Uvođenje analize nelinearnih vremenskih nizova u dodiplomski studij

This article is written for undergraduate students and teachers who would like to get familiar with basic nonlinear time series analysis methods. We present a step-by-step study of a simple example and provide user-friendly programs that allow an easy reproduction of presented results. In particular, we study an artificial time series generated by the Lorenz system. The mutual information and false nearest neighbour method are explained in detail, and used to obtain the best possible attractor reconstruction. Subsequently, the times series is tested for stationarity and determinism, which are both important properties that assure correct interpretation of invariant quantities that can be extracted from the data set. Finally, as the most prominent invariant quantity that allows distinguishing between regular and chaotic behaviour, we calculate the maximal Lyapunov exponent. By following the above steps, we are able to convincingly determine that the Lorenz system is chaotic directly from the generated time series, without the need to use the differential equations. Throughout the paper, emphasis on clear-cut guidance and a hands-on approach is given in order to make the reproduction of presented results possible also for undergraduates, and thus encourage them to get familiar with the presented theory.Ovaj smo članak napisali za dodiplomske studente i nastavnike koji se žele upoznati s osnovnim metodama analize nelinernih vremenskih nizova. Postupno proučavamo jednostavan primjer takvog niza i dajemo programe za lako ponavljanje izloženih ishoda računa. Taj je primjer umjetan vremenski niz stvoren Lorenzovim sustavom jednadžbi. Podrobno objašnjavamo metode uzajamnih informacija i krivog najbližeg susjeda, koje se primjenjuju za najbolje nalaženje nakupinskih točaka. Zatim se ispituju stacionarnost i determinizam vremenskih nizova, koji su važna svojstva za ispravno tumačenje nepromjenljivih veličina koje se mogu izvesti iz skupa podataka. Na kraju računamo najveći Ljapunovljev eksponent koji je najvažnija nepromjenljiva veličina za razlikovanje pravilnog i kaotičnog ponašanja niza. Slijedom ovih koraka utvrđujemo uvjerljivo da je Lorenzov sustav kaotičan, izravno iz izvedenog niza, bez upotrebe diferencijalnih jednadžbi. U radu se poklanja velika pažnja naputcima i izravnoj primjeni metoda kako bi i dodiplomski studenti mogli ponoviti izložene račune i tako se potaknuli da bolje upoznaju prikazanu teoriju

Statistically significant features improve binary and multiple Motor Imagery task predictions from EEGs

In recent studies, in the field of Brain-Computer Interface (BCI), researchers have focused on Motor Imagery tasks. Motor Imagery-based electroencephalogram (EEG) signals provide the interaction and communication between the paralyzed patients and the outside world for moving and controlling external devices such as wheelchair and moving cursors. However, current approaches in the Motor Imagery-BCI system design require effective feature extraction methods and classification algorithms to acquire discriminative features from EEG signals due to the non-linear and non-stationary structure of EEG signals. This study investigates the effect of statistical significance-based feature selection on binary and multi-class Motor Imagery EEG signal classifications. In the feature extraction process performed 24 different time-domain features, 15 different frequency-domain features which are energy, variance, and entropy of Fourier transform within five EEG frequency subbands, 15 different time-frequency domain features which are energy, variance, and entropy of Wavelet transform based on five EEG frequency subbands, and 4 different Poincare plot-based non-linear parameters are extracted from each EEG channel. A total of 1,364 Motor Imagery EEG features are supplied from 22 channel EEG signals for each input EEG data. In the statistical significance-based feature selection process, the best one among all possible combinations of these features is tried to be determined using the independent t-test and one-way analysis of variance (ANOVA) test on binary and multi-class Motor Imagery EEG signal classifications, respectively. The whole extracted feature set and the feature set that contain statistically significant features only are classified in this study. We implemented 6 and 7 different classifiers in multi-class and binary (two-class) classification tasks, respectively. The classification process is evaluated using the five-fold cross-validation method, and each classification algorithm is tested 10 times. These repeated tests provide to check the repeatability of the results. The maximum of 61.86 and 47.36% for the two-class and four-class scenarios, respectively, are obtained with Ensemble Subspace Discriminant among all these classifiers using selected features including only statistically significant features. The results reveal that the introduced statistical significance-based feature selection approach improves the classifier performances by achieving higher classifier performances with fewer relevant components in Motor Imagery task classification. In conclusion, the main contribution of the presented study is two-fold evaluation of non-linear parameters as an alternative to the commonly used features and the prediction of multiple Motor Imagery tasks using statistically significant features

