Combining the Frisch scheme and Yule Walker equations for identifying multivariable errors in variables models

Transcription

1 Poceedings of the 9th Intenational Syposiu on Matheatical Theoy of Netwoks and Systes MTNS July, 00 Budapest, Hungay Cobining the Fisch schee and Yule Walke equations fo identifying ultivaiable eos in vaiables odels Robeto Divesi and Robeto Guidozi Abstact Eos in Vaiables (EIV) odels, i.e. odels whose stochastic envionent consides easueent eos on both inputs and outputs ae intinsically oe ealistic than epesentations assuing an exact knowledge of the input but ae also oe difficult to estiate. The difficulties incease in a non tivial way passing fo the SISO and MISO cases to the MIMO one. This pape poposes a pocedue fo EIV identification of MIMO pocesses based on the Fisch schee that assues additional white noises on all inputs and outputs and shows its effectiveness by eans of Monte Calo siulations. I. INTRODUCTION The identification of eos in vaiables (EIV) odels is a challenging poble that, despite the advantages that it can offe in any contexts, has eceived an inceasing attention only in the last thee decades [] [6]. The EIV envionent enjoys soe peculiaities not shaed by othe classes of odels; aong the, the syetical pesence of additive noise on all vaiables (inputs and outputs if a odel oientation is consideed). EIV odels ae paticulaly suitable when the inteest is ainly focused on the inne laws that descibe the consideed pocess and/o on filteing applications athe than on the pediction of its futue behavio. This class of odels has thus found any applications in syste engineeing, signal pocessing and econoetics [7], [8]. The identification of eos in vaiables odels is oe coplex than that of taditional equation eo odels like, fo instance, ARX and ARMAX whee the input is consideed as exactly known. An inteesting EIV identification appoach is based on the Fisch schee, i.e. on the stochastic context oiginally intoduced by Ragna Fish in the static case fo evaluating the faily of linea elations linking sets of noisy obsevations copatible with the data covaiance atix [], [9]. The extension of the oiginal algebaic context to the dynaic one leads to the use of EIV odels whee the easues of the inputs and outputs ae affected by additive and utually uncoelated white noises. The dynaic case shaes any inteesting algebaic and geoetic popeties with the static one like the existence of convex singulaity hypesufaces defining the faily of solutions that can be associated with a set of obsevations. Moeove, the dynaic popeties of the syste allow the asyptotic deteination of the exact odel of the pocess that has geneated the data, not just the definition of a class Robeto Divesi and Robeto Guidozi ae with the Depatent of Electonics, Copute Science and Systes, Univesity of Bologna, Viale del Risogiento, 4036 Bologna, Italy obeto.divesi@unibo.it, obeto.guidozi@unibo.it of copatible odels [4]. This nice popety is based on the existence of a coon point between the singulaity hypesufaces associated with all odels with odes geate than that of the pocess that has geneated the data. Unfotunately this popety cannot be obseved on finite sequences so that it is necessay to intoduce suitable cost functions in ode to extact a single odel fo the data [8], [9], [0]. The extension of the Fisch schee to the identification of ulti input ulti output (MIMO) eos in vaiables odels intoduces futhe pobles not pesent in the SISO and MISO cases [6], [0]. In fact, the whole systes can be viewed, in this case, as a set of ( = nube of outputs) inteconnected subsystes, that can be associated with singulaity hypesufaces so that these