Weighing the odds: a course in probability and statistics
Weighing the odds: a course in probability and statistics
Quantifying Uncertainty for Non-Gaussian Ensembles in Complex Systems
SIAM Journal on Scientific Computing
Journal of Computational Physics
Mathematical test criteria for filtering complex systems: Plentiful observations
Journal of Computational Physics
Mathematical strategies for filtering complex systems: Regularly spaced sparse observations
Journal of Computational Physics
Journal of Computational Physics
Interpolating Irregularly Spaced Observations for Filtering Turbulent Complex Systems
SIAM Journal on Scientific Computing
Filtering skill for turbulent signals for a suite of nonlinear and linear extended Kalman filters
Journal of Computational Physics
Optimal filtering of complex turbulent systems with memory depth through consistency constraints
Journal of Computational Physics
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Many situations in complex systems require quantitative estimates of the lack of information in one probability distribution relative to another. In short term climate and weather prediction, examples of these issues might involve the lack of information in the historical climate record compared with an ensemble prediction, or the lack of information in a particular Gaussian ensemble prediction strategy involving the first and second moments compared with the non-Gaussian ensemble itself. The relative entropy is a natural way to quantify the predictive utility in this information, and recently a systematic computationally feasible hierarchical framework has been developed. In practical systems with many degrees of freedom, computational overhead limits ensemble predictions to relatively small sample sizes. Here the notion of predictive utility, in a relative entropy framework, is extended to small random samples by the definition of a sample utility, a measure of the unlikeliness that a random sample was produced by a given prediction strategy. The sample utility is the minimum predictability, with a statistical level of confidence, which is implied by the data. Two practical algorithms for measuring such a sample utility are developed here. The first technique is based on the statistical method of null-hypothesis testing, while the second is based upon a central limit theorem for the relative entropy of moment-based probability densities. These techniques are tested on known probability densities with parameterized bimodality and skewness, and then applied to the Lorenz '96 model, a recently developed ''toy'' climate model with chaotic dynamics mimicking the atmosphere. The results show a detection of non-Gaussian tendencies of prediction densities at small ensemble sizes with between 50 and 100 members, with a 95% confidence level.