http://hdl.handle.net/10023/101 2013-06-19T08:33:20Z 2013-06-19T08:33:20Z Nightingale, Glenna Faith http://hdl.handle.net/10023/3710 2013-06-17T16:06:27Z 2013-06-28T00:00:00Z

Abstract: The modelling of biological communities is important to further the understanding of species coexistence and the mechanisms involved in maintaining biodiversity. This involves considering not only interactions between individual biological organisms, but also the incorporation of covariate information, if available, in the modelling process. This thesis explores the use of point processes to model interactions in bivariate point patterns within a Bayesian framework, and, where applicable, in conjunction with covariate data. Specifically, we distinguish between symmetric and asymmetric species interactions and model these using appropriate point processes. In this thesis we consider both pairwise and area interaction point processes to allow for inhibitory interactions and both inhibitory and attractive interactions. It is envisaged that the analyses and innovations presented in this thesis will contribute to the parsimonious modelling of biological communities.

2013-06-28T00:00:00Z Nightingale, Glenna Faith The modelling of biological communities is important to further the understanding of species coexistence and the mechanisms involved in maintaining biodiversity. This involves considering not only interactions between individual biological organisms, but also the incorporation of covariate information, if available, in the modelling process. This thesis explores the use of point processes to model interactions in bivariate point patterns within a Bayesian framework, and, where applicable, in conjunction with covariate data. Specifically, we distinguish between symmetric and asymmetric species interactions and model these using appropriate point processes. In this thesis we consider both pairwise and area interaction point processes to allow for inhibitory interactions and both inhibitory and attractive interactions. It is envisaged that the analyses and innovations presented in this thesis will contribute to the parsimonious modelling of biological communities. Gormley, Richard http://hdl.handle.net/10023/3655 2013-06-10T14:12:09Z 2012-01-01T00:00:00Z

Abstract: Mark-recapture is a method of population estimation that involves capturing a number of animals from a population of unknown size on several occasions, and marking those animals that are caught each time. By observing the number of marked animals that are subsequently seen, estimates of the total population size can be made. There are various subclasses of the mark-recapture method called the Otis-class of models (Otis, Burnham, White & Anderson 1978). These relate to the assumed behaviour of the individuals in the target population. More recent work has generalised the theory of mark-recapture to the so-called plant-capture, where a known number of animals are pre-inserted into the target population. Sampling is then carried out as normal, but with additional information coming from knowledge of the number of planted individuals. The theory underpinning plant-capture is less well-developed than mark-recapture, with the difference on population estimation of the former over the latter not often tested. This thesis shows that, under fixed and random sample-size models, the inclusion of plants can improve the mean point population estimation of various estimators. The estimator of Pathak (1964) is generalised to allow for the inclusion of plants into the target population. The results show that mean estimates from most estimators, under most models, can be improved with the inclusion of plants, and the sample standard deviations of the simulations can be reduced. This improvement in mean point population estimation is particularly pronounced when the number of animals captured is low. Sample coverage, which is the proportion of distinct animals caught during sampling, is also often sought by practitioners. Given here is a generalisation of the inverse population estimator of Pathak (1964) to plant-capture and a proposed new inverse population estimator, which can be used as estimates of the coverage of a sample.

2012-01-01T00:00:00Z Gormley, Richard Mark-recapture is a method of population estimation that involves capturing a number of animals from a population of unknown size on several occasions, and marking those animals that are caught each time. By observing the number of marked animals that are subsequently seen, estimates of the total population size can be made. There are various subclasses of the mark-recapture method called the Otis-class of models (Otis, Burnham, White & Anderson 1978). These relate to the assumed behaviour of the individuals in the target population. More recent work has generalised the theory of mark-recapture to the so-called plant-capture, where a known number of animals are pre-inserted into the target population. Sampling is then carried out as normal, but with additional information coming from knowledge of the number of planted individuals. The theory underpinning plant-capture is less well-developed than mark-recapture, with the difference on population estimation of the former over the latter not often tested. This thesis shows that, under fixed and random sample-size models, the inclusion of plants can improve the mean point population estimation of various estimators. The estimator of Pathak (1964) is generalised to allow for the inclusion of plants into the target population. The results show that mean estimates from most estimators, under most models, can be improved with the inclusion of plants, and the sample standard deviations of the simulations can be reduced. This improvement in mean point population estimation is particularly pronounced when the number of animals captured is low. Sample coverage, which is the proportion of distinct animals caught during sampling, is also often sought by practitioners. Given here is a generalisation of the inverse population estimator of Pathak (1964) to plant-capture and a proposed new inverse population estimator, which can be used as estimates of the coverage of a sample. Yuan, Yuan http://hdl.handle.net/10023/3652 2013-06-10T13:13:40Z 2012-01-01T00:00:00Z

Abstract: The content of this thesis was motivated by the need to estimate anglerfish abundance from stratified random trawl surveys of the anglerfish stock which occupies the northern European shelf (Fernandes et al., 2007). The survey was conducted annually from 2005 to 2010 in order to obtain age-structured estimates of absolute abundance for this stock. An estimation method is considered to incorporate statistical models for herding, length-based net retention probability and missing age data and uncertainty from all of these sources in variance estimation. A key component of abundance estimation is the estimation of capture probability. Capture probability is estimated from the experimental survey data using various logistic regression models with haul as a random effect. Conditional on the estimated capture probability, a number of abundance estimators are developed and applied to the anglerfish data. The abundance estimators differ in the way that the haul effect is incorporated. The performance of these estimators is investigated by simulation. An estimator with form similar to that conventionally used to estimate abundance from distance sampling surveys is found to perform best. The estimators developed for the anglerfish survey data which incorporate random effects in capture probability have wider application than trawl surveys. We examine the analytic properties of these estimators when the capture/detection probability is known. We apply these estimators to three different types of survey data in addition to the anglerfish data, with different forms of random effects and investigate their performance by simulation. We find that a generalization of the form of estimator typically used on line transect surveys performs best overall. It has low bias, and also the lowest bias and mean squared error among all the estimators we considered.

