Asymptotic Change-Point Analysis of the Dependencies in Time Series

In recent papers, Wied and his coauthors have introduced change-point procedures to detect and estimate structural breaks in the correlation between time series. To prove the asymptotic distribution of the test statistic and stopping time as well as the change-point estimation rate, they use an extended functional Delta method and assume nearly constant expectations and variances of the time series. In this thesis, we allow asymptotically infinitely many structural breaks in the means and variances of the time series. For this setting, we present test statistics and stopping times which are used to determine whether or not the correlation between two time series is and stays constant, respectively. Additionally, we consider estimates for change-points in the correlations. The employed nonparametric statistics depend on the means and variances. These (nuisance) parameters are replaced by estimates in the course of this thesis. We avoid assuming a fixed form of these estimates but rather we use "blackbox" estimates, i.e. we derive results under assumptions that these estimates fulfill. These results are supplement with examples. This thesis is organized in seven sections. In Section 1, we motivate the issue and present the mathematical model. In Section 2, we consider a posteriori and sequential testing procedures, and investigate convergence rates for change-point estimation, always assuming that the means and the variances of the time series are known. In the following sections, the assumptions of known means and variances are relaxed. In Section 3, we present the assumptions for the mean and variance estimates that we will use for the mean in Section 4, for the variance in Section 5, and for both parameters in Section 6. Finally, in Section 7, a simulation study illustrates the finite sample behaviors of some testing procedures and estimates.

Time series analysis of a Web search engine transaction log

In this paper, we use time series analysis to evaluate predictive scenarios using search engine transactional logs. Our goal is to develop models for the analysis of searchers’ behaviors over time and investigate if time series analysis is a valid method for predicting relationships between searcher actions. Time series analysis is a method often used to understand the underlying characteristics of temporal data in order to make forecasts. In this study, we used a Web search engine transactional log and time series analysis to investigate users’ actions. We conducted our analysis in two phases. In the initial phase, we employed a basic analysis and found that 10% of searchers clicked on sponsored links. However, from 22:00 to 24:00, searchers almost exclusively clicked on the organic links, with almost no clicks on sponsored links. In the second and more extensive phase, we used a one-step prediction time series analysis method along with a transfer function method. The period rarely affects navigational and transactional queries, while rates for transactional queries vary during different periods. Our results show that the average length of a searcher session is approximately 2.9 interactions and that this average is consistent across time periods. Most importantly, our findings shows that searchers who submit the shortest queries (i.e., in number of terms) click on highest ranked results. We discuss implications, including predictive value, and future research.

Stochastic modelling of financial time series with memory and multifractal scaling

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.

Evaluating Specification Tests for Markov-Switching Time-Series Models

We evaluate the performance of several specification tests for Markov regime-switching time-series models. We consider the Lagrange multiplier (LM) and dynamic specification tests of Hamilton (1996) and Ljung–Box tests based on both the generalized residual and a standard-normal residual constructed using the Rosenblatt transformation. The size and power of the tests are studied using Monte Carlo experiments. We find that the LM tests have the best size and power properties. The Ljung–Box tests exhibit slight size distortions, though tests based on the Rosenblatt transformation perform better than the generalized residual-based tests. The tests exhibit impressive power to detect both autocorrelation and autoregressive conditional heteroscedasticity (ARCH). The tests are illustrated with a Markov-switching generalized ARCH (GARCH) model fitted to the US dollar–British pound exchange rate, with the finding that both autocorrelation and GARCH effects are needed to adequately fit the data.

Understanding the Aquifier : 3D Visualisation and Groundwater Time-Series Animation in Lockyer Valley

The Lockyer Valley, southeast Queensland, hosts intensive irrigated agriculture using groundwater from over 5000 alluvial bores. A current project is considering introduction of PRW (purified recycled water) to augment groundwater supplies. To assess this, a valley-wide MODFLOW simulation model is being developed plus a new unsaturated zone flow model. To underpin these models and provide a realistic understanding of the aquifer framework a 3D visualisation model has been developed using Groundwater Visualisation System (GVS) software produced at QUT.

