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The focus of this thesis is on the employment of theoretical and practical quantile methods in addressing prediction, risk measurement and inference problems. From a prediction perspective, a problem of creating model-free prediction intervals for a future unobserved value of a random variable drawn from a sample distribution is considered. With the objective of reducing prediction coverage error, two common distribution transformation methods based on the normal and exponential distributions are presented and they are theoretically demonstrated to attain exact and error-free prediction intervals respectively. The second problem studied is that of estimation of expected shortfall via kernel smoothing. The goal here is to introduce methods that will reduce the estimation bias of expected shortfall. To this end, several one-step bias correction expected shortfall estimators are presented and investigated via simulation studies and compared with one-step estimators. The third problem is that of constructing simultaneous confidence bands for quantile regression functions when the predictor variables are constrained within a region is considered. In this context, a method is introduced that makes use of the asymmetric Laplace errors in conjunction with a simulation based algorithm to create confidence bands for quantile and interquantile regression functions. Furthermore, the simulation approach is extended to an ordinary least square framework to build simultaneous bands for quantiles functions of the classical regression model when the model errors are normally distributed and when this assumption is not fulfilled. Finally, attention is directed towards the construction of prediction intervals for realised volatility exploiting an alternative volatility estimator based on the difference of two extreme quantiles. The proposed approach makes use of AR-GARCH procedure in order to model time series of intraday quantiles and forecast intraday returns predictive distribution. Moreover, two simple adaptations of an existing model are also presented.

Master of Science / Department of Statistics / Paul I. Nelson / Suppose that based on data consisting of independent repetitions of an experiment a researcher wants to predict the outcome of the next independent outcome of the experiment. The researcher models the data as being realizations of independent, identically distributed random variables { Xi, i=1,2,..n} having density f() and the next outcome as the value of an independent random variable Y , also having density f() . We assume that the density f() lies in one of three location-scale families: standard normal (symmetric); Cauchy (symmetric, heavy-tailed); extreme value (asymmetric.). The researcher does not know the values of the location and scale parameters. For f() = f0() lying in one of these families, an exact prediction interval for Y can be constructed using equivariant estimators of the location and scale parameters to form a pivotal quantity based on { Xi, i=1,2,..n} and Y. This report investigates via a simulation study the performance of these prediction intervals in terms of coverage rate and length when the assumption that f() = f0() is correct and when it is not. The simulation results indicate that prediction intervals based on the assumption of normality perform quite well with normal and extreme value data and reasonably well with Cauchy data when the sample sizes are large. The heavy tailed Cauchy assumption only leads to prediction intervals that perform well with Cauchy data and is not robust when the data are normal and extreme value. Similarly, the asymmetric extreme value model leads to prediction intervals that only perform well with extreme value data. Overall, this study indicates robustness with respect to a mismatch between the assumed and actual distributions in some cases and a lack of robustness in others.

Gripen E, a fighter jet developed by Saab, has to fulfill a number of specifications and is therefore tested thoroughly. This project is about detecting anomalies in such tests and thereby improving the automation of the test data evaluation. The methodology during this project was to model the expected deviation between the measured signals and the corresponding signals from a fuel system model using machine learning methods. This methodology was applied to the mass in one of the fuel tanks. The challenge lies in the fact that the expected deviation is unknown and dependent on the operating conditions of the fuel system in the aircraft. Furthermore, two different machine learning approaches to estimate a prediction interval, within which the residual was expected to be, were tested. These were quantile regression and a variance estimation based method. The machine learning models used in this project were LSTM, Ridge Regression, Random Forest Regressor and Gradient Boosting Regressor. One of the problems encountered was imbalanced data, since different operating modes were not equally represented. Also, whether the time dependency of the signals had to be taken into account was investigated. Moreover, choosing which input signals to use for the machine learning methods had a large impact on the result. The concept appears to work well. Known anomalies were detected, and with a low degree of false alarms. The variance estimation based approach seems superior to quantile regression. For data containing anomalies, the target signal drifted away significantly outside the boundaries of the prediction interval. Such test flights were flagged for anomaly. Furthermore, the concept was also successfully verified for another fuel tank, with only minor and obvious adaptations, such as replacing the target signal with the new one.

Application of multisource feedback (MSF) increased dramatically and became widespread globally in the past two decades, but there was little conceptual work regarding self-other agreement and few empirical studies investigated self-other agreement in other cultural settings. This study developed a new conceptual framework of self-other agreement and used three samples to illustrate how national culture affected self-other agreement. These three samples included 428 participants from China, 818 participants from the US, and 871 participants from globally dispersed teams (GDTs). An EQS procedure and a polynomial regression procedure were used to examine whether the covariance matrices were equal across samples and whether the relationships between self-other agreement and performance would be different across cultures, respectively. The results indicated MSF could be applied to China and GDTs, but the pattern of relationships between self-other agreement and performance was different across samples, suggesting that the results found in the U.S. sample were the exception rather than rule. Demographics also affected self-other agreement disparately across perspectives and cultures, indicating self-concept was susceptible to cultural influences. The proposed framework only received partial support but showed great promise to guide future studies. This study contributed to the literature by: (a) developing a new framework of self-other agreement that could be used to study various contextual factors; (b) examining the relationship between self-other agreement and performance in three vastly different samples; (c) providing some important insights about consensus between raters and self-other agreement; (d) offering some practical guidelines regarding how to apply MSF to other cultures more effectively.