2012-01-01T00:00:00Z Yuan, Yuan The content of this thesis was motivated by the need to estimate anglerfish abundance from stratified random trawl surveys of the anglerfish stock which occupies the northern European shelf (Fernandes et al., 2007). The survey was conducted annually from 2005 to 2010 in order to obtain age-structured estimates of absolute abundance for this stock. An estimation method is considered to incorporate statistical models for herding, length-based net retention probability and missing age data and uncertainty from all of these sources in variance estimation. A key component of abundance estimation is the estimation of capture probability. Capture probability is estimated from the experimental survey data using various logistic regression models with haul as a random effect. Conditional on the estimated capture probability, a number of abundance estimators are developed and applied to the anglerfish data. The abundance estimators differ in the way that the haul effect is incorporated. The performance of these estimators is investigated by simulation. An estimator with form similar to that conventionally used to estimate abundance from distance sampling surveys is found to perform best. The estimators developed for the anglerfish survey data which incorporate random effects in capture probability have wider application than trawl surveys. We examine the analytic properties of these estimators when the capture/detection probability is known. We apply these estimators to three different types of survey data in addition to the anglerfish data, with different forms of random effects and investigate their performance by simulation. We find that a generalization of the form of estimator typically used on line transect surveys performs best overall. It has low bias, and also the lowest bias and mean squared error among all the estimators we considered. Oedekoven, Cornelia Sabrina http://hdl.handle.net/10023/3618 2013-06-07T08:20:33Z 2013-01-01T00:00:00Z

Abstract: Recently, much effort has been expended for improving conventional distance sampling methods, e.g. by replacing the design-based approach with a model-based approach where observed counts are related to environmental covariates (Hedley and Buckland, 2004) or by incorporating covariates in the detection function model (Marques and Buckland, 2003). While these models have generally been limited to include fixed effects, we propose four different methods for analysing distance sampling data using mixed effects models. These include an extension of the two-stage approach (Buckland et al., 2009), where we include site random effects in the second-stage count model to account for correlated counts at the same sites. We also present two integrated approaches which include site random effects in the count model. These approaches combine the analysis stages for the detection and count models and allow simultaneous estimation of all parameters. Furthermore, we develop a detection function model that incorporates random effects. We also propose a novel Bayesian approach to analysing distance sampling data which uses a Metropolis-Hastings algorithm for updating model parameters and a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm for assessing model uncertainty. Lastly, we propose using hierarchical centering as a novel technique for improving model mixing and hence facilitating an RJMCMC algorithm for mixed models. We analyse two case studies, both large-scale point transect surveys, where the interest lies in establishing the effects of conservation buffers on agricultural fields. For each case study, we compare the results from one integrated approach to those from the extended two-stage approach. We find that these may differ in parameter estimates for covariates that were both in the detection and the count model and in model probabilities when model uncertainty was included in inference. The performance of the random effects based detection function is assessed via simulation and when heterogeneity in the data is present, one of the new estimators yields improved results compared to conventional distance sampling estimators.

2013-01-01T00:00:00Z Oedekoven, Cornelia Sabrina Recently, much effort has been expended for improving conventional distance sampling methods, e.g. by replacing the design-based approach with a model-based approach where observed counts are related to environmental covariates (Hedley and Buckland, 2004) or by incorporating covariates in the detection function model (Marques and Buckland, 2003). While these models have generally been limited to include fixed effects, we propose four different methods for analysing distance sampling data using mixed effects models. These include an extension of the two-stage approach (Buckland et al., 2009), where we include site random effects in the second-stage count model to account for correlated counts at the same sites. We also present two integrated approaches which include site random effects in the count model. These approaches combine the analysis stages for the detection and count models and allow simultaneous estimation of all parameters. Furthermore, we develop a detection function model that incorporates random effects. We also propose a novel Bayesian approach to analysing distance sampling data which uses a Metropolis-Hastings algorithm for updating model parameters and a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm for assessing model uncertainty. Lastly, we propose using hierarchical centering as a novel technique for improving model mixing and hence facilitating an RJMCMC algorithm for mixed models. We analyse two case studies, both large-scale point transect surveys, where the interest lies in establishing the effects of conservation buffers on agricultural fields. For each case study, we compare the results from one integrated approach to those from the extended two-stage approach. We find that these may differ in parameter estimates for covariates that were both in the detection and the count model and in model probabilities when model uncertainty was included in inference. The performance of the random effects based detection function is assessed via simulation and when heterogeneity in the data is present, one of the new estimators yields improved results compared to conventional distance sampling estimators. Studeny, Angelika C. http://hdl.handle.net/10023/3414 2013-03-22T09:32:00Z 2012-11-30T00:00:00Z

Abstract: The global loss of biodiversity calls for robust large-scale diversity assessment. Biological diversity is a multi-faceted concept; defined as the “variety of life”, answering questions such as “How much is there?” or more precisely “Have we succeeded in reducing the rate of its decline?” is not straightforward. While various aspects of biodiversity give rise to numerous ways of quantification, we focus on temporal (and spatial) trends and their changes in species diversity. Traditional diversity indices summarise information contained in the species abundance distribution, i.e. each species' proportional contribution to total abundance. Estimated from data, these indices can be biased if variation in detection probability is ignored. We discuss differences between diversity indices and demonstrate possible adjustments for detectability. Additionally, most indices focus on the most abundant species in ecological communities. We introduce a new set of diversity measures, based on a family of goodness-of-fit statistics. A function of a free parameter, this family allows us to vary the sensitivity of these measures to dominance and rarity of species. Their performance is studied by assessing temporal trends in diversity for five communities of British breeding birds based on 14 years of survey data, where they are applied alongside the current headline index, a geometric mean of relative abundances. Revealing the contributions of both rare and common species to biodiversity trends, these "goodness-of-fit" measures provide novel insights into how ecological communities change over time. Biodiversity is not only subject to temporal changes, but it also varies across space. We take first steps towards estimating spatial diversity trends. Finally, processes maintaining biodiversity act locally, at specific spatial scales. Contrary to abundance-based summary statistics, spatial characteristics of ecological communities may distinguish these processes. We suggest a generalisation to a spatial summary, the cross-pair overlap distribution, to render it more flexible to spatial scale.

