This study applies statistical models to mood in patients with bipolar disorder. Three analyses of telemonitored mood data are reported, each corresponding to a journal paper by the author. The first analysis reveals that patients whose sleep varies in quality tend to return mood ratings more sporadically than those with less variable sleep quality. The second analysis finds that forecasting depression with weekly data is not feasible using weekly mood ratings. A third analysis shows that depression time series cannot be distinguished from their linear surrogates, and that nonlinear forecasting methods are no more accurate than linear methods in forecasting mood. An additional contribution is the development of a new k-nearest neighbour forecasting algorithm which is evaluated on the mood data and other time series. Further work is proposed on more frequently sampled data and on system identification. Finally, it is suggested that observational data should be combined with models of brain function, and that more work is needed on theoretical explanations for mental illnesses.

Thesis (MScIng)--University of Stellenbosch, 2003. / ENGLISH ABSTRACT: Physical systems encountered in process engineering are invariably ill-defined, multivariate, and exhibit complex nonlinear dynamical behaviour. The increasing demands for better process efficiency and high product quality have led to the development and implementation of advanced control strategies in process plants. These modern control strategies are based on the use of a mathematical model defined for the process. Traditionally, linear models have been used to approximate the dynamics of processes whereas most processes are governed by nonlinear mechanisms. Since linear systems theory is well-established whereas nonlinear systems theory is not, recent developments in nonlinear dynamical systems theory present opportunities for improved approaches in modelling these process systems. It is now known that a nonlinear description of a process can be obtained from using time-delayed copies reconstructed from measurements taken from the process. Due to low signal to noise ratios associated with measured data it is logical to exploit redundant information in multivariate time signals taken from the systems in reconstructing the underlying dynamics. This study investigated the extension of univariate nonlinear time series analysis to the situation where multivariate measurements are available. Using simulated data from a coupled continuously stirred tank reactor and measured data from a flotation process system, the comparative advantages of using multivariate and univariate state space reconstructions were investigated. With respect to detection of nonlinearity multivariate surrogate analysis were found to give potentially robust results because of preservation of cross-correlations among components in the surrogate data. Multivariate local linear models showed a deterministic structure in both small and large neighbourhood sizes whereas for scalar embeddings determinism was defined only in smaller neighbourhood sizes. Non-uniform multivariate embeddings gave local linear models that resembled models from a trivial reconstruction of the original state space variables. With regard to global nonlinear modelling, multivariate embeddings gave models with better predictability irrespective of the model class used. Further improvements in the performance of models were obtained for multivariate non-uniform embeddings. A relatively new statistical learning algorithm, the least-squares support vector machine (LSSVM), was evaluated using multilayer perceptrons (MLP) as a benchmark in modelling nonlinear time series using simulated and plant data. It was observed that in the absence of autocorrelations in the variables and sparse data LSSVMs performed better than MLPs. Simulation of trained models gave consistent results for the LSSVMs, which was not the case for MLPs. However, the computational costs incurred in training the LSSVM model was significantly higher than for MLPs. LSSVMs were found to be insensitive to dimensionality reduction methods whereas the performance of MLPs degraded with increasing complexity of the dimension reduction method. No relative