54 research outputs found
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An illustration oriented approach in statistics education
This paper explores an illustration oriented approach to assist students in learning Statistics. The importance of providing graphical insight into statistical ideas is emphasized as it helps students in many ways in their learning process. How this approach is practised at the University of Greenwich is described and students’ views, which give very positive indications, are summarised
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The role of Minitab in teaching and learning statistics
This paper describes how the statistical package Minitab is used in teaching statistics in our undergraduate programmes in Mathematics and Statistics to enhance student learning. How the sophisticated recent versions of Minitab can be used to help students understand statistical concepts, develop their statistical thinking and gain valuable skills in performing statistical analysis are discussed
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Some issues on assessment methods and learning in mathematics and statistics
This paper looks at the application of some of the assessment methods in practice with the view to enhance students’ learning in mathematics and statistics. It explores the effective application of assessment methods and highlights the issues or problems, and ways of avoiding them, related to some of the common methods of assessing mathematical and statistical learning. Some observations made by the author on good assessment practice and useful approaches employed at his institution in designing and applying assessment methods are discussed. Successful strategies in implementing assessment methods at different levels are described
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Some hidden Markov models with additional dependence
Hidden Markov models can be used to provide a modelling framework for many environmental processes. They can be modified in several ways to accommodate more dependence among observations and this can provide a rich class of flexible models that are useful in many environmental applications. We consider hidden Markov models that incorporate additional dependence among observations in two different ways (Ramesh and Onof, 2014). One approach is to introduce additional dependence between the state level and the observation level of the process and the other is to incorporate dependence at observation level of the process.
Formulation of the additional dependence models and the construction of their likelihood functions, for both approaches, are described. Some special cases of the models and the associated second-order properties of the process are studied. We employ the maximum likelihood methods to estimate the parameters of the models. Proposed models are used to analyse winter season daily rainfall data from England. Results of the analysis show that the models incorporating additional dependence between the state level and observation level of the process capture the structure of the rainfall distribution well whereas the class of models that incorporate dependence at observation level reproduced the autocorrelation structure of the process better than the other models considered.
Reference:
Ramesh, N.I, Onof, C. (2014). A class of hidden Markov models for regional average rainfall. Hydrological Sciences Journal. Vol.59 (9), 1704 -1717.
DOI: 10.1080/02626667.2014.88148
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Modelling multi-site rainfall time series using stochastic point process models
We consider stochastic point process models, based on doubly stochastic Poisson process, to analyse rainfall data collected in the form of raingauge bucket tip-times over a network of stations in a catchment. Multi-site doubly stochastic point process models are constructed whereby the arrival rate of the process varies according to a finite-state Markov chain thought to be representing the underlying environmental weather conditions. The tip-time series at a station is viewed as a univariate stochastic point process evolving in time and its multi-variate generalisation is studied to analyse data from multiple sites across the network. The likelihood function of this class of multi-site models, which is not usually available for most point process models, is derived by conditioning on the underlying Markov chain of the process. This allows us to make use of the likelihood approach for parameter estimation and inference.
The application of the proposed class of multi-site models, a useful alternative to the well known Poisson cluster models based on either Bartlett-Lewis or Neyman-Scott processes, in rainfall modelling is explored. We use data from the Hydrological Radar Experiment (HYREX) project, supplied by the British Atmospheric Data Centre (BADC), over a dense raingauge network in Brue experimental catchment in Somerset, South-West England. The models are used to make inferences about the properties of accumulated rainfall in discrete time intervals of equal length with the focus on fine time-scale. The proposed models and their variant that incorporate local covariate information such as elevation, temperature, sea-level pressure and relative humidity are utilised to study properties of rainfall time series from multiple sites. Results of the models that incorporate covariates are compared with the results of the model that does not take account of any covariates. The analysis shows the potential of this class of models in reproducing temporal and spatial variability of rainfall characteristics over the catchment area.
References:
Ramesh, N.I., Thayakaran, R. and Onof, C. 2013: Multi-site doubly stochastic Poisson process models for fine-scale rainfall, Stochastic Environmental Research and
Risk Assessment, 1-14. DOI: 10.1007/s00477-012-0674-
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Point process models for short-term rainfall
Point process theory has been widely used to model the stochastic structure of rainfall occurrences, and of hourly and daily rainfall depths. Cluster-based models are found to be useful for modelling short-term rainfall, as they preserve the clustering properties of the rain generating mechanism. Smith and Karr [Water Resour. Res. 19 (1983), 95-103] highlighted the applicability of Cox processes with Markovian intensity in rainfall modelling as they have an appealing physical interpretation. We study two point process models based on the Cox process. These marked Cox process models are used to describe the probabilistic structure of the rainfall intensity process. Different mechanisms for the process of marks are employed. For the first model the marks are rainfall volumes (depth) per event, whereas for the second model the marks are volumes and durations of events. The models we discuss here are similar in form to those described in Rodriguez-Iturbe et al. [Water Resour. Res. 20 (1984), 1611-1619] but are somewhat different in their basic structure. Mathematical formulation of the models is described and some second- moment characteristics of the rainfall depth, and aggregated processes are considered. The derived second-order properties of the accumulated rainfall amounts at different levels of aggregation are used in order to examine the model fit. A brief data analysis is presented
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Modelling seasonal rainfall at fine time-scales using doubly stochastic Poisson processes
Point process theory lends itself to the modelling of rainfall data and has been widely used for this purpose. The doubly stochastic Poisson process or Cox process, introduced in a seminal paper by Cox (1955), is a point process whose rate of occurrence is determined by a stochastic process. Models based on the doubly stochastic Poisson process provide a solid framework for analysing fine time-scale rainfall data. One form of the model arises when the underlying stochastic process becomes a continuous-time irreducible Markov process X(t) on a finite state space. Models of this form have been used to analyse rainfall data by several authors, since their likelihood can be calculated and maximized numerically.
