The novel classes of finite dimensional filters with non-maximal rank estimation algebra on state dimension four and rank of one

2019 ◽  
pp. 1-10 ◽  
Author(s):  
Wenhui Dong ◽  
Xiuqiong Chen ◽  
Stephen S.-T. Yau
Keyword(s):  
Maximal Rank ◽  
The Novel ◽  
Rank Estimation ◽  
Finite Dimensional ◽  
Estimation Algebra

On generalized inverses of matrices

1966 ◽  
Vol 62 (4) ◽  
pp. 673-677 ◽  
Author(s):  
M. H. Pearl
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Maximal Rank ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Finite Dimensional ◽  
The Matrix ◽  
Generalized Inverses Of Matrices ◽  
Necessary And Sufficient

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

scholarly journals MCMC METHODS FOR DIFFUSION BRIDGES

2008 ◽  
Vol 08 (03) ◽  
pp. 319-350 ◽  
Author(s):  
ALEXANDROS BESKOS ◽  
GARETH ROBERTS ◽  
ANDREW STUART ◽  
JOCHEN VOSS
Keyword(s):  
Mcmc Methods ◽  
Recent Theory ◽  
The Novel ◽  
Infinite Dimensional ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
It Diffusion ◽  
Dimensional Approximation ◽  
Independence Sampler ◽  
Random Walk Metropolis Algorithm

We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parametrised by θ ∈ [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

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