subsystes could be sepaately identified by eans of the sae pocedues used fo the MISO case. An appoach of this kind equies sepaate seaches and, even wose, leads to diffeent estiates fo evey input and output noise vaiance with a consequent loss of conguence fo the global odel. It would be thus pefeable to select cost functions teating the odel as a whole and not as a set of sepaate odels. A geoetic appoach to the solution of this poble has been poposed in [6], [5] whee ultivaiable EIV odels have been no longe associated to single points on singulaity hypesufaces but to diections in the noise space. This appoach, that elies, fo a coputational standpoint, on the easy coputation of the intesection between a staight line fo the oigin and a singulaity hypesuface [], offes the additional advantage of an easy intoduction of conguent cost functions [6]. The identification pocedue intoduced in this pape takes advantage of the popeties of both the dynaic Fisch schee and high ode Yule Walke equations and is based on the afoeentioned geoetic appoach so that conguent estiations of the syste paaetes and noise vaiances can be obtained. The ethod is chaacteized by a good estiation accuacy and gives bette esults than the appoach intoduced in [6], as shown by the pefoed Monte Calo siulations. The oganization of the pape is as follows. Section II defines the EIV MIMO identification poble. Section III descibes the popeties of the dynaic Fisch schee in the ultivaiable case while Section IV intoduces the new identification pocedue. The esults obtained in soe Monte Calo siulations ae epoted in Section V. Shot concluding eaks ae finally given in Section VI. ISBN

2 R. Divesi and R. Guidozi Cobining the Fisch Schee and Yule-Walke Equations fo Identifying Multivaiable Eos-in-Vaiables Models u 0 (t) ũ(t) Q(z) P(z) + + u(t) Fig.. y(t) MIMO eos in vaiables odel II. PROBLEM STATEMENT y 0 (t) ỹ(t) Conside the MIMO EIV odel epesented in Fig., descibed by the equations Q(z)y 0 (t) = P(z)u 0 (t) () u(t) = u 0 (t) + ũ(t) () y(t) = y 0 (t) + ỹ(t), (3) whee u 0 (t) R, y 0 (t) R ae the noise fee input and output u 0 (t) = [ u 0 (t)u 0 (t)... u 0 (t)] T (4) y 0 (t) = [ y 0 (t)y 0 (t)...y 0 (t)] T, (5) u(t), y(t) ae the available obsevations affected by the additive noises ũ(t), ỹ(t) and Q(z), P(z) ae ( ) and ( ) polynoial atices in the unitay advance opeato z (z y(t) = y(t + )). By selecting a suitable inial paaetization [], [3], odel () can be patitioned into the set of elations y 0i (t + ν i ) = + ν ij α ijk y 0j (t + k ) (6) j= k= ν i+ j= k= β ijk u 0j (t + k ) i =,..., whee the integes ν i (i =,...,) ae the Konecke obsevability invaiants of the syste while the integes ν ij ae copletely defined by these invaiants though the elations ν ij = ν i ν ij = in (ν i +, ν j ) fo i = j fo i > j ν ij = in (ν i, ν j ) fo i < j. The axial obsevability invaiant will be denoted as ν M i.e. ν M = ax i {ν i } i =,..., ; the syste ode n, i.e. the McMillan degee of Q(z) P(z), is given by n = ν i. i= Assuptions A. Syste () is asyptotically stable. A. Q(z) and P(z) ae left copie. A3. The noiseless input u 0 (t) can be eithe a zeo ean egodic pocess o a quasi stationay bounded deteinistic signal, i.e. such that the liit li N N N u 0 (t)u T 0 (t τ) t= exists τ [4]. Moeove, u 0 (t) is assued as pesistently exciting of sufficiently high ode. A4. ũ(t) and ỹ(t) ae zeo ean egodic white pocesses with unknown diagonal covaiance atices Σ u = diag [ σ u, σ u,..., σ u ] Σ y = diag [ σ y, σ y,..., σ y ]. A5. ũ(t) and ỹ(t) ae utually uncoelated and uncoelated with the noise fee input u 0 (t). A6. The stuctual indexes ν, ν,..., ν will be consideed a pioi known. Unde these assuptions, the