Non-normally distributed data and non-parametric statistics

Different types of numerical data can be collected in a scientific investigation and the choice of statistical analysis will often depend on the distribution of the data. A basic distinction between variables is whether they are ‘parametric’ or ‘non-parametric’. When a variable is parametric, the data come from a symmetrically shaped distribution known as the ‘Gaussian’ or ‘normal distribution’ whereas non-parametric variables may have a distribution which deviates markedly in shape from normal. This article describes several aspects of the problem of non-normality including: (1) how to test for two common types of deviation from a normal distribution, viz., ‘skew’ and ‘kurtosis’, (2) how to fit the normal distribution to a sample of data, (3) the transformation of non-normally distributed data and scores, and (4) commonly used ‘non-parametric’ statistics which can be used in a variety of circumstances.

Application of the manakov-PMD equation to a computational investigation of low-PMD fibres

The phenomenon of low-PMD fibres is examined through numerical simulations. Instead of the coarse-step method we are using an algorithm developed through the Manakov-PMD equation. With the integration of the Manakov-PMD equation we have access to the fibre spin which relates to the orientation of the birefringence. The simulation results produced correspond to the behaviour of a low-PMD spun fibre. Furthermore we provide an analytical approximation compared to the numerical data. © 2005 Optical Society of America.

Expert system design for cold form sections

The concept of an Expert System (ES) has been acknowledged as a very useful tool, but few studies have been carried out in its application to the design of cold rolled sections. This study involves primarily the use of an ES as a tool to improve the design process and to capture the draughtsman's knowledge. Its main purpose is to reduce substantially the time taken to produce a section drawing, thereby facilitating a speedy feedback to the customer. In order to communicate with a draughtsman, it is necessary to use sketches, symbolic representations and numerical data. This increases the complexity of programming an ES, as it is necessary to use a combination of languages so that decisions, calculations, graphical drawings and control of the system can be effected. A production system approach is used and a further step has been taken by introducing an Activator which is an autoexecute operation set up by the ES to operate an external program automatically. To speed up the absorption of new knowledge into the knowledge base, a new Learning System has been constructed. In addition to developing the ES, other software has been written to assist the design process. The section properties software has been introduced to improve the speed and consistency of calculating the section properties. A method of selecting or comparing the most appropriate section for a given specification is also implemented. Simple loading facilities have been introduced to guide the designer as to the loading capacity of the section. This research has concluded that the application of an ES is beneficial and with the activator approach, automated designing can be achieved. On average a complex drawing can be displayed on the screen in about 100 seconds, where over 95% of the initial section design time for repetitive or similar profile can be saved.

Relationship status and identity construction in heterosexual women:A discourse analytic and personal construct study

This thesis reports the findings of three studies examining relationship status and identity construction in the talk of heterosexual women, from a feminist and social constructionist perspective. Semi-structured interviews were conducted with 12 women in study 1 and 13 women for study 2, between the ages of twenty and eighty-seven, discussing their experiences of relationships. All interviews were transcribed and analysed using discourse analysis, by hand and using the Nudist 6 program. The resulting themes create distinct age-related marital status expectations. Unmarried women were aware they had to marry by a ‘certain age’ or face a ‘lonely spinsterhood’. Through marriage women gained a socially accepted position associated with responsibility for others, self-sacrifice, a home-focused lifestyle and relational identification. Divorce was constructed as the consequence of personal faults and poor relationship care, reassuring the married of their own control over their status. Older unmarried women were constructed as deviant and pitiable, occupying social purgatory as a result of transgressing these valued conventions. Study 3 used repertory grid tasks, with 33 women, analysing transcripts and notes alongside numerical data using Web Grid II internet analysis tool, to produce principle components maps demonstrating the relationships between relationship terms and statuses. This study illuminated the consistency with which women of different ages and status saw marriage as their ideal living situation and outlined the domestic responsibilities associated. Spinsters and single-again women were defined primarily by their lack of marriage and by loneliness. This highlighted the devalued position of older unmarried women. The results of these studies indicated a consistent set of age-related expectations of relationship status, acknowledged by women and reinforced by their families and friends, which render many unmarried women deviant and fail to acknowledge the potential variety of women’s ways of living.

