A Method of Correcting Estimation Failure in Latent Differential Equations with Comparisons to Kalman Filtering

Abstract

Studies have used the latent differential equation (LDE) model to estimate the parameters of damped oscillation in various phenomena, but it has been shown that correct, non-zero parameter estimates are only obtained when the latent series exhibits little or no process noise. Consequently, LDEs are limited to modeling deterministic processes with measurement error rather than those with random behavior in the true latent state. The reasons for these limitations are considered, and a piecewise deterministic approximation (PDA) algorithm is proposed to treat process noise outliers as functional discontinuities and obtain correct estimates of the damping parameter. Comprehensive, random-effects simulations were used to compare results with those obtained using a state-space model (SSM) based on the Kalman filter. The LDE with the PDA algorithm (LDEPDA) successfully recovered the simulated damping parameter under a variety of conditions when process noise was present in the latent state. The LDEPDA had greater precision and accuracy than the SSM when estimating parameters from data with sparse jump discontinuities, but worse performance for diffusion processes overall. All three methods were applied to a sample of postural sway data. The basic LDE estimated zero damping, while the LDEPDA and SSM estimated moderate to high damping. The SSM estimated the smallest standard errors for both frequency and damping parameter estimates.

Keywords:

• Differential equation models
• structural equation modeling
• latent differential equation
• Kalman filtering
• time series analysis

Notes

1 The OpenMx default control options for Nelder–Mead include $\alpha =1,\mathit{bet}{a}_o=0.5,{\beta }_i=0.5,\gamma =2,\sigma =0.5.$

2 Equivalent to $2\pi$ times the natural frequency, as data were generated using the natural frequency.