Impact of critical mass on the evolution of cooperation in spatial public goods games

We study the evolution of cooperation under the assumption that the collective benefits of group membership can only be harvested if the fraction of cooperators within the group, i.e. their critical mass, exceeds a threshold value. Considering structured populations, we show that a moderate fraction of cooperators can prevail even at very low multiplication factors if the critical mass is minimal. For larger multiplication factors, however, the level of cooperation is highest at an intermediate value of the critical mass. The latter is robust to variations of the group size and the interaction network topology. Applying the optimal critical mass threshold, we show that the fraction of cooperators in public goods games is significantly larger than in the traditional linear model, where the produced public good is proportional to the fraction of cooperators within the group.Comment: 4 two-column pages, 4 figures; accepted for publication in Physical Review

Uvođenje analize nelinearnih vremenskih nizova u dodiplomski studij

This article is written for undergraduate students and teachers who would like to get familiar with basic nonlinear time series analysis methods. We present a step-by-step study of a simple example and provide user-friendly programs that allow an easy reproduction of presented results. In particular, we study an artificial time series generated by the Lorenz system. The mutual information and false nearest neighbour method are explained in detail, and used to obtain the best possible attractor reconstruction. Subsequently, the times series is tested for stationarity and determinism, which are both important properties that assure correct interpretation of invariant quantities that can be extracted from the data set. Finally, as the most prominent invariant quantity that allows distinguishing between regular and chaotic behaviour, we calculate the maximal Lyapunov exponent. By following the above steps, we are able to convincingly determine that the Lorenz system is chaotic directly from the generated time series, without the need to use the differential equations. Throughout the paper, emphasis on clear-cut guidance and a hands-on approach is given in order to make the reproduction of presented results possible also for undergraduates, and thus encourage them to get familiar with the presented theory.Ovaj smo članak napisali za dodiplomske studente i nastavnike koji se žele upoznati s osnovnim metodama analize nelinernih vremenskih nizova. Postupno proučavamo jednostavan primjer takvog niza i dajemo programe za lako ponavljanje izloženih ishoda računa. Taj je primjer umjetan vremenski niz stvoren Lorenzovim sustavom jednadžbi. Podrobno objašnjavamo metode uzajamnih informacija i krivog najbližeg susjeda, koje se primjenjuju za najbolje nalaženje nakupinskih točaka. Zatim se ispituju stacionarnost i determinizam vremenskih nizova, koji su važna svojstva za ispravno tumačenje nepromjenljivih veličina koje se mogu izvesti iz skupa podataka. Na kraju računamo najveći Ljapunovljev eksponent koji je najvažnija nepromjenljiva veličina za razlikovanje pravilnog i kaotičnog ponašanja niza. Slijedom ovih koraka utvrđujemo uvjerljivo da je Lorenzov sustav kaotičan, izravno iz izvedenog niza, bez upotrebe diferencijalnih jednadžbi. U radu se poklanja velika pažnja naputcima i izravnoj primjeni metoda kako bi i dodiplomski studenti mogli ponoviti izložene račune i tako se potaknuli da bolje upoznaju prikazanu teoriju