merits were found for using complex subspace dimension reduction methods for the data used. No general conclusions could be drawn with respect to the relative superiority of one class of models method over the other. Spatiotemporal structures are routinely observed in many chemical systems, such as reactive-diffusion and other pattern forming systems. We investigated the modelling of spatiotemporal time series using the coupled logistic map lattice as a case study. It was found that including both spatial and temporal information improved the performance of the fitted models. However, the superiority of spatiotemporal embeddings over individual time series was found to be defined for certain choices of the spatial and temporal embedding parameters. / AFRIKAANSE OPSOMMING: Fisiese stelsels wat in prosesingenieurswese voorkom is dikwels nie goed gedefinieer nie, multiveranderlik en vertoon komplekse nie-lineˆere gedrag. Toenemende vereistes vir ho¨e prosesdoeltreffendheid en produkgehalte het gelei tot die ontwikkeling en implementering van gevorderde beheerstrategie¨e vir prosesaanlegte. Hierdie morderne beheerstrategie¨e is gebaseer op die gebruik van wiskundige prosesmodelle. Lineˆere modelle word gewoonlik ontwikkel, al is die onderliggende prosesmeganismes in die algemeen nie-lineˆere, aangesien lineˆere stetselteorie goed gevestig is, en nie-line¨ere stelselteorie nie. Onlangse verwikkelinge in die teorie van nie-lineˆeredinamiese stelsels bied egter geleenthede vir verbeterde modellering van prosesstelsels. Dit is bekend dat ‘n nie-lineˆere beskrywing van ‘n progses verkry kan word deur tydvertraagde kopie¨e van metings van die prosesse te rekonstrueer. Met die lae seintot- geraasverhoudings wat met gemete data geassosieer word, is dit logies om die oortollige informasie in meerveranderlike seine te benut tydens die rekonstruksie van die onderliggende prosesdinamika. In die tesis is die uitbreiding van enkel-veranderlike nie-lineˆere tydreeksontleding na meer-veranderlike stelsels ondersoek. Met data van twee aaneengeskakelde gesimuleerde geroerde tenkreaktore en werklike data van ‘n flottasieproses, is die meriete van enkel- en meerveranderlike rekonstruksies van toestandruimtes ondersoek. Meerveranderlike surrogaatdata-ontleding het nie-lineariteite in die data op ‘n meer robuuste wyse ge¨ıdentifiseer, a.g.v. die behoud van kruis-korrelasies in die komponente van die data. Meerveranderlike lokale lineˆere modelle het ‘n deterministiese struktuur in beide klein en groot naasliggende omgewings ge¨ıdentifiseer, terwyl enkelveranderlike metodes dit slegs vir klein naasliggende omgewings kon doen. Nie-uniforme meerveranderlike inbeddings het lokale lineˆere modelle gegenereer wat soos globale modelle afkomstig van triviale rekonstruksies van die data gelyk het. M.b.t globale nie-lineˆere modellering, het meerveranderlike inbedding deurgaans beter modelle opgelewer. Verdere verbetering in die prestasie van modelle kon verkry word d.m.v. meerveranderlike nie-uniforme inbedding. ‘n Relatief nuwe statistiese algoritme, die kleinste-kwadrate-steunvektormasjien (KKSVM) is ge¨evalueer teenoor multilaag-perseptrons (MLP) as ‘n standaard vir die modellering van nie-lineˆere tydreekse, deur gebruik te maak van gesimuleerde en werklike aanlegdata. Daar is gevind dat die KKSVM beter presteer het as die MLPs wanneer die opeenvolgende waarnemings swak gekorreleer en min was relatief tot die aantal veranderlikes. Die KKSVMs het beduidend langer geneem as die MLPs om te ontwikkel. Hulle was ook minder sensitief vir die metodes wat gevolg is om die dimensionaliteit van die data te verlaag, anders as die MLPs. Ook is gevind dat meer komplekse metodes tot die verlaging van die dimensionaliteit weinig nut gehad het. Geen algemene gevolgtrekkings kan egter gemaak word m.b.t die verskillende modelle nie. Ruimtelik-temporale strukture word algemeen waargeneem in baie chemiese stelsels, soos reaktiewe diffusie e.a. patroonvormende sisteme. Die modellering van ruimtelik-temporale stelsels is bestudeer aan die hand van ‘n gekoppelde logistiese projeksierooster. Insluiting van beide die ruimtelike en temporale inligting het tot beduidend beter modelle gelei, solank as wat di´e inligting op die regte wyse ontsluit is.

