Given two continuous random variables,
X
and
Y
, we study the relationship between their statistical dependence and the Young tableau of the permutation defined from the graph of a bivariate sample ...coming from (
X
,
Y
). From a sample of size
n
of (
X
,
Y
), we identify the Young tableau of the permutation which maps the ranks of the
X
observations on the ranks of the
Y
observations. Procedures to detect statistical dependence between pairs of random variables, based on statistics calculated on the permutation defined by the graph of a bivariate sample have been developed, see García and González-López (2020) Symmetry 12, 9, 1415.
https://doi.org/10.3390/sym12091415
and García and González-López (2014) J Multivar Anal 127, 126–146.
https://doi.org/10.1016/j.jmva.2014.02.010
. In those papers, the information used is the length of the longest increasing (decreasing) subsequence, identified as the first line (the first column) of the Young tableau of the permutation. In this paper, we expose the information captured by the shape of the Young tableau of the permutation.
The article discusses examples of strong (SV > 0.7) simplest nonlinear dependencies in a problem for 114 indicators of 9 psychodiagnostic techniques, which represent exceptions in the context of many ...specific problems for studying statistical relationships, when two reciprocal dependencies, Y(X) and X(Y), are strong. There were only four such dependencies in the model for quintas of the independent variable within the framework of very weak and weak correlations (a total of 180 strong simplest nonlinear dependencies). The author quantitatively analysed and qualitatively interpreted the dependencies for three pairs of variables: “16PF-E: Submissive – Assertive” of R.B. Cattell’s questionnaire and “Competition” of K.W. Thomas’s methodology (SV = 0.78 and SV’ = 0.72 at r = 0.15); “16PF-Q3: Low self-control – High self-control” and “16PF-L: Trusting – Suspicious” of R.B. Cattell’s questionnaire (SV = 1.17 and SV’ = 0.91 at r = 0.28); “Psychasthenia” of the Minnesota Multiphasic Personality Inventory and “Suspicious type” of T.F. Leary’s methodology (SV = 0.84 and SV’ = 0.73 at r = 0.19). For the pair of variables “Low self-control – High self-control” and “Trusting – Suspicious”, models of linear regression are also considered. It is built on the basis of a dependence that is far from linear, as shown by Pearson’s coefficient of weak correlation equal to 0.28. At the same time, founded on the rule for interpreting the absolute value of the correlation coefficient for a sample of 120 subjects (widely used in the psychological community), it indicates the significance of the relationship at the p = 0.01 level, which inevitably requires a linear interpretation. For clarity, the information discussed in the article is illustrated by graphical representations of the dependencies under consideration.
The sources and characters of uncertainties in engineering modeling for risk and reliability analyses are discussed. While many sources of uncertainty may exist, they are generally categorized as ...either aleatory or epistemic. Uncertainties are characterized as epistemic, if the modeler sees a possibility to reduce them by gathering more data or by refining models. Uncertainties are categorized as aleatory if the modeler does not foresee the possibility of reducing them. From a pragmatic standpoint, it is useful to thus categorize the uncertainties within a model, since it then becomes clear as to which uncertainties have the potential of being reduced. More importantly, epistemic uncertainties may introduce dependence among random events, which may not be properly noted if the character of uncertainties is not correctly modeled. Influences of the two types of uncertainties in reliability assessment, codified design, performance-based engineering and risk-based decision-making are discussed. Two simple examples demonstrate the influence of statistical dependence arising from epistemic uncertainties on systems and time-variant reliability problems.
•The IAFLSM performance degradation model is established that realizes adaptive synchronous dynamic update of diffusion coefficient and nonlinear drift coefficient.•The dynamic updating mechanism for ...statistical dependence measurement error based on fractional Lévy stable motion is constructed.•The hidden variables are estimated through parameter estimation method and characteristic function, and the uncertainty is quantified by stability theorem and Monte Carlo technique.•The novel truck rear axle dataset conducted by our research group and the benchmark rolling bearing datasets are used to validate the proposed RUL prediction framework.
Prediction of remaining useful life (RUL) is a critical component of prognostics and health management (PHM) strategies for mechanical systems. Although some degradation models based on stochastic processes have played a pivotal role in the field of RUL prediction, the diffusion coefficients associated with these models are often fixed values that remain independent of the drift coefficient. Additionally, the measurement error is typically assumed to follow a Gaussian distribution that is independently and identically distributed with respect to the level of degradation. To address these constraints, an RUL prediction framework is proposed based on improved adaptive fractional Lévy stable motion (IAFLSM) with statistical dependence measurement error. The established IAFLSM model is capable of effectively characterizing the individual differences and time-varying uncertainties inherent in equipment degradation processes, with the nonlinear drift and diffusion coefficients exhibiting positive correlation in their variability. Moreover, the state space model is employed to realize the synchronous adaptive dynamic update of the drift coefficient and diffusion coefficient, thus accommodating the mechanical equipment degradation trajectory. Furthermore, a statistical dependence measurement error based on fractional Lévy stable motion is constructed, and the corresponding scale parameter incorporates a dynamic update mechanism with statistical correlation of degradation incremental behavior. The hidden variables related to performance degradation model are estimated through parameter estimation method and characteristic function. Expanding upon the proposed framework for RUL prediction, the quantification of the uncertainty intrinsic in the forecasting results is accomplished by employing the stability theorem of stable distribution and Monte Carlo technique. The RUL prediction framework is validated through the use of authentic truck rear axle full-life data and benchmark rolling bearing data. The comparative analysis results demonstrate the effectiveness and superiority of the proposed methodology.