2012-11-30T00:00:00Z Studeny, Angelika C. The global loss of biodiversity calls for robust large-scale diversity assessment. Biological diversity is a multi-faceted concept; defined as the “variety of life”, answering questions such as “How much is there?” or more precisely “Have we succeeded in reducing the rate of its decline?” is not straightforward. While various aspects of biodiversity give rise to numerous ways of quantification, we focus on temporal (and spatial) trends and their changes in species diversity. Traditional diversity indices summarise information contained in the species abundance distribution, i.e. each species' proportional contribution to total abundance. Estimated from data, these indices can be biased if variation in detection probability is ignored. We discuss differences between diversity indices and demonstrate possible adjustments for detectability. Additionally, most indices focus on the most abundant species in ecological communities. We introduce a new set of diversity measures, based on a family of goodness-of-fit statistics. A function of a free parameter, this family allows us to vary the sensitivity of these measures to dominance and rarity of species. Their performance is studied by assessing temporal trends in diversity for five communities of British breeding birds based on 14 years of survey data, where they are applied alongside the current headline index, a geometric mean of relative abundances. Revealing the contributions of both rare and common species to biodiversity trends, these "goodness-of-fit" measures provide novel insights into how ecological communities change over time. Biodiversity is not only subject to temporal changes, but it also varies across space. We take first steps towards estimating spatial diversity trends. Finally, processes maintaining biodiversity act locally, at specific spatial scales. Contrary to abundance-based summary statistics, spatial characteristics of ecological communities may distinguish these processes. We suggest a generalisation to a spatial summary, the cross-pair overlap distribution, to render it more flexible to spatial scale. Munday, Sara Ann http://hdl.handle.net/10023/3220 2012-10-24T14:10:53Z 2011-11-30T00:00:00Z

Abstract: The first main topic of this thesis is the thorough analysis of two families of piecewise linear maps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinite partition of the unit interval whose atoms only accumulate at the origin. The basic properties of these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to a series expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The first example of such an expansion was given by Lüroth. The map Lα is the jump transformation of the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as the classical Farey and Gauss maps which give rise to the continued fraction expansion of a real number. We also consider the topological properties of Fα and some Diophantine-type sets of numbers expressed in terms of the α-Lüroth expansion. Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turn out that the Lebesgue measure λ is invariant for every map Lα and that there exists a unique Lebesgue-absolutely continuous invariant measure for Fα. We will give a precise expression for the density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic. The interest in the invariant measure for Fα lies in the fact that under a particular condition on the underlying partition α, the invariant measure associated to the map Fα is infinite. Then we proceed to introduce and examine the sequence of α-sum-level sets arising from the α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically in terms of Fα. The main result concerning the α-sum-level sets is to establish weak and strong renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate estimates in renewal theory. Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometric description of the Lyapunov spectra associated with each of the maps Lα and Fα. The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the systems we consider are always piecewise linear. However, since our analysis is based on a large family of different partitions of U , the class of maps which we consider in this paper allows us to detect a variety of interesting new phenomena, including that of phase transitions. Finally, we come to the conformal systems of the title. These are the limit sets of discrete subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type sets contained in the limit set that are similar to those considered for the maps Lα. These sets are then used in our second main result to analyse the more geometrically defined strict-Jarník limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson measure associated to the Fuchsian group.

2011-11-30T00:00:00Z Munday, Sara Ann The first main topic of this thesis is the thorough analysis of two families of piecewise linear maps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinite partition of the unit interval whose atoms only accumulate at the origin. The basic properties of these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to a series expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The first example of such an expansion was given by Lüroth. The map Lα is the jump transformation of the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as the classical Farey and Gauss maps which give rise to the continued fraction expansion of a real number. We also consider the topological properties of Fα and some Diophantine-type sets of numbers expressed in terms of the α-Lüroth expansion. Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turn out that the Lebesgue measure λ is invariant for every map Lα and that there exists a unique Lebesgue-absolutely continuous invariant measure for Fα. We will give a precise expression for the density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic. The interest in the invariant measure for Fα lies in the fact that under a particular condition on the underlying partition α, the invariant measure associated to the map Fα is infinite. Then we proceed to introduce and examine the sequence of α-sum-level sets arising from the α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically in terms of Fα. The main result concerning the α-sum-level sets is to establish weak and strong renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate estimates in renewal theory. Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometric description of the Lyapunov spectra associated with each of the maps Lα and Fα. The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the systems we consider are always piecewise linear. However, since our analysis is based on a large family of different partitions of U , the class of maps which we consider in this paper allows us to detect a variety of interesting new phenomena, including that of phase transitions. Finally, we come to the conformal systems of the title. These are the limit sets of discrete subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type sets contained in the limit set that are similar to those considered for the maps Lα. These sets are then used in our second main result to analyse the more geometrically defined strict-Jarník limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson measure associated to the Fuchsian group. Brown, Calum http://hdl.handle.net/10023/3084 2012-10-24T11:38:02Z 2012-11-01T00:00:00Z

Abstract: The use of spatial statistics to investigate ecological processes in plant communities is becoming increasingly widespread. In diverse communities such as tropical rainforests, analysis of spatial structure may help to unravel the various processes that act and interact to maintain high levels of diversity. In particular, a number of contrasting mechanisms have been suggested to explain species coexistence, and these differ greatly in their practical implications for the ecology and conservation of tropical forests. Traditional first-order measures of community structure have proved unable to distinguish these mechanisms in practice, but statistics that describe spatial structure may be able to do so. This is of great interest and relevance as spatially explicit data become available for a range of ecological communities and analysis methods for these data become more accessible. This thesis investigates the potential for inference about underlying ecological processes in plant communities using spatial statistics. Current methodologies for spatial analysis are reviewed and extended, and are used to characterise the spatial signals of the principal theorised mechanisms of coexistence. The sensitivity of a range of spatial statistics to these signals is assessed, and the strength of such signals in natural communities is investigated. The spatial signals of the processes considered here are found to be strong and robust to modelled stochastic variation. Several new and existing spatial statistics are found to be sensitive to these signals, and offer great promise for inference about underlying processes from empirical data. The relative strengths of particular processes are found to vary between natural communities, with any one theory being insufficient to explain observed patterns. This thesis extends both understanding of species coexistence in diverse plant communities and the methodology for assessing underlying process in particular cases. It demonstrates that the potential of spatial statistics in ecology is great and largely unexplored.