Time series data mining is one branch of data mining. Time series analysis and prediction have always played an important role in human activities and natural sciences. A Pseudo-Periodical time series has a complex structure, with fluctuations and frequencies of the times series changing over time. Currently, Pseudo-Periodicity of time series brings new properties and challenges to time series analysis and prediction. This thesis proposes two original computational approaches for time series analysis and prediction: Moving Average of nth-order Difference (MANoD) and Series Features Extraction (SFE). Based on data-driven methods, the two original approaches open new insights in time series analysis and prediction contributing with new feature detection techniques. The proposed algorithms can reveal hidden patterns based on the characteristics of time series, and they can be applied for predicting forthcoming events. This thesis also presents the evaluation results of proposed algorithms on various pseudo-periodical time series, and compares the predicting results with classical time series prediction methods. The results of the original approaches applied to real world and synthetic time series are very good and show that the contributions open promising research directions.

published_or_final_version / Environmental Management / Master / Master of Science in Environmental Management

published_or_final_version / Business Administration / Master / Master of Business Administration

published_or_final_version / abstract / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

The Manchar Lake in Pakistan is in danger. So are the native fisher folk that populate the area and lake. The lake is undergoing water quality degradation due to both a decrease in the amount of freshwater inflow and an increase in the polluted agricultural run-off. The fish in the lake are dying and some fish species are becoming extinct, the people living on and around the lake are facing severe health risks, migratory birds are no longer stopping at the Manchar Lake and agriculture in the area is also suffering. This report focuses on time-series modeling and analysis of water quality data from Manchar Lake. We evaluate data for three sites within the Manchar Lake and complete time series models and analysis for two sites. Particular attention is given to the Electrical Conductivity data of the lake. The approach to modeling and time series analysis provide an example of potential uses of measured data to recognize shifts in water quality within the context of potential insights and recommendations about lake management in the area. / text

Signal extraction in tree-ring research is considered as a general time series decomposition problem. A linear aggregate model for a hypothetical ring-width series is proposed, which allows the problem to be reduced to the estimation and extraction of five discrete classes of signals. These classes represent the signals due to trend, climate, endogenous disturbance, exogenous disturbance, and random error. For each class of signal, some mathematical/statistical techniques of estimation are described and reviewed. Except for the exogenous disturbance signal, the techniques only require information contained within the ring-width series, themselves. A unified mathematical framework for solving this decomposition problem has not yet been explicitly formulated. However, the general applicability of ARMA time series models to this problem and the power and flexibility of state space modelling suggest that these techniques will provide the closest thing to a unified framework in the future.

Time domain properties of 23 tree-ring chronologies derived from a large sample of Douglas-fir and ponderosa pine trees growing in closed-canopy forests of Colorado and New Mexico were analyzed using Box-Jenkins models. A variety of statistical criteria were employed during the identification and validation stages for evaluating the performance of different significant models, and the "best" Box-Jenkins model and its immediate "competitor" were reported for each tree-ring chronology. All series were stationary, and only one was approximately a white noise series. Overall, the ARMA(1,1) model was judged the best for 11 series, and the second for 7 of the remaining 12 series. The AR(2) model was considered the best for 6 series, and the second for 4 of the remaining 17 series. No statistical evidence was found for moving average models, nor for models with more than three different parameters. However, both cyclical (or seasonal) models and third-order autoregressive models with a null second-order parameter were chosen for some series. Fitted models explained from 7 to 51% of the variance of the original ring-index series, with an average of about 22 %. All parameter estimates were positive, and they varied within a relatively small range. From a comparison of all employed criteria, Akaike's Information Criterion (AIC) was the one that performed best in identifying Box-Jenkins models for tree-ring chronologies. Possible distinctions were recognized that would separate the selected models according to species and /or standardization option. Among the 12 chronologies from Colorado sites, all derived using the same standardization option, most Douglas-fir series were best fitted by the ARMA(1,1) model, while most ponderosa pine series were best fitted by the AR(2) model, suggesting a difference in the biological persistence of the two species. On the other hand, most of New Mexico chronologies, developed using various standardization options, were best fitted by the ARMA(1,1) model, and no difference was found between Douglas-fir and ponderosa pine series. Also, models fitted to Colorado chronologies explained a lower amount of variance than those for New Mexico chronologies (averages of 17 versus 29% respectively), and cyclical models were mainly selected for New Mexico series. Although periodicities in Douglas-fir series were probably caused by western spruce budworm outbreaks, similar periodic patterns in ponderosa pine series were more difficult to explain because pine trees in the study area had not been defoliated by that insect. Compared to the original tree-ring chronologies, prewhitened series showed similar short-term growth patterns, reduced long-term growth fluctuations, lower standard deviations, and higher mean sensitivities. Also, cross-correlations between chronologies from the same area usually increased after prewhitening. Since the autocorrelation problem is crucial in analyzing the relationships between different time series, and in removing the biological persistence included in tree-ring chronologies, the Box-Jenkins approach should facilitate the analysis of the dynamic relationships between tree growth and environmental variables.