Statistical inference in time series analysis has been an important subject in various fields including climate science, economics, finance and industrial engineering among others. Numerous problems of research interest include statistical inference about unknown quantities, assessing structural stability and forecasting. These problems have been widely studied in the literature, but mainly for independent data, while in many applications involving time series data dependence is not unusual and in fact quite common. In this thesis, we incorporate serial dependence into the analysis by involving self-normalization in time series analysis. We start with the problem of testing whether there are change-points in a given time series. The method we propose does not require the number of change-points to be predefined, and thus is unsupervised. It does not require any tuning parameters and can be applied to a wide class to quantities of interest. The asymptotic distribution of the test statistic is studied and an approximation scheme is proposed to reduce testing procedure complexity. We then consider the problem of construction of confidence intervals, for which the conventional self-normalizer exhibits certain degrees of asymmetry when applied to quantities other than the mean. The method we propose provides a time-symmetric generalization to the conventional self-normalizer and leads to improved finite sample performance for quantities other than the mean.

Evidence of networks in the resting-brain reflecting the spontaneous brain activity is perhaps the most significant discovery to understand intrinsic brain functionality. Moreover, subsequent detection of dynamics in these networks can be milestone in differentiating the normal and disordered brain functions. However, capturing the correct dynamics is a challenging task since no ground truths' are present for comparison of the results. The change points of these networks can be different for different subjects even during normal brain functions. Even for the same subject and session, dynamics can be different at the start and end of the session based on the fatigue level of the subject scanned. Despite the absence of ground truths, studies have analyzed these dynamics using the existing methods and some of them have developed new algorithms too. One of the most commonly used method for this purpose is sliding window correlation. However, the result of the sliding window correlation is dependent on many parameters and without the ground truth there is no way of validating the results. In addition, most of the new algorithms are complicated, computationally expensive, and/or focus on just one aspect on these dynamics. This study applies the algorithms and concepts from signal processing, image processing, video processing, information theory, and machine learning to analyze the results of the sliding window correlation and develops a novel algorithm to detect change points of these networks adaptively. The findings in this study are divided into three parts: 1) Analyzing the extent of variability in well-defined networks of rodents and humans with sliding window correlation applying concepts from information theory and machine learning domains. 2) Analyzing the performance of sliding window correlation using simulated networks as ground truths for best parameters’ selection, and exploring its dependence on multiple frequency components of the correlating signals by processing the signals in time and Fourier domains. 3) Development of a novel algorithm based on image similarity measures from image and video processing that maybe employed to identify change points of these networks adaptively.