Accounting for derivatives:an evaluation of reporting practice by UK banks

In 1998 the Accounting Standards Board (ASB) published FRS 13, ‘Derivatives and other Financial Instruments: Disclosures’. This laid down the requirements for disclosures of an entity’s policies, objectives and strategies in using financial instruments, their impact on its risk, performance and financial condition, and details of how risks are managed. FRS 13 became effective in March 1999, and this paper uses the 1999 annual reports of UK banks to evaluate the usefulness of disclosures from a user’s perspective. Usefulness is measured in terms of the criteria of materiality, relevance, reliability, comparability and understandability as defined in the ASB’s Statement of Principles (ASB, 1999). Our findings suggest that the narrative disclosures are generic in nature, the numerical data incomplete and not always comparable, and that it is difficult for the user to combine both narrative and numerical information in order to assess the banks’ risk profile. Our overall conclusion is therefore that current UK financial reporting practices are of limited help to users wishing to assess the scale of an institution’s financial risk exposure.

Stochastic anti-resonance in a fibre Raman amplifier

Stochastic anti-resonance, that is resonant enhancement of randomness caused by polarization mode beatings, is analyzed both numerically and analytically on an example of fibre Raman amplifier with randomly varying birefringence. As a result of such anti-resonance, the polarization mode dispersion growth causes an escape of the signal state of polarization from a metastable state corresponding to the pulling of the signal to the pump state of polarization.This phenomenon reveals itself in abrupt growth of gain fluctuations as well as in dropping of Hurst parameter and Kramers length characterizing long memory in a system and noise induced escape from the polarization pulling state. The results based on analytical multiscale averaging technique agree perfectly with the numerical data obtained by direct numerical simulations of underlying stochastic differential equations. This challenging outcome would allow replacing the cumbersome numerical simulations for real-world extra-long high-speed communication systems.

Universal profile of the vortex condensate in two-dimensional turbulence

An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a condensate—a pair of coherent system-size vortices. We perform extensive numerical simulations of this system and carry out theoretical analysis based on momentum and energy exchanges between the turbulence and the vortices. We show that the vortices have a universal internal structure independent of the type of small-scale dissipation, small-scale forcing, and boundary conditions. The theory predicts not only the vortex inner region profile, but also the amplitude, which both perfectly agree with the numerical data.

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.

Numerical modelling of dispersion managed soliton transmission

This thesis presents the results of numerical modelling of the propagation of dispersion managed solitons. The theory of optical pulse propagation in single mode optical fibre is introduced specifically looking at the use of optical solitons for fibre communications. The numerical technique used to solve the nonlinear Schrödinger equation is also introduced. The recent developments in the use of dispersion managed solitons are reviewed before the numerical results are presented. The work in this thesis covers two main areas; (i) the use of a saturable absorber to control the propagation of dispersion managed solutions and (ii) the upgrade of the installed standard fibre network to higher data rates through the use of solitons and dispersion management. Saturable absorbe can be used to suppress the build up of noise and dispersive radiation in soliton transmission lines. The use of saturable absorbers in conjunction with dispersion management has been investigated both as a single pulse and for the transmission of a 10Gbit/s data pattern. It is found that this system supports a new regime of stable soliton pulses with significantly increased powers. The upgrade of the installed standard fibre network to higher data rates through the use of fibre amplifiers and dispersion management is of increasing interest. In this thesis the propagation of data at both 10Gbit/s and 40Gbit/s is studied. Propagation over transoceanic distances is shown to be possible for 10Gbit/s transmission and for more than 2000km at 40Gbit/s. The contribution of dispersion managed solitons in the future of optical communications is discussed in the thesis conclusions.