2012-11-01T00:00:00Z Brown, Calum The use of spatial statistics to investigate ecological processes in plant communities is becoming increasingly widespread. In diverse communities such as tropical rainforests, analysis of spatial structure may help to unravel the various processes that act and interact to maintain high levels of diversity. In particular, a number of contrasting mechanisms have been suggested to explain species coexistence, and these differ greatly in their practical implications for the ecology and conservation of tropical forests. Traditional first-order measures of community structure have proved unable to distinguish these mechanisms in practice, but statistics that describe spatial structure may be able to do so. This is of great interest and relevance as spatially explicit data become available for a range of ecological communities and analysis methods for these data become more accessible. This thesis investigates the potential for inference about underlying ecological processes in plant communities using spatial statistics. Current methodologies for spatial analysis are reviewed and extended, and are used to characterise the spatial signals of the principal theorised mechanisms of coexistence. The sensitivity of a range of spatial statistics to these signals is assessed, and the strength of such signals in natural communities is investigated. The spatial signals of the processes considered here are found to be strong and robust to modelled stochastic variation. Several new and existing spatial statistics are found to be sensitive to these signals, and offer great promise for inference about underlying processes from empirical data. The relative strengths of particular processes are found to vary between natural communities, with any one theory being insufficient to explain observed patterns. This thesis extends both understanding of species coexistence in diverse plant communities and the methodology for assessing underlying process in particular cases. It demonstrates that the potential of spatial statistics in ecology is great and largely unexplored. Potts, Joanne M. http://hdl.handle.net/10023/2068 2013-04-15T12:04:48Z 2011-01-01T00:00:00Z

Abstract: Estimates of animal abundance or density are fundamental quantities in ecology and conservation, but for many species such as rare, small mammals, obtaining robust estimates is problematic. In this thesis, I combine elements of two standard abundance estimation methods, capture-recapture and distance sampling, to develop a method called trapping point transects (TPT). In TPT, a "detection function", g(r) (i.e. the probability of capturing an animal, given it is r m from a trap when the trap is set) is estimated using a subset of animals whose locations are known prior to traps being set. Generalised linear models are used to estimate the detection function, and the model can be extended to include random effects to allow for heterogeneity in capture probabilities. Standard point transect methods are modified to estimate abundance. Two abundance estimators are available. The first estimator is based on the reciprocal of the expected probability of detecting an animal, ^P, where the expectation is over r; whereas the second estimator is the expectation of the reciprocal of ^P. Performance of the TPT method under various sampling efforts and underlying true detection probabilities of individuals in the population was investigated in a simulation study. When underlying probability of detection was high (g(0) = 0:88) and between-individual variation was small, survey effort could be surprisingly low (c. 510 trap nights) to yield low bias (c. 4%) in the two estimators; but under certain situations, the second estimator can be extremely biased. Uncertainty and relative bias in population estimates increased with decreasing detectability and increasing between-individual variation. Abundance of the Key Largo woodrat (Neotoma floridana smalli), an endangered rodent with a restricted geographic range, was estimated using TPT. The TPT method compared well to other viable methods (capture-recapture and spatially-explicit capture-recapture), in terms of both field practicality and cost. The TPT method may generally be useful in estimating animal abundance in trapping studies and variants of the TPT method are presented.

2011-01-01T00:00:00Z Potts, Joanne M. Estimates of animal abundance or density are fundamental quantities in ecology and conservation, but for many species such as rare, small mammals, obtaining robust estimates is problematic. In this thesis, I combine elements of two standard abundance estimation methods, capture-recapture and distance sampling, to develop a method called trapping point transects (TPT). In TPT, a "detection function", g(r) (i.e. the probability of capturing an animal, given it is r m from a trap when the trap is set) is estimated using a subset of animals whose locations are known prior to traps being set. Generalised linear models are used to estimate the detection function, and the model can be extended to include random effects to allow for heterogeneity in capture probabilities. Standard point transect methods are modified to estimate abundance. Two abundance estimators are available. The first estimator is based on the reciprocal of the expected probability of detecting an animal, ^P, where the expectation is over r; whereas the second estimator is the expectation of the reciprocal of ^P. Performance of the TPT method under various sampling efforts and underlying true detection probabilities of individuals in the population was investigated in a simulation study. When underlying probability of detection was high (g(0) = 0:88) and between-individual variation was small, survey effort could be surprisingly low (c. 510 trap nights) to yield low bias (c. 4%) in the two estimators; but under certain situations, the second estimator can be extremely biased. Uncertainty and relative bias in population estimates increased with decreasing detectability and increasing between-individual variation. Abundance of the Key Largo woodrat (Neotoma floridana smalli), an endangered rodent with a restricted geographic range, was estimated using TPT. The TPT method compared well to other viable methods (capture-recapture and spatially-explicit capture-recapture), in terms of both field practicality and cost. The TPT method may generally be useful in estimating animal abundance in trapping studies and variants of the TPT method are presented. Reynolds, Toby J. http://hdl.handle.net/10023/1635 2010-12-13T16:48:23Z 2010-11-30T00:00:00Z

Abstract: Integrated data analyses are becoming increasingly popular in studies of wild animal populations where two or more separate sources of data contain information about common parameters. Here we develop an integrated population model using abundance and demographic data from a study of common guillemots (Uria aalge) on the Isle of May, southeast Scotland. A state-space model for the count data is supplemented by three demographic time series (productivity and two mark-recapture-recovery (MRR)), enabling the estimation of prebreeder emigration rate - a parameter for which there is no direct observational data, and which is unidentifiable in the separate analysis of MRR data. A Bayesian approach using MCMC provides a flexible and powerful analysis framework. This model is extended to provide predictions of future population trajectories. Adopting random effects models for the survival and productivity parameters, we implement the MCMC algorithm to obtain a posterior sample of the underlying process means and variances (and population sizes) within the study period. Given this sample, we predict future demographic parameters, which in turn allows us to predict future population sizes and obtain the corresponding posterior distribution. Under the assumption that recent, unfavourable conditions persist in the future, we obtain a posterior probability of 70% that there is a population decline of >25% over a 10-year period. Lastly, using MRR data we test for spatial, temporal and age-related correlations in guillemot survival among three widely separated Scottish colonies that have varying overlap in nonbreeding distribution. We show that survival is highly correlated over time for colonies/age classes sharing wintering areas, and essentially uncorrelated for those with separate wintering areas. These results strongly suggest that one or more aspects of winter environment are responsible for spatiotemporal variation in survival of British guillemots, and provide insight into the factors driving multi-population dynamics of the species.