EIV MIMO identification poble can be stated as follows. Poble : Given the set of noisy input output obsevations u(),..., u(n), y(),..., y(n), estiate the coefficients α ijk, β ijk and the noise covaiance atices Σ u, Σ y. Reak : The consideed class of odels is quite geneal and can descibe any MIMO syste; by contast the coonly used full polynoial fo epesentations, despite thei pactical usefulness, eet any liits (e.g. the ode of the syste ust always be a ultiple of the nube of outputs). Anothe advantage of the consideed odels is associated with thei inial paaetization that assues not only a lowe covaiance of the estiate but the uch oe ipotant identifiability of evey paaete (no paaete can be expessed as a function of eaining ones); this popety can be exteely elevant in fault diagnosis applications. Reak : The integes ν,..., ν that descibe the stuctue of the odel have been assued as known. Thei deteination (stuctual identification) will be descibed in anothe pape. III. PROPERTIES OF THE DYNAMIC FRISCH SCHEME IN Let us define the vectos THE MIMO CASE ϕ 0i (t) = [ y 0 (t)... y 0 (t + ν i )... y 0i (t)... y 0i (t + ν i ) y 0 (t) y 0 (t + ν i ) u 0 (t)... u 0 (t + ν i )... u 0 (t)... u 0 (t + ν i )] T (7) ϕ i (t) = [ y (t)... y (t + ν i )... y i (t) y i (t + ν i )...u (t)...u (t + ν i )] T (8) ϕ i (t) = [ ỹ (t)... ỹ (t + ν i )... ỹ i (t) ỹ i (t + ν i )...ũ (t)...ũ (t + ν i )] T, (9) 408

3 Poceedings of the 9th Intenational Syposiu on Matheatical Theoy of Netwoks and Systes MTNS July, 00 Budapest, Hungay fo i =,...,, whose enties ae noise fee, noisy and noise saples espectively. By defining also the paaete vectos u θi = [ α i... α iνi α i... α iνi α ii α iiνi α i α iνi ] T β i β i(νi+) β i β i(νi+) (0) fo i =,...,, and taking into account (6), (), (3) it is possible to wite ϕ T 0i(t)θ i = 0 () ϕ i (t) = ϕ 0i (t) + ϕ i (t), () fo i =,...,. With efeence to the i th subsyste, define the covaiance atices of noise fee data, noisy data and noise Σ 0i = E [ϕ 0i (t)ϕ T 0i (t)] (3) Σ i = E [ϕ i (t)ϕ T i (t)] (4) Σ i = E [ ϕ i(t) ϕ T i (t)]. (5) Fo (), () and assuption A5 it follows that Σ 0i θ i = 0 (6) Σ i = Σ 0i + Σ i (7) whee, because of assuptions A4 and A5 Σ i = diag [ σ y I νi,..., σ y i I νi+,..., σ y I νi, σ u I νi+,..., σ u i I νi+,..., σ u I νi+]. (8) Note that Σ 0i is a positive seidefinite atix whose null space has diension one and defines the paaete vecto θi. This atix can be obtained fo the noisy data covaiance atix Σ i if the noise vaiances σ y,..., σ y, σ u,..., σ u ae known. Conside now the set of all diagonal atices Σ i (P) of type (8) that exhibit the sae popeties of Σ i, i.e. the set of points P = ( σ,..., σ, σ +,..., σ + ) R+ such that whee Σ i Σ i (P) 0, in eig ( Σ i Σ i (P)) = 0, (9) Σ i (P) = diag [ σ I νi,..., σ i I νi+,..., σ I νi, σ + I ν i+,..., + I ν i+]. (0) It is possible to pove the following esult [4]. Theoe : The set of points P satisfying (9) defines an hypesuface S(Σ i ) belonging to the fist othant of R +. Evey point P of S(Σ i ) is an adissible solution in the noise space and can thus be associated with a possible vecto of paaetes θ i (P) given by ( Σi Σ i (P) ) θ i (P) = 0. () As an exaple, Fig. shows a typical hypesuface fo a syste with one input and two outputs. Fig.. y y Typical shape of S(Σ i ) fo a one input two output EIV odel. The following coollay is a diect consequence of (6) (7). Coollay : The point P = ( σ y, σ y,..., σ y, σ u,..., σ u ), defined by the tue noise vaiances, belongs to S(Σ i ) and is associated with the tue paaete vecto θ i (P ) = θi. The above consideations ae valid fo each subsyste so that it is possible to define hypesufaces S(Σ ), S(Σ ),..., S(Σ ) that