Mathematical and numerical modelling of dispersion-managed solitons, autosolitons and self-similar optical pulses

This thesis presents theoretical investigation of three topics concerned with nonlinear optical pulse propagation in optical fibres. The techniques used are mathematical analysis and numerical modelling. Firstly, dispersion-managed (DM) solitons in fibre lines employing a weak dispersion map are analysed by means of a perturbation approach. In the case of small dispersion map strengths the average pulse dynamics is described by a perturbation approach (NLS) equation. Applying a perturbation theory, based on the Inverse Scattering Transform method, an analytic expression for the envelope of the DM soliton is derived. This expression correctly predicts the power enhancement arising from the dispersion management.Secondly, autosoliton transmission in DM fibre systems with periodical in-line deployment of nonlinear optical loop mirrors (NOLMs) is investigated. The use of in-line NOLMs is addressed as a general technique for all-optical passive 2R regeneration of return-to-zero data in high speed transmission system with strong dispersion management. By system optimisation, the feasibility of ultra-long single-channel and wavelength-division multiplexed data transmission at bit-rates ³ 40 Gbit s-1 in standard fibre-based systems is demonstrated. The tolerance limits of the results are defined.Thirdly, solutions of the NLS equation with gain and normal dispersion, that describes optical pulse propagation in an amplifying medium, are examined. A self-similar parabolic solution in the energy-containing core of the pulse is matched through Painlevé functions to the linear low-amplitude tails. The analysis provides a full description of the features of high-power pulses generated in an amplifying medium.

Combination of altimetry data from different satellite missions

Substantial altimetry datasets collected by different satellites have only become available during the past five years, but the future will bring a variety of new altimetry missions, both parallel and consecutive in time. The characteristics of each produced dataset vary with the different orbital heights and inclinations of the spacecraft, as well as with the technical properties of the radar instrument. An integral analysis of datasets with different properties offers advantages both in terms of data quantity and data quality. This thesis is concerned with the development of the means for such integral analysis, in particular for dynamic solutions in which precise orbits for the satellites are computed simultaneously. The first half of the thesis discusses the theory and numerical implementation of dynamic multi-satellite altimetry analysis. The most important aspect of this analysis is the application of dual satellite altimetry crossover points as a bi-directional tracking data type in simultaneous orbit solutions. The central problem is that the spatial and temporal distributions of the crossovers are in conflict with the time-organised nature of traditional solution methods. Their application to the adjustment of the orbits of both satellites involved in a dual crossover therefore requires several fundamental changes of the classical least-squares prediction/correction methods. The second part of the thesis applies the developed numerical techniques to the problems of precise orbit computation and gravity field adjustment, using the altimetry datasets of ERS-1 and TOPEX/Poseidon. Although the two datasets can be considered less compatible that those of planned future satellite missions, the obtained results adequately illustrate the merits of a simultaneous solution technique. In particular, the geographically correlated orbit error is partially observable from a dataset consisting of crossover differences between two sufficiently different altimetry datasets, while being unobservable from the analysis of altimetry data of both satellites individually. This error signal, which has a substantial gravity-induced component, can be employed advantageously in simultaneous solutions for the two satellites in which also the harmonic coefficients of the gravity field model are estimated.

The application of contact mechanics to the numerical simulation of particulate material

