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The D-optimal design of blocked experiments with mixture components
No full textSo far, the optimal design of blocked experiments involving mixture components has received scant
attention in the literature. This paper describes the algorithmic approach to designing such experiments.
For constrained and unconstrained experimental regions, the resulting experimental designs are shown to
be statistically much more efficient than the orthogonally blocked design options presented in the literature.
As an alternative to the algorithmic approach, a simple two-stage procedure to construct highly efficient
blocked mixture experiments for unconstrained design regions in the presence of fixed and/or random blocks
is presented. Finally, the similarities and differences between the design of blocked mixture experiments
and mixture experiments in the presence of qualitative process variables are discussed in detail
Structured Factorizations in Scalar Product Spaces
Get PDFLet belong to an automorphism group, Lie algebra, or Jordan algebra of a scalar product. When is factored, to what extent do the factors inherit structure from ? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general , we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors, and singular values that persists across a wide range of scalar products.
A key feature of our analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of different characterizations of these scalar product classes are given
Higher Order Cumulants of Random Vectors and Applications to Statistical Inference and Time Series
Get PDFThis paper provides a unified and comprehensive approach that is useful in deriving expressions for higher order cumulants of random vectors. The use of this methodology is then illustrated in three
diverse and novel contexts, namely: (i) in obtaining a lower bound (Bhattacharya bound) for the variance-
covariance matrix of a vector of unbiased estimators where the density depends on several parameters,
(ii) in studying the asymptotic theory of multivariable statistics when the population is not necessarily
Gaussian and finally, (iii) in the study of multivariate nonlinear time series models and in obtaining higher
order cumulant spectra. The approach depends on expanding the characteristic functions and cumulant
generating functions in terms of the Kronecker products of di¤erential operators. Our objective here is
to derive such expressions using only elementary calculus of several variables and also to highlight some
important applications in statistics
Testing Nonstationary Time Series for Gaussianity and Linearity using the Evolutionary Bispectrum: An application to Internet Traffic Data
Get PDFWe propose statistical tests for Gaussianity and linearity of nonstationary time series based on the evolutionary bispectrum. These tests can be applied to a particular subclass of nonstationary processes, the so-called oscillatory (also known as slowly-varying) processes. We then apply these tests to time series of network measurements arising from internet traffic. Recent works by several researchers have demonstrated that such internet trafic processes are typically nonstationary. Also, the question of whether such processes can be described by some model whose parameters vary with time has been raised and studied at some length. We use the tests developed in this paper to show that there is evidence of non Gaussianity and nonlinearity in such processes uner the assumption that they are described by a model whose parameters (and so its spectral characteristics) vary slowly with time
Numerical Solution of the Equations Governing 2D Eulerian Gas-Solid Two-Phase Flow
Get PDFThis paper extends the 1D model discussed in Hudson &
Harris (MIMS EPrint 2006.10) to 2D. A high resolution scheme is used to discretise two formulations of the 2D model and a variety of test cases are investigated to determine how the model behaves and the accuracy of the scheme
Backward Stochastic Partial Differential Equations with Jumps and Application to Optimal Control of Random Jump Fields
Get PDFWe prove an existence and uniqueness result for a general class of backward stochastic
partial differential equations with jumps. This is a type of equations which appear as adjoint
equations in the maximum principle approach to optimal control of systems described by
stochastic partial differential equations driven by Levy processes
A Functional Limit Theorem for Random Walk Conditioned to Stay Non-negative
Get PDFIn this paper we consider an aperiodic integer-valued random walk S and a
process S* which is an harmonic transform of S killed when it first enters the
negative half; informally S* is "S conditioned to stay non-negative". If S is in
the domain of attraction of the standard Normal law, without centring, a suitably normed and linearly interpolated version of S converges weakly to standard
Brownian motion, and our main result is that under the same assumptions a corresponding statement holds for S*; the limit of course being the 3-dimensional
Bessel process. Since this process can be thought of as Brownian motion conditioned to stay non-negative, in essence we our result shows that the interchange
of the two limit operations is valid.
We also establish some related results, including a local limit theorem for S*;
and a bivariate renewal theorem for the ladder time and height process, which
may be of independent interest
Cache Efficient Bidiagonalization Using BLAS 2.5 Operators
Get PDFOn cache based computer architectures using current standard algorithms, Householder bidiagonalization requires a significant portion of the execution time for computing matrix singular values and vectors In this paper we reorganize the sequence of operations for Householder bidiagonalization of a general m × n matrix, so that two
(_GEMV) vector-matrix multiplications can be done with one pass of the unreduced trailing part of the matrix through cache. Two new BLAS 2.5 operations approximately cut in half the transfer of data from main memory to cache. We give detailed algorithm descriptions and compare timings with the current LAPACK bidiagonalization algorithm
Structured Linearizations for Matrix Polynomials
Get PDFThe classical approach to investigating polynomial eigenvalue problems is linearization, where the underlying matrix polynomial is converted into a larger matrix pencil
with the same eigenvalues. For any polynomial there are infinitely many linearizations with widely varying properties, but in practice the companion forms are typically used. However, these companion forms are not always entirely satisfactory, and
linearizations with special properties may sometimes be required.
Given a matrix polynomial P, we develop a systematic approach to generating
large classes of linearizations for P. We show how to simply construct two vector
spaces of pencils that generalize the companion forms of P, and prove that almost all
of these pencils are linearizations for P. Eigenvectors of these pencils are shown to
be closely related to those of P. A distinguished subspace, denoted DL(P), is then
isolated, and the special properties of these pencils are investigated. These spaces of
pencils provide a convenient arena in which to look for structured linearizations of
structured polynomials, as well as to try to optimize the conditioning of linearizations.
Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial; perhaps the most well-known are symmetric and
Hermitian polynomials. In this thesis we also identify several less well-known types
of structured polynomial (e.g., palindromic, even, odd), explore the relationships
between them, and illustrate their appearance in a variety of applications. Special
classes of linearizations that respect the structure of these polynomials, and therefore
preserve symmetries in their spectra, are introduced and investigated. We analyze
the existence and uniqueness of such linearizations, and show how they may be systematically constructed.
The infinitely many linearizations of any given polynomial P can have widely
varying eigenvalue condition numbers. We investigate the conditioning of linearizations from DL(P), looking for the best conditioned linearization in that space and
comparing its conditioning with that of the original polynomial. We also analyze the
eigenvalue conditioning of the widely used first and second companion linearizations,
and find that they can potentially be much more ill conditioned than P. Our results
are phrased in terms of both the standard relative condition number and the condition number of Dedieu and Tisseur for the problem in homogeneous form, this latter
condition number having the advantage of applying to zero and infinite eigenvalues
Quasi-Monte Carlo estimation in generalized linear mixed models
Get PDFGeneralized linear mixed models (GLMMs) are useful for modelling longitudinal and clustered data, but parameter estimation is very challenging because the likelihood may involve high-dimensional integrals that are analytically intractable. Gauss–Hermite quadrature (GHQ) approximation can be applied but is only suitable for low-dimensional random effects. Based on the Quasi-Monte Carlo (QMC) approximation, a heuristic approach is proposed to calculate the maximum likelihood estimates of parameters in the GLMM. The QMC points scattered uniformly on the high-dimensional integration domain are generated to replace the GHQ nodes. Compared to the GHQ approximation, the proposed method has many advantages such as its affordable computation, good approximation and fast convergence. Comparisons to the penalized quasi-likelihood estimation and Gibbs sampling are made using a real dataset and a simulation study. The real dataset is the salamander mating dataset whose modelling involves six 20-dimensional intractable integrals in the likelihood