Distance correlation coefficients for Lancaster distributions

Dueck, Johannes; Edelmann, Dominic; Richards, Donald St. P.

doi:10.1016/j.jmva.2016.10.012

Cited by 11 publications

(8 citation statements)

References 24 publications

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“…This is for the reason that, even for two quantitative variables, the Pearson correlation and the distance correlation between two variables measure two different quantities and can in general not be converted into each other. We note that for the bivariate normal, the distance correlation is always smaller than or equal to the Pearson correlation [ 10 , Theorem 7], the same holds for many other parametric distributions [ 39 ].…”

Section: Results: Simulation Studiesmentioning

confidence: 97%

Clustering of samples and variables with mixed-type data

2017

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Analysis of data measured on different scales is a relevant challenge. Biomedical studies often focus on high-throughput datasets of, e.g., quantitative measurements. However, the need for integration of other features possibly measured on different scales, e.g. clinical or cytogenetic factors, becomes increasingly important. The analysis results (e.g. a selection of relevant genes) are then visualized, while adding further information, like clinical factors, on top. However, a more integrative approach is desirable, where all available data are analyzed jointly, and where also in the visualization different data sources are combined in a more natural way. Here we specifically target integrative visualization and present a heatmap-style graphic display. To this end, we develop and explore methods for clustering mixed-type data, with special focus on clustering variables. Clustering of variables does not receive as much attention in the literature as does clustering of samples. We extend the variables clustering methodology by two new approaches, one based on the combination of different association measures and the other on distance correlation. With simulation studies we evaluate and compare different clustering strategies. Applying specific methods for mixed-type data proves to be comparable and in many cases beneficial as compared to standard approaches applied to corresponding quantitative or binarized data. Our two novel approaches for mixed-type variables show similar or better performance than the existing methods ClustOfVar and bias-corrected mutual information. Further, in contrast to ClustOfVar, our methods provide dissimilarity matrices, which is an advantage, especially for the purpose of visualization. Real data examples aim to give an impression of various kinds of potential applications for the integrative heatmap and other graphical displays based on dissimilarity matrices. We demonstrate that the presented integrative heatmap provides more information than common data displays about the relationship among variables and samples. The described clustering and visualization methods are implemented in our R package CluMix available from https://cran.r-project.org/web/packages/CluMix.

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Section: Results: Simulation Studiesmentioning

confidence: 97%

Clustering of samples and variables with mixed-type data

2017

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“…In this particular case, the squared affinely invariant distance covariance is given as

{\tilde{V}}^{2} (X, Y) = 4 π \frac{c_{p - 1}}{c_{p}} \frac{c_{q - 1}}{c_{q}} \times (_{3} F_{2} (\frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}; \frac{p}{2}, \frac{q}{2}; Λ) - 2_{3} F_{2} (\frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}; \frac{p}{2}, \frac{q}{2}; \frac{1}{4} Λ) + 1),

where c p has been defined immediately after , Λ denotes the squared cross covariance matrix of X and Y and 3 F 2 denotes a generalised hypergeometric function of matrix argument (see Dueck et al ., , for the details). Dueck et al () outlined a general approach for calculating the distance correlation for the so‐called Lancaster class of distributions. The authors illustrate this result by calculating the distance correlation for the bivariate gamma, the bivariate Poisson and the bivariate negative binomial because all of those distributions belong to the Lancaster class.…”

Section: Estimation Testing and Further Propertiesmentioning

confidence: 99%

“…Dueck et al (2017) outlined a general approach for calculating the distance correlation for the so-called Lancaster class of distributions. The authors illustrate this result by calculating the distance correlation for the bivariate gamma, the bivariate Poisson and the bivariate negative binomial since all of those distributions belong to the Lancaster class.…”

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confidence: 99%

An Updated Literature Review of Distance Correlation and Its Applications to Time Series

Edelmann

Fokianos

Pitsillou

2018

Int Statistical Rev

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Summary The concept of distance covariance/correlation was introduced recently to characterise dependence among vectors of random variables. We review some statistical aspects of distance covariance/correlation function, and we demonstrate its applicability to time series analysis. We will see that the auto‐distance covariance/correlation function is able to identify non‐linear relationships and can be employed for testing the i.i.d. hypothesis. Comparisons with other measures of dependence are included.

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“…Several extensions of Distance correlation have been introduced such as Invariant Distance correlation [ 4 ], Conditional Distance correlation [ 5 ], Distance Correlation of Lancaster distributions [ 6 ], Distance Standard Deviation [ 7 ], Distance Correlation for locally stationary processes [ 8 ], Distance correlation coefficient for multivariate functional data [ 9 ], Partial Distance correlation [ 10 ], and Distance correlation t -test [ 11 ]. Distance correlation has been broadly used for different applications such as time series [ 12 , 13 ], clinical data analysis [ 14 ], genomics [ 15 ], and biomedical data analysis [ 16 ].…”

Section: Introductionmentioning

confidence: 99%

Association Factor for Identifying Linear and Nonlinear Correlations in Noisy Conditions

Kachouie

Deebani

2020

Entropy

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Background: In data analysis and machine learning, we often need to identify and quantify the correlation between variables. Although Pearson’s correlation coefficient has been widely used, its value is reliable only for linear relationships and Distance correlation was introduced to address this shortcoming. Methods: Distance correlation can identify linear and nonlinear correlations. However, its performance drops in noisy conditions. In this paper, we introduce the Association Factor (AF) as a robust method for identification and quantification of linear and nonlinear associations in noisy conditions. Results: To test the performance of the proposed Association Factor, we modeled several simulations of linear and nonlinear relationships in different noise conditions and computed Pearson’s correlation, Distance correlation, and the proposed Association Factor. Conclusion: Our results show that the proposed method is robust in two ways. First, it can identify both linear and nonlinear associations. Second, the proposed Association Factor is reliable in both noiseless and noisy conditions.

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Distance correlation coefficients for Lancaster distributions

Cited by 11 publications

References 24 publications

Clustering of samples and variables with mixed-type data

Clustering of samples and variables with mixed-type data

An Updated Literature Review of Distance Correlation and Its Applications to Time Series

Association Factor for Identifying Linear and Nonlinear Correlations in Noisy Conditions

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