  The unscented transform uses a weighted set of samples called sigma points to
propagate the means and covariances of nonlinear transformations of random
variables. However, unscented transforms developed using either the Gaussian
assumption or a minimum set of sigma points typically fall short when the
random variable is not Gaussian distributed and the nonlinearities are
substantial. In this paper, we develop the generalized unscented transform
(GenUT), which uses adaptable sigma points that can be positively constrained,
and accurately approximates the mean, covariance, and skewness of an
independent random vector of most probability distributions, while being able
to partially approximate the kurtosis. For correlated random vectors, the GenUT
can accurately approximate the mean and covariance. In addition to its superior
accuracy in propagating means and covariances, the GenUT uses the same order of
calculations as most unscented transforms that guarantee third-order accuracy,
which makes it applicable to a wide variety of applications, including the
assimilation of observations in the modeling of the coronavirus (SARS-CoV-2)
causing COVID-19.