Despite their success and widespread usage in industry and business, ES methods have received little attention from the statistical community. We investigate three types of statistical models that have been found to underpin ES methods. They are ARIMA models, state space models with multiple sources of error (MSOE), and state space models with a single source of error (SSOE). We establish the relationship among the three classes of models and conclude that the class of SSOE state space models is broader than the other two and provides a formal statistical foundation for ES methods. To better understand ES methods, we investigate the behaviors of ES methods for time series generated from different processes. We mainly focus on time series of ARIMA type. ES methods forecast a time series using only the series own history. To include covariates into ES methods for better forecasting a time series, we propose a new forecasting method, Exponential Smoothing with Covariates (ESCov). ESCov uses an ES method to model what left unexplained in a time series by covariates. We establish the optimality of ESCov, identify SSOE state space models underlying ESCov, and derive analytically the variances of forecasts by ESCov. Empirical studies show that ESCov outperforms ES methods and regression with ARIMA errors. We suggest a model selection procedure for choosing appropriate covariates and ES methods in practice. Computer models have been commonly used to investigate complex systems for which physical experiments are highly expensive or very time-consuming. Before using a computer model, we need to address an important question ``How well does the computer model represent the real system?" The process of addressing this question is called computer model validation that generally involves the comparison of computer outputs and physical observations. In this thesis, we propose a Bayesian approach to computer model validation. This approach integrates together computer outputs and physical observation to give a better prediction of the real system output. This prediction is then used to validate the computer model. We investigate the impacts of several factors on the performance of the proposed approach and propose a generalization to the proposed approach.

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For this project, we use a modified version of a previously developed mathematical model, which describes the relationships among matrix metalloproteinases (MMPs), their tissue inhibitors (TIMPs), and extracellular matrix (ECM). Our ultimate goal is to quantify and understand differences in parameter estimates between patients in order to predict future responses and individualize treatment for each patient. By analyzing parameter confidence intervals and confidence and prediction intervals for the state variables, we develop a parameter space reduction algorithm that results in better future response predictions for each individual patient. Moreover, use of another subset selection method, namely Structured Covariance Analysis, that considers identifiability of parameters, has been included in this work. Furthermore, to estimate parameters more efficiently and accurately, the standard error (SE- )optimal design method is employed, which calculates optimal observation times for clinical data to be collected. Finally, by combining different parameter subset selection methods and an optimal design problem, different cases for both finding optimal time points and intervals have been investigated.

This thesis discusses two aspects of spatial statistics: sampling and prediction. In spatial statistics, we observe some phenomena in space. Space is typically of two or three dimensions, but can be of higher dimension. Questions in mind could be; What is the total amount of gold in a gold-mine? How much precipitation could we expect in a specific unobserved location? What is the total tree volume in a forest area? In spatial sampling the aim is to estimate global quantities, such as population totals, based on samples of locations (papers III and IV). In spatial prediction the aim is to estimate local quantities, such as the value at a single unobserved location, with a measure of uncertainty (papers I, II and V). In papers III and IV, we propose sampling designs for selecting representative probability samples in presence of auxiliary variables. If the phenomena under study have clear trends in the auxiliary space, estimation of population quantities can be improved by using representative samples. Such samples also enable estimation of population quantities in subspaces and are especially needed for multi-purpose surveys, when several target variables are of interest. In papers I and II, the objective is to construct valid prediction intervals for the value at a new location, given observed data. Prediction intervals typically rely on the kriging predictor having a Gaussian distribution. In paper I, we show that the distribution of the kriging predictor can be far from Gaussian, even asymptotically. This motivated us to propose a semiparametric method that does not require distributional assumptions. Prediction intervals are constructed from the plug-in ordinary kriging predictor. In paper V, we consider prediction in the presence of left-censoring, where observations falling below a minimum detection limit are not fully recorded. We review existing methods and propose a semi-naive method. The semi-naive method is compared to one model-based method and two naive methods, all based on variants of the kriging predictor.

