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1.

Borysov, Stanislav S.

et al.

Stockholm University, Nordic Institute for Theoretical Physics (Nordita). KTH Royal Institute of Technology, Sweden.

Roudi, Yasser

Stockholm University, Nordic Institute for Theoretical Physics (Nordita). The Kavli Institute for Systems Neuroscience, NTNU, Norway.

Balatsky, Alexander V.

Stockholm University, Nordic Institute for Theoretical Physics (Nordita). Institute for Materials Science, Los Alamos National Laboratory, Los Alamos, USA .

We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented results show that binarization preserves the correlation structure of the market. Properties of distributions of external fields and couplings as well as the market interaction network and industry sector clustering structure are studied for different historical dates and moving window sizes. We demonstrate that the observed positive heavy tail in distribution of couplings is related to the sparse clustering structure of the market. We also show that discrepancies between the model's parameters might be used as a precursor of financial instabilities.

We study inference and reconstruction of couplings in a partially observed kinetic Ising model. With hidden spins, calculating the likelihood of a sequence of observed spin configurations requires performing a trace over the configurations of the hidden ones. This, as we show, can be represented as a path integral. Using this representation, we demonstrate that systematic approximate inference and learning rules can be derived using dynamical mean-field theory. Although naive mean-field theory leads to an unstable learning rule, taking into account Gaussian corrections allows learning the couplings involving hidden nodes. It also improves learning of the couplings between the observed nodes compared to when hidden nodes are ignored. DOI: 10.1103/PhysRevE.87.022127

Stockholm University, Nordic Institute for Theoretical Physics (Nordita). University of Copenhagen, Denmark.

Roudi, Yasser

Stockholm University, Nordic Institute for Theoretical Physics (Nordita). Kavli Institute for Systems Neuroscience and Centre for Neural Computation, NTNU, Norway; Institute for Advanced Study, Princeton, NJ, USA.

We review some of the techniques used to study the dynamics of disordered systems subject to both quenched and fast (thermal) noise. Starting from the Martin-Siggia-Rose/Janssen-De Dominicis-Peliti path integral formalism for a single variable stochastic dynamics, we provide a pedagogical survey of the perturbative, i.e. diagrammatic, approach to dynamics and how this formalism can be used for studying soft spin models. We review the supersymmetric formulation of the Langevin dynamics of these models and discuss the physical implications of the supersymmetry. We also describe the key steps involved in studying the disorder-averaged dynamics. Finally, we discuss the path integral approach for the case of hard Ising spins and review some recent developments in the dynamics of such kinetic Ising models.

Stockholm University, Nordic Institute for Theoretical Physics (Nordita). Norwegian University of Science & Technology.

On sampling and modeling complex systems2013In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, E-ISSN 1742-5468, p. P09003-Article in journal (Refereed)

Abstract [en]

The study of complex systems is limited by the fact that only a few variables are accessible for modeling and sampling, which are not necessarily the most relevant ones to explain the system behavior. In addition, empirical data typically undersample the space of possible states. We study a generic framework where a complex system is seen as a system of many interacting degrees of freedom, which are known only in part, that optimize a given function. We show that the underlying distribution with respect to the known variables has the Boltzmann form, with a temperature that depends on the number of unknown variables. In particular, when the influence of the unknown degrees of freedom on the known variables is not too irregular, the temperature decreases as the number of variables increases. This suggests that models can be predictable only when the number of relevant variables is less than a critical threshold. Concerning sampling, we argue that the information that a sample contains on the behavior of the system is quantified by the entropy of the frequency with which different states occur. This allows us to characterize the properties of maximally informative samples: within a simple approximation, the most informative frequency size distributions have power law behavior and Zipf's law emerges at the crossover between the under sampled regime and the regime where the sample contains enough statistics to make inferences on the behavior of the system. These ideas are illustrated in some applications, showing that they can be used to identify relevant variables or to select the most informative representations of data, e.g. in data clustering.

Neurons subject to a common nonstationary input may exhibit a correlated firing behavior. Correlations in the statistics of neural spike trains also arise as the effect of interaction between neurons. Here we show that these two situations can be distinguished with machine learning techniques, provided that the data are rich enough. In order to do this, we study the problem of inferring a kinetic Ising model, stationary or nonstationary, from the available data. We apply the inference procedure to two data sets: one from salamander retinal ganglion cells and the other from a realistic computational cortical network model. We show that many aspects of the concerted activity of the salamander retinal neurons can be traced simply to the external input. A model of non-interacting neurons subject to a nonstationary external field outperforms a model with stationary input with couplings between neurons, even accounting for the differences in the number of model parameters. When couplings are added to the nonstationary model, for the retinal data, little is gained: the inferred couplings are generally not significant. Likewise, the distribution of the sizes of sets of neurons that spike simultaneously and the frequency of spike patterns as a function of their rank (Zipf plots) are well explained by an independent-neuron model with time-dependent external input, and adding connections to such a model does not offer significant improvement. For the cortical model data, robust couplings, well correlated with the real connections, can be inferred using the nonstationary model. Adding connections to this model slightly improves the agreement with the data for the probability of synchronous spikes but hardly affects the Zipf plot.

We describe how the couplings in an asynchronous kinetic Ising model can be inferred. We consider two cases: one in which we know both the spin history and the update times and one in which we know only the spin history. For the first case, we show that one can average over all possible choices of update times to obtain a learning rule that depends only on spin correlations and can also be derived from the equations of motion for the correlations. For the second case, the same rule can be derived within a further decoupling approximation. We study all methods numerically for fully asymmetric Sherrington-Kirkpatrick models, varying the data length, system size, temperature, and external field. Good convergence is observed in accordance with the theoretical expectations.