Earthquakes are amongst the most destructive of natural phenomena and have been the subject of significant research effort over many decades, to predict the onset of seismic events. Electromagnetic emissions detected prior to earthquakes provide a potential data source for seismic predictions and research suggests that specific pre-seismic electromagnetic activity can be directly related to specific earthquakes although it is still an open issue as to the precise links between these electromagnetic emissions and subsequent earthquakes. In this research, findings of the long memory or the self-organization of several pre-earthquake MHz electromagnetic time-series provide significant outcomes regarding the earthquake prediction. It is also recognised that enhanced radon gas emission has an equally long history as being associated with seismic activity. In general, several anomalous soil radon emissions have been observed prior to earthquakes and this has been recorded all over the world. The abnormal soil radon exhalation from the interior of the earth has been associated with earthquakes and is considered as an important field of research. The research reported in this thesis compared and contrasted the merits of combining electromagnetic emission data and radon exhalation data as precursors of earthquakes with the aim of enhancing earthquake prediction methodology. The findings from the long-memory analysis of radon disturbances in the soil indicated a very significant issue: the radon disturbances in the soil prior to earthquakes exhibit similar behaviour as the MHz RF disturbances of general failure. So, the radon precursors and the MHz electromagnetic correspond to the same pre-earthquake phase. Geological explanations were proposed in view of the asperity model. Persistent and anti-persistent MHz anomalies were due to the micro-cracking of the heterogeneous medium of the earth's crust which may have led the system's evolution towards the global failure. Fractal methods have been used on historical data, to investigate MHz electromagnetic time-series spectra on emissions preceding major earthquakes over the period 2007 to 2014 and the characteristics of enhanced radon emissions have been studied over the period 2008 to 2015 for seismic events occurring in the Aegean Region. It has been found that both the electromagnetic emissions and the radon exhalation data exhibit similar fractal behaviour and are associated with impending seismic activity. Hence both phenomena are relevant to earthquake predictions and should both be employed in any systematic approach to this problem as the varying geological and geographic conditions under which earthquakes can occur, might preclude one or other data from being measurable. According to the several techniques applied in this thesis, all should be employed in sequential steps, albeit the power-law spectral fractal analysis is the most significant to trace long-memory patterns of 1/f processes as those of the processes of earthquakes.