Thesis (MScEng)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: In this research work, auto-associative neural networks have been used for changepoint detection. This is a nonlinear technique that employs the use of artificial neural networks as inspired among other by Frank Rosenblatt’s linear perceptron algorithm for classification. An auto-associative neural network was used successfully to detect change-points for various types of time series data. Its performance was compared to that of singular spectrum analysis developed by Moskvina and Zhigljavsky. Fraction of Explained Variance (FEV) was also used to compare the performance of the two methods. FEV indicators are similar to the eigenvalues of the covariance matrix in principal component analysis. Two types of time series data were used for change-point detection: Gaussian data series and nonlinear reaction data series. The Gaussian data had four series with different types of change-points, namely a change in the mean value of the time series (T1), a change in the variance of the time series (T2), a change in the autocorrelation of the time series (T3), and a change in the crosscorrelation of two time series (T4). Both linear and nonlinear methods were able to detect the changes for T1, T2 and T4. None of them could detect the changes in T3. With the Gaussian data series, linear singular spectrum analysis (LSSA) performed as well as the NLSSA for the change point detection. This is because the time series was linear and the nonlinearity of the NLSSA was therefore not important. LSSA did even better than NLSSA when comparing FEV values, since it is not subject to suboptimal solutions as could sometimes be the case with autoassociative neural networks. The nonlinear data consisted of the Belousov-Zhabotinsky (BZ) reactions, autocatalytic reaction time series data and data representing a predator-prey system. With the NLSSA methods, change points could be detected accurately in all three systems, while LSSA only managed to detect the change-point on the BZ reactions and the predator-prey system. The NLSSA method also fared better than the LSSA method when comparing FEV values for the BZ reactions. The LSSA method was able to model the autocatalytic reactions fairly accurately, being able to explain 99% of the variance in the data with one component only. NLSSA with two nodes on the bottleneck attained an FEV of 87%. The performance of both NLSSA and LSSA were comparable for the predator-prey system, both systems, where both could attain FEV values of 92% with a single component. An auto-associative neural network is a good technique for change point detection in nonlinear time series data. However, it offers no advantage over linear techniques when the time series data are linear. / AFRIKAANSE OPSOMMING: In hierdie navorsing is outoassosiatiewe neurale netwerk gebruik vir veranderingspuntwaarneming. Dis is ‘n nielineêre tegniek wat neurale netwerke gebruik soos onder andere geïnspireer deur Frank Rosnblatt se lineêre perseptronalgoritme vir klassifikasie. ‘n Outoassosiatiewe neurale netwerk is suksesvol gebruik om veranderingspunte op te spoor in verskeie tipes tydreeksdata. Die prestasie van die outoassosiatiewe neurale netwerk is vergelyk met singuliere spektrale oontleding soos ontwikkel deur Moskvina en Zhigljavsky. Die fraksie van die verklaarde variansie (FEV) is ook gebruik om die prestasie van die twee metodes te vergelyk. FEV indikatore is soortgelyk aan die eiewaardes van die kovariansiematriks in hoofkomponentontleding. Twee tipes tydreeksdata is gebruik vir veranderingspuntopsporing: Gaussiaanse tydreekse en nielineêre reaksiedatareekse. Die Gaussiaanse data het vier reekse gehad met verskillende veranderingspunte, naamlik ‘n verandering in die gemiddelde van die tydreeksdata (T1), ‘n verandering in die variansie van die tydreeksdata (T2), ‘n verandering in die outokorrelasie van die tydreeksdata (T3), en ‘n verandering in die kruiskorrelasie van twee tydreekse (T4). Beide lineêre en nielineêre metodes kon die veranderinge in T1, T2 en T4 opspoor. Nie een het egter daarin geslaag om die verandering in T3 op te spoor nie. Met die Gaussiaanse tydreeks het lineêre singuliere spektrumanalise (LSSA) net so goed gevaar soos die outoassosiatiewe neurale netwerk of nielineêre singuliere spektrumanalise (NLSSA), aangesien die tydreekse lineêr was en die vermoë van die NLSSA metode om nielineêre gedrag te identifiseer dus nie belangrik was nie. Inteendeel, die LSSA metode het ‘n groter FEV waarde getoon as die NLSSA metode, omdat LSSA ook nie blootgestel is aan suboptimale oplossings, soos wat soms die geval kan wees met die afrigting van die outoassosiatiewe neural netwerk nie. Die nielineêre data het bestaan uit die Belousov-Zhabotinsky (BZ) reaksiedata, ‘n outokatalitiese reaksietydreeksdata en data wat ‘n roofdier-prooistelsel verteenwoordig het. Met die NLSSA metode kon veranderingspunte betroubaar opgespoor word in al drie tydreekse, terwyl die LSSA metode net die veranderingspuntin die BZ reaksie en die roofdier-prooistelsel kon opspoor. Die NLSSA metode het ook beter gevaaar as die LSSA metode wanneer die FEV waardes vir die BZ reaksies vergelyk word. Die LSSA metode kon die outokatalitiese reaksies redelik akkuraat modelleer, en kon met slegs een komponent 99% van die variansie in die data verklaar. Die NLSSA metode, met twee nodes in sy bottelneklaag, kon ‘n FEV waarde van slegs 87% behaal. Die prestasie van beide metodes was vergelykbaar vir die roofdier-prooidata, met beide wat FEV waardes van 92% kon behaal met hulle een komponent. ‘n Outoassosiatiewe neural netwerk is ‘n goeie metode vir die opspoor van veranderingspunte in nielineêre tydreeksdata. Dit hou egter geen voordeel in wanneer die data lineêr is nie.

Title: Statistical analysis of historical temperature series Author: Václav Gergelits Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jaromír Antoch CSc. Supervisor's e-mail address: antoch@karlin.mff.cuni.cz Abstract: In the present work we deal with the statistical analysis of time-series of a mean-temperature obtained from seven European cities from the Europe Union project "IMPROVE". Properties of the time series are analyzed by means of descriptive statistics, being assessing their homoscedasticity, autocorrelation and normality. We report the ways in which the data has been adjusted, including consideration of the impact of the urban heat island and we discuss the availability of additional data. The theoretical part presents a theory of change point detection for a one change model as well as more than one change model taking an autocorrelation into account. In the practical part we analyze the data using change point detection method. The significant increase was not detected for time series of Cadiz and Uppsala. The significant increase was rather detected for the rest of the time series. The increase of temperature could be in a relation to the adjustment for the urban heat island. Keywords: change point detection, temperature time series 1

<p>The distribution of asset returns may lead to structural breaks. Thesebreaks may result in changes of the optimal portfolio weights. For a port-folio investor, the ability of timely detection of any systematic changesin the optimal portfolio weights is of a great interest.In this master thesis work, the use of the Shewhart method, as amethod for detecting a sudden parameter change, the implied changein the multivariate portfolio weights and its performance is reviewed.</p><p> </p>

