sigmapoint

Sigma point, also written sigmapoint, is a deterministically chosen sample point used in probabilistic estimation of nonlinear systems. It represents a probability distribution, typically Gaussian, by capturing its mean and covariance under nonlinear transformations. The concept lies at the heart of the Unscented Transform and related sigma-point Kalman filters, providing an alternative to linearization-based methods such as the extended Kalman filter.

In a common construction for an n-dimensional state with mean vector x and covariance matrix P, a set of 2n+1 sigma points is used. One point is the mean x0 = x, and the remaining points are placed symmetrically around the mean using the square root of (n+λ)P, where λ = α^2(n+κ) − n and α, κ are scaling parameters. Specifically, for i = 1 to n, xi = x + [sqrt((n+λ)P)]_i and x_{i+n} = x − [sqrt((n+λ)P)]_i. Each point carries an associated weight, and the mean and covariance of the transformed distribution are approximated by propagating the sigma points through a nonlinear function f and forming weighted sums of the transformed points.

Sigma points are propagated through the system’s nonlinear model to approximate the posterior statistics without explicit linearization. This approach is used in the Unscented Kalman Filter and related methods, and has variants such as the Cubature Kalman Filter, which uses a different, often 2n-point, scheme. The method is widely applied in robotics, navigation, signal processing, and data assimilation because it can offer higher accuracy than linearization for moderately nonlinear systems while remaining computationally efficient compared with full Monte Carlo methods.

Limitations include sensitivity to the choice of scaling parameters and diminished accuracy when the true distribution deviates strongly from Gaussian assumptions.