Ramesh et al. (2012) explored this class of models for analysing tipping-bucket rainfall data at a single-site. The purpose of this paper is to extend the univariate class of models for fine time-scale rainfall to accommodate seasonality in the analysis of winter season rainfall data. Seasonal doubly stochastic Poisson process models are developed and their application is illustrated in an analysis of tipping-bucket rain gauge data from Bracknell, England. One advantage of using this class of models is that their likelihood can be calculated by conditioning on the underlying Markov process. As a result, the maximum likelihood method is utilised to estimate the parameters of the proposed model. Second-order properties of the sub-hourly rainfall aggregations in discrete time intervals are used for model assessment.
Ramesh, N.I., Onof, C. and Xie, D. (2012). Doubly Stochastic Poisson Process models for precipitation at fine time-scales, Advances in Water Resources, 2012; 45: 58-64
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Extracurricular activities to enhance the employment outcomes of Mathematics graduates
The employability skills of Mathematics graduates have been an area of concern for the mathematics community, employers and stake holders nationwide. Higher education institutions are addressing these skills in many ways, embarking on different strategies, to enhance the employment outcomes of their graduates. Although this topic has received a good deal of attention lately, it is useful to explore different ways to enhance students’ employability skills as they can impact positively on their employment outcomes. This may take the form of specific employability initiatives at department level, appropriate for the programme portfolio, to complement traditional careers advice services provided centrally by the university. This paper describes how the employability skills of Mathematics students can be enhanced by providing relevant extracurricular activities throughout their degree programmes, and discusses the implementation at our institution. Student feedback on the activities has been very positive and the implementation appears to have enhanced our graduates’ employability outcomes
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Some stochastic models for seasonal rainfall at fine time-scales
Stochastic point process models have been widely used to model rainfall time series. Doubly stochastic Poisson processes provide a rich class of models for analysing fine time-scale rainfall data. Models of this type have been used by several authors to describe fine-scale rainfall characteristics at a single site as well as at multiple sites. Ramesh et al. (2013) developed a class of multisite models for analysing tipping-bucket rainfall data recorded over a number of stations in a catchment area. In this paper, we extend the univariate class of models for fine time-scale rainfall to accommodate seasonality and study a number of seasonal doubly stochastic Poisson process models. This includes models incorporating atmospheric covariates in the analysis. The application of these models is illustrated in the modelling of sub-hourly rain gauge data from England. One of the advantages of this class of models, when compared with similar models, is that their likelihood function can be calculated in a tractable form suitable for numerical optimisation. This allows us to use the maximum likelihood approach to estimate the parameters of the proposed stochastic models. We use some of the second-order properties of the fine-scale rainfall aggregations in discrete time intervals for model assessment
NEUROPSYCHIATRIC MANIFESTATIONS OF COLLOID CYSTS: A REVIEW OF THE LITERATURE
Colloid cysts account for approximately 2% of primary brain tumours and the majority of cases are identified in the fourth and
fifth decade. They are small, gelatinous neoplasms lined by a single layer of mucin-secreting columnar epithelium that are thought to
arise from errors in folding of the primitive neuroepithelium. They develop in the rostral aspect of the third ventricle in the foramen
of Monro in 99% of cases and despite their benign histology carry a poor prognosis, with a mortality greater than 10% in
symptomatic cases.
The location of colloid cysts within the ventricular system results in obstruction of the foramen of Monro as the cyst grows,
disrupting the circulation of cerebrospinal fluid (CSF) and causing hydrocephalus. This is the mechanism behind the most common
presenting symptoms of postural headache, nausea and vomiting - a clinical picture synonymous with hydrocephalus and
intracranial pathology.
In addition to these classical neurological symptoms, there is a high prevalence of psychiatric symptoms in the patient population,
with symptoms ranging from anterograde amnesia to gustatory hallucination. These symptoms can occur with or without the
presence of hydrocephalus, and are thought to be secondary to compression of connecting pathways between the mesocortices and
subcortical limbic regions. These symptoms have been shown to be comparative in frequency to the classical symptoms, yet are
rarely the reason for referral to a neurological or neurosurgical service for investigation
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