A Bayesian approach to tilted-ring modelling of galaxies
- Authors: Maina, Eric Kamau
- Date: 2020
- Subjects: Bayesian statistical decision theory , Galaxies , Radio astronomy , TiRiFiC (Tilted Ring Fitting Code) , Neutral hydrogen , Spectroscopic data cubes , Galaxy parametrisation
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/145783 , vital:38466
- Description: The orbits of neutral hydrogen (H I) gas found in most disk galaxies are circular and also exhibit long-lived warps at large radii where the restoring gravitational forces of the inner disk become weak (Spekkens and Giovanelli 2006). These warps make the tilted-ring model an ideal choice for galaxy parametrisation. Analysis software utilizing the tilted-ring-model can be grouped into two and three-dimensional based software. Józsa et al. (2007b) demonstrated that three dimensional based software is better suited for galaxy parametrisation because it is affected by the effect of beam smearing only by increasing the uncertainty of parameters but not with the notorious systematic effects observed for two-dimensional fitting techniques. TiRiFiC, The Tilted Ring Fitting Code (Józsa et al. 2007b), is a software to construct parameterised models of high-resolution data cubes of rotating galaxies. It uses the tilted-ring model, and with that, a combination of some parameters such as surface brightness, position angle, rotation velocity and inclination, to describe galaxies. TiRiFiC works by directly fitting tilted-ring models to spectroscopic data cubes and hence is not affected by beam smearing or line-of-site-effects, e.g. strong warps. Because of that, the method is unavoidable as an analytic method in future Hi surveys. In the current implementation, though, there are several drawbacks. The implemented optimisers search for local solutions in parameter space only, do not quantify correlations between parameters and cannot find errors of single parameters. In theory, these drawbacks can be overcome by using Bayesian statistics, implemented in Multinest (Feroz et al. 2008), as it allows for sampling a posterior distribution irrespective of its multimodal nature resulting in parameter samples that correspond to the maximum in the posterior distribution. These parameter samples can be used as well to quantify correlations and find errors of single parameters. Since this method employs Bayesian statistics, it also allows the user to leverage any prior information they may have on parameter values.
- Full Text:
- Authors: Maina, Eric Kamau
- Date: 2020
- Subjects: Bayesian statistical decision theory , Galaxies , Radio astronomy , TiRiFiC (Tilted Ring Fitting Code) , Neutral hydrogen , Spectroscopic data cubes , Galaxy parametrisation
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/145783 , vital:38466
- Description: The orbits of neutral hydrogen (H I) gas found in most disk galaxies are circular and also exhibit long-lived warps at large radii where the restoring gravitational forces of the inner disk become weak (Spekkens and Giovanelli 2006). These warps make the tilted-ring model an ideal choice for galaxy parametrisation. Analysis software utilizing the tilted-ring-model can be grouped into two and three-dimensional based software. Józsa et al. (2007b) demonstrated that three dimensional based software is better suited for galaxy parametrisation because it is affected by the effect of beam smearing only by increasing the uncertainty of parameters but not with the notorious systematic effects observed for two-dimensional fitting techniques. TiRiFiC, The Tilted Ring Fitting Code (Józsa et al. 2007b), is a software to construct parameterised models of high-resolution data cubes of rotating galaxies. It uses the tilted-ring model, and with that, a combination of some parameters such as surface brightness, position angle, rotation velocity and inclination, to describe galaxies. TiRiFiC works by directly fitting tilted-ring models to spectroscopic data cubes and hence is not affected by beam smearing or line-of-site-effects, e.g. strong warps. Because of that, the method is unavoidable as an analytic method in future Hi surveys. In the current implementation, though, there are several drawbacks. The implemented optimisers search for local solutions in parameter space only, do not quantify correlations between parameters and cannot find errors of single parameters. In theory, these drawbacks can be overcome by using Bayesian statistics, implemented in Multinest (Feroz et al. 2008), as it allows for sampling a posterior distribution irrespective of its multimodal nature resulting in parameter samples that correspond to the maximum in the posterior distribution. These parameter samples can be used as well to quantify correlations and find errors of single parameters. Since this method employs Bayesian statistics, it also allows the user to leverage any prior information they may have on parameter values.
