Literature Review of the Generalised Additive Model for location, scale and shape

Το βασικό αντικείμενο αυτής της μελέτης είναι να παρουσιάσει διαφορετικά στατιστικά μοντέλα και να συζητήσει την συνεισφορά τους στην ερμηνεία των δεδομένων. Το πρώτο μοντέλο που αναλύεται είναι το Γενικευμένο Γραμμικό Μοντέλο (ΓΓΜ) το οποίο είναι μία γενίκευση του γραμμικού μοντέλου θεωρώντας η κατανομή της επεξηγηματικής μεταβλητής να είναι μέλος της εκθετικής οικογένειας κατανομών. Η φύση των δεδομένων καθορίζει σε μεγάλο βαθμό την μορφή του γενικευμένου γραμμικού μοντέλου που θα εφαρμοστεί, μέσω της επιλογής της συνάρτησης συνδέσμου του μοντέλου. Οι επαναληπτικές μέθοδοι που επιτρέπουν την πρακτική εφαρμογή του κάθε μοντέλου καθώς και οι αντίστοιχες μέθοδοι στατιστικής συμπερασματολογίας αναλύονται. Η υπόθεση της εκθετικής οικογένειας κατανομών για την εξαρτημένη μεταβλητή ισχύει και για το Γενικευμένο Προσθετικό Μοντέλο (ΓΠΜ). Για την εξαρτημένη μεταβλητή θεωρείται μία κατανομή που ανήκει στην εκθετική οικογένεια κατανομών μαζί με την εισαγωγή μη παραμετρικών συναρτήσεων που αναμιγνύουν τις εγγενείς ιδιότητες των ΓΓΜ και των προσθετικών μοντέλων. Η απαντητική μεταβλητή εξαρτάται γραμμικώς από γνωστές μη παραμετρικές συναρτήσεις ορισμένων εκ των ανεξάρτητων μεταβλητών, η συμπερασματολογία επικεντρώνεται στις μη παραμετρικές συναρτήσεις. Μία γενική κατηγορία στατιστικών μοντέλων για μία απαντητική μεταβλητή παρουσιάζεται, η οποία ονομάζεται Γενικευμένο Προσθετικό Μοντέλο ως προς τις τέσσερις ροπές. Η επιλογή της κατανομής της επεξηγηματικής μεταβλητής στο ΓΠΜ ως προς τις τέσσερις ροπές γίνεται μέσα από μία ευρεία οικογένεια κατανομών συμπεριλαμβάνοντας έντονα ασύμμετρες ή κυρτωμένες συνεχείς και διακριτές κατανομές. Το συστηματικό κομμάτι των ΓΠΜ ως προς τις τέσσερις ροπές διευρύνεται ώστε να επιτρέπει τη μοντελοποίηση του μέσου και άλλων παραμέτρων της κατανομής της επεξηγηματικής μεταβλητής, είτε ως παραμετρικές είτε ως προσθετικές μη παραμετρικές συναρτήσεις είτε ως όροι τυχαίων γεγονότων.The main objective of this study is to present different statistical models and discuss their contribution to data fit. The first model that is analysed is the Generalised Linear Model(GLM) which is a generalisation of the linear model assuming some member of the exponential family of distributions for the response variable. The nature of the data determines to a great extent the form of the generalised linear model that will be applied, through the choice of the link function of the model. The iterative methods which allow for the practical implementation of each particular model and the respective statistical inference procedures are discussed, as well. The assumption of the exponential family distribution for the response is relaxed in the Generalised Additive Model(GAM). A general distribution is assumed for the dependent variable, with the introduction of smoothing functions that blend the inherent properties of the GLM with the additive models. The response variable depends linearly on unknown smooth functions of some predictor variables, and the inference is focused on these smoothers. A general class of statistical models for a univariate response variable is presented, which is called the Generalized Additive Model for Location, Scale and Shape (GAMLSS). The choice of the distribution for the response variable in GAMLSS is made from a very general family of distributions including highly skewed or kurtotic continuous and discrete distributions. The GAMLSS systematic part is expanded to permit modelling of the mean (or location) and other distributional parameters of the response, as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms

Line transect abundance estimation with uncertain detection on the trackline

Bibliography: leaves 225-233.After critically reviewing developments in line transect estimation theory to date, general likelihood functions are derived for the case in which detection probabilities are modelled as functions of any number of explanatory variables and detection of animals on the trackline (i.e. directly in the observer's path) is not certain. Existing models are shown to correspond to special cases of the general models. Maximum likelihood estimators are derived for some special cases of the general model and some existing line transect estimators are shown to correspond to maximum likelihood estimators for other special cases. The likelihoods are shown to be extensions of existing mark-recapture likelihoods as well as being generalizations of existing line transect likelihoods. Two new abundance estimators are developed. The first is a Horvitz-Thompson-like estimator which utilizes the fact that for point estimation of abundance the density of perpendicular distances in the population can be treated as known in appropriately designed line transect surveys. The second is based on modelling the probability density function of detection probabilities in the population. Existing line transect estimators are shown to correspond to special cases of the new Horvitz-Thompson-like estimator, so that this estimator, together with the general likelihoods, provides a unifying framework for estimating abundance from line transect surveys

Mathematics in Software Reliability and Quality Assurance

This monograph concerns the mathematical aspects of software reliability and quality assurance and consists of 11 technical papers in this emerging area. Included are the latest research results related to formal methods and design, automatic software testing, software verification and validation, coalgebra theory, automata theory, hybrid system and software reliability modeling and assessment

Explicit alternating direction methods for problems in fluid dynamics

Recently an iterative method was formulated employing a new splitting strategy for the solution of tridiagonal systems of difference equations. The method was successful in solving the systems of equations arising from one dimensional initial boundary value problems, and a theoretical analysis for proving the convergence of the method for systems whose constituent matrices are positive definite was presented by Evans and Sahimi [22]. The method was known as the Alternating Group Explicit (AGE) method and is referred to as AGE-1D. The explicit nature of the method meant that its implementation on parallel machines can be very promising. The method was also extended to solve systems arising from two and three dimensional initial-boundary value problems, but the AGE-2D and AGE-3D algorithms proved to be too demanding in computational cost which largely reduces the advantages of its parallel nature. In this thesis, further theoretical analyses and experimental studies are pursued to establish the convergence and suitability of the AGE-1D method to a wider class of systems arising from univariate and multivariate differential equations with symmetric and non symmetric difference operators. Also the possibility of a Chebyshev acceleration of the AGE-1D algorithm is considered. For two and three dimensional problems it is proposed to couple the use of the AGE-1D algorithm with an ADI scheme or an ADI iterative method in what is called the Explicit Alternating Direction (EAD) method. It is then shown through experimental results that the EAD method retains the parallel features of the AGE method and moreover leads to savings of up to 83 % in the computational cost for solving some of the model problems. The thesis also includes applications of the AGE-1D algorithm and the EAD method to solve some problems of fluid dynamics such as the linearized Shallow Water equations, and the Navier Stokes' equations for the flow in an idealized one dimensional Planetary Boundary Layer. The thesis terminates with conclusions and suggestions for further work together with a comprehensive bibliography and an appendix containing some selected programs

Combustion analysis and particulate mutagenicity characterization for a single-cylinder diesel engine fueled by Fischer -Tropsch derived liquids

