quanttes and t s vsble n nteval-mathematcal popetes lke, e. g., sub-dstbutvty. A futhe poblem s caused by the nteval ncluson of the ange of values of a lnea mappng of a vecto consstng of nteval data whch usually geneates addtonal values; see, e. g., Schön and Kuttee (2005) fo a dscusson of the two- and thee-dmensonal case. Snce ecusve fomulatons patculaly explot such dependences fo the sake of a compact and effcent notaton a sgnfcant oveestmaton s expected. hs study s based on pevous wok on the nteval and fuzzy extenson of the Kalman flte (Kuttee and Neumann, 2009). Hee, two man dffeences have to be mentoned. Fst, the appoach s smplfed as the system-state paametes ae consdeed as statc quanttes whch do not change wth tme (o foces). Second, the effcency of the devaton of the measues of the mpecson of the estmated paametes s nceased due to a new fomulaton. he uncetanty of the obsevaton data s fomulated n a compehensve way efeng to physcally meanngful detemnstc nfluence paametes. he pape s oganzed as follows. In Secton 2 least-squaes paamete estmaton s evewed wheeas n Secton 3 the ecusve fomulaton s ntoduced and dscussed. In Secton 4 the appled model of mpecson s motvated and descbed. Secton 5 povdes the nteval fomulaton of the nteval- 02 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)

3 Recusve Least-Squaes Estmaton n Case of Inteval Obsevaton Data mathematcal extenson of ecusve least-squaes state-space estmaton. In Secton 6 the ecusve estmaton of state-space paametes based on the obsevaton of a damped hamonc oscllaton s dscussed as an llustatve example. Secton 7 concludes the pape. 2. Least-Squaes Paamete Estmaton n Lnea Models Recusve least-squaes state-space estmaton s based on the efomulaton of the least-squaes estmaton usng all avalable obsevaton data; see, e. g., Koch (999). he model wth obsevaton equatons s consdeed n the followng. It s a typcal lnea model whch s also known as Gauss-Makov model. It conssts of a functonal pat E l Ax () whch elates the expectaton vecto E(l) of the n -dmensonal vecto l of the obsevatons wth a lnea combnaton of the unknown u -dmensonal vecto of the paametes x wth n u. he n u - dmensonal matx A s called confguaton matx o desgn matx, espectvely. Note that the matx A can be ethe column-egula o column-sngula. he dffeence n u (o n u d n case of column-sngula models wth d the ank defcency) s called edundancy; t quantfes the degee of ovedetemnaton of the lnea estmaton model. In case of an ognally non-lnea model a lneazaton based on a multdmensonal aylo sees expanson of the n vecto-valued functon f s deved as f E l f x f x0 x x0 x E x0 f l f x x x 0 0 x x0 whch yelds a fully analogous epesentaton to Eq. () f the sgn s neglected: f E l A x, wth l : l f x0, x : x x0, A :. x Fo the sake of a smple epesentaton only the lnea case accodng to Eq. () s dscussed n the followng. ypcally, the functonal model pat s gven though the esdual equatons v A x l wth v E l l. (3) he Gauss-Makov model also compses a second model pat whch efes to uncetanty n tems of the egula vaance-covaance matx (vcm) of the obsevatons and esduals ll vv, espectvely, as 2 2 V l Q P (4) ll vv 0 ll 0 2 wth the (theoetcal) vaance of the unt weght, the cofacto matx of the obsevatons 0 Q ll and the weght matx of the obsevatons P Q ll. he unknown vecto of paametes s estmated based on the pncple of weghted least-squaes va the nomal equatons systems A PA A P l (5) x0 (2) 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200) 03

