Thesis (MComm)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The Southern Ocean is a complex system, where the relationship between CO2 concentrations and its drivers varies intra- and inter-annually. Due to the lack of readily available in situ data in the Southern Ocean, a model approach was required which could predict the CO2 concentration proxy variable, fCO2. This must be done using predictor variables available via remote measurements to ensure the usefulness of the model in the future. These predictor variables were sea surface temperature, log transformed chlorophyll-a concentration, mixed layer depth and at a later stage altimetry. Initial exploratory analysis indicated that a non-parametric approach to the model should be taken. A parametric multiple linear regression model was developed to use as a comparison to previous studies in the North Atlantic Ocean as well as to compare with the results of the non-parametric approach. A non-parametric kernel regression model was then used to predict fCO2 and nally a combination of the parametric and non-parametric regression models was developed, referred to as the mixed regression model. The results indicated, as expected from exploratory analyses, that the non-parametric approach produced more accurate estimates based on an independent test data set. These more accurate estimates, however, were coupled with zero estimates, caused by the curse of dimensionality. It was also found that the inclusion of salinity (not available remotely) improved the model and therefore altimetry was chosen to attempt to capture this e ect in the model. The mixed model displayed reduced errors as well as removing the zero estimates and hence reducing the variance of the error rates. The results indicated that the mixed model is the best approach to use to predict fCO2 in the Southern Ocean and that altimetry's inclusion did improve the prediction accuracy. / AFRIKAANSE OPSOMMING: Die Suidelike Oseaan is 'n komplekse sisteem waar die verhouding tussen CO2 konsentrasies en die drywers daarvoor intra- en interjaarliks varieer. 'n Tekort aan maklik verkrygbare in situ data van die Suidelike Oseaan het daartoe gelei dat 'n model benadering nodig was wat die CO2 konsentrasie plaasvervangerveranderlike, fCO2, kon voorspel. Dié moet gedoen word deur om gebruik te maak van voorspellende veranderlikes, beskikbaar deur middel van afgeleë metings, om die bruikbaarheid van die model in die toekoms te verseker. Hierdie voorspellende veranderlikes het ingesluit see-oppervlaktetemperatuur, log getransformeerde chloro l-a konsentrasie, gemengde laag diepte en op 'n latere stadium, hoogtemeting. 'n Aanvanklike, ondersoekende analise het aangedui dat 'n nie-parametriese benadering tot die data geneem moet word. 'n Parametriese meerfoudige lineêre regressie model is ontwikkel om met die vorige studies in die Noord-Atlantiese Oseaan asook met die resultate van die nieparametriese benadering te vergelyk. 'n Nie-parametriese kern regressie model is toe ingespan om die fCO2 te voorspel en uiteindelik is 'n kombinasie van die parametriese en nie-parametriese regressie modelle ontwikkel vir dieselfde doel, wat na verwys word as die gemengde regressie model. Die resultate het aangetoon, soos verwag uit die ondersoekende analise, dat die nie-parametriese benadering meer akkurate beramings lewer, gebaseer op 'n onafhanklike toets datastel. Dié meer akkurate beramings het egter met "nul"beramings gepaartgegaan wat veroorsaak word deur die vloek van dimensionaliteit. Daar is ook gevind dat die insluiting van soutgehalte (nie beskikbaar oor via sateliet nie) die model verbeter en juis daarom is hoogtemeting gekies om te poog om hierdie e ek in die model vas te vang. Die gemengde model het kleiner foute getoon asook die "nul"beramings verwyder en sodoende die variasie van die foutkoerse verminder. Die resultate het dus aangetoon dat dat die gemengde model die beste benadering is om te gebruik om die fCO2 in die Suidelike Oseaan te beraam en dat die insluiting van altimetry die akkuraatheid van hierdie beraming verbeter.