PhD (Statistics) / Department of Statistics / There has been a growing interest in investment opportunities in Africa. The net foreign direct investment (FDI) to Sub-Saharan Africa has increased from $13 billion in 2004 to about $54 billion in 2015. Investing on the stock markets is one of such investment opportunities. Stock markets in Africa have realised growth in market capitalization, membership, value and volume traded due to an increase in investments. This level of growth in African stock markets has raised questions about their efficiency. This thesis examined the weak-form informational efficiency of African stock markets. The aim therefore of this thesis is to test the efficiency of African stock markets in the weak-form of the Efficient Market Hypothesis (EMH) for eight countries, namely, Botswana, Egypt, Kenya, Mauritius, Morocco, Nigeria, South Africa and Tunisia. Since, the researcher will be testing the weak-form of the EMH, the data to be used is on past price information on the markets of the eight countries. Data for the eight countries were obtained from DataStream for the period between August 28, 2000 to August 28, 2015. The data is for a period of 180 months which resulted in 3915 data points. Although there have been studies on the weak-form market efficiency of African stock markets, the efficiency conclusions on the markets have been mixed. This problem might be due to the methods used in the analyses. First, most of the methods used were linear in nature although the data generating process of stock market data is nonlinear and hence nonlinear methods maybe more appropriate in its analysis. Also these linear methods tested the efficiency of African markets in absolute form, however, an efficiency conclusion relying solely on absolute efficiency might be misleading because, stock markets become efficient with time due to improvements in the quality of information processing from reforms on the markets. The researcher solved this problem of using absolute frequency by comparing the results when the presence of long-memory in frequency and time domains of the markets were examined. The researcher used a semi-parametric estimator, the Local Whittle estimator to test for long-memory in frequency domain and the Detrended Fluctuation Analysis (DFA) to test for long-memory in time domain. The DFA method is suitable for both stationary and nonstationary time series which makes it to have more power over methods like the rescaled range analysis (R/S) in the estimation of Hurst exponent. Second, the researcher examined whether the markets were predictable under the Adaptive Market Hypothesis (AMH). The researcher employed the Generalised Spectral (GS) test to examine the Martingale difference hypothesis (MDH) of the markets. The Generalised spectral (GS) test is a non-parametric ii test designed to detect the presence of linear and nonlinear dependencies in a stationary time series. The GS test considers dependence at all lags. Third, because of the nonlinear nature in the data-generating process on the markets, the stationarity of the market returns under a nonlinear Exponential Smooth Threshold Autoregressive (ESTAR) model was examined. A nonlinear ADF unit root test against ESTAR and a modified Wald-type test against ESTAR in the analysis were employed. Fourth, the self-exciting threshold Autoregressive (SETAR) method was employed to model the returns when non-linear patterns were observed as a result of nonlinear data generating process on the markets. The literature on market efficiency of African stock markets has shown that variations exist in the study characteristics. There are variations in the method of analysis, type of test, type of data employed, time period chosen and the scope of analysis for the studies. The researcher therefore quantitatively reviewed previous studies by means of meta-analysis to identify which study characteristics affects efficiency conclusions of African markets using the mixed effects model. The findings showed the presence of long-memory in the returns of the stock markets when the whole sample was used. This made the markets weak-form inefficient, however, when the researcher tested for the persistence of long-memory through time, there were periods the markets were efficient in the weak-form. The memory effect was low in the South African market but high in the Mauritian market. Furthermore, it was observed that, the returns for Egypt, which were highly predictable when the whole data was analysed became not highly predictable when the rolling window approach of the GS test was used. Egypt had one of the lowest percentages of the windows that had a p-value less than 0.05 after South Africa. The results obtained from using the non-linear unit root tests on the logarithmic price series of the markets under study showed that, the markets were non-stationary and hence weak-form efficient under an ESTAR framework but for Botswana. Thus the markets were weak-form efficient when analysed using a non-linear method. This observation means that Africa’s foreign direct investment would have been increased over the years if the appropriate methods are used. This is because, over the years, studies on the weak-form efficiency African stock markets have ended with mixed conclusions with most of the markets being concluded to be weak-form inefficient as a result of the use of linear methods in the analysis. This finding, to us, has had an effect on investors commitments to Africa because the right methodology was not employed. iii The findings from modelling the returns under the non-linear SETAR model showed that, the SETAR model performs better than the standard AR(1) and AR(2) model for all the markets under study after the non-linear patterns were identified in the returns series. The SETAR (2,2,2) model is a threshold model, therefore, investors are able to move freely in search of higher opportunities between the low and high regimes. Investors main aim is to make profits, hence, the threshold model of SETAR gives them the freedom to move to a regime where the rate of returns is increasing unlike the standard AR(1) and AR(2) linear models where there are no switching of regimes. Finally, none of the study characteristics in the market efficiency studies was found to be significant in efficiency conclusions of African stock markets but the indicator for publication bias was significant. This means that there has been a change in attitude in recent years towards studies on informational market efficiency whose results do not support the Efficient Market Hypothesis (EMH), unlike the earlier years when the EMH was formulated and acclaimed to be one of the best propositions in economics. It was therefore concluded that when time-varying methods are used in analysing weak-form efficiency, the dynamics of the markets become known to investors for proper decision-making. Also, nonlinear methods should be used in order to reflect the nonlinear nature of data capturing on the stock markets / NRF

<p>Behaviour occurs in time, and precise timing in the range of seconds and fractions of seconds is for most living organisms necessary for successful interaction with the environment. Our ability to time discrete actions and to predict events on the basis of prior events indicates the existence of an internal timing mechanism. The nature of this mechanism provides essential constraints on models of the functional organisation of the brain. </p><p>The present work indicates that there are discontinuities in the function of time close to 1 s and 1.4 s, both in the amount of drift in a series of produced intervals (Study I) and in the detectability of drift in a series of sounds (Study II). The similarities across different tasks further suggest that action and perceptual judgements are governed by the same (kind of) mechanism. Study III showed that series of produced intervals could be characterised by different amounts of positive fractal dependency related to the aforementioned discontinuities. </p><p>In conjunction with other findings in the literature, these results suggest that timing of intervals up to a few seconds is strongly dependent on previous intervals and on the duration to be timed. This argues against a clock-counter mechanism, as proposed by scalar timing theory, according to which successive intervals are random and the size of the timing error conforms to Weber's law. </p><p>A functional model is proposed, expressed in an autoregressive framework, which consists of a single-interval timer with error corrective feedback. The duration-specificity of the proposed model is derived from the order of error correction, as determined by a semi-flexible temporal integration span. </p>

