Chaos Analysis Framework: How to Safely Identify and Quantify Time-Series Dynamics

Within this chapter, a practical introduction to a nonlinear analysis framework tailored for time-series data is provided, enabling the safe quantification of underlying evolutionary dynamics, which describe the referring empirical data generating process. Furthermore, its application provides the possibility to distinct between underlying chaotic versus stochastic dynamics. In addition, an optional combination with (strange) attractor reconstruction algorithms to visualize the denoted system’s dynamics is possible. Since the framework builds upon a large variety of algorithms and methods, its application is by far trivial, especially, in hindsight of reconstruction algorithms for (strange) attractors. Therefore, a general implementation and application guideline for the correct algorithm specifications and avoidance of pitfalls or other unfavorable settings is proposed and respective (graphical) empirical examples are shown. It is intended to provide the readers the possibility to incorporate the proposed analysis framework themselves and to conduct the analyses and reconstructions properly with correct specifications and to be knowledgeable about misleading propositions or parameter choices. Finally, concluding remarks, future avenues of research and future refinements of the framework are proposed

The characterisation of international stock markets using signal processing techniques.

Investors are constantly asking whether beating the market on a consistent basis is possible. There is probably no definitive answer to the question of how to make a guaranteed profit (or return) because index prices can fluctuate at any time. The aim of most investors, therefore, is to predict the stock market return and the volatility, (a measure of investment nsk) and this requires an understanding of stock market behaviour. In this research, diierent techniques, both previously existing and newly developed here (and associated specifically with the discrete wavelet transform (DWT)), are applied to study the behav~our of global stock market indices We consider type of memory, mterrelationships between stock markets, market reaction to crashes and events, and the best indicators of market types (short-term, long-term or mixed). The unifylng aim is to provide a baseline set of characteristic features which typify behaviors of given market type Principal remarks include the fact that the DWT, alone or with other methods, can succeed in providing an in-depth view of these data, in particular when confronted with non-stationary, non-normal and noisy characteristics. The approach provides an important method for the aualysis and interpretation of financial market time series. Our principal findings on volatility measures, moreover, show strong evidence of long-term memory effects, which are not evident in the returns themselves. Emerging and Mature markets are found to deal differently with crashes and events with the latter taking a shorter time to recover from crises on average, compared to the former. Furthermore, we conclude that this binary classification is too simple and stock markets can now be demonstrated to fall into more than two groups, with the designation L'emerging" ("developing") and "mature" ("developed") proving imprecise. Additionally, and in the context of the global market, from Chapter 5, we note that international co-movements and volatility (or nsk) have increased markedly since the middle of the 20th century and that cloclnuzse transmtssion between global stock markets is observed, i.e from Asaa to h o p e to Amerzca back to Asia). The combination of ~nternadl ependencies and external influences provide the impacts for stock market volatility. The ultimate goal, of course, would be to anticipate these Impacts to be able to make the rlght investment decision

Long Memory and Correlation Structures of Select Stock Returns Using Novel Wavelet and Fractal Connectivity Networks

This study investigates the long range dependence and correlation structures of some select stock markets. Using novel wavelet methods of long range dependence, we show presence of long memory in the stock returns of some emerging economies and the lack of it in developed markets of Europe and the United States. Moreover, we conduct a wavelet based fractal connectivity analysis, which is the first application in economics and financial studies, to segregate markets into fractally similar groups and find that developed markets have similar fractal structures. Similarly stock returns of emerging markets exhibiting long-memory tend to follow similar fractal structures. Furthermore, network analyses of fractal connectivity support our findings on market efficiency which has theoretical roots in both fractal and adaptive market hypothesis

Dependence structure in financial time series: Applications and evidence from wavelet analysis

Conventional time series theory and spectral analysis have independently achieved significant popularity in mainstream economics and finance research over long periods. However, the fact remains that each is somewhat lacking if the other is absent. To overcome this problem, a new methodology, wavelet analysis, has been developed to capture all the information localized in time and in frequency, which provides us with an ideal tool to study non-stationary time series. This paper aims to explore the application of a variety of wavelet-based methodologies in conjunction with conventional techniques, such as the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models and long-memory parameter estimates, in analysing the short and long term dependence structure of financial returns and volatility. Specifically, by studying the long-memory property of these time series we hope to identify the source of their possible predictability. Above all else, we document the indispensable role of trading activities associated with low frequencies in determining the long-run dependence of volatility. It follows that GARCH models incorporating long-memory and asymmetric returns-volatility dynamics can provide reasonably accurate volatility forecasts. Additionally, the persistence parameter of returns, represented by the Hurst index, is observed to be correlated to trading profits obtained from typical technical rules designed to detect and capitalize on existing trending behaviour of stock prices. This implies that the Hurst index can be used as a good indicator of the long-memory characteristic of the market, which in turn drives such trending behaviour

