In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF) The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems The paper also includes the derivation of a square-root version of the CKF for improved numerical stability The CKF is tested experimentally in two nonlinear state estimation problems In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable The second problem addresses the use of the CKF for tracking a maneuvering aircraft The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters

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Cubature Kalman Filters

Maximum likelihood estimation of item parameters in the marginal distribution, integrating over the distribution of ability, becomes practical when computing procedures based on an EM algorithm are used By characterizing the ability distribution empirically, arbitrary assumptions about its form are avoided The Em procedure is shown to apply to general item-response models lacking simple sufficient statistics for ability This includes models with more than one latent dimension

Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm.

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

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Calculation of Gauss quadrature rules

Preface to the classics edition Preface Acknowledgments Part I. The Method of Weighted Residuals: 1. Introduction 2. Boundary-value problems in heat and mass transfer 3. Eigenvalue and initial-value problems in heat and mass transfer 4. Applications to fluid mechanics 5. Chemical reaction systems 6. Convective instability problems Part II. Variational Principles: 7. Introduction to variational principles 8. Variational principles in fluid mechanics 9. Variational principles for heat and mass transfer problems 10. On the search for variational principles 11. Convergence and error bounds Author index Subject index.

The method of weighted residuals and variational principles

The partial credit model (PCM) with a varying slope parameter is developed and called the generalized partial credit model (GPCM). The item step parameter of this model is decomposed to a location and a threshold parameter, following Andrich's (1978) rating scale formulation. The EM algorithm for estimating the model parameters is derived. The performance of this generalized model is compared on both simulated and real data to a Rasch family of polytomous item response models. Simulated data were generated and then analyzed by the various polytomous item response models. The results demonstrate that the rating formulation of the GPCM is quite adaptable to the analysis of polytomous item responses. The real data used in this study consisted of the National Assessment of Educational Progress (Johnson & Allen, 1992) mathematics data that used both dichotomous and polytomous items. The PCM was applied to these data using both constant and varying slope parameters. The GPCM, which provides for varying slope pa...

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Arthur H Stroud

Papers

Gaussian quadrature formulas