On the identifiability of ternary forms

We describe a new method to determine the minimality and identifiability of a Waring decomposition $A$ of a specific form (symmetric tensor) $T$ in three variables. Our method, which is based on the Hilbert function of $A$, can distinguish between forms in the span of the Veronese image of $A$, which in general contains both identifiable and not identifiable points, depending on the choice of coefficients in the decomposition. This makes our method applicable for all values of the length $r$ of the decomposition, from $2$ up to the generic rank, a range which was not achievable before. Though the method in principle can handle all cases of specific ternary forms, we introduce and describe it in details for forms of degree $8$

Identifiability for a class of symmetric tensors

We use methods of algebraic geometry to find new, effective methods for detecting the identifiability of symmetric tensors. In particular, for ternary symmetric tensors T of degree 7, we use the analysis of the Hilbert function of a finite projective set, and the Cayley-Bacharach property, to prove that, when the Kruskal's rank of a decomposition of T are maximal (a condition which holds outside a Zariski closed set of measure 0), then the tensor T is identifiable, i.e. the decomposition is unique, even if the rank lies beyond the range of application of both the Kruskal's and the reshaped Kruskal's criteria

Invalid proxies and volatility changes

When in proxy-SVARs the covariance matrix of VAR disturbances is subject to exogenous, permanent, nonrecurring breaks that generate target impulse response functions (IRFs) that change across volatility regimes, even strong, exogenous external instruments can result in inconsistent estimates of the dynamic causal effects of interest if the breaks are not properly accounted for. In such cases, it is essential to explicitly incorporate the shifts in unconditional volatility in order to point-identify the target structural shocks and possibly restore consistency. We demonstrate that, under a necessary and sufficient rank condition that leverages moments implied by changes in volatility, the target IRFs can be point-identified and consistently estimated. Importantly, standard asymptotic inference remains valid in this context despite (i) the covariance between the proxies and the instrumented structural shocks being local-to-zero, as in Staiger and Stock (1997), and (ii) the potential failure of instrument exogeneity. We introduce a novel identification strategy that appropriately combines external instruments with "informative" changes in volatility, thus obviating the need to assume proxy relevance and exogeneity in estimation. We illustrate the effectiveness of the suggested method by revisiting a fiscal proxy-SVAR previously estimated in the literature, complementing the fiscal instruments with information derived from the massive reduction in volatility observed in the transition from the Great Inflation to the Great Moderation regimes

On the description of identifiable quartics

In this paper we study the identifiability of specific forms (symmetric tensors), with the target of extending recent methods for the case of $3$ variables to more general cases. In particular, we focus on forms of degree $4$ in $5$ variables. By means of tools coming from classical algebraic geometry, such as Hilbert function, liaison procedure and Serre's construction, we give a complete geometric description and criteria of identifiability for ranks $\geq 9$, filling the gap between rank $\leq 8$, covered by Kruskal's criterion, and $15$, the rank of a general quartic in $5$ variables. For the case $r=12$, we construct an effective algorithm that guarantees that a given decomposition is unique

PARX model for football matches predictions

We propose an innovative approach to model and predict the outcome of football matches based on the Poisson Autoregression with eXogenous covariates (PARX) model recently proposed by Agosto, Cavaliere, Kristensen and Rahbek (2016). We show that this methodology is particularly suited to model the goals distribution of a football team and provides a good forecast performance that can be exploited to develop a profitable betting strategy. The betting strategy is based on the idea that the odds proposed by the market do not reflect the true probability of the match because they may incorporate also the betting volumes or strategic price settings in order to exploit bettors’ biases. The out-of-sample performance of the PARX model is better than the reference approach by Dixon and Coles (1997). We also evaluate our approach in a simple betting strategy which is applied to the English football Premier League data for the 2013/2014 and 2014/2015 seasons. The results show that the return from the betting strategy is larger than 35% in all the cases considered and may even exceed 100% if we consider an alternative strategy based on a predetermined threshold which allows to exploit the inefficiency of the betting market

Synergetic and redundant information flow detected by unnormalized Granger causality: application to resting state fMRI

