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.

The method of weighted residuals and variational principles : with application in fluid mechanics, heat and mass transfer

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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A Generalized Partial Credit Model: Application of an EM Algorithm

A multivariate logistic latent trait model for items scored in two or more nominal categories is proposed. Statistical methods based on the model provide 1) estimation of two item parameters for each response alternative of each multiple choice item and 2) recovery of information from “wrong” responses when estimating latent ability. An application to a large sample of data for twenty vocabulary items shows excellent fit of the model according to a chi-square criterion. Item and test information curves are compared for estimation of ability assuming multiple category and dichotomous scoring of these items. Multiple scoring proves substantially more precise for subjects of less than median ability, and about equally precise for subjects above the median.

Estimating item parameters and latent ability when responses are scored in two or more nominal categories

Gaussian Quadrature Formulas

where the As are constants and the Pi(Pil , Pin) are points in the, space. Each formula is exact for all polynomials up to and including a specified degree k where, as usual, k is called the degree of the formula. In Section 2 we give formulas of degrees 2, 3, 5 and 7 which are similar to previously developed formulas for other regions. The formulas of degree 2 are discussed by Stroud [14] for an arbitrary region; the formulas of degrees 3, 5 and 7 are similar to formulas for spheres given by Hammer and Stroud [4] and Ditkin [3]. In Sections 3, 4 and 5 we develop spherical product type formulas of arbitrarily high degree for U and V. These formulas have degree 2h 1 (h 1, 2, . ) and use hn points for even h and h' h -1+ 1 points for odd h. These formulas are obtained by products of one-dimensional formulas and are similar to formulas for the circle and 3-sphere given by Peirce [10, 11] and recently extended to the nsphere by Hetherington [6]. The spherical product formulas are most useful in 2 and 3 dimensions since in higher dimensions the numbers of points in the formulas become very large. In Section 4 we tabulate one-dimensional formulas, which are particular to the integrals U and V, from which product formulas for n = 2, 3, 4 may be constructed. From the formulas we give for U and V we can obtain formulas for any integrals of the form

A. H. Stroud

Papers

Gaussian Quadrature Formulas

Approximate integration formulas for certain spherically symmetric regions