Le mouvement brownien fractionnaire (mbf) constitue un important outil de modélisation utilisé dans plusieurs domaines (biologie, économie, finance, géologie, hydrologie, télécommunications, etc.) ; toutefois, ce modèle ne parvient pas toujours à donner une description suffisamment fidèle de la réalité, à cause, entre autres, des deux limitations suivantes : d'une part le mbf est un processus gaussien, et d'autre part, sa rugosité locale (mesurée par un exposant de Hölder) reste la même tout le long de sa trajectoire, puisque cette rugosité est partout égale au paramètre de Hurst H qui est une constante. En vue d'y remédier, S. Stoev et M.S. Taqqu (2004 et 2005) ont introduit le mouvement multifractionnaire stable linéaire (mmsl) ; ce processus stochastique strictement α-stable (StαS), désigné par {Y(t)}, est obtenu en remplaçant la mesure brownienne par une mesure StαS et le paramètre de Hurst H par une fonction H(.) dépendant de t. On se place systématiquement dans le cas où cette fonction est continue et à valeurs dans l'intervalle ouvert ]1/α,1[. Il convient aussi de noter que l'on a pour tout t, Y(t)=X(t,H(t)), où {X(u,v)} est le champ stochastique StαS, tel que pour tout v fixé, le processus {X(u,v)} est un mouvement fractionnaire stable linéaire. L'objectif de la thèse est de mener une étude approfondie du mmsl, au moyen de méthodes d'ondelettes ; elle consiste principalement en trois parties. (1) On détermine de fins modules de continuité, globaux et locaux de {Y(t)} ; cela repose essentiellement sur une nouvelle représentation de {X(u,v)}, sous la forme d'une série aléatoire, dont on montre la convergence presque sûre dans certains espaces de Hölder. (2) On introduit, via la base de Haar, une autre représentation de {X(u,v)} en série aléatoire ; cette dernière permet la mise en place d'une méthode de simulation efficace du mmsl, ainsi que de ses parties hautes et basses fréquences. (3) On construit des estimateurs par ondelettes du paramètre fonctionnel H(.) du mmsl, ainsi que de son paramètre de stabilité α.

Behaviour occurs in time, and precise timing in the range of seconds and fractions of seconds is for most living organisms necessary for successful interaction with the environment. Our ability to time discrete actions and to predict events on the basis of prior events indicates the existence of an internal timing mechanism. The nature of this mechanism provides essential constraints on models of the functional organisation of the brain. The present work indicates that there are discontinuities in the function of time close to 1 s and 1.4 s, both in the amount of drift in a series of produced intervals (Study I) and in the detectability of drift in a series of sounds (Study II). The similarities across different tasks further suggest that action and perceptual judgements are governed by the same (kind of) mechanism. Study III showed that series of produced intervals could be characterised by different amounts of positive fractal dependency related to the aforementioned discontinuities. In conjunction with other findings in the literature, these results suggest that timing of intervals up to a few seconds is strongly dependent on previous intervals and on the duration to be timed. This argues against a clock-counter mechanism, as proposed by scalar timing theory, according to which successive intervals are random and the size of the timing error conforms to Weber's law. A functional model is proposed, expressed in an autoregressive framework, which consists of a single-interval timer with error corrective feedback. The duration-specificity of the proposed model is derived from the order of error correction, as determined by a semi-flexible temporal integration span.

This work aims at combining the Chaos theory postulates and Artificial Neural Networks classification and predictive capability, in the field of financial time series prediction. Chaos theory, provides valuable qualitative and quantitative tools to decide on the predictability of a chaotic system. Quantitative measurements based on Chaos theory, are used, to decide a-priori whether a time series, or a portion of a time series is predictable, while Chaos theory based qualitative tools are used to provide further observations and analysis on the predictability, in cases where measurements provide negative answers. Phase space reconstruction is achieved by time delay embedding resulting in multiple embedded vectors. The cognitive approach suggested, is inspired by the capability of some chartists to predict the direction of an index by looking at the price time series. Thus, in this work, the calculation of the embedding dimension and the separation, in Takens‘ embedding theorem for phase space reconstruction, is not limited to False Nearest Neighbor, Differential Entropy or other specific method, rather, this work is interested in all embedding dimensions and separations that are regarded as different ways of looking at a time series by different chartists, based on their expectations. Prior to the prediction, the embedded vectors of the phase space are classified with Fuzzy-ART, then, for each class a back propagation Neural Network is trained to predict the last element of each vector, whereas all previous elements of a vector are used as features.

