The Composite Covariance of a Kalman Filter Estimation

A Kalman filter estimation of the state of a system is merely a random vector that has a normal, also called Gaussian, distribution. Elementary statistics teaches any Gaussian distribution is completely and uniquely characterized by its mean and covariance (variance if univariate). Such characterization is required for statistical inference problems on a Gaussian random vector. This mean and composite covariance of a Kalman filter estimate of a system state will be derived here. The derived covariance is in recursive form. One must not confuse it with the “error covariance” output of a Kalman filter. Potential applications, including geological ones, of the derivation are described and illustrated with a simple example.

Reliability analysis of geogrid material with random nonlinear viscoelastic characteristics

Introduction. The behavior in the course of a time of geogrid material with random nonlinear viscoelastic characteristics under tension is analysed. Parameters of viscoelasticity are represented in form of Gaussian random vector. The components of this vector are taken from experimental data. Aim of the research. The objective of this research is the analysis of influence of different factors (value of applied load and the application of load in the form of random value instead of dead one, number of realizations, change of given level of strain) on providing of needed service life of geogrid material with given reliability level. Here reliability is interpreted as function of probability of non-failure. The first crossing of some given level by random strain is considered as a failure. The strain value corresponding to yield limit of geogrid material is accepted as the given level of longitudinal strain. Methods. The realizations of Gaussian random vector of viscoelastic parameters of material with given correlation matrix were imitated by means of linear transformation method. Results. It is demonstrated that longitudinal strain is Gaussian nonstationary random process which stochastic analysis can be made on base of 10 000 realizations. The dependencies on time of mathematical expectation and standard deviation of random longitudinal strain as well as function of probability of non-failure are found. Conclusion. It is shown that durability estimation found on base of the deterministic problem solution is overestimated in comparison with stochastic problem solution if the condition of given service life providing with some reliability level is set up.

Cubature rules and expected value of some complex functions

The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for nite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments are determined.

On the Geometry and Shape of Brain Sub-Manifolds

This paper develops mathematical representations for neuro-anatomically significant substructures of the brain and their variability in a population. The focus of the paper is on the neuro-anatomical variation of the geometry and the "shape" of two-dimensional surfaces in the brain. As examples, we focus on the cortical and hippocampal surfaces in an ensemble of Macaque monkeys and human MRI brains. The "shapes" of the substructures are quantified via the construction of templates; the variations are represented by defining probabilistic deformations of the template. Methods for empirically estimating probability measures on these deformations are developed by representing the deformations as Gaussian random vector fields on the embedded sub-manifolds. The Gaussian random vector fields are constructed as quadratic mean limits using complete orthonormal bases on the sub-manifolds. The complete orthonormal bases are generated using modes of vibrations of the geometries of the brain sub-manifolds. The covariances are empirically estimated from an ensemble of brain data. Principal component analysis is presented for characterizing the "eigen-shape" of the hippocampus in an ensemble of MRI-MPRAGE whole brain images. Clustering based on eigen-shape is presented for two sub-populations of normal and schizophrenic.