Parameter Estimation of Linear Sensorimotor Synchronization Models: Phase Correction, Period Correction, and Ensemble Synchronization

in Timing & Time Perception
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Linear models have been used in several contexts to study the mechanisms that underpin sensorimotor synchronization. Given that their parameters are often linked to psychological processes such as phase correction and period correction, the fit of the parameters to experimental data is an important practical question. We present a unified method for parameter estimation of linear sensorimotor synchronization models that extends available techniques and enhances their usability. This method enables reliable and efficient analysis of experimental data for single subject and multi-person synchronization. In a previous paper (Jacoby et al., 2015), we showed how to significantly reduce the estimation error and eliminate the bias of parameter estimation methods by adding a simple and empirically justified constraint on the parameter space. By applying this constraint in conjunction with the tools of matrix algebra, we here develop a novel method for estimating the parameters of most linear models described in the literature. Through extensive simulations, we demonstrate that our method reliably and efficiently recovers the parameters of two influential linear models: Vorberg and Wing (1996), and Schulze et al. (2005), together with their multi-person generalization to ensemble synchronization. We discuss how our method can be applied to include the study of individual differences in sensorimotor synchronization ability, for example, in clinical populations and ensemble musicians.

Parameter Estimation of Linear Sensorimotor Synchronization Models: Phase Correction, Period Correction, and Ensemble Synchronization

in Timing & Time Perception



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Figure 1 of Jacoby et al. (2015) provides an explanation why constraining the parameter space could dramatically reduce estimation errors. A detailed analysis of this issue can be found in the classical works of Cramér and Rao (Cramér 1999 originally 1946; Rao 1992 originally 1945). Intuitively the interdependence comes from points in the parameter space that do not satisfy eqn. (4). When these points are not considered for parameter estimation the problem disappears and the original method performs well for all cases. This is explained in detail in Jacoby et al. (2015).


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    OkR and OkS are the onsets of the response and stimulus at beat k. Sk=OkSOk1S and Rk=OkROk1R are the inter-stimulus and inter-response intervals, respectively. Ak=OkROkS is the asynchrony. This figure is published in color in the online version.

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    Comparison of the unbounded acvf method [circles] and the unbounded Generalized Least Squares method [thin line]. The correct underlying parameters are plotted as the dashed line. The figure shows the mean estimates based on 100 simulated renditions of the phase correction model of eqn. (3), for different α values (x-axis) with parameters σT2=100 , σM2=25 , nseq = 15, and N = 30. Error bars represent the standard deviation of the estimates. Both methods show extremly large estimation errors for large α values. This demonstrates the necessity of using further constraints in order to obtain a reliable estimates. This figure is published in color in the online version.

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    The bGLS method with a non-isochronous sequence. Ideal performance is marked in dashed line. We simulated 1000 iteration for different α values (x-axis) with parameters ( σT2=100 and σM2=25 , nseq = 15, N = 30). The top, middle, and bottom graphs show the mean and standard deviation of the estimates for phase correction, timekeeper variance, and motor variance, respectively. This figure is published in color in the online version.

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    Comparison of mean and standard deviation of the estimations based on motor noise and least square estimation (LS) that ignores motor noise. Motor variance was fixed ( σM2=25 ), and the different lines present different σT2=100,300,500 timekeeper variance. In all simulations nseq = 15 and N = 30. This figure is published in color in the online version.

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    Schematic illustration of onsets for synchronizing in multi-person cases. We denote the onsets of the responses of the current synchronizer as OkR . We denote by OkSi the onsets of the P other partners. The asynchronies between response and partner i are given by Aki . This figure is published in color in the online version.

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    Simulated string quartet with a setting similar to the data in Wing et al. (2014); namely, the same number of synchronizers and the same number of trials and blocks. The top graph shows the phase constants of all four simulated players. The bottom graph shows the different timekeeper standard deviations (σT = 10, 13, 15, and 18 for violin 1, violin 2, viola, and cello, respectively) and motor variances (σM = 5, 6, 7, 8). Estimation of 100 iterations of the simulations with nseq = 16, and N = 40 are depicted. Error bars represents standard deviation of the estimation error. This figure is published in color in the online version.

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    Reanalyis of real string quartets using the data from Wing et al. (2014). The top figures show the estimated coupling constants for quartet 1 (left) and quartet 2 (right). The bottom figures show the estimated timekeeper and motor variance for each of the players. This figure is published in color in the online version.

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    Simulation results for the period correction experiment in Jacoby and Repp (2012). Note that because of the high accuracy of the method there is a large overlap between the simulated results. (a): mean and standard deviation of the estimated α based on 1000 simulations for the three possible β values and six α values. (b): mean and standard deviation of the estimated β based on 1000 simulations for the six possible α values and three β values. (c): mean and standard deviation of the estimated σT based on 1000 simulations. (d): mean and standard deviation of the estimated σM based on 1000 simulations. This figure is published in color in the online version.

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    Simulated string quartet with period and phase correction. The top graphs show the phase and period constants of all four simulated players. The bottom graphs show the different timekeeper standard deviations (σT =10, 13, 15, and 18 for violin 1, violin 2, viola, and cello, respectively) and motor variances (σM = 5, 6, 7, 8). Estimation of 1000 iterations of simulations with nseq = 16, and N = 40 are depicted in circles; error bars show the standard deviation of the estimates. Note the small bias in the estimation of the timekeeper, probably due a lack of sufficient data points in each of the simulated datasets. This figure is published in color in the online version.


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