Recently there have been reports that certain physiological data seem to have the properties of long-range correlation and self-similarity. These two properties can be characterized by a long-range dependent parameter d, as well as a self-similar parameter H. In Peng et al (1995), the alteration of long-range correlations with life-threatening pathologies are studied by analyzing the heart rate data of different groups of subjects. The self-similarity properties of two well-known processes, namely the Fractional Brownian Motion (FBM) and the Fractional ARIMA (FARIMA), are of interest to see if it is suitable to be used to model the heart rate data in order to examine the health conditions of some patients. The Embedded Branching Process (EBP) method for estimating parameter $H$ and a goodness of fit test for examining the self-similarity of a process based on the EBP method are proposed in Jones and Shen (2004). In this work, the performance of the goodness of fit test are examined using simulated data from the FBM and FARIMA processes. A modification of the distribution of the test statistics under null hypothesis is proposed and has been modified to be more appropriate. Some simulation comparisons of different estimation methods of the parameter $H$ for some FARIMA processes are also presented and applied to heart rate data obtained from Kaohsiung Veterans General Hospital.

Hurst exponent and variance are two quantities that often characterize real-life, highfrequency observations. We develop the method for simultaneous estimation of a timechanging Hurst exponent H(t) and constant scale (variance) parameter C in a multifractional Brownian motion model in the presence of white noise based on the asymptotic behavior of the local variation of its sample paths. We also discuss the accuracy of the stable and simultaneous estimator compared with a few selected methods and the stability of computations that use adapted wavelet filters. Multifractals have become popular as flexible models in modeling real-life data of high frequency. We developed a method of testing whether the data of high frequency is consistent with monofractality using meaningful descriptors coming from a wavelet-generated multifractal spectrum. We discuss theoretical properties of the descriptors, their computational implementation, the use in data mining, and the effectiveness in the context of simulations, an application in turbulence, and analysis of coding/noncoding regions in DNA sequences. The wavelet thresholding is a simple and effective operation in wavelet domains that selects the subset of wavelet coefficients from a noised signal. We propose the selection of this subset in a semi-supervised fashion, in which a neighbor structure and classification function appropriate for wavelet domains are utilized. The decision to include an unlabeled coefficient in the model depends not only on its magnitude but also on the labeled and unlabeled coefficients from its neighborhood. The theoretical properties of the method are discussed and its performance is demonstrated on simulated examples.

Ce travail de thèse porte sur la modélisation et l'estimation de paramètres pertinents pour les signaux de fréquences cardiaques (FC) instantanées. Nous nous intéressons à un paramètre (appelé grossièrement "fractal"), qui témoigne de la régularité locale de la trajectoire et de la dépendance entre les données. Les propriétés asymptotiques de la fonction DFA (Detrended Fluctuation Analysis) et de l'estimateur de H sont étudiées pour le bruit gaussien fractionnaire (FGN) et plus généralement pour une classe semi-paramétrique de processus stationnaires à longue mémoire avec ou sans tendance. On montre que cette méthode n'est pas robuste. On propose la modélisation des séries de FC par une généralisation du FGN, appelée bruit gaussien localement fractionnaire. Un tel processus stationnaire est construit à partir du paramètre dit de fractalité locale (une sorte de paramètre de Hurst avec des valeurs dans IR) sur une bande de fréquences. L'estimation du paramètre est faite par une analyse par ondelettes, tout comme le test d'adéquation. On montre la pertinence du modèle et une évolution du paramètre pendant la course. Une détection des changements de ce paramètre pourrait être extrêmement appropriée. On propose alors une méthode de détection de multiples ruptures du paramètre de longue mémoire (respectivement d'autosimilarité, de fractalité locale). Un estimateur des points de changements est construit, il vérifie un théorème limite. Un théorème de la limite centrale est établi pour l'estimateur des paramètres et un test d'ajustement est mis en place dans chaque zone où le paramètre est inchangé. Enfin, on montre la même évolution du paramètre de fractalité locale sur les FC.

傳統效率市場假設股價的波動是隨機的，亦即股價是無法預測。 近來的文獻指出股價的波動是不完全是隨機的，且股價的波動具有緩長記憶(long memory)的特性。在本文中我們以R/S分析發現臺灣股市的Hurst指數為0.68，即具有趨勢持續性(trend persistent)之效果，根據此依特性，我們根據Necula(2002)的研究，來評價台股選擇權，發現此新評價模式產生之價格較接近市場價格。