exhibit the following ipotant popety. Theoe : All hypesufaces S(Σ ), S(Σ ),..., S(Σ ) have the coon point P coesponding to the actual input and output noise vaiances. Define now the vectos ϕ 0 (t) = [ y 0 (t)... y 0 (t + ν ) () y 0 (t) y 0 (t + ν ) y 0 (t) y 0 (t + ν ) u 0 (t)... u 0 (t + ν M )... u 0 (t)... u 0 (t + ν M )] T ϕ(t) = [ y (t)... y (t + ν ) (3) y (t) y (t + ν ) y (t) y (t + ν ) u (t)... u (t + ν M )... u (t)... u (t + ν M )] T ϕ(t) = [ ỹ (t)... ỹ (t + ν ) (4) ỹ (t) ỹ (t + ν ) ỹ (t) ỹ (t + ν ) ũ (t)... ũ (t + ν M )... ũ (t)... ũ (t + ν M )] T and the atix of paaetes Θ = [ θ θ θ ], (5) whee θ i = [ α i... α iνi 0 }. {{..0 }... (6) (ν + ν i) α ii... α iiνi... α i... α iνi }{{} (ν + ν i) β i... β i(νi+) }{{} (ν M ν i)... β i... β i(νi+) }{{} (ν M ν i) ] T, fo i =,...,. Note that θ i diffes fo θ i only fo the pesence of soe stuctual zeos. Because of (6) and () 409

4 R. Divesi and R. Guidozi Cobining the Fisch Schee and Yule-Walke Equations fo Identifying Multivaiable Eos-in-Vaiables Models (3) it is easy to show that Intoduce also the covaiance atices ϕ T 0 (t)θ = 0 (7) ϕ(t) = ϕ 0 (t) + ϕ(t). (8) Σ 0 = E [ϕ 0 (t)ϕ T 0 (t)] (9) Σ = E [ϕ(t)ϕ T (t)] (30) Σ = E [ ϕ(t) ϕ T (t)]. (3) Fo (7) and (8) it follows iediately that whee Σ 0 Θ = 0 (3) Σ = Σ 0 + Σ (33) Σ = diag [ y I ν+,..., y I ν+, u I νm+,..., u I νm+]. (34) Matix Σ 0 is thus positive seidefinite with a null space of diension. Conside now the faily of all diagonal atices Σ(P) of type (34) which exhibit the sae popeties of Σ, i.e. the set of points P R + such that Σ Σ(P) 0, in eig ( Σ Σ(P)) = 0, (35) whee Σ(P) has the stuctue (34). This set is descibed by the following theoe that constitutes a siple extension of the esult epoted in [4]. Theoe 3: The set of points satisfying (35) defines an hypesuface S(Σ) belonging to the fist othant of R + and lying unde S(Σ ),..., S(Σ ). Because of (3) and (33), the point P, chaacteized by the tue noise vaiances, belongs to S(Σ) and Σ Σ(P ) = Σ 0. The popeties of S(Σ) play an ipotant ole in the solutionof Poble, as it will be shown in the next section. IV. IDENTIFICATION OF MIMO EIV MODELS Although of geat theoetical inteest, the popeties descibed by Theoes and 3 cannot be diectly used in pactice. In fact, in pesence of a finite nube N of noisy data, the covaiance atices Σ i, i =,..., and Σ ust be eplaced by thei saple estiates so that the associated hypesufaces do no longe shae any coon point. It is of couse possible to identify the subsystes sepaately by using one of the selection citeia poposed fo the SISO o MISO cases [8], [9], [0]. The geneic subsyste i would thus be teated as a MISO odel whee all outputs except the i th one play the ole of inputs. Nevetheless, an appoach of this kind is chaacteized by two ain dawbacks. Fistly, it is necessay to solve optiization pobles of coplexity +. This inceases the coputational buden but it is uch oe ipotant to obseve that this pocedue leads to diffeent estiates fo evey input and output noise vaiance with a consequent loss of conguence. To ovecoe these pobles, the seach pocedue and the selection citeion should teat the subsystes as a whole MIMO syste. Fo this eason, the solution of Poble will be based on the geoetic appoach poposed in [6], [5], whee ultivaiable EIV odels have been no longe associated to single points on singulaity hypesufaces but to diections in the noise space. This appoach elies on the coputation of the intesection between a staight line fo the oigin and a singulaity hypesuface descibed by the following theoe []. Theoe 4: Let ξ = (ξ, ξ,..., ξ + ) be a geneic point of the fist othant of R + and ρ the staight line fo the oigin tough ξ. Its intesection with