Finite connectivity systems as error-correcting codes

We investigate the performance of parity check codes using the mapping onto spin glasses proposed by Sourlas. We study codes where each parity check comprises products of K bits selected from the original digital message with exactly C parity checks per message bit. We show, using the replica method, that these codes saturate Shannon's coding bound for K?8 when the code rate K/C is finite. We then examine the finite temperature case to asses the use of simulated annealing methods for decoding, study the performance of the finite K case and extend the analysis to accommodate different types of noisy channels. The analogy between statistical physics methods and decoding by belief propagation is also discussed.

Error-correcting code on a cactus:A solvable model

An exact solution to a family of parity check error-correcting codes is provided by mapping the problem onto a Husimi cactus. The solution obtained in the thermodynamic limit recovers the replica-symmetric theory results and provides a very good approximation to finite systems of moderate size. The probability propagation decoding algorithm emerges naturally from the analysis. A phase transition between decoding success and failure phases is found to coincide with an information-theoretic upper bound. The method is employed to compare Gallager and MN codes.

Analysis and optimisation of the performance of nonlinear optical communication systems

We investigate the feasibility of simultaneous suppressing of the amplification noise and nonlinearity, representing the most fundamental limiting factors in modern optical communication. To accomplish this task we developed a general design optimisation technique, based on concepts of noise and nonlinearity management. We demonstrate the immense efficiency of the novel approach by applying it to a design optimisation of transmission lines with periodic dispersion compensation using Raman and hybrid Raman-EDFA amplification. Moreover, we showed, using nonlinearity management considerations, that the optimal performance in high bit-rate dispersion managed fibre systems with hybrid amplification is achieved for a certain amplifier spacing – which is different from commonly known optimal noise performance corresponding to fully distributed amplification. Required for an accurate estimation of the bit error rate, the complete knowledge of signal statistics is crucial for modern transmission links with strong inherent nonlinearity. Therefore, we implemented the advanced multicanonical Monte Carlo (MMC) method, acknowledged for its efficiency in estimating distribution tails. We have accurately computed acknowledged for its efficiency in estimating distribution tails. We have accurately computed marginal probability density functions for soliton parameters, by numerical modelling of Fokker-Plank equation applying the MMC simulation technique. Moreover, applying a powerful MMC method we have studied the BER penalty caused by deviations from the optimal decision level in systems employing in-line 2R optical regeneration. We have demonstrated that in such systems the analytical linear approximation that makes a better fit in the central part of the regenerator nonlinear transfer function produces more accurate approximation of the BER and BER penalty. We present a statistical analysis of RZ-DPSK optical signal at direct detection receiver with Mach-Zehnder interferometer demodulation

Statnote 10: the two-way analysis of variance

The two-way design has been variously described as a matched-sample F-test, a simple within-subjects ANOVA, a one-way within-groups ANOVA, a simple correlated-groups ANOVA, and a one-factor repeated measures design! This confusion of terminology is likely to lead to problems in correctly identifying this analysis within commercially available software. The essential feature of the design is that each treatment is allocated by randomization to one experimental unit within each group or block. The block may be a plot of land, a single occasion in which the experiment was performed, or a human subject. The ‘blocking’ is designed to remove an aspect of the error variation and increase the ‘power’ of the experiment. If there is no significant source of variation associated with the ‘blocking’ then there is a disadvantage to the two-way design because there is a reduction in the DF of the error term compared with a fully randomised design thus reducing the ‘power’ of the analysis.

Statnote 11: the two-factor analysis of variance

Experiments combining different groups or factors are a powerful method of investigation in applied microbiology. ANOVA enables not only the effect of individual factors to be estimated but also their interactions; information which cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more efficient and often reduces the number of replications required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the degrees of freedom (DF) of the error term in the ANOVA is a more important indicator of the ‘power’ of the experiment than simply the number of replicates. A good method is to ensure, where possible, that sufficient replication is present to achieve 15 DF for each error term of the ANOVA. Finally, in a factorial experiment, it is important to define the design of the experiment in detail because this determines the appropriate type of ANOVA. We will discuss some of the common variations of factorial ANOVA in future statnotes. If there is doubt about which ANOVA to use, the researcher should seek advice from a statistician with experience of research in applied microbiology.

