Practical stability of approximating discrete-time filters with respect to model mismatch using relative entropy concepts

This paper establishes practical stability results for an important range of approximate discrete-time filtering problems involving mismatch between the true system and the approximating filter model. Using local consistency assumption, the practical stability established is in the sense of an asymptotic bound on the amount of bias introduced by the model approximation. Significantly, these practical stability results do not require the approximating model to be of the same model type as the true system. Our analysis applies to a wide range of estimation problems and justifies the common practice of approximating intractable infinite dimensional nonlinear filters by simpler computationally tractable filters.

Practical stability with respect to model mismatch of approximate discrete-time output feedback control

This paper establishes a practical stability result for discrete-time output feedback control involving mismatch between the exact system to be stabilised and the approximating system used to design the controller. The practical stability is in the sense of an asymptotic bound on the amount of error bias introduced by the model approximation, and is established using local consistency properties of the systems. Importantly, the practical stability established here does not require the approximating system to be of the same model type as the exact system. Examples are presented to illustrate the nature of our practical stability result.

Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.

A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems

Biochemical reactions underlying genetic regulation are often modelled as a continuous-time, discrete-state, Markov process, and the evolution of the associated probability density is described by the so-called chemical master equation (CME). However the CME is typically difficult to solve, since the state-space involved can be very large or even countably infinite. Recently a finite state projection method (FSP) that truncates the state-space was suggested and shown to be effective in an example of a model of the Pap-pili epigenetic switch. However in this example, both the model and the final time at which the solution was computed, were relatively small. Presented here is a Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines. This allows larger-scale models to be studied and solutions for larger final times to be computed in a realistic execution time. Additionally the new method computes the solution at intermediate times at virtually no extra cost, since it is derived from Krylov-type methods for computing matrix exponentials. For the purpose of comparison the new algorithm is applied to the model of the Pap-pili epigenetic switch, where the original FSP was first demonstrated. Also the method is applied to a more sophisticated model of regulated transcription. Numerical results indicate that the new approach is significantly faster and extendable to larger biological models.

Accelerated leap methods for simulating discrete stochastic chemical kinetics

Biologists are increasingly conscious of the critical role that noise plays in cellular functions such as genetic regulation, often in connection with fluctuations in small numbers of key regulatory molecules. This has inspired the development of models that capture this fundamentally discrete and stochastic nature of cellular biology - most notably the Gillespie stochastic simulation algorithm (SSA). The SSA simulates a temporally homogeneous, discrete-state, continuous-time Markov process, and of course the corresponding probabilities and numbers of each molecular species must all remain positive. While accurately serving this purpose, the SSA can be computationally inefficient due to very small time stepping so faster approximations such as the Poisson and Binomial τ-leap methods have been suggested. This work places these leap methods in the context of numerical methods for the solution of stochastic differential equations (SDEs) driven by Poisson noise. This allows analogues of Euler-Maruyuma, Milstein and even higher order methods to be developed through the Itô-Taylor expansions as well as similar derivative-free Runge-Kutta approaches. Numerical results demonstrate that these novel methods compare favourably with existing techniques for simulating biochemical reactions by more accurately capturing crucial properties such as the mean and variance than existing methods.

Sequential Monte Carlo for Bayesian sequentially designed experiments for discrete data

In this paper we present a sequential Monte Carlo algorithm for Bayesian sequential experimental design applied to generalised non-linear models for discrete data. The approach is computationally convenient in that the information of newly observed data can be incorporated through a simple re-weighting step. We also consider a flexible parametric model for the stimulus-response relationship together with a newly developed hybrid design utility that can produce more robust estimates of the target stimulus in the presence of substantial model and parameter uncertainty. The algorithm is applied to hypothetical clinical trial or bioassay scenarios. In the discussion, potential generalisations of the algorithm are suggested to possibly extend its applicability to a wide variety of scenarios

Variational Bayes and the reduced dependence approximation for the autologistic model on an irregular grid with applications

Discrete Markov random field models provide a natural framework for representing images or spatial datasets. They model the spatial association present while providing a convenient Markovian dependency structure and strong edge-preservation properties. However, parameter estimation for discrete Markov random field models is difficult due to the complex form of the associated normalizing constant for the likelihood function. For large lattices, the reduced dependence approximation to the normalizing constant is based on the concept of performing computationally efficient and feasible forward recursions on smaller sublattices which are then suitably combined to estimate the constant for the whole lattice. We present an efficient computational extension of the forward recursion approach for the autologistic model to lattices that have an irregularly shaped boundary and which may contain regions with no data; these lattices are typical in applications. Consequently, we also extend the reduced dependence approximation to these scenarios enabling us to implement a practical and efficient non-simulation based approach for spatial data analysis within the variational Bayesian framework. The methodology is illustrated through application to simulated data and example images. The supplemental materials include our C++ source code for computing the approximate normalizing constant and simulation studies.

Solder joint defects classification using the Log-Gabor Filter, the Discrete Wavelet Transform and the Discrete Cosine Transform

Inspection of solder joints has been a critical process in the electronic manufacturing industry to reduce manufacturing cost, improve yield, and ensure project quality and reliability. This paper proposes the use of the Log-Gabor filter bank, Discrete Wavelet Transform and Discrete Cosine Transform for feature extraction of solder joint images on Printed Circuit Boards (PCBs). A distance based on the Mahalanobis Cosine metric is also presented for classification of five different types of solder joints. From the experimental results, this methodology achieved high accuracy and a well generalised performance. This can be an effective method to reduce cost and improve quality in the production of PCBs in the manufacturing industry.

Practical stability of approximating discrete-time filters with respect to model mismatch

This paper establishes practical stability results for an important range of approximate discrete-time filtering problems involving mismatch between the true system and the approximating filter model. Practical stability is established in the sense of an asymptotic bound on the amount of bias introduced by the model approximation. Our analysis applies to a wide range of estimation problems and justifies the common practice of approximating intractable infinite dimensional nonlinear filters by simpler computationally tractable filters.

Synthesis of discrete allyl-functional polyethylene glycol for efficient hydrophilization of dihydr (polydimethylsiloxane)

Amphiphilic poly(ethylene glycol)-block-pol (dimethylsiloxane)-block-poly(ethylene glycol)(PEG-block-PDMS block-PEG) triblock copolymers have been successfully prepared via hydrosilylation using discrete and polydisperse PEG of various chain lengths. Facile synthesis of discrete PEG (dPEG) is achieved via systematic tosylation and etherification of lower glycols. Amphiphilicity of the dPEG block-PDMS-block-dPEG triblock copolymer is illustrated by dynamic light scattering (DLS) and measurement of the critical micelle concentration (CMC).