Verified inclusions for a nearest matrix of specified rank deficiency via a generalization of Wedin’s $$\sin (\theta )$$ theorem

2020 ◽  
Author(s):  
Marko Lange ◽  
Siegfried M. Rump
Keyword(s):  
Rank Deficiency

scholarly journals $$\ell _2$$-penalized approximate likelihood inference in logit mixed models for regional prevalence estimation under covariate rank-deficiency

Metrika ◽  
2021 ◽  
Author(s):  
Joscha Krause ◽  
Jan Pablo Burgard ◽  
Domingo Morales
Keyword(s):  
Mixed Model ◽  
Mean Squared Error ◽  
Routine Data ◽  
Parametric Bootstrap ◽  
Monte Carlo Simulation Study ◽  
Rank Deficiency ◽  
Prevalence Estimation ◽  
Fitting Algorithm ◽  
Individual Contributions ◽  
The Individual

AbstractRegional prevalence estimation requires the use of suitable statistical methods on epidemiologic data with substantial local detail. Small area estimation with medical treatment records as covariates marks a promising combination for this purpose. However, medical routine data often has strong internal correlation due to diagnosis-related grouping in the records. Depending on the strength of the correlation, the space spanned by the covariates can become rank-deficient. In this case, prevalence estimates suffer from unacceptable uncertainty as the individual contributions of the covariates to the model cannot be identified properly. We propose an area-level logit mixed model for regional prevalence estimation with a new fitting algorithm to solve this problem. We extend the Laplace approximation to the log-likelihood by an $$\ell _2$$ ℓ 2 -penalty in order to stabilize the estimation process in the presence of covariate rank-deficiency. Empirical best predictors under the model and a parametric bootstrap for mean squared error estimation are presented. A Monte Carlo simulation study is conducted to evaluate the properties of our methodology in a controlled environment. We further provide an empirical application where the district-level prevalence of multiple sclerosis in Germany is estimated using health insurance records.

Domains of Interference Between Working Bodies in Mechanisms and Manipulators

1995 ◽  
Author(s):  
Edward J. Haug ◽  
Frederick A. Adkins ◽  
Chi-Mei Luh ◽  
Jia-Yi Wang
Keyword(s):  
Numerical Methods ◽  
Jacobian Matrix ◽  
Underlying Mechanism ◽  
Rank Deficiency ◽  
Ground Vehicle ◽  
Complementary Problems ◽  
Stewart Platforms ◽  
Working Bodies ◽  
Set Of Points

Abstract Criteria for the set of all points in a pair of working bodies in a mechanism or manipulator that can coincide for any kinematically admissible configuration of the underlying mechanism, called the domain of interference between the bodies, are formulated. Kinematic equations for the mechanism and parameterizations of the domains of the working bodies are used to derive analytical criteria for domains of interference. Three complementary problems are formulated and analyzed to characterize (1) the set of points in one of the interfering bodies that are occupied by any point in the second body, (2) the set of points in one of the interfering bodies that are occupied by any point on the boundary of the second body, and (3) the set of all points in space that are simultaneously occupied by points in the interfering bodies; each condition occurring for any kinematically admissible configuration of the mechanism. Analytical criteria for the boundaries of domains of interference for each of the three problems arc derived, based on row-rank deficiency of a sub-Jacobian matrix associated with the kinematic equations for each of the problems. Numerical methods for mapping boundaries of domains of interference are presented and illustrated for planar Stewart platforms with domes attached that are characteristic of flight or ground vehicle simulators.

Row and Column Inclusion and the Distribution of Rank

2011 ◽  
Author(s):  
Shaun M. Fallat ◽  
Charles R. Johnson
Keyword(s):  
Detailed Account ◽  
Related Phenomenon ◽  
Rank Deficiency ◽  
Matrix Rank ◽  
General Matrix ◽  
Important Notion

This chapter contains a detailed account of the distribution of rank deficient submatrices within a TN matrix, including a discussion of the important notion of row and column inclusion. The distribution of ranks among submatrices of a TN matrix is much less free than in a general matrix. Rank deficiency of submatrices in certain positions requires rank deficiency elsewhere. Whereas, in the case of general matrices, rank deficiency of a large submatrix can imply rank deficiency of smaller, included submatrices, in the TN case rank deficiency of small submatrices can imply that of much larger ones. The chapter discusses this and other related phenomenon.

Workspace of a Gimbal With Joint Limits

1996 ◽  
Author(s):  
Har-Jou Yeh ◽  
Karim A. Abdel-Malek
Keyword(s):  
Degrees Of Freedom ◽  
Spherical Surface ◽  
Perturbation Technique ◽  
Underlying Mechanism ◽  
Singular Curve ◽  
Constraint Function ◽  
Rank Deficiency ◽  
Analytical Criterion ◽  
Joint Limits ◽  
Three Degrees Of Freedom

Abstract An analytical formulation for determining the workspace of a point on a body suspended in a Gimbal mechanism is presented. Although the gimbal mechanism comprises three degrees of freedom, the resulting workspace is a region on a spherical surface. The constraint function of the underlying mechanism is studied for singularities using a row-rank deficiency condition of its constraint Jacobian. Singular curves on the resultant spherical surface are determined by a similar analytical criterion imposed on the system’s subjacobian, to compute a set of two joint singularities. These singular curves define regions on the spherical surface that may or may not be accessible. A perturbation technique is then used to identify singular curve segments that are boundary to the workspace region. The methodology is illustrated through a numerical example.

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