ROBUST FREQUENCY DOMAIN ARMA MODELLING. Jonas Gillberg Fredrik Gustafsson Rik Pintelon

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1 ROBUST FREQUENCY DOMAIN ARMA MODELLING Jonas Gillerg Fredrik Gustafsson Rik Pintelon Department of Electrical Engineering, Linköping University SE Linköping, Sweden Department ELEC, Vrije Universiteit Brussel Pleinlaan 2, 15 Brussel, Belgium Astract: In this paper a method for the rejection of frequency domain outliers is proposed. The algorithm is ased on the work y Huer on M-estimators and the concept of influence function introduced y Hampel. The estimation takes place in the context of frequency domain continuous-time ARMA modelling, ut the method can e also e applied to the discrete time case.copyright c 26SYSID Keywords: roust estimation, outlier rejection, continuous-time, ARMA, CARMA, frequency Domain, influence function, M-estimator 1. INTRODUCTION A characteristic industrial signal processing prolem is viration analysis using tachometer measurements on rotating axles. Motivated y recent advances in system identification in the frequency domain Pintelon and Schoukens, 21; Ljung, 1999), a frequency domain approach for such analysis was outlined in an earlier paper Gillerg and Gustafsson, 25) y the authors and compared at a theoretical level to the time domain algorithm proposed in Persson, 22; Persson and Gustafsson, 21). Among other things, the new method involved virational modelling using a continuous-time AR model and the restriction of the frequency interval to e used in the estimation. Interpolation was also performed in order to counter the effect of non-uniform sampling. In the same paper the main specifications on a procedure aimed at high-sensitivity viration analysis were listed: 1) Being ased on parametric physical models of the viration. 2) Operate on short data atches in a prespecified speed interval where the data pass several quality checks. 3) Potential to reject wide and disturances that are non-interfering with the viration. 4) Potential to reject narrow and disturances that are interfering with the viration. The methods given in Persson, 22; Persson and Gustafsson, 21) and Gillerg and Gustafsson, 25) successfully solves the first three specifications, ut not the last one. The prolem of narrow-and disturances occurs in several applications. For instance in an automotive drive-line, see Figure 1, where the virations can e due to engine knock or gear shifts. In rootics there are virations from the load, just to mention a few. In this paper the focus will therefore e on roustness against these narrow and disturances.

2 where the model parameter vector θ includes the noise variance λ whose true value is σ 2 ). The transfer function G can e parameterized y θ in an aritrary way, for example y the conventional numerator and denominator parameters: Gp, θ) = Bp) Ap) Ap) = p n + a 1 p n 1 + a 2 p n a n Bp) = p m + 1 p m m θ = [a 1 a 2... a n m λ] T. 4) f [Hz] Fig. 1. Smoothed spectrum for virations extracted from ABS system signals 2. OUTLINE The outline of the paper will e the following. First in Section 3 the frequency domain continuous-time ARMA CARMA) modelling and identification approach will e descried. Then in Section 4 the identification criterion is modified in order to facilitate rejection of frequency domain outliers. The concept of M-estimators and the influence function are also introduced. In Section 5 the asymptotic variance and gross sensitivity of the estimator are calculated, and measures are taken in order to assure consistency in the case of correct modelling. A certain choice of criterion is also proved to make an optimal tradeoff etween decreased ias due to outliers and increased variance due to a more insensitive method. Finally in Section 6 the algorithm is illustrated y a numerical example. 3. FREQUENCY DOMAIN CARMA ESTIMATION In this paper we shall consider continuous-time ARMA models represented as yt) = G c p)et) 1) where et) is continuous time white noise such that E[et)] = E[et)es)] = σ 2 δt s) The operator p is here the differentiation operator. We assume that Gp) is strictly proper, so yt) itself does not have a white-noise component, ut is a well defined second order, stationary process. Its spectrum spectral density) can e written as Φ c ω) = σ 2 G c iω) 2 2) We shall consider a general model parameterization G c p, θ) 3) Let us define the truncated Fourier transformation of the continuous time output {yt) : t [, T ]} in expression 1) as Y T iω) = 1 T T yt)e iωt dt. A complicating element in a practical estimation procedure is that we do not have access to the entire continuous time realization of the output. Instead we have a finite numer of samples of yt) at time instances {t 1, t 2,..., t N }. This is an important issue which have een dealt with in the case of equidistant sampling in a previous paper y the first author Gillerg and Ljung, 25). The case of non-uniform sampling we refer the reader to a technical reports associated with two other papers sumitted to this conference Gillerg and Ljung, 25a) Gillerg and Ljung, 24a). However, if we have an estimate of the continuoustime Fourier transform, the continuous-time periodogram is ˆΦiω n ) = Y T iω n ) 2 Exp Φω n, θ ) 5) and the transform will e asymptotically independent at the frequencies ω k, k = 1,..., N ω where ω k = 2πk/T, k N. In this paper we will focus treat the continuous-time periodogram estimates as our measurements and will consider them to e completely uncorrelated. When an estimate of the power spectrum is availale a model can e identified y the following Maximum-Likelihood ML) procedure descried in Gillerg and Ljung, 24) and Gillerg, 24) ˆθ = arg min θ k=1 ˆΦiω k ) Φiω k, θ) + log Φiω k, θ). 6) where the asymptotic Fisher information at the true parameters θ is Jθ ) = Φ θ ω n, θ ) Φω n, θ ) Φ θ ω n, θ ) T Φω n, θ ). 7) This approach forms the asis for the work presented in the remaining part of the paper.