Application of data mining techniques in the prediction of coronary artery disease : use of anaesthesia time-series and patient risk factor data

The high morbidity and mortality associated with atherosclerotic coronary vascular disease (CVD) and its complications are being lessened by the increased knowledge of risk factors, effective preventative measures and proven therapeutic interventions. However, significant CVD morbidity remains and sudden cardiac death continues to be a presenting feature for some subsequently diagnosed with CVD. Coronary vascular disease is also the leading cause of anaesthesia related complications. Stress electrocardiography/exercise testing is predictive of 10 year risk of CVD events and the cardiovascular variables used to score this test are monitored peri-operatively. Similar physiological time-series datasets are being subjected to data mining methods for the prediction of medical diagnoses and outcomes. This study aims to find predictors of CVD using anaesthesia time-series data and patient risk factor data. Several pre-processing and predictive data mining methods are applied to this data. Physiological time-series data related to anaesthetic procedures are subjected to pre-processing methods for removal of outliers, calculation of moving averages as well as data summarisation and data abstraction methods. Feature selection methods of both wrapper and filter types are applied to derived physiological time-series variable sets alone and to the same variables combined with risk factor variables. The ability of these methods to identify subsets of highly correlated but non-redundant variables is assessed. The major dataset is derived from the entire anaesthesia population and subsets of this population are considered to be at increased anaesthesia risk based on their need for more intensive monitoring (invasive haemodynamic monitoring and additional ECG leads). Because of the unbalanced class distribution in the data, majority class under-sampling and Kappa statistic together with misclassification rate and area under the ROC curve (AUC) are used for evaluation of models generated using different prediction algorithms. The performance based on models derived from feature reduced datasets reveal the filter method, Cfs subset evaluation, to be most consistently effective although Consistency derived subsets tended to slightly increased accuracy but markedly increased complexity. The use of misclassification rate (MR) for model performance evaluation is influenced by class distribution. This could be eliminated by consideration of the AUC or Kappa statistic as well by evaluation of subsets with under-sampled majority class. The noise and outlier removal pre-processing methods produced models with MR ranging from 10.69 to 12.62 with the lowest value being for data from which both outliers and noise were removed (MR 10.69). For the raw time-series dataset, MR is 12.34. Feature selection results in reduction in MR to 9.8 to 10.16 with time segmented summary data (dataset F) MR being 9.8 and raw time-series summary data (dataset A) being 9.92. However, for all time-series only based datasets, the complexity is high. For most pre-processing methods, Cfs could identify a subset of correlated and non-redundant variables from the time-series alone datasets but models derived from these subsets are of one leaf only. MR values are consistent with class distribution in the subset folds evaluated in the n-cross validation method. For models based on Cfs selected time-series derived and risk factor (RF) variables, the MR ranges from 8.83 to 10.36 with dataset RF_A (raw time-series data and RF) being 8.85 and dataset RF_F (time segmented time-series variables and RF) being 9.09. The models based on counts of outliers and counts of data points outside normal range (Dataset RF_E) and derived variables based on time series transformed using Symbolic Aggregate Approximation (SAX) with associated time-series pattern cluster membership (Dataset RF_ G) perform the least well with MR of 10.25 and 10.36 respectively. For coronary vascular disease prediction, nearest neighbour (NNge) and the support vector machine based method, SMO, have the highest MR of 10.1 and 10.28 while logistic regression (LR) and the decision tree (DT) method, J48, have MR of 8.85 and 9.0 respectively. DT rules are most comprehensible and clinically relevant. The predictive accuracy increase achieved by addition of risk factor variables to time-series variable based models is significant. The addition of time-series derived variables to models based on risk factor variables alone is associated with a trend to improved performance. Data mining of feature reduced, anaesthesia time-series variables together with risk factor variables can produce compact and moderately accurate models able to predict coronary vascular disease. Decision tree analysis of time-series data combined with risk factor variables yields rules which are more accurate than models based on time-series data alone. The limited additional value provided by electrocardiographic variables when compared to use of risk factors alone is similar to recent suggestions that exercise electrocardiography (exECG) under standardised conditions has limited additional diagnostic value over risk factor analysis and symptom pattern. The effect of the pre-processing used in this study had limited effect when time-series variables and risk factor variables are used as model input. In the absence of risk factor input, the use of time-series variables after outlier removal and time series variables based on physiological variable values’ being outside the accepted normal range is associated with some improvement in model performance.