Statnote 13: a more complex analysis of variance incorporating a 'repeated measures' factor

Experiments combining different groups or factors and which use ANOVA are a powerful method of investigation in applied microbiology. ANOVA enables not only the effect of individual factors to be estimated but also their interactions; information which cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more efficient and often reduces the sample size required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the degrees of freedom (DF) of the error term in the ANOVA is a more important indicator of the ‘power’ of the experiment than the number of replicates. A good method is to ensure, where possible, that sufficient replication is present to achieve 15 DF for the error term of the ANOVA testing effects of particular interest. Finally, it is important to always consider the design of the experiment because this determines the appropriate ANOVA to use. Hence, it is necessary to be able to identify the different forms of ANOVA appropriate to different experimental designs and to recognise when a design is a split-plot or incorporates a repeated measure. If there is any doubt about which ANOVA to use in a specific circumstance, the researcher should seek advice from a statistician with experience of research in applied microbiology.

The use of analysis of variance (ANOVA) in applied microbiology

Experiments combining different groups or factors and which use ANOVA are a powerful method of investigation in applied microbiology. ANOVA enables not only the effect of individual factors to be estimated but also their interactions; information which cannot be obtained readily when factors are investigated separately. In addition, combining different treatments or factors in a single experiment is more efficient and often reduces the number of replications required to estimate treatment effects adequately. Because of the treatment combinations used in a factorial experiment, the DF of the error term in the ANOVA is a more important indicator of the ‘power’ of the experiment than the number of replicates. A good method is to ensure, where possible, that sufficient replication is present to achieve 15 DF for each error term of the ANOVA. Finally, it is important to consider the design of the experiment because this determines the appropriate ANOVA to use. Some of the most common experimental designs used in the biosciences and their relevant ANOVAs are discussed by. If there is doubt about which ANOVA to use, the researcher should seek advice from a statistician with experience of research in applied microbiology.

Data analysis methods in optometry Part 4: an introduction to analysis of variance (ANOVA)

In any investigation in optometry involving more that two treatment or patient groups, an investigator should be using ANOVA to analyse the results assuming that the data conform reasonably well to the assumptions of the analysis. Ideally, specific null hypotheses should be built into the experiment from the start so that the treatments variation can be partitioned to test these effects directly. If 'post-hoc' tests are used, then an experimenter should examine the degree of protection offered by the test against the possibilities of making either a type 1 or a type 2 error. All experimenters should be aware of the complexity of ANOVA. The present article describes only one common form of the analysis, viz., that which applies to a single classification of the treatments in a randomised design. There are many different forms of the analysis each of which is appropriate to the analysis of a specific experimental design. The uses of some of the most common forms of ANOVA in optometry have been described in a further article. If in any doubt, an investigator should consult a statistician with experience of the analysis of experiments in optometry since once embarked upon an experiment with an unsuitable design, there may be little that a statistician can do to help.

Data analysis methods in optometry. Part 5: correlation

1. Pearson's correlation coefficient only tests whether the data fit a linear model. With large numbers of observations, quite small values of r become significant and the X variable may only account for a minute proportion of the variance in Y. Hence, the value of r squared should always be calculated and included in a discussion of the significance of r. 2. The use of r assumes that a bivariate normal distribution is present and this assumption should be examined prior to the study. If Pearson's r is not appropriate, then a non-parametric correlation coefficient such as Spearman's rs may be used. 3. A significant correlation should not be interpreted as indicating causation especially in observational studies in which there is a high probability that the two variables are correlated because of their mutual correlations with other variables. 4. In studies of measurement error, there are problems in using r as a test of reliability and the ‘intra-class correlation coefficient’ should be used as an alternative. A correlation test provides only limited information as to the relationship between two variables. Fitting a regression line to the data using the method known as ‘least square’ provides much more information and the methods of regression and their application in optometry will be discussed in the next article.

Data methods in optometry Part 9: experimental design and analysis of variance

The key to the correct application of ANOVA is careful experimental design and matching the correct analysis to that design. The following points should therefore, be considered before designing any experiment: 1. In a single factor design, ensure that the factor is identified as a 'fixed' or 'random effect' factor. 2. In more complex designs, with more than one factor, there may be a mixture of fixed and random effect factors present, so ensure that each factor is clearly identified. 3. Where replicates can be grouped or blocked, the advantages of a randomised blocks design should be considered. There should be evidence, however, that blocking can sufficiently reduce the error variation to counter the loss of DF compared with a randomised design. 4. Where different treatments are applied sequentially to a patient, the advantages of a three-way design in which the different orders of the treatments are included as an 'effect' should be considered. 5. Combining different factors to make a more efficient experiment and to measure possible factor interactions should always be considered. 6. The effect of 'internal replication' should be taken into account in a factorial design in deciding the number of replications to be used. Where possible, each error term of the ANOVA should have at least 15 DF. 7. Consider carefully whether a particular factorial design can be considered to be a split-plot or a repeated measures design. If such a design is appropriate, consider how to continue the analysis bearing in mind the problem of using post hoc tests in this situation.