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Bayesian accelerated life tests for the Weibull distribution under non-informative priors
- Authors: Mostert, Philip
- Date: 2020
- Subjects: Accelerated life testing -- Statistical methods , Accelerated life testing -- Mathematical models , Failure time data analysis , Bayesian statistical decision theory , Monte Carlo method , Weibull distribution
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/172181 , vital:42173
- Description: In a competitive world where products are designed to last for long periods of time, obtaining time-to-failure data is both difficult and costly. Hence for products with high reliability, accelerated life testing is required to obtain relevant life-data quickly. This is done by placing the products under higher-than-use stress levels, thereby causing the products to fail prematurely. Part of the analysis of accelerated life-data requires a life distribution that describes the lifetime of a product at a given stress level and a life-stress relationship – which is some function that describes the way in which the life distribution changes across different stress levels. In this thesis it is assumed that the underlying life distribution is the wellknown Weibull distribution, with shape parameter constant over all stress levels and scale parameter as a log-linear function of stress. The primary objective of this thesis is to obtain estimates from Bayesian analysis, and this thesis considers five types of non-informative prior distributions: Jeffreys’ prior, reference priors, maximal data information prior, uniform prior and probability matching priors. Since the associated posterior distribution under all the derived non-informative priors are of an unknown form, the propriety of the posterior distributions is assessed to ensure admissible results. For comparison purposes, estimates obtained via the method of maximum likelihood are also considered. Finding these estimates requires solving non-linear equations, hence the Newton-Raphson algorithm is used to obtain estimates. A simulation study based on the time-to-failure of accelerated data is conducted to compare results between maximum likelihood and Bayesian estimates. As a result of the Bayesian posterior distributions being analytically intractable, two methods to obtain Bayesian estimates are considered: Markov chain Monte Carlo methods and Lindley’s approximation technique. In the simulation study the posterior means and the root mean squared error values of the estimates under the symmetric squared error loss function and the two asymmetric loss functions: the LINEX loss function and general entropy loss function, are considered. Furthermore the coverage rates for the Bayesian Markov chain Monte Carlo and maximum likelihood estimates are found, and are compared by their average interval lengths. A case study using a dataset based on accelerated time-to-failure of an insulating fluid is considered. The fit of these data for the Weibull distribution is studied and is compared to that of other popular life distributions. A full simulation study is conducted to illustrate convergence of the proper posterior distributions. Both maximum likelihood and Bayesian estimates are found for these data. The deviance information criterion is used to compare Bayesian estimates between the prior distributions. The case study is concluded by finding reliability estimates of the data at use-stress levels.
- Full Text:
- Authors: Mostert, Philip
- Date: 2020
- Subjects: Accelerated life testing -- Statistical methods , Accelerated life testing -- Mathematical models , Failure time data analysis , Bayesian statistical decision theory , Monte Carlo method , Weibull distribution
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/172181 , vital:42173
- Description: In a competitive world where products are designed to last for long periods of time, obtaining time-to-failure data is both difficult and costly. Hence for products with high reliability, accelerated life testing is required to obtain relevant life-data quickly. This is done by placing the products under higher-than-use stress levels, thereby causing the products to fail prematurely. Part of the analysis of accelerated life-data requires a life distribution that describes the lifetime of a product at a given stress level and a life-stress relationship – which is some function that describes the way in which the life distribution changes across different stress levels. In this thesis it is assumed that the underlying life distribution is the wellknown Weibull distribution, with shape parameter constant over all stress levels and scale parameter as a log-linear function of stress. The primary objective of this thesis is to obtain estimates from Bayesian analysis, and this thesis considers five types of non-informative prior distributions: Jeffreys’ prior, reference priors, maximal data information prior, uniform prior and probability matching priors. Since the associated posterior distribution under all the derived non-informative priors are of an unknown form, the propriety of the posterior distributions is assessed to ensure admissible results. For comparison purposes, estimates obtained via the method of maximum likelihood are also considered. Finding these estimates requires solving non-linear equations, hence the Newton-Raphson algorithm is used to obtain estimates. A simulation study based on the time-to-failure of accelerated data is conducted to compare results between maximum likelihood and Bayesian estimates. As a result of the Bayesian posterior distributions being analytically intractable, two methods to obtain Bayesian estimates are considered: Markov chain Monte Carlo methods and Lindley’s approximation technique. In the simulation study the posterior means and the root mean squared error values of the estimates under the symmetric squared error loss function and the two asymmetric loss functions: the LINEX loss function and general entropy loss function, are considered. Furthermore the coverage rates for the Bayesian Markov chain Monte Carlo and maximum likelihood estimates are found, and are compared by their average interval lengths. A case study using a dataset based on accelerated time-to-failure of an insulating fluid is considered. The fit of these data for the Weibull distribution is studied and is compared to that of other popular life distributions. A full simulation study is conducted to illustrate convergence of the proper posterior distributions. Both maximum likelihood and Bayesian estimates are found for these data. The deviance information criterion is used to compare Bayesian estimates between the prior distributions. The case study is concluded by finding reliability estimates of the data at use-stress levels.