3 The method in 6) can also e descried as the θ which is the solution to the vector equation ) Φ θ ω n, θ) ˆΦωn ) Φω n, θ) Φω n, θ) 1 =. where Φ θ ω n, θ) is the gradient of the spectrum with respect to the parameters. 4. M-ESTIMATORS AND OUTLIER REJECTION If the model is the correct one, the maximumlikelihood method is known to e the est possile optimal) estimator with consistency and asymptotic efficiency Wald, 1949)Cramér, 146). If there, on the other hand, are unmodelled outliers present, the method might not e optimal. This may lead us to consider roust estimators which give up efficiency at the true model in exchange for reasonale performance if the model is not the truecasella and Berger, 22). Therefore we introduce the following method ΨˆΦ, θ) = Φ θ ω n, θ) ˆΦωn ) Φω n, θ) ) =. 8) inspired y the M-estimators introduced y Huer Huer, 1981). The unknown here is the function ψ which will e found using the influence function approach introduced y Hampel Hampel et al., 1986). Certain measures must however e taken in order to assure that the estimates are consistent when periodogram originate from the assumed model in 5). This implies that ψ ˆΦiωn ) Φiω n, θ ) ) ) df n ˆΦiωn ) = 9) ψ x) dg x) = 1) where we from now on define the distriutions F n Exp Φiω n, θ ) 11) G Exp 1 12) In this context we define the solution to equation 8) as a vector valued stochastic variale T F ) with a distriution dependent on F = F 1,..., F Nω ), the distriution of the periodogram ˆΦiω n ). The so called influence function introduced y Hampel is then defined as IF ˆΦ; T, F ) = lim h T 1 h)f + hδˆφ) T F ) h and measures the asymptotic ias caused y contamination in the data. Here is the multivariale measure which puts the mass 1 at ˆΦ δˆφ = ˆΦiω1 ),..., ˆΦωNω ). 5. INFLUENCE FUNCTION AND ASYMPTOTIC VARIANCE In the case of the estimator in 8) it is possile to show that see p.11 in Hampel et al., 1986)) IF ˆΦ; T, F ) = MΨ, T F )) 1 ΨˆΦ, T F )) where ) MΨ, F ) = ΨˆΦ, θ) θ=t F ) df ˆΦ). It is also possile to show see p.85 in Hampel et al., 1986)) that the asymptotic variance of the parameter estimates will e V T, F ) = IF ˆΦ; T, F )IF ˆΦ; T, F )df ˆΦ) =Mψ, F ) 1 Qψ, F )Mψ, F ) T where QΨ, F ) = ΨˆΦ, T F ))Ψ T ˆΦ, T F ))df ˆΦ) Here = and ) MΨ, F ) = ΨˆΦ, θ) Φ θ ω n, θ) ˆΦωn ) Φω n, θ) Φ θ ω n, θ ) Φ θ ω n, θ ) T Φω n, θ ) Φω n, θ ) + Φ θ ω n, θ ) Φ θ ω n, θ ) T Φω n, θ ) Φω n, θ ) df ˆΦ) θ=θ )) θ=θ df ˆΦ) Φ )) θ ω n, θ) ˆΦωn ) = Φω n, θ) θ=θ Φ θ ω ) ) n, θ ) ˆΦωn ) ψ Φω n, θ ) Φω n, θ ) ) ˆΦω n ) ˆΦωn ) Φω n, θ ) ψ. Φω n, θ ) Because of 9) and 1) the expected value of first expression on the right hand side of the equation aove will e zero. Therefore we will have MΨ, F ) = Jθ) xψ x)dgx). Since the periodogram at different frequencies are assumed independent we also have QΨ, F ) = ΨˆΦ, T F ))Ψ T ˆΦ, T F ))df ˆΦ) =Jθ) ψ 2 x)dgx) where J is the Fischer information in 7) and the asymptotic variance will e ψ 2 x)dgx) V T, F ) = xψ x)dgx) ) 2 J 1 θ)

