Dankers, A.G.

Van den Hof, P.M.J.

Weerts, H.H.M.

English

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 Maximum likelihood identification in dynamic networks withrank-reduced noiseCitation for published version (APA):Weerts, H. H. M., Van den Hof, P. M. J., & Dankers, A. G. (2017). Maximum likelihood identification in dynamicnetworks with rank-reduced noise. Abstract from 36th Benelux Meeting on Systems and Control, 28-30 March2017, Spa, Belgium, Spa, Belgium.Document status and date:Published: 01/01/2017Document Version:Accepted manuscript including changes made at the peer-review stagePlease check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023Maximum likelihood identification in dynamic networkswith rank-reduced noiseH.H.M. WeertsEindhoven University of Technology1P.M.J. Van den HofEindhoven University of TechnologyA.G. DankersUniversity of Calgary1 IntroductionA typical assumption in multivariable and dynamic net-work system identification is that process noise is fullrank. When a network becomes large-scale, e.g. whenmany sensors are placed physically close to each other,we have to consider the possibility that noise is rank-reduced2. Many of the typical steps in a predictionerror identification framework can not immediately betaken in case noise is rank-reduced. Recently in a pre-diction error setting we have derived conditions underwhich consistent estimates of the network can be ob-tained in case noise is rank-reduced [1]. Our objectiveis to extend those results towards maximum likelihoodestimates of the network dynamics, even when noise isof a reduced rank.2 ApproachIn order to estimate the network a prediction error ap-proach is taken, leading to the multivariable predictionerror ε(t, θ) of dimension L, the number of measurednode variables. The typical identification criterion isthe weighted least squares (WLS)θ∗ := arg minθĒ εT (t, θ)Qε(t, θ). (1)If Q is chosen as the identity matrix this criterion leadsto consistent estimates under rather standard condi-tions, even when noise is rank-reduced [1]. A questionthat needs to be answered is whether this criterion leadsto maximum likelihood estimates. Typically, taking Qas the inverse of the covariance matrix of the innovationprocess leads to a maximum likelihood estimate. Whennoise is rank-reduced the covariance matrix is singular,and therefore can not be inverted.Due to the rank-reduced situation the covariance of theinnovation can be written as BΛBT , with B an L × pdimensional matrix and Λ the p×p dimensional covari-ance matrix of e(t). We use the left-inverse B† of B and1Contact details: h.h.m.weerts@tue.nl or P.O. Box 513,5600 MB Eindhoven, The Netherlands.2A noise process v(t) of dimension L is called rank-reducedif it can be generated by filtering a white noise process e(t) ofdimension p < L.the nullspace of B spanned by the rows of matrix N toconstruct a generalized inverse of BΛBT as weightingmatrixQλ = (B†)TΛ−1B† + λNTN. (2)The B and N terms of the weight actually representtwo orthogonal components of the innovation process.λ governs the balance between the two orthogonal com-ponents. When λ → ∞ the variance of the estimatereduces to some lower bound, possibly 0.3 Maximum likelihoodWhen the innovation ε is of rank p < L then it does nothave a probability density function (pdf) in the usualsense. However a pdf for the innovation exists on a pdimensional subspace [2], which is then mapped to theL dimensional space by matrix B. This means thatB†ε(t) = e(t) does have a pdf, and moreover Nε(t) = 0with probability 1 for all t. The distribution of ε isspecified by a pdf on the hyperplane Nε(t) = 0.When weight Qλ is used in (1) with λ → ∞ then thesecond term of Qλ ensures that the estimate is takenon the hyperplane Nε(t) = 0. The first term of Qλmaps the prediction error to dimension p via B†, andthen applies the appropriately weight for estimation.When the appropriate B, N and Λ are chosen thenthe estimate (1) with weight Q = Qλ is a maximumlikelihood estimate.References[1] H.H.M. Weerts, P.M.J. Van den Hof, A. Dankers,“Identification of dynamic networks with rank-reducedprocess noise,” IFAC World Congress, 2017, to appear.[2] C.R. Rao, “Linear Statistical Inference and ItsApplications,” John Wiley & Sons, 1973.