Objectives: We develop a framework for the analysis of synergy and redundancy in the pattern of information flow between subsystems of a complex network. Methods: The presence of redundancy and/or synergy in multivariate time series data renders difficult to estimate the neat flow of information from each driver variable to a given target. We show that adopting an unnormalized definition of Granger causality one may put in evidence redundant multiplets of variables influencing the target by maximizing the total Granger causality to a given target, over all the possible partitions of the set of driving variables. Consequently we introduce a pairwise index of synergy which is zero when two independent sources additively influence the future state of the system, differently from previous definitions of synergy. Results: We report the application of the proposed approach to resting state fMRI data from the Human Connectome Project, showing that redundant pairs of regions arise mainly due to space contiguity and interhemispheric symmetry, whilst synergy occurs mainly between non-homologous pairs of regions in opposite hemispheres. Conclusions: Redundancy and synergy, in healthy resting brains, display characteristic patterns, revealed by the proposed approach. Significance: The pairwise synergy index, here introduced, maps the informational character of the system at hand into a weighted complex network: the same approach can be applied to other complex systems whose normal state corresponds to a balance between redundant and synergetic circuits.Comment: 6 figures. arXiv admin note: text overlap with arXiv:1403.515

Vi(E)va LLM! A Conceptual Stack for Evaluating and Interpreting Generative AI-based Visualizations

The automatic generation of visualizations is an old task that, through the years, has shown more and more interest from the research and practitioner communities. Recently, large language models (LLM) have become an interesting option for supporting generative tasks related to visualization, demonstrating initial promising results. At the same time, several pitfalls, like the multiple ways of instructing an LLM to generate the desired result, the different perspectives leading the generation (code-based, image-based, grammar-based), and the presence of hallucinations even for the visualization generation task, make their usage less affordable than expected. Following similar initiatives for benchmarking LLMs, this paper copes with the problem of modeling the evaluation of a generated visualization through an LLM. We propose a theoretical evaluation stack, EvaLLM, that decomposes the evaluation effort in its atomic components, characterizes their nature, and provides an overview of how to implement and interpret them. We also designed and implemented an evaluation platform that provides a benchmarking resource for the visualization generation task. The platform supports automatic and manual scoring conducted by multiple assessors to support a fine-grained and semantic evaluation based on the EvaLLM stack. Two case studies on GPT3.5-turbo with Code Interpreter and Llama2-70-b models show the benefits of EvaLLM and illustrate interesting results on the current state-of-the-art LLM-generated visualizations

Bootstrapping DSGE models

This paper explores the potential of bootstrap methods in the empirical evalu- ation of dynamic stochastic general equilibrium (DSGE) models and, more generally, in linear rational expectations models featuring unobservable (latent) components. We consider two dimensions. First, we provide mild regularity conditions that suffice for the bootstrap Quasi- Maximum Likelihood (QML) estimator of the structural parameters to mimic the asymptotic distribution of the QML estimator. Consistency of the bootstrap allows to keep the probability of false rejections of the cross-equation restrictions under control. Second, we show that the realizations of the bootstrap estimator of the structural parameters can be constructively used to build novel, computationally straightforward tests for model misspecification, including the case of weak identification. In particular, we show that under strong identification and boot- strap consistency, a test statistic based on a set of realizations of the bootstrap QML estimator approximates the Gaussian distribution. Instead, when the regularity conditions for inference do not hold as e.g. it happens when (part of) the structural parameters are weakly identified, the above result is no longer valid. Therefore, we can evaluate how close or distant is the esti- mated model from the case of strong identification. Our Monte Carlo experimentations suggest that the bootstrap plays an important role along both dimensions and represents a promising evaluation tool of the cross-equation restrictions and, under certain conditions, of the strength of identification. An empirical illustration based on a small-scale DSGE model estimated on U.S. quarterly observations shows the practical usefulness of our approach

Waring decompositions of special ternary forms with different Hilbert functions

We prove the existence of ternary forms admitting apolar sets of points of cardinality equal to the Waring rank, but having different Hilbert function and different regularity. This is done exploiting liaison theory and Cayley-Bacharach properties for sets of points in the projective planeComment: 12 pages. Comments are welcome