Consider a time series model which is stationary apart from a single shift in mean. If the time of a level shift is known, the least squares estimator of the magnitude of this level shift is a minimum variance unbiased estimator. If the time is unknown, however, this estimator is biased. Here, we first carry out extensive simulation studies to determine the relationship between the bias and three parameters of our time series model: the true magnitude of the level shift, the true time point and the autocorrelation of adjacent observations. Thereafter, we use two generalized additive models to generalize the simulation results. Finally, we examine to what extent the bias can be reduced by multiplying the least squares estimator with a shrinkage factor. Our results showed that the bias of the estimated magnitude of the level shift can be reduced when the level shift does not occur close to the beginning or end of the time series. However, it was not possible to simultaneously reduce the bias for all possible time points and magnitudes of the level shift.

For the last several decades, sequential change point problems have been studied in both the theoretical area (sequential analysis) and the application area (industrial SPC). In the conventional application, the baseline process is assumed to be stationary, and the shift pattern is a step function that is sustained after the shift. However, in biosurveillance, the underlying assumptions of problems are more complicated. This thesis investigates several issues in biosurveillance such as non-homogeneous populations, spatiotemporal surveillance methods, and correlated structures in regional data. The first part of the thesis discusses popular surveillance methods in sequential change point problems and off-line problems based on count data. For sequential change point problems, the CUSUM and the EWMA have been used in healthcare and public health surveillance to detect increases in the rates of diseases or symptoms. On the other hand, for off-line problems, scan statistics are widely used. In this chapter, we link the method for off-line problems to those for sequential change point problems. We investigate three methods--the CUSUM, the EWMA, and scan statistics--and compare them by conditional expected delay (CED). The second part of the thesis pertains to the on-line monitoring problem of detecting a change in the mean of Poisson count data with a non-homogeneous population size. The most common detection schemes are based on generalized likelihood ratio statistics, known as an optimal method under Lodern's criteria. We propose alternative detection schemes based on the weighted likelihood ratios and the adaptive threshold method, which perform better than generalized likelihood ratio statistics in an increasing population. The properties of these three detection schemes are investigated by both a theoretical approach and numerical simulation. The third part of the thesis investigates spatiotemporal surveillance based on likelihood ratios. This chapter proposes a general framework for spatiotemporal surveillance based on likelihood ratio statistics over time windows. We show that the CUSUM and other popular likelihood ratio statistics are the special cases under such a general framework. We compare the efficiency of these surveillance methods in spatiotemporal cases for detecting clusters of incidence using both Monte Carlo simulations and a real example. The fourth part proposes multivariate surveillance methods based on likelihood ratio tests in the presence of spatial correlations. By taking advantage of spatial correlations, the proposed methods can perform better than existing surveillance methods by providing the faster and more accurate detection. We illustrate the application of these methods with a breast cancer case in New Hampshire when observations are spatially correlated.

The distribution of asset returns may lead to structural breaks. Thesebreaks may result in changes of the optimal portfolio weights. For a port-folio investor, the ability of timely detection of any systematic changesin the optimal portfolio weights is of a great interest.In this master thesis work, the use of the Shewhart method, as amethod for detecting a sudden parameter change, the implied changein the multivariate portfolio weights and its performance is reviewed.

Electromyogram (EMG) is an established tool to study operation of neuromuscular systems. In analysing EMG signals, accurate detection of the movement-related events in the signal is frequently necessary. I explored the application of change-point detection algorithm proposed by Moskvina et. al., 2003 to EMG event detection, and evaluated the technique’s performance comparing it to two common threshold-based event detection methods and to the visual estimates of the EMG events performed by trained practitioners in the field. The algorithm was implemented in MATLAB and applied to EMG segments recorded from wrist and trunk muscles. The quality and frequency of successful detection were assessed for all methods, using the average visual estimate as the baseline, against which techniques were evaluated. The application showed that the change-point detection can successfully locate multiple changes in the EMG signal, but the maximum value of the detection statistic did not always identify the muscle activation onset.

Electromyogram (EMG) is an established tool to study operation of neuromuscular systems. In analysing EMG signals, accurate detection of the movement-related events in the signal is frequently necessary. I explored the application of change-point detection algorithm proposed by Moskvina et. al., 2003 to EMG event detection, and evaluated the technique’s performance comparing it to two common threshold-based event detection methods and to the visual estimates of the EMG events performed by trained practitioners in the field. The algorithm was implemented in MATLAB and applied to EMG segments recorded from wrist and trunk muscles. The quality and frequency of successful detection were assessed for all methods, using the average visual estimate as the baseline, against which techniques were evaluated. The application showed that the change-point detection can successfully locate multiple changes in the EMG signal, but the maximum value of the detection statistic did not always identify the muscle activation onset.