Maximum likelihood identification in dynamic networks with rank-reduced noise

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 Maximum likelihood identification in dynamic networks withrank-reduced noiseCitation for published version (APA):Weerts, H. H. M., Van den Hof, P. M. J., & Dankers, A. G. (2017). Maximum likelihood identification in dynamicnetworks with rank-reduced noise. Abstract from 36th Benelux Meeting on Systems and Control, 28-30 March2017, Spa, Belgium, Spa, Belgium.Document status and date:Published: 01/01/2017Document Version:Accepted manuscript including changes made at the peer-review stagePlease check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023Maximum likelihood identification in dynamic networkswith rank-reduced noiseH.H.M. WeertsEindhoven University of Technology1P.M.J. Van den HofEindhoven University of TechnologyA.G. DankersUniversity of Calgary1 IntroductionA typical assumption in multivariable and dynamic net-work system identification is that process noise is fullrank. When a network becomes large-scale, e.g. whenmany sensors are placed physically close to each other,we have to consider the possibility that noise is rank-reduced2. Many of the typical steps in a predictionerror identification framework can not immediately betaken in case noise is rank-reduced. Recently in a pre-diction error setting we have derived conditions underwhich consistent estimates of the network can be ob-tained in case noise is rank-reduced [1]. Our objectiveis to extend those results towards maximum likelihoodestimates of the network dynamics, even when noise isof a reduced rank.2 ApproachIn order to estimate the network a prediction error ap-proach is taken, leading to the multivariable predictionerror ε(t, θ) of dimension L, the number of measurednode variables. The typical identification criterion isthe weighted least squares (WLS)θ∗ := arg minθĒ εT (t, θ)Qε(t, θ). (1)If Q is chosen as the identity matrix this criterion leadsto consistent estimates under rather standard condi-tions, even when noise is rank-reduced [1]. A questionthat needs to be answered is whether this criterion leadsto maximum likelihood estimates. Typically, taking Qas the inverse of the covariance matrix of the innovationprocess leads to a maximum likelihood estimate. Whennoise is rank-reduced the covariance matrix is singular,and therefore can not be inverted.Due to the rank-reduced situation the covariance of theinnovation can be written as BΛBT , with B an L × pdimensional matrix and Λ the p×p dimensional covari-ance matrix of e(t). We use the left-inverse B† of B and1Contact details: h.h.m.weerts@tue.nl or P.O. Box 513,5600 MB Eindhoven, The Netherlands.2A noise process v(t) of dimension L is called rank-reducedif it can be generated by filtering a white noise process e(t) ofdimension p < L.the nullspace of B spanned by the rows of matrix N toconstruct a generalized inverse of BΛBT as weightingmatrixQλ = (B†)TΛ−1B† + λNTN. (2)The B and N terms of the weight actually representtwo orthogonal components of the innovation process.λ governs the balance between the two orthogonal com-ponents. When λ → ∞ the variance of the estimatereduces to some lower bound, possibly 0.3 Maximum likelihoodWhen the innovation ε is of rank p < L then it does nothave a probability density function (pdf) in the usualsense. However a pdf for the innovation exists on a pdimensional subspace [2], which is then mapped to theL dimensional space by matrix B. This means thatB†ε(t) = e(t) does have a pdf, and moreover Nε(t) = 0with probability 1 for all t. The distribution of ε isspecified by a pdf on the hyperplane Nε(t) = 0.When weight Qλ is used in (1) with λ → ∞ then thesecond term of Qλ ensures that the estimate is takenon the hyperplane Nε(t) = 0. The first term of Qλmaps the prediction error to dimension p via B†, andthen applies the appropriately weight for estimation.When the appropriate B, N and Λ are chosen thenthe estimate (1) with weight Q = Qλ is a maximumlikelihood estimate.References[1] H.H.M. Weerts, P.M.J. Van den Hof, A. Dankers,“Identification of dynamic networks with rank-reducedprocess noise,” IFAC World Congress, 2017, to appear.[2] C.R. Rao, “Linear Statistical Inference and ItsApplications,” John Wiley & Sons, 1973.

Maximum likelihood identification in dynamic networks with rank-reduced noise

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