The short-term effect of air pollution on cardiovascular mortality in Tianjin, China : comparison of time series and case–crossover analyses

Background: Many studies have illustrated that ambient air pollution negatively impacts on health. However, little evidence is available for the effects of air pollution on cardiovascular mortality (CVM) in Tianjin, China. Also, no study has examined which strata length for the time-stratified case–crossover analysis gives estimates that most closely match the estimates from time series analysis. Objectives: The purpose of this study was to estimate the effects of air pollutants on CVM in Tianjin, China, and compare time-stratified case–crossover and time series analyses. Method: A time-stratified case–crossover and generalized additive model (time series) were applied to examine the impact of air pollution on CVM from 2005 to 2007. Four time-stratified case–crossover analyses were used by varying the stratum length (Calendar month, 28, 21 or 14 days). Jackknifing was used to compare the methods. Residual analysis was used to check whether the models fitted well. Results: Both case–crossover and time series analyses show that air pollutants (PM10, SO2 and NO2) were positively associated with CVM. The estimates from the time-stratified case–crossover varied greatly with changing strata length. The estimates from the time series analyses varied slightly with changing degrees of freedom per year for time. The residuals from the time series analyses had less autocorrelation than those from the case–crossover analyses indicating a better fit. Conclusion: Air pollution was associated with an increased risk of CVM in Tianjin, China. Time series analyses performed better than the time-stratified case–crossover analyses in terms of residual checking.

The impact of nonlinear exposure risk relationships on seasonal time series data: modelling danish neonatal birth anthropometric data

Background Birth weight and length have seasonal fluctuations. Previous analyses of birth weight by latitude effects identified seemingly contradictory results, showing both 6 and 12 monthly periodicities in weight. The aims of this paper are twofold: (a) to explore seasonal patterns in a large, Danish Medical Birth Register, and (b) to explore models based on seasonal exposures and a non-linear exposure-risk relationship. Methods Birth weight and birth lengths on over 1.5 million Danish singleton, live births were examined for seasonality. We modelled seasonal patterns based on linear, U- and J-shaped exposure-risk relationships. We then added an extra layer of complexity by modelling weighted population-based exposure patterns. Results The Danish data showed clear seasonal fluctuations for both birth weight and birth length. A bimodal model best fits the data, however the amplitude of the 6 and 12 month peaks changed over time. In the modelling exercises, U- and J-shaped exposure-risk relationships generate time series with both 6 and 12 month periodicities. Changing the weightings of the population exposure risks result in unexpected properties. A J-shaped exposure-risk relationship with a diminishing population exposure over time fitted the observed seasonal pattern in the Danish birth weight data. Conclusion In keeping with many other studies, Danish birth anthropometric data show complex and shifting seasonal patterns. We speculate that annual periodicities with non-linear exposure-risk models may underlie these findings. Understanding the nature of seasonal fluctuations can help generate candidate exposures.

Exposure to hot and cold temperatures and ambulance attendances in Brisbane, Australia: A time series study

ABSTRACT Objectives: To investigate the effect of hot and cold temperatures on ambulance attendances. Design: An ecological time series study. Setting and participants: The study was conducted in Brisbane, Australia. We collected information on 783 935 daily ambulance attendances, along with data of associated meteorological variables and air pollutants, for the period of 2000–2007. Outcome measures: The total number of ambulance attendances was examined, along with those related to cardiovascular, respiratory and other non-traumatic conditions. Generalised additive models were used to assess the relationship between daily mean temperature and the number of ambulance attendances. Results: There were statistically significant relationships between mean temperature and ambulance attendances for all categories. Acute heat effects were found with a 1.17% (95% CI: 0.86%, 1.48%) increase in total attendances for 1 °C increase above threshold (0–1 days lag). Cold effects were delayed and longer lasting with a 1.30% (0.87%, 1.73%) increase in total attendances for a 1 °C decrease below the threshold (2–15 days lag). Harvesting was observed following initial acute periods of heat effects, but not for cold effects. Conclusions: This study shows that both hot and cold temperatures led to increases in ambulance attendances for different medical conditions. Our findings support the notion that ambulance attendance records are a valid and timely source of data for use in the development of local weather/health early warning systems.

