Lower Bound on the Accuracy of Parameter Estimation Methods for Linear Sensorimotor Synchronization Models

In: Timing & Time Perception

The mechanisms that support sensorimotor synchronization — that is, the temporal coordination of movement with an external rhythm — are often investigated using linear computational models. The main method used for estimating the parameters of this type of model was established in the seminal work of Vorberg and Schulze (2002), and is based on fitting the model to the observed auto-covariance function of asynchronies between movements and pacing events. Vorberg and Schulze also identified the problem of parameter interdependence, namely, that different sets of parameters might yield almost identical fits, and therefore the estimation method cannot determine the parameters uniquely. This problem results in a large estimation error and bias, thereby limiting the explanatory power of existing linear models of sensorimotor synchronization. We present a mathematical analysis of the parameter interdependence problem. By applying the Cramér–Rao lower bound, a general lower bound limiting the accuracy of any parameter estimation procedure, we prove that the mathematical structure of the linear models used in the literature determines that this problem cannot be resolved by any unbiased estimation method without adopting further assumptions. We then show that adding a simple and empirically justified constraint on the parameter space — assuming a relationship between the variances of the noise terms in the model — resolves the problem. In a follow-up paper in this volume, we present a novel estimation technique that uses this constraint in conjunction with matrix algebra to reliably estimate the parameters of almost all linear models used in the literature.

  • CramérH. (1999). Mathematical methods of statistics. Princeton, NJ, USA: Princeton University Press.

  • DiedrichsenJ., IvryR. B., & PressingJ. (2003). Cerebellar and basal ganglia contributions to interval timing. In MeckW. H. (Ed.), Functional and neural mechanisms of interval timing (pp. 457–481). Boca Raton, FL, USA: CRC Press.

    • Search Google Scholar
    • Export Citation
  • FellerW. (2008). An introduction to probability theory and its applications. New York, NY, USA: John Wiley & Sons.

  • FriedlanderB. (1984). On the computation of the Cramer–Rao bound for ARMA parameter estimation. IEEE Trans. Acoust., 32, 721–727.

  • HaryD., & MooreG. (1987a). On the performance and stability of human metronome — Synchronization strategies. Br. J. Math. Stat. Psychol., 40, 109–124.

    • Search Google Scholar
    • Export Citation
  • HaryD., & MooreG. (1987b). Synchronizing human movement with an external clock source. Biol. Cybern., 56, 305–311.

  • JacobyN., TishbyN., ReppB. H., AhissarM., & KellerP. E. (2015). Parameter estimation of linear sensorimotor synchronization models: Phase correction, period correction and ensemble synchronization. Timing Time Percept., 3, XX–YY.

    • Search Google Scholar
    • Export Citation
  • LiS., LewandowskyS., & DeBrunnerV. E. (1996). Using parameter sensitivity and interdependence to predict model scope and falsifiability. J. Exp. Psychol. Gen., 125, 360–369.

    • Search Google Scholar
    • Export Citation
  • LjungL. (1998). System identification. Berlin, Germany: Springer.

  • MatesJ. (1994a). A model of synchronization of motor acts to a stimulus sequence. Biol. Cybern., 70, 463–473.

  • MatesJ. (1994b). A model of synchronization of motor acts to a stimulus sequence. II. Stability analysis, error estimation and simulations. Biol. Cybern., 70, 475–484.

    • Search Google Scholar
    • Export Citation
  • MichonJ. (1967). Timing in temporal tracking. Soesterberg, The Netherlands: Institute for Perception RVO-TNO.

  • PearsonK. (1896). Mathematical contributions to the theory of evolution. III. Regression, heredity and panmixia. Philos. Trans. R. Soc. Lond., 187, 253–318.

    • Search Google Scholar
    • Export Citation
  • RaoC. R. (1992). Information and the accuracy attainable in the estimation of statistical parameters. In KotzS. & JohnsonN. L. (Eds), Breakthroughs in statistics (pp. 235–247). New York, NY, USA: Springer.

    • Search Google Scholar
    • Export Citation
  • ReppB. H. (2005). Sensorimotor synchronization: A review of the tapping literature. Psychonom. Bull. Rev., 12, 969–992.

  • ReppB. H., & SuY. (2013). Sensorimotor synchronization: A review of recent research (2006–2012). Psychonom. Bull. Rev., 20, 403–452.

    • Search Google Scholar
    • Export Citation
  • ReppB., KellerP., & JacobyN. (2012). Quantifying phase correction in sensorimotor synchronization: Empirical comparison of three paradigms. Acta Psychol., 139, 281–290.

    • Search Google Scholar
    • Export Citation
  • SchulzeH., CordesA., & VorbergD. (2005). Keeping synchrony while tempo changes: Accelerando and ritardando. Music Percept., 22, 461–477.

    • Search Google Scholar
    • Export Citation
  • StevensL. T. (1886). On the time-sense. Mind, 11, 393–404.

  • StrangG. (2006). Linear algebra and its applications. Belmont, CA, USA: Thomson Brooks Cole/ Boston, MA, USA: Cengage Learning.

    • Export Citation
  • VorbergD., & SchulzeH. (2002). Linear phase-correction in synchronization: Predictions, parameter estimation, and simulations. J. Math. Psychol., 46, 56–87.

    • Search Google Scholar
    • Export Citation
  • VorbergD., & SchulzeH. H. (2013). Modeling synchronization in musical ensemble playing: Parameter estimation and sensitivity to assumptions. Fourteenth Rhythm Production and Perception Workshop, University of Birmingham, Birmingham, UK.

    • Search Google Scholar
    • Export Citation
  • VorbergD., & WingA. (1996). Modeling variability and dependence in timing. In HeuerH. & KeeleS.W. (Eds), Handbook of perception and action, Vol. 2 (pp. 181–262). London, UK: Academic Press.

    • Search Google Scholar
    • Export Citation
  • WingA. M. (2002). Voluntary timing and brain function: An information processing approach. Brain Cogn., 48, 7–30.

  • WingA. M., & KristoffersonA. B. (1973). Response delays and the timing of discrete motor responses. Percept. Psychophys., 14, 5–12.

  • WingA. M., EndoS., BradburyA., & VorbergD. (2014). Optimal feedback correction in string quartet synchronization. J. R. Soc. Interface, 11, 1125. DOI: 10.1098/rsif.2013.1125.

    • Search Google Scholar
    • Export Citation

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