2010-11-30T00:00:00Z Reynolds, Toby J. Integrated data analyses are becoming increasingly popular in studies of wild animal populations where two or more separate sources of data contain information about common parameters. Here we develop an integrated population model using abundance and demographic data from a study of common guillemots (Uria aalge) on the Isle of May, southeast Scotland. A state-space model for the count data is supplemented by three demographic time series (productivity and two mark-recapture-recovery (MRR)), enabling the estimation of prebreeder emigration rate - a parameter for which there is no direct observational data, and which is unidentifiable in the separate analysis of MRR data. A Bayesian approach using MCMC provides a flexible and powerful analysis framework. This model is extended to provide predictions of future population trajectories. Adopting random effects models for the survival and productivity parameters, we implement the MCMC algorithm to obtain a posterior sample of the underlying process means and variances (and population sizes) within the study period. Given this sample, we predict future demographic parameters, which in turn allows us to predict future population sizes and obtain the corresponding posterior distribution. Under the assumption that recent, unfavourable conditions persist in the future, we obtain a posterior probability of 70% that there is a population decline of >25% over a 10-year period. Lastly, using MRR data we test for spatial, temporal and age-related correlations in guillemot survival among three widely separated Scottish colonies that have varying overlap in nonbreeding distribution. We show that survival is highly correlated over time for colonies/age classes sharing wintering areas, and essentially uncorrelated for those with separate wintering areas. These results strongly suggest that one or more aspects of winter environment are responsible for spatiotemporal variation in survival of British guillemots, and provide insight into the factors driving multi-population dynamics of the species. Cave, Vanessa M. http://hdl.handle.net/10023/885 2010-12-13T16:13:06Z 2010-06-25T00:00:00Z

Abstract: To underpin and improve advice given to government and other interested parties on the state of Britain’s common songbird populations, new models for analysing ecological data are developed in this thesis. These models use data from the British Trust for Ornithology’s Constant Effort Sites (CES) scheme, an annual bird-ringing programme in which catch effort is standardised. Data from the CES scheme are routinely used to index abundance and productivity, and to a lesser extent estimate adult survival rates. However, two features of the CES data that complicate analysis were previously inadequately addressed, namely the presence in the catch of “transient” birds not associated with the local population, and the sporadic failure in the constancy of effort assumption arising from the absence of within-year catch data. The current methodology is extended, with efficient Bayesian models developed for each of these demographic parameters that account for both of these data nuances, and from which reliable and usefully precise estimates are obtained. Of increasing interest is the relationship between abundance and the underlying vital rates, an understanding of which facilitates effective conservation. CES data are particularly amenable to an integrated approach to population modelling, providing a combination of demographic information from a single source. Such an integrated approach is developed here, employing Bayesian methodology and a simple population model to unite abundance, productivity and survival within a consistent framework. Independent data from ring-recoveries provide additional information on adult and juvenile survival rates. Specific advantages of this new integrated approach are identified, among which is the ability to determine juvenile survival accurately, disentangle the probabilities of survival and permanent emigration, and to obtain estimates of total seasonal productivity. The methodologies developed in this thesis are applied to CES data from Sedge Warbler, Acrocephalus schoenobaenus, and Reed Warbler, A. scirpaceus.

2010-06-25T00:00:00Z Cave, Vanessa M. To underpin and improve advice given to government and other interested parties on the state of Britain’s common songbird populations, new models for analysing ecological data are developed in this thesis. These models use data from the British Trust for Ornithology’s Constant Effort Sites (CES) scheme, an annual bird-ringing programme in which catch effort is standardised. Data from the CES scheme are routinely used to index abundance and productivity, and to a lesser extent estimate adult survival rates. However, two features of the CES data that complicate analysis were previously inadequately addressed, namely the presence in the catch of “transient” birds not associated with the local population, and the sporadic failure in the constancy of effort assumption arising from the absence of within-year catch data. The current methodology is extended, with efficient Bayesian models developed for each of these demographic parameters that account for both of these data nuances, and from which reliable and usefully precise estimates are obtained. Of increasing interest is the relationship between abundance and the underlying vital rates, an understanding of which facilitates effective conservation. CES data are particularly amenable to an integrated approach to population modelling, providing a combination of demographic information from a single source. Such an integrated approach is developed here, employing Bayesian methodology and a simple population model to unite abundance, productivity and survival within a consistent framework. Independent data from ring-recoveries provide additional information on adult and juvenile survival rates. Specific advantages of this new integrated approach are identified, among which is the ability to determine juvenile survival accurately, disentangle the probabilities of survival and permanent emigration, and to obtain estimates of total seasonal productivity. The methodologies developed in this thesis are applied to CES data from Sedge Warbler, Acrocephalus schoenobaenus, and Reed Warbler, A. scirpaceus. O'Loan, Caleb J. http://hdl.handle.net/10023/869 2010-11-10T14:20:54Z 2010-06-23T00:00:00Z

Abstract: A quantum channel is a mapping which sends density matrices to density matrices. The estimation of quantum channels is of great importance to the field of quantum information. In this thesis two topics related to estimation of quantum channels are investigated. The first of these is the upper bound of Sarovar and Milburn (2006) on the Fisher information obtainable by measuring the output of a channel. Two questions raised by Sarovar and Milburn about their bound are answered. A Riemannian metric on the space of quantum states is introduced, related to the construction of the Sarovar and Milburn bound. Its properties are characterized. The second topic investigated is the estimation of unitary channels. The situation is considered in which an experimenter has several non-identical unitary channels that have the same parameter. It is shown that it is possible to improve estimation using the channels together, analogous to the case of identical unitary channels. Also, a new method of phase estimation is given based on a method sketched by Kitaev (1996). Unlike other phase estimation procedures which perform similarly, this procedure requires only very basic experimental resources.