4 as H. Kuttee, and I. Neumann A PA A Pl (6) fo a column-egula desgn matx A. In case of a column-sngula desgn matx a genealzed matx nvese s used leadng to A PA A Pl. (7) he cofacto matx and the vcm of the estmated paametes ae deved by the law of vaance popagaton as and 2 xx ˆˆ and xx ˆˆ 0 xx ˆˆ, (8) Q A PA Q Q A PA and Q, (9) 2 xx ˆˆ xx ˆˆ 0 xx ˆˆ espectvely. Note that thee ae seveal othe quanttes of nteest such as the estmated vectos of obsevatons ˆl and esduals ˆv, the coespondng cofacto matces and vcms, and the estmated value of the vaance of the unt weght 2 vˆ Pvˆ ˆ 0. (0) Due to the estcted space these quanttes ae not teated n ths pape. he dscusson s lmted to the ecusve estmaton of the paamete vecto and on the detemnaton of ts vcm. 3. Recusve Paamete Estmaton n Lnea Models he dea behnd ecusve paamete estmaton s the optmal combnaton of the most ecent estmated paamete vecto and of obsevaton data whch wee not ncluded n the pevous estmaton due to, e. g., the late avalablty. hs s a typcal stuaton n contnuously opeatng montong systems whee the state of the consdeed object s obseved epeatedly n defned ntevals. he set of paamete vecto components can be undestood as state-space epesentaton. Wth each newly ncomng set of obsevatons the estmated state of the object s updated as a bass fo futhe analyss and possbly equed decsons such as, e. g., n alam systems. Note that the algothms pesented hee just ely on the ndces of the obsevaton data whch ae not necessaly elated wth tme. Hence, by ntoducng negatve weghts t s also possble to elmnate obsevaton data fom the estmaton whch s equed n case of eoneous data. hs combnaton s consdeed as optmal n the meanng of the least-squaes pncple. hus, the equed equatons ae deved fom the equatons gven n Secton 2. he obsevaton vecto s sepaated nto two pats, the fst one contanng the set of all old obsevatons l and the second one contanng the new obsevatons l. he esdual vecto v, the desgn matx A and the weght matx P ae dvded nto coespondng pats accodng to v A l P 0 x, P v A l 0 P () 04 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)

5 Recusve Least-Squaes Estmaton n Case of Inteval Obsevaton Data whee the old and the new obsevaton vectos ae consdeed as uncoelated whch leads to the 0 matces at the off-dagonal blocks of P. he least-squaes soluton x ˆ of the paamete vecto can be obtaned usng Eq. (6) o (7), espectvely. Note that the uppe ndces n backets ndcate the ecuson step. he ecuson algothm eques the soluton x ˆ and ts cofacto matx Q xx ˆˆ whch ae assumed to be deved n the pevous ecuson step. he exstence of ths soluton s guaanteed n geneal snce an 0 ntal soluton ˆx 0 can always be deved at least fom a fst consstent set of obsevatons l wth 0 0 dm n u. l In the followng, only column-egula desgn matces ae assumed whch yeld egula nomal equatons matces. Note that compaable equatons can be deved fo columnsngula matces. Applcaton of the least-squaes pncple on Eq. () leads to the extended nomal equatons system A P A A P A A P l A P l (2) and hence to the new, updated vecto of estmated paametes A P A A P A A P l A P l (3) whch s based on all avalable obsevaton nfomaton. he coespondng cofacto matx consequently eads as xx ˆˆ. Q A P A A P A (4) he ecuson s ntoduced though the matx dentty accodng to, e. g., Koch (999, p. 37), A BD C A AB D CAB CA (5) whch allows to efomulate Eq. (4) and thus Eq. (3). hs yelds the updated vecto of estmated paametes wth ˆ xx ˆˆ ww (6) x Q A Q w xx ˆˆ xx ˆˆ xx ˆˆ ww xx ˆˆ, (7) Q Q Q A Q A Q Q Q A Q A ww ll xx ˆˆ, (8) ˆ. w l A x (9) he vecto w quantfes the dscepancy between the new obsevatons and the obsevaton values whch can be pedcted fom the avalable paamete values x ˆ. In total, Eq. (6) to Eq. (9) ae vey compact as they avod calculate the nvese of the nomal equatons matx n total. Instead, the nvese of the cofacto matx Q ww s needed whch has the same dmenson as the numbe of new obsevatons l. If the numbe of new obsevatons n each step s athe small, the ecuson sequence s qute effcent and hence well-suted fo eal-tme applcatons. Computaton tme can be saved addtonally f the matx poduct Q A s stoed n an auxlay matx. In ode to summaze the devatons n ths secton t xx ˆˆ 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200) 05

