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.
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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).