2010-06-23T00:00:00Z O'Loan, Caleb J. A quantum channel is a mapping which sends density matrices to density matrices. The estimation of quantum channels is of great importance to the field of quantum information. In this thesis two topics related to estimation of quantum channels are investigated. The first of these is the upper bound of Sarovar and Milburn (2006) on the Fisher information obtainable by measuring the output of a channel. Two questions raised by Sarovar and Milburn about their bound are answered. A Riemannian metric on the space of quantum states is introduced, related to the construction of the Sarovar and Milburn bound. Its properties are characterized. The second topic investigated is the estimation of unitary channels. The situation is considered in which an experimenter has several non-identical unitary channels that have the same parameter. It is shown that it is possible to improve estimation using the channels together, analogous to the case of identical unitary channels. Also, a new method of phase estimation is given based on a method sketched by Kitaev (1996). Unlike other phase estimation procedures which perform similarly, this procedure requires only very basic experimental resources. New, Leslie Frances http://hdl.handle.net/10023/864 2013-06-07T13:45:26Z 2010-06-23T00:00:00Z

Abstract: State-space modelling is a powerful tool to study ecological systems. The direct inclusion of uncertainty, unification of models and data, and ability to model unobserved, hidden states increases our knowledge about the environment and provides new ecological insights. I extend the state-space framework to create multi-species models, showing that the ability to model ecosystem interactions is limited only by data availability. State-space models are fit using both Bayesian and Frequentist methods, making them independent of a statistical school of thought. Bayesian approaches can have the advantage in their ability to account for missing data and fit hierarchical structures and models with many parameters to limited data; often the case in ecological studies. I have taken a Bayesian model fitting approach in this thesis. The predator-prey interactions between the hen harrier (Circus cyaneus) and red grouse (Lagopus lagopus scoticus) are used to demonstrate state-space modelling’s capabilities. The harrier data are believed to be known without error, while missing data make the cyclic dynamics of the grouse harder to model. The grouse-harrier interactions are modelled in a multi-species state-space model, rather than including one species as a covariate in the other’s model. Finally, models are included for the harriers’ alternate prey. The single- and multi-species state-space models for the predator-prey interactions provide insight into the species’ management. The models investigate aspects of the species’ behaviour, from the mechanisms behind grouse cycles to what motivates harrier immigration. The inferences drawn from these models are applicable to management, suggesting actions to halt grouse cycles or mitigate the grouse-harrier conflict. Overall, the multi-species models suggest that two popular ideas for grouse-harrier management, diversionary feeding and habitat manipulation to reduce alternate prey densities, will not have the desired effect, and in the case of reducing prey densities, may even increase the harriers’ impact on grouse chicks.

2010-06-23T00:00:00Z New, Leslie Frances State-space modelling is a powerful tool to study ecological systems. The direct inclusion of uncertainty, unification of models and data, and ability to model unobserved, hidden states increases our knowledge about the environment and provides new ecological insights. I extend the state-space framework to create multi-species models, showing that the ability to model ecosystem interactions is limited only by data availability. State-space models are fit using both Bayesian and Frequentist methods, making them independent of a statistical school of thought. Bayesian approaches can have the advantage in their ability to account for missing data and fit hierarchical structures and models with many parameters to limited data; often the case in ecological studies. I have taken a Bayesian model fitting approach in this thesis. The predator-prey interactions between the hen harrier (Circus cyaneus) and red grouse (Lagopus lagopus scoticus) are used to demonstrate state-space modelling’s capabilities. The harrier data are believed to be known without error, while missing data make the cyclic dynamics of the grouse harder to model. The grouse-harrier interactions are modelled in a multi-species state-space model, rather than including one species as a covariate in the other’s model. Finally, models are included for the harriers’ alternate prey. The single- and multi-species state-space models for the predator-prey interactions provide insight into the species’ management. The models investigate aspects of the species’ behaviour, from the mechanisms behind grouse cycles to what motivates harrier immigration. The inferences drawn from these models are applicable to management, suggesting actions to halt grouse cycles or mitigate the grouse-harrier conflict. Overall, the multi-species models suggest that two popular ideas for grouse-harrier management, diversionary feeding and habitat manipulation to reduce alternate prey densities, will not have the desired effect, and in the case of reducing prey densities, may even increase the harriers’ impact on grouse chicks. Bishop, Jonathan R. B. http://hdl.handle.net/10023/718 2010-12-14T09:07:50Z 2009-06-24T00:00:00Z

Abstract: Mark-recapture methods use repeated captures of individually identifiable animals to provide estimates of properties of populations. Different models allow estimates to be obtained for population size and rates of processes governing population dynamics. State-space models consist of two linked processes evolving simultaneously over time. The state process models the evolution of the true, but unknown, states of the population. The observation process relates observations on the population to these true states. Mark-recapture models specified within a state-space framework allow population dynamics models to be embedded in inference ensuring that estimated changes in the population are consistent with assumptions regarding the biology of the modelled population. This overcomes a limitation of current mark-recapture methods. Two alternative approaches are considered. The "conditional" approach conditions on known numbers of animals possessing capture history patterns including capture in the current time period. An animal's capture history determines its state; consequently, capture parameters appear in the state process rather than the observation process. There is no observation error in the model. Uncertainty occurs only through the numbers of animals not captured in the current time period. An "unconditional" approach is considered in which the capture histories are regarded as observations. Consequently, capture histories do not influence an animal's state and capture probability parameters appear in the observation process. Capture histories are considered a random realization of the stochastic observation process. This is more consistent with traditional mark-recapture methods. Development and implementation of particle filtering techniques for fitting these models under each approach are discussed. Simulation studies show reasonable performance for the unconditional approach and highlight problems with the conditional approach. Strengths and limitations of each approach are outlined, with reference to Soay sheep data analysis, and suggestions are presented for future analyses.

2009-06-24T00:00:00Z Bishop, Jonathan R. B. Mark-recapture methods use repeated captures of individually identifiable animals to provide estimates of properties of populations. Different models allow estimates to be obtained for population size and rates of processes governing population dynamics. State-space models consist of two linked processes evolving simultaneously over time. The state process models the evolution of the true, but unknown, states of the population. The observation process relates observations on the population to these true states. Mark-recapture models specified within a state-space framework allow population dynamics models to be embedded in inference ensuring that estimated changes in the population are consistent with assumptions regarding the biology of the modelled population. This overcomes a limitation of current mark-recapture methods. Two alternative approaches are considered. The "conditional" approach conditions on known numbers of animals possessing capture history patterns including capture in the current time period. An animal's capture history determines its state; consequently, capture parameters appear in the state process rather than the observation process. There is no observation error in the model. Uncertainty occurs only through the numbers of animals not captured in the current time period. An "unconditional" approach is considered in which the capture histories are regarded as observations. Consequently, capture histories do not influence an animal's state and capture probability parameters appear in the observation process. Capture histories are considered a random realization of the stochastic observation process. This is more consistent with traditional mark-recapture methods. Development and implementation of particle filtering techniques for fitting these models under each approach are discussed. Simulation studies show reasonable performance for the unconditional approach and highlight problems with the conditional approach. Strengths and limitations of each approach are outlined, with reference to Soay sheep data analysis, and suggestions are presented for future analyses. Brewer, Ciara http://hdl.handle.net/10023/488 2008-06-18T09:51:15Z 2008-06-01T00:00:00Z