6 H. Kuttee, and I. Neumann can be stated that the ecusve fomulaton of least-squaes estmaton n a lnea model s equvalent to the all-at-once estmaton usng the completely avalable obsevaton data. he obtaned algothm s thooughly based on the effcent update of a matx nvese. Besdes the ecusve update of least-squaes estmates n sequental obsevaton pocedues, ecusve elmnaton of ncoect obsevatons fom the estmaton pocess s possble as well. In combnaton, the two technques can be appled fo polynomal flteng as a genealzaton of the movng-aveage technque. 4. Obsevaton Impecson he algothm fo ecusve paamete estmaton deved n Secton 3 eles on obsevaton uncetanty of andom vaablty type only. If, howeve, mpecson has to be taken nto account, the estmaton equatons have to be extended n a pope way; see, e. g., Kuttee and Neumann (2009) fo the Kalman flte. he statng pont s the entepetaton of the obsevaton vecto l as g g l y g s y g s0 s s0 y G s wth y : y g s0, G : (20) s s s0 s0 and wth y the andom vecto of ognally obtaned obsevatons whch have to be educed egadng physcal o geometcal effects. hese eductons ae consdeed as addtve; they ae descbed as a functon g of basc nfluence paametes s such as tempeatue o a pessue. he numecal values s 0 of these nfluence paametes ae based on, e. g., actual obsevaton, long-tem expeence, conventon, expets opnon o just ough estmates. As the values of the paametes s ae fxed though all calculatons the nfluence on the estmaton s detemnstc. Remanng devatons ae to be expected; ths effect s compsed n the lnea appoxmaton G s of the elaton between the basc nfluence paametes and the obsevaton values whch ae used n the model. Eq. (20) allows the sepaate ntoducton of andom vaablty and mpecson. Random vaablty s assocated wth the andom vecto y, manly though the vcm yy ll. Impecson efes to s and s modeled by means of a eal nteval vecto s wth s s, s 0, s and s the nteval adus as a measue of mpecson. Note that the tem n backets denotes the nteval epesentaton wth lowe and uppe bounds wheeas the tem n angle backets denotes the mdpont-adus epesentaton. Fom the vewpont of applcatons t s easonable to assume s m 0 fo the nteval mdpont snce justfed knowledge about any devaton would mply moe efned coectons leadng to the consequent valdty of the assumpton. hs sepaaton allows the dentfcaton of the coected obsevaton values y wth the mean pont of the nteval vecto lm y and of the emanng detemnstc eos wth s whch ae bounded by s. he total ange of the obsevaton vecto l wth espect to s s gven as l l y G s s s. (2) hs convex polyhedon s geneally a tue subset of the nteval vecto l lm, l y, G s. he opeato appled to a matx convets the matx coeffcents to the absolute values. Due to the 06 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)

