Assessment of causal influences is a ubiquitous and important subject across diverse research fields. Drawn from consciousness studies, integrated information is a measure that defines integration as the degree of causal influences among elements. Whereas pairwise causal influences between elements can be quantified with existing methods, quantifying multiple influences among many elements poses two major mathematical difficulties. First, overestimation occurs due to interdependence among influences if each influence is separately quantified in a part-based manner and then simply summed over. Second, it is difficult to isolate causal influences while avoiding noncausal confounding influences. To resolve these difficulties, we propose a theoretical framework based on information geometry for the quantification of multiple causal influences with a holistic approach. We derive a measure of integrated information, which is geometrically interpreted as the divergence between the actual probability distribution of a system and an approximated probability distribution where causal influences among elements are statistically disconnected. This framework provides intuitive geometric interpretations harmonizing various information theoretic measures in a unified manner, including mutual information, transfer entropy, stochastic interaction, and integrated information, each of which is characterized by how causal influences are disconnected. In addition to the mathematical assessment of consciousness, our framework should help to analyze causal relationships in complex systems in a complete and hierarchical manner.integrated information | mutual information | transfer entropy | information geometry | consciousness Q uantitative assessment of causal influences among elements in a complex system is a fundamental problem in many fields of science, including physics (1), economics (2), gene networks (3), social networks (4), ecosystems (5), and neuroscience (6). There have been many previous attempts to quantify causal influences between elements in stochastic systems. Information theory has played a pivotal role in these endeavors, leading to various measures, including predictive information (7), transfer entropy (8), and stochastic interaction (9). Drawn from consciousness studies involving measurement of integration of neural activity (10, 11), the mathematical concept of integrated information is also useful as a framework for analyzing causal relationships in complex systems with multiple elements.Recent research suggests that the brain loses the ability to integrate information when consciousness is lost during dreamless sleep (12), general anesthesia (13), or vegetative states (14), suggesting that quantifying integration of information can serve as a neurophysiological marker of consciousness (10,11,15). The integrated information theory (IIT) of consciousness (16,17) proposes a measure of integration called integrated information that quantifies multiple causal influences among elements of a system. Integrated inform...

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In this letter, we present a minor subspace rule that extracts the subspace that spans the m minor components of a n-dimensional vector stationary random process, m < n: The algorithm is self-stabilizing such that the subspace vectors do not need to be periodically normalized to unit modulus, and the algorithm does not require matrix inversions or divides to maintain its stable behavior.Index Terms-Adaptive algorithm, minor component analysis, subspace methods.

Background: The four heterogeneous childhood cancers, neuroblastoma, non-Hodgkin lymphoma, rhabdomyosarcoma, and Ewing sarcoma present a similar histology of small round blue cell tumor (SRBCT) and thus often leads to misdiagnosis. Identification of biomarkers for distinguishing these cancers is a well studied problem. Existing methods typically evaluate each gene separately and do not take into account the nonlinear interaction between genes and the tools that are used to design the diagnostic prediction system. Consequently, more genes are usually identified as necessary for prediction. We propose a general scheme for finding a small set of biomarkers to design a diagnostic system for accurate classification of the cancer subgroups. We use multilayer networks with online gene selection ability and relational fuzzy clustering to identify a small set of biomarkers for accurate classification of the training and blind test cases of a well studied data set.

Mathematical and statistical models have played important roles in neuroscience, especially by describing the electrical activity of neurons recorded individually, or collectively across large networks. As the field moves forward rapidly, new challenges are emerging. For maximal effectiveness, those working to advance computational neuroscience will need to appreciate and exploit the complementary strengths of mechanistic theory and the statistical paradigm.