Abstract: Surveying animal populations is an important aspect of wildlife management. Distinguishing trend from random fluctuations and quantifying trend are key goals in any analysis. The aim of this thesis is to review analyses of Butterfly Monitoring Survey (BMS) data and to develop new methods which address some flaws in previous studies. The BMS was established in 1976 at Monks Wood, Cambridgeshire and sites were added over time throughout Britain in order to monitor butterfly population trends. Weekly counts are made over the monitoring season and the main aims are to produce annual indices and compare these indices over time for any particular species. Originally, weekly counts were summed to produce relative indices and missing counts were estimated using linear interpolation. This thesis discusses the weaknesses of this basic method and suggests possible improvements. In recent years, with advancements in statistical methods and increased computer power, new methods can be applied to accommodate the longitudinal and flexible nature of ecological data. Mixed Models, Generalized Estimating Equations and Generalized Additive Models are used and the relative merits of each modelling approach discussed. These methods allow for correlation and non-linearity in data. Model selection is an important consideration when modelling and different tests are introduced and compared. Once a model is selected, site-level indices are estimated, which can be collated to produce regional and national indices. Different methods of estimating precision around indices are also contrasted. Bootstrapping is found to be a convenient and dependable approach. Abundance is difficult to disentangle from detectability when only counts of species are carried out. Methods for dealing with this problem are suggested. Once reliable annual abundance estimates are found, they can be compared over time using a variety of statistical techniques. The chain-ratio method is applied to a subset of real data.

2008-06-01T00:00:00Z Brewer, Ciara Surveying animal populations is an important aspect of wildlife management. Distinguishing trend from random fluctuations and quantifying trend are key goals in any analysis. The aim of this thesis is to review analyses of Butterfly Monitoring Survey (BMS) data and to develop new methods which address some flaws in previous studies. The BMS was established in 1976 at Monks Wood, Cambridgeshire and sites were added over time throughout Britain in order to monitor butterfly population trends. Weekly counts are made over the monitoring season and the main aims are to produce annual indices and compare these indices over time for any particular species. Originally, weekly counts were summed to produce relative indices and missing counts were estimated using linear interpolation. This thesis discusses the weaknesses of this basic method and suggests possible improvements. In recent years, with advancements in statistical methods and increased computer power, new methods can be applied to accommodate the longitudinal and flexible nature of ecological data. Mixed Models, Generalized Estimating Equations and Generalized Additive Models are used and the relative merits of each modelling approach discussed. These methods allow for correlation and non-linearity in data. Model selection is an important consideration when modelling and different tests are introduced and compared. Once a model is selected, site-level indices are estimated, which can be collated to produce regional and national indices. Different methods of estimating precision around indices are also contrasted. Bootstrapping is found to be a convenient and dependable approach. Abundance is difficult to disentangle from detectability when only counts of species are carried out. Methods for dealing with this problem are suggested. Once reliable annual abundance estimates are found, they can be compared over time using a variety of statistical techniques. The chain-ratio method is applied to a subset of real data. Marques, Tiago Andre Lamas Oliveira http://hdl.handle.net/10023/391 2011-06-17T13:18:04Z 2007-11-01T00:00:00Z

Abstract: Distance sampling is one of the most commonly used methods for estimating density and abundance. Conventional methods are based on the distances of detected animals from the center of point transects or the center line of line transects. These distances are used to model a detection function: the probability of detecting an animal, given its distance from the line or point. The probability of detecting an animal in the covered area is given by the mean value of the detection function with respect to the available distances to be detected. Given this probability, a Horvitz-Thompson- like estimator of abundance for the covered area follows, hence using a model-based framework. Inferences for the wider survey region are justified using the survey design. Conventional distance sampling methods are based on a set of assumptions. In this thesis I present results that extend distance sampling on two fronts. Firstly, estimators are derived for situations in which there is measurement error in the distances. These estimators use information about the measurement error in two ways: (1) a biased estimator based on the contaminated distances is multiplied by an appropriate correction factor, which is a function of the errors (PDF approach), and (2) cast into a likelihood framework that allows parameter estimation in the presence of measurement error (likelihood approach). Secondly, methods are developed that relax the conventional assumption that the distribution of animals is independent of distance from the lines or points (usually guaranteed by appropriate survey design). In particular, the new methods deal with the case where animal density gradients are caused by the use of non-random sampler allocation, for example transects placed along linear features such as roads or streams. This is dealt with separately for line and point transects, and at a later stage an approach for combining the two is presented. A considerable number of simulations and example analysis illustrate the performance of the proposed methods.

2007-11-01T00:00:00Z Marques, Tiago Andre Lamas Oliveira Distance sampling is one of the most commonly used methods for estimating density and abundance. Conventional methods are based on the distances of detected animals from the center of point transects or the center line of line transects. These distances are used to model a detection function: the probability of detecting an animal, given its distance from the line or point. The probability of detecting an animal in the covered area is given by the mean value of the detection function with respect to the available distances to be detected. Given this probability, a Horvitz-Thompson- like estimator of abundance for the covered area follows, hence using a model-based framework. Inferences for the wider survey region are justified using the survey design. Conventional distance sampling methods are based on a set of assumptions. In this thesis I present results that extend distance sampling on two fronts. Firstly, estimators are derived for situations in which there is measurement error in the distances. These estimators use information about the measurement error in two ways: (1) a biased estimator based on the contaminated distances is multiplied by an appropriate correction factor, which is a function of the errors (PDF approach), and (2) cast into a likelihood framework that allows parameter estimation in the presence of measurement error (likelihood approach). Secondly, methods are developed that relax the conventional assumption that the distribution of animals is independent of distance from the lines or points (usually guaranteed by appropriate survey design). In particular, the new methods deal with the case where animal density gradients are caused by the use of non-random sampler allocation, for example transects placed along linear features such as roads or streams. This is dealt with separately for line and point transects, and at a later stage an approach for combining the two is presented. A considerable number of simulations and example analysis illustrate the performance of the proposed methods. Asseburg, Christian http://hdl.handle.net/10023/174 2012-07-26T12:58:31Z 2006-01-01T00:00:00Z