4 5.1 Optimal B-Roust Estimator Assume now that we want to find a function ψ that limits the gross sensitivity measured as γut, F ) = sup{ IF ˆΦ; T, F ) } 13) ˆΦ where. means taking the asolute value of each vector component, while at the same time minimizes the trace of the variance T r V T, F ). In our case we will have ψ 2 x)dgx) T r V T, F ) =T r xψ x)dgx) ) 2 J 1 θ) and ψ 2 x)dgx) = xψ x)dgx) ) 2 T r J 1 θ) IF ˆΦ; T, F ) = MΨ, T G)) 1 ΨˆΦ, T F )) ) Jθ) 1 Φ = θ ω n, θ) xψ x)dgx) Φω n, θ) ψ ˆΦωn ) Φω n, θ) xψ x)dgx) Jθ) 1 Φ θ ω n, θ) Φω n, θ) 5.2 Computing a The parameter in 14) can e considered a tuning factor which is selected y the user in order to strike a alance etween the asymptotic ias and variance of the parameter estimates. The factor a is then selected such that the estimator is asymptotically uniased when the data is produced y the model. That is ψx)dgx) = [x 1 a] e x dx =. This means that three different cases have to e considered. 5.3 Case 1 The first, when 1 a which is trivial since yields =. [x 1 a] e x dx = = 16) According to Theorem 1 and Equation 2.4.1) on p. 122 in Hampel et al., 1986) a version of the so called Huer function ψx) =[x 1 a] 14) if < x 1 a = x 1 a if < x 1 a if x 1 a minimizes ψ 2 x)dgx) xψ x)dgx) ) 2 among all mappings ψ that satisfy ψx)dgx) = sup ψx) x xψ x)dgx) xψ x)dgx) Therefore, using the function in 14) in ) Φ θ ω n, θ) ˆΦωn ) = 15) Φω n, θ) will minimize while T r V T, F ) = sup ˆΦ ψ 2 x)dgx) xψ x)dgx) ) 2 T r J 1 θ) Nω Jθ) 1 Φ θ ω n,θ) Φω n,θ) IF ˆΦ; T, F ) xψ x)dgx) 5.4 Case 2 The second case is when < 1 a. Here [x 1 a] e x dx = 1+a+ + 1+a+ x 1 a)e x dx The first integral in this expression will e 1+a+ x 1 a)e x dx = [ e x xe x a) ] 1+a+ = [ a x)e x] 1+a+ = 1 )e e 1+a) a The second integral is 1+a+ e x dx = [ e x] 1+a+ = e e 1+a) x 1 a)e x dx. This means that given we have to chose a) such that 1 )e e 1+a) a + e e 1+a) = This means that in this case { e e 1+a) + a = 1 + a e e 1+a) a =

5 5.5 Case 3 In the third case we have 1 a and 1+a+ 1+a [x 1 a] e x dx = x 1 a)e x dx + 1+a 1+a+ The first integral will in this case e 1+a while the second is 1+a+ 1+a x 1 a)e x dx = e x dx e x dx = [ e x] 1+a = e e 1+a) x 1 a)e x dx. = [ a x)e x] 1+a+ 1+a = a 1 a )e e 1+a) a 1 a + )e e 1+a) = 1)e 1)e )e 1+a) The third integral is the same as previously 1+a+ This means that we have x 1 a)e x = e e 1+a) [x 1 a] e x dx = e e 1+a) + 1)e 1)e )e 1+a) + e e 1+a) = e e )e 1+a) = and the solution is a = ln e e 1 + a 1 a Fig. 2. The parameter a as a function of the parameter. The reakpoint etween the two solution occurs at.7968 The ojective is to estimate the parameters of the continuous-time spectrum Φiω) = λ iω) 2 + a 1 iω) + a ) where a 1 = 3 and a 2 = 2. The underlying continuous-time system is the autoregressive model σ yt) = p 2 et) 18) + a 1 p + a 2 where et) is continuous-time Gaussian white noise. In Figure 3 we have computed the standard deviation for the parameter estimates of the continuous-time model found in 18). Both the new and the original ML method have een used. Simulations have een performed using N MC = 1 Monte-Carlo runs for each value of. As expected the standard deviation of the roust method will increase as decrease, ut it is interesting to note that it only approximately doule for such a small value as =.1. In the estimation we have used frequencies w = { :.1 : 2π} 5.6 Equations σ a1.4.2 This means that a) is defined y the equations e e 1+a) + a = if 1 + a e 1+a = ln e e if 1 + a In Figure 2 the parameter a is presented as a function of the parameter. The reakpoint etween the two sets of solutions occurs when NUMERICAL EXAMPLE In this section we will compare the efficiency and roustness of the proposed estimation method in 8) and 14) with that of the ML approach in 6). σ a2 σ λ Fig. 3. The standard deviations of the parameter estimates. Roust dotted) and ML method dashed). In Figure 4 we have estimated the same parameters as in the example prolem aove. In order to simulate outliers, we have also introduced an