7 Recusve Least-Squaes Estmaton n Case of Inteval Obsevaton Data constucton pocedue the nteval vecto l epesents the closest nteval ncluson of l whch s exact component by component. Moe nfomaton on ntevals, nteval vectos, athmetc ules, etc., can be found n standad textbooks such as Alefeld and Hezbege (983) o Jauln et al. (2000). Fo a bette undestandng of possble models fo the basc nfluence paametes thee examples fo Eq. (20) ae gven hee whch ae also elevant fo the applcaton example n Secton 6. One possblty s the modelng of an ndvdual addtve paamete fo each obsevaton n tems of l y 0 0 s l y 0 0 s l y 0 0 s n n n An altenatve s the modelng of one common addtve paamete as an unknown obsevaton offset as l y l l 2 2 n y y n As a second altenatve a common multplcatve paamete can be modeled descbng the effect of an unknown dft wth tme t o step ndex as l y t t 0 l y t t l y t t n n n It s also possble to efe the multplcatve paamete to the magntude of the obseved value y whch can be equed n case of an unknown scale facto n dstance obsevatons wth espect to a efeence length such as l y y l y y l y y n n n In addton, all models can be composed fo jont use. Many othe models can be meanngful; see, e. g., Schön and Kuttee (2006) fo a study on a efned nteval modelng of obsevaton and paamete uncetanty n GPS (Global Postonng System) data analyss. Note that the modelng of obsevaton mpecson n tems of eal ntevals can be extended to fuzzy numbes and ntevals, espectvely, n a staghtfowad manne. It s well-known that due to the convexty of fuzzy ntevals the espectve -cuts can be dentfed as eal ntevals; see, e. g., Mölle and Bee (2004). he technque of -cut dscetzaton explots ths popety. Fo ths eason the pesent dscusson can easly be seen as a specal case of a fuzzy appoach whch s dscussed hee as an nteval appoach fo the sake of smplcty but wthout loss of genealty. 0 s. s. s.. (22) (23) (24) (25) 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200) 07

8 H. Kuttee, and I. Neumann 5. Inteval Extenson of Recusve Estmaton If ecusve estmaton as ntoduced n Secton 3 s appled to nteval obsevaton data as defned n Secton 4, oveestmaton s the key poblem whch has to be solved. Oveestmaton ases fom seveal causes. A fst one was ndcated n the dscusson of Eq. (2) snce the ange of values of a lnea mappng z F x, x x, (26) s a convex polyhedon n geneal but usually not an nteval vecto. Hence, nteval mathematcs s not closed wth espect to a lnea mappng. Moeove, the sub-dstbutvty popety MF x M F x (27) holds whch eflects lackng assocatvty n case of matx multplcatons and nteval vectos. Fnally, aleady fo sngle ntevals the ange of values can be oveestmated such as, e. g., x x 2 x, 2x 0,0 y y x x, x x (28) n case of dependences between the ntevals. hs shows that the naïve applcaton of the fundamental ules of nteval athmetc s not a pope way fo evaluatng the ange of paamete values n ecusve estmaton snce t s cucal to avod any possble cause of oveestmaton. Actually, the tghtest nteval ncluson of the actual ange of values s always gven as z z z MFx, x x z MF x. (29) 0 In case of Eq. (28) ths yelds the coect ange of values z x x x x 0 x 0 z z 0,0. (30) 0 If ecusve least-squaes estmaton s consdeed as descbed by Eq. (6) to Eq. (9) the extenson s staghtfowad fo the nteval mdponts x ˆ m and l m whch yelds ˆ m m xx ˆˆ ww m (3) x Q A Q w n a compact and effcent epesentaton wth w ˆ m lm A x m (32) whch s possble because of the symmety of the ntevals wth espect to the mdponts. Howeve, fo the calculaton of the nteval adus x ˆ an altenatve method s equed because n Eq. (3) oveestmaton occus snce the tue ange of values xx ˆˆ ww,, (33) Q A Q l A l l s a convex polyhedon whch s ncluded by a nteval vecto. hough ths ncluson the set x ˆ s enlaged and the addtonal values ae taken nto account n the next ecuson step. hus, the effect of oveestmaton accumulates vey quckly. hs poblem s ovecome effectvely f the ecuson s esolved by efeng the ecuson equatons to the complete set of obsevatons whch ae avalable at a espectve ecuson step. In ode to explan and educe the effect of dependences the obsevatons on the pat ae efeed to the ognal, ndependent values of the basc nfluence paametes s. 08 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)