Particulate solids are complex redundant systems which consist of discrete particles. The interactions between the particles are complex and have been the subject of many theoretical and experimental investigations. Invetigations of particulate material have been restricted by the lack of quantitative information on the mechanisms occurring within an assembly. Laboratory experimentation is limited as information on the internal behaviour can only be inferred from measurements on the assembly boundary, or the use of intrusive measuring devices. In addition comparisons between test data are uncertain due to the difficulty in reproducing exact replicas of physical systems. Nevertheless, theoretical and technological advances require more detailed material information. However, numerical simulation affords access to information on every particle and hence the micro-mechanical behaviour within an assembly, and can replicate desired systems. To use a computer program to numerically simulate material behaviour accurately it is necessary to incorporte realistic interaction laws. This research programme used the finite difference simulation program `BALL', developed by Cundall (1971), which employed linear spring force-displacement laws. It was thus necessary to incorporate more realistic interaction laws. Therefore, this research programme was primarily concerned with the implementation of the normal force-displacement law of Hertz (1882) and the tangential force-displacement laws of Mindlin and Deresiewicz (1953). Within this thesis the contact mechanics theories employed in the program are developed and the adaptations which were necessary to incorporate these laws are detailed. Verification of the new contact force-displacement laws was achieved by simulating a quasi-static oblique contact and single particle oblique impact. Applications of the program to the simulation of large assemblies of particles is given, and the problems in undertaking quasi-static shear tests along with the results from two successful shear tests are described.

Numerical modelling of complex femtosecond laser inscribed fiber gratings - comparison wih experiment

We present experimental studies and numerical modeling based on a combination of the Bidirectional Beam Propagation Method and Finite Element Modeling that completely describes the wavelength spectra of point by point femtosecond laser inscribed fiber Bragg gratings, showing excellent agreement with experiment. We have investigated the dependence of different spectral parameters such as insertion loss, all dominant cladding and ghost modes and their shape relative to the position of the fiber Bragg grating in the core of the fiber. Our model is validated by comparing model predictions with experimental data and allows for predictive modeling of the gratings. We expand our analysis to more complicated structures, where we introduce symmetry breaking; this highlights the importance of centered gratings and how maintaining symmetry contributes to the overall spectral quality of the inscribed Bragg gratings. Finally, the numerical modeling is applied to superstructure gratings and a comparison with experimental results reveals a capability for dealing with complex grating structures that can be designed with particular wavelength characteristics.

Verification and validation of numerical models of the transport of insulation debris

Damage to insulation materials located near to a primary circuit coolant leak may compromise the operation of the emergency core cooling system (ECCS). Insulation material in the form of mineral wool fiber agglomerates (MWFA) maybe transported to the containment sump strainers, where they may block or penetrate the strainers. Though the impact of MWFA on the pressure drop across the strainers is minimal, corrosion products formed over time may also accumulate in the fiber cakes on the strainers, which can lead to a significant increase in the strainer pressure drop and result in cavitation in the ECCS. An experimental and theoretical study performed by the Helmholtz-Zentrum Dresden-Rossendorf and the Hochschule Zittau/Görlitz is investigating the phenomena that maybe observed in the containment vessel during a primary circuit coolant leak. The study entails the generation of fiber agglomerates, the determination of their transport properties in single and multi-effect experiments and the long-term effect that corrosion and erosion of the containment internals by the coolant has on the strainer pressure drop. The focus of this paper is on the verification and validation of numerical models that can predict the transport of MWFA. A number of pseudo-continuous dispersed phases of spherical wetted agglomerates represent the MWFA. The size, density, the relative viscosity of the fluid-fiber agglomerate mixture and the turbulent dispersion all affect how the fiber agglomerates are transported. In the cases described here, the size is kept constant while the density is modified. This definition affects both the terminal velocity and volume fraction of the dispersed phases. Note that the relative viscosity is only significant at high concentrations. Three single effect experiments were used to provide validation data on the transport of the fiber agglomerates under conditions of sedimentation in quiescent fluid, sedimentation in a horizontal flow and suspension in a horizontal flow. The experiments were performed in a rectangular column for the quiescent fluid and a racetrack type channel that provided a near uniform horizontal flow. The numerical models of sedimentation in the column and the racetrack channel found that the sedimentation characteristics are consistent with the experiments. For channel suspension, the heavier fibers tend to accumulate at the channel base even at high velocities, while lighter phases are more likely to be transported around the channel.

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.