Further growth of diesel engines in the light-duty and heavy-duty vehicular markets is closely linked to the potential health risks of diesel exhaust. Cleaner burning fuels, such as those derived from natural gas via the Fischer-Tropsch (FT) process, offer a potentially economically viable alternative to standard diesel fuel. As part of this study, a two-liter, single-cylinder, four-stroke direct-injected engine was instrumented for in-cylinder pressure measurements. The emissions and performance data from engine operation with Federal low sulfur No. 2 diesel fuel (DF) and natural gas derived FT fuel were compared. Also as part of the study, an investigation was carried out on the mutagenic characteristics of particulate matter (PM) derived from FT and DF fuel combustion by relating the in-vitro mutagenic activity of the particulate matter to engine operating conditions and particle size via the Ames Salmonella typhimurium bioassay (Maron and Ames, 1983). Particulate matter from two engine conditions were gathered using a Micro-Orifice Uniform Deposition Impactor (MOUDI) for size selective mutagenic analysis.;Results of the mutagenicity study indicate differences in the mutagenic response of the PM soluble organic fraction (SOF) of both Federal diesel No. 2 and FT fuel as functions of engine operating conditions, fuel type and particle size. The extracted solubles from particles of aerodynamic diameters greater than 100 nm were found to exhibit significantly greater mutagenic effect than their smaller counterparts (\u3c100 nm) for both fuels. Results of the combustion and emission study revealed a general trend for lower emissions for FT fuel compared to DF fuel. NOx emissions correlated well with ignition delay and the amount of heat released in the premixed combustion phase. With the exception of two high load engine conditions, lower CO and total hydrocarbon (THC) emissions were the general trend for FT fuel.;Engine test facilities were located at the U.S. DOE\u27s National Energy Technology Laboratory (NETL) in Morgantown, WV. Particulate matter samples were collected in the NETL engine test cell. Measurement and extractions were also performed at NETL. The extracted PM was analyzed at the National Institute of Occupational Safety and Health (NIOSH), also in Morgantown, WV, to determine particulate matter in-vitro mutagenicity via the AMES bioassay method

Modelling CD4 count and mortality in a cohort of patients initiated on HAART.

Master of Science in Statistics. University of KwaZulu-Natal. Durban, 2018.Longitudinally measured data and time-to-event or survival data are often associated in some ways, and are traditionally analyzed separately (Asar et al., 2015). However, separate analyses are not applicable in this case because they may lead to inefficient or biased results. To remedy this, joint models optimally incorporate all available information (longitudinal and survival data) simultaneously (Wulfsohn & Tsiatis, 1997). Furthermore incorporating all sources of data improves the predictive capability of the joint model and lead to more informative inferences for the purpose of decision-making (Seyoum & Temesgen, 2017). The primary goal of this analysis was to determine the effect of repeatedly measured CD4 counts on mortality. The standard time-to-event models require that the time-dependent covariates of interest are external; where the value of the covariate at a future time point is not affected by the occurrence of the event. This requirement would not be fulfilled in this setting, since the repeatedly measured outcome is directly related to the mortality mechanism. Hence, a joint modeling approach was required. We applied the methods developed in this thesis to the CAPRISA AIDS Treatment program (CAT). We also sought to determine if the patients’ baseline BMI (Body mass index), baseline age, gender, baseline viral load, baseline CD8 count, baseline TB status and clinic site, influence the evolution of the CD4 count over time. Various linear mixed models were fitted to the CD4 count, adjusting for repeated measurements, as well as including intercept and slope as random effects. Different types of covariance structures were assessed and the spatial spherical correlation structure was found to be the best fit. The Cox PH model was employed to model mortality. Finally the joint model for longitudinal and time-to-event data was fitted. Out of the 4014 patients, 1457 (36.30%) were male. There were more patients presenting without TB at ART initiation, 3042 (75.78%) compared to those with prevalent TB, 972 (24.22%). Results from the multivariable random effects model showed that the patients gender, age, baseline viral load and baseline CD8 cell count had statistically significant influences on the rate of change in CD4 cell count over time. The un-adjusted and adjusted hazards regression both found CD4:CD8 ratio, viral load, gender and age of patients to be significant predictors of mortality. The result from the joint model in this study indicated that CD4 count change due to HAART and mortality had been influenced jointly by gender, age, baseline viral load, baseline CD8 count, time (in years) , CD4:CD8 ratio and by the interaction effects of time (in years) with TB status, baseline viral load and baseline CD8 cell count. CD4 count proved to be significantly associated with mortality, after adjusting for age, gender and other potential confounders Model diagnostics were performed for validating model assumptions, and our joint model fitted quite well with fairly good diagnostic attributes. The methods that were developed in this thesis were applied to the CAPRISA AIDS Treatment program (CAT) between June 2004 to December 2013