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It can be argued that the identification of sound mathematical models is the ultimate goal of any scientific endeavour. On the other hand, particularly in the investigation of complex systems and nonlinear phenomena, discriminating between alternative models can be a very challenging task. Quite sophisticated model selection criteria are available but their deployment in practice can be problematic. In this work, the Akaike Information Criterion is reformulated with the help of purely information theoretic quantities, namely, the Gibbs-Shannon entropy and the Mutual Information. Systematic numerical tests have proven the improved performances of the proposed upgrades, including increased robustness against noise and the presence of outliers. The same modifications can be implemented to rewrite also Bayesian statistical criteria, such as the Schwartz indicator, in terms of information-theoretic quantities, proving the generality of the approach and the validity of the underlying assumptions.

The inadequacies of basic physics models for disruption prediction have induced the community to increasingly rely on data mining tools. In the last decade, it has been shown how machine learning predictors can achieve a much better performance than those obtained with manually identified thresholds or empirical descriptions of the plasma stability limits. The main criticisms of these techniques focus therefore on two different but interrelated issues: poor “physics fidelity” and limited interpretability. Insufficient “physics fidelity” refers to the fact that the mathematical models of most data mining tools do not reflect the physics of the underlying phenomena. Moreover, they implement a black box approach to learning, which results in very poor interpretability of their outputs. To overcome or at least mitigate these limitations, a general methodology has been devised and tested, with the objective of combining the predictive capability of machine learning tools with the expression of the operational boundary in terms of traditional equations more suited to understanding the underlying physics. The proposed approach relies on the application of machine learning classifiers (such as Support Vector Machines or Classification Trees) and Symbolic Regression via Genetic Programming directly to experimental databases. The results are very encouraging. The obtained equations of the boundary between the safe and disruptive regions of the operational space present almost the same performance as the machine learning classifiers, based on completely independent learning techniques. Moreover, these models possess significantly better predictive power than traditional representations, such as the Hugill or the beta limit. More importantly, they are realistic and intuitive mathematical formulas, which are well suited to supporting theoretical understanding and to benchmarking empirical models. They can also be deployed easily and efficiently in real-time feedback systems.

In the last years the reputation of medical, economic, and scientific expertise has been strongly damaged by a series of false predictions and contradictory studies. The lax application of statistical principles has certainly contributed to the uncertainty and loss of confidence in the sciences. Various assumptions, generally held as valid in statistical treatments, have proved their limits. In particular, since some time it has emerged quite clearly that even slightly departures from normality and homoscedasticity can affect significantly classic significance tests. Robust statistical methods have been developed, which can provide much more reliable estimates. On the other hand, they do not address an additional problem typical of the natural sciences, whose data are often the output of delicate measurements. The data can therefore not only be sampled from a nonnormal pdf but also be affected by significant levels of Gaussian additive noise of various amplitude. To tackle this additional source of uncertainty, in this paper it is shown how already developed robust statistical tools can be usefully complemented with the Geodesic Distance on Gaussian Manifolds. This metric is conceptually more appropriate and practically more effective, in handling noise of Gaussian distribution, than the traditional Euclidean distance. The results of a series of systematic numerical tests show the advantages of the proposed approach in all the main aspects of statistical inference, from measures of location and scale to size effects and hypothesis testing. Particularly relevant is the reduction even of 35% in Type II errors, proving the important improvement in power obtained by applying the methods proposed in the paper. It is worth emphasizing that the proposed approach provides a general framework, in which also noise of different statistical distributions can be dealt with.

The increasingly sophisticated investigations of complex systems require more robust estimates of the correlations between the measured quantities. The traditional Pearson correlation coefficient is easy to calculate but sensitive only to linear correlations. The total influence between quantities is, therefore, often expressed in terms of the mutual information, which also takes into account the nonlinear effects but is not normalized. To compare data from different experiments, the information quality ratio is, therefore, in many cases, of easier interpretation. On the other hand, both mutual information and information quality ratio are always positive and, therefore, cannot provide information about the sign of the influence between quantities. Moreover, they require an accurate determination of the probability distribution functions of the variables involved. As the quality and amount of data available are not always sufficient to grant an accurate estimation of the probability distribution functions, it has been investigated whether neural computational tools can help and complement the aforementioned indicators. Specific encoders and autoencoders have been developed for the task of determining the total correlation between quantities related by a functional dependence, including information about the sign of their mutual influence. Both their accuracy and computational efficiencies have been addressed in detail, with extensive numerical tests using synthetic data. A careful analysis of the robustness against noise has also been performed. The neural computational tools typically outperform the traditional indicators in practically every respect.

The most widely used forms of model selection criteria, the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC), are expressed in terms of synthetic indicators of the residual distribution: the variance and the mean-squared error of the residuals respectively. In many applications in science, the noise affecting the data can be expected to have a Gaussian distribution. Therefore, at the same level of variance and mean-squared error, models, whose residuals are more uniformly distributed, should be favoured. The degree of uniformity of the residuals can be quantified by the Shannon entropy. Including the Shannon entropy in the BIC and AIC expressions improves significantly these criteria. The better performances have been demonstrated empirically with a series of simulations for various classes of functions and for different levels and statistics of the noise. In presence of outliers, a better treatment of the errors, using the Geodesic Distance, has proved essential.

Understanding the details of the correlation between time series is an essential step on the route to assessing the causal relation between systems. Traditional statistical indicators, such as the Pearson correlation coefficient and the mutual information, have some significant limitations. More recently, transfer entropy has been proposed as a powerful tool to understand the flow of information between signals. In this paper, the comparative advantages of transfer entropy, for determining the time horizon of causal influence, are illustrated with the help of synthetic data. The technique has been specifically revised for the analysis of synchronization experiments. The investigation of experimental data from thermonuclear plasma diagnostics proves the potential and limitations of the developed approach.