Evoplex: A platform for agent-based modeling on networks

Agent-based modeling and network science have been used extensively to advance our understanding of emergent collective behavior in systems that are composed of a large number of simple interacting individuals or agents. With the increasing availability of high computational power in affordable personal computers, dedicated efforts to develop multi-threaded, scalable and easy-to-use software for agent-based simulations are needed more than ever. Evoplex meets this need by providing a fast, robust and extensible platform for developing agent-based models and multi-agent systems on networks. Each agent is represented as a node and interacts with its neighbors, as defined by the network structure. Evoplex is ideal for modeling complex systems, for example in evolutionary game theory and computational social science. In Evoplex, the models are not coupled to the execution parameters or the visualization tools, and there is a user-friendly graphical interface which makes it easy for all users, ranging from newcomers to experienced, to create, analyze, replicate and reproduce the experiments.Comment: 6 pages, 5 figures; accepted for publication in SoftwareX [software available at https://evoplex.org

Evolution of extortion in structured populations

Extortion strategies can dominate any opponent in an iterated prisoner’s dilemma game. But if players are able to adopt the strategies performing better, extortion becomes widespread and evolutionary unstable. It may sometimes act as a catalyst for the evolution of cooperation, and it can also emerge in interactions between two populations, yet it is not the evolutionarily stable outcome. Here we revisit these results in the realm of spatial games. We find that pairwise imitation and birth-death dynamics return known evolutionary outcomes. Myopic best response strategy updating, on the other hand, reveals counterintuitive solutions. Defectors and extortioners coarsen spontaneously, which allows cooperators to prevail even at prohibitively high temptations to defect. Here extortion strategies play the role of a Trojan horse. They may emerge among defectors by chance, and once they do, cooperators become viable as well. These results are independent of the interaction topology, and they highlight the importance of coarsening, checkerboard ordering, and best response updating in evolutionary games

Decelerated invasion and waning-moon patterns in public goods games with delayed distribution

We study the evolution of cooperation in the spatial public goods game, focusing on the effects that are brought about by the delayed distribution of goods that accumulate in groups due to the continuous investments of cooperators. We find that intermediate delays enhance network reciprocity because of a decelerated invasion of defectors, who are unable to reap the same high short-term benefits as they do in the absence of delayed distribution. Long delays, however, introduce a risk because the large accumulated wealth might fall into the wrong hands. Indeed, as soon as the curvature of a cooperative cluster turns negative, the engulfed defectors can collect the heritage of many generations of cooperators and by doing so start a waning-moon pattern that nullifies the benefits of decelerated invasion. Accidental meeting points of growing cooperative clusters may also act as triggers for the waning-moon effect, thus linking the success of cooperators with their propensity to fail in a rather bizarre way. Our results highlight that “investing in the future” is a good idea only if that future is sufficiently near and not likely to be burdened by inflation

Focus on multilayer networks

Multilayer networks have in recent years emerged as an important new paradigm of network science. Groundbreaking research has shown that processes that unfold on different but interdependent network layers can not be simply reduced to a conglomerate of additive processes on a single network. On the contrary, small and seemingly unimportant changes in one network layer can have far-reaching and indeed catastrophic consequences in other network layers. Such cascades of failures can lead to concurrent malfunctions in electrical power grids, they can gridlock traffic, and accelerate epidemics, to name just some examples. In the light of this functional relevance, network science has had to redefine structural measures, rethink growth processes, and come up with new mathematical formulations for multilayer networks. The field is still very much alight and vibrant, and with the focus on multilayer networks, the New Journal of Physics has given due space to the forefront research along these lines

Noise-guided evolution within cyclical interactions

We study a stochastic predator-prey model on a square lattice, where each of the six species has two superior and two inferior partners. The invasion probabilities between species depend on the predator-prey pair and are supplemented by Gaussian noise. Conditions are identified that warrant the largest impact of noise on the evolutionary process, and the results of Monte Carlo simulations are qualitatively reproduced by a four-point cluster dynamical mean-field approximation. The observed noise-guided evolution is deeply routed in short-range spatial correlations, which is supported by simulations on other host lattice topologies. Our findings are conceptually related to the coherence resonance phenomenon in dynamical systems via the mechanism of threshold duality. We also show that the introduced concept of noise-guided evolution via the exploitation of threshold duality is not limited to predator-prey cyclical interactions, but may apply to models of evolutionary game theory as well, thus indicating its applicability in several different fields of research.Comment: to be published in New J. Phy