Estimating Dynamical Systems: Derivative Estimation Hints From Sir Ronald A. Fisher

Area of Science:

• Psychology
• Statistics
• Dynamical Systems Analysis

Background:

• Fitting dynamical systems to psychological data allows novel questions about human change over time.
• Estimating derivatives of time series is key, but Local Linear Approximation (LLA) introduces correlated errors.
• Correlated errors from LLA can severely bias differential equation model parameter estimates.

Purpose of the Study:

• To improve the fitting of dynamical systems to psychological data.
• To present a novel method for estimating derivatives that avoids the correlated errors of LLA.
• To demonstrate the benefits of this new method in estimating differential equation model parameters.

Main Methods:

• Developed a novel derivative estimation method inspired by orthogonal polynomial fitting.
• Compared the proposed method against a generalized Local Linear Approximation (LLA).
• Applied both methods to simulated data for estimating derivatives and differential equation parameters.

Main Results:

• The proposed method demonstrated improved derivative estimation compared to LLA.
• Differential equation model parameter estimates were less biased using the new method.
• Successfully applied the method to real-world data, estimating oscillation frequency in UK mortality data.

Conclusions:

• The novel derivative estimation technique enhances dynamical systems fitting in psychological research.
• This method offers a more accurate approach to analyzing time series data and modeling change.
• The provided R functions facilitate the application of this improved methodology.