As a fundamental concept in information theory, mutual information (MI) has been commonly applied to quantify association between random vectors. Most existing nonparametric estimators of MI have ...unstable statistical performance since they involve parameter tuning. We develop a consistent and powerful estimator, called fastMI, that does not incur any parameter tuning. Based on a copula formulation, fastMI estimates MI by leveraging Fast Fourier transform-based estimation of the underlying density. Extensive simulation studies reveal that fastMI outperforms state-of-the-art estimators with improved estimation accuracy and reduced run time for large data sets. fastMI provides a powerful test for independence that exhibits satisfactory type I error control. Anticipating that it will be a powerful tool in estimating mutual information in a broad range of data, we develop an R package fastMI for broader dissemination.
Distance covariance and distance correlation have been widely adopted in measuring dependence of a pair of random variables or random vectors. If the computation of distance covariance and distance ...correlation is implemented directly accordingly to its definition then its computational complexity is O(n
2
), which is a disadvantage compared to other faster methods. In this article we show that the computation of distance covariance and distance correlation of real-valued random variables can be implemented by an O(nlog n) algorithm and this is comparable to other computationally efficient algorithms. The new formula we derive for an unbiased estimator for squared distance covariance turns out to be a U-statistic. This fact implies some nice asymptotic properties that were derived before via more complex methods. We apply the fast computing algorithm to some synthetic data. Our work will make distance correlation applicable to a much wider class of problems. A supplementary file to this article, available online, includes a Matlab and C-based software that realizes the proposed algorithm.
Previous research has shown that treating dependent effect sizes as independent inflates the variance of the mean effect size and introduces bias by giving studies with more effect sizes more weight ...in the meta-analysis. This article summarizes the different approaches to handling dependence that have been advocated by methodologists, some of which are more feasible to implement with education research studies than others. A case study using effect sizes from a recent meta-analysis of reading interventions is presented to compare the results obtained from different approaches to dealing with dependence. Overall, mean effect sizes and variance estimates were found to be similar, but estimates of indexes of heterogeneity varied. Meta-analysts are advised to explore the effect of the method of handling dependence on the heterogeneity estimates before conductiong moderator analyses and to choose the approach to dependence that is best suited to their research question and their data set.
Unique information via dependency constraints James, Ryan G; Emenheiser, Jeffrey; Crutchfield, James P
Journal of physics. A, Mathematical and theoretical,
01/2019, Letnik:
52, Številka:
1
Journal Article
Recenzirano
Odprti dostop
The partial information decomposition (PID) is perhaps the leading proposal for resolving information shared between a set of sources and a target into redundant, synergistic, and unique ...constituents. Unfortunately, the PID framework has been hindered by a lack of a generally agreed-upon, multivariate method of quantifying the constituents. Here, we take a step toward rectifying this by developing a decomposition based on a new method that quantifies unique information. We first develop a broadly applicable method-the dependency decomposition-that delineates how statistical dependencies influence the structure of a joint distribution. The dependency decomposition then allows us to define a measure of the information about a target that can be uniquely attributed to a particular source as the least amount which the source-target statistical dependency can influence the information shared between the sources and the target. The result is the first measure that satisfies the core axioms of the PID framework while not satisfying the Blackwell relation, which depends on a particular interpretation of how the variables are related. This makes a key step forward to a practical PID.
Causal Inference for Social Network Data Ogburn, Elizabeth L.; Sofrygin, Oleg; Díaz, Iván ...
Journal of the American Statistical Association,
2024, Letnik:
119, Številka:
545
Journal Article
Recenzirano
Odprti dostop
We describe semiparametric estimation and inference for causal effects using observational data from a single social network. Our asymptotic results are the first to allow for dependence of each ...observation on a growing number of other units as sample size increases. In addition, while previous methods have implicitly permitted only one of two possible sources of dependence among social network observations, we allow for both dependence due to transmission of information across network ties and for dependence due to latent similarities among nodes sharing ties. We propose new causal effects that are specifically of interest in social network settings, such as interventions on network ties and network structure. We use our methods to reanalyze an influential and controversial study that estimated causal peer effects of obesity using social network data from the Framingham Heart Study; after accounting for network structure we find no evidence for causal peer effects.
Supplementary materials
for this article are available online.
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