Improved parameter estimation by noise compensation in the time-scale domain James R. McCusker, Todd Currier, Kourosh Danai à Department of Mechanical and Industrial Engineering, University of Massachusetts Amherst, Amherst MA 01003, USA article info Article history: Received 19 February 2009 Received in revised form 30 March 2010 Accepted 7 June 2010 Available online 12 June 2010 Keywords: System identification Noise suppression Denoising Parameter estimation Wavelet transforms abstract It was shown recently that parameter estimation can be performed directly in the time- scale domain by isolating regions wherein the prediction error can be attributed to the error of individual dynamic model parameters [1]. Based on these single-parameter equations of the prediction error, individual model parameters error can be estimated for iterative parameter estimation. An added benefit of this parameter estimation method, besides its unique convergence characteristics, is the added capacity for direct noise compensation in the time-scale domain. This paper explores this benefit by introducing a noise compensation method that estimates the distortion by noise of the prediction error in the time-scale domain and incorporates that as a confidence factor to bias the estimation of individual parameters error. This method is shown to improve the precision of the estimated parameters when the confidence factors accurately represent the noise distortion of the prediction error. & 2010 Elsevier B.V. All rights reserved. 1. Introduction Dynamics are germane to most natural systems and artifacts. As such, dynamic models are the natural choice for representing most systems. But in addition to their form, the parameters of these models need to be estimated so as to improve the accuracy of the model relative to the observations. A major difficulty in para- meter estimation is the noise present in the observations. Parameter estimation is generally based on the prediction error of the model, M Y , defined as eðtÞ¼ yðtÞb yðtÞ ð1Þ where y(t) is an individual measured output and b yðtÞ¼ M Y ðuðtÞÞ is the model output obtained with the same inputs u(t) as those applied to the process. Parameter estimation has traditionally entailed seeking the true parameters H  ¼½y  1 , ... , y  Q  T 2 R Q so as to minimize a loss function V in terms of the prediction error as [2] b H ¼ b HðY N Þ¼ arg min H V ðe N Þ ð2Þ where Y N ¼½yðt 1 Þ, ... , yðt N Þ T comprises the measured out- puts sampled at t k 2½t 1 , t N  and V ðe N Þ¼ X N k ¼ 1 Lðeðt k ÞÞ ð3Þ is a scalar-valued (typically positive) function of the prediction error, defined by L. The implicit assumption in parameter estimation is that the model M Y accurately represents the process. As such, yðt, uðtÞÞ  b yðt, uðtÞjH ¼ H  Þþ n ð4Þ where n denotes measurement noise. Furthermore, if the model is identifiable [3] then having the model output b yðt, HÞ match the measured output y(t) (in mean-square sense) will be equivalent to the model parameters H matching the true parameters H  ; i.e., yðtÞ b yðt, HÞþ n¼)H ¼ H  ð5Þ Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/sigpro Signal Processing 0165-1684/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2010.06.008 à Corresponding author. Tel.: + 1 413 545 1561; fax: + 1 413 545 1027. E-mail address: danai@ecs.umass.edu (K. Danai). Signal Processing 91 (2011) 72–84