S(Σ i ) is the point P i given by whee P i = ξ λ Mi, λ Mi = ax eig ( Σ i Σ ξ i = diag [ ξ I νi,..., ξ i I νi+,...,ξ I νi, Σ ξ i ) (36) ξ + I νi+,...,ξ + I νi+]. (37) Reak 3: To copute the intesection P between S(Σ) and ρ, (36) will be eplaced with P = ξ λ M, λ M = ax eig ( Σ Σξ ), (38) whee Σ ξ has the stuctue (34). It is thus possible to conside the intesections P, P,..., P of a line ρ fo the oigin with the singulaity sufaces S(Σ ),..., S(Σ ) that define univocally the paaetes θ (P ), θ (P ),..., θ (P ) i.e. a whole MIMO odel. The use of this appoach, which is valid also fo finite nubes of data, still leads to diffeent values of each noise vaiance. Howeve, since P, P,..., P belong to the sae staigth line fo the oigin, the atios between the noise vaiances ae univocally defined. It is now necessay to intoduce a selection citeion to identify the tue diection in the noise space, i.e. that associated with the tue odel. The citeion intoduced in the next subsection takes advantage of the popeties of both the high ode Yule Walke equations and the hypesuface S(Σ). A. The selection citeion Let us define the q (being q a use chosen paaete) vecto of delayed inputs ϕ d u (t) = [ u (t q)... u (t ) (39) u (t q)... u (t )... u (t q)... u (t )] T, that, because of (), satisfies the condition ϕ d u(t) = ϕ d u0(t) + ϕ d u(t), (40) 40

5 Poceedings of the 9th Intenational Syposiu on Matheatical Theoy of Netwoks and Systes MTNS July, 00 Budapest, Hungay whee ϕ d u0(t) = [ u 0 (t q)... u 0 (t ) (4) u 0 (t q)... u 0 (t )... u 0 (t q)... u 0 (t )] T, ϕ d u (t) = [ ũ (t q)... ũ (t ) (4) ũ (t q)... ũ (t )... ũ (t q)... ũ (t )] T. Define also the q [n + + ( + )ν M ] atix Σ d = E [ϕ d u (t)ϕt (t)] (43) = E [(ϕ d u0(t) + ϕ d u(t))(ϕ 0 (t) + ϕ(t)) T ], (44) that can be estiated fo the noisy data. Because of assuptions A4 A5 we have so that,fo (7) Σ d = E [ϕ d u0(t)ϕ T 0 (t)], (45) Σ d Θ = 0. (46) Relation (46) epesents a set of high ode Yule Walke equations that can be used to define a selection citeion. Fo this pupose, conside a diection ρ in the noise space, its intesections P,..., P with S(Σ ),..., S(Σ ) and the associated MIMO odel θ (P ), θ (P ),..., θ (P ). Relation (46) allows the intoduction of the cost function J(P,..., P ) = J(ρ) (47) = tace ( Θ T (P,..., P )(Σ d ) T Σ d Θ(P,..., P ) ), whee Θ(P,..., P ) = Θ(ρ) has the sae stuctue of Θ and has been constucted with the enties of θ (P ),..., θ (P ). This function satisfies the conditions ) J(ρ) 0 ) J(ρ) = 0 if P = = P = P, whee the second condition follows diectly fo (46). Poble can thus be solved by iniizing J(ρ) in the noise space. Of couse, when the covaiance atices ae eplaced by thei saple estiates elation (46) does no longe hold exactly and the iniu of the cost function (47) will be geate than zeo. B. The identification algoith The whole identification pocedue is suaized by the following algoith. Algoith. ) Copute, on the basis of the noisy obsevations u(),..., u(n), y(),...,y(n), the saple estiates of the covaiance atices Σ,...,Σ and Σ: ˆΣ i = N ν M ˆΣ = N ν M ϕ i (t)ϕ T i (t), N ν M t= ϕ(t)ϕ T (t). N ν M t= i =,..., ) Choose a value of the delay q in (39) and copute the saple estiate of the covaiance atix Σ d : ˆΣ d = N ν M ϕ d N ν M q u(t)ϕ T (t). t=q+ 3) Stat fo a geneic diection ρ in the fist othant of R +. 4) Copute the intesections P,...,P between ρ and S(ˆΣ ),..., S(ˆΣ ). 5) Constuct the diagonal atices Σ(P ),..., Σ(P ) and copute the paaete vectos θ (P ), θ (P ),..., θ (P ) by eans of the elations (ˆΣi Σ i (P i ) ) θ i (P i ) = 0 i =,...,. 6) Constuct, with the enties of θ (P ),..., θ (P ) the atix Θ(ρ) of type (5). 7) Copute the cost function J(ρ) (47). 