This master’s thesis is engaged in usage of cluster analysis for ECG signal to separate normal QRS complexes from abnormal ones. For this, it is used two algorithms created in professional computing interface MATLAB. The outputs from this master’s thesis are dendrograms, which divide QRS complexes into abnormal and normal clusters, and Pearson correlation coefficients.

Electrocardiography (ECG) is a non-invasive method used in medicine to track the electrical pulses sent by the heart. The time between two subsequent electrical impulses and hence the heartbeat of a subject, is referred to as an RR interval. Previous studies show that RR intervals can be used for identifying sleep patterns and cardiovascular diseases. Additional research indicates that RR intervals can be used to predict the cardiovascular age of a subject. This thesis investigates, if this assumption is true, based on two different datasets as well as simulated data based on Gaussian Processes. The datasets used are Holter recordings provided by the University of Gdańsk as well as a dataset provided by Physionet. The former represents a balanced dataset of recordings during nocturnal sleep of healthy subjects whereas the latter one describes an imbalanced dataset of records of a whole day of subjects that suffered from myocardial infarction. Feature-based models as well as a deep learning architecture called DeepSleep, based on a paper for sleep stage detection, are trained. The results show, that the prediction of a subject's age, only based in RR intervals, is difficult. For the first dataset, the highest obtained test accuracy is 37.84 per cent, with a baseline of 18.23 per cent. For the second dataset, the highest obtained accuracy is 42.58 per cent with a baseline of 39.14 per cent. Furthermore, data is simulated by fitting Gaussian Processes to the first dataset and following a Bayesian approach by assuming a distribution for all hyperparameters of the kernel function in use. The distributions for the hyperparameters are continuously updated by fitting a Gaussian Process to a slices of around 2.5 minutes. Then, samples from the fitted Gaussian Process are taken as simulated data, handling impurity and padding. The results show that the highest accuracy achieved is 31.12 per cent with a baseline of 18.23 per cent. Concludingly, cardiovascular age prediction based on RR intervals is a difficult problem and complex handling of impurity does not necessarily improve the results.

A commonly used tool in data analysis is to compute a sample mean. Assuming a uni-modal distribution, its mean provides valuable information about which value is typically found in an observation. Also, it is one of the simplest and therefore very robust statistics to compute and suffers much less from sampling effects of tails of the distribution than estimates of higher moments. In the context of a time series, the sample mean is a time average. Due to correla- tions among successive data points, the information stored in a time series might be much less than the information stored in a sample of independently drawn data points of equal size, since correlation always implies redundancy. Hence, the issue of how close the sample estimate of a time average is to the true mean value of the process depends on correlations in data. In this thesis, we will study the proba- bility that a single time average deviates by more than some threshold value from the true process mean. This will be called the Large Deviation Probability (LDP), and it will be a function of the time interval over which the average is taken: The longer the time interval, the smaller will this probability be. However, it is the precise functional form of this decay which will be in the focus of this thesis. The LDP is proven to decay exponentially for identically independently distributed data. On the other hand we will see in this thesis that this result does not apply to long-range correlated data. The LDP is found to decay slower than exponential for such data. It will be shown that for intermittent series this exponential decay breaks down severely and the LDP is a power law. These findings are outlined in the methodological explanations in chapter 3, after an overview of the theoretical background in chapter 2. In chapter 4, the theoretical and numerical results for the studied models in chapter 3 are compared to two types of empirical data sets which are both known to be long- range correlated in the literature. The earth surface temperature of two stations of two climatic zones are modelled and the error bars for the finite time averages are estimated. Knowing that the data is long-range correlated by estimating the scaling exponent of the so called fluctuation function, the LDP estimation leads to noticeably enlarged error bars of time averages, based on the results in chapter 3. The same analysis is applied on heart inter-beat data in chapter 5. The contra- diction to the classical large deviation principle is even more severe in this case, induced by the long-range correlations and additional inherent non-stationarity. It will be shown that the inter-beat intervals can be well modeled by bounded fractional Brownian motion. The theoretical and numerical LDP, both for the model and the data, surprisingly indicates no clear decay of LDP for the time scales under study.

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