A survey of generalized inverses and their use in stochastic modelling

In many stochastic models, in particular Markov chains in discrete or continuous time and Markov renewal processes, a Markov chain is present either directly or indirectly through some form of embedding. The analysis of many problems of interest associated with these models, eg. stationary distributions, moments of first passage time distributions and moments of occupation time random variables, often concerns the solution of a system of linear equations involving I – P, where P is the transition matrix of a finite, irreducible, discrete time Markov chain. Generalized inverses play an important role in the solution of such singular sets of equations. In this paper we survey the application of generalized inverses to the aforementioned problems. The presentation will include results concerning the analysis of perturbed systems and the characterization of types of generalized inverses associated with Markovian kernels

Simple procedures for finding mean first passage times in Markov chains

The derivation of mean first passage times in Markov chains involves the solution of a family of linear equations. By exploring the solution of a related set of equations, using suitable generalized inverses of the Markovian kernel I – P, where P is the transition matrix of a finite irreducible Markov chain, we are able to derive elegant new results for finding the mean first passage times. As a by-product we derive the stationary distribution of the Markov chain without the necessity of any further computational procedures. Standard techniques in the literature, using for example Kemeny and Snell’s fundamental matrix Z, require the initial derivation of the stationary distribution followed by the computation of Z, the inverse I – P + eπT where eT = (1, 1, …,1) and πT is the stationary probability vector. The procedures of this paper involve only the derivation of the inverse of a matrix of simple structure, based upon known characteristics of the Markov chain together with simple elementary vectors. No prior computations are required. Various possible families of matrices are explored leading to different related procedures

Bounds on expected coupling times in Markov chains

In the author’s paper “Coupling and Mixing Times in Markov Chains” (RLIMS, 11, 1- 22, 2007) it was shown that it is very difficult to find explicit expressions for the expected time to coupling in a general Markov chain. In this paper simple upper and lower bounds are given for the expected time to coupling in a discrete time finite Markov chain. Extensions to the bounds under additional restrictive conditions are also given with detailed comparisons provided for two and three state chains

Stationary distributions and mean first passage times of perturbed Markov chains

Stationary distributions of perturbed finite irreducible discrete time Markov chains are intimately connected with the behaviour of associated mean first passage times. This interconnection is explored through the use of generalized matrix inverses. Some interesting qualitative results regarding the nature of the relative and absolute changes to the stationary probabilities are obtained together with some improved bounds

Coupling and mixing times in a Markov Chains [sic]

The derivation of the expected time to coupling in a Markov chain and its relation to the expected time to mixing (as introduced by the author in “Mixing times with applications to perturbed Markov chains” Linear Algebra Appl. (417, 108-123 (2006)) are explored. The two-state cases and three-state cases are examined in detail

Markovian queues with correlated arrival processes

In an attempt to examine the effect of dependencies in the arrival process on the steady state queue length process in single server queueing models with exponential service time distribution, four different models for the arrival process, each with marginally distributed exponential interarrivals to the queueing system, are considered. Two of these models are based upon the upper and lower bounding joint distribution functions given by the Fréchet bounds for bivariate distributions with specified marginals, the third is based on Downton’s bivariate exponential distribution and fourthly the usual M/M/1 model. The aim of the paper is to compare conditions for stability and explore the queueing behaviour of the different models

Detection of leukocytes stained with acridine orange using unique spectral features acquired from an image-based spectrometer

A leukocyte differential count can be used to diagnosis a myriad blood disorders, such as infections, allergies, and efficacy of disease treatments. In recent years, attention has been focused on developing point-of-care (POC) systems to provide this test in global health settings. Acridine orange (AO) is an amphipathic, vital dye that intercalates leukocyte nucleic acids and acidic vesicles. It has been utilized by POC systems to identify the three main leukocyte subtypes: granulocytes, monocytes, and lymphocytes. Subtypes of leukocytes can be characterized using a fluorescence microscope, where the AO has a 450 nm excitation wavelength and has two peak emission wavelengths between 525 nm (green) and 650 nm (red), depending on the cellular content and concentration of AO in the cells. The full spectra of AO stained leukocytes has not been fully explored for POC applications. Optical instruments, such as a spectrometer that utilizes a diffraction grating, can give specific spectral data by separating polychromatic light into distinct wavelengths. The spectral data from this setup can be used to create object-specific emission profiles. Yellow-green and crimson microspheres were used to model the emission peaks and profiles of AO stained leukocytes. Whole blood was collected via finger stick and stained with AO to gather preliminary leukocyte emission profiles. A MATLAB algorithm was designed to analyze the spectral data within the images acquired using the image-based spectrometer. The algorithm utilized watershed segmentation and centroid location functions to isolate independent spectra from an image. The output spectra represent the average line intensity profiles for each pixel across a slice of an object. First steps were also taken in processing video frames of manually translated microspheres. The high-speed frame rate allowed objects to appear in multiple consecutive images. A function was applied to each image cycle to identify repeating centroid locations. The yellow-green (515 nm) and crimson (645 nm) microspheres exhibited a distinct separation in colorimetric emission with a peak-to-peak difference of 36 pixels, which is related to the 130 nm peak emission difference. Two AO stained leukocytes exhibited distinct spectral profiles and peaks across different wavelengths. This could be due to variations in the staining method (incubation period and concentration) effecting the emissions or variations in cellular content indicating different leukocyte subtypes. The algorithm was also effective when isolating unique centroids between video frames. We have demonstrated the ability to extract spectral information from data acquired from the image-based spectrometer of microspheres, as a control, and AO stained leukocytes. We determined that the spectral information from yellow-green and crimson microspheres could be used to represent the wavelength range of AO stained leukocytes, thus providing a calibration tool. Also, preliminary spectral information was successfully extracted from yellow-green microspheres translated under the linear slit using stationary images and video frames, thus demonstrating the feasibility of collecting data from a large number of objects

Common trends, cointegration and competitive price behaviour

This article describes a characterisation of competitive market behaviour using the concepts of cointegration analysis. It requires all (n) rms to set prices to follow a single stochastic trend (equivalently the vector of n prices should have cointegrating rank n - 1). This implies that, in the long run, prices are driven by the shocks that impact on all companies, ruling out the possibility that the price set by any one rm is weakly exogenous