A frequent problem that appears in practical survival data analysis is censoring. A censored observation occurs when the observation of the event time (duration or survival time) may be prevented by the occurrence of an earlier competing event (censoring time). Censoring may be due to different causes. For example, the loss of some subjects under study, the end of the follow-up period, drop out or the termination of the study and the limitation in the sensitivity of a measurement instrument. The literature about censored data focuses on the i.i.d. case. However in many real applications the data are collected sequentially in time or space and so the assumption of independence in such case does not hold. Here we only give some typical examples from the literature involving correlated data which are subject to censoring. In the clinical trials domain it frequently happens that the patients from the same hospital have correlated survival times due to unmeasured variables like the quality of the hospital equipment. Censored correlated data are also a common problem in the domain of environmental and spatial (geographical or ecological) statistics. In fact, due to the process being used in the data sampling procedure, e.g. the analytical equipment, only the measurements which exceed some thresholds, for example the method detection limits or the instrumental detection limits, can be included in the data analysis. Many other examples can also be found in other fields like econometrics and financial statistics. Observations on duration of unemployment e.g., may be right censored and are typically correlated. When the data are not independent and are subject to censoring, estimation and inference become more challenging mathematical problems with a wide area of applications. In this context, we propose here some new and flexible tools based on a nonparametric approach. More precisely, allowing dependence between individuals, our main contribution to this domain concerns the following aspects. First, we are interested in developing more suitable confidence intervals for a general class of functionals of a survival distribution via the empirical likelihood method. Secondly, we study the problem of conditional mean estimation using the local linear technique. Thirdly, we develop and study a new estimator of the conditional quantile function also based on the local linear method. In this dissertation, for each proposed method, asymptotic results like consistency and asymptotic normality are derived and the finite sample performance is evaluated in a simulation study.

A frequent problem that appears in practical survival data analysis is censoring. A censored observation occurs when the observation of the event time (duration or survival time) may be prevented by the occurrence of an earlier competing event (censoring time). Censoring may be due to different causes. For example, the loss of some subjects under study, the end of the follow-up period, drop out or the termination of the study and the limitation in the sensitivity of a measurement instrument. The literature about censored data focuses on the i.i.d. case. However in many real applications the data are collected sequentially in time or space and so the assumption of independence in such case does not hold. Here we only give some typical examples from the literature involving correlated data which are subject to censoring. In the clinical trials domain it frequently happens that the patients from the same hospital have correlated survival times due to unmeasured variables like the quality of the hospital equipment. Censored correlated data are also a common problem in the domain of environmental and spatial (geographical or ecological) statistics. In fact, due to the process being used in the data sampling procedure, e.g. the analytical equipment, only the measurements which exceed some thresholds, for example the method detection limits or the instrumental detection limits, can be included in the data analysis. Many other examples can also be found in other fields like econometrics and financial statistics. Observations on duration of unemployment e.g., may be right censored and are typically correlated. When the data are not independent and are subject to censoring, estimation and inference become more challenging mathematical problems with a wide area of applications. In this context, we propose here some new and flexible tools based on a nonparametric approach. More precisely, allowing dependence between individuals, our main contribution to this domain concerns the following aspects. First, we are interested in developing more suitable confidence intervals for a general class of functionals of a survival distribution via the empirical likelihood method. Secondly, we study the problem of conditional mean estimation using the local linear technique. Thirdly, we develop and study a new estimator of the conditional quantile function also based on the local linear method. In this dissertation, for each proposed method, asymptotic results like consistency and asymptotic normality are derived and the finite sample performance is evaluated in a simulation study.

No description available.