PhDH / Department of Hydrology and Water Resources / Demand for water resources has been on the increase and is compounded by population growth and related development demands. Thus, numerous sectors are affected by water scarcity and therefore effective management of drought-induced water deficit is of importance. Luvuvhu River Catchment (LRC), a tributary of the Limpopo River Basin in South Africa has witnessed an increasing frequency of drought events over the recent decades. Drought impacts negatively on communities’ livelihoods, development, economy, water resources, and agricultural yields. Drought assessment in terms of frequency and severity using Drought Indices (DI) in different parts of the world has been reported. However, the forecasting and prediction component which is significant in drought preparedness and setting up early warning systems is still inadequate in several regions of the world. This study aimed at characterising, assessing, and predicting drought conditions using DI as a drought quantifying parameter in the LRC. This was achieved through the application of hybrid statistical and machine learning models including predictions via a combination of hybrid models. Rainfall and temperature data were obtained from South African Weather Service, evapotranspiration, streamflow, and reservoir storage data were obtained from the Department of Water and Sanitation while root zone soil moisture data was derived from the NASA earth data Giovanni repository. The Standardised Precipitation Index (SPI), Standardised Precipitation Evapotranspiration Index (SPEI), Standardised Streamflow Index (SSI), and Nonlinear Aggregated Drought Index (NADI) were selected to assess and characterise drought conditions in the LRC. SPI is precipitation based, SPEI is precipitation and evapotranspiration based, SSI is based on streamflow while NADI is a multivariate index based on rainfall, potential evapotranspiration, streamflow, and storage reservoir volume. All indices detected major historical drought events that have occurred and reported over the study area, although the precipitation based indices were the only indices that categorised the 1991/1992 drought as extreme at 6- and 12- month timescales while the streamflow index and multivariate NADI underestimated the event. The most recent 2014/16 drought was also categorised to be extreme by the standardised indices. The study found that the multivariate index underestimates most historical drought events in the catchment. The indices further showed that the most prevalent drought events in the LRC were mild drought. Extreme drought events were the least found at 6.42%, 1.08%, 1.56%, and 4.4% for SPI, SPEI, SSI, and NADI, respectively. Standardised indices and NADI showed negative trends and positive upward trends, respectively. The positive trend showed by NADI depicts a decreased drought severity over the study period. Drought events were characterised based on duration, intensity, severity, and frequency of drought events for each decade of the 30 years considered in this study i.e. between 1986 – 1996, 1996 – 2006, 2006 – 2016. This was done to get finer details of how drought characteristics behaved at a 10-year interval over the study period. An increased drought duration was observed between 1986 - 1996 while the shortest duration was observed between 1996 - 2006 followed by 2006 - 2016. NADI showed an overall lowest catchment duration at 1- month timescale compared to the standardised indices. The relationship between drought severity and duration revealed a strong linear relationship across all indices at all timescales (i.e. an R2 of between 0.6353 and 0.9714, 0.6353 and 0.973, 0.2725 and 0.976 at 1-, 6- and 12- month timescales, respectively). In assessing the overall utilisation of an index, the five decision criteria (robustness, tractability, transparency, sophistication, and extendibility) were assigned a raw score of between one and five. The sum of the weighted scores (i.e. raw scores multiplied by the relative importance factor) was the total for each index. SPEI ranked the highest with a total weight score of 129 followed by the SSI with a score of 125 and then the SPI with a score of 106 while NADI scored the lowest with a weight of 84. Since SPEI ranked the highest of all the four indices evaluated, it is regarded as an index that best describes drought conditions in the LRC and was therefore used in drought prediction. Statistical (GAM-Generalised Additive Models) and machine learning (LSTM-Long Short Term Memory) based techniques were used for drought prediction. The dependent variables were decomposed using Ensemble Empirical Mode Decomposition (EEMD). Model inputs were determined using the gradient boosting, and all variables showing some relative off importance were considered to influence the target values. Rain, temperature, non-linear trend, SPEI at lag1, and 2 were found to be important in predicting SPEI and the IMFs (Intrinsic Mode Functions) at 1, 6- and 12- month timescales. Seven models were applied based on the different learning techniques using the SPEI time series at all timescales. Prediction combinations of GAM performed better at 1- and 6- month timescales while at 12- month, an undecomposed GAM outperformed the decomposition and the combination of predictions with a correlation coefficient of 0.9591. The study also found that the correlation between target values, LSTM, and LSTM-fQRA was the same at 0.9997 at 1- month and 0.9996 at 6- and 12- month timescales. Further statistical evaluations showed that LSTM-fQRA was the better model compared to an undecomposed LSTM (i.e. RMSE of 0.0199 for LSTM-fQRA over 0.0241 for LSTM). The best performing GAM and LSTM based models were used to conduct uncertainty analysis, which was based on the prediction interval. The PICP and PINAW results indicated that LSTM-fQRA was the best model to predict SPEI timeseries at all timescales. The conclusions drawn from drought predictions conducted in this study are that machine learning neural networks are better suited to predict drought conditions in the LRC, while for improved model accuracy, time series decomposition and prediction combinations are also implementable. The applied hybrid machine learning models can be used for operational drought forecasting and further be incorporated into existing early warning systems for drought risk assessment and management in the LRC for better water resources management. Keywords: Decomposition, drought, drought indices, early warning system, frequency, machine learning, prediction intervals, severity, water resources. / NRF