Unscented Kalman filter (UKF)

The unscented Kalman filter also targets nonlinear models but avoids Jacobians. It deterministically chooses a small set of sigma points that match the mean and covariance of the current estimate, pushes each one through the true nonlinear function, and reconstructs the transformed mean and covariance from the propagated points (the unscented transform). This captures the effect of the nonlinearity to higher order than the EKF's single tangent-plane approximation and is often more accurate and more stable, at comparable computational cost, without requiring any derivatives.

In BCI the UKF became notable for decoders that use an expressive nonlinear tuning model and, in some formulations, incorporate several time lags of the kinematic state, letting firing rate depend nonlinearly on position, velocity, and their recent history. It remains a fully approximate filter — it still summarizes the posterior by a mean and covariance and is not exact for strongly non-Gaussian posteriors — but it is the usual first choice when the observation model's nonlinearity matters and a particle filter would be too costly for real-time use.