本研究探討常用的兩個無母數迴歸方法，核迴歸與樣條迴歸，在具有傘型限制式下，對於傘型函數的估計與不具限制式下的傘型函數估計比較，同時也探討不同誤差變異對估計結果的影響，並進一步探討受限制下兩方法的估計比較。本研究採用「估計頂點位置與實際頂點位置差」及「誤差平方和」作為衡量估計結果的指標。在帶寬及節點的選取上，本研究採用逐一剔除交互驗證法來篩選。模擬結果顯示，受限制的核函數在誤差變異較大的頂點位置估計較佳，誤差變異縮小時反而頂點位置估計較差，受限制的B-樣條函數也有類似的狀況。而在兩方法的比較上，對於較小的誤差變異，核函數的頂點位置估計能力不如樣條函數，但在整體的誤差平方和上卻沒有太大劣勢，當誤差變異較大時，核函數的頂點位置估計能力有所提升，整體誤差平方和仍舊維持還不錯的結果。 / In this study, we give an umbrella order constraint on kernel and spline regression model. We compare their estimation in two measurements, one is the difference of estimate peak and true peak, the other one is the sum of square difference on predict and the true value. We use leave-one-out cross validation to select bandwidth for kernel function and also to decide the number of knots for spline function. The effect of different error size is also considered. Some of R packages are used when doing simulation. The result shows that when the error size is bigger, the prediction of peak location is better in both constrained kernel and spline estimation. The constrained spline regression tends to provide better peak location estimation compared to constrained kernel regression.

Le développement des approches robustes et non paramétriques pour l’analyse et le traitement statistique de gros volumes de données présentant une forte variabilité,comme dans les domaines de l’environnement et de la génétique, est fondamental.Nous modélisons ici des données complexes de biologie appliquées à l’étude du comportement de bivalves et à l’analyse de liaison génétique. L’application des mathématiques à l’analyse du comportement de mollusques bivalves nous a permis d’aller vers une quantification et une traduction mathématique de comportements d’animaux in-situ, en milieu proche ou lointain. Nous avons proposé un modèle de régression non paramétrique et comparé 3 estimateurs non paramétriques, récursifs ou non,de la fonction de régression pour optimiser le meilleur estimateur. Nous avons ensuite caractérisé des rythmes biologiques, formalisé l’évolution d’états d’ouvertures,proposé des méthodes de discrimination de comportements, utilisé la méthode des shot-noises pour caractériser différents états d’ouverture-fermetures transitoires et développé une méthode originale de mesure de croissance en ligne.En génétique, nous avons abordé un cadre plus général de statistiques robustes pour l’analyse de liaison génétique. Nous avons développé des estimateurs robustes aux hypothèses de normalités et à la présence de valeurs aberrantes, nous avons aussi utilisé une approche statistique, où nous avons abordé la dépendance entre variables aléatoires via la théorie des copules. Nos principaux résultats ont montré l’intérêt pratique de ces estimateurs sur des données réelles de QTL et eQTL. / The development of robust and nonparametric approaches for the analysis and statistical treatment of high-dimensional data sets exhibiting high variability, as seen in the environmental and genetic fields, is instrumental. Here, we model complex biological data with application to the analysis of bivalves’ behavior and to linkage analysis. The application of mathematics to the analysis of mollusk bivalves’behavior gave us the possibility to quantify and translate mathematically the animals’behavior in situ, in close or far field. We proposed a nonparametric regression model and compared three nonparametric estimators (recursive or not) of the regressionfunction to optimize the best estimator. We then characterized the biological rhythms, formalized the states of opening, proposed methods able to discriminate the behaviors, used shot-noise analysis to characterize various opening/closing transitory states and developed an original approach for measuring online growth.In genetics, we proposed a more general framework of robust statistics for linkage analysis. We developed estimators robust to distribution assumptions and the presence of outlier observations. We also used a statistical approach where the dependence between random variables is specified through copula theory. Our main results showed the practical interest of these estimators on real data for QTL and eQTL analysis.

No description available.