9 Recusve Least-Squaes Estmaton n Case of Inteval Obsevaton Data An equvalent esult s avalable f Eq. (3) and Eq. (4) ae dectly used. hs s possble because of the fomal dentty of the least-squaes soluton pesented n Secton 2 whch uses all obsevatons at once and the ecusve soluton gven n Secton 3. Statng wth xx ˆˆ Q A P l A P l (34) the ecuson s only needed fo the update of the cofacto matx vecto s s assumed, Eq. (34) can be ewtten as xx ˆˆ Q xx ˆˆ. If fo all ecuson steps an dentcal Q A P y G s A P y G s (35) usng Eq. (20). Note that matx G elates the new obsevatons n the -th ecuson step wth the constant vecto of basc nfluence paametes wheeas matx G ecusvely comples the espectve matces G of all pevous steps. Reodeng of Eq. (35) yelds and and fnally xx ˆˆ Q A P y A P y A P G s A P G s (36) xx ˆˆ xx ˆˆ Q A P y A P y Q A P G A P G s (37) ˆ ˆ m xx ˆˆ. x x Q A P G A P G s (38) hus, the nteval vecto adus of the estmated paametes n the -th ecuson step s effcently deved as o espectvely, wth the ecusvely calculated matx xx ˆˆ Q A P G A P G s (39) Q M s, (40) xx ˆˆ :. M A P G A P G (4) 6. Applcaton example Recusve estmaton s always elevant f the values of the estmated paametes ae needed n eal-tme o f the avalable data stoage s lmted. In ode to demonstate effcent ecusve estmaton n case of both data andom vaablty and mpecson usng nteval mathematcs, the obsevaton of a damped hamonc oscllaton s pesented and dscussed exemplaly. he pncpal obsevaton confguaton s shown n Fgue. he mathematcal model s defned as 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200) 09

10 H. Kuttee, and I. Neumann 2 y t y0 Aexp t sn t y( t) spng length at tme t A y 0 oscllaton ampltude (42) oscllaton phase dampng paamete oscllaton peod offset paamete wth the appoxmately known paametes A,,,, and y 0 whch have to be estmated fom the obsevatons y at dscete tmes t. he paamete s functonally elated wth the spng constant. Fgue. Spng-dampng model. he values of the paametes chosen fo the smulatons n ths secton ae pesented n able I; denotes the constant ndvdual standad devaton of all sngle obsevatons y. he esultng oscllaton s shown n Fgue 2. It s dentcal fo all followng thee smulatons. able I. A po values fo the smulaton of a damped hamonc oscllaton A y 0 - / Fgue 2. Damped spng oscllaton obseved wth 00 ponts ove 0 peods. 0 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)