Reconstructing complex system dynamics from time series: a method comparison

Modeling complex systems with large numbers of degrees of freedom has become a grand challenge over the past decades. In many situations, only a few variables are actually observed in terms of measured time series, while the majority of variables - which potentially interact with the observed ones - remain hidden. A typical approach is then to focus on the comparably few observed, macroscopic variables, assuming that they determine the key dynamics of the system, while the remaining ones are represented by noise. This naturally leads to an approximate, inverse modeling of such systems in terms of stochastic differential equations (SDEs), with great potential for applications from biology to finance and Earth system dynamics. A well-known approach to retrieve such SDEs from small sets of observed time series is to reconstruct the drift and diffusion terms of a Langevin equation from the data-derived Kramers-Moyal (KM) coefficients. For systems where interactions between the observed and the unobserved variables are crucial, the Mori-Zwanzig formalism (MZ) allows to derive generalized Langevin equations that contain non-Markovian terms representing these interactions. In a similar spirit, the empirical model reduction (EMR) approach has more recently been introduced. In this work we attempt to reconstruct the dynamical equations of motion of both synthetical and real-world processes, by comparing these three approaches in terms of their capability to reconstruct the dynamics and statistics of the underlying systems. Through rigorous investigation of several synthetical and real-world systems, we confirm that the performance of the three methods strongly depends on the intrinsic dynamics of the system at hand. For instance, statistical properties of systems exhibiting weak history-dependence but strong state-dependence of the noise forcing, can be approximated better by the KM method than by the MZ and EMR approaches. In such situations, the KM method is of a considerable advantage since it can directly approximate the state-dependent noise. However, limitations of the KM approximation arise in cases where non-Markovian effects are crucial in the dynamics of the system. In these situations, our numerical results indicate that methods that take into account interactions between observed and unobserved variables in terms of non-Markovian closure terms (i.e., the MZ and EMR approaches), perform comparatively better. © 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaf

Forecasting Financial Time Series using Linear Predictive Filters

Forecasting financial time series is regarded as one of the most challenging applications of time series prediction due to their dynamic nature. However, it is the fundamental element of most investment activities thus attracting the attention of practitioners and researchers for many decades. The purpose of this research is to investigate and develop novel methods for the prediction of financial time series considering their dynamic nature. The predictive performance of asset prices time series themselves is exploited by applying digital signal processing methods to their historical observations. The novelty of the research lies in the design of predictive filters by maximising their spectrum flatness of forecast errors. The filters are then applied to forecast linear combinations of daily open, high, low and close prices of financial time series. Given the assumption that there are no structural breaks or switching regimes in a time series, the sufficient and necessary conditions that a time series can be predicted with zero errors by linear filters are examined. It is concluded that a band-limited time series can be predicted with zero errors by a predictive filter that has a constant magnitude response and constant group delay over the bandwidth of the time series. Because real world time series are not band-limited thus cannot be forecasted without errors, statistical tests of spectrum flatness which evaluate the departure of the spectral density from a constant value are introduced as measures of the predictability of time series. Properties of a time series are then investigated in the frequency domain using its spectrum flatness. A predictive filter is designed by maximising the error spectrum flatness that is equivalent to maximise the “whiteness” of forecast errors in the frequency domain. The focus is then placed on forecasting real world financial time series. By applying spectrum flatness tests, it is found that the property of the spectrum of a linear combination of daily open, high, low and close prices, which is called target prices, is different from that of a random walk process as there are much more low frequency components than high frequency ones in its spectrum. Therefore, an objective function is proposed to derive the target price time series from the historical observations of daily open, high, low and close prices. A predictive filter is then applied to obtain the one-step ahead forecast of the target prices, while profitable trading strategies are designed based on the forecast of target prices series. As a result, more than 70% success ratio could be achieved in terms of one-step ahead out-of-sample forecast of direction changes of the target price time series by taking the S&P500 index for example