8) Move to a new diection ρ ± ρ coesponding to a decease of J(ρ). 9) Repeat steps 3 8 until the diection ρ coesponding to the iniu of J(ρ) is found. Algoith involves the coputation of the points P,...,P and P in the noise space so that it gives + diffeent estiations fo each noise vaiance σ y,..., σ y, σ u,..., σ u. Even if othe choices ae possible we will conside as noise vaiance estiates the values given by the point P on S(ˆΣ). Fo Theoe 3 it is easy to show that these values lead to atices ˆΣ i Σ i ( P), i =,...,, that ae all positive definite. The algoith is thus concluded as follows 0) Copute the intesection P between ρ and S(ˆΣ) and conside the enties of P as estiates of the tue noise vaiances. Reak 4: It is ipotant to obseve that the whole pocedue equies only the coputation of a vey liited nube of points of the consideed singulaity hypesufaces, i.e. only thei intesections with the diections tested by the adopted optiization algoith. The whole pocedue is thus vey efficient fo a nueical point of view. V. NUMERICAL RESULTS The effectiveness of the poposed identification ethod has been tested by eans of Monte Calo siulations and copaed to that of the identification pocedue descibed in [6]. The nueical exaple concens the two input two output odel consideed in [6], chaacteized by the polynoial atices [ ] z 0.4 z z + 0. Q(z) = 0. z + 0. z 0.4 [ ] 0.6 z z z + P(z) =. 0. z z z z The tue inputs u 0 (t), u 0 (t) ae pseudo ando binay sequences of unit vaiance and length N = 000. The tue 4

6 R. Divesi and R. Guidozi Cobining the Fisch Schee and Yule-Walke Equations fo Identifying Multivaiable Eos-in-Vaiables Models TABLE I TRUE AND ESTIMATED VALUES OF THE COEFFICIENTS OF Q(z). α α α α tue YW (Exp.) ± ± ± ± NS (Exp.) ± ± ± ± 0.07 YW (Exp.) ± ± ± ± NS (Exp.) ± ± ± ± α α α α tue YW (Exp.) 0.07 ± ± ± ± 0.0 NS (Exp.) ± ± ± ± YW (Exp.) ± ± ± ± NS (Exp.) ± ± ± ± TABLE II TRUE AND ESTIMATED VALUES OF THE COEFFICIENTS OF P(z). β β β 3 β β β 3 tue YW (Exp.) ± ± ± ± ± ± NS (Exp.) ± ± ± ± ± ± 0.07 YW (Exp.) 0.45 ± ± ± ± ± ± NS (Exp.) 0.98 ± ± ± ± ± ± β β β 3 β β β 3 tue YW (Exp.) ± ± ± ± ± ± 0.03 NS (Exp.) ± ± ± ± ± ± YW (Exp.) 0.50 ± ± ± ± ± ± NS (Exp.) ± ± ± ± ± ± 0.0 TABLE III TRUE AND ESTIMATED VALUES OF THE NOISE VARIANCES. σ u σ u σ y σ y tue YW (Exp.) ± ± ± ± NS (Exp.) ± ± ± ± YW (Exp.) ± ± ± ± NS (Exp.) ± ± ± ± values of the noise vaiances ae u = 0. u = 0. y = 0.75 u = 0., coesponding to signal-to-noise atios of about 0 db on both inputs and outputs. Two expeients have been pefoed. In the fist expeient the tested algoiths ae initialized by choosing a diection ρ nea the tue one while in the second expeient the initial diection is fa fo the tue one. Fo both cases a Monte Calo siulation of 00 independent uns has been caied out. The value of q in step has been set to 5 since it has been obseved that geate values do not lead to significant ipoveents. The obtained esults ae shown in Tables 3 whee the tue value of paaetes and noise vaiances, the eans of thei estiates and the coesponding standad deviations ae epoted. The poposed Yule Walke based algoith is denoted as YW while the appoach [6], based on the null space of Σ 0 is denoted as NS. It can be obseved that the YW ethod is chaacteized by bette accuacy and highe obustness with espect to the initial estiate of the noise space diection. 4

