Measuring Interactions in Categorical Datasets Using Multivariate Symmetrical Uncertainty

Interaction between variables is often found in statistical models, and it is usually expressed in the model as an additional term when the variables are numeric. However, when the variables are categorical (also known as nominal or qualitative) or mixed numerical-categorical, defining, detecting, and measuring interactions is not a simple task. In this work, based on an entropy-based correlation measure for n nominal variables (named as Multivariate Symmetrical Uncertainty (MSU)), we propose a formal and broader definition for the interaction of the variables. Two series of experiments are presented. In the first series, we observe that datasets where some record types or combinations of categories are absent, forming patterns of records, which often display interactions among their attributes. In the second series, the interaction/non-interaction behavior of a regression model (entirely built on continuous variables) gets successfully replicated under a discretized version of the dataset. It is shown that there is an interaction-wise correspondence between the continuous and the discretized versions of the dataset. Hence, we demonstrate that the proposed definition of interaction enabled by the MSU is a valuable tool for detecting and measuring interactions within linear and non-linear models.

Exploratory Graph Analysis for Factor Retention: Simulation Results for Continuous and Binary Data

Exploratory graph analysis (EGA) is a commonly applied technique intended to help social scientists discover latent variables. Yet, the results can be influenced by the methodological decisions the researcher makes along the way. In this article, we focus on the choice regarding the number of factors to retain: We compare the performance of the recently developed EGA with various traditional factor retention criteria. We use both continuous and binary data, as evidence regarding the accuracy of such criteria in the latter case is scarce. Simulation results, based on scenarios resulting from varying sample size, communalities from major factors, interfactor correlations, skewness, and correlation measure, show that EGA outperforms the traditional factor retention criteria considered in most cases in terms of bias and accuracy. In addition, we show that factor retention decisions for binary data are preferably made using Pearson, instead of tetrachoric, correlations, which is contradictory to popular belief.

Engineering non-classical correlation and teleportation with Robust fidelity using Jaynes–Cummings model

In this paper, we propose a model to describe the geometry of quantum correlations and entanglement through their distinct physical significance in quantum information processing and modern communications. However, geometric discord, using trace, Hilbert–Schmidt distances, and entanglement of formation, is engineered to be a well-defined non-classical correlation measure of an atomic field system. It consists of employing Jaynes–Cummings model to study the interaction between an excited atom at two levels and a single electromagnetic field mode inside an electrodynamic cavity in two cases, namely resonance and non-resonance. In fact, the dynamics of these measures depends decisively on the atom-field initial parameters where, importantly, the field parameters can be specified as control settings to implement an optimal teleportation protocol. The obtained results reveal that the behaviors of teleported geometric quantum discord and entanglement are similar to those displayed for maximum fidelity in terms of fully entanglement fraction. Therefore, since fidelity always exceeds the classical limit, one can design a quantum teleportation scheme with robust fidelity superior to any conventional communication protocol.

The Markov gap for geometric reflected entropy

The reflected entropy SR(A : B) of a density matrix ρAB is a bipartite correlation measure lower-bounded by the quantum mutual information I(A : B). In holographic states satisfying the quantum extremal surface formula, where the reflected entropy is related to the area of the entanglement wedge cross-section, there is often an order-N2 gap between SR and I. We provide an information-theoretic interpretation of this gap by observing that SR− I is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity SR− I the Markov gap. We then prove that for time-symmetric states in pure AdS3 gravity, the Markov gap is universally lower bounded by log(2)ℓAdS/2GN times the number of endpoints of the cross-section. We provide evidence that this lower bound continues to hold in the presence of bulk matter, and comment on how it might generalize above three bulk dimensions. Finally, we explore the Markov recovery problem controlling SR− I using fixed area states. This analysis involves deriving a formula for the quantum fidelity — in fact, for all the sandwiched Rényi relative entropies — between fixed area states with one versus two fixed areas, which may be of independent interest. We discuss, throughout the paper, connections to the general theory of multipartite entanglement in holography.