A study of cluster analysis techniques and their applications

This thesis seeks to describe the development of an inexpensive and efficient clustering technique for multivariate data analysis. The technique starts from a multivariate data matrix and ends with graphical representation of the data and pattern recognition discriminant function. The technique also results in distances frequency distribution that might be useful in detecting clustering in the data or for the estimation of parameters useful in the discrimination between the different populations in the data. The technique can also be used in feature selection. The technique is essentially for the discovery of data structure by revealing the component parts of the data. lhe thesis offers three distinct contributions for cluster analysis and pattern recognition techniques. The first contribution is the introduction of transformation function in the technique of nonlinear mapping. The second contribution is the us~ of distances frequency distribution instead of distances time-sequence in nonlinear mapping, The third contribution is the formulation of a new generalised and normalised error function together with its optimal step size formula for gradient method minimisation. The thesis consists of five chapters. The first chapter is the introduction. The second chapter describes multidimensional scaling as an origin of nonlinear mapping technique. The third chapter describes the first developing step in the technique of nonlinear mapping that is the introduction of "transformation function". The fourth chapter describes the second developing step of the nonlinear mapping technique. This is the use of distances frequency distribution instead of distances time-sequence. The chapter also includes the new generalised and normalised error function formulation. Finally, the fifth chapter, the conclusion, evaluates all developments and proposes a new program. for cluster analysis and pattern recognition by integrating all the new features.

IN VIVO analysis of ocular morphological changes during phakic accommodation

The principal theme of this thesis is the in vivo examination of ocular morphological changes during phakic accommodation, with particular attention paid to the ciliary muscle and crystalline lens. The investigations detailed involved the application of high-resolution imaging techniques to facilitate the acquisition of new data to assist in the clarification of aspects of the accommodative system that were poorly understood. A clinical evaluation of the newly available Grand Seiko Auto Ref/ Keratometer WAM-5500 optometer was undertaken to assess its value in the field of accommodation research. The device was found to be accurate and repeatable compared to subjective refraction, and has the added advantage of allowing dynamic data collection at a frequency of around 5 Hz. All of the subsequent investigations applied the WAM-5500 for determination of refractive error and objective accommodative responses. Anterior segment optical coherence tomography (AS-OCT) based studies examined the morphology and contractile response of youthful and ageing ciliary muscle. Nasal versus temporal asymmetry was identified, with the temporal aspect being both thicker and demonstrating a greater contractile response. The ciliary muscle was longer in terms of both its anterior (r = 0.49, P <0.001) and overall length (r = 0.45, P = 0.02) characteristics, in myopes. The myopic ciliary muscle does not appear to be merely stretched during axial elongation, as no significant relationship between thickness and refractive error was identified. The main contractile responses observed were a thickening of the anterior region and a shortening of the muscle, particularly anteriorly. Similar patterns of response were observed in subjects aged up to 70 years, supporting a lensocentric theory of presbyopia development. Following the discovery of nasal/ temporal asymmetry in ciliary muscle morphology and response, an investigation was conducted to explore whether the regional variations in muscle contractility impacted on lens stability during accommodation. A bespoke programme was developed to analyse AS-OCT images and determine whether lens tilt and decentration varied between the relaxed and accommodated states. No significant accommodative difference in these parameters was identified, implying that any changes in lens stability with accommodation are very slight, as a possible consequence of vitreous support. Novel three-dimensional magnetic resonance imaging (MRI) and analysis techniques were used to investigate changes in lens morphology and ocular conformation during accommodation. An accommodative reduction in lens equatorial diameter provides further evidence to support the Helmholtzian mechanism of accommodation, whilst the observed increase in lens volume challenges the widespread assertion that this structure is incompressible due to its high water content. Wholeeye MRI indicated that the volume of the vitreous chamber remains constant during accommodation. No significant changes in ocular conformation were detected using MRI. The investigations detailed provide further insight into the mechanisms of accommodation and presbyopia, and represent a platform for future work in this field.

Orbit and altimetric corrections for the ERS satellites through analysis of single and dual satellite crossovers

