Smoothing-independent estimation of a linear differential equation model

Change over time has been addressed in many fields using differential equation modeling and dynamical systems concepts. Application of these methodologies to much of psychology has been difficult, in part due to the presence of significant amounts of measurement error, relatively slow sampling rates and difficulties associated with collecting long time series. Currently several methods can estimate derivatives for time series with measurement error. Although these methods can produce unbiased derivative estimates, parameters of differential equation models fit to univariate time series are frequently biased. The degree of the bias frequently depends on parameters that smooth time series data, balancing the need to reduce error without obscuring true changes over time. The current work examines estimation of a linear oscillator by modeling data that is simultaneously smoothed at multiple time scales. By doing so, bias that occurs when estimating the linear oscillator model using differential equation modeling and univariate time series can be eliminated. The concepts regarding smoothing seem likely to be applicable to many contexts in which parameter bias is related to the amount time series data have been smoothed.