Institute of Measurement Science
Topic
Rényi dimensions and entropies in causal analysis of measured time series
PhD. program
Applied mathematics
Name of the supervisor
RNDr. Anna Krakovská, CSc.
Contact:
Receiving school
Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava
Annotation
Detecting causal relationships from time series is an emerging topic in many scientific disciplines. It brings theoretical challenges as well as the opportunity to design new methods and test them, for example, on signals from multifractal processes as well as on real measurements. Diverse application areas include extensive sets of climate measurements, multi-channel electroencephalographic recordings from the human brain, temporal evolution of a set of macro-economic indicators and other real challenges of finding causal relationships from measured time series. Open problems to be explored include distinguishing between direct and indirect effects and revealing the presence of unobserved confounders.
A prospective avenue in causal analysis involves assessing the differences in complexity between driving and driven systems. As a tool for estimating the uncertainty or information complexity, we will use Rényi's generalized dimensions and entropies, which are an extension of the classical Shannon information. However, numerical approaches to estimating these measures are computationally demanding, and their reliability is constantly questioned. Therefore, methods for estimating dimensions and entropies are constantly increasing. Our proposal is a generalization of a simple and fast technique, which is based on the evaluation of the distances of two nearest neighbors of points in the state spaces of the investigated systems. It was published in the context of second-order Rényi dimension estimation, but the generalization for Rényi entropies and subsequent use in causal time series analysis is a natural next step that needs to be explored.
The topic is suitable for graduates interested in creative application and development of relevant mathematical approaches. The successful candidate must also have good English skills, as well as experience in creating and testing software in the MatLab environment. As part of the study, the doctoral student will become familiar with selected methods that draw from the theory of dynamical systems, including chaos and fractal theory, and partially also from statistics and information theory.
The dissertation will be completed in a partner external educational institution, at the workplace of Institute of Measurement Science of the Slovak Academy of Sciences, v. v. i. in Bratislava.
A prospective avenue in causal analysis involves assessing the differences in complexity between driving and driven systems. As a tool for estimating the uncertainty or information complexity, we will use Rényi's generalized dimensions and entropies, which are an extension of the classical Shannon information. However, numerical approaches to estimating these measures are computationally demanding, and their reliability is constantly questioned. Therefore, methods for estimating dimensions and entropies are constantly increasing. Our proposal is a generalization of a simple and fast technique, which is based on the evaluation of the distances of two nearest neighbors of points in the state spaces of the investigated systems. It was published in the context of second-order Rényi dimension estimation, but the generalization for Rényi entropies and subsequent use in causal time series analysis is a natural next step that needs to be explored.
The topic is suitable for graduates interested in creative application and development of relevant mathematical approaches. The successful candidate must also have good English skills, as well as experience in creating and testing software in the MatLab environment. As part of the study, the doctoral student will become familiar with selected methods that draw from the theory of dynamical systems, including chaos and fractal theory, and partially also from statistics and information theory.
The dissertation will be completed in a partner external educational institution, at the workplace of Institute of Measurement Science of the Slovak Academy of Sciences, v. v. i. in Bratislava.