Due to the failure of PRARE the orbital accuracy of ERS-1 is typically 10-15 cm radially as compared to 3-4cm for TOPEX/Poseidon. To gain the most from these simultaneous datasets it is necessary to improve the orbital accuracy of ERS-1 so that it is commensurate with that of TOPEX/Poseidon. For the integration of these two datasets it is also necessary to determine the altimeter and sea state biases for each of the satellites. Several models for the sea state bias of ERS-1 are considered by analysis of the ERS-1 single satellite crossovers. The model adopted consists of the sea state bias as a percentage of the significant wave height, namely 5.95%. The removal of ERS-1 orbit error and recovery of an ERS-1 - TOPEX/Poseidon relative bias are both achieved by analysis of dual crossover residuals. The gravitational field based radial orbit error is modelled by a finite Fourier expansion series with the dominant frequencies determined by analysis of the JGM-2 co-variance matrix. Periodic and secular terms to model the errors due to atmospheric density, solar radiation pressure and initial state vector mis-modelling are also solved for. Validation of the dataset unification consists of comparing the mean sea surface topographies and annual variabilities derived from both the corrected and uncorrected ERS-1 orbits with those derived from TOPEX/Poseidon. The global and regional geographically fixed/variable orbit errors are also analysed pre and post correction, and a significant reduction is noted. Finally the use of dual/single satellite crossovers and repeat pass data, for the calibration of ERS-2 with respect to ERS-1 and TOPEX/Poseidon is shown by calculating the ERS-1/2 sea state and relative biases.

Optimising the clinical analysis of retinal image quality in the human eye

Visual perception is dependent on both light transmission through the eye and neuronal conduction through the visual pathway. Advances in clinical diagnostics and treatment modalities over recent years have increased the opportunities to improve the optical path and retinal image quality. Higher order aberrations and retinal straylight are two major factors that influence light transmission through the eye and ultimately, visual outcome. Recent technological advancements have brought these important factors into the clinical domain, however the potential applications of these tools and considerations regarding interpretation of data are much underestimated. The purpose of this thesis was to validate and optimise wavefront analysers and a new clinical tool for the objective evaluation of intraocular scatter. The application of these methods in a clinical setting involving a range of conditions was also explored. The work was divided into two principal sections: 1. Wavefront Aberrometry: optimisation, validation and clinical application The main findings of this work were: • Observer manipulation of the aberrometer increases variability by a factor of 3. • Ocular misalignment can profoundly affect reliability, notably for off-axis aberrations. • Aberrations measured with wavefront analysers using different principles are not interchangeable, with poor relationships and significant differences between values. • Instrument myopia of around 0.30D is induced when performing wavefront analysis in non-cyclopleged eyes; values can be as high as 3D, being higher as the baseline level of myopia decreases. Associated accommodation changes may result in relevant changes to the aberration profile, particularly with respect to spherical aberration. • Young adult healthy Caucasian eyes have significantly more spherical aberration than Asian eyes when matched for age, gender, axial length and refractive error. Axial length is significantly correlated with most components of the aberration profile. 2. Intraocular light scatter: Evaluation of subjective measures and validation and application of a new objective method utilising clinically derived wavefront patterns. The main findings of this work were: • Subjective measures of clinical straylight are highly repeatable. Three measurements are suggested as the optimum number for increased reliability. • Significant differences in straylight values were found for contact lenses designed for contrast enhancement compared to clear lenses of the same design and material specifications. Specifically, grey/green tints induced significantly higher values of retinal straylight. • Wavefront patterns from a commercial Hartmann-Shack device can be used to obtain objective measures of scatter and are well correlated with subjective straylight values. • Perceived retinal stray light was similar in groups of patients implanted with monofocal and multi focal intraocular lenses. Correlation between objective and subjective measurements of scatter is poor, possibly due to different illumination conditions between the testing procedures, or a neural component which may alter with age. Careful acquisition results in highly reproducible in vivo measures of higher order aberrations; however, data from different devices are not interchangeable which brings the accuracy of measurement into question. Objective measures of intraocular straylight can be derived from clinical aberrometry and may be of great diagnostic and management importance in the future.

Numerical and statistical analysis of long-haul undersea optical communication systems

Firstly, we numerically model a practical 20 Gb/s undersea configuration employing the Return-to-Zero Differential Phase Shift Keying data format. The modelling is completed using the Split-Step Fourier Method to solve the Generalised Nonlinear Schrdinger Equation. We optimise the dispersion map and per-channel launch power of these channels and investigate how the choice of pre/post compensation can influence the performance. After obtaining these optimal configurations, we investigate the Bit Error Rate estimation of these systems and we see that estimation based on Gaussian electrical current systems is appropriate for systems of this type, indicating quasi-linear behaviour. The introduction of narrower pulses due to the deployment of quasi-linear transmission decreases the tolerance to chromatic dispersion and intra-channel nonlinearity. We used tools from Mathematical Statistics to study the behaviour of these channels in order to develop new methods to estimate Bit Error Rate. In the final section, we consider the estimation of Eye Closure Penalty, a popular measure of signal distortion. Using a numerical example and assuming the symmetry of eye closure, we see that we can simply estimate Eye Closure Penalty using Gaussian statistics. We also see that the statistics of the logical ones dominates the statistics of the logical ones dominates the statistics of signal distortion in the case of Return-to-Zero On-Off Keying configurations.