7 Poceedings of the 9th Intenational Syposiu on Matheatical Theoy of Netwoks and Systes MTNS July, 00 Budapest, Hungay VI. CONCLUSIONS This pape has poposed a MIMO EIV identification pocedue based on the use of a geneal class of inially paaeteized odels paticulaly suitable fo diagnosis and filteing applications. This pocedue is based on a geoetic appoach that associates EIV odels with diections in the noise space and leads to the coputation of both odel paaetes and noise vaiances. The pefoed Monte Calo siulations show the effectiveness of this appoach. [] R. Guidozi and M. Pieantoni, A new paaetization of Fisch schee solutions, Poc. of the XII Int. Conf. on Systes Science, Woclaw, 995, pp [] R. Guidozi, Invaiants and canonical fos fo syste stuctual and paaetic identification, Autoatica, vol. 7, 98, pp [3] R. Guidozi, Equivalence, invaiance and dynaical syste canonical odelling Pat I, Kybenetika, vol. 5, 989, pp [4] L. Ljung, Syste Identification Theoy fo the Use, Pentice Hall, Englewood Cliffs, NJ, 999. REFERENCES [] T. Södestö, Identification of stochastic linea systes in pesence of input noise, Autoatica, vol. 7, 98, pp [] R. E. Kalan, Syste identification fo noisy data, in A. R. Bednaek and L. Cesai (Eds.), Dynaical Systes II, Acadeic Pess, 98, pp [3] B. D. O. Andeson and M. Deistle, Identifiability of dynaic eos in vaiables odels, Jounal of Tie Seies Analysis, vol. 5, 984, pp. 3. [4] S. Beghelli, R. Guidozi and U. Soveini, The Fisch schee in dynaic syste identification, Autoatica vol. 6, 990, pp [5] P. Stoica, M. Cedevall and A. Eiksson, Cobined instuental vaiable and subspace fitting appoach to paaete estiation of noisy input output systes, IEEE Tansactions on Signal Pocessing, vol. 43, 995, pp [6] R. Guidozi, Identification of ultivaiable pocesses in the Fisch schee context, MTNS 96, St. Louis, June 996. [7] W. X. Zheng, Tansfe function estiation fo noisy input and output data, Intenational Jounal of Adaptive Contol and Signal Pocessing, vol., 998, pp [8] S. Beghelli, P. Castaldi, R.P. Guidozi and U. Soveini, A obust citeion fo odel selection in identification fo noisy data, Poc. of Ninth Intenational Confeence on Systes Engineeing, Las Vegas, Nevada, 993, pp [9] R. Divesi, R. Guidozi and U. Soveini, Fisch schee based algoiths fo EIV identification, Poc. of the th IEEE Mediteanean Confeence on Contol and Autoation, Kusadasi, Tukey, 004. [0] R. Divesi, R. Guidozi and U. Soveini, Yule Walke equations in the Fisch schee solution of eos in vaiables identification pobles, Poc. of the 7th Intenational Syposiu on Matheatical Theoy of Netwoks and Systes, Kyoto, Japan, 006, pp [] R. Divesi, R. Guidozi and U. Soveini, Maxiu likelihood identification of noisy input output odels, Autoatica, vol. 43, 007, pp [] T. Södestö, Eos in Vaiables ethods in syste identification, Autoatica, vol. 43, 007, pp [3] K. Mahata, An ipoved bias copensation appoach fo eos in vaiables odel identification, Autoatica, vol. 43, 007, pp [4] J. C. Agueo and G. C. Goodwin, Identifiability of eos in vaiables dynaic systes, Autoatica, vol. 44, 008, pp [5] R. Guidozi and R. Divesi, A geoetic appoach to ultivaiable eos in vaiables identification, Pep. of the 5th IFAC Syposiu on Syste Identification, St. Malo, Fance, 009, pp [6] R. Divesi and R. Guidozi, Fisch schee based identification of ultivaiable eos in vaiables odels, Pep. of the 5th IFAC Syposiu on Syste Identification, St. Malo, Fance, 009, pp [7] S. Van Huffel (Ed.), Recent Advances in Total Least Squaes Techniques and Eos in Vaiables Modelling, SIAM, Philadelphia, PA, 997. [8] S. Van Huffel and P. Leeling(Eds.), Total Least Squaes Techniques and Eos in Vaiables Modelling: Analysis, Algoiths and Applications, Kluwe Acadeic Publishes, Dodecht, The Nethelands, 00. [9] R. Fisch, Statistical confluence analysis by eans of coplete egession systes, Pub. No. 5, Econoic Institute, Oslo Univesity, Noway, 934. [0] R. Guidozi, R. Divesi and U. Soveini, The Fisch schee in algebaic and dynaic identification pobles, Kybenetika, vol. 44, 008, pp