The time lag between deformation process and seismic activity in El Hierro Island during the eruptive process (2011–2014): a functional phased approach

On 10 October 2011, a submarine eruption occurred in El Hierro island. Thus, the eruptive process in the Canary islands was reactivated after 40 years of inactivity. The main objective of this work is to evaluate, using Functional Data Analysis, how the surface deformation phenomenon explains the seismic–volcanic activity in the island. The GNSS-GPS data are from the FRON (GRAFCAN) station, located in Frontera. These data measure, each 4 h, the distance between the FRON station and the reference station LPAL (La Palma island) from August, 2010 to December, 2013. In this study a functional correlation measure is employed to establish the relation between the deformation curve and the curve of cumulative energy released. The period of time analysed has been divided into four phases to avoid the mix of phenomena. For each phase, the correlation measure and the time lag between deformation curve and the curve of cumulative energy released have been estimated. These values show a strong relation between these curves. With respect to time lag period, the only significant lag, of about 1 month, occurred in Phase 1, which was after a long period without seismic activity. The later phases had very short, insignificant, lags. After a long period without seismic and volcanic activity in El Hierro island, the time lag between the deformation process and the beginning of the seismic activity takes approximately 1 month. In a similar situation a method to predict in real time the beginning of the seismic activity is proposed. This method, based on the changes produced in the derivative curves when there is a rapid descent in the deformation curve, could activate a warning system approximately 13 days before the beginning of seismicity.

Assessing Disparity Using Measures of Racial and Educational Isolation

We develop a local, spatial measure of educational isolation (EI) and characterize the relationship between EI and our previously developed measure of racial isolation (RI). EI measures the extent to which non-college educated individuals are exposed primarily to other non-college educated individuals. To characterize how the RI-EI relationship varies across space, we propose a novel measure of local correlation. Using birth records from the State of Michigan (2005–2012), we estimate associations between RI, EI, and birth outcomes. EI was lower in urban communities and higher in rural communities, while RI was highest in urban areas and parts of the southeastern United States (US). We observed greater heterogeneity in EI in low RI tracts, especially in non-urban tracts; residents of high RI tracts are likely to be both educationally and racially isolated. Associations were also observed between RI, EI, and gestational length (weeks) and preterm birth (PTB). For example, moving from the lowest to the highest quintile of RI was associated with a 1.11 (1.07, 1.15) and 1.16 (1.10, 1.22) increase in odds of PTB among NHB and NHW women, respectively. Moving from the lowest to the highest quintile of EI was associated with a 1.07 (1.02, 1.12) and 1.03 (1.00, 1.05) increase in odds of PTB among NHB and NHW women, respectively. This work provides three tools (RI, EI, and the local correlation measure) to researchers and policymakers interested in how residential isolation shapes disparate outcomes.

A consensus reaching process based on the concordance correlation measure of intuitionistic fuzzy sets in multi-criteria decision making

Although some correlation measure of intuitionistic fuzzy sets(IFSs) have been proposed, some of them cannot express the consistency of information or satisfy the axioms of similarity measure. In this paper, we present a consensus reaching process based on the concordance correlation measure of IFSs in multi-criteria decision making problems. Firstly, we define an innovative concordance correlation measure of IFSs, which not only takes the average information deviation of IFSs into account but also overcomes the disadvantages of previous correlation measures. In addition, its properties and the relationship between the defined new concordance correlation measure and Pearson correlation coefficient of IFSs are discussed. Secondly, considering that the classical TOPSIS method cannot be applied to the correlation measure with negative values, we continue to introduce the concept of relative concordance correlation measure and propose a consensus reaching process with minimum adjustment for an innovative behavioral TOPSIS method. Furthermore, a detailed numerical example and the comparison analyses are provided to verify the advantages of the proposed method. At last, we discuss the sensitivity and stability of the method.

Multiple Ordinal Correlation Based on Kendall’s Tau Measure: A Proposal

The joint analysis of various ordinal variables is necessary in many experimental studies within research fields such as sociology and psychology. Therefore, the necessary measures of multiple ordinal dependence must be easy to interpret and facilitate the interpretation of multivariate models that fit ordinal data. The main objective of this article is to propose a multiple ordinal correlation measure based on a bivariate correlation measure: Kendall’s tau. A sample version of the measure is proposed for its estimation. Furthermore, a confidence interval and a multiple ordinal independence test are proposed. The measure is applied to various simulations, covering a wide range of multiple ordinal dependency scenarios, in order to illustrate the adequacy of the measure and the proposed inferential techniques. Finally, the measure is applied to a real-world study based on a social survey of the levels of life satisfaction and the happiness index of a population.

Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue–Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this paper we prove a lower bound for the maximum order complexity of the sum of digits function in Zeckendorf base which is an example of a morphic sequence. We also prove that the polynomial subsequences of this sequence keep large maximum order complexity, such as the Thue–Morse sequence.