11 Recusve Least-Squaes Estmaton n Case of Inteval Obsevaton Data able II gves the numecal paametes of the mpecson models fo the thee smulatons of the damped hamonc oscllaton. he smulatons wee calculated based on ecusve estmaton wth nteval data. he dffeences between the smulatons le n the modelng of mpecson. Model I assumes ndvdual nteval ad of dentcal sze fo all obsevatons; cf. Eq. (22). Model II assumes two nteval components whch ae common fo all obsevatons: an (addtve) offset eflectng the uncetanty about the zeo efeence (cf. Eq. (23)) and a (multplcatve) facto popotonal to the obseved spng length y whch efes to the epstemc uncetanty wth espect to an etalon o a dffeent length efeence (cf. Eq. (25)). hee ae no ndvdual tems as n Model I. Model III s based on Model II but compses an addtonal (multplcatve) facto popotonal to tme t whch epesents a dft; cf. Eq. (24). able II. Impecson models fo the smulatons Model I Model II Indvdual mpecson tems fo all obsevatons wo common mpecson tems fo all obsevatons, no ndvdual tems: addtve tem 3 s 0 and tem popotonal to spng length y s 0 4 s 0 4 Model III As mpecson model II wth an addtonal facto pop. to tme t: s 0 4 Fgue 3 shows the esults fo the ecusve estmaton usng Model I, Fgue 4 fo Model II and Fgue 5 fo Model III. In each case the fst ten epochs wee combned fo the estmaton of the ntal soluton of the ecuson. 00 obsevatons wee used n total; they ae ndcated n Fgue 2. Based on the ntal soluton the next obsevaton was ntoduced to the estmaton, and the estmated paametes and the cofacto matx wee updated usng Eq. (3), Eq. (32), Eq. (7) and Eq. (8). he nteval ad of the estmated paametes wee calculated usng Eq. (39). Random nose was added to each obsevaton value as ndcated n able I; fo all thee smulaton the same nosy obsevaton data wee used. In all thee fgues the ecusvely estmated paametes ae ndcated by lght gay damonds. he standad devatons of the estmated paametes ae shown wth dak gay damonds fo all epochs symmetc to 0. he nteval ad of the estmated paametes ae shown wth black damonds symmetc to 0. All fgues show the decease of the standad devatons of all estmated paametes tendng towads zeo wth nceasng numbe of obsevatons and epochs, espectvely. Lke the estmated paametes these values ae dentcal fo all thee smulatons snce the model fo the standad devatons of the obsevatons was dentcal as well. hus, they confm the geneal expectaton of successvely mpoved nfomaton about the non-obsevable system state nfomaton. In contast to the decease of the standad devatons thee ae seveal effects whch eflect the systematc, detemnstc chaacte of the modeled mpecson tems. All gven values ae exact component by component as explaned n Secton 5 meanng that they epesent the coect ange of values. In Fgue 3 the mpecson of the dampng paamete and of the oscllaton peod ae educed when moe obsevatons ae avalable. Howeve, ths does not hold fo the ampltude, the phase and the offset paamete. Due to the 4 ndvdually modeled obsevaton nteval ad thee s a emanng epstemc uncetanty: A ˆ 2.5 0, ˆ 4 ˆ0, , 0 y. Lookng at Fgue 4, the stuaton changes completely. Phase, dampng and peod do not suffe fom the modeled nteval data uncetanty whch eflects two systematc effects: unknown offset and scale 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)

12 H. Kuttee, and I. Neumann of the obsevaton of the spng length. Obvously, ths type of uncetanty s elmnated n total aleady n the ntal soluton of these thee paametes. he modeled mpecson s absobed by the ampltude ˆ 4 3 A 0 and by the offset y ˆ 0. A possble explanaton s that these two paametes 0, epesent absolute nfomaton wheeas fo the othe thee paametes only elatve nfomaton s equed. In such a case effect dffeences ae elevant whch ae elmnated n case of dentcal effect magntudes. Of couse a smla effect also occus n case of Model I whee obsevaton dffeences lead to dentcal nteval ad dffeent fom 0. hs easonng s also suppoted by the esults shown n Fgue 5. Hee, an addtonal scale mpecson component s modeled wth espect to tme. Hence, the obsevaton nteval ad ae addtonally nceasng lnealy wth tme. hs s dectly popagated to the offset mpecson. All othe paamete mpecson measues show peodc effects whch ndcate that dependng on the patcula tme of estmaton wth espect to the completeness of a peod the modeled systematc components ae moe o less elmnated o not. Obvously, fo the detemnaton of the paametes thee ae bette and wose condtons whch have to be known when mpecson s consdeed accodng to Model III. In any case, the pesented methods povde a mathematcal and algothmc famewok whch allows adequate decsons. 7. Conclusons Recusve paamete estmaton based on the least-squaes pncple s an mpotant task n the engneeng dscplnes f, e. g., the paametes have to be estmated and updated n eal tme, espectvely. Although well known and well establshed as a classcal estmaton technque, poblems ase n case of ecusvely popagatng data uncetanty compsng both effects of andom vaablty and mpecson due to emanng systematc eos. Impecse data can be effectvely modeled usng eal ntevals. Hence, f ntevals ae gven fo the ognal obsevatons, the detemnaton of the coespondng ntevals of the estmated paametes can be consdeed as the task to calculate the ange of values. If standad ntevalmathematcal ules ae appled, the poblem of oveestmaton s elevant. It can be ovecome f the computatons ae efeed to ndependent bass nfluence paametes whch ae assumed to cause mpecson. In ths pape a method was ntoduced whch allows ecusve estmaton usng nteval data n a vey effcent way. he actual obsevaton values ae used as mdponts of symmetc ntevals. Hence, ths yelds the same esults as n classcal least-squaes estmaton. Fo the computaton of the nteval ad of the estmated paametes the ecuson based on the obsevaton data s esolved. Instead, all obsevaton ntevals ae ntoduced smultaneously to the algothm. he ecuson s efeed to the update of the cofacto matx of the estmated paametes and of a matx poduct. Both can be acheved effcently so that the fnal devaton of the nteval ad s staghtfowad. he method was demonstated usng the applcaton of a damped hamonc oscllaton. Based on thee smulaton uns wth dffeent models of data mpecson the ways of popagatng andom vaablty and mpecson wee shown and dscussed. Futue wok has to extend the pesented algothm to state-space flteng such as the Kalman flte. Although such an extenson s aleady avalable though the esoluton of the ecuson t s fa fom beng effcent. Besdes, the use of asymmetc obsevaton ntevals has to be consdeed whch s moe appopate than symmetc ntevals fo a ealstc modelng of systematc eos due to, e. g., atmosphec efacton. 2 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)