Abstract: Multi-species functional response models are required to model the predation of generalist preda- tors, which consume more than one prey species. In chapter 2, a new model for the multi-species functional response is presented. This model can describe generalist predators that exhibit func- tional responses of Holling type II to some of their prey and of type III to other prey. In chapter 3, I review some of the theoretical distinctions between Bayesian and frequentist statistics and show how Bayesian statistics are particularly well-suited for the fitting of functional response models because uncertainty can be represented comprehensively. In chapters 4 and 5, the multi- species functional response model is fitted to field data on two generalist predators: the hen harrier Circus cyaneus and the harp seal Phoca groenlandica. I am not aware of any previous Bayesian model of the multi-species functional response that has been fitted to field data. The hen harrier's functional response fitted in chapter 4 is strongly sigmoidal to the densities of red grouse Lagopus lagopus scoticus, but no type III shape was detected in the response to the two main prey species, field vole Microtus agrestis and meadow pipit Anthus pratensis. The impact of using Bayesian or frequentist models on the resulting functional response is discussed. In chapter 5, no functional response could be fitted to the data on harp seal predation. Possible reasons are discussed, including poor data quality or a lack of relevance of the available data for informing a behavioural functional response model. I conclude with a comparison of the role that functional responses play in behavioural, population and community ecology and emphasise the need for further research into unifying these different approaches to understanding predation with particular reference to predator movement. In an appendix, I evaluate the possibility of using a functional response for inferring the abun- dances of prey species from performance indicators of generalist predators feeding on these prey. I argue that this approach may be futile in general, because a generalist predator's energy intake does not depend on the density of any single of its prey, so that the possibly unknown densities of all prey need to be taken into account.

2006-01-01T00:00:00Z Asseburg, Christian Multi-species functional response models are required to model the predation of generalist preda- tors, which consume more than one prey species. In chapter 2, a new model for the multi-species functional response is presented. This model can describe generalist predators that exhibit func- tional responses of Holling type II to some of their prey and of type III to other prey. In chapter 3, I review some of the theoretical distinctions between Bayesian and frequentist statistics and show how Bayesian statistics are particularly well-suited for the fitting of functional response models because uncertainty can be represented comprehensively. In chapters 4 and 5, the multi- species functional response model is fitted to field data on two generalist predators: the hen harrier Circus cyaneus and the harp seal Phoca groenlandica. I am not aware of any previous Bayesian model of the multi-species functional response that has been fitted to field data. The hen harrier's functional response fitted in chapter 4 is strongly sigmoidal to the densities of red grouse Lagopus lagopus scoticus, but no type III shape was detected in the response to the two main prey species, field vole Microtus agrestis and meadow pipit Anthus pratensis. The impact of using Bayesian or frequentist models on the resulting functional response is discussed. In chapter 5, no functional response could be fitted to the data on harp seal predation. Possible reasons are discussed, including poor data quality or a lack of relevance of the available data for informing a behavioural functional response model. I conclude with a comparison of the role that functional responses play in behavioural, population and community ecology and emphasise the need for further research into unifying these different approaches to understanding predation with particular reference to predator movement. In an appendix, I evaluate the possibility of using a functional response for inferring the abun- dances of prey species from performance indicators of generalist predators feeding on these prey. I argue that this approach may be futile in general, because a generalist predator's energy intake does not depend on the density of any single of its prey, so that the possibly unknown densities of all prey need to be taken into account. Yeh, Shu-Ying http://hdl.handle.net/10023/158 2007-03-14T10:43:14Z 2007-06-01T00:00:00Z

Abstract: In many areas of science, especially geophysics, geography and meteorology, the data are often directions or axes rather than scalars or unrestricted vectors. Directional statistics considers data which are mainly unit vectors lying in two- or three-dimensional space (R² or R³). One way in which directional data arise is as normals to foliations. A (codimension-1) foliation of {R}^{d} is a system of non-intersecting (d-1)-dimensional surfaces filling out the whole of {R}^{d}. At each point z of {R}^{d}, any given codimension-1 foliation determines a unit vector v normal to the surface through z. The problem considered here is that of reconstructing the foliation from observations ({z}{i}, {v}{i}), i=1,...,n. One way of doing this is rather similar to fitting smooth splines to data. That is, the reconstructed foliation has to be as close to the data as possible, while the foliation itself is not too rough. A tradeoff parameter is introduced to control the balance between smoothness and closeness. The approach used in this thesis is to take the surfaces to be surfaces of constant values of a suitable real-valued function h on {R}^{d}. The problem of reconstructing a foliation is translated into the language of Schwartz distributions and a deep result in the theory of distributions is used to give the appropriate general form of the fitted function h. The model parameters are estimated by a simplified Newton method. Under appropriate distributional assumptions on v{1},...,v{n}, confidence regions for the true normals are developed and estimates of concentration are given.

2007-06-01T00:00:00Z Yeh, Shu-Ying In many areas of science, especially geophysics, geography and meteorology, the data are often directions or axes rather than scalars or unrestricted vectors. Directional statistics considers data which are mainly unit vectors lying in two- or three-dimensional space (R² or R³). One way in which directional data arise is as normals to foliations. A (codimension-1) foliation of {R}^{d} is a system of non-intersecting (d-1)-dimensional surfaces filling out the whole of {R}^{d}. At each point z of {R}^{d}, any given codimension-1 foliation determines a unit vector v normal to the surface through z. The problem considered here is that of reconstructing the foliation from observations ({z}{i}, {v}{i}), i=1,...,n. One way of doing this is rather similar to fitting smooth splines to data. That is, the reconstructed foliation has to be as close to the data as possible, while the foliation itself is not too rough. A tradeoff parameter is introduced to control the balance between smoothness and closeness. The approach used in this thesis is to take the surfaces to be surfaces of constant values of a suitable real-valued function h on {R}^{d}. The problem of reconstructing a foliation is translated into the language of Schwartz distributions and a deep result in the theory of distributions is used to give the appropriate general form of the fitted function h. The model parameters are estimated by a simplified Newton method. Under appropriate distributional assumptions on v{1},...,v{n}, confidence regions for the true normals are developed and estimates of concentration are given.