Effects of extreme temperatures on years of life lost for cardiovascular deaths : a time series study in Brisbane, Australia

Background: Extreme temperatures are associated with cardiovascular disease (CVD) deaths. Previous studies have investigated the relative CVD mortality risk of temperature, but this risk is heavily influenced by deaths in frail elderly persons. To better estimate the burden of extreme temperatures we estimated their effects on years of life lost due to CVD. Methods and Results: The data were daily observations on weather and CVD mortality for Brisbane, Australia between 1996 and 2004. We estimated the association between daily mean temperature and years of life lost due to CVD, after adjusting for trend, season, day of the week, and humidity. To examine the non-linear and delayed effects of temperature, a distributed lag non-linear model was used. The model’s residuals were examined to investigate if there were any added effects due to cold spells and heat waves. The exposure-response curve between temperature and years of life lost was U-shaped, with the lowest years of life lost at 24 °C. The curve had a sharper rise at extremes of heat than of cold. The effect of cold peaked two days after exposure, whereas the greatest effect of heat occurred on the day of exposure. There were significantly added effects of heat waves on years of life lost. Conclusions: Increased years of life lost due to CVD are associated with both cold and hot temperatures. Research on specific interventions is needed to reduce temperature-related years of life lost from CVD deaths.

Pore formation during dehydration of polycrystalline gypsum observed and quantified in a time-series synchrotron radiation based X-ray micro-tomography experiment

We conducted an in-situ X-ray micro-computed tomography heating experiment at the Advanced Photon Source (USA) to dehydrate an unconfined 2.3 mm diameter cylinder of Volterra Gypsum. We used a purpose-built X-ray transparent furnace to heat the sample to 388 K for a total of 310 min to acquire a three-dimensional time-series tomography dataset comprising nine time steps. The voxel size of 2.2 μm3 proved sufficient to pinpoint reaction initiation and the organization of drainage architecture in space and time. We observed that dehydration commences across a narrow front, which propagates from the margins to the centre of the sample in more than four hours. The advance of this front can be fitted with a square-root function, implying that the initiation of the reaction in the sample can be described as a diffusion process. Novel parallelized computer codes allow quantifying the geometry of the porosity and the drainage architecture from the very large tomographic datasets (20483 voxels) in unprecedented detail. We determined position, volume, shape and orientation of each resolvable pore and tracked these properties over the duration of the experiment. We found that the pore-size distribution follows a power law. Pores tend to be anisotropic but rarely crack-shaped and have a preferred orientation, likely controlled by a pre-existing fabric in the sample. With on-going dehydration, pores coalesce into a single interconnected pore cluster that is connected to the surface of the sample cylinder and provides an effective drainage pathway. Our observations can be summarized in a model in which gypsum is stabilized by thermal expansion stresses and locally increased pore fluid pressures until the dehydration front approaches to within about 100 μm. Then, the internal stresses are released and dehydration happens efficiently, resulting in new pore space. Pressure release, the production of pores and the advance of the front are coupled in a feedback loop.

Learning hierarchical prototypes of motion time series for interactive systems

Abstract. For interactive systems, recognition, reproduction, and generalization of observed motion data are crucial for successful interaction. In this paper, we present a novel method for analysis of motion data that we refer to as K-OMM-trees. K-OMM-trees combine Ordered Means Models (OMMs) a model-based machine learning approach for time series with an hierarchical analysis technique for very large data sets, the K-tree algorithm. The proposed K-OMM-trees enable unsupervised prototype extraction of motion time series data with hierarchical data representation. After introducing the algorithmic details, we apply the proposed method to a gesture data set that includes substantial inter-class variations. Results from our studies show that K-OMM-trees are able to substantially increase the recognition performance and to learn an inherent data hierarchy with meaningful gesture abstractions.

Diurnal temperature range and childhood asthma : a time-series study

BACKGROUND: Hot and cold temperatures have been associated with childhood asthma. However, the relationship between daily temperature variation and childhood asthma is not well understood. This study aimed to examine the relationship between diurnal temperature range (DTR) and childhood asthma. METHODS: A Poisson generalized linear model combined with a distributed lag non-linear model was used to examine the relationship between DTR and emergency department admissions for childhood asthma in Brisbane, from January 1st 2003 to December 31st 2009. RESULTS: There was a statistically significant relationship between DTR and childhood asthma. The DTR effect on childhood asthma increased above a DTR of 10[degree sign]C. The effect of DTR on childhood asthma was the greatest for lag 0--9 days, with a 31% (95% confidence interval: 11% -- 58%) increase of emergency department admissions per 5[degree sign]C increment of DTR. Male children and children aged 5--9 years appeared to be more vulnerable to the DTR effect than others. CONCLUSIONS: Large DTR may trigger childhood asthma. Future measures to control and prevent childhood asthma should include taking temperature variability into account. More protective measures should be taken after a day of DTR above10[degree sign]C.