13 Recusve Least-Squaes Estmaton n Case of Inteval Obsevaton Data Fgue 3. Uncetanty popagaton fo the fve model paametes ove 00 epochs mpecson model I (black: mpecson measues, dak gay: standad devatons, lght gay: paamete vaatons due to smulated obsevaton values) 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200) 3

14 H. Kuttee, and I. Neumann Fgue 4. Uncetanty popagaton fo the fve model paametes ove 00 epochs mpecson model II (black: mpecson measues, dak gay: standad devatons, lght gay: paamete vaatons due to smulated obsevaton values) 4 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)

15 Recusve Least-Squaes Estmaton n Case of Inteval Obsevaton Data Fgue 5. Uncetanty popagaton fo the fve model paametes ove 00 epochs mpecson model III (black: mpecson measues, dak gay: standad devatons, lght gay: paamete vaatons due to smulated obsevaton values) 4th Intenatonal Wokshop on Relable Engneeng Computng (REC 200) 5

16 H. Kuttee, and I. Neumann Refeences Alefeld, G. and J. Hezbege. Intoducton to Inteval Computatons. Academc Pess, Boston, San Dego & New Yok, 983. Gelb, A. Appled Optmal Estmaton. MI Pess, Cambdge, MA, 974 Jauln, L., E. Walte, O. Ddt and M. Keffe: Appled Inteval Analyss. Spnge, Beln, Koch, K. R. Paamete Estmaton and Hypothess estng n Lnea Models. Spnge, Beln & New Yok, 999. Koch, K. R. Intoducton to Bayesan Statstcs. Spnge, Beln, Mölle, B. and M. Bee. Fuzzy Randomness. Spnge, Beln & New Yok, Kuttee, H. and I. Neumann. Fuzzy extensons n state-space flteng. Poc. ICOSSAR 2009, aylo and Fancs Goup, London, ISBN , , Schön, S. and H. Kuttee. Usng zonotopes fo oveestmaton-fee nteval least-squaes -some geodetc applcatons-. Relable Computng (2):37-55, Schön, S. and H. Kuttee. Uncetanty n GPS netwoks due to emanng systematc eos: the nteval appoach. Jounal of Geodesy. 80(3):50-62, th Intenatonal Wokshop on Relable Engneeng Computng (REC 200)