Comparison of Some Improved Estimators for Linear Regression Model under Different Conditions

Multiple linear regression model plays a key role in statistical inference and it has extensive applications in business, environmental, physical and social sciences. Multicollinearity has been a considerable problem in multiple regression analysis. When the regressor variables are multicollinear, it becomes difficult to make precise statistical inferences about the regression coefficients. There are some statistical methods that can be used, which are discussed in this thesis are ridge regression, Liu, two parameter biased and LASSO estimators. Firstly, an analytical comparison on the basis of risk was made among ridge, Liu and LASSO estimators under orthonormal regression model. I found that LASSO dominates least squares, ridge and Liu estimators over a significant portion of the parameter space for large dimension. Secondly, a simulation study was conducted to compare performance of ridge, Liu and two parameter biased estimator by their mean squared error criterion. I found that two parameter biased estimator performs better than its corresponding ridge regression estimator. Overall, Liu estimator performs better than both ridge and two parameter biased estimator.

Results of analysis and model simulation of the bed profiles

Bedforms such as dunes and ripples are ubiquitous in rivers and coastal seas, and commonly described as triangular shapes from which height and length are calculated to estimate hydrodynamic and sediment dynamic parameters. Natural bedforms, however, present a far more complicated morphology; the difference between natural bedform shape and the often assumed triangular shape is usually neglected, and how this may affect the flow is unknown. This study investigates the shapes of natural bedforms and how they influence flow and shear stress, based on four datasets extracted from earlier studies on two rivers (the Rio Paraná in Argentina, and the Lower Rhine in The Netherlands). The most commonly occurring morphological elements are a sinusoidal stoss side made of one segment and a lee side made of two segments, a gently sloping upper lee side and a relatively steep (6 to 21°) slip face. A non-hydrostatic numerical model, set up using Delft3D, served to simulate the flow over fixed bedforms with various morphologies derived from the identified morphological elements. Both shear stress and turbulence increase with increasing slip face angle and are only marginally affected by the dimensions and positions of the upper and lower lee side. The average slip face angle determined from the bed profiles is 14°, over which there is no permanent flow separation. Shear stress and turbulence above natural bedforms are higher than above a flat bed but much lower than over the often assumed 30° lee side angle.

Toy model to describe the effect of positional blocklike disorder in metamaterials composites

We study theoretically the effect of a new type of blocklike positional disorder on the effective electromagnetic properties of one-dimensional chains of resonant, high-permittivity dielectric particles, where particles are arranged into perfectly well-ordered blocks whose relative position is a random variable. This creates a finite order correlation length that mimics the situation encountered in metamaterials fabricated through self-assembled techniques, whose structures often display short-range order between near neighbors but long-range disorder, due to stacking defects. Using a spectral theory approach combined with a principal component statistical analysis, we study, in the long-wavelength regime, the evolution of the electromagnetic response when the composite filling fraction and the block size are changed. Modifications in key features of the resonant response (amplitude, width, etc.) are investigated, showing a regime transition for a filling fraction around 50%.

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero-and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent a of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.

Bone morphology of the femur and tibia captured by statistical shape modelling predicts rapid bone loss in acute spinal cord injury patients

We would like to thank the study participants and the clinical and research staff at the Queen Elizabeth National Spinal Injury Unit, as without them this study would not have been possible. We are grateful for the funding received from Glasgow Research Partnership in Engineering for the employment of SC during data collection for this study. We would like to thank the Royal Society of Edinburgh's Scottish Crucible scheme for providing the opportunity for this collaboration to occur. We are also indebted to Maria Dumitrascuta for her time and effort in producing inter-repeatability results for the shape models.