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Overshoots and undershoots of Lévy Processes
Get PDFWe obtain a new fluctuation identity for a general Lévy process
giving a quintuple law describing the time of first passage, the time of the
last maximum before first passage, the overshoot, the undershoot and the
undershoot of the last maximum. With the help of this identity, we revisit
the results of Klüppelberg et al. (2004) concerning asymptotic overshoot
distribution of a particular class of Lévy processes with semi-heavy tails
and refine some of their main conclusions. In particular we explain how
different types of first passage contribute to the form of the asymptotic
overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that
the underlying Lévy process is spectrally one sided
Large Modelling Conditional Covariance in the Linear Mixed Model
Get PDFWe provide a data-driven method for modelling the conditional, within
subject, covariance matrix arising in linear mixed models (Laird and Ware, 1982).
Given an agreed structure for the between subject covariance matrix we use a regression equation approach to model the within subject covariance matrix. Using an EM
algorithm we estimate all of the parameters in the model simultaneously and obtain
analytical expressions for the standard errors. By re-analyzing Kenward's (1987) cattle
data, we compare our new model with classical menu-selection-based modelling techniques, demonstrating its superiority using the Bayesian Information Criterion (BIC).
We also conduct a simulation study which confirms our observational findings. The paper extends our previous covariance modelling work (Pan and MacKenzie, 2003, 2006)
to the conditional covariance space of the linear mixed model (LMM)
Identification of Vessels from Engine Sounds by Spectral Comparison and Verification
No full textWe consider the problem of identifying a vessel from its engine sound. A database of known vessels is used for comparison of acoustic characteristics in the frequency domain. Only one training sample is required from each class for classification. With more training samples, a vessel is not only classified but also verified whether it belongs to the same class as its closest match in the database. The acoustic signals are regarded as stationary time series with mixed spectra. We boost their power by adding a small amount of noise in order to assist classification. Three nonparametric estimators of spectra
are compared in classification and verification experiments. The case of a vessel not being represented in the database is also considered with the intention to have it rejected at the verification stage
Automatic semigroups and categories
No full textWe consider various automata-theoretic properties of semigroupoids and small categories and their relationship to the corresponding properties in semigroups and monoids. We introduce natural definitions of finite automata and regular languages over finite graphs, generalising the usual notions over finite alphabets. These allow us to introduce a definition of automaticity for semigroupoids and small categories, which generalises those introduced for semigroups by Hudson and for groupoids by Epstein. We also introduce a definition of prefix-automaticity for semigroupoids and small categories, generalising that for certain monoids introduced by Silva and Steinberg.
We study the relationship between automaticity properties in a semigroupoid and in a certain associated semigroup. This allows us to extend to semigroupoids and small categories a number of results about automatic and prefix-automatic semigroups and monoids. In the course of our study, we also prove some new results about automaticity and prefix-automaticity in semigroups and monoids. These include the fact that prefix-automaticity is preserved under the taking of cofinite subsemigroups
LAPACK-Style Codes for Pivoted Cholesky and QR Updating
Get PDFRoutines exist in LAPACK for computing the Cholesky
factorization of a symmetric positive definite
matrix and in LINPACK there is a pivoted routine for positive semidefinite matrices. We present new higher level BLAS LAPACK-style codes for computing this pivoted factorization. We show that these can be many times faster than the LINPACK code. Also, with a new stopping criterion, there is more reliable rank detection and smaller normwise backward error. We also present algorithms that update the QR factorization of a matrix after it has had a block of rows or columns added or a block of columns deleted.
This is achieved by updating the factors Q and R of the original matrix. We present some LAPACK-style codes and show these can be much faster than computing the factorization from scratch
Pattern selection by a granular wave in a rotating drum
No full textThe results of an experimental investigation of granular segregation in a thin rotating drum are presented. A mechanism based on the presence of an uphill wave of particles has been found to govern the observed pattern of petals. Specifically we develop a simple model that captures the essential physics of the segregation and yields an algebraic expression that predicts the number of petals in the pattern
The problem of differentiation of an Abelian function over its parameters
Get PDFTheory of Abelian functions was a central topic of the 19th century mathematics. In mid-seventies of the last century a new wave arose of investigation in this field in response to the discovery that Abelian functions provide solutions of a number of challenging problems of modern Theoretical and Mathematical Physics.
In a cycle of our joint papers published in 2000–05, we have developed a theory of multivariate sigma-function, an analogue of the classic Weierstrass sigma-function. A sigma-function is defined on a cover of U , where U is the space of a bundle p : U → B defined by a family of plane algebraic curves of fixed genus. The base B of the bundle is the space of the family parameters and a fiber J_b over
b ∈ B is the Jacobi variety of the curve with the parameters b. A second logarithmic derivative of
the sigma-function along the fiber is an Abelian
function on J_b.
Thus, one can generate a ring F of fiber-wise Abelian functions on U. The problem to find derivations of the ring F along the base B is a reformulation of the classic problem of differentiation of Abelian functions over parameters. Its solution is relevant to a number of topical applications.
This work presents a solution of this problem recently found by the authors. Our method of solution essentially employs the results from Singularity Theory about vector fields tangent to the discriminant of a singularity y^n -x^s, gcd(n, s) = 1
Elastic wave radiation from a high frequency finite-length transducer
No full textA simple model of a high frequency piezoelectric transducer affixed to an elastic half-space is analysed. The problem is reformulated as a modified matrix Wiener–Hopf equation, containing a kernel for which there is no known exact factorisation. An approximate factorisation is obtained, and the resulting integral equation is solved by iteration, in limit that the length of the transducer is very much larger than a typical wavelength
Stochastic Resonance in Vision: Models and Data
No full textStochastic resonance (SR) is a phenomenon whereby small amounts of additive noise can greatly enhance the performance of a non-linear signal processing system. It is well known that many sensory systems are non-linear in nature and the phenomenon of SR has been widely studied in them. Perhaps one underexplored area of research is the visual system, in particular the retina and this pro ject aims to explore to what extent SR is or could be utilised in the retina. The thesis is organised as follows: In chapter one I look at the historical development of SR from its conception as an explanation of the periodic occurrence of ice ages to applications in sensory systems. The structure of the retina is explored in chapter two and I explain why SR might be expected to occur in visual systems. The mathematics of the problem is detailed in chapter three, showing why SR arises in systems containing a threshold type non-linearity. The phenomenon is extensively studied experimentally in chapter four, with many different stimuli being tested. These are broadly organised into two types of task, detection and discrimination, with each stimulus carefully designed to probe a different aspect of the phenomenon. In the final chapter overall conclusions are drawn, the scope for future work is explored, including the description of a retina model and the possibilities for practical applications are raised.
The thesis contains novel receiver operating characteristic (ROC) curves, used to demonstrate that SR can occur in systems containing a threshold type non-linearity. I go on to show that noise can be used to enhance the perceptibility of some signals. However, this enhanced perception is shown to be limited to low level visual tasks