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Bayesian hierarchical modelling with application in spatial epidemiology
- Authors: Southey, Richard Robert
- Date: 2018
- Subjects: Bayesian statistical decision theory , Spatial analysis (Statistics) , Medical mapping , Pericarditis , Mortality Statistics
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/59489 , vital:27617
- Description: Disease mapping and spatial statistics have become an important part of modern day statistics and have increased in popularity as the methods and techniques have evolved. The application of disease mapping is not only confined to the analysis of diseases as other applications of disease mapping can be found in Econometric and financial disciplines. This thesis will consider two data sets. These are the Georgia oral cancer 2004 data set and the South African acute pericarditis 2014 data set. The Georgia data set will be used to assess the hyperprior sensitivity of the precision for the uncorrelated heterogeneity and correlated heterogeneity components in a convolution model. The correlated heterogeneity will be modelled by a conditional autoregressive prior distribution and the uncorrelated heterogeneity will be modelled with a zero mean Gaussian prior distribution. The sensitivity analysis will be performed using three models with conjugate, Jeffreys' and a fixed parameter prior for the hyperprior distribution of the precision for the uncorrelated heterogeneity component. A simulation study will be done to compare four prior distributions which will be the conjugate, Jeffreys', probability matching and divergence priors. The three models will be fitted in WinBUGS® using a Bayesian approach. The results of the three models will be in the form of disease maps, figures and tables. The results show that the hyperprior of the precision for the uncorrelated heterogeneity and correlated heterogeneity components are sensitive to changes and will result in different results depending on the specification of the hyperprior distribution of the precision for the two components in the model. The South African data set will be used to examine whether there is a difference between the proper conditional autoregressive prior and intrinsic conditional autoregressive prior for the correlated heterogeneity component in a convolution model. Two models will be fitted in WinBUGS® for this comparison. Both the hyperpriors of the precision for the uncorrelated heterogeneity and correlated heterogeneity components will be modelled using a Jeffreys' prior distribution. The results show that there is no significant difference between the results of the model with a proper conditional autoregressive prior and intrinsic conditional autoregressive prior for the South African data, although there are a few disadvantages of using a proper conditional autoregressive prior for the correlated heterogeneity which will be stated in the conclusion.
- Full Text:
- Authors: Southey, Richard Robert
- Date: 2018
- Subjects: Bayesian statistical decision theory , Spatial analysis (Statistics) , Medical mapping , Pericarditis , Mortality Statistics
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/59489 , vital:27617
- Description: Disease mapping and spatial statistics have become an important part of modern day statistics and have increased in popularity as the methods and techniques have evolved. The application of disease mapping is not only confined to the analysis of diseases as other applications of disease mapping can be found in Econometric and financial disciplines. This thesis will consider two data sets. These are the Georgia oral cancer 2004 data set and the South African acute pericarditis 2014 data set. The Georgia data set will be used to assess the hyperprior sensitivity of the precision for the uncorrelated heterogeneity and correlated heterogeneity components in a convolution model. The correlated heterogeneity will be modelled by a conditional autoregressive prior distribution and the uncorrelated heterogeneity will be modelled with a zero mean Gaussian prior distribution. The sensitivity analysis will be performed using three models with conjugate, Jeffreys' and a fixed parameter prior for the hyperprior distribution of the precision for the uncorrelated heterogeneity component. A simulation study will be done to compare four prior distributions which will be the conjugate, Jeffreys', probability matching and divergence priors. The three models will be fitted in WinBUGS® using a Bayesian approach. The results of the three models will be in the form of disease maps, figures and tables. The results show that the hyperprior of the precision for the uncorrelated heterogeneity and correlated heterogeneity components are sensitive to changes and will result in different results depending on the specification of the hyperprior distribution of the precision for the two components in the model. The South African data set will be used to examine whether there is a difference between the proper conditional autoregressive prior and intrinsic conditional autoregressive prior for the correlated heterogeneity component in a convolution model. Two models will be fitted in WinBUGS® for this comparison. Both the hyperpriors of the precision for the uncorrelated heterogeneity and correlated heterogeneity components will be modelled using a Jeffreys' prior distribution. The results show that there is no significant difference between the results of the model with a proper conditional autoregressive prior and intrinsic conditional autoregressive prior for the South African data, although there are a few disadvantages of using a proper conditional autoregressive prior for the correlated heterogeneity which will be stated in the conclusion.
- Full Text:
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