A análise de regressão kernel não paramétrica desconsidera qualquer influência das formas funcionais geralmente empregadas em análises de regressões paramétricas, permitindo os dados \"falarem por si mesmos\". Enquanto os estimadores paramétricos são considerados globais, os kernels não paramétricos usam uma amostra de dados próximas (definida pela largura da janela) a um ponto para ajustar a estimação, o que permite focar em peculiaridades locais dos dados. Ambas as análises foram aplicadas aos dados do Censo Agropecuário de 2006 realizado pelo IBGE, agregados municipalmente e em dezessete faixas de áreas, para estimar uma função de produção com o objetivo de estabelecer a relação entre o tamanho das propriedades agrícolas e o valor da produção por hectare (produtividade). A relação constatada foi inversa, porém a análise local feita pelos estimadores kernels explicitou uma relação direta entre as elasticidades de produção dos insumos e o tamanho das propriedades agrícolas, o que não justifica uma política de redistribuição de terras no sentido do aumento da produtividade. Além disto, análises gráficas contra fatuais (que manteve os insumos, exceto a área, constantes em seus valores médios) mostraram que a relação não é linear, não é monotônica, e difere dentre as regiões, o que é um desafio para a elaboração de políticas de redistribuição de terras. / Nonparametric kernel regression analysis disregards any influence of the functional forms commonly employed in parametric regression analyzes, allowing the data to \"speak for itself.\" While parametric estimators are considered global, nonparametric kernels use a sample of nearby data (defined by the bandwidth) at a point to adjust the estimation, which allows focusing on local peculiarities of the data. Both analyzes were applied to data from the 2006 IBGE Census of Agriculture, aggregated in municipalities and in seventeen areas, to estimate a production function with the objective of establishing the relationship between the size of agricultural properties and the value of production by hectare (productivity). The observed relationship was reversed, but the local analysis made by the kernels estimators explained a direct relationship between the elasticities of production of the inputs and the size of the agricultural properties, which does not justify a policy of redistribution of land in order to increase productivity. In addition, graphical analyzes against factors (which kept the inputs, except the area, constant in their mean values) showed that the relationship is not linear, is not monotonic, and differs among regions, which is a challenge for the elaboration of land redistribution policies.

Nesta tese consideramos um modelo de regressão não paramétrica, quando a variável explicativa e um processo estritamente estacionário e alpha-mixing. São estudadas as condições sobre o processo Xt e sua estrutura de dependência, assim como do domínio da função f a ser estimada. Também são feitas as adaptações necessárias aos procedimentos para obter as taxas de convergência do risco para a norma Lp, no caso de ondaletas deformadas. Em relação às ondaletas adaptativas de Haar, obtêm-se as taxas de convergência do risco do estimador proposto. Mediante estudos de simulação, e avaliado o desempenho dos procedimentos propostos quando aplicados a amostras finitas sob diferentes níveis de perturbação do sinal e diferentes tamanhos da amostra. Também são feitas aplicações a dados reais. / In this thesis we consider a nonparametric regression model, when the exploratory variables are alpha-mixing stationary processes. We obtain convergence rates for risk for Lp norm, via warped wavelets, under suitable regularity conditions. For estimation using design adapted Haar wavelets we obtain convergence rates for the risk of the proposed estimator. The performance of the estimators are assessed via simulation studies with dierent sample sizes and dierent signal-to-noise ratios. Applications to real data are also given.

Nesta tese consideramos um modelo de regressão não paramétrica, quando a variável explicativa e um processo estritamente estacionário e alpha-mixing. São estudadas as condições sobre o processo Xt e sua estrutura de dependência, assim como do domínio da função f a ser estimada. Também são feitas as adaptações necessárias aos procedimentos para obter as taxas de convergência do risco para a norma Lp, no caso de ondaletas deformadas. Em relação às ondaletas adaptativas de Haar, obtêm-se as taxas de convergência do risco do estimador proposto. Mediante estudos de simulação, e avaliado o desempenho dos procedimentos propostos quando aplicados a amostras finitas sob diferentes níveis de perturbação do sinal e diferentes tamanhos da amostra. Também são feitas aplicações a dados reais. / In this thesis we consider a nonparametric regression model, when the exploratory variables are alpha-mixing stationary processes. We obtain convergence rates for risk for Lp norm, via warped wavelets, under suitable regularity conditions. For estimation using design adapted Haar wavelets we obtain convergence rates for the risk of the proposed estimator. The performance of the estimators are assessed via simulation studies with dierent sample sizes and dierent signal-to-noise ratios. Applications to real data are also given.