6 additive, random and exponentially distriuted disturance at 5 % of all frequencies, with a variance which is twice the maximum value of the power spectrum in 17). From the figure one can a 1 a 2 λ Fig. 4. The ias for the parameter estimates. True solid), ML method dashed) and roust dotted). see that the outliers cause ias in the parameter estimates when the ML method is utilized. This ias is then reduced y approximately 5 % with the use of the more roust criterion. 7. CONCLUSIONS In this paper a method for the rejection of frequency domain outliers is proposed. The algorithm is ased on the work y Huer on M- estimators and the concept of influence function introduced y Hampel. The estimation takes place in the context of frequency domain continuoustime ARMA modelling, ut the method can e also e applied to the discrete time case. It is also proved that a certain choice of criterion will produce an optimal tradeoff etween ias and variance under certain assumptions. Finally the method is illustrated y a numerical example which shows that the ias can e reduced y approximately 5%. REFERENCES Casella, G. and R.L. Berger 22). Statistical Inference. Duxury. Pacific Grove, CA. Cramér, H. 146). Mathematical Methods of Statistics. Princeton University Press. Princeton, NJ. Gillerg, J. 24). Methods for frequency domain estimation of continuous-time models. Lic. Dissertation Linköping Studies in Science and Technology Thesis No Department of Electrical Engineering, Linköping University. SE Linköping, Sweden. Gillerg, J. and F. Gustafsson 25). Frequencydomain continuous-time AR modelling using non-uniformly sampled measurements. In: Proceedings of IEEE Int Conf on Acoustics, Speech and Signal Processing in Philadelphia, USA. Gillerg, J. and L. Ljung 24a). Frequencydomain ARMA models: Interpolation and nonuniform sampling. Technical Report LiTH- ISY-R Department of Electrical Engineering, Linköping University. SE Linköping, Sweden. Gillerg, J. and L. Ljung 24). Frequencydomain ARMA models: Interpolation and nonuniform sampling. Technical Report LiTH- ISY-R Department of Electrical Engineering, Linköping University. SE Linköping, Sweden. Gillerg, J. and L. Ljung 25a). Frequencydomain ARMA models from non-uniformly sampled data. Technical Report LiTH-ISY-R Department of Electrical Engineering, Linköping University. SE Linköping, Sweden. Gillerg, J. and L. Ljung 25). Frequencydomain ARMA models from sampled data. In: Proc. 16th IFAC World Congress. Hampel, F. R., E.M. Ronchetti, P.J. Rousseeuw and W.A. Stael 1986). Roust Statistics: the approach ased on influence function. John Wiley & Sons. Huer, P. 1981). Roust Statistics. John Wiley & Sons. Ljung, L. 1999). System Identification: Theory for the User. Prentice-Hall. Upper Saddle River. Persson, N. 22). Event ased sampling with applicaion to spectral estimation. Lic. Dissertation Linköping Studies in Science and Technology Thesis No Department of Electrical Engineering, Linköping University. SE Linköping, Sweden. Persson, N. and F. Gustafsson 21). Event ased sampling with application to viration analysis in pneumatic tires. In: Proceedings of IEEE Int Conf on Acoustics, Speech and Signal Processing in Salt Lake City, USA. Pintelon, R. and J. Schoukens 21). System Identification - A Frequency Domain Approach. IEEE Press. Piscataway, NJ. Wald, A. 1949). Note on the consistency of the maximum likelihood estimate. Anti. Math. Statis. 2,